id
stringlengths 10
10
| text
stringlengths 0
8.91M
|
|---|---|
1511.00099
|
Sketch-based Image Retrieval from Millions of Images
Sarthak Parui Anurag Mittal Computer Vision Lab,
Dept. of Computer Science & Engineering
Indian Institute of Technology Madras
Tel.: +91-44-22575352
Proliferation of touch-based devices has made sketch-based image retrieval practical.
While many methods exist for sketch-based object detection/image retrieval on small datasets, relatively less work has been done on large (web)-scale image retrieval. In this paper, we present an efficient approach for image retrieval from millions of images based on user-drawn sketches.
Unlike existing methods for this problem which are sensitive to even translation or scale variations, our method handles rotation, translation, scale (i.e. a similarity transformation) and small deformations.
The object boundaries are represented as chains of connected segments and the database images are pre-processed to obtain such chains that have a high chance of containing the object. This is accomplished using two approaches in this work: a) extracting long chains in contour segment networks and b) extracting boundaries of segmented object proposals.
For efficient online processing, each database image is preprocessed to extract long sequences of contour segments (chains) using two complimentary methods. These chains are then represented by similarity-invariant variable length descriptors.
Descriptor similarities are computed by a fast Dynamic Programming-based partial matching algorithm.
This matching mechanism is used to generate a hierarchical k-medoids based indexing structure for the extracted chains of all database images in an offline process which is used to efficiently retrieve a small set of possible matched images for query chains.
Finally, a geometric verification step is employed to test geometric consistency of multiple chain matches to improve results.
Qualitative and quantitative results clearly demonstrate superiority of the approach over existing methods.
§ INTRODUCTION
The explosive growth of digital images on the web has substantially increased the need for an accurate, efficient and user-friendly large-scale image retrieval system.
With the growing popularity of touch-based smart computing devices and the consequent ease and simplicity of querying images via hand-drawn sketches on touch screens <cit.>, sketch-based image retrieval has emerged as an interesting application.
The standard mechanism of text-based querying could be imprecise due to wide demographic variations. It also faces the issue of availability, authenticity and ambiguity in the tag and text information surrounding an image <cit.>, which necessitates us to exploit image content for better search.
Although various popular image search engines such as Google[<https://images.google.com/>] and TinEye[<https://www.tineye.com/>] provide an interface for similar image search using an exemplar image as the query, a user may not have access to such an image every time. Instead, a hand-drawn sketch may be used for querying since sketching is a fundamental mechanism for humans to conceptualize and render visual content as also suggested by various Neuroscience studies <cit.>.
Thus, Sketch-based image retrieval, being a far more expressive way of image search, either alone or in conjunction with other retrieval mechanisms such as text, may yield better results, which makes it an important and interesting problem to study.
Apart from online web-scale image retrieval, an efficient sketch-based image retrieval mechanism has numerous other applications. For instance, it can be used to efficiently retrieve intended images from a constrained dataset, viz. a user's personal photo-album in a touch-sensitive camera/tablet for which no text/tag information is available. Observing the expressive power of free-hand user sketches, a few methods have been proposed for searching/designing apparels <cit.>, accessories <cit.> or generic 3D objects such as home appliances <cit.> using user-sketches. These applications require efficient shape representation and fast sketch-to-image matching to facilitate a smooth user experience. Furthermore, sketch-based retrieval can also be used to improve existing text-based image retrieval systems.
For instance, it may be possible to build a sketch in an on-line manner using the first few results of a text query system <cit.> and use this sketch for retrieving images that may not have any associated tag information. Image tag information may also be improved for a database in an off-line process using sketch-based retrieval.
Sketch-based Object Detection:
Several approaches have been considered in the literature for describing shapes and measuring their similarity.
To represent the shape structure of the objects present in an image, typically edge-detection/boundary detection is performed as a pre-processing task and the location and orientation information contained in the edge pixels (edgels) are utilized for determining shape correspondence. Basic approaches for measuring the similarity of rigid shapes include Chamfer Matching <cit.>, <cit.> proposed Chamfer Matching in which the binary template of a target shape is efficiently matched in an edge-image by calculating the distances to the nearest edge pixels. This process can be sped up using a pre-computed Distance Transform of such an edge-image.a priori for each pixel in the image (Distance Transform), which helps in capturing the possible geometric transformations of the reference template for efficient edgemap alignment. <cit.> used a closely related point correspondence measure called the Hausdorff distance which examines the fraction of points in one set that lie within some $\varepsilon$ distance of points in the other set (and vice versa). <cit.> improved the accuracy and efficiency of edgemap alignment substantially by incorporating image edge-orientation information and using a three-dimensional distance transform to match over possible locations and orientations.
To handle non-rigidity of shapes and speed up matching, <cit.> proposed Shape Context in which a shape is described by a set of descriptors, each of which captures the spatial distributions of other points along the shape contour in a log-polar space. <cit.> extend Shape Context by using the inner-distance instead of the Euclidean distance which restricts the paths between any two contour points to remain within the shape, thus making the descriptors quite robust to articulations. <cit.> further normalize each part affinely before computing the innner distance for achieving even greater invariance to shape variations, especially in the case of perspective distortions.
Shape Context-based methods have shown a good promise for clutter free shape-to-shape matchings. However, for matching shapes in images where an image has lots of extra and/or clutter edges, these are difficult to apply.
Several approaches have been proposed in the literature for sketch-based Object Detection/retrieval. These methods primarily address the issue of effective shape representation and accurate matching.
To represent the shape structure of the objects present in an image, typically edge-detection/boundary detection is performed as a pre-processing task. Then utilizing the location and orientation information contained in the edge pixels (edgels), a wide range of object representation and subsequently sketch-to-image matching mechanisms are proposed.
To determine the correspondence between two shapes, Huttenlocher et al. <cit.> use Hausdorff distance which examines the fraction of points in one set that lie near points in the other set (and vice versa).
Many methods have been proposed in the literature for sketch-based Object Detection/Retrieval in images although the running time for detection and retrieval is not necessarily a consideration.
With a goal of identifying discriminative shape elements, <cit.> detect long segments of edge chains as key curves. They describe an image by counting the number of such curves in each image patch which makes the representation susceptible to even a small amount of clutter. <cit.> propose local contour features thatencapsulate the spatial distribution of points in annular regions and attempt to handle clutter by excluding points inside the circle. However, since their feature detector responds to only circular/arc-like structures, very few object categories (typically only circular shapes) can be accurately represented by this method. <cit.> use a shape-tree to form a hierarchical structure of contour segments for representing each object, which helps in capturing local geometric properties along with the global shape structure. Deformation is allowed at individual nodes of the tree and an efficient Dynamic Programming-based matching algorithm is used to match two curves.
Many methods have been proposed in the literature for more sophisticated sketch-based Object Detection/Retrieval in images that handle more shape variations, although the running time for detection and retrieval is often compromised. <cit.> use a shape-tree to form a hierarchical structure of contour segments for representing each object, which helps in capturing local geometric properties along with the global shape structure. Deformation is allowed at individual nodes of the tree and an efficient Dynamic Programming-based matching algorithm is used to match two curves.In order to address the issues of background clutter and intra-class shape variations, <cit.> create a Contour Segment Network (CSN) by connecting nearby edge pixels in an edge-map. For matching a sketch to an image, they find paths through the constructed CSN that resemble the outline of the sketch. In a later approach, <cit.> propose groups of k (typically $\leq 4$) Adjacent approximately straight Segments (kAS) as a local feature and describe them in a scale insensitive way. These kASs are extracted for a large number of image windows and finally the object boundary is traced by linking individual small matched kASs in a multi-scale detection phase. In further work, they learn a codebook of Pairs of Adjacent Segments (PAS) <cit.> which is used in combination with Hough-based centroid voting and non-rigid thin-plate spline matching for detecting the sketched object in cluttered images.
In contrast to kAS matching, <cit.> propose a multi-stage contour-based detection approach, where Dynamic Programming is used to match segments directly to the edge pixels.
In contrast to straight segments in kAS, Ravishankar et al. <cit.> prefer relatively curved segments to achieve high discriminability. They propose a multi-stage contour based detection approach, where to handle substantial amount of deformation, shapes are decomposed into small segments at high curvature points. These small segments are then scaled, rotated, deformed and then matched sequentially in the gradient image using a Dynamic Programming-based approach. A sophisticated scoring mechanism considering scale, orientation and local deformation is employed to form groups of $k$ adjacent segments. To maintain a trade-off between occlusion handling and computational cost, triples of adjacent segments are used instead of pairs of adjacent segments unlike <cit.>.
Along with local geometric properties, to capture the global shape structure, <cit.> use a shape-tree to form a hierarchical structure of contour segments for representing each object. The deformation of an object in this representation is determined by applying deformation at individual nodes of the tree. An efficient Dynamic Programming-based matching algorithm is devised to match two curves.
Numerous methods have been proposed to learn object shape model and possible deformations from the training images and exploit various optimization frameworks to match the sketch-based model to an image for Object Detection.
<cit.> and <cit.> propose a part-based object representation by learning class-specific codebooks of local contour fragments and their spatial arrangements. For matching a query sketch, they perform Chamfer matching <cit.> and then find detections using a star-shaped voting model.
Although, their approach perform well for the learned classes, it is hard to use their method in a class-generic image matching and recognition/retrieval framework. <cit.> use a set of salient contours <cit.> to represent an image and formulate shape matching as a set-to-set matching problem. To make the alignment robust to background clutter, both image and sketch contours are jointly selected by approximately solving a hard combinatorial problem using Linear Programming.
To handle large intra-class appearance variations, <cit.> model an object within a bandwidth of its contour/sketch and search for the image contours using a shape-specific window called Shape-band.
<cit.> exploit skeleton information (medial axis) to capture the structure of an object and learn a tree-union structure to model the non-rigid object deformation. They use Oriented Chamfer Matching for computing shape similarity.
Recently, several approaches have shown a good Object Detection performance using self-contained angle-based descriptors for representing the object contours. <cit.> propose a shape descriptor based on a three-dimensional histogram of angles and distances for triples of consecutive sample points along the object contours. To explicitly handle varying local shape distortions, they exploit a particle filter framework to jointly solve the contour fragment grouping and matching problems.
Donoser et al. <cit.> analyze angles between any two points and a fixed third point on a close contour and design an integral image-based mechanism to match whole object boundary. Similar to <cit.>, <cit.> sample a contour into a fixed number of points and calculate the angles between a line connecting any two sampled points and a line to a third point relative to the position of the first two points. This representation is insensitive to translation and rotation but not scale. They employ a partial matching mechanism between two such angle-based descriptors by efficiently choosing the range of consecutive points using an integral image-based approach. To achieve robustness with respect to scale, object detection is performed over a range of scales.
All of these methods and many other state-of-the-art methods for Object Detection and Retrieval <cit.> typically represent objects using small contour fragments and trace the object boundary by linking them at a costly multi-scale detection phase. Furthermore, they employ expensive online matching operations based on complex shape features to enhance the detection performance and typically show results on relatively small-sized datasets such as ETHZ <cit.> and MPEG-7 <cit.> containing only a few hundreds or thousands of images while taking a considerable time to parse through each image at detection/search time. However, for a dataset with millions of images with a desired retrieval time of at most a few seconds, these methods are inapplicable/insufficient. Efficient Image pre-processing and a mechanism for fast online retrieval are necessary for large (web)-scale Image Retrieval.
Large-Scale Image Retrieval using Sketches:
Only a few attempts exist in the literature for the problem of sketch-based image retrieval on large databases. <cit.> decompose an image or sketch into different spatial regions and measure the correlation between the direction of strokes in the sketch and the direction of gradients in the image by proposing two types of descriptors, viz. an Edge Histogram Descriptor and a Tensor Descriptor. Histograms of prominent gradient orientations are encoded in the Edge Histogram Descriptor, whereas the Tensor Descriptor determines a single representative vector per cell that captures the main orientation of the image gradients of that cell. Descriptors at corresponding positions in the sketch and the image are correlated for matching.
They assume that the descriptors for individual spatial regions relative to the overall image will have a similar footprint if an image is similar to a sketch. Due to this strong spatial assumption, they fail to retrieve images if the sketched object is present at a different scale, orientation and/or position. To determine similar images corresponding to a sketch, a linear scan over all database images is performed. This further limits the scalability of the
method for large databases.
To address the issue of scalability, <cit.> propose an indexing-friendly raw contour based method. Given a sketch query, the primary objective of this method is to retrieve database images which closely correlate with the shape and the position of the sketched object. For every possible image location and a few orientations, they generate an inverted list of images that have edge pixels (edgels) at that particular location and orientation. For a sketch query, similar images are determined by counting the number of similar edgels in both the sketch and images, which makes this method susceptible to scale, translation and rotation changes. <cit.> introduce a hashing based framework with a strong assumption that a user only wants spatially consistent images as the search result. They extract HoG features <cit.> for overlapping spatial patches in an image and represent them using binary vectors by thresholding the HoG responses. The similarity between corresponding patches in the sketch and the image is estimated using the Min-hash algorithm <cit.> that exploits the set-overlap similarity of these binarized descriptors.
They generate HoG features <cit.> for overlapping spatial patches in an image and the similarity between corresponding patches in the sketch and in the image is estimated using Min-hash algorithm. Similar to <cit.>, a reverse indexing structure on the hash keys is built to facilitate fast retrieval.
In order to encapsulate the spatial structure, <cit.> describe both images and sketches using Gradient field HoG (GF-HOG) which encodes a sparse orientation field computed from the gradients of the edge pixels. To facilitate retrieval, a Bag of Visual Words model is used and sketch-to-image similarity is measured by computing the distance between corresponding frequency histograms representing the distribution of GF-HOG derived “visual words”. However, this representation is noisy in presence of even small amount of background clutter. Moreover, since the experiments were performed only on small datasets that contain less than $400$ images, the usability of the method on large scale retrieval is not very clear.
<cit.> extend their prior idea <cit.> to large scale retrieval by building a vocabulary tree <cit.> on the descriptors.
They extract heavily overlapping contour fragments (allowing only one point shift) from a sketch and the edge-map of an image. In their approach, each contour is composed of a fixed number of points $L$ and described as a matrix of $\binom{L}{2}$ angles that denote the orientation of the lines joining any two such points with respect to a vertical line. This makes their method sensitive to scale and orientation changes. Furthermore, due to the use of dense descriptors, the computational complexity is still very high for large datasets. Moreover, the experiments were performed only on very small datasets consisting of less than $300$ images and therefore the applicability of this method in large scale is again not very clear.
To alleviate the issue of scale, translation and rotation sensitivity, <cit.> determine the most “salient” object in the image and measure image similarity based on a descriptor built on the object. However, determining saliency is a very hard problem and the accuracy of even the state-of-the-art saliency methods in natural images is low <cit.>, thus rendering the method possibly quite unreliable. which makes this method possibly quite unreliable.
Proposed method in context:
In this paper, we develop a system for large scale sketch-based image retrieval that can handle scale, translation and rotation (similarity) variations without compromising on efficiency, which we believe has not been addressed earlier in the literature.
First, the essential shape information of all the database images is captured by extracting sequences/chains of contour segments in an offline process using two efficient and often complimentary methods: (a) finding long connected segments in contour segment networks (Sec. <ref>) and (b) using boundaries of segmented object proposals (Sec. <ref>).
Such chains are represented using a similarity-invariant variable length descriptors (Sec. <ref>). These chain descriptors are matched using an efficient Dynamic Programming-based approximate substring matching algorithm. Note that, variability in the length of the descriptors makes the formulation unique and more challenging. Furthermore, partial matching is allowed to accommodate intra-class variations, small occlusions and the presence of non-object portions in the chains (Sec. <ref>).
A hierarchical indexing tree structure of the chain descriptors of the entire image database is built offline to facilitate fast online search by matching the chains down the tree (Sec. <ref>).
Finally, a geometric verification scheme is used for refining the retrievals by considering the geometric consistency among multiple chain matchings (Sec. <ref>).
Results on several datasets indicate superior performance and advantages of our approach compared to prior work , along with its strengths and weaknesses(Sec. <ref>). Further a detailed analysis is performed to evaluate the strengths and weaknesses of the proposed algorithm (Sec. <ref>).
In this paper, we systematically investigate the problem of large scale sketch-based image retrieval and address the issue of scale, translation and rotation (similarity) invariance without compromising on efficiency, which we believe has not been addressed earlier in the literature. We propose a similarity-invariant, variable length compact shape descriptor that enables efficient deformation handling matching even for datasets with millions of images. unlike any relevant existing work. First, the essential shape information of all the database images is represented in a similarity-invariant way in an offline process by extracting sequences/chains of contour segments from each image. This is accomplished by proposing two efficient and often complimentary methods for chain extraction that use i) contour segment network (Sec. <ref>), ii) segmented object proposals (Sec. <ref>).
Second, to facilitate efficient similarity invariant matching, the extracted chains are represented using succinct variable length descriptors that are locally insensitive to scale, translation and rotation (Sec. <ref>).
Third, an efficient Dynamic Programming-based approximate substring matching algorithm is proposed for fast matching of such chains between a sketch and an image or between two images. Note that, variability in the length of the descriptors makes the formulation unique and more challenging. Furthermore, partial matching is allowed to accommodate intra-class variations, small occlusions and the presence of non-object portions in the chains (Sec. <ref>).
Fourth, a hierarchical indexing tree structure of the chain descriptors of the entire image database is built offline to facilitate fast online search by matching the chains down the tree (Sec. <ref>).
Finally, a geometric verification scheme is devised for an on-the-fly elimination of false positives that may accidentally receive a high matching score due to partial shape similarity (Sec. <ref>).
Effect of different components of the proposed method is carefully examined to understand their relative implications (Sec. <ref>). Qualitative and quantitative comparisons with the state-of-the-art on a dataset of $1.2$ million images clearly indicate superior performance and advantages of our approach (Sec. <ref>). Moreover, a detailed analysis is performed in a challenging shape dataset <cit.> that allows us to evaluate the advantages and disadvantages of the proposed method (Sec. <ref>).
§ FROM IMAGES TO CONTOUR CHAINS
In this section, we describe offline preprocessing of database images with an objective of having a compact representation which can be used to efficiently match the images with a query sketch. We first note that a user typically draws an object along its boundary <cit.> and a sketch of the object boundaries can more or less capture the distinctive object shape information <cit.>. Thus, an image representation based on contour information of the object boundaries would be quite appropriate in this scenario. <cit.> construct an unweighted Contour Segment Network (CSN) which links nearby edge segments, and then extract $k(\leq4)$ adjacent segments($k$AS) from such a network. Full contour segment networks <cit.> capture a good amount of information about the image edge segments, but are difficult to represent compactly and match efficiently in a large database. Shape Descriptors can be utilized; but they can be quite noisy, especially in the presence of image clutter. In this work, we represent shape information using (long) chains of contour segments which we believe contain a good amount of information for capturing the shape perspective and are efficient for storage and matching at the same time. While extraction of contour chains in sketches is trivial, doing so in database images is non-trivial due to the exponential number of chain possibilities. In this work, we devise two efficient and often complimentary methods for extracting and encoding the essential object boundary information in a database image by (a) finding long chains in contour segment networks, and (b) using boundaries of segmented object proposals.
§.§ Finding Long Chains in Contour Segment Network
It has been observed that long sequences of contour segments typically have a good amount of intersection with the important object boundaries. Therefore, in our approach, we try to extract long sequences of segments or chains. Furthermore, these segments must be connected to each other strongly, i.e. their connecting end points should be close to each other for them to be considered in the same chain. To this end, we extract a set of salient contours for each image that have been experimentally found to be good candidates for object boundaries and then propose a technique to group these contours into meaningful sequences that have a good chance of overlapping with the boundaries of the objects present in an image.
§.§.§ Obtaining Salient Contours
At first, in order to use fixed parameters, all the database images are normalized to a standard size (we consider the size of the longest side as 256 pixels). Then, the Berkeley Edge Detector <cit.> is used to generate a probabilistic edge-map corresponding to the object boundaries in the image. This gives superior object boundaries compared to traditional edge detection approaches such as Canny <cit.> by considering texture along with color and brightness in an image and specifically learning for object boundaries. However such a boundary-map typically still contains a lot of clutter (Fig. <ref>(b)). Therefore, an intelligent grouping of edge pixels is done to yield better contours that have a higher chance of belonging to an object boundary. The method proposed by <cit.> groups edge pixels by considering long connected edge sequences that have as little bends as possible, especially at the junction
points as it has been found that the object boundaries mostly follow a straight path at the junction points. Contours that satisfy such a constraint are called salient contours in their work and this method is used by us to extract a set of salient contours from a database image (Fig. <ref>(c)).
Grouping Salient Contours into Chains: (a) Image, (b) Edgemap <cit.>, (c) Salient contours <cit.>, (d) Illustrative snapshot of a constructed graph, (e) Maximum Spanning Tree for one component, (f) A long chain
§.§.§ Segmenting Salient Contours
Given the salient contours of an image, we consider bends in them (which remain since the object shape has bends).
The salient contours thus obtained may still contain some bends in them. Some articulation should be allowed at such bends since it has been observed that the object shape perspective remains relatively unchanged under articulations at such bend points <cit.>. These bend points along the contour are determined as the local maxima of the curvature. Although curvature has been defined in the literature in many ways <cit.>, we use a simple formulation that is fast and robust. The curvature of a point $p_c$ is obtained using $m$ points on either side of it as:
\begin{equation}
\kappa_ {p_c} = \sum_{i=1}^m w_i \cdot \angle p_{c-i}p_cp_{c+i}
\label{HCPDetection}
\end{equation}
where $w_i$ is the weight defined by a Gaussian function centered at $p_c$. This function robustly estimates the curvature at point $p_c$ at a given scale $m$.
The salient contours thus obtained are split into multiple segments at such high curvature points and as a result, a set of straight line-like segments are obtained for an image.(Fig. <ref>(d)).
§.§.§ Chaining the Segments
Given a set of straight line-like segments in an image, we try to connect them. into chains that might be able to cover the object boundaries.The connectivity among the segments suggests an underlying graph structure. Thus, a weighted graph/contour segment network is constructed where each end of a contour segment is considered as a vertex/joint and the edge weight between any two vertices is equal to the length of segment. Vertices from two different contour segments are merged if they are spatially close. However, since the gaps between such vertices are very small, we do not assign any weight for these connections. Fig. <ref>(b) shows the graph corresponding to the illustrative set of straight line-like segments in Fig. <ref>(a).
Given a set of straight line-like contour segments in an image, we design compact representation of an image by considering ordered sequences of segments that utilize the connectedness of the object boundary. The connectivity among the segments suggests an underlying graph structure. Ferrari et al. <cit.> utilize this by constructing an unweighted contour segment network which links nearby edge segments, and then extracting $k(\leq4)$ adjacent contour segments($k$-AS) for a large number of image windows. They trace the object boundary by linking individual small $k$-AS at the multi-scale detection phase.
Shotton et al. <cit.> builds a codebook of small contour fragments and and combine them using a boosted sliding window classifier while detecting the objects.
To account for substantial object deformations, Ravishankar et al. <cit.> also breaks the object contours at high curvature points and these small segments are separately searched in a gradient image and then grouped at the online object detection phase.
Although such approaches perform well in clutter, it leads to a costly online matching operation, which motivates us to represent an object with much longer segment chains a priori for each image rather than with very small contour fragments.
It has been observed that long sequences of segments typically have a large intersection with important object boundaries. Therefore, in our approach, we try to extract the long sequences. To obtain such long sequences, in contrast to <cit.>, a weighted (rather than an unweighted) graph is constructed where each end of a contour segment is considered as a vertex and the edge weight is equal to the length of the segment. Vertices from two different contour segments are also joined by an edge if they are spatially close. The weight of such an edge is taken as $\lambda_{d}\cdot\exp\left(-d/D\right)$, where $d$ is the spatial distance between the two end points, $D$ is the diagonal length of the normalized image and $\lambda_{d}$ is a constant factor that provides a trade-off between the segment length and the inter-segment gap (Fig. <ref>(d)).
Graph of Joints: (a) Set of segments after breaking the salient contours (shown in different colors) at high curvature points (e.g joint b is a high curvature point). (b) Nearby endpoints of two different segments are merged if they are sufficiently close and a graph of joints ($j$) is created.
The weight of an edge in the graph/contour segment network represents the spatial extent of the segments. and the connectedness between them. Therefore, a long path in the graph based on the edge weights (higher weight is better) relates to a long connected sequence of contour segments or chains in an image. As the graph may contain cycles, to get non-cyclical paths, the maximum spanning tree[Maximum spanning tree of a graph can be computed by negating the edge weights and computing the minimum spanning tree.] is constructed for each connected component in the contour segment network (Fig. <ref>(e)) using a standard minimum spanning tree algorithm <cit.> and paths are extracted from these trees. Note that the Maximum Spanning Tree algorithm removes the minimum weight edges in any cycle (Fig. <ref>(a)).
(a) An illustrative snapshot of a network of joints. The red colored edge joining $J_6$ and $J_7$, being the least weight edge in the induced cycle, is removed in the Maximum Spanning Tree of the network. (b) Two possible long paths/chains in the constructed tree.
Since we consider only long chains, minimum weight edges which correspond to the small segments would not be typically picked anyway in a long chain. For instance, in Fig. <ref>(a), a long chain passing through the joints $J_7$ and $J_6$ follows the path $< ..., J_7, J_4, J_5, J_6, ...>$ and therefore, removal of the smallest segment $\overline{J_7J_6}$ in the cycle does not lead to any change in the extracted long chain. Fig. <ref>(b) illustrates possible chains for the contour segment network in Fig. <ref>(a) while Fig. <ref>(f) shows an extracted chain for an image (Fig. <ref>(a)).
A long path or chain thus obtained may deviate from the object boundary at the junction points (Fig. <ref>(f)). Ideally, to capture maximum shape information, all possible sequences through such junction points should be considered. However, this drastically increases the number of chains in the representation and is therefore impractical for a database of millions of images.
Hence, as a trade-off between the representative power and compactness, a greedy approach may be followed by considering only edge-disjoint long sequences in the graph. These can be determined by sequentially finding and removing the longest paths in the graph. As a result, an image may be represented by a set of strictly non-overlapping chains.
This leads to an image representation by a set of strictly non-overlapping chains. This representation is compact and typically results in distinctive long chains as shown in our prior work <cit.>.
(a) A single chain does not capture the object boundary, which can be addressed by considering multiple overlapping chains allowing certain overlap among them(b).
A single long chain from this tree typically cannot cover the distinctive portion of any object (Fig. <ref>(a)). Therefore, to extract informative chains, the best $N_{OC}$ chains are determined by allowing certain overlaps among the chains. The object boundary is typically smooth. Therefore, in the case of multiple possibilities at the junction points, we measure the smoothness of possible paths at that junction and prefer an almost straight path compared to a curve. Fig. <ref> illustrates two possibilities at the junction $v$ and the sequence $\left\langle ... uvw ... \right\rangle$ is preferred to $\left\langle ... uvw^\prime ...\right\rangle$. To this end, the angle between three successive joints $u, v\text{ and } w$ is calculated. Since a straight line is preferred, the deviation of $\angle uvw$ from 180°is determined as a measure of smoothness.
Furthermore, to use this measure only in the case of multiple possibilities at junction points, smoothness at a joint of a chain is normalized by the smoothness terms of all possible chains at that joint.
Let $t, u, v, w$ be four consecutive joints along a chain $C$ and $\dis(u, v)$ be the distance between any two joints $u$ and $v$. Then the score of the chain $C$ is determined as:
\begin{dmath}
Score(C) = \sum_{\substack{u, v \in C}} \dis(u, v) \cdot \left[ 1 +
\frac{\lambda_{l}}{2} \cdot \frac{\exp\left(-\lambda_{s}\cdot \left| \pi - \angle tuv \right|\right)}{\sum\limits_{x \mid t,u,x \in C^\prime} \exp\left(-\lambda_{s}\cdot \left| \pi - \angle tux \right|\right)} \\
+ \frac{\lambda_{l}}{2} \cdot \frac{\exp\left(-\lambda_{s}\cdot \left| \pi - \angle uvw \right|\right)}{\sum\limits_{y \mid u,v,y \in C^{\prime\prime}} \exp\left(-\lambda_{s}\cdot \left| \pi - \angle uvy \right|\right)}
\right]
\label{OverlappingChainScore}
\end{dmath}
Here $\lambda_{l}$ and $\lambda_{s}$ are two scalar constants. $C^\prime$ and $C^{\prime\prime}$ are possible chains through the joints $t, u$ and $u, v$ respectively. Negative exponential functions are used since only the values close to the desired value (i.e. $\pi$) can be considered a good candidate and anything beyond a certain limit should be given a low score. Note that the smoothness term is weighted by the length of the segment in order to achieve robustness with respect to the number of intermediate joints.
A tree with $N_{l}$ number of leaves has $\binom{N_{l}}{2}$ paths and there is a substantial overlap of joints among many paths.
We exploit this and use a Least Common Ancestor-based implementation <cit.> to efficiently score all $\binom{N_{l}}{2}$ paths and to sequentially select some top $N_{OC} ( = 5)$ chains such that the relative overlap among them is less than $\lambda_{thresh_{chain}} (=60\%)$.
Fig. <ref>(b) illustrates the usefulness of considering multiple overlapping chains where only the third chain encapsulates the informative shape information of a swan.
Quite reasonable results may be achieved using such long chains in a contour segment network alone <cit.>. We next propose another technique that can often provide complimentary chains in the case this algorithm fails.
$\left\langle ... uvw ... \right\rangle$, being comparatively smoother, is preferred to $\left\langle ... uvw^\prime ...\right\rangle$ for scoring possible paths through the junction $v$
We exploit this and use a Least Common Ancestor-based implementation <cit.> to efficiently score all $\binom{N_{l}}{2}$ paths. Finally to reduce redundancy in chain representation, top $N_{oc} ( = 5)$ chains are chosen such that the relative overlap among them is less than $\lambda_{thresh_{chain}} (=60\%)$ . Fig <ref> illustrates the usefulness of considering multiple overlapping chains where the third chain encapsulates the informative shape information of a swan. However, since every chains are not
considered for succinct representation, it is possible to miss the chain which actually encloses the object boundary (Fig <ref>(a)). To compensate, further chains may be extracted by segmenting an image into object regions. Moreover image segmentation helps us in perceptual grouping of image pixels considering color, brightness and texture, which could lead to better chain extraction.
However, due to the strict non-overlapping criterion, a long chain may not cover the distinctive portion of a particular object. Fig <ref>(a) illustrates such a situation where the chains did not cover the distinctive portion of any swan. Therefore, to extract informative chains, the best $N_{C}$ chains are determined by allowing certain overlaps among them. Object boundary is typically smooth. Therefore, along with the length of the chain, the smoothness at the junctions are also considered for calculating individual chain score. Fig <ref> illustrates two termination possibilities of same chain and the sequence $\left\langle ... uvw \right\rangle$ is preferred to $\left\langle ... uvw^\prime \right\rangle$ if the length of the segment $vw$ is same as that of $vw^\prime$. To measure smoothness, the angle between three successive joints $u, v \text{and } w$ is calculated. Since a smoother curve is preferred, the deviation of $\angle uvw$ from 180textdegree is determined. Let $u,v \text{and} w$ be three consecutive joints along a chain $C$ and $\dis(u, v)$ represent the distance between any two joints $u$ and $v$. Then the score of the chain $C$ is determined as:
\begin{equation}
Score(C) = \sum_{\substack{u, v \in C}} \dis(u, v) +
\sum_{\substack{u, v, w\in C}} \lambda_{lsc} \cdot \dis(v, w) \cdot \exp\left(-\lambda_{smooth}\cdot \left| \pi - \angle uvw \right|\right)
\label{OverlappingChainScore}
\end{equation}
\begin{equation}
Score(C) = \sum_{\substack{u, v \in C\\
v = u + 1}} \dis(u, v) +
\sum_{\substack{u, v, w\in C\\
v = u + 1\\
w = v + 1}} \lambda_{lsc} \cdot \dis(v, w) \cdot \exp\left(-\lambda_{smooth}\cdot \left| \pi - \angle uvw \right|\right)
\label{OverlappingChainScore}
\end{equation}
Here $\lambda_{lsc}$ and $\lambda_{smooth}$ are two scalar constants. Note that the smoothness term is weighted by the length of the segment in order to achieve robustness with respect to the number of intermediate joints. To this end, $\dis(v, w)$ is considered instead of $\dis(u,w)$ to account for over-counting $\dis(u,v)$. A tree with $N_{l}$ number of leaves has $\binom{N_{l}}{2}$ paths and there is a substantial overlap of joints among many paths. We exploit this and use a Least Common Ancestor-based implementation <cit.> to efficiently score all $\binom{N_{l}}{2}$ paths. Finally to reduce redundancy in chain representation, top $N_{oc} ( = 5)$ chains are chosen such that the relative overlap among them is less than $\lambda_{thresh_{chain}} (=60\%)$ . Fig <ref> illustrates the usefulness of considering multiple overlapping chains where the third chain encapsulates the informative shape information of a swan. However, since every chains are not
considered for succinct representation, it is possible to miss the chain which actually encloses the object boundary (Fig <ref>(a)). To compensate, further chains may be extracted by segmenting an image into object regions. Moreover image segmentation helps us in perceptual grouping of image pixels considering color, brightness and texture, which could lead to better chain extraction.
[Multiple possibilities at junction point $v$]
[Top $N_{GOP}$ proposals may fail to capture the object shape (above), whereas chaining the salient contours leads to successful object boundary extraction (below)]
§.§ Using Segmented Object Proposals
Partitioning an image into semantically interpretable regions is an interesting topic in Computer Vision and image segmentation techniques are widely used to obtain possibly connected regions.
plays important role in human perception <cit.>. Therefore to obtain object regions, image segmentation techniques are widely used. These perceptual groupingtechniques typically consider different image cues, such as brightness, color, texture over local image patches and then cluster these features using statistical techniques such as mixture models <cit.>, finding modes <cit.>, region-based split-and-merge techniques <cit.>, global optimization approaches <cit.> or graph partitioning algorithms <cit.>. To combine pixels into perceptually meaningful regions, many superpixel-based grouping techniques have been proposed as well <cit.> which typically produce an over-
segmentation of the image. However, for the purpose of object boundary representation, the unit of interest is a single object region for each object present in an image.
Therefore, instead of considering the algorithms that exhaustively and uniquely label every pixel or superpixel in an image, we consider segmented object proposals.
The primary objective of popular segmented object proposal techniques <cit.> is to provide locations and boundaries of the possible objects in an image. Since we need to extract boundary information for millions of images, we use a very fast method called Geodesic Object Proposals (GOP) <cit.> for extracting a set of possible object regions. This method typically produces many overlapping scored object regions. To limit the number of chains for each image, we consider only some top $N_{GOP} (= 20)$ proposals based on their scores.
In GOP, the authors first over-segment the image <cit.> into a set of superpixels.
With an objective of automatically selecting object seeds, a geodesically central superpixel is chosen as the initial seed and then the next seeds are placed far from the existing seeds.
RankSVM <cit.> is used as a classifier to train the system for better seed placement. For an individual seed, a foreground and background mask is generated in order to compute a geodesic distance transform. Level sets of each of the distance transforms define an object segment. This method typically produces many overlapping object regions. These regions are scored based on the growth of the region and the overlap with other regions. To limit the number of chains for each image, we consider only some top $N_{GOP} (= 20)$ proposals based on their scores.
Note that while it is possible to use image segmentation techniques such as <cit.> and <cit.> to obtain good connected regions, they are typically very slow <cit.>.
However, the state-of-the-art segmentation techniques are typically very slow <cit.>. To combine pixels into perceptually meaningful regions, many superpixel-based grouping techniques have been proposed as well <cit.> which typically produce an over-segmentation of the image. However, Furthermore, for the purpose of object boundary representation, the unit of interest is a single object region for each object present in an image. Therefore, Geodesic Object Proposal <cit.> is used in this work. that generates a set of object region proposals. The method is based on superpixel growing and each proposal corresponds to a segment in the image. Thus, the object shape information can be easily extracted by considering the boundaries of the proposed segments. We remove the segments that mostly touch the image boundary as such segments have incomplete boundaries and a possible object is only partially present in the image. Furthermore, to extract only distinctive object boundary, very small regions are also discarded (Fig. <ref>(b) and Fig. <ref>(a)). Finally the boundaries of the remaining regions from the top $N_{GOP}$ proposals are taken as chains.
Fig. <ref>(b) shows a chain successfully obtained from the segmented proposals where the chains extracted using contour segment network were inferior (Fig. <ref>(a)).
Limitation of considering only top $N_{oc}$ chains from contour segment network: (a) None of the chains encloses sufficient object boundary, (b) A distinctive chain is extracted (below) considering segmented object proposals <cit.> (above).
Fig. <ref> illustrates a reverse example where the top object proposals do not contain informative object boundary information, while the overlapping chains extracted from the contour segment network covers the desired object boundary. Therefore we consider both the approaches for extracting chains for database images in an offline process.
[Top segmented proposals obtained using the default parameters of <cit.> fail to capture the object shape]
[Overlapping chains extracted using contour segment network cover distinctive object boundary]
Limitation of extracting chains only from boundaries of segmented object proposals
Fig. <ref> demonstrates the entire chain creation framework and Fig. <ref> shows the chains thus obtained in some common images. Note that in our framework, it is easy to adapt other possibly better boundary extraction mechanisms as well due to the flexibility of the framework in terms of chain length and the number of chains.
Entire chain creation framework
Object shape information can be easily extracted by considering the boundaries of the proposed segments. We observed that for few proposals, most of the proposed region are in the image boundary. Such regions are ignored as they have a higher chance of belonging to the background. Furthermore, to extract only distinctive object boundary, very small regions are also discarded. Finally the boundaries of the remaining regions are taken as chains. and use these along with previously extracted chains.Fig. <ref>(b) shows that a chain obtained from the segmented proposals capture better boundary information as compared to the chains obtained by grouping the salient contours. However, since only top $N_{GOP}$ proposals are considered, it is possible to miss the one that actually contains an object. Fig. <ref> illustrates such an example where top proposals do not contain informative object boundary information. In contrast, the overlapping chains formed by
grouping salient contours enclose essential object boundaries. Therefore, we consider both salient contour grouping and segmented object proposals for extracting chains for database images and these chains are extracted in an offline process. Fig. <ref> demonstrates the entire chain creation framework and Fig. <ref>(a) shows the chains thus obtained in some common images.
Chains extracted for some images. Different chains are represented using different colors.
The chain for the curve SE is composed of three line segments. The descriptor for this chain is $\varPsi=\left\langle \gamma_{i}=\frac{l_{seg_{i}}}{l_{seg_{i+1}}}, \theta_i \left|\right. i \in \lbrace1, 2\rbrace \right\rangle$.
§ CREATING DESCRIPTORS FOR EACH CHAIN
In order to efficiently match two chains in a similarity-invariant way, we require a compact descriptor that captures the shape information of the extracted chains in a similarity-invariant way. Towards this goal, the local shape information is captured at the joints in a scale, in-plane rotation and translation invariant way.
For the $i^{th}$ joint of chain $k$ ($J_i^k$), the segment length ratio $\gamma_i=\frac{l_{seg_{i}}}{l_{seg_{i+1}}}$ ($l_{seg_{i}}$ denotes the length of the $i^{th}$ segment) and the anti-clockwise angle $\theta_i$ (range: [0, 2$\pi$]) between the adjacent pair of segments $seg_i$ and $seg_{i+1}$ are determined, as shown in Fig. <ref>.
The descriptor $\varPsi^{k}$ for a chain $k$ with $N$ segments is then defined as an ordered sequence of such similarity-invariant quantities:
\begin{equation}
\varPsi^{k} = \left\langle \gamma_i, \theta_i \left|\right. i \in \lbrace 1\ldots N-1\rbrace \right\rangle
\label{eqnDesc}
\end{equation}
Note that <cit.> also use joint information by measuring the relative angles among all pairs of sampled points along a contour. However, their representation is not scale invariant which leads to a costly online multi-scale matching phase. In contrast, our proposed descriptor is insensitive to similarities and is suitable for efficiently representing and matching contour chain information in millions of images.
Having extracted chains from images and compactly represented them in a similarity-invariant way, we next describe an approach for efficiently matching two such chains.
§ MATCHING TWO CHAINS
Standard vectorial type of distance measures are not applicable for matching two chains due to the variability in the lengths of the chains in our case.
This constraint makes the task more challenging since most of the fast indexing mechanisms for large scale retrieval exploit a metric structure <cit.>. Further, note that the object boundary is typically captured by only a portion of the chain in the database image (Fig. <ref>). Therefore, a partial matching strategy of such chains needs to be devised which can be smoothly integrated with an indexing structure to efficiently determine object shape similarity.
Since image chains are typically noisy, it is not uncommon to obtain a chain that captures an object boundary and has non-object contour segments on either side of the object boundary portion. Furthermore, we assume that the object boundary is typically captured by a more or less contiguous portion of the chain without large gaps in between. Although such large split-ups may occur in certain circumstances, allowing such matches leads to a lot of false matches of images due to too much relaxation of the matching criteria. This is illustrated in Fig. <ref>(a), where the split matches are individually good matches but put together do not match with the intended shape structure at all. Thus, in our work, the similarity between two chains is measured by determining the maximum (almost) contiguous matching portions of the sequences while leaving out the non-matching portions on either side from consideration (Fig. <ref>(b)).
This is quite similar to the Longest Common Substring[Substring, unlike subsequence, does not allow gap between successive tokens.] problem <cit.>, with some modifications that can be solved efficiently using Dynamic Programming. We first consider the individual scores for matching two joints across two chains.
(a) A match when fragmented skips are allowed. (b) A match when only almost-contiguous matches are allowed. Matched joints are shown with the same marker in the sketch and the image. Unmatched portions of the chains are indicated by dashed lines.
§.§ Joint Similarity
Since exact correspondence of the joints does not capture the deformation that an object may undergo, we provide a slack while matching and score the match between a pair of joints based on the deviation from exact correspondence. The score $S_{jnt}(x,y)$ for matching the $x^{th}$ joint of chain $C_1$ to the $y^{th}$ joint of chain $C_2$ is taken to be the product of two constituent scores:
\begin{equation}
S_{jnt}(x, y) = S_{lr}(x, y) \cdot S_{ang}(x, y)
\label{jointMatchingScore}
\end{equation}
$S_{lr}(x, y)$ is the closeness in the segment length ratio of the two adjacent segments at the $x^{th}$ and $y^{th}$ joints of the two descriptors:
\begin{equation}
%S_{lr}(x, y) = \exp \left( \lambda_{lr} \cdot \left(1 - 1\middle/ \Omega\left(\gamma^{C_1}_{x}, \gamma^{C_2}_{y}\right) \right) \right)
S_{lr}(x, y) = \exp \left( \lambda_{lr} \cdot \left(1 - \middle. \Omega\left(\gamma^{C_1}_{x}, \gamma^{C_2}_{y}\right) \right) \right)
\label{segmentLengthRatio}
\end{equation}
where $\gamma_{x} = \frac{l_{seg_x}}{l_{seg_{x+1}}}$ is as defined in Sec. <ref>, $\Omega\left(a, b\right) = \min\left(a/b, b/a\right), a,b\in\mathbb{R_{\text{\textgreater 0}}}$ measures the relative similarity between two ratios ($\Omega\left(a, b\right) \in (0,1]$) and $\lambda_{lr}(=0.5)$ is a constant. $S_{ang}(x, y)$ determines the closeness of the angles at the $x^{th}$ and $y^{th}$ joints and is defined as:
\begin{equation}
S_{ang}(x, y) = \exp \left(- \lambda_{ang} \cdot \left|\theta^{C_1}_{x} - \theta^{C_2}_{y}\right|\right)
\label{jointAngleDifference}
\end{equation}
where $\lambda_{ang}\left(=2\right)$ is a constant.
These two components measure the structure similarity between a pair of joints.
Due to the consideration of length ratios and relative angles, the joint matching score $S_{jnt}(.,.)$ is also invariant to scale, translation and rotation.
However, lengthy segments are more relevant to an object and should get a higher score. Thus, it is desirable to give a higher score to a pair of matched joints if the segment lengths corresponding to the joints are large.
In general database image chains are noisy, whereas sketch chains are correctly extracted.
However, lengthy segments are more relevant to an object and should get a higher score. Thus, it is desirable to give a higher score to a pair of matched joints if the segment lengths corresponding to the joints are large. This is captured by $S_{sz}$ and is defined as:
\begin{equation}
S_{sz} (x, y) = \min \left( \left(\left. l_{seg_x}^{C_1} + l_{seg_{x+1}}^{C_1}\right)\middle.\right., \left(\left. l_{seg_y}^{C_2}+ l_{seg_{y+1}}^{C_2}\right)\middle. \right. \right)
\label{weightFactorInJointMatchScore}
\end{equation}
where, $seg_x$ and $seg_{x+1}$ are the two segments on either side of a joint $x$. The information about individual segment lengths is also retained in the chain extraction stage for such a calculation.
§.§ Handling Skips
Given the scoring mechanism between a pair of joints, the match score between two chains can be determined by calculating the cumulative joint matching score of contiguous portions in the two chains. Although exact matching of such portions can be considered, due to intra-class shape variations, small partial occlusion or noise, a few non-object joints may occur in the object boundary portion of the chain. To handle these non-object portions, some skips need to be allowed. Thus, the problem is formulated as one that finds the longest almost-contiguous matching portion of the two chains that are to be matched. Since only descriptors are available at this stage, this matching is performed in the space of chain descriptors.
To find the longest match while minimizing the number of skips,To this end, a skip penalty $\alpha$ is considered for the skipped joints. Note that, the loss of shape information due to a skip depends on the complexity of the skipped joints. It has been observed that a sharper angle captures more shape information than a smoother one. Hence, a skipped joint with a sharper angle should be penalized more.
The sharpness ($S_x$) of any joint $x$ can be calculated by taking the deviation of the joint angle ($\theta_x$) from 180°(Fig. <ref>):
\begin{equation}
S_x = 1-\exp(-\left| \pi - \theta_x \right|)
\label{eqn:sharpnessOfAngle}
\end{equation}
Sharpness ($S_x$) of joint $x$ is calculated by determining the difference of joint angle ($\theta_x$) from $\pi$.
Furthermore, lengthier skips typically cause more loss in shape information. Therefore, to penalize skips based on its complexity, along with the sharpness of the skipped angle, the skip penalty ($\omega_x$) is also weighted by the average length of the segments on either side of a skipped joint $x$:
\begin{equation}
\omega_x = S_x + \lambda_{skc} \cdot \frac{\left(l_{seg_x} + l_{seg_{x+1}}\right)}{2}
\label{eqn:skipPenaltyWeight}
\end{equation}
where, to determine the penalty of the skipped segments relative to the chain, the length of each segment is normalized by the length of the chain. $\lambda_{skc}$ is a scalar constant that determines the relative effect of the two components.
§.§ Matching using Dynamic Programming
Towards finding almost-contiguous matches, one can formulate the match score $M(p_1,\allowbreak q_1, p_2, q_2)$ for the portion of the chain between joints $p_1$ and $q_1$ in chain $C_1$ and joints $p_2$ and $q_2$ in chain $C_2$. Let the set $\text{J}_1$ and $\text{J}_2$ denote the set of joints of chains $C_1$ and $C_2$ respectively in this interval. Also let $\text{JM}$ be a matching between $\text{J}_1$ and $\text{J}_2$ in this interval. We restrict $\text{JM}$ to obey the order constraint on the matches, i.e., if the joints $a_1$ and $b_1$ of the first chain are matched to the joints $a_2$ and $b_2$ respectively in the second chain, then $a_1$ occurring before $b_1$ implies that $a_2$ also occurs before $b_2$ and vice versa.
Also let $X(\text{JM})=\left\lbrace x \lvert (x,y) \in \text{JM}\right\rbrace$ and $Y(\text{JM})=\left\lbrace y \lvert (x,y) \in \text{JM}\right\rbrace$ be the set of joints covered by $\text{JM}$.
Then $M(p_1, q_1, p_2, q_2)$ is defined as:
\begin{equation}
\begin{split}
M(p_1, q_1, p_2, q_2) = \max_{\substack{\text{JM}\in \text{ ordered }\\
\text{matchings in} \\
\text{the interval} \\
(p_1,q_1) \text{and} \\
(p_2,q_2)}} \left( \sum_{\substack{(x, y) \in \\
\text{JM}}} S_{jnt}(x, y) - \sum_{\substack{x \in \\
\text{J}_1\setminus X(\text{JM})}} \omega^{1}_{x}\alpha^1 - \sum_{\substack{y \in \\
\text{J}_2\setminus Y(\text{JM})}} \omega^{2}_{y}\alpha^2 \right) \\
\end{split}
\label{matchingScoreWithStartAndEnd}
\end{equation}
Note that $\alpha^1$ and $\alpha^2$ may be different since while matching a sketch chain to an image chain, more penalty is given to a skip in the sketch chain ($\alpha=0.07$) since it is considered cleaner and relatively more free from clutter compared to an image chain ($\alpha=0.03$).
Now, the maximum matching score ending at the joint $q_{1}$ of $C_1$ and $q_{2}$ of $C_2$ from any pair of starting joints, is defined as:
\begin{equation}
M(q_1, q_2) = \max_{p_1, p_2} M(p_1, q_1, p_2, q_2)
\label{matchingScoreWithEnd}
\end{equation}
We also take the matching score of a null set ($p_1$$q_1$ or $p_2$$q_2$) as zero which constrains $M(q_1, q_2)$ to take only non-negative values.
Then, it is not difficult to prove that $M$ can be rewritten using the following recurrence relation:
\begin{equation}
\begin{split}
M(q_1, q_2)=\begin{cases}
0, & \text{if } q_1, q_2 = 0 \\
\max \begin{cases}
M(q_1-1, q_2-1) + S_{jnt}(q_1, q_2) \\
M(q_1-1, q_2) -\omega^{1}_{q_1} \alpha^{1} \\
M(q_1, q_2-1) - \omega^{2}_{q_2} \alpha^{2}\\
\end{cases}, &\text{otherwise}
\end{cases}
\end{split}
\label{matchScore}
\end{equation}
This formulation immediately leads to an efficient Dynamic Programming solution that computes $M$ for all possible values of $q_1$ and $q_2$ starting from the first joints to the last ones. A search for the largest value of $M(q_1, q_2)$ over all possible $q_1$ and $q_2$ will then give us the best almost-contiguous matched portions between two chains $C_1$ and $C_2$ in terms of the highest matching score. Fig. <ref> visually illustrates the Dynamic Programming-based matching procedure, where the chains are partially matched and a few joints are skipped while matching. This approach helps us to efficiently obtain a matching score between a pair of chains. Furthermore, to handle an object flip, we match by flipping one of the chains as well and determine the best matching score as the one that gives the highest score between the two directions. We call the final score between two chains $C_1$ and $C_2$ as the Chain Matching Score $CMS(C_1, C_2)$.
Partial matching of chains with small skips. Matched joints are indicated by the same marker and colored the same in the table (Best viewed in color).
Skipping of important joints can lead to a false positive match. However $\angle2\overline{C_1}3, \angle3\overline{C_1}4, \angle4\overline{C_1}5$ in sketch chain and corresponding angles $\angle2\overline{C_2}3, \angle3\overline{C_2}4, \angle4\overline{C_2}5$ in image chain is highly dissimilar leading to a low Global Angle Consistency score for these two falsely matched chains.
[Skipping of important joints can lead to a false positive match]
[$\angle2\bar{J_1}3, \angle3\bar{J_1}4, \angle4\bar{J_1}5$ in sketch chain and corresponding angles $\angle2\bar{J_2}3, \angle3\bar{J_2}4, \angle4\bar{J_2}5$ in image chain is highly dissimilar leading to a low Global Angle Consistency score for these two falsely matched chains]
(a) Skipping of important joints leads to false positive match. (b) Angle between two consecutive matched joints and the centroid of the matched portion of the chain ($CEN$) is highly dissimilar for the falsely matched chain
The entire operation of matching two chains takes $\mathcal{O}(n_{C_1}*n_{C_2})$ time, where $n_{C_1}$ and $n_{C_2}$ are the number of joints in chains $C_1$ and $C_2$ respectively. It has been observed that a chain typically consists of 12-17 joints leading to a running time of approximately 100-400 units of joint matching, which is not very high.
Note that, this DP formulation is similar to the Smith-Waterman algorithm (SW) <cit.>, which aligns two protein sequences based on a fixed alphabet-set and predefined matching costs. <cit.> use SW to perform matching between two images under wide-baseline viewpoint changes.
Our method is a slight variation from this since it performs matching based on a continuous-space formulation that measures the deviation from exact correspondence to handle deformation.
However, matching two chains by determining local joint correspondence alone sometimes leads to a globally inconsistent match as both deformation and skips of individual joints are allowed while matching. In Fig. <ref>, most of the joints are locally matched correctly in a similarity invariant way; but a few joints are skipped in the sketch chain and in the database chain, which leads to a globally inconsistent matching. This necessitates consideration of global consistency of the matched portions of chains for improved matching.
However, matching two chains by determining local joint correspondence may lead to a match of globally different shape. This necessitates us to consider global consistency of the matched portions of a pair of chains.
§.§ Intra-chain Angular consistency
Individual joints are matched in a scale, translation and rotation invariant way. Furthermore, while matching skipping of joints are allowed in both sketch and in the image. Therefore if some distinctive portion of the object shape gets skipped, it may lead to a false matching of globally dissimilar shape.
In Fig. X, most of the joints are locally matched in a similarity invariant way. However, a,b,cth joint in database chain and a,b,c th joint in the sketch chain are skipped, which leads to a globally different matching. Thus to maintain global shape similarity, it is necessary to consider the consistency of all the matched joints together.
§.§ Global Angle Consistency of the Matched Chains
The angle that any two consecutive matched joints make with respect to some global reference point will be similar for two correctly matched chains and different for falsely-matched chains.
However, the difference in angle will be high for most of the joints, if two shapes are globally dissimilar. The centroid of the matched portion of a chain is a robust point that can be used as a reference. Thus, to determine the Global Angle Consistency (GAC) between any two matched chains $C_1$ and $C_2$, we consider the centroid of the matched portion of the chain ($\overline{C}$) as the reference point and calculate the differences in angles that any two consecutive joints make:
\begin{equation}
GAC(C_1, C_2) = \exp\left(-\lambda_{ac}\cdot\frac{1}{N^{J}}\sum\limits_{i=1}^{N^{J}-1} \angle J_{1}^{i}\overline{C_{1}}J_{1}^{i+1} - \angle J_{2}^{i}\overline{C_{2}}J_{2}^{i+1}\right)
\label{eqn:intraChainGeometricConsistency}
\end{equation}
where, $N^J$ is the total number of matched joints between $C_1$ and $C_2$ and $\lambda_{ac}$ is a scalar constant. A higher value of $\lambda_{ac}$ indicates a harder constraint on the global shape similarity.
Fig. <ref> shows an example where a false match is rejected due to a low global angle consistency score. Note that the computation of the global angle consistency is quite fast as it involves only the matched joints and can be done from the descriptor directly without referring back to the corresponding images.
The chain matching score (CMS) is weighted by the global angle consistency score (GAC) to obtain the final chain similarity score of a pair of chains $C_1 \text{and } C_2$:
\begin{equation}
CS(C_1,C_2) = GAC(C_1, C_2) \cdot CMS(C_1, C_2)
\label{eqn:finalChainScore}
\end{equation}
This chain-to-chain matching strategy is used to match two image chains during indexing as well as a sketch chain to an image chain during image retrieval.
Offline Image Indexing for faster retrieval is considered next.
the differences in angles that any two consecutive joints make with respect to the centroid of the matched portion of the chain in both
If two shapes are gloabally similar, the angle that any two consecutive matched joints make with resepct to some reference point will be similar for both postively matched chains. However, in case of false positives, even though individual joints may get matched due to relaxation in our matching criterion,constraining
Note that each joint is intersection of two straight line-like segments. If two shapes are globally similar (in two positively matched chains), the angle that every matched segment makes with respect to a reference point should be similar.
Since only small skips are allowed, the centroid of the matched portion of a chain remains relatively incensitive to noise. Therefore we consider angle difference of every matched segment in two chains for calculating the global consistency:
If the difference of the angle that the endpoints of individual segment makes with the centroid of the chain is high,
Thus to maintain global shape similarity,
§ IMAGE INDEXING
Given a chain descriptor, matching it online with all chains obtained from millions of images will take a considerable amount of time. Therefore, for fast retrieval of images from a large dataset, an indexing mechanism is required.
Different indexing techniques have been considered in the literature for content-based image retrieval, viz. tree-based approaches using $k$d tree and its variants <cit.>, hierarchical k-means <cit.>, hashing <cit.> etc. These approaches exploit the vectorial representation of the extracted features and perform either exact or approximate nearest neighbor search. However, in our representation, the length of each chain is not fixed. Furthermore the matching score cannot be obtained as a direct accumulation of the scores of individual dimensions. It is also not possible to use metric-based indexing techniques in our case due to a violation of the triangle inequality <cit.>. These considerations rule out most of the possibilities such as $k$d tree, hashing etc. Therefore, in this work, an approach similar to hierarchical k-means <cit.> but using medoids instead of means is used, which has been found to perform comparable to the state-of-the-art indexing techniques <cit.>.
Due to the variability in the length of the descriptors, it is difficult to use metric-based data structures, such as k-d tree <cit.> or Vantage-Point tree <cit.>. Therefore, in this work, an approach similar to hierarchical k-means <cit.> is used,
[][] [][]
(a) Similar chains are clustered at the leaf nodes of the hierarchical k-medoid-based indexing structure. (b) Considering individual chain matching without global consistency check can lead to a false positive retrieval.
All the database chains are considered for indexing and a hierarchical structure is constructed by splitting the set of extracted chains into $k$ different clusters using the k-medoids algorithm <cit.>. Note that, because of the variable-length chain descriptors, k-means is inapplicable. At first, $k$ chains are chosen as the cluster centroids probabilistically using the initialization mechanism of k-means++ <cit.> which increases both speed and accuracy.
The remaining chains are matched to each medoid chain using Dynamic Programming-based partial matching algorithm and assigned to the closest one based on the matching score (Eqn. <ref>). However, due to partial matching, it is possible to get a high matching score for more than one medoids. Therefore, a chain is assigned to all the medoids for which the matching score is greater than some $Th_{ms}=(80\%)$ of the score of the closest medoid. This operation is then recursively performed on the individual clusters to determine the clusters at different levels of the tree. A leaf node of such a tree maintains a list of images of which at least one chain matches to the corresponding medoid chain.
Note that, since an image has multiple chains and even one chain can belong to multiple nodes in our approach, the same image can be present at multiple leaves (Fig. <ref>).
Given such a hierarchical chain tree constructed offline during indexing, we next discuss how to search in the Image Database given a query sketch.
Similar chains are clustered at the leaf nodes of the hierarchical k-medoid-based indexing structure.
§ IMAGE RETRIEVAL GIVEN A QUERY SKETCH
A user typically draws an object along its boundary <cit.>. From a touch-based device, the input order of the contour points of the object boundary is usually available. Therefore, sketch chains can be trivially obtained in an online retrieval system breaking them at turns in the drawing.
Offline line drawings can be decomposed in a manner similar to the edge-detected Images (Sec. <ref>).
However, chains with less than some $Th_{nj} (= 5)$ joints are discarded as they are very simple and can match to any non-informative portion of another chain. Finally, descriptors are determined in a manner similar to image chains (Eqn. <ref>).
§.§ Search in the Hierarchical Chain Tree
For each of these sketch chain descriptors, a search is performed in the hierarchical k-medoids tree. At every level of the tree, the query chain is matched with all the medoid chains and then the subtree of the best matched medoid is explored in a best-bin-first manner <cit.>. At first, a single traversal is performed through the tree following the best matched medoids at every level. This yields a small set of images corresponding to the best matched leaf medoid.
Since at every level the query chain can get a good match with more than one medoids, to consider those possible matches, all the unexplored branches along the path are added to a priority queue. After the first traversal, the branch closest to the query chain is extracted from this priority queue and explored further. The search procedure stops once a pre-determined number of database images are retrieved. For all these retrieved images, at least one chain of each image matches with the query chain. Note that, for multiple sketch chains, we get multiple sets of images from the leaf nodes of the search tree, all of which are taken for the next step.
For each of these sketch chain descriptors, a search in the hierarchical k-medoids tree yields a small set of images in which at least one chain for each image matches with the query chain in the tree (Fig. X). Note that, for multiple sketch chains, we get multiple sets of images from the leaf nodes of the search tree, all of which are taken for the next step. Given a set of retrieved images with corresponding matched chains, we devise a sketch-to-image matching strategy to rank the images. Since all chain matchings between a sketch and an image may not be retrieved from the hierarchical tree due to low similarity scores, we try to match the remaining chains of the sketch also with other chains of a shortlisted image to obtain the complete chain-matching information between the corresponding sketch and image.
The matching score of an image for a given sketch is then calculated based on the cumulative matching scores of individual matched chain pairs between the sketch and the image.
However, the actual object boundary may be split across multiple chains. Therefore it is necessary to consider all such matchings while determining the match score between a sketch and an image. However, such multiple matches may not be geometrically consistent with each other. Fig. <ref> shows a case where two chains individually match well in both the sketch and the image, but the matches are not geometrically consistent with each other. This necessitates us to consider the geometric consistency of the matched chains to discard false positive retrievals.
Since the actual object boundary may be split across multiple chains, it is necessary to consider geometric consistency of the matched portions of multiple chains for correct retrievals. Although such geometric consistency has been studied previously in the literature <cit.>, it is considered in a new context in this work.
Considering only individual chain matches without global consistency check can lead to a false positive retrieval.
§.§ Geometric consistency across multiple matched chains
The geometric consistency of the matched portions of a pair of chains $\mathbf{p}=(m(C_S),\allowbreak m(C_I))$ with respect to that of another chain pair $\mathbf{p^\prime}=(m(C^\prime_S), m(C^\prime_I))$, where $C_S$ and $C^\prime_S$ are the sketch chains and $C_I$, $C^\prime_I$ are the image chains, is measured based on two factors: a) distance-consistency $G_d(\mathbf{p}, \mathbf{p^\prime})$ and b) angular-consistency $G_a(\mathbf{p}, \mathbf{p^\prime})$.
The centroids of the matched chain portions can be obtained in a manner that is relatively robust to the presence of noise. Therefore, $G_d(\mathbf{p}, \mathbf{p^\prime})$ is defined in terms of the closeness of the distances between the chain centroids $d(m(C_S), m(C_S^\prime))$ in the sketch and $d(m(C_I), m(C_I^\prime))$ in the database image (Fig. <ref>). These distances are normalized by the total length of the matched portions of the corresponding chains in order to achieve scale insensitivity:
\begin{eqnarray}
%&G_{d}(\mathbf{p}, \mathbf{p^\prime}) = \exp \left( \lambda_{c} \cdot \left(1 - 1\middle/ \Omega\left( \frac{d(m(C_S), m(C_S^\prime))}{L_{S}}, \frac{d(m(C_I), m(C_I^\prime))}{L_{I}} \right) \right) \right)
&G_{d}(\mathbf{p}, \mathbf{p^\prime}) = \exp \left( \lambda_{c} \cdot \left(1 - \middle. \Omega\left( \frac{d(m(C_S), m(C_S^\prime))}{L_{S}}, \frac{d(m(C_I), m(C_I^\prime))}{L_{I}} \right) \right) \right)
\label{spatialConsistency}
\end{eqnarray}
where, $L_{S}$=length $(m(C_S)) +$ length$(m(C_S^\prime))$, $L_{I}$=length $(m(C_I))$+ length $(m(C_I^\prime))$ , $\lambda_{c}$(=1) is a scalar constant and $\Omega$ is defined in Eqn. <ref>.
Pairwise geometric consistency of the matched portions of a chain pair $\mathbf{p}=(C_S,C_I)$ with respect to $\mathbf{p^\prime}=(C^\prime_S, C^\prime_I)$ uses (i) the distances $d(C_S, C_S^\prime)$ and $d(C_I, C_I^\prime)$ between their centroids ($\overline{C}$) and (ii) the difference of angles $\left|\phi^{C_S}_{i} - \phi^{C_I}_{i}\right|$.
The next factor $G_a$ measures angular-consistency. To achieve rotational invariance, the line joining the corresponding chain centers is considered as the reference axis and the relative angle difference at the $i^{th}$ joint is determined (Fig. <ref>).
$G_{a}(\mathbf{p}, \mathbf{p^\prime})$ is defined using the average difference of such relative angles of all the individual matched joints in a chain:
\begin{equation}
G_{a}(\mathbf{p}, \mathbf{p^\prime}) = \exp \left( - \lambda_a \cdot \frac{1}{N^{J_{\mathbf{p}}}} \sum\limits_{i=1}^{N^{J_{\mathbf{p}}}} \left|\phi^{C_S}_{i} - \phi^{C_I}_{i}\right| \right)
\label{segmentwiseAngularConsistency}
\end{equation}
where $N^{J_{\mathbf{p}}}$ is the number of matched joints between $C_S$ and $C_I$ and $\lambda_{a}$(=2) is a scalar constant.
Since, both $G_d$ and $G_a$ should be high for consistent matching, we consider the pairwise geometric consistency $G(\mathbf{p}, \mathbf{p^\prime})$ as a product of the constituent factors:
$G(\mathbf{p}, \mathbf{p^\prime}) = G_d(\mathbf{p}, \mathbf{p^\prime}) \cdot G_a(\mathbf{p}, \mathbf{p^\prime})$.
Erroneously matched chains are typically geometrically inconsistent with others and one may have both geometrically consistent and inconsistent pairs in a group of matched pairs between a sketch and an image. Therefore, the geometric consistency $GC(\mathbf{p})$ for a matched pair $\mathbf{p}$ is taken to be the maximum of $G(\mathbf{p}, \mathbf{p^\prime})$ with respect to all other matched pairs $\mathbf{p^\prime}$: $GC(\mathbf{p}) = \operatorname*{max}_{\substack{\mathbf{p^\prime}}} G(\mathbf{p}, \mathbf{p^\prime})$. The max operator allows us to neglect the falsely matched pairs while considering only the consistent matched pairs.
Finally, the similarity score of a database image $I$ with respect to a sketch query $S$ is determined as:
\begin{equation}
Score(S, I) = \sum\limits_{\mathbf{p}\in \text{P}} GC(\mathbf{p}) \cdot CS( \mathbf{p})
\label{finalImageScore}
\end{equation}
where $CS(\mathbf{p})$ is the Chain Score for the chain pair $\mathbf{p}$ (Eqn. <ref>) and $\text{P}$ is the set of all matched pairs of chains between a sketch $S$ and an image $I$.
Since erroneously matched chains get very low score for consistency, effectively only the geometrically consistent chains are given weight for scoring an image.
This score is used to determine the final ranking of the database images, which can be used for ranked display of such images.
Fig. <ref> shows the complete retrieval framework. Results of experiments are considered next.
The entire Retrieval Framework
§ EXPERIMENTS
To evaluate the performance of our system, we created a database of $1.2$ million images, which contains $1$ million Flickr images taken from the MIRFLICKR-1M image collection <cit.>. In addition, we included $0.2$ million images from the Imagenet <cit.> database in order to have some common object images in our database.
In the experiments, the hierarchical index for 1.2 million images is generated with a branching factor of $32$ and a maximum leaf node size of $100$, which leads to a maximum tree depth of 6. Using this tree, we obtain around $1500$ similar images for a given sketch, for which geometric consistency (Eqn. <ref>) is applied to finally rank the list of retrieved images from the chain tree.
The whole operation for a given sketch typically takes $1-5$ seconds on a single thread running on an Intel Core i7-3770 3.40GHz CPU. The running time typically depends on the number of chains in the sketch and most of the processing time is consumed by the geometric verification phase.
However, this geometric consistency check can be trivially parallelized. Therefore, the time consumed by this stage can be scaled down almost linearly with the number of cores.
The hierarchical index for our dataset required only around $250$ MB of memory.
For fast online access of database chain descriptors during geometric verification and ranking of retrievals, all the descriptors for $1.2$ million images are loaded a priori in the memory, which additionally required approximately $9$ GB of memory. Note that, the chain descriptors can be distributed across multiple CPUs if such geometric consistency check is performed parallely. Furthermore, to make our approach work in a memory-constrained environment, for every sketch, only the descriptors corresponding to ˜$1500$ selected images may be loaded each time in the memory at runtime although this may slow down the process somewhat due to online disk access. Although the running time may increase slightly, this drastically reduces the additional memory requirement to only a few MBs. Note that either only requried chain descriptors can be loaded in the memory on the fly while measuring the geometric consistency or all the chain descriptors may be loaded a priori for fast online access. We observed a memory footprint of approximately 9 GB while also loading the chain descriptors for all 1.2M images.
Top retrieved images for 14 sketches from 1.2 million images. Retrieved images indicate similarity insensitivity and deformation handling of our approach. Chains are embedded on the retrieved images to illustrate the locations of the matchings. Multiple matched chains are shown using different colors. Correct, similar and false matches are illustrated by green, yellow and red boxes respectively (Best viewed in color).
Visual results for $14$ sketches of different categories of varying complexity are shown in Fig. <ref>. These clearly indicate insensitivity of our approach to similarity transforms (e.g positive retrievals of the swan sketch). Furthermore, due to our partial matching scheme, an object is retrieved even under a viewpoint change if a portion of the distinguishing shape structure of the object is matched (e.g $4^{th}$ image for swan). Global invariance to similarities as well as matching with flipped objects can be seen in the results for the sketches of swan and bi-cycle ($2^{nd}$ and $10^{th}$ retrieved image for swan, $2^{nd}$ and $9^{th}$ retrieved image for bi-cycle etc.).
It can be easily observed that the performance of our approach depends on the complexity/distinctiveness of the shape structure. False matches (e.g cross for the sketch of airplane; two adjacent clocks/cups for the sketch of spectacle; keychain, brush for the sketch of bottle in Fig. <ref>) typically occur due to some shape similarity between the sketch and an object in the image, the probability of which is higher when the sketch is simple and/or contains only one chain (e.g bottle).
The performance of our approach depends on the complexity/distinctiveness of the shape structure and it can be easily observed from To understand the characteristics of the missed retrievals as well in a controlled dataset, we also tested our system on the ETHZ extended shape dataset consisting of $385$ images of $7$ different categories with significant scale, translation and rotation variations. Fig. <ref> shows top retrieval results from the ETHZ extended shape dataset <cit.> for a few sketches. It can be observed that the accuracy of retrieval heavily depends on the quality of the sketch.
Fig. <ref> shows top retrieval results from ETHZ extended shape dataset <cit.> for some of the sketches, whereas Fig. <ref> details the performance of our approach for different categories.
As discussed earlier, the performance of our approach depends on the complexity/distinctiveness of the shape structure and the quality of the user sketch. In Fig. <ref>, a significant portion of the first swan ($7^{th}$ sketch) is circular and thus it matches to locally circular shapes. However, for a relatively better sketch of swan ($8^{th}$ sketch), the number of positive retrievals is higher.
Sometimes, the matched portions of two different shapes appear to be globally similar, which again leads to false positives at a few top positions (e.g matched portion of giraffe and mug for the sketch of hat in Fig. <ref>).
Top retrievals from ETHZ extended shape dataset <cit.> for few sketches.
In general, our approach performs worse when the object shape is very simple, i.e. the amount of distinctiveness in the shape structure is less. In Fig. X, the sketch of applelogo, due to simplicity in the shape structure, matches to lot of locally circular objects.
Furthermore, the retireval accuracy highly depends on the complexity of the drawn sketch. In Fig. X, it can be observed that significant portion of the Xth swan is circular and thus it matches to locally circular shapes.
Even when the object shape is well captured in both the sketch and the image, sometimes our approach retrieves an incorrect object if the distinctive object boundary portion(s) get skipped. Fig. <ref>(a) illustrates such a situation where skipping of very important object boundary portions in the image of “star” leads to a wrong retrieval. Furthermore, the matched portions of corresponding chains are globally similar. Therefore, even global angular consistency check fails to identify the false retrieval in this case.
This case cannot be easily addressed if we are to allow skips to handle noise in boundary extraction.
Note that, for a sketch of “star”, however, an image of apple does not get a high matching score because of asymmetricity in our chain matching criteria where we assume that the sketch chains are much less noisy than the image chains and hence skips are more heavily penalized in the sketch chains.
We assume that sketch chains are not noisy and it contains only object shape information. Therefore, to retrieve the intended object, we penalize skips in sketch chain heavily. In contrast, database chains are typically noisy and sometimes non-object boundary portion becomes part of the chain even within the object boundary portion. Thus, with a goal to match the object portion, we allow comparatively more skips in database chains.
Therefore, in case of star(sketch)-to-apple(image) matching, the chances that the distinctive portions of a star (i.e. the successive turning points) getting skipped is less. This also explains the reason behind very good retrievals for the objects whose shape is (almost) unique.
Fig. <ref>(b) also shows an example where allowing local deformation and skipping an informative joint lead to a false positive.
Further, It can be observed that for highly deformable objects, viz. giraffe, the performance is quite poor. Sofisticated algorithms can be used for handling high amount of deformation and is considered as a future work.
[][] [][]
Skipping of important joints and allowing local deformations lead to false positive retrievals. Matched joints are numbered same in both the sketch and the image.
Quantitative measurement of the performance of a large scale retrieval system is not easy due to the difficulty in obtaining ground truth data, which is currently unavailable for a web-scale dataset. Some common metrics to measure retrieval performances (F-measure, Mean Average Precision <cit.> etc.) use recall which is impossible to compute without a full annotation of the dataset. Therefore, to evaluate the performance of our approach quantitatively, we use the Precision-at-$K$ measure for different rank levels ($K$) for the retrieval results <cit.>. This is an acceptable measure since an end-user of a large scale Image Retrieval system typically cares only about the top results which must be good.
First, we separately evaluate the major components of our method to understand their effects. Then, we discuss the retrieval performance of the proposed algorithm in comparison to prior work on our large dataset. To study the properties of our algorithm in detail, the proposed algorithm is also evaluated on the ETHZ shape dataset <cit.>, on which it is possible to compute the recall as well. Finally, using these evaluations, we discuss the strengths and weaknesses of our approach.
§.§ Evaluation of major components
User sketches are highly subjective <cit.> and the retrieval performance depends on the quality of the user sketch. Therefore, to obtain a robust estimate of the performance, the system must be tested using a diverse set of user sketches of varying complexity. To this end, we use a dataset (<cit.>) of $50$ user-drawn sketches consisting of $10$ sketches for each of the five shape categories in the ETHZ shape dataset <cit.>, viz. applelogo, bottle, giraffe, mug and swan. This dataset provides a wide variety in terms of the quality of the sketch and is therefore an appropriate choice for evaluation purposes. For testing these sketches, we add the images of the ETHZ shape dataset <cit.> to our dataset of $1.2$ million images. Finally, for each of these sketches, we determine the number of correct matches in the top $50$ retrievals, where such counting has to be done manually as we do not possess any prior categorical information of the dataset images.
Two chain extraction strategies are used in our work to capture the object shape information: a) overlapping long chains from contour segment networks and b) the boundaries of the segmented object proposals.
To understand the relative benefits of these, we first measure the retrieval performance considering only a single method of chain extraction for all the database images at a time.
Table <ref> shows the retrieval scores for different object categories.
Allowing overlaps while extracting long chains from contour segment networks (OC+GC) compared to using only non-overlapping chains (NOC+GC) <cit.> increases the chance of covering the entire object boundary and hence gives superior results. Therefore its retrieval performance is better than using only non-overlapping chains (NOC+GC) <cit.>. When the chains are extracted using segmented object proposals (GOP+GC) <cit.>, improved results can be observed for a few categories. For some objects, segmentation produces better result and if the entire object is covered in a single segmented proposal, then an accurate chain corresponding to the object boundary can be extracted. Therefore, for such categories, viz. appelogo, swan etc., one obtains a good accuracy when chains extracted using segmented proposals are used.
Considerable performance improvement is observed for few categories when the chains extracted using segmented object proposals <cit.> are used instead (GOP+GC). This is primarily because very accurate and precise object boundary can be extracted for comparatively simpler shapes using these proposals.
Typically simpler shape categories have lesser distinctive portions in their shape structure.
Some object categories have comparatively lesser distinctive portions in their shape structure. For example, there is a subtle difference between any circular object and apple/applelogo. Therefore, for these categories, boundary representation without any deviation due to noise is essential.
If the entire object is covered in a segmented proposal, then an accurate chain corresponding to the object boundary can be extracted. Therefore, for few categories,viz. applelogo, swan retrieval accuracy increases when chains extracted using segmented proposals are used.
However, for other categories, the top proposals cover only small and/or ambiguous portions of the object(s). In such cases (giraffe, bottle), a better retrieval score is obtained when chains are extracted from the contour segment networks.
To utilize the benefits of both these mechanisms, chains extracted from these two methods are combined for all the database images (OC+GOP+GC) and significant improvement on retrieval performance can be observed as compared to the different ideas considered in isolation or compared to only using the non-overlapping chains <cit.>.
In case of apple/applelogo, proposals are typically better because smaller regions tend to get higher score among different proposals <cit.>.
Method Applelogo Bottle Giraffe Mug Swan
NOC+GC $21.4\pm2.6$ $10.2\pm2.4$ $7.8\pm2.7$ $84.4\pm3.2$ $50.9\pm2.8$
OC+GC $22.2\pm2.4$ $13.8\pm3.6$ $12.2\pm4.4$ $90.6\pm1.8$ $58.4\pm3.4$
GOP+GC $26.8\pm3.5$ $13\pm2.6$ $7.2\pm2.3$ $91\pm1.9$ $64\pm2.9$
OC+GOP $25.6\pm1.8$ $14.6\pm3.2$ $6.2\pm1.4$ $70.4\pm2.9$ $64.7\pm2.1$
OC+GOP+GC $27.6\pm2.83$ $14.6\pm3.2$ $12.4\pm4.3$ $93.8\pm1.5$ $65.1\pm2.2$
(ETHZ Models) 1|c|$54$ 1|c|$16$ 1|c|$20$ 1|c|$94$ 1|c|$80$
Percentage of true positive images in top 50 retrievals with the corresponding standard deviation using different chain extraction mechanisms. NOC: Non-Overlapping Chains <cit.>, OC: Chains with Overlap allowed, GOP: Chains extracted using Geodesic Object Proposal <cit.>. OC+GOP uses chains extracted from both methods. GC indicates performance with geometric verification. Last row details the performance when ETHZ Models <cit.> are used.
Table <ref> also shows the importance of considering geometric consistency of the matched chains (OC+GOP+GC vs. OC+GOP).
Recall that individual chains are matched in a scale, translation and rotation insensitive way.
Therefore considering geometric consistency between the matched portion helps us to reduce the score of falsely matched chains (Fig. <ref>(b)) and thereby eliminate them. Although significant performance gain is observed after applying geometric consistency, it is not possible to apply this step when only one chain is matched between a sketch and a database image.
In that case, the chain matching score (Eqn. <ref>) is considered as the final matching score of the image for a given sketch. Due to this, for some of the categories (viz. bottle, swan etc.) in Table <ref>, applying geometric consistency does not make much difference in the retrieval score.
Object categories vary in the complexity and uniqueness of their shape information.
Therefore, a wide variation in the retrieval accuracy can be observed in Table <ref> for different categories. For highly deformable objects,viz. giraffe, the performance is poor since our approach can only handle a similarity transform.
Furthermore, there is a considerable amount of texture variation and background clutter for the giraffe images in the ETHZ shape dataset <cit.>, making
it hard to extract good chains for this category.
Thus the extracted chains from either method fails to cover substantial and/or important portions of the object boundary, which also attributes to low retrieval score for the sketches of giraffe. For simpler shapes, viz. apple, bottle etc., many false positives get a good matching score as these shapes are relatively simpler and thus easy to match.
Typically, our approach performs well for object categories with a distinctive shape and where the chains can be extracted easily.
The influence of the sketch quality is also evident from the standard deviation in the retrieval accuracies.
Table <ref> also lists the performance of our approach for different categories using the fairly good quality ETHZ dataset model sketches <cit.>, for which much better performance was obtained.
Table <ref> also lists the performance of our approach for different categories when ETHZ model shapes <cit.> are used. In this case, the performance is better than the average performance on user-drawn sketches (<cit.>), which is expected since those sketches are computer generated and are therefore much cleaner.
Next, we show much more comprehensive retrieval results of the proposed algorithm and compare them to prior work.
§.§ Comparisons with Prior Work
First, both qualitative and quantitative comparison is performed on our large dataset. Then, to understand the behavior of our algorithm better, results are shown on a standard shape dataset (ETHZ shape dataset <cit.>). A comparative study is further performed on this small dataset and challenging cases of our approach are highlighted.
§.§.§ Million Image Dataset
To evaluate our system for large scale retrieval on different object categories, we asked $5$ random subjects to draw sketches for a variety of objects on a touch-based tablet and collected $75$ sketches. These sketches, along with $100$ sketches from a crowd-sourced sketch database <cit.>, containing $24$ different categories in total, are used for retrieval.
Non-availability of a public implementation of any prior work makes it difficult to have a comparative study with prior work.
Even though a Windows phone App () based on <cit.> is available, the database is not available to make a fair comparison to other algorithms. Hence, we re-implemented this algorithm <cit.> (EI) as well as another by <cit.> (TENSOR) and tested their algorithms on our database for the purpose of comparison. <cit.> did not provide complete implementation details in their publication and it is not trivial to make the method proposed by <cit.> run efficiently on a very large database. Furthermore, <cit.> did not show any result on a large scale dataset and <cit.> showed results only for 3 sketches. Hence, these methods were not compared against.
Precision (expressed as Percentage of true positives) at different ranks for $175$ retrieval tasks in $24$ categories on a dataset of $1.2$ million images. B: Best, W: Worst, A: Average performances are computed among sketches for each category and then averaged.
Method 3cTop 5 3c|Top 10 3c|Top 25 3c|Top 50 3c|Top 100 3c|Top 250
B W A B W A B W A B W A B W A B W A
<cit.> 30.8 7.5 14.7 30 7.1 13.7 24.8 7 12.9 20.8 7 12.3 16.5 5.8 10.2 9.4 3 5.7
<cit.> 36.7 20.8 23.4 34.2 17.9 21.5 30 15.3 19.5 27 13.8 17.5 22.2 11.2 14.8 15.7 7.8 10.5
<cit.> 80.8 42.5 60.8 72.5 38.3 53.6 54.7 29.3 39.5 40.3 20.7 28.5 31.8 16.3 22.2 23 12.5 16.5
OC+GOP+GC (Ours) 86.5 43.7 65.5 78.1 38.4 58.2 62.4 30.5 44.6 51.7 23.8 34.6 46.7 21.5 30.8 34.1 15 21.7
Table <ref> shows the performances of our algorithm in comparison with TENSOR <cit.> and EI <cit.> at different retrieval levels.
First, the precision is computed for all sketches of a given object category and then the best, worst and average retrieval scores of different categories are averaged over all $24$ categories. The significant deviation between the best and the worst retrieval performances indicate the diversity in the quality of the user sketches and the system response to it. It can be observed from Table <ref> that our method significantly outperforms the other two methods on this large dataset. Both TENSOR <cit.> and EI <cit.> consider edge matchings approximately at the same location in an image as that of the sketch and therefore, the retrieved images from their system contain the sketched shape only at the same position, scale and orientation while images containing the
sketched object at a different scale, orientation and/or position are missed leading to false retrievals in top few matches (Fig. <ref>).
Similar performance was observed by us on the Sketch Match app (), although a direct comparison with it is inappropriate since the databases are different.
To evaluate the advantage of the geometry-based verification step, we also show the retrieval performance with and without this step and it can be observed that the geometric consistency check improves our results substantially. It can also be observed that using two types of chain extraction strategy and considering global angular consistency improved the performance compared to only using the non-overlapping chains <cit.>.
Note that, due to the non-availability of a fully annotated dataset of a million images, it is extremely hard to use an automated parameter learning algorithm. Hence, parameters are chosen empirically by trying out a few variations. Better parameter learning/tuning could possibly improve the results further.
Top 4 results by (b) <cit.>, (c) <cit.> and (d) our system on a 1.2 million image dataset for some sample sketches (a).
§.§.§ ETHZ Dataset
tableComparison of % of true positive retrievals in top 20 using our $63$ sketches and ETHZ models <cit.> on ETHZ dataset <cit.>
Method 3cOur Sketches ETHZ
Best Worst Avg Models <cit.>
TENSOR <cit.> 17 10 13.5 13.6
EI <cit.> 46 6 26.7 27.9
NOC+GC <cit.> 60 28 42.9 49.3
OC+GOP+GC 76 37 56.3 76.4
figureRetrieval performance of the proposed algorithm for different categories of the ETHZ shape dataset <cit.>
To provide comparisons on a standard dataset and study the recall characteristics which is difficult for a large dataset, we tested our system on the ETHZ shape dataset <cit.>. consisting of 283 images of 5 different categories with significant scale, translation and rotation variation. We used the user-drawn sketches of <cit.> for evaluation purposes.
Although standard sketch-to-image matching algorithms for Object Detection that perform time consuming online processing certainly perform better than our approach on this small dataset, such comparison would be unfair since the objectives are different. Hence, we compare only against TENSOR <cit.> and EI <cit.>. In this dataset, we measure the percentage of positive retrievals in top $20$ retrieved results which also gives an idea of recall of various approaches since the number of true positives is fixed.
Table <ref> shows the best, worst and average performance for the different sketches in a category (as for the previous dataset). It can be seen that our method performs much better than other methods on this dataset as well.
The retrieval performance on ETHZ models <cit.> further indicates substantial advantage of using very good sketches.
The performance on ETHZ models <cit.> is better than the average performance, which is expected since those sketches are computer generated and are therefore cleaner.
Comparison of Percentage of true positive retrievals in top $20$ using $50$ sketches of <cit.> and ETHZ models <cit.> on ETHZ dataset <cit.>.
Method 3cOur Sketches ETHZ Models
Best Worst Avg <cit.>
TENSOR <cit.> 17 10 13.5 13.6
EI <cit.> 46 6 26.7 27.9
NOC+GC <cit.> 60 28 42.9 49.3
OC+GOP+GC (Ours) 76 37 56.3 76.4
§ CONCLUSIONS
We have proposed an efficient image retrieval approach via hand-drawn sketches for large datasets. To the best of our knowledge, this is the first major work in the field of large scale sketch-based image retrieval that handles rotation, translation, scale and small variations of the object shape even for a dataset consisting of millions of images. This is accomplished by representing the images using chains of contour segments that have a high probability of containing the object boundary. using two complimentary methods. We have argued the flexibility of the system for adapting different boundary representation mechanisms.A similarity-invariant variable length descriptor is proposed that is used to partially match two chains in a hierarchical indexing framework. We also proposed a geometric verification scheme for improving the search accuracy. Experimental results shown on different datasets clearly indicate the benefits of our approach compared to the existing methods.
Due to similarity-invariance of our approach as compared to other relevant work, our method could be used to efficiently search in large natural image databases, which typically have a lot of variations. The proposed method could also open the window for efficiently searching in constrained image databases, viz. personal photo albums, which typically do not contain any tag/text information. Furthermore, our method could be augmented by other techniques such as text for tagging images in an offline fashion. or for improving online results.
One major issue of our approach is the difficulty in extracting “good” representative chains in the presence of considerable background clutter. Newer and better boundary extraction mechanisms can be easily adapted to our framework for improving the quality of the chains.
Furthermore, the proposed sketch-to-image matching approach is only similarity-invariant and also fails to handle substantial deformation and major viewpoint changes. A sophisticated affine or even projective-invariant matching mechanism could possibly help retrieve images with such variations as well and can be considered in future work.
We have proposed a sketch-based fast image retrieval approach for large datasets that, unlike any prior work, handles rotation, translation, scale and small deformations of the object shape.
This is accomplished by first extracting the essential shape information from all database images in an offline stage. To achieve this, two efficient and often complementary methods are proposed using a) contour segment network, b) segmented object proposals and multiple but a small number of long sequences of contour segments or chains are extracted. These chains are then represented using a variable length descriptor that is locally insensitive to scale, translation and rotation. An efficient Dynamic Programming-based partial matching algorithm is proposed to match these chain descriptors while handling small amount of deformation and noise.
This matching mechanism is used to generate a hierarchical k-medoids based indexing structure for the extracted chains of all database images in an offline process.
Exploiting our partial matching strategy, an image chain is assigned to one or more clusters at every level of the tree to which it is close in terms of chain matching score. Given a sketch query, this indexing structure facilitates very fast online retrieval of possible candidate images.
Finally a geometric verification step for these candidate images is used to reduce the number of false positives. Extensive experiments performed on a $1.2$ million Image Database clearly indicate significant performance improvement over other existing
One major issue of our approach is the extraction of “good” representative chains in the presence of considerable background clutter. Moreover, the proposed sketch-to-image matching approach fails to handle substantial deformation and major viewpoint changes. While we believe that it is difficult to address all these issues for an efficient web-scale retrieval with current processing power, it would be an interesting future work for further improving the accuracy of the system.
Furthermore, our method could be augmented by other techniques for tagging images in an offline fashion or for improving online results.
To understand the intricacies of our algorithm, a detailed analysis is performed using a standard shape dataset. Our method, augmented by other techniques, could also be used for tagging images in an offline fashion or for improving online results.
|
1511.00427
|
Edwards et al.
Influence of Non-Potential Coronal Magnetic Topology
$^{1}$ Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK
email: <[email protected]>
$^{2}$ Met Office, FitzRoy Road, Exeter, EX1 3PB, UK
$^{3}$ School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK
By comparing a magneto-frictional model of the low coronal magnetic field to a potential-field source-surface model, we investigate the possible impact of non-potential magnetic structure on empirical solar-wind models. These empirical models (such as Wang–Sheeley–Arge) estimate the distribution of solar-wind speed solely from the magnetic-field structure in the low corona. Our models are computed in a domain between the solar surface and 2.5 solar radii, and are extended to 0.1 AU using a Schatten current-sheet model. The non-potential field has a more complex magnetic skeleton and quasi-separatrix structures than the potential field, leading to different sub-structure in the solar-wind speed proxies. It contains twisted magnetic structures that can perturb the separatrix surfaces traced down from the base of the heliospheric current sheet. A significant difference between the models is the greater amount of open magnetic flux in the non-potential model. Using existing empirical formulae this leads to higher predicted wind speeds for two reasons: partly because magnetic flux tubes expand less rapidly with height, but more importantly because more open field lines are further from coronal-hole boundaries.
§ INTRODUCTION
The structure of the magnetic field in the solar corona plays an important role in many solar phenomena, not least the interplanetary magnetic field <cit.> and the solar wind <cit.>. On a global scale, the coronal magnetic topology has been studied in terms of the magnetic skeleton <cit.> and also in terms of quasi-separatrix structures <cit.>. It has been found that topological structures such as pseudostreamers and streamers can play an important role in the solar wind <cit.>. The network of streamers and pseudostreamers known as the “S-web” is thought to be connected to the acceleration of the slow solar wind via the release of plasma held in closed loops under these structures into coronal holes through magnetic reconnection <cit.>.
Operational forecasts by NOAA and the UK Met Office use the Wang–Sheeley–Arge model for the background solar wind <cit.>. This provides an extrapolation of the magnetic field out to $21.5\,\mathrm{R}_\odot$ (0.1 AU), along with an estimate of the radial solar-wind speed on that spherical surface. Since the model is based solely on static magnetic-field extrapolations, the speed is estimated purely from the three-dimensional magnetic-field structure, using an empirical relation (see Section <ref>). The topology of the low-coronal magnetic field is therefore an important element of these solar-wind forecasts. Once the magnetic-field components and speed at $21.5\,\mathrm{R}_\odot$ have been determined, these then act as the inner boundary conditions for the Enlil solar-wind model <cit.>, which simulates the Parker spiral to calculate the solar-wind speed and field direction at 1 AU or out even further to include Mars, Jupiter, or Saturn.
The standard Wang–Sheeley–Arge magnetic field is determined in two stages: the inner stage is a potential-field source-surface (PFSS) extrapolation to $2.5\,\mathrm{R}_\odot$ from an observed boundary condition at $1\,\mathrm{R}_\odot$ <cit.>. It assumes a purely radial magnetic field at $2.5\,\mathrm{R}_\odot$. This means that any change of polarity on the upper boundary (known at the source surface) will form a line of null points or “null line”. This null line forms the base of the heliospheric current sheet (HCS). The second stage of the model continues the HCS out to 0.1 AU ($21.5\,\mathrm{R}_\odot$). To do this, a second potential field is generated between $2.5\,\mathrm{R}_\odot$ and $21.5\,\mathrm{R}_\odot$, whose lower boundary condition is the absolute value of radial magnetic field ($|B_r|$) on the source surface of the inner PFSS extrapolation. A potential field is then extrapolated assuming that it decays to zero at infinity. Once this field is generated, it is reversed where it connects to a patch of field that was negative in the inner part, thus creating infinitesimally thin current sheets between oppositely directed fields. This is known as the Schatten current sheet model <cit.>. It is implemented primarily to smooth out latitudinal gradients in $|\mathbfit{B}|$, producing a latitudinally uniform heliospheric magnetic field in closer accordance with Ulysses observations <cit.>.
The aim of this article is to study the effect of replacing the innermost PFSS extrapolation in the WSA model (up to $2.5\,\mathrm{R}_\odot$) with a more sophisticated, non-potential, magnetic-field model. We focus on the qualitative differences that may be expected when the potential-field assumption is removed, as preparation for future validation using the Enlil model. The work is motivated by significant shortcomings of the potential-field assumption, which does not allow electric currents to form. Yet currents are manifestly present in the low corona: not only do we observe twisted magnetic structures directly, but the associated free energy required to power flares or coronal mass ejections is (by definition) not present in potential fields.
Recent years have seen the development of several non-potential models for the global coronal magnetic field <cit.>, ranging from full-MHD models including plasma thermodynamics <cit.> to static nonlinear force-free field extrapolations <cit.>.
<cit.> compared a full MHD model with the corresponding PFSS model for four Carrington rotations during Solar Cycle 23. They found that many features were similar between the two approaches, such as the boundaries of the coronal holes, although less open flux was found in the PFSS models. Notable differences were found in the heights reached by closed coronal loops, suggesting that the “source surface" used in the PFSS model should not be spherical.
An alternative, but less computationally expensive, model known as the current sheet source surface (CSSS) model has been presented by <cit.>. This sets a cusp surface (normally at $2.5\,\mathrm{R}_\odot$) and an outer source surface (normally $15\,\mathrm{R}_\odot$) where the field lines are forced to be radial. The model is based on a magnetohydrostatic solution that includes large-scale horizontal currents, in addition to sheet currents above the cusp surface (similar to the Schatten current-sheet model). This model has been validated against the PFSS model and also against solar-wind observations <cit.> and has been found to provide better correlation with in situ solar-wind observations than the PFSS model.
Here, we compare PFSS extrapolations with the magneto-frictional model, in which electric currents and free magnetic energy are built up quasi-statically as the coronal magnetic field is sheared by surface-footpoint motions <cit.>. The model is sufficiently simple to simulate continuously months to years of coronal evolution, but sufficiently detailed to include the time-dependent build up of electric currents, magnetic helicity, and free magnetic energy. The resulting magnetic topology is substantially different from PFSS extrapolations, with the formation of twisted magnetic flux ropes and very different magnetic connectivities. It was previously shown by <cit.> that the presence of current causes magnetic-field structures to expand and so influences the amount of open field present. Near solar maximum the effect is particularly important, since the global magnetic field is dominated by active region fields, which can be highly non-potential. Thus we expect a significant difference in the predicted solar wind compared to the PFSS-based WSA model, as we will demonstrate in this article.
§ CORONAL MAGNETIC FIELD MODELS
In this article, we compare a potential-field source-surface (PFSS) model and a non-potential (NP) model based on the magneto-frictional method for two dates: 4 April 2000 and 30 April 2013. Figure <ref> shows that the two models lead to different three-dimensional magnetic-field structures, which we will analyse in Section <ref>. Both dates are taken from solar maximum, the first from the Cycle 23 maximum and the second from the Cycle 24 maximum. We chose solar maximum dates as these are expected to show more significant differences between the two models, as well as being the phase when the solar-wind structure is least well understood.
We have used identical boundary conditions for the photospheric radial magnetic field in the PFSS and NP models. The NP model for a particular day must be generated by evolution over a sufficient integration time, during which the evolution of $B_r$ on the full solar photosphere is required. Accordingly, the NP model has been evolved in tandem with a surface flux-transport simulation for the photospheric radial field <cit.>, as described by <cit.>. The simulation was initiated on 15 June 1996 and new bipolar magnetic regions assimilated during the evolution, based on US National Solar Observatory/Kitt Peak and SOLIS (Synoptic Optical Long-term Investigations of the Sun) data[<http://solis.nso.edu/vsm/vsm_maps.php>]. This integration time is sufficient to allow reasonable electric currents to build-up in the NP model in a self-consistent way. The minimum integration time necessary for i) the total coronal current and ii) the open magnetic flux to reach steady levels is only about two months, although <cit.> found that the topology of high-latitude magnetic structures can have a memory of two years or more in the NP model.
§.§ Potential-Field Source-Surface Model
A potential field is extrapolated from the boundary condition at $1\,\mathrm{R}_\odot$ where the radial component of the magnetic field is taken from the flux transport model. Additionally the assumption is made that at $2.5\,\mathrm{R}_\odot$ the field is purely radial. The potential field is found by solving Laplace's equation in spherical coordinates and the solution is given in terms of spherical harmonics. Ideally, an infinite number of harmonics should be included; however, for numerical reasons it is necessary to truncate this sum and in this case we sum up to a maximum harmonic number $l_{\mathrm{max}}=85$. This model is static and is extrapolated from the simulated photospheric magnetic field on a particular day.
§.§ Magneto-frictional Model
The model is described in more detail by <cit.>. In contrast to static potential field extrapolations, the magneto-frictional model follows the continuous time-evolution of the coronal magnetic field, driven by the photospheric evolution. The developing magnetic field structure is significantly more complex than the potential field, with large-scale electric currents both in active regions and in the quiet Sun <cit.>.
The time evolution of the magnetic field is found by solving the uncurled induction equation for the vector potential [$\mathbfit{A}$],
\begin{equation}
\frac{\partial \mathbfit{A}}{\partial t}=\mathbfit{v} \times \mathbfit{B}-\mathbf{\mathcal{E}},
\label{eq:induction}
\end{equation}
where $\mathbfit{B}=\nabla \times \mathbfit{A}$. Ohmic diffusion is neglected but we include a hyperdiffusion of the form
\begin{equation}
\mathbf{\mathcal{E}}=-\frac{\mathbfit{B}}{B^2}\nabla \cdot (\eta_4B^2\nabla \alpha),
\end{equation}
\begin{equation}
\alpha = \frac{\mathbfit{B} \cdot \mathbfit{j}}{B^2}
\end{equation}
is the current helicity density, $\mathbfit{j}=\nabla \times \mathbfit{B}$ is the current density and $\eta_4=10^{11}$ km$^4$ s$^{-1}$ <cit.>. The hyperdiffusion simulates the mean effect of small-scale turbulence in the coronal magnetic field, allowing magnetic reconnection but preserving magnetic helicity in the volume.
The velocity is determined using the magneto-frictional technique <cit.> and is given by
\begin{equation}
\mathbfit{v}=\frac{1}{\nu}\frac{\mathbfit{j}\times \mathbfit{B}}{B^2}+v_{\mathrm{out}}(r)\mathbfit{e}_r.
\end{equation}
The first term causes the system to relax to a force-free equilibrium, while the second term is a radial outflow imposed near $r=2.5\,\mathrm{R}_\odot$ to ensure that the field is approximately radial there, whilst allowing horizontal magnetic structures such as flux ropes to be ejected through the boundary. It is important to note that this outflow velocity is uniform in $(\theta,\phi)$ and is imposed, so it cannot be used to model the spatial and temporal distribution of solar wind speed. Rather, in Section <ref>, we consider an empirical model of the solar wind speed based only on the magnetic structure, as in the WSA model.
Magnetic field lines for 4 April 2000 (top row) and 30 April 2013 (bottow row) for the NP model (left column) and PFSS model (right column). The solar surface is shaded with $B_r(R_\odot)$ (white positive, black negative).
§.§ Schatten Current-Sheet Model
As mentioned earlier, in the forecasting of the solar wind it is necessary to know how the magnetic field behaves between the $2.5\,\mathrm{R}_\odot$ source surface and the 0.1 AU inner boundary of the Enlil code. In the volume between $2.5\,\mathrm{R}_\odot$ and $21.5\,\mathrm{R}_\odot$ a potential field is extrapolated outward from the absolute value $|B_r|$ at $2.5\,\mathrm{R}_\odot$, assuming all magnetic-field components decay to zero at infinity. The original field direction is then restored along field lines originating in regions of negative $B_r$ on the $2.5\,\mathrm{R}_\odot$ source surface. This produces infinitesimally thin current sheets and is known as the Schatten current-sheet model <cit.>. The main advantage of this model is that it spreads the distribution of magnetic flux across the outer boundary more evenly than that found from a simple radial extrapolation of the field.
We implement the Schatten current-sheet model in the same way for both the PFSS and NP models, using their respective distributions of $B_r$ on the source surface at $2.5\,\mathrm{R}_\odot$. Note that our model contains only sheet currents in the outer region, unlike the Current Sheet Source Surface model of <cit.>, which also includes horizontal volume currents, controlled by a single additional free parameter. In principle the latter may allow for improved agreement with solar-wind observations if the part of this model below the cusp surface ($2.5\,\mathrm{R}_\odot$) was replaced by the NP model, but here we simply employ both of our lower coronal models in conjunction with the original Schatten model.
Magnetic-field lines traced in the plane of sky for the combined NP and Schatten current-sheet model for 30 April 2013. Red-dashed lines show field directed out of the Sun, blue-solid lines show field directed into the Sun. The NP model is in the inner shell, while the current sheet model is in the much larger outer shell.
Figure <ref> shows some field lines traced in the plane of sky for the combined NP and Schatten current-sheet model for 30 April 2013. There are infinitesimally thin current sheets between the inwardly directed (blue solid) field lines and the outwardly directed (red dashed) field lines. In the Schatten current-sheet part of the model, the field lines are mostly radial except near to the $2.5\,\mathrm{R}_\odot$ boundary where they spread out around streamer and pseudostreamer structures.
§ EFFECT ON MAGNETIC TOPOLOGY
We investigate the magnetic topology in several ways. Firstly we consider the regions of open field on the photosphere by tracing field lines. This will highlight any differences in the sizes and shapes of the footpoints of open-field regions. Secondly, we consider the magnetic skeleton <cit.>, which is a network of null points and their associated separatrix structures that divide space into topologically distinct flux domains. In addition, we look for regions where field lines are highly divergent. This is often known as the quasi-skeleton and is measured by a quantity known as the “squashing factor" or $Q$ <cit.>.
§.§ Open-Field Regions
The difference in size and shape of photospheric open-field regions between the two models is evident in Figure <ref>, which maps the location of coronal-hole footpoints on the photospheric boundary for each case.
Map of open magnetic field regions on the photosphere for 4 April 2000 (top) and 30 April 2013 (bottom) in the NP field model (left) and the PFSS model (right). Black indicates closed field line footpoints, while white indicates open field line footpoints.
We can see that on both dates in the PFSS extrapolations (right column) the coronal holes are much smaller than in the NP model (left column). Also the polar coronal holes are not present in the PFSS model for 4 April 2000 (top right) whereas they are present in the NP model for the same date (top left). Otherwise, we can see that some coronal holes are seen in similar positions and with similar shapes in the PFSS and NP models; this is to be expected since they have a common lower boundary condition.
§.§ Magnetic Skeleton
In three dimensions the field lines that pass through a null point form two structures: a two-dimensional separatrix surface and a one-dimensional spine line <cit.>. When two separatrix surfaces intersect they form a separator. In global models, additional features of the magnetic skeleton are HCS curtains. These are special separatrix surfaces that are traced down from the null line on the $2.5\,\mathrm{R}_\odot$ boundary in PFSS models <cit.>.
We find the magnetic nulls using the trilinear method of <cit.>. From these null points we trace out the magnetic skeleton of separatrix surfaces using the method described by <cit.>. This method works by taking a ring of points in the fan plane of the null close to the null point, and mapping these points out along the field line until they hit either another null or the boundary. If they hit another null a separator is traced back to the initial null and the ring is broken and the points mapped along the spine.
The null point finding method <cit.> and similarly the skeleton finding method <cit.> have both been used to find the global coronal topology in PFSS extrapolations. We now expand on this work by applying these same methods in the global NP model of the solar corona.
Three-dimensional representation of the magnetic skeleton between $1\,\mathrm{R}_\odot$ and $2.5\,\mathrm{R}_\odot$ for 4 April 2000. Left shows the NP model and right the PFSS model. Red and Blue dots represent positive and negative null points, respectively. Thin blue, pink, and green lines represent field lines in the separatrix surfaces from negative nulls, positive nulls, and the HCS null line respectively. Yellow lines represent separators. Purple and orange lines represent the spines of negative and positive nulls, respectively. The solar surface is shaded with $B_r(R_\odot)$ (white positive, black negative). (Animated versions of these figures are available in the Electronic Supplementary Materials.)
Cuts through the magnetic skeleton at $2.5\,\mathrm{R}_\odot$ (top) and $21.5\,\mathrm{R}_\odot$ (bottom) for the NP model (left) and PFSS model (right) for 4 April 2000. The green lines represent the intersection of the HCS with the surface; the pink and blue lines represent the intersection of separatrix surfaces from positive and negative nulls, respectively, with the surface. The background is shaded with $B_r$ (white positive, black negative).
Three-dimensional representation of the magnetic skeleton between $1\,\mathrm{R}_\odot$ and $2.5\,\mathrm{R}_\odot$ for 30 April 2013. Left shows the NP model and right the PFSS model. Colours and symbols are as in Figure <ref>. (Animated versions of these figures are available in the Electronic Supplementary Materials.)
Figure <ref> shows the magnetic skeletons for the solar corona between $1\,\mathrm{R}_\odot$ and $2.5\,\mathrm{R}_\odot$ for the NP (left) and PFSS (right) models for 4 April 2000. Correspondingly, Figure <ref> shows the intersections of the separatrix surfaces with the spherical boundary surface at $2.5\,\mathrm{R}_\odot$. We see that in both models the heliospheric current sheet (hereafter HCS; thick-green line) is very warped, and in the case of the NP field (left) the HCS has split into two disjoint loops as is typical for solar maximum <cit.>.
Otherwise, the large-scale structure is broadly similar in the two models except that the PFSS field is much smoother than the NP field. There are, however, more separators and separatrix curtains (separatrix surfaces that reach the source surface) in the NP field. The twisting of the magnetic field when currents are present can cause more separators to form and multiple separators connecting the same pairs of nulls <cit.>. The greater number of separatrix curtains is associated with the greater amount of open field in the NP case. Again, this is due to the presence of currents, which cause the structures to expand, possibly past the $2.5\,\mathrm{R}_\odot$ boundary <cit.>. When structures expand close to this boundary, they tend to become open due to the outflow boundary condition imposed in the NP model (which mimicks the effect of the real solar wind). The additional open field is evident in the field-line plots in Figure <ref> and in the maps of the coronal holes in Figure <ref>.
Figure <ref> also shows the separatrix surfaces mapped out to $21.5\,\mathrm{R}_\odot$ for the NP and PFSS models. Although this results in a topologically equivalent pattern, the separatrices map to different latitudes and/or longitudes. For example, the region of open field inside the detached HCS loop (in the NP model) occupies a smaller proportion of the spherical surface at $21.5\,\mathrm{R}_\odot$ than at $2.5\,\mathrm{R}_\odot$, implying that field in this region is expanding sub-radially.
Similarly, in Figures <ref> and <ref> we can compare the magnetic skeletons for 30 April 2013. Here it is clear that the PFSS model is again much smoother, both in the shape of the HCS and also in the shapes and positions of the separatrix curtains. The positive and negative separatrix surfaces in the northern hemisphere interact much more in the NP model, creating many more and also longer separators than in the PFSS model. The HCS itself is much more distorted in the NP model, crossing some longitudes many times. The magnetic field in these folds then contracts sub-radially out to $21.5\,\mathrm{R}_\odot$. Several of the open field regions also contract sub-radially in the PFSS model, with some becoming too small to distinguish at $21.5\,\mathrm{R}_\odot$.
A notable difference between the NP and PFSS models is in the magnetic surfaces traced down from the HCS. In the PFSS model, field lines in the surfaces traced from the HCS all map down to the photosphere and divide open-field regions from closed-field regions. In the NP model this is not always the case. If, for instance, a flux rope is sitting below the base of the HCS and aligned along it, then field lines traced down from the HCS can pass below the flux rope and loop back up to the $2.5\,\mathrm{R}_\odot$ boundary. Such a structure can clearly be seen in the example from 30 April 2013. A 3D image showing example field lines of this type is shown in Figure <ref> (field lines highlighted in purple are “U-shaped"). In Figure <ref> (left), these “U-shaped” field lines create an apparent excursion of the green HCS curve into the negative polarity region. This excursion is not a polarity-inversion line but merely indicates the opposite end of these field lines traced from the true HCS.
Cuts through the magnetic skeleton at $2.5\,\mathrm{R}_\odot$ (top) and $21.5\,\mathrm{R}_\odot$ (bottom) for the NP model (left) and PFSS model (right) for 30 April 2013. The green lines represent the intersection of the HCS with the surface; the pink and blue lines represent the intersection of separatrix surfaces from positive and negative nulls, respectively, with the surface. The background is shaded with $B_r$ (white positive, black negative).
Field lines traced from HCS in the NP model for 30th April 2013. Field lines that map down to the photosphere are coloured green. Field lines in the HCS curtains that wrap under the flux rope and map back up to the outer boundary are coloured purple. Field lines are shaded with height above the photosphere with white being closest to the solar surface.
Observationally, the HCS and associated separatrix surfaces are identified as streamers or helmet streamers. Separatrix curtains from null points also form streamer-like structures. These structures differ from helmet streamers because there is no polarity change across the streamer and so they are known as pseudostreamers. The difference is illustrated in Figures <ref>a and b. In Figures <ref> to <ref>, the green magnetic surfaces traced down from the HCS correspond to streamers, whereas the separatrix curtains (pink and blue surfaces) cutting the outer boundary correspond to pseudostreamers. It should be noted that not all of these streamer structures would be observed in coronagraphs as the streamer needs to be aligned along the line of sight in order to be visible.
Sketch of the cross section of a streamer (a), a pseudostreamer (b) and a double streamer (c).
The radial inflation of the field in the NP model can lead to structures that appeared as pseudostreamers in the PFSS model becoming double streamers in the NP model (see Figure <ref>b and c). Effectively the null point has risen out of the domain. A good example of this is seen in Figure <ref>: where the HCS has split into two loops in the NP model, there is a separatrix surface in the PFSS model. We note that it is possible to have double streamers also in the PFSS model, as would occur if the central photospheric polarity is strong enough <cit.>.
§.§ Squashing Factor
As well as examining the magnetic skeleton we also examine the “quasi-skeleton”. We trace field lines down from the $21.5\,\mathrm{R}_\odot$ outer boundary to the photosphere, $1\,\mathrm{R}_\odot$, and calculate the squashing factor [$Q$] of this mapping, using the definition of <cit.> appropriate for spherical coordinates:
\begin{equation}
Q = N^2/|\Delta|
\end{equation}
\begin{equation}
N^2=\frac{R_*^2}{R^2}\left[\left(\frac{\sin(\Theta)}{\sin(\theta)}\frac{\partial \Phi}{\partial \phi}\right)^2+\left(\sin(\Theta)\frac{\partial \Phi}{\partial \theta}\right)^2+\left(\frac{1}{\sin(\theta)}\frac{\partial \Theta}{\partial \phi} \right)^2+\left(\frac{\partial \Theta}{\partial \theta}\right)^2\right],
\end{equation}
\begin{equation}
\Delta=B_r/B^*_r,
\end{equation}
where $(\Theta(\theta,\phi),\Phi(\theta,\phi))$ is the field line mapping from a sphere of radius $R$ to one of radius $R_*$ and $B_r$ and $B^*_r$ are the components of the magnetic field normal to the boundary at each end of the field line.
This is a measure of gradients in the field-line mapping; so-called quasi-separatrix layers (QSLs) of high $Q$ are locations where nearby magnetic-field lines diverge strongly.
Squashing factor [$Q$] calculated at $21.5\,\mathrm{R}_\odot$ boundary for 4 April 2000 (top row) and 30 April 2013 (bottom row) for the NP (left column) and PFSS (right column) models.
Figure <ref> shows the squashing factor on this upper boundary. We see that many of the structures match the locations of the separatrix-surface cuts through the $21.5\,\mathrm{R}_\odot$ boundary shown in Figure <ref> and Figure <ref> (bottom rows). These structures are true discontinuities in the field line mapping. However, we see extra features in $Q$ that do not appear in the skeleton. While separatrix curtains in the skeleton separate disconnected coronal holes at the photosphere, the additional QSLs seen in the map of $Q$ represent locations where open field lines undergo rapid but continuous changes in footpoint location from one part of the photosphere to another. They typically divide open-field regions of the same polarity between different photospheric coronal holes that are connected by narrow corridors of open field (since the mapping is continuous). This is the basis of the “S-web” model of <cit.>, who suggest that the presence of these narrow open-field corridors can explain why slow solar wind is found across a wide angular range on the Sun, not just at the boundaries of polar coronal holes. In this scenario, our comparison indicates that the NP model provides more source regions for the slow solar wind than the PFSS model.
We note that the spiral features of high $Q$ seen near the South Pole at approximately 100$^\circ$ longitude on 4 April 2000 and approximately 340$^\circ$ longitude on 30 April 2013 are associated with field lines twisting around the poles as the Sun rotates; the simulation domain of the NP model extends to only $89.5^\circ$ so the mapping cannot be continuous across the poles.
§ EFFECT ON EMPIRICAL WIND SPEEDS
In the WSA model, radial wind speed at $21.5\,\mathrm{R}_\odot$ is determined by an empirical formula based solely on the modeled magnetic-field structure. For example, one current implementation (based on GONG and NSO magnetograms) uses the formula (C.N. Arge, private communication)
\begin{equation}
v_r(\theta,\phi)=240.0+\frac{675}{(1+f_\mathrm{s})^{1/4.5}}\left[1.0-0.8e^{(-(\theta_\mathrm{b}/1.9)^{2})}\right]^{3} \mathrm{km}\,\mathrm{s}^{-1},
\label{eqn:vr}
\end{equation}
where $f_\mathrm{s}(\theta,\phi)$ measures the local radial expansion of a magnetic flux tube, and $\theta_\mathrm{b}(\theta,\phi)$ is the angular distance of the field line's photospheric footpoint from the nearest coronal-hole boundary. This is a slight modification of the formula discussed by <cit.>. Here we study how the factors $f_\mathrm{s}$ and $\theta_\mathrm{b}$ differ between the NP and PFSS models. For completeness, we show the corresponding $v_r$ distributions computed with Equation (<ref>), but we caution that the numerical factors in this formula have been optimised to best match solar-wind observations for the PFSS model. It is known that these empirical wind models are sensitive to the type of magnetic-field model used <cit.>. Optimizing such an empirical formula for the NP model will be carried out in the future.
§.§ Flux Tube Expansion
Flux-tube expansion [$f_\mathrm{s}$] at $21.5\,\mathrm{R}_\odot$ for 4 April 2000 (top) and 30 April 2013 (bottom) for the NP model (left) and the PFSS model (right).
Figure <ref> shows the flux-tube expansion [$f_\mathrm{s}$] calculated between $1\,\mathrm{R}_\odot$ and $2.5\,\mathrm{R}_\odot$ and then mapped along the field lines to $21.5\,\mathrm{R}_\odot$ for each of the cases. This quantity was introduced by <cit.> as a proxy for the solar-wind speed in purely magnetic models. Greater flux-tube expansion was shown to correlate with slower wind speed. The flux-tube expansion is defined as the ratio of the magnetic-field components at each end of the field line normal to the two boundaries,
\begin{equation}
\end{equation}
where, in our case, $\mathrm{R}_\mathrm{s}=2.5\,\mathrm{R}_\odot$.
We see that in all cases the pattern of $f_\mathrm{s}$ on the $21.5\,\mathrm{R}_\odot$ surface is structured by the separatrix-surface cuts (Figures <ref> and <ref>), but clearly also reflects the variations in field-line divergence shown by the squashing factor [$Q$] (Figure <ref>). Overall there is a greater expansion in the PFSS models (right-hand column) than in the corresponding NP models (left-hand column). This is because the photospheric open-field regions that correspond to the base of coronal holes are much smaller in the PFSS model (see Figure <ref>). Since all field must be open at $2.5\,\mathrm{R}_\odot$, smaller photospheric open-field regions will imply greater expansion factors.
§.§ Coronal Hole Boundary Distance
<cit.> proposed a relationship between the wind speed on an open magnetic field-line and the distance of that field line's footpoint from the nearest coronal-hole boundary. It was incorporated into the WSA model by <cit.> to account for discrepancies in wind speeds along field lines that had the same flux-tube expansion factor. In fact, it is a much more important factor in determining the wind speed in Equation (<ref>) than the flux-tube expansion <cit.>. We measure it as the angular separation of the photospheric footpoints from their nearest coronal-hole boundary. We describe the boundaries of the coronal holes (as shown in Figure <ref>) using a series of points, and for each open-field-line footpoint we calculate the great-circle distance to the nearest coronal-hole boundary point. From this we then calculate the angular separation. Figure <ref> shows maps of this quantity for field lines traced down from $21.5\,\mathrm{R}_\odot$ to $1\,\mathrm{R}_\odot$. The largest distances from the edges of coronal holes are seen on 30 April 2013 in the NP model (bottom left Figure <ref>) since this frame contains the largest coronal holes, as seen from Figure <ref> (bottom-left).
Distance [$\theta_\mathrm{b}$] from coronal hole boundary of footpoints of fieldlines traced from $21.5\,\mathrm{R}_\odot$ for 4 April 2000 (top) and 30 April 2013 (bottom) for the NP model (left) and the PFSS model (right).
§.§ Empirical Wind Speed
Distribution [$v_r(\theta,\phi)$] of empirical solar wind speed at $21.5\,\mathrm{R}_\odot$ for 4 April 2000 (top) and 30 April 2013 (bottom) for the NP model (left) and the PFSS model (right).
Empirical solar-wind speed in the plane of sky viewed from Carrington longitude $0^\circ$ on 4 April 2000 (left) and 30 April 2013 (right). Red-dashed shows NP model, Blue-solid shows PFSS model.
Figure <ref> shows the empirical solar-wind speed at $21.5\,\mathrm{R}_\odot$ calculated using Equation (<ref>). The highest wind speeds occur in the NP field model and out of the two examples given the wind was faster on 30 April 2013. The wind speed calculated using the potential field for 4 April 2000 shows the overall slower wind speed. Visual comparison of Figure <ref> with Figures <ref> and <ref> shows that the distribution of empirical wind speed derives primarily from the coronal-hole boundary distance [$\theta_\mathrm{b}$], rather than the flux-tube expansion [$f_\mathrm{s}$]. This is a result of the formula in Equation (<ref>), and is in accordance with <cit.>.
Figure <ref> shows a polar plot of the empirical solar-wind speed in the plane of sky, from one particular viewing angle. The speeds for the NP (red) and PFSS (blue) models are overlaid. The additional solar-wind structure predicted by the NP model at lower latitudes is evident, particularly on 30 April 2013.
The highest wind speeds correspond to locations where there are large coronal holes. The southern polar coronal hole shows speeds of over 650$\,{\rm km}\,{\rm s}^{-1}$ in the NP model on 4 April 2000, but this coronal hole is not present in the PFSS model for the same date. This is clearly visible in Figure <ref> (left) where we see fast wind at both poles in the NP model but not in the PFSS model. On 30 April 2013, both models have substantial polar coronal holes, but the wind speeds in the NP model are higher owing to the reduced horizontal expansion of the coronal holes with height, compared with the PFSS model (cf. Figure <ref>).
At lower latitudes, Figure <ref> clearly shows additional sub-structure in the NP wind speed as compared to the PFSS wind speed. Such differences would be observed at 1 AU as additional temporal fluctuations in the predicted wind speed, as compared to current WSA forecasts. The difference reflects both the larger number of low-latitude coronal holes in the NP model, but also the more complex magnetic structure of the NP model. The latter was demonstrated by the more complex skeleton and quasi-skeleton found in Section <ref>.
§ CONCLUSIONS
In summary, we have shown that removing the potential field (current-free) assumption used in solar-wind models such as WSA could have an important impact on predicted wind-speed distributions. In comparing magneto-frictional non-potential (NP) and potential field source surface (PFSS) models on two dates near to solar maximum, we have found significant differences in the latitude–longitude distribution of predicted solar wind speed at 0.1 AU. If extrapolated using, for example, the Enlil model, these differences would in turn lead to significant differences in temporal variations of predicted wind speed at 1 AU. Since we used identical photospheric boundary conditions (magnetic maps) in both models, our results suggest that the uncertainty due to omission of coronal electric currents in existing models is likely to be at least as large as that due to the use of magnetogram data from different observatories <cit.>. To give one example, we found substantial polar coronal holes in the NP model for 4 April 2000, despite their absence in the PFSS model.
Although the formula we use for empirical wind speed (Equation (<ref>)) has been optimized specifically for the PFSS model, the differences between the predicted speed in the NP and PFSS models arise from differences in the basic physical quantities used in the formula, namely, the expansion rate of open magnetic flux tubes and the distance of their footpoints from coronal-hole boundaries on the solar photosphere. Ultimately the cause is a difference in the topological structure of the coronal magnetic field. The key difference is that “inflation” of the coronal field by the presence of electric currents in the NP model leads to additional open magnetic field and more coronal holes (open-field regions). This in turn leads to lower flux-tube expansion factors than in the PFSS model, at the same time as shorter footpoint distances to coronal hole boundaries, particularly at lower latitudes.
While specific details will vary (and will in general be quite sensitive to model input and parameters), these general conclusions are not specific to the NP model. We would expect to see similar differences in full MHD models where non-potential magnetic structure is built up through continuous driving. Indeed, <cit.> already found more open field in their MHD model compared to a PFSS model, although the difference in that case was less pronounced because the MHD model was initiated from a PFSS extrapolation and relaxed to equilibrium, rather than driving the photospheric field continuously over a longer period. When continuous driving of the photospheric field is included, the NP model suggests that more complex topologies may form in the magnetic field, such as twisted structures. We have seen how the HCS separatrix surfaces no longer always separate closed field from open field. Moreover, in the presence of coronal currents, open field lines may no longer have an anchoring point at the photosphere (cf. the U-shaped field lines in Figure <ref>). All of these differences will have an impact on solar-wind speed predictions.
In the future, we plan to couple output from this model to the Enlil model, in order that we can test the model against time series of in-situ wind speed (and polarity) measurements at 1 AU. This will require us to optimize the empirical formula for wind speed to the NP model; in this article, we used the existing PFSS-based formula to study the differences between the PFSS and NP models. An important uncertainty to quantify will be the sensitivity of the predicted wind speeds to differences in the photospheric input.
A.R. Yeates and S.J. Edwards were supported by STFC through consortium grant ST/K001043/1 and the Durham University Impact Acceleration Account, as well as by the US Air Force Office for Scientific Research. D.H. Mackay would like to thank the Leverhulme Trust and STFC for financial support. The authors thank Andrew L. Haynes for the use of his separatrix surface and null-point finding codes. Numerical simulations used the SRIF and STFC funded HPC cluster at the University of St Andrews.
§ DISCLOSURE OF POTENTIAL CONFLICTS OF INTEREST
The authors declare that they have no conflicts of interest.
|
1511.00185
|
Additive functions on co-compact coverings]A short note on additive functions on Riemannian co-compact coverings
Kha]Minh Kha
M.K., Department of Mathematics, Texas A&M University,
College Station, TX 77843-3368, USA
The main purpose of this note is to provide a topological approach to defining additive functions on Riemannian co-compact normal coverings.
§ INTRODUCTION
Let $X$ be a connected smooth Riemannian manifold of dimension $n$ equipped with an isometric, properly discontinuous, free, and co-compact action of a discrete deck group $G$. Notice that the deck group $G$ is finitely generated due to the Švarc-Milnor lemma and hence, $\Hom(G, \mathbb{R})$ is finite dimensional. Furthermore, the orbit space $M:=X/G$ is a compact Riemannian manifold when equipped with the metric pushed down from $X$. The action of an element $g \in G$ on $x \in X$ is denoted by $g\cdot x$. Let $\pi$ be the covering map from $X$ onto $M$. Thus, $\pi(g\cdot x)=\pi(x)$ for any $(g,x) \in G \times X$.
Following <cit.>, we define the class of additive functions on the covering $X$ as follows:
$ $
* A real smooth function $u$ on $X$ is said to be additive if there is a homomorphism $\alpha: G \rightarrow \mathbb{R}$ such that
\begin{equation*}
u(g\cdot x)=u(x)+\alpha(g), \quad \mbox{for all} \quad (g,x) \in G \times X.
\end{equation*}
We denote by $\mathcal{A}(X)$ the space of all additive functions on $X$.
A map $h$ from $X$ to $\mathbb{R}^m$ ($m \in \mathbb{N}$) is called a vector-valued additive function on $X$ if every component of $h$ belongs to $\mathcal{A}(X)$.
We remark that additive functions on co-compact covers appeared in various results such as studying the structure of positive $G$-multiplicative type solutions <cit.>, describing the off-diagonal long time asymptotics of the heat kernel <cit.> and the Green's function asymptotics of periodic elliptic operators <cit.> on a noncompact abelian cover of a compact Riemannian manifold.
A direct construction of additive functions on $X$ can be found in either <cit.> or <cit.>. However, this construction depends on the choice of a fundamental domain for the base $M$ in $X$. A more invariant approach to defining additive functions on covers was mentioned briefly in <cit.>. Our goal in this note is to present the full details of this approach for any co-compact covering.
§ ADDITIVE FUNCTIONS ON CO-COMPACT NORMAL COVERINGS
We begin with the following notion (see <cit.>):
Let $H_{DR}^1(M), H_{DR}^1(X)$ be De Rham cohomologies of $M$ and $X$, respectively. We denote by $\Omega^1(M; G)$ the image in $H_{DR}^1(M)$ of the set of all closed differential 1-forms $\omega$ on $M$ (modulo the exact ones) such that their lifts $\omega$ to $X$ are exact.
In other words, $\Omega^1(M; G)$ is the kernel of the homomorphism
$$\pi^{*}: H_{DR}^1(M) \rightarrow H_{DR}^1(X),$$
where $\pi^{*}$ is the induced homomorphism of the covering map $\pi: X \rightarrow M$.
By De Rham's theorem, $\Omega^1(M; G)$ is a finite dimensional vector space. Indeed, more is true:
$\Omega^1(M, G) \cong \Hom(G, \mathbb{R})$.
By Hurewicz's theorem (see e.g., <cit.>), the homologies $H_1(M)$ and $H_1(X)$ are isomorphic to the abelianizations of the fundamental groups $\pi_1(M)$ and $\pi_1(X)$, respectively. Therefore, we can identify De Rham cohomologies $H_{DR}^1(M)$ and $H_{DR}^1(X)$ with $\Hom(\pi_1(M), \mathbb{R})$ and $\Hom(\pi_1(X), \mathbb{R})$, correspondingly. Since $X$ is a normal covering of $M$, $\pi_1(X)$ is a normal subgroup of $\pi_1(M)$ and moreover, the sequence
$$0 \rightarrow \pi_1(X) \rightarrow \pi_1(M) \rightarrow G \rightarrow 0$$
is exact.
Because $\Hom(\cdot, \mathbb{R})$ is a contravariant exact functor, we deduce the exactness of the following sequence of vector spaces:
$$0 \rightarrow \Hom(G, \mathbb{R}) \rightarrow H_{DR}^1(M) \rightarrow H_{DR}^1(X) \rightarrow 0.$$
Hence, $\Omega^1(M, G)$ is isomorphic to $\Hom(G, \mathbb{R})$.
Fixing a base point $x_0 \in X$. For any closed 1-form $\omega$ on $M$ such that its lift to $X$ is exact,
there exists a unique function $f_{\omega} \in C^{\infty}(X, \mathbb{R})$ such that $\pi^* \omega=df_{\omega}$ and $f_{\omega}(x_0)=0$. Equivalently,
$$f_{\omega}(x)=\int_{x_0}^x \pi^*\omega, \quad \forall x \in X.$$
For such 1-form $\omega$, we have:
* Fix any $g \in G$, then $f_{\omega}(g \cdot x)-f_{\omega}(x)$ is independent of $x \in X$.
* If $\pi^*\omega=0$ then $\omega=0$.
$ $
* For each $g \in G$, let $L_{g}$ be the diffeomorphism of $X$ that maps $x$ to $g \cdot x$. Since $\pi \circ L_{g}=\pi$, we get $df_{\omega}=\pi^*\omega=L_{g}^*\pi^*\omega=L_{g}^*df_{\omega}=dL_{g}^*f_{\omega}=d(f_{\omega}\circ L_{g})$. Thus, $d(f_{\omega}-f_{\omega}\circ L_{g})=0$ and so, $f_{\omega}\circ L_{g}-f_{\omega}$ is constant since $X$ is connected.
* Fix any point $p \in M$. We pick an evenly covered open subset $U$ of $M$ such that it contains $p$. Then there is a smooth local section $\sigma: U \rightarrow X$, i.e., $\pi \circ \sigma=id_{|U}$ (see e.g., <cit.>). Hence, $\omega(p)=\sigma^{*}\pi^{*}\omega(p)=0$.
On $\mathcal{A}(X)$, we introduce an equivalent relation $\sim$ as follows: $f_1 \sim f_2$ in $\mathcal{A}(X)$ if and only if $f_1-f_2=f \circ \pi$ for some function $f\in C^{\infty}(M,\mathbb{R})$.
By Lemma <ref> (i), the map $\omega \mapsto f_{\omega}$ induces the following linear map
\begin{equation}
\label{Lambda}
\begin{split}
\Lambda: \Omega^1(M, G)&\rightarrow \mathcal{A}(X)/\sim
\\ [\omega] &\mapsto [f_{\omega}],
\end{split}
\end{equation}
where $[\omega]$, $[f_{\omega}]$ are the equivalent classes of $\omega$, $f_{\omega}$ in $\Omega^1(M, G)$ and $\mathcal{A}(X)/\sim$, correspondingly. We now claim that $[\omega]=0$ if and only if $[f_{\omega}]=0$, and hence $\Lambda$ is an injective linear map. Indeed, due to Lemma <ref> (ii), the condition that $\omega$ is exact is equivalent to $\pi^*\omega=d(f \circ \pi)$ for some $f \in C^{\infty}(M,\mathbb{R})$. But this is the same as $df_{\omega}=d(f\circ \pi)$, or $[f_{\omega}]=0$.
Consider an additive function $f$ on $X$. According to the definition, there exists a unique group homomorphism $\ell_f: G \rightarrow \mathbb{R}$ such that $f(g \cdot x)=f(x)+\ell_f(g)$ for any $g \in G$, $x \in X$. Then the map $f \mapsto \ell_f$ induces the linear map
\begin{equation}
\label{Upsilon}
\begin{split}
\Upsilon: \mathcal{A}(X)/\sim &\rightarrow \Hom{(G, \mathbb{R})}
\\ [f] &\mapsto \ell_f,
\end{split}
\end{equation}
which is injective.
Then the composition $\Upsilon \circ \Lambda$ is also injective. By Lemma <ref>, $\dim_{\mathbb{R}}\Omega^1(M,G)=\dim_{\mathbb{R}}\Hom{(G, \mathbb{R})}<\infty$. These facts together imply that the linear maps $\Upsilon$ and $\Lambda$ are isomorphism. We conclude:
The three vector spaces $\Omega^1(M, G)$, $\mathcal{A}(X)/\sim$ and $\Hom{(G, \mathbb{R})}$ are isomorphic to each other.
In particular, we obtain:
Assume that $G=\bZ^d$. Then there is a smooth $\mathbb{R}^d$-valued function $h$ on $X$ such that for any $(g,x) \in \bZ^d \times X$,
\begin{equation}
\label{additivity}
h(g\cdot x)=h(x)+g.
\end{equation}
The following standard proposition says that given any additive function $u$ on $X$, then one can pick a harmonic additive function $f$ such that $f-u$ is $G$-periodic.
For any $\ell$ in $\Hom(G, \mathbb{R})$, there exists a unique (modulo a real constant) harmonic function $f$ on $X$ such that for any $(g, x) \in G \times X$, we have
\begin{equation}
\label{harmonic_additive}
f(g \cdot x)=f(x)+\ell(g).
\end{equation}
First, we show the existence part. Due to Theorem <ref>, let $\tilde{f}$ be a function on $X$ satisfying $\tilde{f}(g \cdot x)=\tilde{f}(x)+\ell(g)$ for any $(g, x) \in G \times X$.
We recall the isomorphism $\Lambda$ defined in Lambda. We put $\alpha:=\Lambda^{-1}([\tilde{f}])\in \Omega^1(M, G)$. By the Hodge theorem, there exists a unique harmonic 1-form $\omega$ on $M$ such that $[\omega]=\alpha$ in $H^1_{DR}(M)$. Let $f$ be a smooth function such that $f \in \mathcal{A}(X)$ and $\pi^*\omega=df$.
Then $f$ satisfies harmonic_additive since $[f]=[\tilde{f}]$ in $\mathcal{A}(X)/\sim$. Thus, it is sufficient to show that $f$ is harmonic on $X$. We denote by $\delta_X$, $\Delta_X$ and $\delta_M$, $\Delta_M$ the codifferential and Laplace-Beltrami operators on $X$ and $M$, respectively. Since the covering map $\pi$ is a local isometry between $X$ and $M$, its pullback $\pi^{*}$ intertwines the codifferential operators, i.e.,
$$\delta_X \pi^*=\pi^* \delta_M.$$
Since $\Delta_M\omega=0$, it follows that $\delta_M \omega=0$. Thus,
\begin{equation*}
\Delta_X f=\delta_X df=\delta_X \pi^*\omega=\pi^* \delta_M\omega=0.
\end{equation*}
For the uniqueness part, let $f_1$ and $f_2$ be any two harmonic functions on $X$ such that harmonic_additive holds for each of these functions. Since $f_1-f_2$ is $G$-periodic, it can be pushed down to a real function $f$ on $M$. Moreover, $\pi^*\Delta_M f=\Delta_X \pi^*f=\Delta_X(f_1-f_2)=0$. Therefore, $f$ must be constant since it is a harmonic function on a compact, connected Riemannian manifold $M$. Thus, $f_1-f_2$ is constant.
Fixing a base point $x_0$ in $X$. Then to each $\alpha \in \Hom(G, \mathbb{R})$, there exists a unique harmonic function $f_{\alpha}$ defined on $X$ such that $f_{\alpha}(x_0)=0$ and $\Upsilon([f_{\alpha}])=\alpha$, where $\Upsilon$ is introduced in Upsilon. Consequently,
\begin{equation*}
\mathcal{A}(X)=\bigsqcup_{\alpha \in \Hom(G, \mathbb{R})}\big\{f_{\alpha}+\varphi \hspace{2pt} \mid \hspace{2pt} \varphi \hspace{3pt} \mbox{is periodic}\hspace{1pt}\big\}.
\end{equation*}
When $G=\bZ^d$,
the Albanese pseudo-metric $d_G$ introduced in <cit.> is actually the pseudo-distance arising from any harmonic vector-valued additive function $h$ satisfying additivity, i.e., $d_G(x,y)=|h(x)-h(y)|$ for any $x,y \in X$.
§ ACKNOWLEDGEMENTS
The author was partially supported by the NSF grant DMS-1517938.
AUTHOR = Agmon, Shmuel,
TITLE = On positive solutions of elliptic equations with periodic
coefficients in ${\bf R}^n$, spectral results and
extensions to elliptic operators on Riemannian manifolds,
BOOKTITLE = Differential equations (Birmingham, Ala., 1983),
SERIES = North-Holland Math. Stud.,
VOLUME = 92,
PAGES = 7–17,
PUBLISHER = North-Holland, Amsterdam,
YEAR = 1984,
MRCLASS = 35J15 (35B05 35P99 58G25),
MRNUMBER = 799327 (87a:35060),
MRREVIEWER = W. Allegretto,
AUTHOR= Kha, Minh,
TITLE = Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds,
note=arXiv:1511.00276, preprint,
AUTHOR = Kotani, Motoko,
AUTHOR = Sunada, Toshikazu,
TITLE = Albanese maps and off diagonal long time asymptotics for the
heat kernel,
JOURNAL = Comm. Math. Phys.,
FJOURNAL = Communications in Mathematical Physics,
VOLUME = 209,
YEAR = 2000,
NUMBER = 3,
PAGES = 633–670,
ISSN = 0010-3616,
CODEN = CMPHAY,
MRCLASS = 58J37 (58J35 58J65),
MRNUMBER = 1743611 (2001h:58036),
MRREVIEWER = Ivan G. Avramidi,
AUTHOR = Kuchment, Peter,
AUTHOR = Pinchover, Yehuda,
TITLE = Liouville theorems and spectral edge behavior on abelian
coverings of compact manifolds,
JOURNAL = Trans. Amer. Math. Soc.,
FJOURNAL = Transactions of the American Mathematical Society,
VOLUME = 359,
YEAR = 2007,
NUMBER = 12,
PAGES = 5777–5815,
ISSN = 0002-9947,
CODEN = TAMTAM,
MRCLASS = 58J05 (35B05 35J15 35P05 58J50),
MRNUMBER = 2336306 (2008h:58037),
MRREVIEWER = Alberto Parmeggiani,
AUTHOR = Lee, John M.,
TITLE = Introduction to smooth manifolds,
SERIES = Graduate Texts in Mathematics,
VOLUME = 218,
EDITION = Second Edition,
PUBLISHER = Springer, New York,
YEAR = 2013,
PAGES = xvi+708,
ISBN = 978-1-4419-9981-8,
MRCLASS = 58-01 (53-01 57-01),
MRNUMBER = 2954043,
AUTHOR = Lin, Vladimir Ya.,
AUTHOR = Pinchover, Yehuda,
TITLE = Manifolds with group actions and elliptic operators,
JOURNAL = Mem. Amer. Math. Soc.,
FJOURNAL = Memoirs of the American Mathematical Society,
VOLUME = 112,
YEAR = 1994,
NUMBER = 540,
PAGES = vi+78,
ISSN = 0065-9266,
CODEN = MAMCAU,
MRCLASS = 58G03 (35C15 35J15),
MRNUMBER = 1230774 (95d:58119),
MRREVIEWER = Vadim A. Kaĭmanovich,
|
1511.00567
|
In a hybrid PON/xDSL access network, multiple Customer Premise Equipment (CPE)
nodes connect over individual Digital Subscriber Lines (DSLs) to a
drop-point device. The drop-point device, which is typically
reverse powered from the customer, is co-located with
an Optical Network Unit (ONU) of the Passive Optical Network (PON).
We demonstrate that the drop-point experiences
very high buffer occupancies when no flow control or standard
Ethernet PAUSE frame flow control is employed.
In order to reduce the buffer occupancies in the drop-point,
we introduce two gated flow control protocols that extend the
polling-based PON medium access control to the DSL segments between
the CPEs and the ONUs.
We analyze the timing of the gated flow control mechanisms to
specify the latest possible time instant when CPEs can start the DSL
upstream transmissions so that the ONU can forward the upstream
transmissions at the full PON upstream transmission bit rate.
Through extensive simulations for a wide range of bursty traffic models,
we find that the gated flow control mechanisms, specifically, the
ONU and CPE grant sizing policies, enable effective control of the maximum
drop-point buffer occupancies.
Buffer occupancy, Digital Subscriber Line (DSL), Flow control,
Medium access control, Optical Network Unit (ONU),
Passive Optical Network (PON), Polling protocol.
§ INTRODUCTION
Access networks are communication networks that interconnect private local
area networks, such as the networks in the homes of individuals, with public
metropolitan and core networks, such as those constructed by service providers
to connect paying subscribers to the Internet.
Private local area networks often employ high speed wired and wireless
communications technologies, such as IEEE 802.3 Ethernet (up to 1 Gbit/sec)
and IEEE 802.11 WiFi (up to 600 Mbit/sec). These high-speed communications
technologies are cost effective in private local area networks due to
the short distances involved and subsequent low installation costs.
Public metropolitan and core networks employ a variety of communication
technologies that include dense wavelength division multiplexed technologies
over fiber optic transmission channels (up to 1 Tbit/sec). These high-speed
communication technologies are cost effective due to the
cost sharing over many paying subscribers. Access networks require
significantly higher installation costs compared to private local area
networks due to larger distances that must be covered. At the same time,
access networks have significantly smaller degrees of cost sharing compared
to public metropolitan and core networks; thereby increasing cost per paying
subscriber. As a result, access network technologies must keep installation
costs low <cit.>. Utilizing existing bandwidth-limited copper wire or
shared optical fiber will keep installation costs low <cit.>.
In this paper we present our study of hybrid access networks that utilize
both copper wire and shared optical fiber. The shared optical fiber extends
from the service provider's central office to a drop-point whereby the final
few hundred meters to the subscriber premise are reached by existing
copper wire. Figure <ref> illustrates this hybrid access network
architecture that leverages the installation cost benefits of existing copper
wire and the latest advances in digital subscriber line (DSL) <cit.>
technology that can realize up to 1 Gbps over short distances of twisted-pair
copper wire.
A hybrid PON/xDSL access network architecture consists of a
passive optical network (PON) connected to multiple copper digital
subscriber lines (DSLs). The PON OLT
connects to several drop-point devices. Each drop-point device is a
combined PON Optical Network Unit (ONU) and DSL Access Multiplexer
(DSLAM). Through the DSLAM, each drop-point device connects to
multiple subscriber DSL customer premise equipment (CPE) nodes.
The optical fiber segment of this hybrid access network is organized as a
shared passive optical network (PON), whereby multiple optical network
units (ONUs)
share a single optical fiber connected to an optical line terminal (OLT) at the
service provider central office. The copper segments begin at each ONU whereby
the fiber is dropped and existing copper wires are utilized via DSL
technology to reach each subscriber premise. Each ONU is coupled with a DSL
access multiplexer (DSLAM) at the fiber drop-point. This so-called drop-point
device is active and therefore requires electric power to operate. However,
service providers want these devices to maintain the deploy-anywhere
property of the optical splitter/combiner in a typical PON. To maintain this
property, each drop-point device is reverse powered using a subscriber's power
source. For this reason, it is of critical importance to reduce the energy
consumption of this device.
Reducing the memory capacity of the drop-point is an option for reducing
its energy consumption. A drop-point with a small memory capacity
translates into a design with a smaller memory device that contains
fewer transistors and capacitors that consume energy. However, reducing
the memory capacity of a drop-point can result in significant packet loss
if measures are not taken to back-pressure the buffering into either the
OLT in the downstream direction, or the DSL customer premise equipment (CPE) in
the upstream direction. The magnitude of buffering that can occur at the
drop-point is quite large due to the transmission bit rate mismatch between
the DSL line and the PON. Flow control mechanisms are, therefore, required to
avoid significant packet loss. In this paper we specifically examine several
upstream polling strategies for controlling the flow of upstream data from
each CPE to its associated drop-point device <cit.>. The objective of
these strategies is to minimize the maximum buffer occupancy required at
each drop-point with very low or no packet loss.
§.§ Background
Providing digital data communication through the access network
emerged with Digital Subscriber Loop or Line (DSL) technology in the
late 1970s and early 1980s <cit.>. At that time, researchers
identified mechanisms to aggregate digital data signals with analog
telephony signals and identified effective power levels and coding
mechanisms to tolerate the transmission impairments of the copper
loops used for analog telephony. These impairments included signal
reflections, cross-talk, and impulse noise <cit.>. Recent
efforts exploit multiple-input-multiple-output (MIMO) or vectoring
techniques to cancel the crosstalk
impairment <cit.>. Systems using these techniques can
achieve approximately 1 Gbps transmission using four twisted pairs
across distances up to 300 meters <cit.>. The recently
developed G.fast <cit.> DSL standard utilizes vectoring
techniques to achieve up to 1 Gbps speeds over these short distances.
Passive optical networks were envisioned in the late 1980s and early
1990s as an alternative to copper transmission between service
provider central offices and subscriber
premises <cit.>. A PON utilizes a shared
fiber optic transmission medium shared by up to a few dozen
subscribers thereby reducing per-subscriber installation
costs. Further, PONs employ passive devices between the service
provider's central office and the subscriber premises to also reduce
recurring operational costs. Standardization of PON technologies began
around the early 2000s (e.g., Ethernet PONs <cit.>) and have
subsequently achieved widespread deployment in the past few
years. Each of the various PON standards has considered the dynamic
bandwidth allocation (DBA) algorithms that decide how various
subscribers share the bandwidth of the optical fiber out of scope. As
a result, research activity on DBA algorithms started around the time
the standards were being
developed <cit.>.
Hybrid access network designs combine several transmission media types
(e.g., fiber, copper,
free space) <cit.> to reach
Hybrid fiber and copper access networks <cit.>
provide a good balance between the increased bandwidth of fiber optic
transmission and the cost benefits of using already deployed copper
transmission lines. Wireless technologies in access networks
add both a very low-cost installation option by using free space transmission
as well as mobility features for users.
§.§ Related Work
Although there is significant literature on the integration of PONs
with wireless transmission media, e.g., WOBAN <cit.> and
FiWi <cit.>, there is a dearth of literature on the
integration of PONs with copper transmission media.
Around the time the various PON standards were being developed,
researchers proposed developing hybrid PON/xDSL access networks. These
hybrid access networks would utilize DSL transmission technologies
with existing twisted-pair copper wire in conjunction with PONs. In
<cit.> an early PON standard called ITU-T 983.1
Broadband PON (BPON) was coupled with VDSL to reach subscribers in a
cost-effective manner. Specifically,
an architecture for a combined ONU/VDSL line card (drop-point) device
that bridged a single VDSL line onto the PON was described in <cit.>.
A full demonstration system for transferring MPEG-2 video through a
BPON/VDSL network using the ONU/VDSL line card <cit.> was
presented in <cit.>. In <cit.>, a mathematical model
of the number of VDSL subscribers that can be serviced by a single
ONU as a function of a few VDSL parameters (e.g., symmetric
operation and bit rates) was presented. This model can help service
providers design their PON/xDSL networks to support the desired
number of subscribers. In a study on QoS-aware intra-ONU scheduling
for PONs <cit.>, hybrid PON/xDSL access networks were noted
as a promising candidate for cost-effective broadband access. This
early work on hybrid PON/xDSL access networks demonstrated its
feasibility and provided some analysis for capacity planning but
ignored detailed design elements of the drop-point device that
bridges the PON with the various DSL lines connecting to
Two physical-layer systems to bridge VDSL signals over a fiber
access network were proposed in <cit.>. Individual VDSL
signals are converted to be spectrally stacked into a composite
signal that modulates an optical carrier. In the first system the
optical carrier is supplied by a laser at the ONU and in the second
system the optical carrier is supplied by a laser in the OLT that is
reflected and modulated by a Reflective Semiconductor Optical
Amplifier (RSOA) at the ONU. The optical carrier provides 1 GHz of
spectral width accommodating 40 VDSL lines without guard bands and
25 VDSL lines with guard bands. Although, this approach to a hybrid
PON/xDSL allows the drop-point device to avoid buffering as well as
contain simple logic by pulling the DSLAM functionality into the
OLT, the design requires the PON to carry the full bandwidth of each
VDSL line even when idle. Designs that operate at the link layer
rather than physical layer can avoid transmission of idle data on
the PON thereby increasing the number of subscribers that can be
supported by capitalizing on statistical multiplexing gains.
The coaxial copper cable deployed by cable companies represents another existing
copper technology that can be used in conjunction with PONs to create
a hybrid access network.
Such a hybrid access network combining an Ethernet PON with an Ethernet over
Coax (EoC) network was proposed in <cit.>.
The proposed network uses EPON protocols on the EoC
segment in isolation from the EPON segment without any coordination between
the segments. A similar network was examined in <cit.> in terms of
the blocking probability and delay for a video-on-demand
service. None of these studies discussed the design of the drop-point device
or explored DBA algorithms for these types of networks.
In November 2011, the IEEE 802.3 working group initiated the creation of
a study to extend the EPON protocol over hybrid fiber-coax cable television
networks; the developing standard is referred to as EPON Protocol over Coax
(EPoC) <cit.>. Developing bandwidth allocation schemes for
EPoC has received little research attention to date.
In particular, a DBA algorithm that increases
channel utilization in spite of increased propagation delays due to
the coaxial copper network was designed in <cit.>.
Mechanisms to map Ethernet frame transmissions to/from
the time division multiplexed channel of the PON to the time and frequency
division multiplexed coaxial network have been studied
in <cit.>.
§.§ Our Contribution
In this paper we contribute the first hybrid PON/xDSL drop-point design
providing lowered
energy consumption by means of reduced buffering requirements. We
mitigate the packet loss effects of the small drop-point buffers by defining
and evaluating several polling strategies that contain flow control.
Although we focus on xDSL as the copper technology in the hybrid
access networks, our proposed flow control polling protocols can be
analogously employed with other copper technologies, such as coax cable.
We define polling mechanisms that place the DSL CPEs under the control
of the PON OLT. With this flow control mechanism the polling MAC protocols that
have been designed for PONs are extended to a second stage of polling in the DSL
segments. We call this mechanism GATED flow control as the OLT on the
PON not only grants transmission access to ONUs on the PON but
determines when DSL CPEs transmit upstream to their attached ONUs. As
far as we know, we are the first to explore joint upstream transmission
coordination for hybrid PON/xDSL access networks.
The work presented in this paper provides significant extensions to the work
we presented at two conferences <cit.>. In <cit.>,
we presented a preliminary form of one of the two Gated flow control
mechanisms along with some initial simulation results. In <cit.>,
we present simulation results for one DBA algorithm,
namely (Online, Limited) <cit.>.
In contrast, in this article we comprehensively specify two
Gated flow control protocols through detailed analysis of the CPE
transmission timing (scheduling) and present extensive simulation results
that include the (Online, Gated) and (Online, Excess) DBA algorithms.
§ PON/XDSL NETWORK
In this section, we briefly describe the PON/xDSL network architecture
and outline flow control based on conventional PON polling in
conjunction with the standard Ethernet PAUSE frame.
§.§ Network Architecture
As illustrated in Figure <ref>, a PON/xDSL hybrid access
network connects multiple CPE devices $c,\ c = 1, 2, \ldots, E$, each
via its own DSL, to a drop-point device. Let $R_d$ [bit/s] denote the
upstream transmission bit rate on each DSL line and $\delta_c$ denote the
one-way propagation delay [s] between CPE $c$ and its drop-point;
the main model notations are summarized in Table <ref>.
Each drop-point consists of a DSLAM combined with a ONU of the
PON. Let $O$ denote the total number of ONUs in the PON; whereby each ONU
is part of a drop-point, $R_p$ be the upstream transmission bit rate [bit/s]
from an ONU to the PON OLT, and $\tau$ be the one-way transmission
delay [s] between an ONU and the OLT. We note that typically $R_p > R_d$.
Model Parameters for PON/xDSL Hybrid Access Network
Param. Meaning
2|c|Network structure
$R_{d}$ xDSL upstream transmission bit rate [bit/s]
$R_{p}$ PON upstream transmission bit rate [bit/s]
$E$ Number of CPEs per ONU; CPE index $c,\ c = 1, 2, \ldots, E$
$\delta_c$ One-way propagation delay from CPE $c$ to drop-point [s]
$\tau$ One-way propagation delay between OLT and ONU [s]
2|c|Polling protocol
$g_p$ Transmission time [s] for grant message on downstream PON
$g_d$ Transmission time [s] for grant message on downstream DSL
$G_c$ Size of upstream transmission window [bits] granted to
CPE $c$
$M$ Maximum packet size [bits]
2|c|Polling analysis for individual CPE $c$
$\sigma_c$ Start time instant of CPE $c$ upstream DSL transmission
(relative to start time of a cycle)
$\alpha_c$ Time instant when CPE $c$ data
starts to arrive at drop-point
$\omega_c$ Time instant when CPE $c$ data is compl. received
at drop-point
$\mu_c$ Time instant when ONU starts to transmit (serve) CPE $c$
(= time instant of max. CPE $c$ drop-point buffer occupancy)
$\beta_c$ Time instant when ONU upstream transm. of CPE $c$ data
$T$ Cycle duration from start instant of OLT grant transmission
to receipt of CPE data by OLT
2|c|Segregated CPE transmissions on PON
$\mu_{(E)}$ Start time of ONU transm. of
CPE $1, 2, \ldots, E$ data
$\sigma_c^s$ Start time of CPE $c$ upstream DSL transmission
2|c|Multiplexed CPE transmissions on PON
$\mu^m$ Start time of ONU transm. of multiplexed CPE data
$\sigma_c^m$ Start time of CPE $c$ upstream DSL transmission
To support the “deploy-anywhere” property, each drop-point device is
remotely powered over the DSL using the power supply of several
subscribers. As a result of the remote powering, the drop-point design
must consume as little energy as possible. We explore reducing
buffering at the drop-point to reduce energy consumption. By reducing
the maximum buffer occupancy, the drop-point can be designed
with a reduced memory capacity that will translate into fewer energy
consuming transistors and/or capacitors.
We utilize flow control
strategies through MAC polling to control buffering at each
drop-point. We introduce three upstream polling strategies that
provide flow control:
* ONU polling with PAUSE frame flow control
* Gated ONU:CPE polling flow control
with segregated CPE transmission on PON (ONU:CPE:seg)
* Gated ONU:CPE polling flow control
with multiplexed CPE transmission on PON (ONU:CPE:mux)
§.§ ONU Polling with PAUSE-Frame Flow Control
Our first proposed upstream polling strategy utilizes OLT media access
control (MAC) through polling only on the PON segment. With this strategy,
each CPE continuously
transmits upstream on its attached DSL. To control the flow of
upstream traffic so as to reduce the maximum buffer occupancy,
we utilize the standard Ethernet PAUSE frame flow control:
When an Ethernet receiver's buffer reaches a certain threshold that
Ethernet node transmits a PAUSE frame to the attached node in a
point-to-point configuration. Upon receipt of the PAUSE frame, an
Ethernet transmitter squelches its transmission for the time period
indicated in the PAUSE frame.
In the PON/xDSL network, the drop-point monitors its upstream
DSL buffer and once its occupancy reaches a certain threshold,
the drop-point transmits
a PAUSE frame downstream to the DSL CPE. When the DSL CPE
receives the PAUSE frame it squelches its transmission for the time
period indicated in the PAUSE frame.
§ GATED ONU:CPE POLLING FLOW CONTROL
§.§ Overview of ONU:CPE Polling Protocol
Our proposed upstream ONU:CPE polling strategies extend the
OLT MAC polling <cit.>
to each DSL CPE.
A DSL CPE transmits upstream only when explicitly
polled by the PON OLT with a GATE message.
The PON OLT conducts two stages of polling,
the first stage polls each ONU and the second stage polls each CPE.
More specifically, in a given cycle, the OLT sends a gate
message to the ONU to grant the ONU an upstream transmission window for
the data and bandwidth requests (reports) from the attached CPEs as well as
$E$ gate messages for the ONU to forward to the attached $E$ CPEs.
We denote $g_p$ for the downstream transmission time of a gate message on
the PON and $g_d$ for the downstream transmission time of a gate message
on a DSL. Moreover, we denote $G_c$ for the size [bit] of the upstream
transmission window granted to CPE $c$.
By controlling the transmission of each DSL CPE, the PON OLT can exercise
tight control over the magnitude of buffering that occurs at the drop-point.
§.§ CPE Grant Sizing
In ONU:CPE polling, the OLT can apply any of the existing ONU grant
sizing strategies <cit.> to assign each ONU an upstream
transmission window duration (grant size) according to the reported
bandwidth requests.
In turn, the OLT allocates a given ONU grant size to grants to the
attached CPEs and other (non-xDSL traffic) at the ONU.
When making a grant sizing decision for an ONU, the OLT knows the
bandwidth requests from all CPEs attached to the ONU.
Thus, the OLT can employ any of the grant sizing approaches requiring
knowledge of all bandwidth requests,
i.e., so-called offline approaches <cit.>,
for sizing the CPE grants.
As specified by the VDSL standard <cit.>, Ethernet frames are
encapsulated in a continuous stream of Packet Transfer Mode (PTM) 65 Byte
codewords, see <cit.>. Each codeword
contains one synchronization byte for every 64 bytes of data as well
as control characters and idle data bytes. The VDSL CPE under study
has been designed to suppress PTM codewords that contain all idle
data bytes. However, Ethernet frames can be encapsulated in PTM codewords
that contain idle data bytes. The number of bytes to be transmitted to
release a certain number of intended Ethernet frames from the CPE
depends on how the individual Ethernet frames expand within the PTM
codewords due to the inclusion of both PTM control characters and idle
data bytes. Modeling the exact number of bytes consumed by PTM codewords for
a given number of Ethernet frames requires knowledge of the individual
Ethernet frame sizes. That information is not available at the
OLT. Therefore, we estimate the CPE grant size to accommodate the PON
grant size with one synchronization byte for every 64 data bytes. We
then assume one extra codeword to contain control characters and idle
data bytes.
Due to the CPE grant size estimation, it is possible
that the CPE grant is too small and therefore does not allow all of
the PTM codewords containing the intended Ethernet frames to be
transmitted. In this case, an intended Ethernet frame will only be
partially received at the ONU with the other part left at the CPE. With
the next grant, the remainder of this Ethernet frame will be
transmitted, along with the other Ethernet frames intended for that
grant. The resulting extra Ethernet frame at the ONU will not
be accommodated by the current PON upstream grant. That Ethernet frame becomes
residue that stays at the drop point until it can be serviced in the next PON
upstream grant.
We also note that if we increased our CPE grant size estimate,
then the grant would be
too large and result in one or more Ethernet frames left as residue at
the ONU because the PON upstream grant would not accommodate them.
In the subsequent analysis of ONU CPE polling in this
Section <ref>, we neglect the drop point buffer residue. The
simulations in Section <ref> consider the full xDSL
and PON framing details and thus include the effects of the residue.
We note that due to neglecting the residue,
the PON delay analysis in Section <ref> is approximate.
However, we emphasize that the timing (scheduling) analyses in
Sections <ref> and <ref> are accurate
for the grant sizes determined by the OLT.
§.§ Basic Polling Timing Analysis for an Individual CPE
In this subsection we examine the timing of the polling of a single
CPE $c$ attached to an ONU. We establish basic timing relationships
of the CPE and ONU upstream data transmissions.
Due to the transmission delays of the ONU and CPE grant messages
and the downstream propagation delays, the CPE can start transmitting
at the earliest at time instant
\begin{eqnarray} \label{sigmac:eqn}
\sigma_c = 2 g_p + \tau + g_d + \delta_c.
\end{eqnarray}
Note that we measure time instants relative to the beginning of the
cycle, i.e., we consider the time instant when the OLT begins to transmit
the gate message downstream as zero.
For the basic analysis we assume that the CPE begins to transmit its
data at this earliest possible time instant $\sigma_c$ to the
Illustration of polling timing for an individual CPE $c$.
As illustrated in Figure <ref>, a CPE upstream
transmission grant of size $G_c$ needs to be transmitted through
both the DSL segment (CPE $\to$ drop-point) and the PON segment
(drop-point $\to$ OLT). To determine when the transmission on the
PON should begin, we must consider that the last bit of a packet
must have arrived at the drop-point device from a CPE before the
first bit of that same packet can be transmitted by the ONU to the
OLT. We let $M$ denote the maximum packet size [in bit] and
conservatively consider maximum size packets in the following
analysis. Focusing on the last packet of the CPE upstream
transmission, we note that the end of the last packet, i.e., the end
of the CPE upstream transmission must be received by the drop-point
before the ONU can forward this last packet over the PON to the OLT.
We denote $\alpha_c$ for the time instant when the CPE upstream
transmission begins to arrive (and occupy buffer space) at the drop
point, i.e.,
\begin{eqnarray}
\alpha_c = \sigma_c + \delta_c.
\end{eqnarray}
After complete receipt of the last packet at time instant
\begin{eqnarray} \label{omega:eqn}
\omega_c = \alpha_c + \frac{G_c}{R_d},
\end{eqnarray}
the ONU can immediately transmit this last packet
to the OLT. We denote $\beta_c$ for the time instant when
the last packet is completely transmitted by the ONU, i.e.,
when the CPE transmission stops to occupy buffer in the drop-point.
\begin{eqnarray}
\beta_c = \omega_c + \frac{M}{R_p}.
\end{eqnarray}
The end of the last packet reaches the OLT after the PON propagation delay,
resulting in the cycle duration $T = \beta_c + \tau$.
For the last packet to be able to start ONU transmission at time instant
$\beta_c - M/R_p$, all preceding packets must have already transmitted
by the ONU by time instant $\beta_c - M/R_p$.
More generally, the ONU finishes the transmission of the
$G_c$ bits of CPE data by instant $\beta_c$,
if the ONU starts the PON upstream transmission
(service) of the CPE data at time instant
\begin{eqnarray} \label{muc:eqn}
\mu_c = \beta_c - \frac{G_c}{R_p}.
\end{eqnarray}
We note that throughout this study we consider polling strategies
that transmit CPE data at the full optical transmission bit rate
$R_p$ on the PON upstream channel from ONU to OLT. Since the xDSL
transmission bit rate $R_d$ is typically lower than the fiber
transmission bit rate $R_p$, the drop point needs to buffer a part
of a CPE data transmission, which is received at rate $R_d < R_p$ at
the drop point, before onward transmission at rate $R_p$ over the
PON. Polling strategies that transmit on the PON upstream channel at
a rate lower than $R_p$ can reduce drop point buffering at the
expense of increased delay. The study of such strategies that only
partially utilize the optical upstream transmission bit rate is left
for future research.
Illustration of buffer occupancy for a given CPE $c$ in drop point:
The CPE buffer is filled at rate $R_d$ until the ONU starts transmitting
the CPE data at time instant $\mu_c$ with rate $R_p > R_d$. Then,
the buffer occupancy decreases at rate $R_p - R_d$ until the CPE data stops
arriving to the drop point at instant $\omega_c$; from then on the
CPE buffer is drained at rate $R_p$.
§.§ Drop-point Buffer Occupancy of a Single CPE
Based on the basic timing analysis in the preceding section,
we characterize the buffer occupancy due to a single CPE $c$
in the drop-point.
The buffer occupancy grows at rate $R_d$ [bit/s]
from arrival instant $\alpha_c$ of the CPE $c$ upstream transmission to
the drop point until the starting instant $\mu_c$ of the ONU upstream
From instant $\mu_c$ on the drop-point buffer drains at
rate $R_p - R_d$ up to instant $\omega_c$,
when the CPE transmission has been completely received at the drop-point.
From instant $\omega_c$ through the end of the ONU upstream transmission
at $\beta_c$, the buffer drains at rate $R_p$.
Since $R_p > R_d$, the maximum buffer occupancy $B_{\max, c}$ occurs
at time instant $\mu_c$ when the ONU starts to serve (transmit)
the CPE traffic.
The drop-point has been receiving CPE data at rate $R_d$ since time instant
$\alpha_c$, resulting in
\begin{eqnarray} \label{Bmax:eqn}
B_{\max, c} = (\mu_c - \alpha_c) R_d = G_c - \frac{R_d}{R_p} (G_c - M).
\end{eqnarray}
Thus, the buffer occupancy $B_c(t)$
of the drop point buffer associated with CPE $c$ is
\begin{eqnarray}
B_c(t)=\begin{cases} %0 & t\leq \alpha_c\\
R_d (t-\alpha_c) & t\in[\alpha_c,\mu_c]\\
B_{\max, c} - (R_p - R_d) \, (t - \mu_c) & t \in [ \mu_c, \omega_c]\\
M - R_p \, (t - \omega_c) & t\in[\omega_c, \beta_c],
% 0 & t\geq \beta_c.
\end{cases}
\end{eqnarray}
and zero otherwise.
For joint CPE buffering in the drop point (ONU),
the superposition of the buffer occupancies
\begin{eqnarray} \label{Bt:eqn}
B(t) := \sum_c B_c(t)
\end{eqnarray}
characterizes the occupancy level of the shared ONU buffer.
The maximum of $B(t)$ is the maximum ONU buffer occupancy.
§.§.§ PON Segment Packet Delay
Considering maximum sized packets,
the first packet of a given CPE upstream transmission is completely received
by the drop point (ONU) at time instant $\alpha_c + M/R_d$.
This first packet has to wait (queue) at the drop point until
its transmission over the PON upstream wavelength channel commences
at time instant $\mu_c$.
Thus, the queueing delay is $\mu_c - \alpha_c -M/R_d$,
which can be expressed in terms of the maximum CPE buffer occupancy
$B_{\max,c}$ (<ref>) as $(B_{\max, c} - M)/ R_d$.
The last packet of the CPE upstream transmission, which is completely
received by the ONU at time instant $\omega_c$ (<ref>),
does not experience any queueing delay.
Each packet experiences that transmission delay $M/R_p$
and propagation delay $\tau$ of the PON.
Summing these delay components gives the total PON delay for a packet;
and averaging over the packets in the CPE upstream transmission
leads to the average packet delay on the PON segment.
§.§ ONU:CPE Polling with Segregated CPE Transmissions on PON
In this section, we specify the Gated ONU:CPE polling protocol with
segregated CPE transmissions on the PON upstream channels.
That is, the data of each DSL CPE is transmitted
upstream in its own sub-window of the overall ONU upstream
transmission window.
We consider $E$ CPEs attached to a given ONU.
The ONU sends the $E$ CPE data transmissions successively
according to a prescribed transmission order, as
specified in Section <ref>, over the
PON upstream wavelength channel.
The CPEs time (schedule) their transmissions
as specified in Section <ref> to ensure that
the CPE data arriving at rate $R_d$ to the ONU can be transmitted
without interruptions at the full PON rate $R_p,\ R_p > R_d,$ to the OLT.
§.§.§ CPE Polling Order
The detailed analysis of the polling time with two CPEs in Appendix 1
indicates that the transmission order of CPE 1 followed by CPE 2 results in
shorter cycle duration if
\begin{eqnarray}
G_1 < G_2 + 2 \frac{\delta_2 - \delta_1}{\frac{1}{R_d} - \frac{1}{R_p}}.
\end{eqnarray}
That is, transmitting the traffic from the CPE
with the smaller grant size $G_1$ on the upstream PON channel before the
CPE with the larger grant $G_2$ generally
reduces the cycle duration, provided the
round-trip propagation delays $\delta_1$ and $\delta_2$
between the ONU and the two CPEs are not too different.
Typically, the CPEs are all in close vicinity of the ONU, thus
the round-trip propagation delay differences are often
negligible, even when scaled by the $1/(\frac{1}{R_d} - \frac{1}{R_p})$ factor.
For the remainder of this study we consider therefore
the CPE transmission order $c = 1,\ c = 2, \ldots, c = E$
with $G_1 \leq G_2 \cdots \leq G_E$ on the PON upstream transmission channel.
§.§.§ CPE Transmission Timing
We derive the earliest time instant $\mu_{1, 2, \ldots, E}$
that the ONU can start
upstream transmission such that all $E$ CPE data sets arrive in
time to the drop-point for the ONU to continuously transmit at rate
Specifically, we prove the following theorem:
In order to meet the constraint of continuous (back-to-back) transmission
of the data from CPEs $1, 2, \ldots, E$ in separate sub-transmission windows
at the PON rate $R_p$, the ONU can start transmission
at the earliest at time instant
\begin{eqnarray}
\mu_{(E)} \!\!\! &=& \!\!\! \mu_{(E-1)} \nonumber \\
&& \!\!\!\!\!\!\!\!\!\!\! \!
+ \max \left( 0 ,\ \mu_E - \mu_{(E-1)} - \frac{\sum_{c = 1}^{E-1} G_c}{R_p}
\right), \label{mu12e:eqn}
\end{eqnarray}
whereby the ONU transmission starting instant $\mu_c,\ c = 1, 2, \ldots, E$,
for an individual CPE $c$ is given by Eqn. (<ref>).
We consider initially two CPEs $c = 1$ and $c = 2$.
Considering each of these two CPEs individually,
Eqn. (<ref>) gives the respective time instants $\mu_1$ and
$\mu_2$ when ONU service could at the earliest commence, when considering
a given CPE in isolation.
There are two cases:
If $\mu_1 + G_1/R_p > \mu_2$, then the earliest instant for the
continuous ONU transmission to commence is $\mu_1$. This is because the
transmission of the data from CPE $c = 1$ takes longer
than CPE $c=2 $ needs to get its data “ready” for ONU transmission.
If, on the other hand, $\mu_1 +G_1/R_p < \mu_2$,
then the ONU transmission of CPE $c = 1$ data must be delayed
in order to avoid a gap between the end of the ONU transmission of the
CPE $c = 1$ data and the start of the ONU transmission of the
CPE $c = 2$ data.
The earliest instant for the continuous ONU transmission to commence is
$\mu_2 - G_1/R_p$, which gives the ONU just enough time to transmit
the CPE $c = 1$ data before the CPE $c = 2$ data is “ready” for ONU
In summary, the two cases for $E = 2$ CPEs result in the earliest
start time
\begin{eqnarray} \label{induct0:eqn}
\mu_{(2)} = \max \left( \mu_1,\ \mu_2 - \frac{G_1}{R_p} \right)
\end{eqnarray}
for continuous ONU transmission at rate $R_p$.
We proceed to the general case of $E, \ E > 2$ CPEs by induction:
Consider the continuous (back-to-back) ONU transmission of
CPE $c = 1$ and CPE $c = 2$ data as one CPE transmission with
earliest ONU transmission instant (when considered individually)
Next, we consider this back-to-back CPE $c = 1$ and $c=2$ data
as well as the CPE $c = 3$ data. Analogous to (<ref>),
we obtain the earliest starting instant of the continuous
ONU transmission of the data from CPEs $c = 1, 2$, and 3:
\begin{eqnarray}
\mu_{(3)} = \max \left( \mu_{(2)} ,\ \mu_3 - \frac{G_1+G_2}{R_p} \right).
\end{eqnarray}
Proceeding to the induction step with the continuous ONU transmission
of the CPE $c = 1, 2, \ldots, E-1$ data with earliest transmission instant
$\mu_{(E-1)}$ as well as the CPE $c = E$ data results in the
earliest transmission instant given by Eqn. (<ref>)
The sub-transmission window of CPE $c = 1$ starts at $\mu_{(E)}$,
while CPE $c = 2$ starts when the ONU transmission of CPE $c = 1$ data is
complete. Generally, the starting instants of the segregated CPE
sub-transmission windows $c = 1, 2, \ldots, E$ are
\begin{eqnarray}
\mu_c^s = \mu_{(E)} + \sum_{i = 1}^{c-1} \frac{G_i}{R_p}.
\end{eqnarray}
From these starting instants $\mu_c^s$ of the segregated CPE sub-transmission
windows, we find the corresponding starting instants $\sigma_c^s$
of the CPE transmissions by re-tracing the analysis in
Section <ref>.
Briefly, for the continuous ONU transmission of the CPE $c$ data at rate
$R_p$ is it sufficient for CPE $c$ to commence transmission
$G_c/R_d + M/R_p + \delta_c$ before the end of the ONU transmission at instant
$\mu_c^s + G_c / R_p$, i.e.,
\begin{eqnarray} \label{sigcs:eqn}
\sigma_c^s = \mu_c^s + \frac{G_c-M}{R_p} - \frac{G_c}{R_d} - \delta_c.
\end{eqnarray}
Starting the CPE transmissions at $\sigma_c^s$ instead of
the earliest possible $\sigma_c$ (<ref>)
for an individual transmission reduces the drop-point buffer occupancy.
§.§ ONU:CPE Polling with Multiplexed CPE Transmissions on PON
In this section, we specify the ONU:CPE polling protocol with statistical
multiplexing of the packets from the individual CPEs
in the ONU upstream transmission window.
All DSL CPEs attached to the same drop-point statistically multiplex their
transmissions into a joint ONU upstream transmission window
(rather than in the separate sub-windows in Section <ref>).
The OLT effectively grants transmission
windows to a given ONU to fit in all the traffic (in randomly
statistically multiplexed order) of the DSL CPEs attached to the
drop-point containing the ONU.
When the aggregate upstream transmission bit rate of the $E$ CPEs at
an ONU is less than the PON upstream transmission bit rate, i.e.,
when $E R_d \leq R_p$, then the ONU can commence the continuous
transmission of the multiplexed CPE data at the earliest at
\begin{eqnarray} \label{mum:eqn}
\mu^m &=& (E+1) g_p+ \tau + g_d + \max_{c}
\left( 2 \delta_c + \frac{ G_c}{R_d} \right) \nonumber \\
&& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+ \frac{ EM - \sum_{c = 1}^E G_c}{R_p}.
\end{eqnarray}
The individual CPE upstream transmissions $c = 1, 2, \ldots, E$,
can at the earliest be completely received by the drop point by time
instants $\sigma_c + \delta_c + G_c / R_d$, whereby $\sigma_c$ is given
by Eqn. (<ref>).
The latest such instant of complete reception of the data from a CPE at
the drop point is
\begin{eqnarray}
\omega = (E+1) g_p+ \tau + g_d + \max_{c}
\left( 2 \delta_c + \frac{ G_c}{R_d} \right).
\end{eqnarray}
If the aggregate transmission bit rate $E R_d$ of the $E$ CPEs does not
exceed the PON upstream transmission bit rate $R_p$,
the ONU can transmit all multiplexed CPE data upstream such that only
one data packet,
from at most each of the $E$ CPEs,
remains to be transmitted after $\omega$.
Thus, the ONU can complete the upstream transmission by
$\omega + E M / R_p$.
Since the ONU has to transmit a total of $\sum_{c = 1}^E G_c$ bits of CPE data,
the corresponding starting time instant of the ONU transmission must be
$\sum_{c = 1}^E G_c / R_p$ before $\omega + E M / R_p$,
resulting in the transmission start instant given by Eqn. (<ref>).
With the ONU transmission starting at instant $\mu^m$,
the ONU transmission is completed at instant
$\mu^m + \sum_{c=1}^E G_c / R_p$.
All CPE data has to arrive to the drop-point at least
$E M/R_p$ before the ONU transmission
completion instant $\mu^m + \sum_c G_c / R_p$.
CPE $c$ data is completely received by the drop point
$G_c / R_d + \delta_c$ after the CPE transmission
starting instant $\sigma_c^m$.
Thus, CPE $c$ can start transmission at the latest at instant
\begin{eqnarray} \label{sigcm:eqn}
\sigma_c^m = \mu^m + \frac{ \sum_{c=1}^E G_c - EM }{R_p} - \frac{G_c}{R_d} - \delta_c.
\end{eqnarray}
§ PERFORMANCE EVALUATION
We conducted a wide set of simulations
to answer three questions of practical interest:
* When is flow control required to provide a specific bound on ONU
buffer occupancy without loss at the ONU?
* When does PAUSE frame flow control fail to provide a specific bound
on ONU buffer occupancy without loss at the ONU?
* What is the range of bounds on ONU buffer occupancy without loss
at the ONU that can be achieved with Gated flow control?
We used a PON/xDSL hybrid access network simulator that we developed
using the CSIM discrete event simulation library. We considered the
XGPON <cit.> protocol for the PON segment and the
VDSL2 <cit.> protocol for the DSL segment as these two
technologies are being actively deployed in real hybrid access
networks. We set the XGPON upstream bit rate to $R_p = 2.488$ Gbps
and the guard time to 30 ns. The XGPON contained $O = 32$ ONUs, each
with $E = 8$ attached VDSL lines (for a total of 256 CPEs). The
upstream bit rate for each VDSL line was set to $R_d = 77$ Mbps to
achieve a realistic worst-case over-subscription rate of 8x. The OLT
to ONU one-way propagation delays $\tau$ were continuously
distributed between 2.5 $\mu$s (i.e., 500 m) and 100 $\mu$s (i.e.,
20 km). The ONU to CPE propagation delays $\delta$ are considered
negligible. We set the maximum cycle length to $Z = 3$ ms. The CPEs
independently generated data packets according to a quad mode packet
size distribution with 60 % 64 Byte packets, 4 % 300 Byte packets,
11 % 580 Byte packets, and 25 % 1518 Byte packets.
Each simulation run for a given traffic load considered $10^8$ packets.
a) No flow control (max. CPE buff.)
b) PAUSE frame flow control (max. CPE buff.)
c) No flow control (max. ONU buff.)
d) PAUSE frame flow control (max. ONU buff.)
e) No flow control (pkt. loss rate)
f) PAUSE frame flow control (pkt. loss rate)
Comparison of no flow control vs. ONU polling PAUSE frame flow
control with CPE buffer capacity of 1 MB.
§.§ No Flow Control
To answer question 1 we forgo the use of any flow control, utilize
large CPE buffer capacities (1 MBytes), and monitor the
maximum buffer occupancy. The DBA algorithm, source traffic
burstiness, and presented traffic load are factors that will affect
buffer occupancy at the ONU. Therefore, we vary these factors. We
consider the (Online, Gated) and (Online, Excess) DBA algorithms that
have been shown to provide good performance in conventional
PONs <cit.>, with a reporting approach akin to <cit.>
for the newly generated traffic.
Gated grant sizing assigns each ONU the full bandwidth
request <cit.>.
The employed (Online, Excess) grant sizing
approach assigns each ONU its request up to
the maximum ONU grant size of Limited grant sizing <cit.>,
i.e., an $1/O$ share of the total PON upstream transmission capacity $Z R_p$
in a cycle,
plus a $1/O$ share of accumulated unused excess bandwidth (which was
also limited to $Z R_p$) <cit.>; thus, the total maximum
ONU grant is $2 Z R_p / O$.
We vary the
burstiness of the traffic by using a self-similar traffic source in
which we vary the Hurst parameter from 0.5 (equivalent to a Poisson
traffic source) to 0.925 (equivalent to very bursty traffic).
Fig. <ref>a), c), and e)
contains plots of the maximum buffer occupancies and packet loss rate
versus presented traffic load without the use of flow
The traffic load is represented as a fraction of the full XGPON upstream
transmission rate of 2.488Gbps.
We define the maximum CPE buffer occupancy as the largest (maximum)
of the maximum CPE buffer occupancies $B_{\max, c}$,
see Fig. <ref> and Eqn. (<ref>), observed during a very long
simulation considering over $10^8$ packet transmissions.
The maximum ONU buffer occupancy is analogously defined as the largest
aggregate of the CPE buffer occupancies, see Eqn. (<ref>).
Our primary observation from these plots is that the maximum buffer
occupancy increases modestly until a certain “knee point”
load value and
then increases very sharply. The “knee point” load value
depends on both the
DBA algorithm and the burstiness of the source traffic.
If the buffer occupancy below the knee point load value meets
requirements, then flow control can be switched on
just when the knee point load value is reached.
As an example, when using the (Online, Excess) DBA algorithm, the
maximum ONU buffer occupancy value before the knee point
is 32 KB or less and the maximum CPE buffer
occupancy is 10 KB or less. If 32 KB was the desired upper bound
on the maximum aggregate ONU buffer occupancy, then flow control need
only be activated once the presented load approached 0.94 for
non-bursty traffic ($H=0.5$) or 0.4 for highly bursty traffic
($H=0.925$). Not surprisingly, bursty traffic will require flow control
under wider load conditions than non-bursty traffic.
§.§ ONU Polling PAUSE Frame Flow Control
To answer question 2 we use PAUSE frame flow control with a
threshold of 35 % buffer capacity to trigger the transmission of
PAUSE frames with a duration of 2 ms. A set of experiments, that we
leave out due to space constraints, were conducted to explore that
two-dimensional parameter space of buffer threshold and PAUSE
duration. Those experiments indicated that (35%, 2 ms) provided
the best performance.
Figure <ref>b), d), and f)
contains plots of the maximum
buffer occupancies and packet loss rates versus presented traffic load
with PAUSE frame flow control.
We observe that the maximum buffer occupancy trends when using PAUSE
frame flow control are similar to when no flow control is used. A
notable exception is that for the (Online, Excess) DBA algorithm,
the maximum CPE buffer occupancy stays below approximately 300 KB
when PAUSE frame flow control is used,
compared to 1 MB (i.e., the full capacity) when no flow control is
used. For the (Online, Gated) DBA algorithm, the maximum CPE
buffer occupancy reaches the 1 MB buffer capacity for highly bursty traffic
($H = 0.8$ and 0.925), regardless of whether PAUSE frame
flow control is used. The unlimited grant sizes of the
(Online, Gated) DBA algorithm appear to undermine the efforts of flow
From the packet loss rate plots in
Figure <ref>e) and f)
we observe that when using the
(Online, Excess) DBA algorithm, PAUSE frame flow control can
eliminate packet losses. On the other hand, for
the (Online, Gated) DBA algorithm with unlimited grant sizes,
PAUSE frame flow control is unable
to lower the packet loss rate for the bursty $H = 0.8$ and 0.925 traffic.
a) Max. CPE buffer occupancy
b) Max. ONU buffer occupancy
Maximum occupancies of CPE and ONU buffers for GATED Flow
Control approaches ONU:CPE:seg and ONU:CPE:mux with (Online, Excess)
dynamic bandwidth allocation (DBA) on PON for different levels of
traffic burstiness (i.e., different Hurst parameters $H$).
§.§ GATED ONU:CPE Polling Flow Control
To answer question 3 we present results for the two Gated ONU:CPE
polling flow control protocols introduced in
Section <ref>, namely segregated (ONU:CPE:seg) and
multiplexing (ONU:CPE:mux) polling flow control. We continue to
consider the (Online, Excess) sizing for the ONU grants. A given ONU
grant is distributed to the CPEs according to the equitable
iterative excess distribution method <cit.>,
which fairly divides the ONU grant among the CPEs, allowing CPEs
with high traffic loads to utilize the unused fair shares of the low
traffic loads.
Figure <ref> contain plots of the maximum buffer occupancies
and average packet delays as a function of load.
§.§.§ Maximum CPE and ONU Buffer Occupancies
We observe from Figure <ref>
that for low loads of bursty traffic with Hurst parameters $H > 0.5$,
the maximum CPE and ONU buffer occupancies are approximately twice
the maximum ONU grant size of a Limited DBA grant
sizing at low traffic loads, i.e., approximately
$2 Z R_p / O$.
At low bursty traffic loads it is likely that only very few CPEs
(that are attached to only a few ONUs)
generate a traffic burst at a given time, while the other CPEs have no
This permits the ONUs with attached CPEs with a traffic burst
through the considered Online
Excess DBA mechanism <cit.>
to utilize the excess bandwidth allocation from the ONUs without
traffic bursts. The considered Online Excess DBA limits
the excess allocation from other ONUs to a given ONU to once the
maximum Limited DBA grant size.
Thus, if a single CPE at an ONU generates a traffic burst,
the CPE is allocated a grant of twice the maximum Limited DBA grant size,
resulting in correspondingly large maximum CPE and ONU buffer occupancies.
(The ONU buffer occupancies slightly above 40 kB are due to small
residual backlog from preceding cycles due to the
different DSL and PON framing mechanisms, see Section <ref>.)
Interestingly, we observe from Figure <ref> that
the maximum CPE and ONU buffer occupancies in the bursty
($H > 0.5$) traffic scenarios decrease with increasing
traffic load.
As the traffic load increases, more and more CPEs
have backlogged (queued) traffic bursts.
When all ONUs have some CPEs with backlogged traffic, there is no more
excess allocation from ONUs with little or no traffic backlog to
ONUs with large traffic backlog.
Thus, the Online Excess DBA mechanism turns into the Online Limited DBA
mechanism and allocates to each ONU the maximum Limited DBA ONU grant size.
Thus, as the traffic load increases, the traffic amount
transmitted upstream on the PON bandwidth is more equally distributed among
the ONUs as more and more ONUs have CPEs with backlogged traffic bursts.
In turn, the grant allocation to a given ONU is more
equally divided among its attached CPEs as the
traffic load increases and more and more CPEs at an ONU have backlogged
traffic bursts.
For the Poisson traffic scenario ($H = 0.5$), we observe from
Fig. <ref> that the CPE and ONU buffer occupancies
continuously increase with increasing traffic load (except for a
drop in CPE buffer occupancy at very high loads). In contrast to
bursty traffic sources that generate bursts of several packets at a
time, Poisson traffic sources generate individual data packets.
These individually generated packets are uniformly distributed
(spread) among the CPEs, and correspondingly the ONUs. Thus, there
is essentially no excess allocation among ONUs at low load levels
and the maximum CPE and ONU buffer occupancies grow gradually with
increasing traffic load. (For high load levels there is some excess
allocation, which decreases at very high loads as all CPEs and ONUs
have backlogged traffic, resulting in the CPE buffer occupancy drop
at very high loads.)
In additional simulations, we observed that for the Online, Gated PON DBA,
which grants the ONUs the full bandwidth requests <cit.>,
the maximum CPE and ONU buffer occupancies depend mainly on the burstiness
of the traffic: around 10 kBytes for Poisson traffic and
on the order of 10 MBytes for bursty traffic with $H > 0.5$,
for the considered network scenario.
In contrast, for the Online, Limited PON DBA, which strictly
limits the grant allocation to an ONU to a prescribed limit $Z R_p/O$
(and does not permit re-allocations among ONUs which are
possibly in the Online, Excess PON DBA) <cit.>,
we have observed that the maximum CPE and ONU buffer occupancies
are generally bounded by the maximum ONU grant size
$Z R_p/O$ <cit.>.
Thus, our extensive simulations have validated that Gated ONU:CPE polling
flow control effectively limits the maximum CPE and ONU buffer occupancies
through the employed grant sizing mechanisms.
a) DSL delay
b) PON delay
Average packet delays on DSL and PON segments for
GATED Flow Control approaches ONU:CPE:seg and ONU:CPE:mux with
(Online, Excess) DBA on PON for different level of
traffic burstiness (i.e., Hurst parameter $H$).
§.§.§ DSL and PON Delay
We observe from Figure <ref> that the DSL
delay component from the instant of packet generation to the
complete packet reception at the drop point (ONU) increases first
slowly for low loads. Then, for moderately high loads above 0.6, we
begin to observe rapidly increasing DSL delays, first for the highly
bursty $H = 0.925$ traffic and then at higher loads above 0.75 for
the $H = 0.8$ and $H = 0.675$ traffic scenarios. The DSL delays for
these $H > 0.5$ scenarios shoot up to values above 18 s (i.e.,
outside the plotted range) as the traffic bursts overwhelm the
system resources. In contrast, for Poisson traffic, we observe
steadily increasing delays that remain below 1 s even for very high
traffic loads. We also observe that the “mux” approach, which
multiplexes upstream transmissions from different CPEs on the
upstream PON wavelength channel achieves lower delays than the
“seg” approach with segregated CPE upstream transmissions on the
PON. The DSL delay reduction achieved with the multiplexing approach
is particularly pronounced for high Poisson traffic loads, where
the multiplexing approaches reduces the DSL delay by over 0.5 s
compared to the corresponding delay with the segregated approach.
The PON segment delay of a packet from the instant of
packet reception at the drop point (ONU) to the instant
of packet reception at the ONU depends on the
CPE buffer occupancies, as analyzed in Section <ref>.
Essentially, for the segregated CPE transmission approach,
the average PON packet delay corresponds directly
(is proportional) to the
average of the maximum CPE buffer occupancies $B_{\max, c}$
across the individual polling cycles.
We observe from Fig. <ref> initially (in the low load
decreasing PON delay with increasing load
for the highly bursty $H = 0.925$ traffic,
The other traffic scenarios give initially slowly increasing
PON delays that rapidly increase for high loads in the 0.75–0.95 load range
and then level out.
For the very bursty $H = 0.925$ traffic, the individual
maximum CPE buffer occupancies $B_{\max,c}$ of payload data packets
(i.e., ignoring the drop point buffer occupancy of CPEs sending only
Report control packets) are initially very large
due to the traffic bursts at individual CPEs and associated ONUs,
which receive excess allocations from the other ONUs
(similar to the dynamics for low loads in Fig. <ref>).
These excess allocations diminish as CPEs at more and more ONUs get backlogged,
resulting in a decrease of the average maximum CPE buffer occupancies,
and correspondingly a decrease of the average PON packet delay.
For the other traffic scenarios with $H = 0.8$ and lower,
the burstiness is less pronounced, avoiding a decrease of the average
maximum CPE buffer occupancy for increasing loads in the low load region,
whereas the
largest (across a long simulation run) maximum CPE buffer occupancy
does exhibit a significant decrease, see Fig. <ref>.
For very high loads, the average PON packet delay,
which is proportional to the average maximum CPE buffer occupancy,
levels out around 0.8 ms.
This leveling out is analogous to the leveling out of the largest
maximum CPE buffer occupancy in Fig. <ref>).
We note that the average PON packet delay of roughly 0.8 ms
is substantially longer than the maximum PON packet delay
obtained with the delay analysis in Section <ref>
for the maximum CPE buffer occupancy of roughly 5 kB for very high loads in
Fig. <ref>.
The analysis in Section <ref> neglects the small residual
drop point buffering. However, the relatively few packets that
make up the residual buffering have to wait approximately the full
cycle length $Z = 3$ ms for upstream transmission in the next cycle;
thus, substantially increasing the mean PON packet delay.
Nevertheless, due to the flow control back-pressuring the data into the CPEs
until an ONU grant can accommodate the CPE data transmissions, the PON segment
delays are minuscule compared to the DSL segment delays.
We observe from Fig. <ref> that
multiplexing CPE transmissions gives generally lower PON delays
than segregating CPE transmissions.
The delay analysis in Section <ref> applies directly
to segregated CPE transmissions in that the CPE buffer
in the drop point is filled
at the rate $R_d$ of a single DSL line. The CPE buffer is filled
until the full optical transmission rate $R_p > R_d$ can be
sustained for the transmission of all $E$ CPE data sets over the PON.
When multiplexing CPE transmissions, multiple DSL lines supply
data to the drop point. Thus, the PON transmission can commence earlier,
resulting in shorter queueing delays for the first packets that arrived from
the CPEs to the drop point.
§ CONCLUSION
We have examined the buffering in the drop-point device connecting
the relatively low-transmission rate xDSL segment to the relatively
high-transmission bit rate PON segment in a hybrid PON/xDSL access
network. We found that the drop-point device experiences very high
buffer occupancies on the order of Mega Bytes or larger when no flow
control or when flow control with the standard Ethernet PAUSE frame
are employed. In an effort to reduce the buffer occupancies in the
drop points and thus to reduce the energy consumption of the drop
point devices, which are typically reverse powered from subscribers,
we introduced Gated ONU:CPE polling flow control protocols. We
specified the timing (scheduling) of these Gated ONU:CPE polling
flow control protocols for two types of upstream transmission:
segregated CPE sub-windows or
multiplexed CPE transmissions within an ONU
upstream transmission window. Through extensive simulations for a
wide range of levels of traffic burstiness, we verified that the
Gated ONU:CPE polling protocols effectively limit the drop-point
buffering in individual CPE buffers or an aggregated ONU buffer. The
maximum CPE and ONU buffer occupancies correspond approximately to
the grant size limits of the polling-based medium access executed at
the OLT. Through adjusting the ONU and CPE grant sizes in the
proposed Gated ONU:CPE polling flow control protocols, the OLT can
effectively control the buffering in the drop-point devices.
One important direction for future research on
hybrid access networks is to extend the hybrid access
network evaluation to the local wired and wireless networks that
interconnect with the access network at the CPE.
Excessive buffering in the CPEs could be mitigated by further back-pressuring
the data transmissions to the gateways or host whose applications
generate large traffic flows.
Another important direction for future research is to examine
control mechanisms through software defined networking in hybrid
access networks <cit.>.
§ ACKNOWLEDGMENT
This material is based upon work supported by the National Science
Foundation under Grant No. CNS-1059430 and by a gift from Huawei
(a) $G_1 < G_1^{\mathrm{th1}}$
(b) $G_1^{\mathrm{th1}} < G_1 < G_1^{\mathrm{th2}}$
(c) $G_1 > G_1^{\mathrm{th2}}$
Illustration of cases for analysis of minimum completion time
for two CPEs (CPE 1 and CPE 2) with segregate sub-windows in the PON grant.
This illustration shows the round-trip propagation delays
$2\delta_1,\ 2\delta_2$ on the DSL networks as well as the
DSL upstream transmission delays $G_1/R_d$ and $G_2/R_d$.
The PON upstream transmission delays $(G_1 - M)/R_p$
and $(G_2 - M)/R_p$ can be masked by the DSL upstream transmissions
and influence when the PON upstream transmissions can commence.
The PON upstream transmission delays $M/R_p$ occur
after the DSL upstream transmission is complete.
§ APPENDIX: ANALYSIS OF CPE TRANSMISSION ORDERING FOR TWO CPES
We assume for the following analysis without loss of
generality that CPE 1 has a smaller propagation delay to
the drop-point device than CPE 2 (i.e., $\delta_1 \leq \delta_2$).
We analyze the minimum delay $T$ for complete reception of both
upstream transmissions at the OLT.
There are three main cases for evaluating $T$,
as illustrated in Fig. <ref>:
Case small $G_1$, see Fig. <ref>(a)
There is a gap between the CPE partitions on the PON since
$G_1$ is too small to mask the time until $G_2$ is ready for PON
upstream transmission.
(In this small $G_1$ case, the transmission
of CPE 1 could be delayed so as to avoid the occurrence of a gap,
and reduce the time that the ONU buffer holds the CPE 1 data.)
Case medium $G_1$, see Fig. <ref>(b)
The partitions of CPE 1 and CPE 2 are transmitted back-to-back on the PON.
Case large $G_1$, see Fig. <ref>(c)
$G_1$ is so large that the PON upstream transmission of $G_2$
is completed before $G_1$ is ready for PON upstream transmission.
We proceed to analyze the transmission order of the CPE transmission
windows on the PON and identify the minimum times for complete
reception of both CPE data transmissions at the OLT.
We denote with 12 the transmission order CPE 1 followed by
CPE 2, and denote 21 for the reverse transmission order.
To reduce clutter in this scheduling analysis, we re-define the
time periods $\beta$ and $\mu$ from Section <ref> with
reference to the end of the downstream gate transmission by the ONU.
In order to identify the threshold $G_1^{\mathrm{th1}}$
that distinguishes the small and
medium $G_1$ cases we
initially consider the transmissions of CPE 1 and CPE 2 as completely
independent, i.e., we initially only consider one of these CPE
transmissions at a time.
From Fig. <ref>(a), we note that
the time period from the ending instant of the gate message transmissions
by the ONU to the time instant that
the ONU transmission of CPE 1 data is completed as
\begin{eqnarray} \label{beta1:eqn}
\beta_1 = 2 \delta_1 + \frac{G_1}{R_d}
+ \frac{M}{R_p}.
\end{eqnarray}
Similarly, we express the time period until the time instant of
the beginning of the CPE 2 data transmission on the PON as
\begin{eqnarray} \label{alpha2:eqn}
\mu_2 = 2 \delta_2 + \frac{G_2}{R_d} - \frac{G_2- M}{R_p}.
\end{eqnarray}
The transmission of CPE 1 data by the ONU is completed before
the ONU transmission of CPE 2 data can commence if
$\mu_2 > \beta_1$, i.e., if
\begin{eqnarray} \label{Gth1:eqn}
G_1 < G_2 \left( 1 - \frac{R_d}{R_p} \right)
+ 2 R_d (\delta_2 - \delta_1 ) =: G_1^{\mathrm{ th1}}.
\end{eqnarray}
Thus, for $G_1 < G_1^{\mathrm th1}$, the transmission of CPE 1 data before
CPE 2 data does not delay the commencement of CPE 2 data
transmission. Hence, the transmission order 12 achieves the
minimum cycle (completion) time
\begin{eqnarray}
T = 3 g_p + g_d + 2 \tau + 2 \delta_2 + \frac{G_2}{R_d}
+ \frac{M}{R_p}.
\end{eqnarray}
Next, we identify the threshold $G_1^{\mathrm{th2}}$ that distinguishes
the medium and large $G_1$ cases.
We note from Fig. <ref>(c) that
the ONU transmission of CPE 2 data is completed by
\begin{eqnarray}
\beta_2 = 2 \delta_2 + \frac{G_2}{R_d} + \frac{M}{R_p}.
\end{eqnarray}
The ONU transmission of CPE 1 data can commence at the
earliest at time
\begin{eqnarray}
\mu_1 = 2 \delta_1 + \frac{G_1}{R_d} - \frac{G_1 - M}{R_p}.
\end{eqnarray}
For $\mu_1 > \beta_2$, or equivalently, for
\begin{eqnarray}
G_1 > G_2 + 2 \frac{\delta_2 - \delta_1}{\frac{1}{R_d}
- \frac{1}{R_p}} =: G_1^{\mathrm{th2} }.
\end{eqnarray}
the ONU transmission of CPE 1 data is completed
before the ONU transmission of CPE 2 data can commence
That is, the CPE 2 data transmission does not delay
the CPE 1 data transmission.
Thus, the 21 transmission order gives the minimum
completion time
\begin{eqnarray}
T = 3 g_p + g_d + 2 \tau
+ 2 \delta_1 + \frac{G_1}{R_d} + \frac{M}{R_p}.
\end{eqnarray}
Note also that $G_1^{\mathrm{th1}} \leq G_1^{\mathrm{th2}} \
\forall \delta_2 \geq \delta_1,\ R_p > R_d$.
We now turn to the medium $G_1$ range illustrated in
Fig. <ref>(b).
We note from the illustration in Fig. <ref>(b) that the completion
time for the 12 transmission order is
\begin{eqnarray} \label{TcsmallG1:eqn}
T^{12} = 3 g_p + g_d + 2 \tau + 2 \delta_1 + \frac{G_1}{R_d}
+ \frac{M + G_2}{R_p}.
\end{eqnarray}
We similarly obtain the completion time $T_c^{21}$ for the
21 transmission order and note that
\begin{eqnarray}
T_c^{12} &\leq& T_c^{21} \\
\Leftrightarrow 2 \delta_1 + \frac{G_1}{R_d} + \frac{ G_2}{R_p}
&\leq & 2 \delta_2 + \frac{G_2}{R_d} + \frac{ G_1}{R_p}
\label{detcrit:eqn}\\
\Leftrightarrow G_1 &\leq& G_1^{\mathrm{th2} }.
\end{eqnarray}
Thus, the transmission order 12 gives the minimum
$T$ if $G_1 \leq G_1^{\mathrm{th2}}$.
In summary, the minimum time period $T$ from the instant
of commencing the transmission of the gate messages from the OLT to
the complete reception of both CPE data transmissions at the OLT is
obtained by the transmission order CPE 1 data followed by CPE 2 data
on the PON for $G_1 \leq G_1^{\mathrm{th2}}$.
For $G_1 \geq G_1^{\mathrm{th2}}$, the reverse transmission order of
CPE 2 data followed by CPE 1 data on the PON minimizes $T$.
T. Koonen, “Fiber to the home/fiber to the premises: What, where, and when?”
Proceedings of the IEEE, vol. 94, no. 5, pp. 911–934, May 2006.
F. Mazzenga, M. Petracca, F. Vatalaro, R. Giuliano, and G. Ciccarella,
“Coexistence of FTTC and FTTDp network architectures in different
VDSL2 scenarios,” Trans. on Emerging Telecommun. Techn., in print,
T. Starr, J. Cioffi, and P. Silverman, Understanding digital subscriber
line technology.1em plus 0.5em minus 0.4emPrentice Hall, 1999.
Y. Luo, “Activities, drivers and benefits of extending PON over other
media,” in Proc. OSA NFOEC, Mar. 2013, pp. 1–3.
S. Ahamed, P. Bohn, and N. Gottfried, “A tutorial on two-wire digital
transmission in the loop plant,” IEEE Transactions on Communications,
vol. 29, no. 11, pp. 1554–1564, Nov. 1981.
G. Ginis and J. Cioffi, “Vectored transmission for digital subscriber line
systems,” IEEE J. Selected Areas in Communications, vol. 20, no. 5,
pp. 1085–1104, Jun 2002.
B. Lee, J. Cioffi, S. Jagannathan, and M. Mohseni, “Gigabit DSL,”
IEEE Transactions on Communications, vol. 55, no. 9, pp. 1689–1692,
Sept. 2007.
“ITU-T G.9701, Fast access to subscriber terminals (G.fast) -
Physical layer specification,”
M. Timmers, M. Guenach, C. Nuzman, and J. Maes, “G.fast: evolving the copper
access network,” IEEE Communications Magazine, vol. 51, no. 8, pp.
74–79, Aug. 2013.
J. Stern, J. Ballance, D. Faulkner, S. Hornung, D. Payne, and K. Oakley,
“Passive optical local networks for telephony applications and beyond,”
IET Electronics Letters, vol. 23, no. 24, pp. 1255–1256, Nov. 1987.
N. Frigo, P. Iannone, P. Magill, T. Darcie, M. Downs, B. Desai, U. Koren,
T. Koch, C. Dragone, H. Presby, and G. Bodeep, “A wavelength-division
multiplexed passive optical network with cost-shared components,” IEEE
Photonics Technology Letters, vol. 6, no. 11, pp. 1365–1367, Nov. 1994.
G. Kramer and G. Pesavento, “Ethernet passive optical network (epon): building
a next-generation optical access network,” IEEE Communications
Magazine, vol. 40, no. 2, pp. 66–73, Feb. 2002.
M. Dias, E. Wong, D. Pham Van, and L. Valcarenghi, “Offline energy-efficient
dynamic wavelength and bandwidth allocation algorithm for TWDM-PONs,” in
Proc. IEEE ICC, 2015, pp. 5018–5023.
I. Gravalos, K. Yiannopoulos, G. Papadimitriou, and E. Varvarigos, “The
max–min fair approach on dynamic bandwidth allocation for XG-PONs,”
Trans. Emerging Telecommun. Techn., vol. 26, no. 10, pp. 1212–1224,
G. Kramer, B. Mukherjee, and G. Pesavento, “IPACT: A dynamic protocol for an
Ethernet PON (EPON),” IEEE Communications Magazine, vol. 40, no. 2,
pp. 74–80, Feb. 2002.
P. Sarigiannidis, G. Papadimitriou, P. Nicopolitidis, E. Varvarigos, M. Louta,
and V. Kakali, “IFAISTOS: A fair and flexible resource allocation policy
for next-generation passive optical networks,” in Proc. IEEE (ICUMT),
2014, pp. 7–14.
B. Skubic, J. Chen, J. Ahmed, L. Wosinska, and B. Mukherjee, “A comparison of
dynamic bandwidth allocation for epon, gpon, and next-generation tdm pon,”
IEEE Communications Magazine, vol. 47, no. 3, pp. S40–S48, Mar. 2009.
Ö. C. Turna, M. A. Aydın, A. H. Zaim, and T. Atmaca, “A new dynamic
bandwidth allocation algorithm based on online–offline mode for EPON,”
Opt. Switching and Netw., vol. 15, pp. 29–43, 2015.
J. Zheng and H. Mouftah, “A survey of dynamic bandwidth allocation algorithms
for Ethernet Passive Optical Networks,” Optical Switching and
Networking, vol. 6, no. 3, pp. 151–162, July 2009.
M. Ahmed, I. Ahmad, and D. Habibi, “Service class resource management for
green wireless-optical broadband access networks (WOBAN),” IEEE/OSA
Journal of Lightwave Techn., vol. 33, no. 1, pp. 7–18, 2015.
L. Fang, L. Zhou, X. Liu, X. Zhang, M. Sui, F. Effenberger, and J. Zhou,
“Demonstration of end-to-end cloud-DSL with a PON-based fronthaul
supporting 5.76-Gb/s throughput with 48 eCDMA-encoded 1024-QAM discrete
multi-tone signals,” OSA Optics Express, vol. 23, no. 10, pp.
13 499–13 504, 2015.
G. Kramer, M. D. Andrade, R. Roy, and P. Chowdhury, “Evolution of optical
access networks: Architectures and capacity upgrades,” Proceedings of
the IEEE, vol. 100, no. 5, pp. 1188–1196, May 2012.
R. Gaudino, R. Giuliano, F. Mazzenga, F. Vatalaro, and L. Valcarenghi,
“Unbundling in optical access networks: Focus on hybrid fiber-VDSL and
TWDM-PON,” in Proc. Fotonica AEIT Italian Conf. on Photonics
Techn., 2014, pp. 1–4.
S. Sarkar, S. Dixit, and B. Mukherjee, “Hybrid wireless-optical
broadband-access network (WOBAN): A review of relevant challenges,”
IEEE/OSA Journal of Lightwave Technology, vol. 25, no. 11, pp.
3329–3340, Nov. 2007.
A. Ahmed and A. Shami, “RPR–EPON–WiMAX hybrid network: A solution for
access and metro networks,” IEEE/OSA Journal of Optical Commun. and
Netw., vol. 4, no. 3, pp. 173–188, 2012.
N. Ghazisaidi, M. Maier, and C. Assi, “Fiber-wireless (FiWi) access
networks: A survey,” IEEE Communications Magazine, vol. 47, no. 2,
pp. 160–167, Feb. 2009.
I. Cooper and M. Bramhall, “ATM passive optical networks and integrated
VDSL,” IEEE Communications Magazine, vol. 38, no. 3, pp. 174–179,
Mar. 2000.
I. Cooper, V. Barker, M. Andrews, M. Bramhall, and P. Ball, “Video over BPON
with integrated VDSL,” Fujitsu Scientific and Technical Journal,
vol. 37, no. 1, pp. 87–96, June 2001.
R. Roka, “The analysis of the effective utilization of PON and VDSL
technologies in the access network,” in Proc. IEEE EUROCON, vol. 1,
Sept. 2003, pp. 216–219.
C. Assi, Y. Ye, and S. Dixit, “Support of QoS in IP-based
Ethernet-PON,” in Proc. IEEE Globecom, vol. 7, Dec. 2003, pp.
J. Lepley, M. Thakur, I. Tsalamanis, C. Bock, C. Arellano, J. Prat, and
S. Walker, “VDSL transmission over a fiber extended-access network,”
OSA Journal of Optical Networking, vol. 4, no. 8, pp. 517–523, Aug.
G. Wu, D. Liu, S. Zhang, C. Zhang, and Y. Chang, “Novel access technology
based on hybrid Ethernet passive optical network and ethernet passive
electronic network,” Frontiers of Optoelectronics in China, vol. 2,
no. 3, pp. 328–333, Sept. 2009.
J. Wei, “The VoD services carried by hybrid PON+EoC networking,” in
Proc. IEEE Int. Conf. on Broadband Network Multimedia Tech. (IC-BNMT),
Oct. 2009, pp. 467–471.
P. Bhaumik, S. Thota, B. Mukherjee, K. Zhangli, J. Chen, H. Elbakoury, and
L. Fang, “EPON protocol over coax (EPoC): overview and design issues from
a MAC layer perspective?” IEEE Communications Magazine, vol. 51,
no. 10, pp. 144–153, Oct. 2013.
P. Bhaumik, S. Thota, K. Zhangli, J. Chen, H. ElBakoury, L. Fang, and
B. Mukherjee, “EPON Protocol over Coax (EPoC): Round-trip time aware
dynamic bandwidth allocation,” in Proc. IEEE Int. Conf. on Optical
Network Design and Modeling (ONDM), Apr. 2013, pp. 287–292.
——, “On downstream transmissions in EPON protocol over coax (EPoC): An
analysis of coax framing approaches and other relevant considerations,”
Photonic Network Communications, vol. 28, no. 2, pp. 178–189, Oct.
M. McGarry, E. Gurrola, and Y. Luo, “On the reduction of ONU upstream
buffering for PON/xDSL hybrid access networks,” in Proc. Globecom,
Dec. 2013, pp. 2667–2673.
A. Mercian, E. Gurrola, M. McGarry, and M. Reisslein, “Improved polling
strategies for efficient flow control for buffer reduction in PON/xDSL
hybrid access networks,” in Proc. IEEE Asilomar Conf. on Signals,
Systems, and Computers, Nov. 2015.
M. Hossen and M. Hanawa, “Multi-OLT and multi-wavelength PON-based open
access network for improving the throughput and quality of services,”
Opt. Switching and Netw., vol. 15, pp. 148–159, 2015.
B. Kantarci and H. T. Mouftah, “Bandwidth distribution solutions for
performance enhancement in long-reach passive optical networks,” IEEE
Commun. Surveys & Tutorials, vol. 14, no. 3, pp. 714–733, 2012.
N. Merayo, T. Jiménez, P. Fernández, R. J. Durán, R. M. Lorenzo,
I. de Miguel, and E. J. Abril, “A bandwidth assignment polling algorithm to
enhance the efficiency in QoS long-reach EPONs,” European Trans.
on Telecommunications, vol. 22, no. 1, pp. 35–44, 2011.
M. McGarry and M. Reisslein, “Investigation of the DBA algorithm design
space for EPONs,” IEEE/OSA Journal of Lightwave Technology,
vol. 30, no. 14, pp. 2271–2280, July 2012.
“ITU-T G.993.2, Very high speed digital subscriber line transceivers
2 (VDSL2),” <http://www.itu.int/rec/T-REC-G.993.2/en>.
“ITU-T G.992.3, Asymmetric digital subscriber line transceivers 2
(ADSL2),” <http://www.itu.int/rec/T-REC-G.992.3/en>.
“ITU-T G.987.3, 10-Gigabit-capable passive optical networks
(XG-PON): Transmission convergence (TC) specifications,”
B. Skubic, J. Chen, J. Ahmed, B. Chen, L. Wosinska, and B. Mukherjee, “Dynamic
bandwidth allocation for long-reach PON: overcoming performance
degradation,” IEEE Communications Magazine, vol. 48, no. 11, pp.
100–108, 2010.
A. Mercian, M. McGarry, and M. Reisslein, “Offline and online multi-thread
polling in long-reach PONs: A critical evaluation,” IEEE/OSA J.
Lightwave Techn., vol. 31, no. 12, pp. 2018–2028, 2013.
C. Assi, Y. Ye, S. Dixit, and M. Ali, “Dynamic bandwidth allocation for
quality-of-service over Ethernet PONs,” IEEE Journal on Selected
Areas in Communications, vol. 21, no. 9, pp. 1467–1477, Nov. 2003.
X. Bai, A. Shami, and C. Assi, “On the fairness of dynamic bandwidth
allocation schemes in Ethernet passive optical networks,” Computer
Communications, vol. 29, no. 11, pp. 2123–2135, 2006.
K. Kerpez, J. Cioffi, G. Ginis, M. Goldburg, S. Galli, and P. Silverman,
“Software-defined access networks,” IEEE Communications Magazine,
vol. 52, no. 9, pp. 152–159, 2014.
|
1511.00073
|
We present the first evidence of clear signatures of tidal distortions in the density distribution of the fascinating open cluster NGC 6791.
We find that the 2D density map shows a clear elongation and an irregular distribution starting from $\sim 300^{\prime\prime}$ from the cluster center and two tails extending in opposite directions beyond the tidal radius. These features are aligned to both the absolute proper motion and to the Galactic centre directions. Accordingly we find that both the surface brightness and star count density profiles reveal a departure from a King model starting from $\sim600^{\prime\prime}$. These observational evidences suggest that NGC 6791 is currently undergoing mass-loss likely due to gravitational shocking and interactions with the tidal field
of the Milky Way. We derive the expected mass-loss due to stellar evolution and tidal interactions and we estimate the initial cluster mass to be $M_{ini} = (1.5-4.0 ) \times 10^5 M_{\odot}$.
§ CONTEXT AND RESULTS
NGC 6791 is one of the most massive open cluster (Carraro 2014) in the MW, possibly harbouring more than a stellar population (Geisler et al. 2012, Bragaglia et al. 2014).We used deep images obtained with the wide field imager MegaCam mounted at the Canada-France-Hawaii Telescope (CFHT) to cover a $2^o \times 2^o$ area around the cluster (Dalessandro et al. 2015). The upper left panel in Fig 1 shows the obtained ($g^{\prime}$, $g^{\prime}-r^{\prime})$ CMDs of the innermost region of NGC 6791 (left) and the external control field (right) including stars located at a distance $r \geq 3000^{\prime\prime}$ from the cluster center. Large scale 2D colour-coded surface density map of NGC6791 obtained by using the optical matched filter technique is shown in the upper right panel of Fig. 1. The contour levels span from 3$\sigma$ to 40$\sigma$ with irregular steps. The solid arrow represents the direction of the absolute proper motion while the dashed ones mark the direction of the Galactic Center and that perpendicular to the Galactic plane (Z=0).The observed star count density profile is shown in the upper left panel of Fig. 1 (open grey squares). The dashed line represents the density value of the background as obtained in the control field. The black filled dots are densities obtained after background subtraction. The best single-mass King model is also over-plotted to the observations (solid line). For $r \geq 600^{\prime\prime}$ the density profile clearly deviates from the King model following a power-law with exponent $\alpha \sim -1.7$.By using a simple analytic approach (Lamers et al. 2005), we estimated the mass likely lost by NGC 6791 during its evolution because of the effect of both stellar evolution and dynamical interactions. On this basis we estimated the cluster initial mass as a function of the dissolution time parameter $t_{0}$. In particular the red curve (lower right panel in Fig. 1) shows the dependence for the current cluster mass (5000 $M_{\odot}$) and its actual age of 8 Gyr.
Upper left: CMD of NGC 6791 and nearby field. Upper right: star counts . Lower left: King profile fitting . Lower right: Mass at birth estimate.
§ CONCLUSIONS
NGC 6791 shows clear evidence of tidal features in its star distribution in the form of irregular but evident elongations and tidal tails.
These features are present also in the star density and surface brightness profile and they represent clear indication of recent mass-loss.
By using the simple recipes we derived the initial mass of NGC6791 to be $M_{ini} = (1.5-4 ) \times 10^5 M_{\odot}$, i.e. several tens larger
than its present day mass. This finding would qualitatively explain why the cluster could have survived for such a long time contrary to the expectations of current estimates of the destruction rate of Galactic open clusters.
[Bragaglia et al.(2014)]bra14
Bragaglia, A., Sneden, C., Carretta, E., Gratton, R., Lucatello, S., Bernath, P., Brooke, J.S.A., Ram, R.S.
ApJ, 796, 68
[Carraro (2014)]Car14
Carraro, G. 1995,
ASPC, 482, 245
[Dalessandro et al. (2015)]Ale15
Dalessandro, E., Miocchi, P., Carraro, G., Jilkova, L., Moitinho, A. 2015,
MNRAS, 449, 1811
[Geisler et al. (2012)]gei12
Geisler, D., Villanova, S., Carraro, G., Pilachowski, C., Cummings, J., Johnson, C.I., Bresolin, F. 2012,
ApJ, 756, 40
[Lamers et al. (2005)]lam05
Lamers, H., Gieles, M., Bastian, N., Baumgartd, H., Kharchenko, N., Portegies Zwart, S. 2005,
A&A, 441, 17
|
1511.00420
|
In the extreme value analysis of time series, not only the tail behavior is of interest, but also the serial dependence plays a crucial role. Drees and Rootzén (2010) established limit theorems for a general class of empirical processes of so-called cluster functionals which can be used to analyse various aspects of the extreme value behavior of mixing time series. However, usually the limit distribution is too complex to enable a direct construction of confidence regions. Therefore, we suggest a multiplier block bootstrap analog to the empirical processes of cluster functionals. It is shown that under virtually the same conditions as used by Drees and Rootzén (2010), conditionally on the data, the bootstrap processes converge to the same limit distribution. These general results are applied to construct confidence regions for the empirical extremogram introduced by Davis and Mikosch (2009). In a simulation study, the confidence intervals constructed by our multiplier block bootstrap approach compare favorably to the stationary bootstrap proposed by Davis et al. (2012).
[Keywords and phrases: bootstrap, cluster
functionals, clustering
of extremes, confidence regions, extremogram, serial dependence, uniform central limit theorem.
AMS 2010 Classification: Primary 62G32; Secondary 60G70, 60F17.
§ INTRODUCTION
Time series of observations in environmetrics, (financial) risk
management and other fields often exhibit a non-negligible serial
dependence between extremes. For example, stable areas of low (or
high) pressure may lead to consecutive days of high precipitation
(or high temperature). Likewise, large losses to a financial
investment tend to occur in clusters.
The statistical analysis of the serial dependence structure between
extreme observations is still a challenging task. Yet even if one is
only interested in marginal parameters, like extreme quantiles, it
is crucial to take into account the serial dependence when
assessing the estimation error; see, e.g., Drees (2003) for a
simulation study which demonstrates how misleading confidence
intervals may be if the serial dependence is ignored.
In most applications, no parametric time series model for the
extremal behavior suggests itself. Hence, one should resort to
non-parametric procedures to avoid the risk of an unquantifiable,
but potentially large modeling error. In this context, a general
class of empirical processes that can capture a wide range of
different aspects of the extremal behavior of time series prove a
powerful tool.
To be more concrete, assume that a stationary time series
$(X_t)_{1\le t\le n}$ with values in $E=\R^d$ is observed, from
which we construct $m_n:=\floor{n/r_n}$ blocks
\begin{equation} \label{eq:Ynjdef}
Y_{n,j} := (X_{n,i})_{(j-1)r_n<i\le jr_n}, \quad 1\le j\le m_n,
\end{equation}
of “standardized extreme observations” $X_{n,i}$, $1\le i\le n$. A
typical choice for univariate time series is
\begin{equation} \label{eq:Xnidefuniv}
X_{n,i}:=(X_i-u_n)^+/a_n:= a_n^{-1}(X_i-u_n)1_{\{X_i>u_n\}}
\end{equation}
for suitable normalizing constants $u_n\in\R$ and $a_n>0$.
Later on, we will use a different notion of extreme
observation in our application to the analysis of the extremogram, for
a multivariate time series.
Denote by $E_\cup := \bigcup_{l\in\N} E^l$ the set of vectors of
arbitrary length with components in $E$, which is equipped with the
$\sigma$-field $\mathbb{E}_\cup$ induced by
the Borel-$\sigma$-fields on $E^l$, $l\in\N$. Let $\FF$ be a
family of so-called cluster functionals, i.e. functions
$f:(E_\cup,\mathbb{E}_\cup)\to(\R,\B)$ such that $f(0)=0$ and
$f(y_1,\ldots,y_l)=f(0,\ldots,0,y_1,\ldots,y_l,0,\ldots,0)$ for all
$(y_1,\ldots,y_l)\in E_\cup$ where the numbers of coordinates equal
to 0 in the beginning and in the end of the argument on the
right-hand side can be arbitrary. Thus the value of the cluster
functional depends only on the core of the argument, which is
the smallest subvector of consecutive coordinates that contains all
non-zero values (resp. it equals 0 if the argument only consists of
zeros). Then, the pertaining empirical process of cluster
functionals is defined by
\begin{equation} \label{eq:Zndef}
Z_n(f) := \frac 1{\sqrt{n v_n}} \sum_{j=1}^{m_n} \big( f(Y_{n,j})-
E f(Y_{n,j})\big), \quad f\in\FF,
\end{equation}
with $v_n:= P\{X_{n,1}\ne 0\}$. Drees and Rootzén (2010) established
sufficient conditions for $Z_n$ to converge to a Gaussian process in
the space $\ell^\infty(\FF)$ of bounded functions on $\FF$. The
following theorem summarizes their main results; the conditions are
recalled in the appendix.
* If the conditions (B1), (B2) and (C1)–(C3) are fulfilled,
the finite-dimensional marginal distributions (fidis) of the
empirical process $Z_n$ converge to the pertaining fidis of a
Gaussian process $Z$ with covariance function $c$ (defined in
* Under the conditions (B1), (B2) and (D1)–(D4) the
empirical process $Z_n$ is asymptotically tight in
$\ell^\infty(\FF)$. If, in addition, the conditions (C1)–(C3)
are met, then $Z_n$ weakly converges to $Z$.
* If the assumptions (B1), (B2), (D1), (D2'), (D3) and (D5)
are satisfied and, in addition, (D6) (or the more restrictive
condition (D6')) holds, then $Z_n$ is asymptotically
equicontinuous. Hence, $Z_n$ weakly converges to $Z$ in
$\ell^\infty(\FF)$ if also the conditions (C1)–(C3) hold.
For certain types of families $\FF$ of cluster functionals, Drees and
Rootzén (2010) also gave sets of conditions that are sufficient for
$(Z_n(f))_{f\in\FF}$ to converge and easier to verify than the
abstract conditions listed in the appendix.
We will demonstrate their usefulness by improving on limit results
on an empirical version of the so-called extremogram introduced by
Davis and Mikosch (2009) in the framework of time series with
regularly varying marginals. To be more precise, assume that
$(X_t)_{t\in\Z}$ is a stationary $\R^d$-valued time series such that
for all $h\in\N$ the vector $(X_0,X_h)\in\R^{2d}$ is regularly
varying. Recall that a random vector $W\in\R^l$ is regularly varying
if there exists a non-null measure $\nu$ on
such that
$$ \frac{P\{W\in xB\}}{P\{\|W\|>x\}}\;\longrightarrow\;
\nu(B)<\infty
for all $\nu$-continuity sets $B\in\B^l$ that are bounded away from
the origin 0. Note that, while this definition of regular variation
does not depend on the choice of the norm $\|\cdot\|$, the specific
form of the limiting measure $\nu$ does. In any case, the limiting measure is homogeneous of order $-\alpha$ for some $\alpha>0$, the so-called index of regular variation.
Then, with $F_{\|X\|}^\leftarrow$ denoting the quantile function of
$\|X_0\|$ and $a_n:= F_{\|X\|}^\leftarrow(1-1/n)\to\infty$, to each
lag $h\in\N$ there exists a measure $\nu_{(0,h)}$ on
$\R^{2d}\setminus\{0\}$ such that
\begin{equation} \label{eq:nuhdef}
nP\{a_n^{-1}(X_0,X_h)\in B\}\;\longrightarrow\;
\nu_{(0,h)}(B)
\end{equation}
for all $\nu_{(0,h)}$-continuity sets $B\in\B^{2d}$ bounded away from the origin.
In particular, for all $A,B\in\B^d$ bounded away from 0 such that
$\nu_h(\partial(A\times B))=0=\nu_h(\partial(A\times \R^d))$ and
$\nu_h(A\times \R^d)>0$ one has
$$ P(X_h\in a_nB\mid X_0\in a_n A) = \frac{P\{a_n^{-1}(X_0,X_h)\in
A\times B\}}{P\{a_n^{-1}X_0\in A\}} \;\longrightarrow\;
\frac{\nu_{(0,h)}(A\times B)}{\nu_{(0,h)}(A\times R^d)} =:
\rho_{A,B}(h).
Davis and Mikosch (2009) called $\rho_{A,B}$ (as a function of $h$)
the extremogram of $(X_t)_{t\in\Z}$ (pertaining to $A,B$). It
is worth mentioning that the extremogram is closely related to the
concept of tail processes introduced by Basrak and Segers (2008).
Based on the observations $X_1,\ldots, X_n$, they proposed the
following empirical counterpart as an estimator of $\rho_{A,B}(h)$:
\begin{equation} \label{eq:empextremodef1}
\hat \rho_{A,B}(h) := \frac{\sum_{i=1}^{n-h} \Ind{X_i\in a_k A,
X_{i+h}\in a_kB}}{\sum_{i=1}^n \Ind{X_i\in a_k A}}.
\end{equation}
Here $k=k_n$ is a sequence that tends to $\infty$ at a slower rate
than $n$ so that $a_k\to \infty$ at a slower rate than $a_n$, and
thus the number of extreme observations used for estimation tends to
$\infty$. Under suitable conditions, $(\hat
\rho_{A,B}(h))_{h\in\{0,\ldots,h_0\}}$ is asymptotically normal (see
Davis and Mikosch, 2009, Corollary 3.4).
This result has two serious drawbacks. First, usually, the
normalizing constants $a_k$ are unknown and must hence be replaced
with an empirical counterpart, like, e.g., the $\floor{n/k}+1$
largest observed norm:
\begin{equation} \label{eq:amhatdef}
\hat a_k := \hat a_{k,n} := \|X\|_{n-\floor{n/k}:n}.
\end{equation}
It is not obvious whether this modification influences the
asymptotic behavior of the empirical extremogram.
Secondly, the extremogram for a fixed pair of sets $A$ and $B$
conveys limited information on the extremal dependence structure, in
particular in a multivariate setting, i.e. if $d>1$. To get a
fuller picture, one should consider the extremogram for a whole
family of sets simultaneously. For example, in the case $d=1$, Drees et al. (2015) considered rays $(-\infty,-x)$ and $(x,\infty)$ for all $x>0$ simultaneously. However, the techniques used by Davis
and Mikosch (2009) are not applicable to infinite families of sets.
We will show that both problems can be neatly solved using the
theory of empirical processes of cluster functionals. Indeed, if the
families of sets $A$ and $B$ are suitably chosen and the bias of
$\hat \rho_{A,B}(h) $ is asymptotically negligible, then the
asymptotic normality of the empirical extremogram with estimated
normalizing sequence $\hat a_k$ follows immediately.
If one wants to construct confidence regions using this limit
theorem, then estimators of the limiting covariance structure are
needed. Since the direct estimation does not look promising, Davis
et al. (2012) proposed to use a so-called stationary bootstrap
instead. Here we follow a somewhat different approach. First, in the
general setting considered by Drees and Rootzén (2010), it is shown
that the convergence of a multiplier block bootstrap version of the
empirical process of cluster functional conditionally given the data
follows under the same conditions as the convergence of $Z_n$
itself. From this powerful result it is easily concluded that a
multiplier block bootstrap version can be used to construct
confidence regions for the extremogram.
Though in the present paper we focus on the extremogram as one
possible measure for the extremal dependence structure of the time
series, the same approach using empirical processes of cluster
functionals can be used in a much wider context. For example, Drees
(2011) analyzed block estimators of the so-called extremal index of
absolutely regular time series using empirical processes of cluster
functionals and suggested a bias corrected version thereof.
The paper is organized as follows. In Section 2 we introduce
multiplier block bootstrap versions of the empirical process $Z_n$.
Moreover, we give sufficient conditions under which, in probability
conditional on the data, this bootstrap processes weakly converge to
the same limiting process as $Z_n$. In Section 3, it is demonstrated
that the theory developed by Drees and Rootzén (2010) yields limit theorems for the empirical extremogram with
estimated normalizing sequence uniformly over suitable families of
sets. In the same setup, a bootstrap result easily follows from the
general theory developed in Section 2. The results of a small simulation study are reported in Section 3. All proofs are postponed to
Section <ref>.
Throughout the paper, we will use the notation $x^{(k)}$ for the
vector $(x_1, \ldots,x_k)$ made up by the first $k$ components in
the vector $x$, if $x$ has at least $k$ components, and otherwise
$x^{(k)}=x$. The maximum norm of a vector $x\in\R^l$ for some
$l\in\N$ is denoted by $\|x\|$. We omit indices of random variables
to denote a generic random variable with the same distribution; for
example, $\xi$ is a generic random variable with the same
distribution as $\xi_j$ and $Y_n$ is a generic random vector with
the same distribution as $Y_{n,j}$.
§ MULTIPLIER PROCESSES
In what follows, $(X_{n,i})_{1\le i\le n, n\in\N}$ is a row-wise
stationary triangular scheme of $E=\R^d$-valued random vectors.
Usually these vectors are derived from some fixed stationary time
series $(X_t)_{t\in\Z}$ by a transformation which depends on the
stage $n$ and which sets all but the “extreme” observations to 0
in such a way that the probability that a transformed observation is
non-zero tends to 0 as $n\to\infty$. For univariate time series,
often definition (<ref>) is used. In our application
to the empirical extremogram instead we define
\begin{equation} \label{eq:Xnidefextremo}
X_{n,i}^{(h,\tilde h)} := a_k^{-1} \big(X_i\Ind{X_i\not\in (-\infty,a_kx_*)^d},X_{i+h}\Ind{X_{i+h}\not\in (-\infty,a_kx_*)^d},X_{i+\tilde h}\Ind{X_{i+\tilde h}\not\in (-\infty,a_kx_*)^d}\big)
\end{equation}
for some $x_*>0$ and $h,\tilde h\in\N_0$.
According to Theorem <ref>, under suitable conditions,
the empirical process $Z_n$ of cluster functionals converge to a
Gaussian process $Z$ with covariance function $c$, which is defined
in (C3) as the limit of the covariance function of the cluster
functionals applied to a block $Y_n$ of $r_n$ consecutive
“standardized extremes” $X_{n,i}$. One may try to estimate this
covariance function by an empirical covariance, but since most of
the blocks $Y_{n,j}$ defined in (<ref>) equal 0, a
bootstrap approach seems more promising.
Because the processes are defined via functionals applied to whole
blocks $Y_{n,j}$ of “standardized extremes”, it suggests itself to
use some block bootstrap. More precisely, we consider the following
two versions of multiplier block bootstrap processes:
\begin{eqnarray} \label{eq:multiplierdef}
Z_{n,\xi}(f) & := & \frac 1{\sqrt{n v_n}} \sum_{j=1}^{m_n} \xi_j \big( f(Y_{n,j})-
E f(Y_{n,j})\big),\\
Z_{n,\xi}^*(f) & := & \frac 1{\sqrt{n v_n}} \sum_{j=1}^{m_n} \xi_j \big( f(Y_{n,j})-
\overline{f(Y_n)}\big), \quad f\in\FF, \label{eq:bootstrapdef}
\end{eqnarray}
where $\overline{f(Y_n)} := m_n^{-1} \sum_{j=1}^{m_n} f(Y_{n,j})$
and $\xi_j, j\in\N$, are i.i.d. random variables with $E(\xi_j)=0$
and $Var(\xi_j)=1$ independent of $(X_{n,i})_{1\le i\le n, n\in\N}$.
Note that in the definition of the multiplier process $Z_{n,\xi}$
expectations $E f(Y_n)$ are used which are usually unknown to the
statistician. Hence, in some applications, it may be useful to
replace them with the estimators $\overline{f(Y_n)}$, which leads to
the bootstrap processes $Z_{n,\xi}^*$.
Our main goal is to prove weak convergence of $Z_{n,\xi}$ and
$Z_{n,\xi}^*$ to $Z$ in probability, conditionally on the data. To
this end, as usual, we metrize weak convergence in
$\ell^\infty(\FF)$ using the bounded Lipschitz metric on the space
of probability measures on $\ell^\infty(\FF)$. That is, for two
probability measures $Q_1$ and $Q_2$ we define
$$ d_{BL(\ell^\infty(\FF))}(Q_1,Q_2) := \sup_{g\in
BL_1(\ell^\infty(\FF))} \Big| \int g\, dQ_1-\int g\, dQ_2\Big|,
\begin{eqnarray*}
BL_1(\ell^\infty(\FF)) & := & \big\{
g:\ell^\infty(\FF)\to \R\mid \|g\|_\infty := \sup_{z\in
\ell^\infty(\FF)}|g(z)|\le 1,\\
& & \hspace*{0.5cm} |g(z_1)-g(z_2)|\le \|z_1-z_2\|_\FF :=
\sup_{f\in\FF}|z_1(f)-z_2(f)| \text{ for all }
\end{eqnarray*}
Likewise, for the convergence of the fidis, we use the distance
$$ d_{BL(\R^l)}(Q_1,Q_2) := \sup_{g\in
BL_1(\R^l)} \Big| \int g\, dQ_1-\int g\, dQ_2\Big|,
between two probability measures $Q_1$ and $Q_2$ on $\R^l$, where
$$ BL_1(\R^l) := \big\{ g:\R^l\to \R\mid
\sup_{v\in \R^l}|g(v)|\le 1, |g(v_1)-g(v_2)|\le \|v_1-v_2\| \text{
for all } v_1,v_2\in\R^l\big\}.
By $E_\xi$ (resp. $E_\xi^*$) we denote the (outer) expectation with
respect to $(\xi_j)_{j\in\N}$, i.e.$E_\xi\big(f(\xi_1,\ldots,\xi_{m_n},X_{n,1},\ldots,
X_{n,n}) \mid X_{n,1},\ldots, X_{n,n}\big)$ is the expectation of
the function conditionally on the observations. Likewise, we denote by $P_\xi$ the
probability measure w.r.t. $(\xi_j)_{j\in\N}$. (Cf. Kosorok, 2003,
for a precise definition using a special construction of probability
Our first result shows that the asymptotic behavior of the fidis of
$Z_{n,\xi}$, conditionally on the data, is the same as the
(unconditional) behavior of the fidis of $Z_n$.
Under the conditions (B1), (B2) and (C1)–(C3) one has for all
$f_1,\ldots, f_l\in\FF$
\begin{equation} \label{eq:bootfidis}
\sup_{g\in BL_1(\R^l)} \Big| E_\xi g\big((Z_{n,\xi}(f_k))_{1\le
k\le l}\big) -E g\big((Z(f_k))_{1\le
k\le l}\big) \Big| \;\longrightarrow\; 0
\end{equation}
in probability.
Since the supremum in (<ref>) is bounded by 2, it readily follows that
\begin{eqnarray*}
\lefteqn{\sup_{g\in BL_1(\R^l)} \Big| E g\big((Z_{n,\xi}(f_k))_{1\le
k\le l}\big) -E g\big((Z(f_k))_{1\le
k\le l}\big) \Big|}\\
& \le & E \sup_{g\in BL_1(\R^l)} \Big| E_\xi g\big((Z_{n,\xi}(f_k))_{1\le
k\le l}\big) -E g\big((Z(f_k))_{1\le
k\le l}\big) \Big| \;\longrightarrow\; 0,
\end{eqnarray*}
that is, the (unconditional) weak convergence of the fidis of
$Z_{n,\xi}=(Z_{n,\xi}(f))_{f\in\FF}$ to the corresponding fidis of
Following the ideas developed by Kosorok (2003), the following
result establishes the asymptotic tightness of $Z_{n,\xi}$ under a
bracketing entropy condition, and thus also the weak convergence of
$Z_{n,\xi}$ under the same conditions as the convergence of the
original empirical process in Theorem <ref>(ii).
Suppose that the conditions (B1), (B2), (D1), (D3) and (D4) hold
* (D2) holds and $\xi$ is bounded, or
* (D2') holds and $E^*(F^2(Y_n))=O(r_n v_n)$.
Then $Z_{n,\xi}$ is asymptotically tight in $l^\infty(\FF)$. Hence
it converges to $Z$ if, in addition, the conditions (C1)–(C3) are
Now a modification of the arguments given in the proof of Theorem 2
of Kosorok (2003) yields the desired convergence result for the
multiplier process conditionally on the data.
If condition (D3) and convergence (<ref>) hold and
$Z_{n,\xi}$ weakly converges to $Z$, then
\begin{equation} \label{eq:bootconv1}
\sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi g(Z_{n,\xi})-
Eg(Z)\big| \;\longrightarrow\; 0
\end{equation}
in outer probability.
A combination of this result with Theorem <ref> and
Proposition <ref> leads to
If the conditions (B1), (B2), (C1)-(C3) and (D1)-(D4) are
satisfied and $\xi$ is bounded, then convergence (<ref>) holds.
According to Theorem <ref>, under (D3) the weak
convergence of the multiplier process $Z_{n,\xi}$ to $Z$
conditionally on the data follows from the weak convergence of the
fidis conditionally on the data and the (unconditional) convergence
of $Z_{n,\xi}$ to $Z$. The latter assertion may also be derived by
establishing the asymptotic equicontinuity of $Z_{n,\xi}$ using a
metric entropy condition (instead of verifying tightness using a
bracketing entropy condition as in Proposition
Suppose that the conditions (B1), (B2), (D1), (D2'), (D3)
(D5') [t]14.9cmFor all $\delta>0, n\in\N, (e_i)_{1\le i\le
\floor{m_n/2}} \in \{-1,0,1\}^{\floor{m_n/2}}$ and $k\in\{1,2\}$ the
map $\sup_{f,g\in\FF, \rho(f,g)<\delta}$
$\sum_{j=1}^{\floor{m_n/2}} e_j\big(\xi_j(
is measurable
are fulfilled and
* (D6) holds and $\xi$ is bounded, or
* (D6') holds.
Then $Z_{n,\xi}$ is asymptotically equicontinuous. Hence,
it converges to $Z$ if, in addition, the conditions (C1)–(C3) are
Using Theorem <ref> and Corollary 2.6.12 of van der
Vaart and Wellner (1996), we obtain as an immediate consequence
If the conditions (B1), (B2), (C1)-(C3), (D1), (D2'), (D3) and
(D5') are met, if $F$ is measurable with $E(F^2(Y_n))=O(r_nv_n)$ and $\FF$ is a VC-hull
class, then convergence (<ref>) holds.
To sum up, we have shown that, roughly under the same conditions as
used in Theorem <ref>, the multiplier process
$Z_{n,\xi}$ shows the same asymptotic behavior conditionally on the
data as the empirical process $Z_n$ unconditionally. The following
result gives conditions under which the convergence of $Z_{n,\xi}$
implies the convergence of the bootstrap process $Z_{n,\xi}^*$
conditionally on the data.
If convergence (<ref>) of the fidis of $Z_{n,\xi}$
holds conditionally on the data, condition (D3) is satisfied and
$Z_n\to Z$ and $Z_{n,\xi}\to Z$ weakly, then
\begin{equation} \label{eq:ZnxiZnxistarapprox}
E_\xi \sup_{f\in\FF}
|Z_{n,\xi}^*(f)-Z_{n,\xi}(f)|\;\longrightarrow\; 0
\end{equation}
in outer probability, $Z_{n,\xi}^*\to Z$ weakly and
\begin{equation} \label{eq:bootconv2}
\sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi g(Z_{n,\xi}^*)-
Eg(Z)\big| \;\longrightarrow\; 0
\end{equation}
in outer probability. In particular, these assertions hold under
the conditions of Corollary <ref> and under the
assumptions of Corollary <ref>.
Note that also the normalizing factor $(nv_n)^{-1/2}$ in the
definition of $Z_{n,\xi}^*$ may be unknown. In most
applications of multiplier processes, though, this is not
problematic, because this factor is not needed to construct
confidence regions. Nevertheless, it is noteworthy that assertion
(<ref>) remains valid if $v_n$ is replaced with some
estimator $\hat v_n$ that is consistent in the sense that $\hat
v_n/v_n\to 1$ in probability.
For specific types of cluster functionals, Drees and Rootzén (2010)
gave simpler sufficient conditions for the convergence of the
corresponding empirical process which carry over to the multiplier
processes considered here. In the next section we will use the
conditions of Corollary 3.6 of that paper, which deals with
so-called generalized tail array sums, i.e. empirical processes
with functionals of the form $f_\phi(y_1,\ldots,y_r)=\sum_{i=1}^r
\phi(y_i)$ for functions $\phi:(E,\mathbb{E})\to(\R,\B)$ such that
§ PROCESSES OF EXTREMOGRAMS
In this section we employ the general theory to analyze the
asymptotic behavior of the empirical extremogram $\hat\rho_{n,A,B}$,
a version with empirical normalization and a bootstrap version
thereof, uniformly over suitable families of sets $A$ and $B$ and
over lags $h\in\{0,\ldots,h_0\}$ for some fixed $h_0\in\N$.
Throughout this section we are only interested in the behavior for
vectors with at least one large component. We thus consider families
$\CC$ of pairs of measurable subsets of $\R^d$ such that
$$ x_* := \inf_{(A,B)\in\CC} \inf_{x\in A} \max_{1\le j\le d}
i.e. $A\subset \R^d\setminus (-\infty,x_*)^d$ for all
$(A,B)\in\CC$. However, the results below can be generalized to
families of sets that are uniformly bounded away from 0 so that
$\inf_{(A,B)\in\CC}\inf_{x\in A} \max_{1\le j\le d} |x_j|>0 $. For
the sake of notational simplicity, we assume that $n+h_0$ (instead
of $n$) $\R^d$-valued random vectors $X_1,\ldots,X_{n+h_0}$ are
To keep the presentation simple, we will assume that $X_0$ is
regularly varying on the full cone
$\R^d\setminus\{0\}$ with a limiting measure $\nu_0$
which is not concentrated on $(-\infty,0]^d$; see Theorem
<ref> below. This assumption could be weakened to
the regular variation on the cone $\R^d\setminus (-\infty,0]^d$
defined in the spirit of Das et al. (2013), i.e. there exists a
normalizing sequence $\tilde a_n>0$ and a measure $\tilde \nu_0$ such that
$$ nP\{ X_0/\tilde a_n\in B\} \;\longrightarrow\; \tilde \nu_0(B) $$
for all $\tilde\nu_0$-continuity sets $B\in\B$ bounded away from
$(-\infty,0]^d$, where the limit has to be finite. Here one may
choose $\tilde a_n$ as the $(1-1/n)$-quantile of $\max_{1\le j\le
d} X_{0,j}$. Under the slightly more restrictive assumption used
in the results below, one has
$$ \frac{P\{\max_{1\le j\le d} X_{0,j}>u\}}{P\{\|X_0\|>u\}}
\;\longrightarrow\;\nu_0\big(\R^d\setminus(-\infty,1]^d\big)
as $u\to\infty$, and hence $\tilde a_n\sim a_n
\big(\nu_0\big(\R^d\setminus(-\infty,1]^d\big)\big)^{1/\alpha}$ and
where $-\alpha$ is the degree of homogeneity of $\nu_0$, i.e. $\nu_0(\lambda B)=\lambda^{-\alpha}\nu_0(B)$.
For some intermediate sequence $k=k_n$ (i.e. $k_n\to\infty$,
$k_n/n\to 0$), we define the empirical extremogram to the sets $A$
and $B$ and lag $h$ as
$$ \hat \rho_{n,A,B}(h) := \frac{\textstyle \sum_{i=1}^n
1_{A\times B}(X_i/a_k,X_{i+h}/a_k)}{\textstyle \sum_{i=1}^n
Note that this is a slight modification of the definition given by
Davis and Mikosch (2009) in that we do not use the maximal number of
summands in the denominator. However, it is easily seen that all
results given below carry over to the original definition.
The uniform asymptotic behavior of the empirical extremogram will
easily follow from that of the stochastic process
\begin{eqnarray*}
\tilde Z_n(h,A,B) & := & \frac 1{\sqrt{nv_n}} \sum_{i=1}^n \Big( 1_{A\times
B}(X_i/a_k,X_{i+h}/a_k)-P\{X_i\in a_kA,X_{i+h}\in a_kB\}\Big),
\end{eqnarray*}
$h\in\{0,\ldots,h_0\}, (A,B)\in\CC,$ with
$$ v_n:= P\{X_0\not\in (-\infty,a_kx_*)^d \}.
This process, in turn, can be analyzed using the theory for
empirical processes of cluster functionals developed by Drees and
Rootzén (2010). In order to use conditions on the joint distribution
of the $X_t$ as weak as possible, it is useful to consider such
processes indexed by $(A,B)\in\CC$ and just two lags $h,\tilde
h\in\{0,\ldots,h_0\}$. Let
\begin{eqnarray*}
\tilde X_{n,i} & := & \frac{X_i}{a_k} 1_{\R^d\setminus
(-\infty,x_*)^d}\Big(\frac{X_i}{a_k}\Big), \quad 1\le i\le
X_{n,i}^{(h,\tilde h)} & := & (\tilde X_{n,i},\tilde X_{n,i+h},\tilde X_{n,i+\tilde h}), \quad 1\le i\le
Y_{n,j}^{(h,\tilde h)} & := & (X_{n,i}^{(h,\tilde
h)})_{(j-1)r_n<i\le jr_n}, \quad 1\le j\le m_n,\\
v_n^{(h,\tilde h)} & := & P\{X_{n,i}^{(h,\tilde h)}\ne 0\} =
P\{(X_0,X_h,X_{\tilde h})\not\in (-\infty,x_*)^{3d}\},\\
\DD & := & \{A\times B\times \R^d, A\times\R^d\times B\mid
(A,B)\in \CC\},\\
f_D(y_1,\ldots, y_r) & := & \sum_{i=1}^r 1_D(y_i), \quad
y_i\in\R^{3d},\quad D\in\DD,\\
\FF & := & \{f_D\mid D\in\DD\}, \quad \text{and}\\
Z_n^{(h,\tilde h)}(f_D) & := & \frac 1{\sqrt{n v_n^{(h,\tilde h)}}}
\sum_{j=1}^{m_n} \big( f_D(Y_{n,j}^{(h,\tilde h)})- E f_D(Y_{n,j}^{(h,\tilde
& = & \frac 1{\sqrt{n v_n^{(h,\tilde h)}}}
\sum_{i=1}^{m_nr_n} \big( 1_D(X_{n,i}^{(h,\tilde h)})- P\{X_{n,i}^{(h,\tilde
h)}\in D\}\big), \quad D\in\DD.
\end{eqnarray*}
Note that, for $n=m_nr_n$, we have $\tilde
Z_n(h,A,B)=(v_n^{(h,\tilde h)}/v_n)^{1/2} Z_n^{(h,\tilde
h)}(f_{A\times B\times \R^d})$ and $\tilde Z_n(\tilde
h,A,B)=(v_n^{(h,\tilde h)}/v_n)^{1/2} Z_n^{(h,\tilde h)}(f_{A\times
\R^d\times B})$; under the conditions of Theorem <ref>
the difference between these processes is asymptotically negligible
even if $m_nr_n<n$.
Using Corollary 3.6 of Drees and Rootzén (2010) and Drees and Rootzén (2015), we
obtain the following set of sufficient conditions for the
convergence of $\tilde Z_n$.
Suppose that all four-dimensional marginal distributions of the
stationary time series $(X_t)_{t\in\N_0}$ are regularly varying, i.e. for all index vectors $I\in\N_0^l$ of dimension $l\le 4$ there exists a measure $\nu_I$ such that
\begin{equation} \label{eq:fourdimregvar}
n P\{a_n^{-1}X_I\in B\} \;\longrightarrow\,
\nu_I(B)<\infty
\end{equation}
for all Borel sets $B$ bounded away from $0\in\R^{ld}$, and that $\nu_0(\R^d\setminus(-\infty,x^*)^d)>0$.
In addition, assume that
the conditions (B1), (B2) and ($\widetilde{\text{B3}}$) are fulfilled, and
$r_n=o(\sqrt{nv_n})$. Finally, assume that there exists a bounded semi-metric $\bar\varrho$
on $\CC$ such that $\CC$ is totally bounded w.r.t. $\bar\varrho$, and a function $u:(0,\infty)\to (0,\infty)$ such that
$\lim_{t\downarrow 0} u(t)=0$ and
\begin{equation} \label{eq:clustermoment}
E\Big(\sum_{i=1}^{r_n} 1_{(A\times B)\Delta (\tilde A\times\tilde
B)}(X_i/a_k,X_{i+h}/a_k)\Big)^2 \le u\big(
\bar\varrho\big((A,B),(\tilde A,\tilde B)\big)\big) r_n v_n
\end{equation}
for all $(A,B),(\tilde A,\tilde B)\in\CC$, $h\in\{0,\ldots,h_0\}$,
and that the conditions (D5) and (D6) hold for
$\varrho\big(f_D,f_{\tilde D}\big):= \bar\varrho\big((A,B),(\tilde A,\tilde
B)\big)$ if $D=A\times B\times \R^d$, $\tilde D=\tilde A\times \tilde B\times
\R^d$, or $D=A\times \R^d\times B$, $\tilde D=\tilde A\times \R^d\times \tilde B$,
and $\varrho\big(f_D,f_{\tilde D}\big):= L$ else for some
sufficiently large constant $L>1$.
(Here $C_1\Delta C_2$ denotes the
symmetric difference of the two sets $C_1$ and $C_2$.)
$\tilde Z_n$ converges weakly to
a Gaussian process $\tilde Z$ with covariance function
$$\tilde c\big((h,A,B),(\tilde h,\tilde A,\tilde B)\big):=
\sum_{i=-\infty}^\infty \frac{\nu_{(0,h,i,i+\tilde h)}(A\times B\times \tilde A\times \tilde B)}{
\nu_{0}\big(\R^d\setminus (-\infty,x_*)^d\big)}<\infty.
Observe that in (<ref>) necessarily the following
consistency condition holds: for vectors $I_0=(i_j)_{1\le j\le l}$
and $I=(i_j)_{1\le j\le 4}$ of indices and $\nu_{I_0}$-continuity
sets $A\in \B^{ld}$ bounded away from the origin one has
Usually the moment condition (<ref>) and the
entropy condition (D6) are most difficult to verify. The proof
of Theorem <ref> shows that the process $\tilde Z_n$
indexed by $\tilde \FF:=\{(h,A,B)\mid h\in\{0,\ldots,h_0\},
(A,B)\in\CC\}$ is asymptotically tight if and only if the empirical
processes $Z_n^{(h,\tilde h)}$ indexed by $\{f_{A\times B\times
\R^d}\mid (A,B)\in\CC\}$ resp. $\{f_{A\times \R^d\times B}\mid
(A,B)\in\CC\}$ are asymptotically tight for all $h,\tilde
h\in\{0,\ldots,h_0\}$. Thus we may replace condition (D6) by the
assumption that these families are VC-subgraph class of functions,
which in turn is equivalent to the assumption that
\begin{equation} \label{eq:defFbar}
\bar\FF:=\{\bar
f_{A\times B}\mid (A,B)\in\CC\} \quad \text{with} \quad \bar f_D(y_1,\ldots, y_r) :=
\sum_{i=1}^r 1_D(y_i) \;\; \text{for} \;\; y_i\in\R^{2d}, 1\le i\le r,
\end{equation}
is a VC-subgraph class of
functions. Likewise, one may divide the family $\CC$ into a finite
number of subfamilies $\CC_j$ and check that $\bar\FF_j:=\{\bar
f_{A\times B}\mid (A,B)\in\CC_j\}$ is a VC-subgraph class of
For applications to the asymptotic analysis of empirical
extremograms, we shall consider families $\CC$ such that for
$(A,B)\in\CC$ also $(A,\R^d)$ belongs to $\CC$. The following simple
example exhibits another closedness property of $\CC$ which is
important to prove convergence of the empirical extremogram with
estimated normalizing constant.
Fix some $\lambda_0>0$ and measurable sets $A_0,B_0\subset \R^d$
bounded away from 0 such that $x\in A_0$ implies $\lambda x\in A_0$
for all $\lambda>1$ and likewise for $x\in B_0$. (In particular, one
may choose a set $A_0\subset [0,\infty)^d\setminus[0,1]^d$ such
that $x\in A_0$ and $y\ge x$ imply $y\in A_0$.) Then, for
$\CC_1:=\{\lambda(A_0,B_0)\mid \lambda> \lambda_0\}$, the family
$\bar\FF_1$ is a VC-subgraph class of functions. To see this, note
that $\bar f_{\lambda (A\times B)} \le f_{\tilde \lambda
(A\times B)}$ if $\lambda> \tilde\lambda$, i.e. the functions are linearly ordered. Hence no set of size 2 can be
shattered by the subgraphs of $\bar\FF_1$. Likewise, the family
$\bar\FF_2$ pertaining to $\CC_2:=\{\lambda(A_0,\R^d)\mid \lambda>
\lambda_0\}$ is a VC-subgraph class of functions.
Condition (<ref>) can be reformulated as follows.
There exists a semi-metric $\tilde\varrho$ on $[\lambda_0,\infty)$
such that $[\lambda_0,\infty)$ is totally bounded w.r.t. $\tilde\varrho$ and $E\big(\sum_{i=1}^{r_n} 1_{(\lambda(A_0\times
B_0))\setminus (\tilde\lambda(A_0\times
u(\tilde\varrho(\lambda,\tilde\lambda))r_nv_n$ and $E\big(\sum_{i=1}^{r_n}
1_{(\lambda A_0)\setminus (\tilde\lambda A_0)}(X_i/a_k)\big)^2\le
u(\tilde\varrho(\lambda,\tilde\lambda))r_nv_n$ hold for all
$\lambda_0< \lambda<\tilde\lambda$ and all $n\in\N$.
The families of sets $A$ and $B$ most widely discussed in the
literature are sets of upper right orthants $(x,\infty)$ and
complements $\R^d\setminus (-\infty,x]$ of lower left orthants.
Consider the family $\CC_1:=\big\{ \big((x_A,\infty),(x_B,\infty)\big) \mid
x_A,x_B \not\in (-\infty,x_*]^d\big\}$ of pairs of upper right
orthants bounded away from the origin. Then condition (D6) holds for
$\CC:=\CC_1\cup\big\{\big((x_A,\infty),\R^d\big)|x_A\not\in (-\infty,x_*]^d\big\}$ if condition (B1) is satisfied and
\begin{equation} \label{eq:extremomomcond}
E\Big( \sum_{i=1}^{r_n} 1_{\{X_i\not\in
(-\infty,a_kx_*)^d\}}\Big)^{2+\delta} = O(r_nv_n),
\end{equation}
for some $\delta>0$.
(see Section <ref>).
By the same arguments one can show that condition (D6) is
fulfilled for the family $\CC:=\big\{ \big(\R^d\setminus(-\infty,x_A],\R^d\setminus(-\infty,x_B]\big),
\big(\R^d\setminus(-\infty,x_A],\R^d\big) \mid x_A,x_B \in (x_*,\infty)^d\big\}$.
From Theorem <ref> one may easily conclude the uniform
asymptotic normality of the empirical extremogram centered at the
pre-asymptotic extremogram
$$ \rho_{t,A,B}(h) := P(X_h/t\in B\mid X_0/t\in A). $$
Suppose that the conditions of Theorem <ref> are met, that
$(A,\R^d)\in\CC$ for all $(A,B)\in\CC$ and
$\inf_{(A,B)\in\CC}\nu_0(A)>0$, and that $\sup_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC}|\rho_{a_k,A,B}(h)-\rho_{A,B}(h)|\to 0$. Then
\begin{eqnarray}
\lefteqn{\sqrt{nv_n} \Big(\hat\rho_{n,A,B}(h)- \rho_{a_k,A,B}(h)\Big)_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC}} \nonumber\\
& \to &
\Big(\frac{\nu_0\big(\R^d\setminus(-\infty,x_*)^d\big)}{\nu_0(A)}\big(\tilde
Z(h,A,B)-\rho_{A,B}(h)\tilde Z(h,A,\R^d)\big)\Big)_{h\in\{0,\ldots,h_0\},
(A,B)\in\CC} \nonumber\\
& =: & (R(h,A,B))_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC}
\label{eq:empextremoconv}
\end{eqnarray}
Hence if, in addition,
\begin{equation} \label{eq:preasympcond1}
\sup_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC}
\sqrt{\frac nk}|\rho_{a_k,A,B}(h)-\rho_{A,B}(h)|\to 0,
\end{equation}
\sqrt{nv_n} \Big(\hat\rho_{n,A,B}(h)- \rho_{A,B}(h)\Big)_{h\in\{0,\ldots,h_0\},
(A,B)\in\CC} \to
(R(h,A,B))_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC}
We have already mentioned in the introduction that the empirical
extremogram $\hat\rho_{n,A,B}(h)$ is not a valid estimator if the
normalizing constants $a_k$ are unknown. In this case we replace
them by some estimator $\hat a_k$ which is consistent in the sense
that $\hat a_k/a_k\to 1$ in probability. Noting that
$$ \hat{\hat \rho}_{n,A,B}(h) := \frac{\textstyle \sum_{i=1}^n
1_{A\times B}(X_i/\hat a_k,X_{i+h}/\hat a_k)}{\textstyle
\sum_{i=1}^n 1_{A}(X_i/\hat a_k)}= \hat \rho_{n,(\hat
a_k/a_k)A,(\hat a_k/a_k)B}(h),
the asymptotic normality of $\hat{\hat \rho}_{n,A,B}(h)$ is an easy
consequence of Corollary <ref>, provided that
$\rho_{t,A,B}(h)$ is a sufficiently regular function of $t$.
Assume that the conditions of Corollary <ref>
(except (<ref>)) are fulfilled and, in addition,
$\hat a_k/a_k\to 1$ in probability, that $(A,B)\in\CC$ implies $(\lambda A,\lambda B)\in\CC$
for all $\lambda$ in a neighborhood of 1 and that $\sup_{(A,B)\in\CC}
\bar\varrho\big((A,B),(\lambda A,\lambda B)\big) \to 0$ as
$\lambda\to 1$. Then
\sqrt{nv_n} \Big(\hat{\hat \rho}_{n,A,B}(h)- \rho_{\hat a_k,A,B}(h)\Big)_{h\in\{0,\ldots,h_0\},
(A,B)\in\CC} \to
(R(h,A,B))_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC}
weakly. Hence, if the following second order condition holds
\begin{equation} \label{eq:secordrho}
\rho_{t,A,B}(h) = \rho_{A,B}(h) + \Phi_h(t)\Psi_h(A,B) +
\end{equation}
uniformly for $h\in\{0,\ldots,h_0\}$, $(A,B)\in\CC$, and some extended
regularly varying function $\Phi_h$ (see Bingham et al., 1987,
Section 2.0) satisfying $\Phi_h(t)\to 0$ as $t\to \infty$ and some
functions $\Psi_h$ such that $\sup_{(A,B)\in\CC}
|\Psi_h(A,B)|<\infty$, then
\sqrt{nv_n} \Big(\hat{\hat \rho}_{n,A,B}(h)- \rho_{a_k,A,B}(h)\Big)_{h\in\{0,\ldots,h_0\},
(A,B)\in\CC} \to
(R(h,A,B))_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC}
weakly, provided $\Phi_h(a_k)=O((k/n)^{1/2})$. If $\Phi_h(a_k)=o((k/n)^{1/2})$, then this
convergence holds with $\rho_{A,B}(h)$ instead of
* If $(X_0,X_h)$ satisfies the second order condition
\begin{equation} \label{eq:regvarsecord}
a^\leftarrow(t) P\{(X_0,X_h)/t\in A\times B\} =
\nu_{(0,h)}(A\times B)+\Phi_h(t)\tilde\Psi_h(A\times B)+
\end{equation}
uniformly for all $(A,B)\in\CC$ with $\sup_{(A,B)\in\CC}|\tilde \Psi_h(A\times B)|<\infty$, then under the conditions of
Corollary <ref> direct calculations show that
$\rho_{t,A,B}(h)=P\{(X_0,X_h)/t\in A\times B\}/P\{(X_0,X_h)/t\in A\times
\R^d\}$ satisfies condition (<ref>) with
$\Psi_h(A, B) = \big(\tilde \Psi_h(A\times
B)-\rho_{A,B}(h)\tilde \Psi_h(A\times\R^d)\big)/\nu_0(A).$
* If the conditions of Theorem <ref> hold
when (<ref>) is replaced with
$$ E\Big(\sum_{i=1}^{r_n} 1_{[-y,y]^d\setminus [-x,x]^d}(X_i/a_k)\Big)^2 \le u(y-x) r_n v_n
for all $1-\delta\le x<y\le 1+\delta$, $n\in\N$, and some function $u$
satisfying $u(t)\to 0$ as $t\downarrow 0$, then the same arguments
as used in the proof of Theorem <ref> show that
$$ \frac 1{\sqrt{n/k}} \sum_{i=1}^n \Big( \Ind{\|X_i\|/a_k>x}-
P\{\|X_i\|/a_k>x\}\Big) = \sqrt{kv_n} \tilde
Z_n(0,\R^d\setminus[-x,x]^d,\R^d), \quad x\in[1-\delta,1+\delta],
converges weakly to a continuous Gaussian process. From
$P\{\|X_0\|/a_k>x\}\sim x^{-\alpha}/k$ and
\begin{eqnarray*}
\lefteqn{ \frac{\|X\|_{n-\floor{n/k}:n}}{a_k}}\\
& = & \inf\Big\{ x\,\Big|\,
\sum_{i=1}^n \Ind{\|X_i\|/a_k>x}\le \floor{n/k}\Big\} \\
& = & \inf\Big\{ x\,\Big|\,
\sqrt{kv_n} \tilde Z_n(0,\R^d\setminus[-x,x]^d,\R^d)\le (k/n)^{1/2}\big(\floor{n/k}-nP\{\|X_0\|/a_k>x\}\big)
\Big\},
\end{eqnarray*}
one can easily conclude that $\|X\|_{n-\floor{n/k}:n}/a_k\to 1$ in
probability, i.e. $\|X\|_{n-\floor{n/k}:n}$ is consistent for
Indeed, a refined analysis shows that under the second order
condition (<ref>) one even has
$\|X\|_{n-\floor{n/k}:n}/a_k-1=O_P((k/n)^{1/2})$ if
As the distribution of the limit process arising in Corollary 3.6
is difficult to estimate, we use the bootstrap approach discussed in
Section <ref> to approximate the distribution of the
empirical extremogram. Let
\begin{eqnarray*}
\hat\rho^*_{n,A,B}(h) & := & \frac{\sum_{j=1}^{m_n} (1+\xi_j)
\sum_{i=1}^{r_n} 1_{A\times B} \big( a_k^{-1}
(X_{(j-1)r_n+i},X_{(j-1)r_n+i+h})\big)}{\sum_{j=1}^{m_n} (1+\xi_j)
\sum_{i=1}^{r_n} 1_{A} \big( a_k^{-1}
\hat{\hat\rho}^*_{n,A,B}(h) & := & \frac{\sum_{j=1}^{m_n} (1+\xi_j)
\sum_{i=1}^{r_n} 1_{A\times B} \big( \hat a_k^{-1}
(X_{(j-1)r_n+i},X_{(j-1)r_n+i+h})\big)}{\sum_{j=1}^{m_n} (1+\xi_j)
\sum_{i=1}^{r_n} 1_{A} \big( \hat a_k^{-1}
& = & \hat\rho^*_{n,(\hat a_k/a_k)A,(\hat a_k/a_k)B}(h)\\
R_{n,\xi}(h,A,B) & := & \sqrt{nv_n}\big( \hat\rho^*_{n,A,B}(h)-
\hat\rho_{n,A,B}(h)\big)\\
\hat R_{n,\xi}(h,A,B) & := & \sqrt{nv_n}\big( \hat{\hat\rho}^*_{n,A,B}(h)-
\hat{\hat\rho}_{n,A,B}(h)\big) = R_{n,\xi}(h,(\hat a_k/a_k)A,(\hat
\end{eqnarray*}
Suppose that all conditions of Corollary <ref> are
fulfilled and that $\xi_j$, $j\in\N$, are i.i.d. random variables
with $E(\xi_1)=0$ and $Var(\xi_1)=1$ independent of $(X_t)_{t\in\N_0}$. Then,
\begin{eqnarray} \label{eq:extremobootstrap1}
\sup_{g\in BL_1(\ell^\infty(\{0,\ldots,h_0\}\times
\CC))} \big| E_\xi g(R_{n,\xi})-E g(R)\big| &\to & 0\\
\sup_{g\in BL_1(\ell^\infty(\{0,\ldots,h_0\}\times
\CC))} \big| E_\xi g(\hat R_{n,\xi})-E g(R)\big| &\to & 0
\quad \text{in probability.} \label{eq:extremobootstrap2}
\end{eqnarray}
Let $\tilde \FF:=\{0,\ldots, h_0\}\times\CC$. In view of Theorem
<ref>, approximate confidence regions for the
extremogram $(\rho_{A,B}(h))_{(h,A,B)\in\tilde\FF}$ can be obtained
from Monte Carlo simulations of $\hat{\hat\rho}^*_{n,A,B}(h)$. To
this end, suppose $\DD$ is a family of subsets of
$\ell^\infty(\tilde\FF)$ such that $\sup_{D\in\DD}P\{R\in
U_\eps(\partial D)\}\to 0$ as $\eps\downarrow 0$, where $U_\eps(A)$
denotes the open $\eps$-neighborhood of a set $A$ w.r.t. the
supremum norm $\|\cdot\|_{\tilde\FF}$ on $\ell^\infty(\tilde\FF)$.
Then all indicator functions $1_D$, $D\in\DD$, can be uniformly well
approximated from above and from below by functions of the form
$g_{\eps,A}:=(1-d_A/\eps)^+$ with $d_A(z):= \inf_{\tilde z\in A}
\|z-\tilde z\|$. Since the functions $\eps g_{\eps,A}$ belong to
$BL_1(\ell^\infty(\tilde\FF))$, it is easily seen that
(<ref>) and (<ref>) imply
$\sup_{D\in\DD} \big| P_\xi\{R_{n,\xi}\in D\}-P\{R\in D\}\big|\to 0$
and $\sup_{D\in\DD} \big| P_\xi\{\hat R_{n,\xi}\in D\}-P\{R\in
D\}\big|\to 0$ as $n\to\infty$, respectively.
In particular, if for sufficiently large $n\in\N$, $D_\alpha$ is a
subset of $\ell^\infty(\tilde\FF)$ such that
\begin{equation} \label{eq:bootconf1}
\hat{\hat\rho}_{n,A,B}(h))_{(h,A,B)\in\tilde\FF}\in
\end{equation}
then under the conditions of Corollary <ref> with
$\Phi(a_k)=o((k/n)^{1/2})$, for sufficiently large $n$, we have
\begin{equation} \label{eq:bootconf2}
(\hat{\hat\rho}_{n,A,B}(h)-\rho_{A,B}(h))_{(h,A,B)\in\tilde\FF} \in
D_\alpha\big\} \approx \alpha.
\end{equation}
To find such a set (or rather an approximation to it), one may
simulate $B$ independent realizations
$(\hat{\hat\rho}^{*(b)}_{n,A,B}(h))_{(h,A,B)\in\tilde\FF}$, $1\le
b\le B$, of the bootstrap version of the empirical extremogram. For
some fixed set $D\subset \ell^\infty(\tilde\FF)$ let
$D_\alpha:=\lambda_\alpha D$ with $\lambda_\alpha$ denoting the
smallest $\lambda\ge 0$ such that
$$ \frac 1B \sum_{b=1}^B 1_{\textstyle
\big\{\big(\hat{\hat\rho}^{*(b)}_{n,A,B}(h)-
\hat{\hat\rho}_{n,A,B}(h)\big)_{(h,A,B)\in\tilde\FF}\in\lambda
Here $D$ ought to be star-shaped, i.e. $z\in D$ implies $\lambda
z\in D$ for all $\lambda\in[0,1]$. The shape of $D$ determines the
emphasis which is laid on particular features of the extremogram.
See Section <ref> for an example.
§ FINITE SAMPLE PERFORMANCE OF BOOTSTRAPPED EXTREMOGRAMS
In this section we investigate the finite sample performance of confidence intervals which are constructed using the multiplier block bootstrap approach, the stationary bootstrap proposed by Davis et al. (2012) and a modified version of the latter.
Davis et al. (2012) suggested to construct bootstrap samples from an observed time series $(X_t)_{1\le t\le n}$ as follows. Let $K_i$, $1\le i\le n$, be iid random variables uniformly distributed on $\{1,\ldots,n\}$, and $L_i$, $1\le i\le n$, iid random block lengths with a geometric distribution with expectation $r$, independent of $(K_i)_{1\le i\le n}$. Define $S_j:=\sum_{i=1}^j L_i$, $0\le j\le n$, $N:=\min\{j| S_j \ge n\}$, and $L_j^*:=L_j$ for $1\le j<N$ and $L_N^*:=n-S_{N-1}$. For $i\in \{S_{j-1}+1,S_{j-1}+2,\ldots, S_j\}$, $1\le j\le N$, let $X_i^*:= X_{K_j-1+i-S_{j-1}}$, where $X_t$ for $t>n$ is interpreted as $X_{(t \text{ mod } (n-1))+1}$. This means that blocks of length $L_j$ starting from the observation at $K_j$ are glued together until one obtains a new time series $(X_t^*)_{1\le t\le n}$ of length $n$; in this process one repeats the original time series after the last observation as often as necessary.
Now denote by $\hat\rho^{(*DMC)}_{n,A,B}(h)$ the bootstrap estimator of $\rho_{A,B}(h) $ calculated from $(X_t^*)_{1\le t\le n}$.
Davis et al. (2012) proved that under suitable conditions, conditionally on the data, the limit distribution of $\hat\rho^{(*DMC)}_{n,A,B}(h)-\hat\rho_{n,A,B}(h)$ is the same as the one of $\hat\rho_{n,A,B}(h)-\rho_{a_k,A,B}(h)$, so that bootstrap confidence intervals can be constructed.
One disadvantage of this approach is that for indices $i$ near the end of a block such that $i\le S_j< i+h$ for some $1\le j<N$ the indicator $\Ind{X^*_i\in a_kA, X^*_{i+h}\in a_kB}$ has a completely different behavior than $\Ind{X_i\in a_kA, X_{i+h}\in a_kB}$, because $(X^*_i, X^*_{i+h})$ does not correspond to a pair of observations with lag $h$ in the original time series.
To overcome this drawback, we suggest the following modification of the stationary bootstrap estimator. For simplicity, we assume that the time series has been observed at $n+h$ time points (in other words, we redefine $n$). Then we define
$$ \hat\rho_{n,A,B}^{(*stat)} (h):= \frac{\sum_{j=1}^N \sum_{i=1}^{L_j^*} \Ind{X_{K_j-1+i}\in a_k A,
X_{K_j-1+i+h}\in a_kB}}{\sum_{j=1}^N \sum_{i=1}^{L_j^*} \Ind{X_{K_j-1+i}\in a_k A}}
which has the same asymptotic behavior as $\hat\rho^{(*DMC)}_{n,A,B}(h)$, but only observations are compared which are lagged by $h$. In essence, this mean that we apply the stationary bootstrap technique to the bivariate time series $(X_t,X_{t+h})_{1\le t\le n}$.
In addition to these two version of stationary bootstrap estimators, we consider the multiplier bootstrap. Here we have drawn multipliers $\xi_j$ from a Student $t$-distribution with 5 degrees of freedom and scale parameter such that $Var(\xi_j)=1$. However, this particular choice is not crucial as in further simulations we have obtained a similar performance of the multiplier block bootstrap for other distributions which are symmetric about 1 with an unbounded support (e.g., for normally distributed multiplier).
Here we report the results for three different models:
* a GARCH model $X_t=\sigma_t\eps_t$, $\sigma_t^2=\alpha_0+\alpha_1X_{t-1}^2+\beta_1\sigma_{t-1}^2$ with $\alpha_0=10^{-4}, \alpha_1=0.08, \beta_1=0.9$ and $t$-distributed innovations $\eps_t$ with 8 degrees of freedom, independent of $\sigma_t$
* an autoregressive model of order 1: $X_t=\varphi X_{t-1}+\eps_t$ with $\varphi=0.8$ and symmetrized Fréchet distribution of the innovations, i.e., $P\{\eps_t>x\}=P\{\eps_t<-x\}=(1-\exp(-x^{-3}))/2$ for all $x>0$
* a moving average time series of order 3, namely $X_t=\eps_{t}+0.5\eps_{t-1}+0.8\eps_{t-2}$ with $\eps_t$ as in (ii).
For each model we simulated $10\,000$ time series of length $n=2000$.
Empirical coverage probability of confidence intervals (<ref>) for $\rho_{a_k,(1,\infty),(1,\infty)}(h)$ as a function of $h$, constructed using multiplier block bootstrap (blue $*$), stationary bootstrap suggested by Davis et al. (red $+$) and the modification thereof (black $\circ$) with (average) block length $r=100$ (left) and $r=20$ (right) for the $t$-GARCH model (i), different thresholds $a_k$ with exceedance probability $p$ are used in the three rows; the nominal coverage probability 0.95 is indicated by the horizontal line.
We consider the extremogram for $A=B=(1,\infty)$, i.e., $\lim_{u\to\infty} P(X_h>u|X_0>u)$, which is often also called tail dependence coefficient, and lags $1\le h\le 10$. As normalizing constants $a_k$ (thresholds) we have chosen the $(1-p)$-quantile of the stationary distribution for $p\in\{0.01,0.025,0.05\}$ which have been estimated by the corresponding empirical quantiles. The true pre-asymptotic extremograms have been determined by simulation (based on 1000 time series of length $10^7$). Analytic expression for the (asymptotic) extremograms are known for the linear models (ii) and (iii) (see e.g., Meinguet and Segers, 2010, Example 9.2). For the GARCH model, they were determined using a simulation algorithm suggested by Ehlert et al. (2015).
In each simulation we have drawn $b=1000$ bootstrap replicas according to each of the three bootstrap procedures. If, for fixed $h$, the upper and lower empirical $\alpha/2$-quantile of the resulting $b$ bootstrap estimates of the extremogram are denoted by $u_b$ and $l_b$ then, according to (<ref>) and (<ref>),
\begin{equation} \label{eq:confint}
\big[2\hat\rho_{n,A,B}(h)-u_b, 2\hat\rho_{n,A,B}(h)-l_b\big]\cap [0,1]
\end{equation}
is a confidence interval for the (pre-asymptotic) extremogram with nominal coverage probability $1-\alpha$.
We first discuss the results for the $t$-GARCH model in detail, before we show the results for the linear time series in abbreviated form. For this model,
Figure <ref> shows the empirical coverage probabilities of all three bootstrap procedures as a function of $h$ for the pre-asymptotic extremogram. The three rows correspond to the three thresholds with ascending exceedance probabilities. The left column shows the results for (average) block length $r=100$, the right column for $r=20$. For all bootstrap procedures, the actual coverage probabilities are much smaller than the nominal value 0.95 if the threshold is chosen too high. For the estimator based on the largest 5% of the observations and blocks of length $r=100$, the coverage probability of the multiplier block bootstrap is reasonably close to the nominal size while both versions of the stationary bootstrap have a considerably lower coverage probability. In all simulations, the multiplier block bootstrap yields the highest coverage probability, while the stationary bootstrap proposed by Davis et al. (2012) performs worst. Moreover, in most cases the performance is better for larger block sizes. In particular, the stationary bootstrap proposed by Davis et al. is sensitive to too small a block size, as was to be expected from the above discussion.
The main reason for the disappointing performance for high thresholds is that then for very few or even none time instants both $X_t$ and $X_{t+h}$ exceed the threshold. If there are no joint exceedances in the original time series (leading to an estimate 0 for the extremogram) then also the bootstrap estimate equals 0 if one uses the multiplier block bootstrap or the modified stationary bootstrap (and it equals 0 for the original stationary bootstrap with very high probability). Hence the confidence intervals do not cover the true value if this is not exactly equal to 0, which is neither the case for the pre-asymptotic nor the asymptotic extremogram, leading to a high non-coverage probability. Indeed, for $p=0.01$, Figure <ref> shows that if one considers only those simulations when the estimated extremogram does not equal 0, then the empirical coverage probability is rather close to the nominal value.
Empirical coverage probability of confidence intervals (<ref>) for the pre-asymptotic extremogram $\rho_{a_k,(1,\infty),(1,\infty)}(h)$ as a function of $h$, constructed using multiplier block bootstrap (blue $*$), stationary bootstrap suggested by Davis et al. (red $+$) and the modification thereof (black $\circ$) with (average) block length $r=100$ (left) and $r=20$ (right) for the $t$-GARCH model (i), based only on those simulations in which for some $t$ both $X_t$ and $X_{t+h}$ exceed the threshold $a_k$.
To overcome this weakness, we suggest to estimate the error distribution using a bootstrap based on a lower threshold if one wants to construct confidence intervals for the pre-asymptotic extremogram for a high threshold (or even the extremogram). Denote by $\hat\rho_{n,p}$ the empirical extremogram based on the exceedances over the threshold with exceedance probability $p$, and by $\hat\rho_{n,p}^*$ some bootstrap version thereof. Then, according to Theorem <ref>, conditional on the data, for $0<p_1<p_2$, the bootstrap error $\hat\rho_{n,p_1}^*-\hat\rho_{n,p_1}$ has approximately the same distribution as $(p_2/p_1)^{1/2}\big(\hat\rho_{n,p_2}^*-\hat\rho_{n,p_2})$. So if $u_b$ and $l_b$ denote the empirical bootstrap quantiles as defined above, calculated from the bootstrap for the threshold with the higher exceedance probability $p_2$, then
\begin{equation} \label{eq:confint2}
\Big[\hat\rho_{n,p_1}-(p_2/p_1)^{1/2}\big(u_b-\hat\rho_{n,p_2}), \hat\rho_{n,p_1}-(p_2/p_1)^{1/2}\big(l_b-\hat\rho_{n,p_2})\Big]\cap [0,1]
\end{equation}
is a confidence interval with nominal coverage probability $1-\alpha$.
Empirical coverage probability of confidence intervals (<ref>) for $\rho_{a_k,(1,\infty),(1,\infty)}(h)$ as a function of $h$, constructed using multiplier block bootstrap (blue $*$), stationary bootstrap suggested by Davis et al. (red $+$) and the modification thereof (black $\circ$) with (average) block length $r=100$ (left) and $r=20$ (right) for the $t$-GARCH model (i).
Figure <ref> displays the empirical coverage probabilities of this confidence interval for the pre-asymptotic extremogram, $p_1\in\{0.01,0.025\}$ and $p_2=0.05$, which are now much closer to the nominal size 0.95. (Indeed, for $p_1=0.01$ the new confidence intervals are a bit too conservative.) As for small $p_1$ the pre-asymptotic extremograms are closer to the limit extremograms, for these thresholds one may also be interested in the coverage probability for the latter, which are shown in Figure <ref>. The confidence intervals based on the threshold with exceedance probabilities $p_1=0.01$ are still a bit conservative, while for $p_1=0.025$, when the bias is larger and the confidence intervals more narrow, the actual coverage probabilities are too low.
Empirical coverage probability of confidence intervals (<ref>) (<ref>) for the exremogram $\rho_{(1,\infty),(1,\infty)}(h)$ as a function of $h$, constructed using multiplier block bootstrap (blue $*$), stationary bootstrap suggested by Davis et al. (red $+$) and the modification thereof (black $\circ$) with (average) block length $r=100$ (left) and $r=20$ (right) for the $t$-GARCH model (i).
Empirical coverage probability of confidence intervals (<ref>) (solid lines) and (<ref>) (dashed lines) constructed using multiplier block bootstrap (blue $*$) and modified stationary bootstrap (black $\circ$) with block length $r=100$ for the AR(1) model (ii) (left) and MA(3) model (iii) (right) and different thresholds with exceedance probability $p$; the nominal coverage probability 0.95 is indicated by the horizontal line.
Finally, we briefly discuss the linear time series models. As overall the conclusions are similar, we present just the most important findings for block size $r=100$. Figure <ref> shows the coverage probabilities for the autoregressive model (ii) in the left column and for the moving average (iii) in the right column, both for the confidence intervals (<ref>) (solid lines) and (<ref>) (dashed lines). In order to not overload the plot, the results for the original stationary bootstrap (which again performed worst) are not shown. Again the multiplier block bootstrap gives the highest coverage probabilities, which are nevertheless not satisfactory if one uses the direct bootstrap interval
(<ref>) for a high threshold for the extremogram at lags not close to 0. This is particularly true, if the true value is small (e.g., for large lags in the autoregressive model). In these cases, it helps a lot to borrow strength from the bootstrap for a lower threshold as in (<ref>).
§ PROOFS
Theorem <ref>. We combine ideas from the proofs of Theorem 2.3 of Drees and
Rootzén (2010) and of Theorem 2 by Kosorok (2003). Denote by
$Y_{n,j}^*$, $1\le j\le m_n$, independent copies of $Y_{n,j}$ that
are independent of $(\xi_i)_{i\in\N}$. As in Drees and
Rootzén (2010), we define $ \Delta^*_{n,j}(f) := f(Y_{n,j}^*) - f((Y_{n,j}^*)^{(r_n-l_n)})$, $1\le j\le
m_n$. (Recall that $x^{(l)}:=(x_1,\ldots,x_l)$ for
$x=(x_1,\ldots,x_r)$ with $r\ge l$.)
We first analyze the asymptotic behavior of
$$ \frac 1{\sqrt{n v_n}} \sum_{j=1}^{\floor{m_n/2}} \big( \Delta_{n,2j}^*(f)-
E \Delta_{n,2j}^*(f)\big)
conditionally given $(Y_{n,j}^*)_{1\le j\le m_n}$. Note that $E_\xi\big(\xi_{2j} (\Delta_{n,2j}^*(f)-
E \Delta_{n,2j}^*(f))\big)=E\big( \xi_{2j} (\Delta_{n,2j}^*(f)-
E \Delta_{n,2j}^*(f))\mid (Y^*_{n,j})_{1\le j\le m_n}\big)=0$. Moreover, because of $E\xi_{2j}^2=1$
\begin{eqnarray}
\lefteqn{ \frac 1{n v_n} \sum_{j=1}^{\floor{m_n/2}} E_\xi \Big( \xi_{2j}^2 (\Delta_{n,2j}^*(f)-
E \Delta_{n,2j}^*(f))^2\Ind{|\xi_{2j} (\Delta_{n,2j}^*(f)-
E \Delta_{n,2j}^*(f))|\le \sqrt{nv_n}}\Big)} \nonumber \\
& \le & \frac 1{n v_n} \sum_{j=1}^{\floor{m_n/2}}(\Delta_{n,2j}^*(f)-E \Delta_{n,2j}^*(f))^2.\hspace*{6cm}
\label{eq:ineq1}
\end{eqnarray}
Now by condition (C1)
\begin{eqnarray}
\lefteqn{P\Big\{ \sum_{j=1}^{\floor{m_n/2}} (\Delta_{n,2j}^*(f)-E \Delta_{n,2j}^*(f))^2
\Ind{|\Delta_{n,2j}^*(f)-E \Delta_{n,2j}^*(f)|> \sqrt{nv_n}} \ne 0\Big\}} \nonumber\\
& \le & \floor{m_n/2} P\{ |\Delta_{n}^*(f)-E \Delta_{n}^*(f)| > \sqrt{nv_n}\} \nonumber\\
& \to & 0 \hspace*{10cm}\label{eq:conv1}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{E\Big( \frac 1{n v_n} \sum_{j=1}^{\floor{m_n/2}}(\Delta_{n,2j}^*(f)-E \Delta_{n,2j}^*(f))^2
\Ind{|\Delta_{n,2j}^*(f)-E \Delta_{n,2j}^*(f)|\le \sqrt{nv_n}}\Big) }\nonumber \\
& \le & \frac {m_n}{2n v_n} E\Big((\Delta_{n}^*(f)-E \Delta_{n}^*(f))^2
\Ind{|\Delta_{n}^*(f)-E \Delta_{n}^*(f)|\le \sqrt{nv_n}}\Big) \nonumber \\
& = & o\Big(\frac{r_n v_n m_n}{n v_n} \Big) \nonumber\\
& = & o(1). \label{eq:conv2}
\end{eqnarray}
Combining (<ref>)–(<ref>), we see that the
left-hand side of (<ref>) tends to 0 in probability.
Next check that from (<ref>) and (<ref>) we may conclude
\begin{eqnarray}
\lefteqn{\sum_{j=1}^{\floor{m_n/2}} P_\xi \big\{|\xi_{2j} (\Delta_{n,2j}^*(f)-
E \Delta_{n,2j}^*(f))|> \sqrt{nv_n}\big\} } \nonumber\\
& \le & \sum_{j=1}^{\floor{m_n/2}} \Big( \frac{E(\xi_{2j}^2) (\Delta_{n,2j}^*(f)-
E \Delta_{n,2j}^*(f))^2}{nv_n} \Ind{|\Delta_{n,2j}^*(f)-E \Delta_{n,2j}^*(f)|\le \sqrt{nv_n}} \nonumber\\
& & \hspace*{2cm} + \Ind{|\Delta_{n,2j}^*(f)-E \Delta_{n,2j}^*(f)|> \sqrt{nv_n}}\Big) \nonumber\\
& \to & 0 \label{eq:conv3}
\end{eqnarray}
in probability. Therefore, to each subsequence $n'$ there exists a subsubsequence
$n''$ such that the convergence of the left-hand side of (<ref>) and the convergence
of the left-hand side of
(<ref>) hold almost surely. By Theorem 4.10 of Petrov (1995), on the corresponding set of probability
1, for all $\eta>0$,
\begin{equation} \label{eq:conv4}
P_\xi\Big\{ \frac 1{\sqrt{n'' v_{n''}}} \Big|\sum_{j=1}^{\floor{m_{n''}/2}} \xi_{2j} \big( \Delta_{n'',2j}^*(f)-
E \Delta_{n'',2j}^*(f)\big) \Big| >\eta\Big\} \;\longrightarrow\; 0.
\end{equation}
We can argue the same way to obtain convergence (<ref>)
uniformly for a finite number of cluster functionals $f_1,\ldots, f_l$ and
for the analogous sum over the odd numbered blocks.
By Lemma 3 of Kosorok (2003) and the conditions (C2) and (C3), the subsubsequence $n''$ can be
chosen such that on a set with probability 1
$$ \sup_{g\in BL_1(\R^l)} \Big| E_\xi g\Big(\frac 1{\sqrt{n'' v_{n''}}}
\sum_{j=1}^{m_{n''}} \xi_j \big( f_k(Y_{n'',j}^*)- E f_k(Y_{n'',j}^*)\big)_{1\le k\le
- Eg(Z((f_k)_{1\le k\le l}))\Big| \to 0.
Because of the aforementioned generalizations of (<ref>) it follows that
\begin{eqnarray*}
\lefteqn{ \sup_{g\in BL_1(\R^l)} \Big| E_\xi g\Big(\frac 1{\sqrt{n'' v_{n''}}}
\sum_{j=1}^{m_{n''}} \xi_j \big( f_k((Y_{n'',j}^*)^{(r_{n''}-l_{n''})})-
E f_k((Y_{n'',j}^*)^{(r_{n''}-l_{n''})})\big)_{1\le k\le l}}\\
& & \hspace*{8cm}
- Eg\big(Z((f_k)_{1\le k\le l})\big)\Big| \to 0. \hspace*{1cm}
\end{eqnarray*}
Since, by (B2),
$$ \big\| P^{(Y_{n,j}^{(r_n-\ell_n)})_{1\le j\le
m_n}} - P^{((Y^*_{n,j})^{(r_n-\ell_n)})_{1\le j\le
m_n}} \big\|_{TV} \le m_n \beta_{n,l_n}
\; \longrightarrow\; 0
(see Drees and Rootzén, 2010, proof of Lemma 5.1), the last convergence in turn implies
\begin{equation} \label{eq:conv5}
\sup_{g\in BL_1(\R^l)} \Big| E_\xi g\Big(\frac 1{\sqrt{n'' v_{n''}}}
\sum_{j=1}^{m_{n''}} \xi_j \big( f_k(Y_{n'',j}^{(r_{n''}-l_{n''})})
- E f_k(Y_{n'',j}^{(r_{n''}-l_{n''})})\big)_{1\le k\le l}\Big)
- Eg(Z((f_k)_{1\le k\le l}))\Big| \; \longrightarrow\;0
\end{equation}
in probability. Hence, along a further subsequence of $n''$, the convergence
holds almost surely and w.l.o.g. we may
assume almost sure convergence along $n''$.
By the above arguments, one easily sees that the analog to (<ref>)
also holds for $\Delta_{n,2j}(f)$ instead of $\Delta_{n,2j}^*(f)$. Together with the same
argument for the odd numbered blocks it follows that $n''$ can be chosen such that on a set with probability
1, for all $\eta>0$,
$$ P_\xi\Big\{ \Big\| \Big(\frac 1{\sqrt{n'' v_{n''}}} \sum_{j=1}^{m_{n''}} \xi_j
\big( \Delta_{n'',j}(f_k)-
E \Delta_{n'',j}(f_k)\big) \Big)_{1\le k\le l}\Big\| >\eta\Big\} \;
\longrightarrow\; 0.
Thus from (<ref>) we can conclude that for all subsequences $n'$
there exists a subsubsequence $n''$ such that almost surely
$$ \sup_{g\in BL_1(\R^l)} \Big| E_\xi g\Big(\frac 1{\sqrt{n'' v_{n''}}}
\sum_{j=1}^{m_{n''}} \xi_j \big( f_k(Y_{n'',j})- E f_k(Y_{n'',j})\big)_{1\le k\le l}\Big)
- Eg\big(Z((f_k)_{1\le k\le l})\big)\Big| \; \longrightarrow\; 0,
which is equivalent to the assertion.
Proposition <ref>. The asymptotic tightness of $Z_{n,\xi}$ follows if we can prove
asymptotic tightness of $\big((nv_n)^{-1/2}
\sum_{j=1}^{\floor{m_n/2}} \xi_{2j}(f(Y_{n,2j})-
Ef(Y_{n,2j}))\big)_{f\in\FF}$ and the analogous assertion for the
sum over the odd numbered blocks. Similarly as in the proof of
Theorem 2.8 of Drees and Rootzén (2010), it suffices to prove
tightness of $\big((nv_n)^{-1/2}
\sum_{j=1}^{\tilde m_n} \xi_{j}(f(Y_{n,j}^*)-
Ef(Y_{n,j}^*))\big)_{f\in\FF}$ with $\tilde
m_n\in\{\floor{m_n/2},\ceil{m_n/2}\}$ and $Y_{n,j}^*$ denoting
independent copies of $Y_{n,j}$, because the total variation
distance between the distribution of the processes with dependent
blocks (which are separated in time)
resp. with independent blocks tends to 0.
To this end, we verify that the conditions of van der Vaart and
Wellner (1996), Theorem 2.11.9, are fulfilled for
$Z_{ni}:=(nv_n)^{-1/2}\xi_i (f(Y_{n,i}^*)- Ef(Y_{n,i}^*))$ which
are centered random variables because of the independence of $\xi_i$
and $Y_{n,i}^*$.
The second displayed formula of this theorem is an immediate
consequence of condition (D3), since $E \xi_i^2=1$ implies
$E\big(\xi_i(f(Y_{n,i}^*)- g(Y_{n,i}^*))\big)^2=E\big(f(Y_{n,i}^*)-
g(Y_{n,i}^*)\big)^2$. Likewise, the bracketing number for the
multiplier process considered here is the same as the bracketing
number for the original process so that the bracketing entropy
condition (i.e. the third displayed formula in Theorem 2.11.9)
follows from (D4).
It remains to verify that
\begin{equation} \label{eq:L1Lindeberg}
\frac{\tilde m_n}{\sqrt{nv_n}} E^*\big| \xi F(Y_n) \Ind{|\xi
F(Y_n)|>\eta\sqrt{nv_n}}\big| \;\longrightarrow\; 0,\quad \forall\,
\eta>0.
\end{equation}
If $\xi$ is bounded, then this convergence is obvious from (D2).
Under the conditions of part (ii), one has for all $u_n>0$
\begin{eqnarray*}
\lefteqn{E^* \big| \xi^2 F^2(Y_n) \Ind{|\xi
F(Y_n)|>\eta\sqrt{nv_n}}\big|} \\
& \le & E\big(\xi^2 \Ind{|\xi|>u_n}\big) E^*(F^2(Y_n)) + E\big(\xi^2 \Ind{|\xi|\le
\end{eqnarray*}
By condition (D2') one can find a sequence $u_n\to\infty$ such that
$$ E\Big(F^2(Y_n)\Ind{F(Y_n)>\eta\sqrt{nv_n}/u_n}\Big) =o(r_nv_n). $$
Moreover, also the first term is of smaller order than $r_nv_n$,
because $E\big(\xi^2 \Ind{|\xi|>u_n}\big)\to 0$ and, by assumption,
$E^*(F^2(Y_n))=O(r_nv_n)$. Now, by the Cauchy-Schwarz inequality and
the Chebyshev inequality, the left-hand side of
(<ref>) can be bounded by
\begin{eqnarray*}
\lefteqn{ \frac{\tilde m_n}{\sqrt{nv_n}}\Big( E^*\big| \xi^2 F^2(Y_n)
\Ind{|\xi F(Y_n)|>\eta\sqrt{nv_n}}\big|\cdot E^*\Ind{|\xi
& \le & o\Big( \frac{\tilde
m_n}{\sqrt{nv_n}} (r_nv_n)^{1/2}\Big)\Big(\frac{E^*(\xi^2
F^2(Y_n))}{\eta^2 nv_n}\Big)^{1/2}
\;\longrightarrow\; 0. \hspace*{4cm}
\end{eqnarray*}
Theorem <ref>. By (D3) the family $\FF$ is totally bounded w.r.t. the metric
$\rho$. Hence there exists a sequence of finite $\delta$-nets
$\FF_\delta$ of $\FF$, i.e. finite sets such that to every
$f\in\FF$ there exists $\pi_\delta(f)\in\FF_\delta$ whose
$\rho$-distance to $f$ is less than $\delta$. Because $Z$ has
continuous sample paths w.r.t. $\rho$ and $g\in
BL_1(\ell^\infty(\FF))$ is bounded and Lipschitz-continuous with
Lipschitz-constant 1, we may conclude
\begin{equation} \label{eq:gZdiscret}
\lim_{\delta\downarrow 0} E^* \sup_{g\in BL_1(\ell^\infty(\FF))}
\big| g(Z(\pi_\delta\circ \cdot))-g(Z(\cdot))\big| = 0.
\end{equation}
For fixed $\delta>0$, denote by $l=\sharp \FF_\delta$ the
cardinality of the $\delta$-net. Theorem <ref> gives
\begin{eqnarray}
\lefteqn{ \sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi
g(Z_{n,\xi}(\pi_\delta\circ \cdot))- E g(Z(\pi_\delta\circ\cdot))\big|}
\nonumber\\
& \le & \sup_{h\in BL_1(\R^l)} \big|E_\xi
h\big((Z_{n,\xi}(f))_{f\in\FF_\delta}\big)- E
& \to & 0 \label{eq:Znxidiskretconv}
\end{eqnarray}
in outer probability (cf. van der Vaart and Wellner, 1996, p.182).
Next note that by the definition of $BL_1(\ell^\infty(\FF))$
\sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi
g(Z_{n,\xi}(\pi_\delta\circ \cdot))- E_\xi g(Z_{n,\xi})\big|
\le E_\xi \min\Big(
\sup_{f\in\FF} |Z_{n,\xi}(\pi_\delta(f))-Z_{n,\xi}(f)|, 2\Big).
Since $Z_{n,\xi}$ weakly converges to $Z$, it is asymptotically
equicontinuous, that is, for all $\eps>0$ and all sequences
$\delta_n\downarrow 0$
$$ P^* \Big\{
\sup_{f,g\in\FF, \rho(f,g)<\delta_n}
|Z_{n,\xi}(f)-Z_{n,\xi}(g)|>\eps\Big\} \;\longrightarrow\; 0.
$$ E^* \min\Big(\sup_{f,g\in\FF, \rho(f,g)<\delta_n}
|Z_{n,\xi}(f)-Z_{n,\xi}(g)|,2\Big) \;\longrightarrow\; 0,
and thus by Fubini's theorem (van der Vaart and Wellner, 1996,
Lemma 1.2.6)
$$ E^* \Big(E_\xi \min\Big(\sup_{f\in\FF}
\;\longrightarrow\; 0.
This in turn implies
\begin{equation} \label{eq:Znxidiscret}
\sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi
g(Z_{n,\xi}(\pi_{\delta_n}\circ \cdot))- E_\xi g(Z_{n,\xi})\big| \;\longrightarrow\; 0
\end{equation}
in outer probability for all $\delta_n\downarrow 0$.
By (<ref>), for all $\eps>0$ and all
$\delta_n\downarrow 0$, one has for sufficiently large $n$ that $E^*
\sup_{g\in BL_1(\ell^\infty(\FF))}
\big| g(Z(\pi_{\delta_n}\circ \cdot))-g(Z(\cdot))\big|<\eps/3$. Therefore, in view of
(<ref>) and (<ref>), for all
$\eps,\eta>0$ and sufficiently large $n$
\begin{eqnarray*}
\lefteqn{P^*\Big\{\sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi
g(Z_{n,\xi})- Eg(Z)| >\eps\Big\}}\\
& \le & P^*\Big\{\sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi
g(Z_{n,\xi}(\cdot))-E_\xi g(Z_{n,\xi}(\pi_{\delta_n}\circ
\cdot))\big|>\eps/3\Big\} \\
& & + P^*\Big\{\sup_{g\in BL_1(\ell^\infty(\FF))} \big| E_\xi g(Z_{n,\xi}(\pi_{\delta_n}\circ
\cdot))- E g(Z(\pi_{\delta_n}\circ \cdot))\big| > \eps/3\Big\} \\
& < & \eta,
\end{eqnarray*}
which proves the assertion.
Proposition <ref>. For $f\in\FF$ define $Tf:\R\times E_\cup \to \R$, $Tf(t,y):=t
f(y)$ and $T\FF:=\{Tf\mid f\in\FF\}$. We are going to apply Theorem 2.11.1 of van der Vaart and
Wellner (1996) to the processes
$$ \tilde Z_n(g) := \frac 1{\sqrt{n v_n}} \sum_{j=1}^{\tilde m_n} \big( g(\xi_j,Y_{n,j}^*)-
E g(\xi_j,Y_{n,j}^*)\big), \quad g\in T\FF,
with $\tilde m_n\in\{\floor{m_n/2},\ceil{m_n/2}\}$ and $Y_{n,j}^*$ denoting
independent copies of $Y_{n,j}$. The assertion then follows by
the same arguments as used in the proof of Drees and Rootzén
(2010), Theorem 2.10 (cf. also the proof of Proposition
<ref> of the present paper).
Because $|TF|$ is an envelope function of $T\FF$ and
\begin{eqnarray*}
\lefteqn{\sum_{j=1}^{\tilde m_n} E^*\Big( \big((nv_n)^{-1/2}
TF(\xi_j,Y_{n,j}^*)\big)^2 \Ind{(nv_n)^{-1/2}
& \le & \frac 1{r_nv_n} E^*\big( \xi^2 F^2(Y_n)
\Ind{|\xi F(Y_n)|>\eta\sqrt{nv_n}}\big),
\end{eqnarray*}
the first condition of Theorem 2.11.1 is obviously fulfilled if
$\xi$ is bounded and (D2') holds, while it follows from the
arguments given at the end of the proof of Proposition
<ref> if $E(F^2(Y_n))=O(r_nv_n)$ holds.
The second condition of Theorem 2.11.1 is equivalent to our
condition (D3), because $E(\xi_j^2)=1$ and the independence of
$\xi_j$ and $Y_{n,j}^*$ imply
It remains to verify the metric entropy condition (2.11.2) of van der Vaart and
Wellner (1996), which is equivalent to
$$ \lim_{\delta\downarrow 0} \limsup_{n\to\infty} P^* \Big\{
\int_0^\delta \sqrt{\log N(\eps,T\FF,\tilde d_n)}\, d\eps>\eta\Big\}=0
for all $\eta>0$ where
$$ \tilde d_n(Tf,Tg) := \Big( \frac 1{n v_n} \sum_{j=1}^{\tilde
m_n} \xi_j^2 \big( f(Y^*_{n,j})-g(Y^*_{n,j})\big)^2\Big)^{1/2}.
If $|\xi|\le c$, then
$ \tilde d_n(Tf,Tg)\le c d_n(f,g)$ so that $N(\eps,T\FF,\tilde
d_n)\le N(\eps/c,\FF,d_n)$ and the entropy condition readily
follows from (D6).
If $\xi$ is not necessarily bounded, but the uniform entropy
condition (D6') holds, then one may proceed similarly as in the
proof of Theorem 2.10 of Drees and Rootzén (2010). Let
$$ Q_{n,\xi} := \frac{\sum_{j=1}^{\tilde m_n} \xi_j^2
\eps_{Y_{n,j}^*}}{\sum_{j=1}^{\tilde m_n} \xi_j^2} \in \mathcal{Q}
with $\eps_y$ denoting the Dirac measure with mass 1 at $y$,
and check that
$$ \tilde d_n(Tf,Tg) = \bigg(\frac{\sum_{j=1}^{\tilde m_n}
\xi_j^2}{nv_n}\bigg)^{1/2} d_{Q_{n,\xi}}(f,g).
Hence $N(\eps,T\FF,\tilde d_n)\le
N\big(\eps(nv_n/\sum_{j=1}^{\tilde m_n} \xi_j^2)^{1/2},\FF,
d_{Q_{n,\xi}}\big)$. Moreover, for all $\tau>0$
\begin{eqnarray*}
P\Big\{ \Big(\int F^2\, d Q_{n,\xi}\Big)^{1/2} > \tau \Big(\frac{nv_n}{\sum_{j=1}^{\tilde m_n}
\xi_j^2}\Big)^{1/2}\Big\}
& = & P\Big\{ \sum_{j=1}^{\tilde m_n} \xi_j^2
F^2(Y_{n,j}^*)>\tau^2 nv_n\Big\} \\
& \le & \frac 1{\tau^2 nv_n} E\Big(\sum_{j=1}^{\tilde m_n} \xi_j^2
& \le & \frac{E(F^2(Y_n))}{\tau^2 r_n v_n}.
\end{eqnarray*}
Since $E(F^2(Y_n))=O(r_nv_n)$, this probability can be made
arbitrarily small for all $n$ by choosing $\tau$ sufficiently
Thus, for all $\eta>0$, there exists $\tau>0$ such that with outer
probability of at least $1-\eta$
\begin{eqnarray*}
\int_0^\delta \sqrt{\log N(\eps,T\FF,\tilde d_n)}\, d\eps
& = & \tau \int_0^{\delta/\tau} \sqrt{\log N(\eps\tau,T\FF,\tilde d_n)}\,
& \le & \tau \int_0^{\delta/\tau} \sqrt{\log N\Big(\eps\tau\Big( \frac{nv_n}{\sum_{j=1}^{\tilde m_n}
\xi_j^2}\Big)^{1/2},\FF,d_{Q_{n,\xi}}\Big)}\,
& \le & \tau \int_0^{\delta/\tau} \sup_{Q\in\mathcal{Q}}
\sqrt{\log N\Big(\eps\Big(\int F^2\, dQ\Big)^{1/2},\FF,
d_Q\Big)}\, d\eps\\
& \to & 0
\end{eqnarray*}
as $\delta\downarrow 0$ by (D6').
Hence, under both sets of conditions, the asymptotic
equicontinuity follows from Theorem 2.11.1 of van der Vaart and
Wellner (1996).
Corollary <ref>. Because
$$ Z_{n,\xi}^*(f)-Z_{n,\xi}(f) = \frac 1{\sqrt{nv_n}}
\sum_{j=1}^{m_n} \xi_j(Ef(Y_{n,j})-\overline{f(Y_n)}) = -\frac
1{m_n} \sum_{j=1}^{m_n} \xi_j \cdot Z_n(f),
$$ E_\xi \Big|\frac 1{m_n} \sum_{j=1}^{m_n} \xi_j\Big| \le \Big(E_\xi \Big(\frac 1{m_n} \sum_{j=1}^{m_n}
\xi_j\Big)^2\Big)^{1/2} = \Big(\frac 1{m_n} Var(\xi)\Big)^{1/2}
= \frac 1{\sqrt{m_n}}
and $Z_n\to Z$ weakly in $\ell^\infty(\FF)$, one has
$$ E_\xi \sup_{f\in\FF} |Z_{n,\xi}^*(f)-Z_{n,\xi}(f)| \le E_\xi
\Big|\frac 1{m_n} \sum_{j=1}^{m_n} \xi_j\Big|\cdot
\sup_{f\in\FF} |Z_n(f)| \;\longrightarrow\; 0,
which implies (<ref>).
Hence the weak convergence $Z_{n,\xi}^*\to Z$ follows from the analogous
convergence of $Z_{n,\xi}$.
Finally, by (<ref>), the definition of $BL_1(\ell^\infty(\FF))$
and Theorem <ref>
$$ \big| E_\xi g(Z_{n,\xi}^*)- Eg(Z)\big| \le \big| E_\xi g(Z_{n,\xi}^*)-E_\xi
g(Z_{n,\xi})\big| + \big|E_\xi g(Z_{n,\xi})- Eg(Z)\big|\;\longrightarrow\; 0
in outer probability uniformly for all $g\in
Theorem <ref>. The convergence of $(Z_n^{(h,\tilde h)}(f))_{f\in\FF}$ follows from Corollary 3.6(ii) and Remark 3.7(i) of Drees and Rootzén (2010); see also Drees and Rootzén (2015). To see this, check that
Condition (D3) is fulfilled since for $\delta<1$
\begin{eqnarray*}
\lefteqn{\sup_{\varrho(f_D,f_{\tilde D})<\delta} \frac 1{r_n v_n} E\Big( \sum_{i=1}^{r_n}
1_{D\Delta\tilde D}(X_{n,i}^{(h,\tilde h)})\Big)^2}\\
& \le & \max_{\bar h\in\{0,\ldots,h_0\}}\sup_{\bar\varrho((A,B),(\tilde A,\tilde
B))<\delta} \frac 1{r_n v_n} E\Big( \sum_{i=1}^{r_n}
1_{(A\times B)\Delta (\tilde A\times \tilde
B)}(X_i/a_k,X_{i+\bar h}/a_k)\Big)^2\\
& \le & \sup_{0<t< \delta} u(t).
\end{eqnarray*}
Since, by ($\widetilde{\text{B3}}$), for $i>\max(h,\tilde h)$
\begin{eqnarray*}
P\big(X_{n,i+1}^{(h,\tilde h)}\ne 0\mid X_{n,1}^{(h,\tilde h)}\ne 0\big)
& \le & \sum_{j,l\in\{0,h,\tilde h\}} P(X_{n,i+1+j}\ne 0\mid X_{n,1+l}\ne 0) \\
& \le & \sum_{j,l\in\{0,h,\tilde h\}} s_n(i+j-l),
\end{eqnarray*}
$\big(X_{n,i}^{(h,\tilde h)}\big)_{1\le i\le n}$ satisfies the analog to ($\widetilde{\text{B3}}$) if $s_n(i)$ is replaced with
$$ \tilde s_n(i) := \left\{ \begin{array}{l@{\quad}l}
\sum_{j,l\in\{0,h,\tilde h\}} s_n(i+j-l), & i>\max(h,\tilde h),\\
1, & i\le \max(h,\tilde h).
\end{array}
\right.
(<ref>) ensures that convergence (3.8) of Drees and Rootzén (2010) holds, because
\begin{eqnarray*}
\frac 1{v_n^{(h,\tilde h)}} E\big(1_{A\times B\times\R^d}(X_{n,0}^{(h,\tilde h)}),1_{\tilde A\times\tilde
B\times\R^d}(X_{n,i}^{(h,\tilde h)})\big) & = & \frac{k P\{a_k^{-1}(X_0,X_h,X_i,X_{i+
h})\in A\times B\times \tilde A\times \tilde B\}}{k P\{a_k^{-1}(X_0,X_h,X_{\tilde h})\in \R^{3d}\setminus (-\infty,x_*)^{3d}\}}\\
& \to & \frac{\nu_{(0,h,i,i+h)}(A\times B\times \tilde A\times \tilde
B)}{\nu_{(0,h,\tilde h)}(\R^{3d}\setminus (-\infty,x_*)^{3d})}\\
& =: & d_i \big(f_{A\times B\times\R^d}, f_{\tilde A\times\tilde
\end{eqnarray*}
and likewise
\begin{eqnarray*}
\frac 1{v_n^{(h,\tilde h)}} E\big(1_{A\times B\times\R^d}(X_{n,0}^{(h,\tilde h)}),1_{\tilde A\times\R^d\times\tilde
B}(X_{n,i}^{(h,\tilde h)})\big)
& \to &
\frac{\nu_{(0,h,i,i+\tilde h)}(A\times B\times \tilde A\times \tilde
B)}{\nu_{(0,h,\tilde h)}(\R^{3d}\setminus (-\infty,x_*)^{3d})}\\
& =: & d_i \big(f_{A\times B\times\R^d}, f_{\tilde A\times\R^d\times\tilde
\frac 1{v_n^{(h,\tilde h)}} E\big(1_{A\times \R^d\times B}(X_{n,0}^{(h,\tilde h)}),1_{\tilde A\times\tilde B\times\R^d}
(X_{n,i}^{(h,\tilde h)})\big)
& \to &
\frac{\nu_{(0,\tilde h,i,i+h)}(A\times B\times \tilde A\times \tilde
B)}{\nu_{(0,h,\tilde h)}(\R^{3d}\setminus (-\infty,x_*)^{3d})}\\
& =: & d_i \big(f_{A\times \R^d\times B}, f_{\tilde A\times\tilde B\times\R^d}\big)\\
\frac 1{v_n^{(h,\tilde h)}} E\big(1_{A\times \R^d\times B}(X_{n,0}^{(h,\tilde h)}),1_{\tilde A\times\R^d\times\tilde B}
(X_{n,i}^{(h,\tilde h)})\big)
& \to &
\frac{\nu_{(0,\tilde h,i,i+\tilde h)}(A\times B\times \tilde A\times \tilde
B)}{\nu_{(0,h,\tilde h)}(\R^{3d}\setminus (-\infty,x_*)^{3d})}\\
& =: & d_i \big(f_{A\times \R^d\times B}, f_{\tilde A\times\R^d\times\tilde B}\big).
\end{eqnarray*}
Hence, by Drees and Rootzén (2015), condition (C3) holds and $Z_n^{(h,\tilde h)}$ converges to a Gaussian process
with the covariance function specified in formula (3.10) of Drees and Rootzén (2010) in terms of the functions $d_i$.
\begin{eqnarray*}
\frac{v_n^{(h,\tilde h)}}{v_n} & = & \frac{k P\{a_k^{-1}(X_0,X_h,X_{\tilde h})\in \R^{3d}\setminus
(-\infty,x_*)^{3d}\}}{k P\{a_k^{-1}X_0\in \R^{d}\setminus
& \to & \frac{\nu_{(0,h,\tilde h)}(\R^{3d}\setminus
(-\infty,x_*)^{3d})}{\nu_0 (\R^d\setminus (-\infty,x_*)^d)},
\end{eqnarray*}
the convergence of $(\tilde Z_n(\bar h,A,B))_{\bar h\in\{h,\tilde
h),(A,B)\in\CC}$ to a Gaussian process with covariance function $\tilde c$
follows from the approximation (3.6) of Drees and
Rootzén (2010). Now the assertion is obvious.
condition (D6) in Example <ref>.
For fixed $r\in\N$ define
functions $f_D^{(r)}: \R^{2rd}\to\R$, $f_D^{(r)}(y_1,\ldots,y_r)
:=\sum_{i=1}^r 1_D(y_i)$ with $D\in\{A\times B \mid
(A,B)\in\CC\}=\{(x,\infty)\mid x=(x_1,\ldots,x_d)\in (x_*,\infty)^{2d}\}$. The
subgraph of $f^{(r)}_{(x,\infty)}$ equals
\begin{eqnarray*}
\lefteqn{\big\{ (t,(y_1,\ldots,y_r))\in\R^{2rd+1} \mid
t<f^{(r)}_{(x,\infty)}(y_1,\ldots, y_r)\big\}}\\
& = & \bigcup_{j=0}^r
(-\infty,j)\times \{y=(y_1,\ldots, y_r)\mid
f^{(r)}_{(x,\infty)}(y)=j\} \\
& =: & M_x.
\end{eqnarray*}
Consider some fixed set $S=\{(t^{(l)},(y_1^{(l)},\ldots,
y_r^{(l)})) \mid 1\le l\le m\}$ of $m$ points in $\R^{2rd+1}$. If
for $x,\tilde x\in \R^{2d}$ the symmetric difference
$(x,\infty)\Delta (\tilde x,\infty)$ does not contain any of the
$y_i^{(l)}=(y_{i,1}^{(l)},\ldots,y_{i,2d}^{(l)})$, $1\le i\le r$, $1\le l\le m$, then the intersections
$S\cap M_x$ and $S\cap M_{\tilde x}$ are identical. Since the
hyperplanes $\{x\in\R^{2d}\mid x_j=y_{i,j}^{(l)}\}$, $1\le j\le
2d$, $1\le i\le r$, $1\le l\le m$, divide $\R^{2d}$ into at most
$(mr+1)^{2d}$ hypercubes and for $x,\tilde x$ belonging
to the same hypercube $(x,\infty)\Delta (\tilde x,\infty)$ does not contain any of the $y_i^{(l)}$,
the family $\CC$ can pick out at most
$(mr+1)^{2d}$ different subsets of $S$. Hence it cannot shatter
$S$ if $(mr+1)^{2d}<2^m$, which is fulfilled if $m\ge 3d\log r$
$r$ is sufficiently large.
To sum up, so far we have shown that, for some $r_0\in\N$ and all $r\ge r_0$, the VC-index of $\FF^{(r)}
:=\{f_{A\times B}^{(r)} \mid (A,B)\in\CC\}$ is less than $3d\log r$.
By Theorem 2.6.7 of van der Vaart and Wellner (1996), we conclude
\begin{equation} \label{eq:entropbd1}
N\Big(\eps \big(\int (F^{(r)})^2\, dQ\big)^{1/2}, \FF^{(r)},
L_2(Q)\Big) \le K_1 r^{K_2} \eps^{-K_3\log r}
\end{equation}
for all $\eps\in (0,1)$, all probability measures $Q$ on
$\R^{2rd}$, and suitable universal constants $K_1$, $K_2$ and
$K_3$ with $F^{(r)}(y):=\sum_{i=1}^r 1_{\R^{2d}\setminus(-\infty
x_*]^{2d}}(y_i)$ denoting the envelope function of $\FF^{(r)}$.
Next let $H(y) := \sum_{i=1}^r 1_{\{y_i\ne 0\}}$ for
$y=(y_1,\ldots,y_r)\in\R^{2rd}$, $X_{n,i}^{(h)}:= (\tilde
X_{n,i},\tilde X_{n,i+h})$ for $1\le i\le n$ and define
independent copies $Y_{n,j}^{(h)*}$ of $Y_{n,j}^{(h)}:=
(X_{n,i}^{(h)})_{(j-1)r_n<i\le jr_n}$, $1\le j\le m_n$. Consider
the non-zero values of the $N_r := \sum_{j=1}^{m_n}1_{\{H(Y_{n,j}^{(h)*})\le r\}}$ of
these blocks with at most $r$ non-zero $X_{n,i}^{(h)}$'s; if
necessary, these are completed by zeros to obtain vectors $\bar
Y_j := \big(Y_{n,j,i_1}^{(h)*}, \ldots, Y_{n,j,i_r}^{(h)*}\big)$,
i.e. $Y_{n,j,i_l}^{(h)*}\ne 0$ for $1\le l\le H(Y_{n,j}^{(h)*})\le
r$ and $Y_{n,j,i_l}^{(h)*}= 0$ for $H(Y_{n,j}^{(h)*})<l\le
r$. Let
$$Q_{n,r} := \frac 1{N_r} \sum_{j=1}^{m_n} \eps_{\bar Y_j}
1_{\{H(Y_{n,j}^{(h)*})\le r\}},
and consider the squared random $L_2$-distance
\begin{eqnarray*}
d_n^2(f_{(x,\infty)},f_{(\tilde x,\infty)})
& = & \frac 1{n v_n^{(h,\tilde h)}} \sum_{j=1}^{m_n}
\big(f_{(x,\infty)}(Y_{n,j}^{(h)*}) - f_{(\tilde
x,\infty)}(Y_{n,j}^{(h)*})\big)^2 \\
& \le & \frac{N_r}{n v_n^{(h,\tilde h)}} \int (f_{(x,\infty)}-f_{(\tilde
x,\infty)})^2 \, dQ_{n,r} + \frac1{n v_n^{(h,\tilde h)}}
\sum_{j=1}^{m_n} H^2(Y_{n,j}^{(h)*})1_{\{H(Y_{n,j}^{(h)*})>r\}}
\end{eqnarray*}
for all $r\in\N$. In particular,
$$ d_n^2(f_{(x,\infty)},f_{(\tilde x,\infty)})
\le \frac{N_{R_{n,\eps}}}{n v_n^{(h,\tilde h)}} \int (f_{(x,\infty)}-f_{(\tilde
x,\infty)})^2 \, dQ_{n,{R_{n,\eps}}} + \frac{\eps^2}2
$$ R_{n,\eps} := \max\bigg(\min \Big\{ r\in\N \;\Big|\; \frac1{n v_n^{(h,\tilde h)}}
\sum_{j=1}^{m_n}
so that a ball with radius $\tilde\eps:=\big(nv_n^{(h,\tilde h)}/(2N_{R_{n,\eps}})\big)^{1/2} \eps$ w.r.t. $L_2(Q_{n,{R_{n,\eps}}})$ is contained in a ball with radius $\eps$ w.r.t. $d_n$.
Note that
$$ \int (F^{(r)})^2\, dQ_{R_{n,\eps}} \le \frac 1{N_{R_{n,\eps}}}
\sum_{j=1}^{m_n} H^2(Y_{n,j}^{(h)*}) 1_{\{H(Y_{n,j}^{(h)*})\le
Hence, in view of (<ref>), $\bar\FF$ (defined in (<ref>)) can be covered by
\begin{eqnarray*}
\lefteqn{N\Big(\tilde\eps ,\FF^{(R_{n,\eps})},
& \le & K_1 R_{n,\eps}^{K_2} \bigg( \eps \Big(\frac{n v_n^{(h,\tilde
h)}}{2\sum_{j=1}^{m_n} H^2(Y_{n,j}^{(h)*}) 1_{\{H(Y_{n,j}^{(h)*})\le
R_{n,\eps}\}}}\Big)^{1/2}\bigg)^{-K_3 \log R_{n,\eps}}\\
& \le & K_1 R_{n,\eps}^{K_2} \bigg( \frac{\eps}{R_{n,\eps}} \Big(\frac{2\sum_{j=1}^{m_n} 1_{\{Y_{n,j}^{(h)*}\ne 0\}}} {n v_n^{(h,\tilde h)}}\Big)^{-1/2}\bigg)^{-K_3 \log R_{n,\eps}}
\end{eqnarray*}
balls with radius $\eps$ w.r.t. $d_n$.
Next observe that (<ref>) implies $E\big(H^{2+\delta}(Y_{n,1}^{(h)*})\big)=O(r_n v_n)$:
\begin{eqnarray}
E\Big( \sum_{i=1}^{r_n} 1_{\{X_{n,i}^{(h)}\ne
& \le & E\Big( 2\max_{l\in\{0,h\}}
\sum_{i=1}^{r_n} 1_{\{\tilde X_{n,i+l}\ne
0\}}\Big)^{2+\delta} \nonumber\\
& \le & 2^{2+\delta} \sum_{l\in\{0,h\}} E\Big(
\sum_{i=1}^{r_n} 1_{\{\tilde X_{n,i+l}\ne
0\}}\Big)^{2+\delta} \nonumber\\
& = & 2^{3+\delta} E\Big(
\sum_{i=1}^{r_n} 1_{\{X_i\not\in
(-\infty,a_kx_*)^d\}}\Big)^{2+\delta}\nonumber \\
& = & O(r_nv_n), \label{eq:extremomomcond2}
\end{eqnarray}
where in the last but one line we have used the stationarity of
the time series. Hence,
(-\infty,x_*)^{2d}\} =: r_n v_n^{(h)}=O(r_nv_n)$ implies that $r_n v_n^{(h)} =O\big( P\{Y_{n,1}^{(h)*}\ne 0\}\big)$, because else
0)=\infty$ and thus
$$ \limsup_{n\to\infty}
\frac{E(H^2(Y_{n,1}^{(h)*}))}{E(H(Y_{n,1}^{(h)*}))} =\limsup_{n\to\infty}
\frac{E(H^2(Y_{n,1}^{(h)*})\mid Y_{n,1}^{(h)*}\ne
0)}{E(H(Y_{n,1}^{(h)*})\mid Y_{n,1}^{(h)*}\ne
0)} \ge \limsup_{n\to\infty} E(H(Y_{n,1}^{(h)*})\mid Y_{n,1}^{(h)*}\ne
0) = \infty,
in contradiction to (<ref>). By Chebyshev's inequality,
$$ P\Big\{ \sum_{j=1}^{m_n} 1_{\{Y_{n,j}^{(h)*}\ne 0\}}> 2 m_nP\{Y_{n,1}^{(h)*}\ne 0\}\Big\} \le \frac 1{m_n P\{Y_{n,1}^{(h)*}\ne 0\}} \to 0.
Since $v_n$, $v_n^{(h)}$ and $v_n^{(h,\tilde h)}$ are all of the same order (by the regular variation of $(X_0,X_h,X_{\tilde h})$), we conclude that with
probability tending to 1
$$ N(\eps,\bar\FF,d_n) \le K_1 R_{n,\eps}^{K_2} (K_4 \eps/R_{n,\eps})^{-K_3
\log R_{n,\eps}}.
Finally, (<ref>) implies that to each $\eta>0$ there
exist constants $M,\tau>0$ such that
\begin{eqnarray*}
\lefteqn{P\Big\{ \frac1{n v_n^{(h,\tilde h)}} \sum_{j=1}^{m_n}
>\frac{\eps^2}2 \text{ for some } 0<\eps\le 1\Big\}}\\
& \le & \sum_{l=0}^\infty P\Big\{ \frac1{n v_n^{(h,\tilde h)}} \sum_{j=1}^{m_n}
>\frac{2^{-2(l+1)}}2\Big\} \\
& \le & \sum_{l=0}^\infty 2^{2l+3} E\Big(\frac1{n v_n^{(h,\tilde h)}} \sum_{j=1}^{m_n} H^2(Y_{n,j}^{(h)*})
1_{\{H(Y_{n,j}^{(h)*})>M2^{l(2+\tau)/\delta}\}}\Big) \\
& \le & \sum_{l=0}^\infty \frac{2^{2l+3}}{n v_n^{(h,\tilde h)}} \cdot
\frac{m_n E(H^{2+\delta}(Y_{n,1}^{(h)*}))}{(M
2^{l(2+\tau)/\delta})^\delta} \\
& \le & \frac{K_6}{M^\delta} \sum_{l=0}^\infty 2^{-2 l\tau}\\
& < & \eta
\end{eqnarray*}
with $K_6$ denoting some universal constant.
Hence $R_{n,\eps} \le M\eps^{-(2+\tau)/\delta}$ with probability
greater than $1-\eta$, so that
$$ \int_0^\xi \big(\log N(\eps,\bar\FF,d_n)\big)^{1/2}\, d\eps \le
\int_0^\xi \big(K_7+K_8|\log\eps|+K_9(\log\eps)^2\big)^{1/2}\,
tends to 0 as $\xi$ tends to 0, which proves condition (D6).
under the additional assumptions of Theorem <ref>,
the process $\tilde Z_n$ converges.
Corollary <ref>. Check that
\begin{eqnarray*}
\hat\rho_{n,A,B}(h) & = & \frac{nP\{X_0\in a_kA,X_h\in a_kB\}+\sqrt{nv_n}\tilde Z_n(h,A,B)}{nP\{X_0\in a_kA\}
+\sqrt{nv_n}\tilde Z_n(h,A,\R^d)}\\
& = & \frac{\rho_{a_k,A,B}(h)+\sqrt{v_n/n}\tilde Z_n(h,A,B)/P\{X_0\in
a_kA\}}{1+\sqrt{v_n/n}\tilde Z_n(h,A,\R^d)/P\{X_0\in a_kA\}}\\
& = & \rho_{a_k,A,B}(h)+\frac{\sqrt{v_n/n}}{P\{X_0\in
a_kA\}}\cdot\frac{\tilde Z_n(h,A,B)-\rho_{a_k,A,B}(h)\tilde
Z_n(h,A,\R^d)}{1+\sqrt{v_n/n}\tilde Z_n(h,A,\R^d)/P\{X_0\in
\end{eqnarray*}
Since by the regular variation of $X_0$
$$ \frac{P\{X_0\in a_kA\}}{v_n}= \frac{P\{X_0\in
a_kA\}}{P\{X_0\not\in (-\infty,x_*)^d\}} \to
\frac{\nu_0(A)}{\nu_0\big(\R^d\setminus(-\infty,x_*)^d\big)},
the first assertion is an immediate consequence of Theorem
<ref> and the second follows from $v_n\sim
\nu_0\big(\R^d\setminus(-\infty,x_*)^d\big)/k$.
Corollary <ref>. The first assertion follows from
$ \hat{\hat\rho}_{n,A,B}(h)-\rho_{\hat a_k,A,B}(h) = $ $\hat\rho_{n,(\hat a_k/a_k)A,(\hat a_k/a_k)B}(h)-\rho_{a_k,(\hat a_k/a_k)A,(\hat
a_k/a_k)B}(h)$, the uniform convergence in
(<ref>) and the continuity of $\tilde
Z(h,\cdot,\cdot)$ w.r.t. $\bar\varrho$.
Under condition (<ref>) we have by the extended
regular variation of $\Phi_h$ and the consistency of $\hat a_k$
\begin{eqnarray*}
\rho_{\hat a_k,A,B}(h) -\rho_{a_k,A,B}(h) & = & (\Phi_h(\hat
a_k)-\Phi_h(a_k)) \Psi_h(A,B)+ o(|\Phi_h(\hat a_k)|+|\Phi(a_k)|)
\\
& \le & |\Phi_h(a_k)| \Big|\frac{\Phi_h(\hat
a_k)}{\Phi_h(a_k)}-1\Big| |\Psi_h(A,B)|+ o(|\Phi_h(a_k)|)\\
& = & o_P((k/n)^{1/2}),
\end{eqnarray*}
which proves the second assertion.
Theorem <ref>. For $ h\in\{0,\ldots, h_0\}$ and $ (A,B)\in\CC$, let
\begin{eqnarray*}
\tilde Z_{n,\xi}(h,A,B) &:= & \frac{\sqrt{nv_n^{(h,\tilde h)}}}{\sqrt{nv_n}}
Z_{n,\xi}^{(h,\tilde h)}(f_{A\times B\times
\R^d})\\
& = & \frac 1{\sqrt{nv_n}} \sum_{j=1}^{m_n} (1+\xi_j)
\sum_{i=1}^{r_n} \Big(1_{A\times B} \big( a_k^{-1}
& & \hspace*{3cm}{ }-P\big\{a_k^{-1}
(X_{(j-1)r_n+i},X_{(j-1)r_n+i+h})\in A\times B\big\}\Big)
\end{eqnarray*}
with $Z_{n,\xi}^{(h,\tilde h)}$ denoting the multiplier process
pertaining to $Z_n^{(h,\tilde h)}$ (cf. (<ref>)).
By Proposition <ref>, Theorem <ref>
and the proof of Theorem <ref> (in particular, the
convergence of $v_n^{(h,\tilde h)}/v_n$)
\begin{equation} \label{eq:extremoprocbootstrap}
\sup_{g\in BL_1 (\ell^\infty(\{0,\ldots,h_0\}\times
\CC) )} \big| E_\xi g(\tilde Z_{n,\xi}) -Eg(\tilde Z)\big|
\;\longrightarrow\; 0
\end{equation}
in outer probability.
$$ g_j(h,A,B) = f_{A\times B\times \R^d}(Y_{n,j}^{(h,\tilde h)} ) =\sum_{i=1}^{r_n} 1_{A\times B} \big( a_k^{-1}
Recall from the proof of Theorem <ref> that
$$ \hat\rho_{n,A,B}(h) = \frac{\sum_{j=1}^{m_n}
g_j(h,A,B)}{\sum_{j=1}^{m_n} g_j(h,A,\R^d)} +
(cf. also Corollary 3.6 of Drees and Rootzén, 2010).
\begin{eqnarray*}
\lefteqn{R_{n,\xi}(h,A,B)}\\
& = & \sqrt{nv_n} \bigg(\frac{\sum_{j=1}^{m_n} (1+\xi_j)
g_j(h,A,B)}{\sum_{j=1}^{m_n} (1+\xi_j) g_j(h,A,\R^d)} - \frac{\sum_{j=1}^{m_n}
g_j(h,A,B)}{\sum_{j=1}^{m_n} g_j(h,A,\R^d)} +
& = & \sqrt{nv_n}\frac{ \sum_{j=1}^{m_n} \xi_j g_j(h,A,B)- \sum_{j=1}^{m_n} \xi_j g_j(h,A,\R^d)\cdot
\sum_{j=1}^{m_n}g_j(h,A,B)/ \sum_{j=1}^{m_n} g_j(h,A,\R^d)}{\sum_{j=1}^{m_n} (1+\xi_j) g_j(h,A,\R^d)}+
\end{eqnarray*}
Note that
$$ \sum_{j=1}^{m_n} \xi_j g_j(h,A,B) = \sqrt{nv_n} \tilde
Z_{n,\xi}(h,A,B)+ \sum_{j=1}^{m_n} \xi_j r_n P\{(X_0,X_h)/a_k\in
A\times B\},
where according to the central limit theorem and the regular variation of $(X_0,X_h)$
the second term is of the order $O_P\big(m_n^{-1/2}
\begin{eqnarray*}
R_{n,\xi}(h,A,B) & = & nv_n \frac{\tilde
Z_{n,\xi}(h,A,\R^d)+o_P(1)}{m_nr_n P\{X_0/a_k\in A\} +
O_P(\sqrt{nv_n})} +o_P(1)\\
& = & \frac{\nu_0\big(\R^d\setminus
(-\infty,x_*)^d\big)}{\nu_0(A)} \Big(\tilde
Z_{n,\xi}(h,A,\R^d)\Big) + o_P(1)
\end{eqnarray*}
uniformly for $h\in\{0,\ldots, h_0\}, (A,B)\in\CC$. In the
last step we have used that by the regular variation of $X_0$ and
the definition of $v_n$
$$ \frac{P\{X_0/a_k\in A\}}{v_n} \,\to\, \frac{\nu_0(A)}{\nu_0\big(\R^d\setminus
where, by assumption, $\nu_0(A)$ is bounded away from 0. Now we
can conclude (<ref>) from
(<ref>) and (<ref>).
Finally, notice that $\big| g(\hat R_{n,\xi})-g(R_{n,\xi})\big| \le
\sup_{h\in\{0,\ldots,h_0\}, (A,B)\in\CC} \big| R_{n,\xi}(h,(\hat a_k/a_k)A,(\hat
a_k/a_k)B)-$ $R_{n,\xi}(h,A,B)\big|\to 0$ in outer probability for all $g\in BL_1\big(\ell^\infty(\{0,\ldots,h_0\}\times
\CC)\big)$, because $\hat
a_k/a_k\to 1$ and $R_{n,\xi}(h,\cdot,\cdot)$ is asymptotically
equicontinuous w.r.t. $\bar\varrho$. Therefore,
(<ref>) is an immediate consequence of
§ APPENDIX
The following conditions were used by Drees and Rootzén (2010, 2015). For
the ease of reference, we use the same numbering as in these papers.
(B1) [t]14.9cmThe rows $(X_{n,i})_{1\le i\le
n}$ are stationary, $ \ell_n = o(r_n), \; \ell_n\to\infty,$ $r_n =
o(n)$, $r_nv_n \to 0,$ $nv_n \to \infty$
(B2) $\beta_{n,l_n} n/r_n \to 0. $
($\widetilde{\text{B3}}$) [t]14.8cmFor all $n\in\N$ and all $1\le i\le r_n$ there exists $s_n(i)\ge P(X_{n,i+1}\ne 0\mid X_{n,1}\ne 0)$ such that $s_\infty(i):=\lim_{n\to\infty} s_n(i)$ exists and $\lim_{n\to\infty} \sum_{i=1}^{r_n} s_n(i) =\sum_{i=1}^\infty s_\infty(i)<\infty$.
Recall that for $y=(y_1,\ldots, y_r)$ and $l<r$ we define
(C1) [t]14.5cmFor $\Delta_n(f) :=
f(Y_n) - f(Y_n^{(r_n - \ell_n)})$
\begin{eqnarray}
E\Big( (\Delta_n(f)- E\Delta_n(f))^2
\Ind{|\Delta_n(f)- E\Delta_n(f)|\le
\sqrt{nv_n}}\Big) & = & o(r_n v_n) \label{necsuffbbsbcond1} \nonumber\\
P \big\{|\Delta_n(f)- E\Delta_n(f)|>
\sqrt{nv_n}\big\} & = & o(r_n/n) \label{necsuffbbsbcond2}
\nonumber
\end{eqnarray}
for all $f\in\FF$.
(C2) [t]17cm
$E\Big( (f(Y_n) - E f(Y_n))^2\Ind{|f(Y_n) - E
f(Y_n)|>\eps\sqrt{nv_n}}\Big) = o(r_n v_n),$ $\forall\,
\eps>0, f\in\FF.$
(C3) $\displaystyle \frac 1{r_nv_n} Cov\big( f(Y_n),
g(Y_n)\big) \to c(f,g), \quad \forall\, f,g\in\FF.$
(D1) [t]14.5cmThe index set $\FF$ consists of
cluster functionals $f$ such that $E(f^2(Y_n))$ is finite for all
$n\geq 1$ and such that the envelope function
$$ F(x) := \sup_{f\in\FF} |f(x)| $$
is finite for all $x\in E_\cup$.
$$E^* \Big( F(Y_{n}) \Ind{F(Y_{n})>\eps\sqrt{nv_n}}\Big) =
o\big(r_n\sqrt{v_n/n}\big), \quad \forall\, \eps>0.$$
\begin{equation*}
E^* \Big( F^2(Y_{n}) \Ind{F(Y_{n})>\eps\sqrt{nv_n}}\Big)
= o(r_nv_n), \quad\forall\, \eps>0.
\end{equation*}
(D3) [t]14.9cmThere exists a semi-metric $\rho$ on
$\FF$ such that $\FF$ is totally bounded (i.e., for all $\eps>0$ the
set $\FF$ can be covered by finitely many balls with radius $\eps$
w.r.t. $\rho$) such that
$$\lim_{\delta\downarrow 0}
\limsup_{n\to\infty}
\sup_{f,g\in\FF, \; \rho(f,g)<\delta} \frac 1{r_nv_n}
E(f(Y_{n})-g(Y_{n}))^2 = 0.$$
Finally, we consider different entropy conditions, which measure the
complexity of the family $\FF$. The bracketing number
$N_{[\cdot]}(\eps,\FF,L_2^n)$ is defined as the smallest number
$N_\eps$ such that for each $n\in\N$ there exists a partition
$(\FF_{n,k}^\eps)_{1\le k\le N_\eps}$ of $\FF$ such that
\begin{equation}
E^* \sup_{f,g\in\FF_{n,k}^\eps} \big(f(Y_{n})-g(Y_{n})\big)^2
\le \eps^2 r_n v_n, \quad \forall\, 1\le k\le N_\eps.
\end{equation}
For a given semi-metric $d$ on $\FF$, the (metric) covering number
$N(\eps,\FF,d)$ is the minimum number of balls with radius $\eps$
w.r.t. $d$ needed to cover $\FF$. The condition (D6) bounds the rate of increase of
$N(\eps,\FF,d_n)$ as $\eps$ tends to 0 for the random semi-metric
$$ d_n(f,g) := \Big( \frac 1{nv_n} \sum_{j=1}^{m_n} \big(
that is the $L_2$-semi-metric w.r.t. to empirical measure
$(nv_n)^{-1}\sum_{j=1}^{m_n} \eps_{Y_{n,j}^*}$,
where $Y_{n,j}^*$, $1\le j\le m_n$, are i.i.d. copies of $Y_{n,1}$.
In (D6') we instead use the supremum of all covering numbers $N(\eps,\FF,d_Q)$
$d_Q(f,g):= \big(\int (f-g)^2\, dQ\big)^{1/2}$ and $Q$
ranges over the set of discrete probability measures
$\mathcal{Q}$ on $(E_\cup,\mathbb{E}_\cup)$.
(D4) [t]15cm
$$\lim_{\delta\downarrow 0}
\limsup_{n\to\infty} \int_0^\delta
\sqrt{\log N_{[\cdot]}(\eps,\FF,L_2^n)}\, d\eps = 0.$$
(D5) [t]14.9cmFor all $\delta>0$, $n\in\N$, $(e_i)_{1\le i\le
\floor{m_n/2}} \in \{-1,0,1\}^{\floor{m_n/2}}$ and $k\in\{1,2\}$ the
map $\sup_{f,g\in\FF, \rho(f,g)<\delta}$
$\sum_{j=1}^{\floor{m_n/2}} e_j\big(
is measurable.
\begin{equation*}
\lim_{\delta\downarrow 0} \limsup_{n\to\infty}P^*\Big\{ \int_0^\delta
\sqrt{ \log N(\eps,\FF,d_n)}\, d\eps > \tau \Big\} = 0, \quad
\forall \tau>0.
\end{equation*}
(D6') [t]14.9cmThe envelope function $F$ is
measurable with $E( F^2(Y_{n}))=O(r_nv_n)$ and
\begin{equation*}
\int_0^1 \sup_{Q\in\mathcal{Q}} \sqrt{ \log
N(\eps{\textstyle(\int F^2dQ)^{1/2}},\FF,d_Q)}\, d\eps < \infty.
\end{equation*}
Acknowledgement: I thank Anja Janßen for helpful
discussions about regular variation on general cones and for providing R-code to calculate the extremogram for $t$-GARCH time series. The financial
support by the German Research foundation DFG via the grant JA
2160/1 is gratefully acknowledged.
1.2ex plus0.2ex minus0.2ex
Basrak, B., and Segers, J. (2008). Regularly varying multivariate time series. Université Catholique de Louvain,
Institut de Statistique discussion paper 0717.
Bingham, N.H., Goldie, C.M., and Teugels, J.L. (1987). Regular
Variation. Cambridge University Press.
Das, B., Mitra, A., and Resnick S. (2013). Living on the
multi-dimensional edge: seeking hidden risks using regular
variation. Adv. Appl. Probab.
45, 139–163.
Davis, R., and Mikosch, T. (2009). The extremogram: A correlogram for
extreme events. Bernoulli 15, 977-–1009.
Davis, R., Mikosch, T., and Cribben, I. (2012). Towards estimating extremal serial dependence via the
bootstrapped extremogram. J. Econometr. 170, 142–152.
Drees, H. (2003). Extreme quantile estimation for dependent data with applications to
finance. Bernoulli 9, 617–657.
Drees, H. (2011). Bias correction for estimators of the extremal index. Preprint, arXiv:1107.0935v1
Drees, H., and Rootzén, H. (2010). Limit Theorems for Empirical Processes of Cluster Functionals.
Ann. Statist. 38, 2145–2186.
Drees, H., and Rootzén, H. (2015). Correction note to “Limit Theorems for Empirical Processes of Cluster Functionals”. Preprint, arXiv:1510.09090v1.
Drees, H., Segers, J., and Warchoł, M. (2015). Statistics for Tail Processes of Markov Chains. Extremes 18, 369–402.
Ehlert, A., Fiebig, U.-R., Janßen, A., and Schlather, M. (2015). Joint extremal behavior of hidden and observable time series with applications to GARCH processes. Extremes 18, 109–140.
Kosorok, M.R. (2003). Bootstraps of sums of independent but not identically distributed stochastic processes.
J. Multiv. Analysis 84, 299–318.
Meinguet, T., and Segers, J. (2010). Regularly varying time series in Banach spaces. Preprint, arXiv:1001.3262v1.
Petrov, V.V. (1995). Limit Theorems of Probability Theory. Oxford Science Publication.
Resnick, S.I. (2007). Heavy-tail Phenomena. Springer.
van der Vaart, A.W., and Wellner, J.A. (1996). Weak Convergence
and Empirical Processes. Springer.
|
1511.00575
|
In order to reduce the energy cost of data centers, recent studies suggest distributing computation workload among multiple geographically dispersed data centers, by exploiting the electricity price difference. However, the impact of data center load redistribution on the power grid is not well understood yet. This paper takes the first step towards tackling this important issue, by studying how the power grid can take advantage of the data centers' load distribution proactively for the purpose of power load balancing. We model the interactions between power grid and data centers as a two-stage problem, where the utility company chooses proper pricing mechanisms to balance the electric power load in the first stage, and the data centers seek to minimize their total energy cost by responding to the prices in the second stage. We show that the two-stage problem is a bilevel quadratic program, which is NP-hard and cannot be solved using standard convex optimization techniques. We introduce benchmark problems to derive upper and lower bounds for the solution of the two-stage problem. We further propose a branch and bound algorithm to attain the globally optimal solution, and propose a heuristic algorithm with low computational complexity to obtain an alternative close-to-optimal solution. We also study the impact of background load prediction error using the
theoretical framework of robust optimization. The simulation results demonstrate that our proposed scheme can not only improve the power grid reliability but also reduce the energy cost of data centers.
Smart grid, data center, demand response, dynamic electricity pricing, load balancing, proactive design.
§ NOMENCLATURE
§.§ Acronyms
PS1 Stage-1 problem
PS2 Stage-2 problem
PI Integrated problem
RS1 Stage-1 of the restricted problem
RS2 Stage-2 of the restricted problem
PE1 Equivalent problem of the Stage-1 problem
PE2 Equivalent problem of the Stage-2 problem
PR1 Relaxed Stage-1 problem
WCP Worst-case performance optimization problem
§.§ Sets
$\mathcal{T}$ Set of time slots
$\mathcal{N}$ Set of data centers
§.§ Indices
$t$ Index of time slots
$i$ Index of data centers
§.§ Parameters
$T$ Number of time slots
$N$ Number of data centers
$L^{t}$ Total incoming workload within time slot $t$
$M_{i}$ Total number of servers in data center $i$
$\mu_{i}$ Service rate of servers in data center $i$
$d_{i}^{t}$ Transmission delay to data center $i$ in time slot $t$
$D$ Delay bound
$P_{idle}$ Average idle power of server
$P_{peak}$ Average peak power of server
$R_{i}$ Power usage effectiveness of data center $i$
$\xi$ Empirical parameter of power consumption
$\alpha_{i}^{t}$ Base price for data center $i$ in time slot $t$
$\beta_{i}$ Sensitivity parameter of price for data center $i$
$Q_{i}^{t}$ Available supply to data center $i$ in time slot $t$
$B_{i}^{t}$ Background load in location $i$ and time slot $t$
$C_{i}$ Power capacity in location $i$
$\underline{\pi}_{i}^{t}$ Price lower bound for data center $i$ in time slot $t$
$\overline{\pi}_{i}^{t}$ Price upper bound for data center $i$ in time slot $t$
$\pi_{\max}^{t}$ Maximum average price in time slot $t$
$\theta_{i}$ Coefficient for energy consumption of data center $i$
$E^{t}$ Total energy required in time slot $t$
$\underline{E}_{i}^{t}$ Energy lower bound for data center $i$ in time slot $t$
$\overline{E}_{i}^{t}$ Energy upper bound for data center $i$ in time slot $t$
$\Delta_{i,\min}^{t}$ Error lower bound in location $i$ and time slot $t$
$\Delta_{i,\max}^{t}$ Error upper bound in location $i$ and time slot $t$
§.§ Variables
$\lambda_{i}^{t}$ Workload assigned to data center $i$ in time slot $t$
$x_{i}^{t}$ Number of active servers in data center $i$ and time $t$
$e_{i}^{t}$ Energy consumption of data center $i$ in time slot $t$
$s_{i}^{t}$ Billing reference for data center $i$ in time slot $t$
$\pi_{i}^{t}$ Unit energy price for data center $i$ in time slot $t$
$r_{i}^{t}$ Electric load ratio in location $i$ and time slot $t$
$\underline{z}_{i}^{t}$ Binary variables
$\overline{z}_{i}^{t}$ Binary variables
$\delta_{i}^{t}$ Load prediction error in location $i$ and time slot $t$
§ INTRODUCTION
Energy management of large and distributed data centers has become an increasingly important problem. With the fast development of cloud computing services, it is now common for a cloud provider (e.g., Google, Microsoft, and Amazon) to build multiple, large, and geographically dispersed data centers across the continent. Each data center may include hundreds of thousands of servers, massive storage equipment, cooling facilities, and power transformers. The energy consumption and cost of data centers hence can be significant <cit.>. For example, Google reported in 2011 that its data centers continuously draw almost 260 MW of power, which is more than what Salt Lake City consumes <cit.>. Microsoft's data center in Washington US consumes 48 MW of power, which is equivalent to the power consumption of about 40,000 households. This has motivated growing research activities toward optimizing the data center operations to reduce the total energy cost. For example, Qureshi et al. in <cit.> proposed an energy cost minimization method for distributed data centers to exploit electricity price difference. The idea is later extended in <cit.>.
However, most existing studies of energy management of distributed data centers have focused on the energy cost minimization from the viewpoint of data centers, but fail to consider the impact of such energy management practice on the power grid. Note that, due to their enormous energy consumption, data centers are expected to have a great influence on the operation of the power grid <cit.>. Without taking such impact into account, these energy management schemes may adversely affect power-grid stability and load balancing.
In this paper, we aim to study the energy cost minimization of distributed data centers based on their impact to the power grid. We seek to benefit from the recent advances in two-way communications that are available in smart grid <cit.> to allow interactions and coordinations between energy suppliers and consumers in real time to improve demand side management. In our proposed framework, the utility company can set dynamic prices to the demand-responsive data centers, and the data centers can dynamically change energy consumption in response to the price changes. This can effectively coordinate demand with supply, and hence avoid unintended power overloading.
The overall framework of our proposed system setup is shown in Fig. <ref>. Cloud service users send computing requests via Internet to the cloud provider. Exploiting various electricity prices at different locations, the cloud provider minimizes the total energy cost by assigning users' requests to different data centers. The utility company utilizes the demand response of data centers, and tries to achieve power load balancing by altering the electricity consumption of data centers through dynamic pricing.
The main contributions of this paper are as follows:
* Data center and smart grid interaction: To the best of our knowledge, this is the first paper that studies the interactions between smart grid and data centers by considering the active decisions on both sides. In particular, how does the utility company properly incentivize data centers to provide demand response services toward a reliable power grid?
* Modeling and solution methods: We formulate the interactions between smart grid and data centers as a two-stage price optimization problem. In its original form, this problem cannot be solved by standard convex programming techniques. Therefore, we reformulate the problem as a mixed integer quadratic program, and design a customized branch-and-bound algorithm to attain the globally optimal solution. We also design a low-complexity descent algorithm to attain a close-to-optimal solution.
* Performance benchmarks: To help characterizing the optimal solution of the two-stage price optimization problem, we construct two single-level optimization problems, namely an Integrated Problem and a Restricted Problem, which correspond to the performance upper and lower bounds of the two-stage price optimization problem.
* Case studies and implications: Our proposed method can not only balance the power load for smart grid but also reduce total energy cost for data centers, hence achieving a win-win result.
The remainder of the this paper is organized as follows. We review the related work in Section II. After that, we formulate the system model as a two-stage price optimization problem in Section III. In Section IV, we study two benchmark problems to provide performance bounds for the formulated two-stage price optimization problem. In Section V, we analyze the solution of the two-stage price optimization problem, design a branch-and-bound algorithm to yield the global optimum, and propose an alternative heuristic algorithm to solve the sub-optimal solution. In Section VI, we analyze the worst-case performance by considering the prediction error in background power load. Performance of the proposed scheme is evaluated in Section VII. This paper is concluded in Section VIII.
Smart grid and data center interaction.
§ RELATED WORK AND MOTIVATION
§.§ Literature Review
There are many existing research results on managing data center's workload to reduce energy cost, such as those studying the energy cost minimization problem with multi-electricity-market environment <cit.>, green renewable generators <cit.>, online optimization <cit.>, service level agreements <cit.>, and deregulated electricity price <cit.>. Zhang et al. <cit.> designed a Vickrey-Clarke-Groves auction mechanism, in which tenants of data centers voluntarily bid for emergency demand response. However, these results did not consider the active response by the utility companies, nor did they consider how the data centers' demand response may bring large load fluctuations across different locations over time. This motivates us to study the interactions between smart grid and geographically dispersed data centers, and examine how smart grid can properly incentivize data centers through dynamic pricing to improve the grid reliability.
There has been a large body of research on demand response of strategic energy consumers <cit.>. For example, in <cit.>, Mohsenian-Rad and Leon-Garcia suggested scheduling household devices based on the predicted prices to minimize the electricity cost. In <cit.>, Nguyen et al. proposed a game theoretic model, in which an electricity provider dynamically updates the energy prices to reduce the peak load, by considering the load profiles of users. In <cit.>, Li et al. studied demand response based on utility maximization, and proposed a distributed algorithm to compute optimal prices and power schedules. In <cit.>, Wong et al. designed a time-dependent price to incentivize users to shift power load so as to relieve stress during peak hours.
§.§ Motivation
Different from traditional residential or industrial consumers, data centers are special electricity consumers. This is not only because of their enormous energy consumption, but also because of flexibility of energy consumptions over multiple locations. The previous studies in <cit.> mainly focused on the workload distribution from the perspective of data centers. As reported in <cit.>, such workload distribution of data centers has great impact on power load balancing in the smart grid.
In the power system, the utility company is responsible for supplying power to meet the demand, and for maintaining the safe operation of the smart grid system. The utility company can utilize the demand response of data centers to manage their energy consumption. However, most of the existing demand response programs focused on the time flexibility of residential demands, without considering the demand side management over multiple locations. The latter is difficult to do for residential demands, but is very suitable in the case of geographically dispersed data centers.[The cloud provider owns multiple data centers located in different geographical locations, and thus gains flexibility of power loads over locations via workload assignment over different data centers. As an example, when Google responds to a user's web search query, the corresponding computation can be done in any of the Google's data centers (as long as certain service quality agreement is satisfied).] This motivates us to design the dynamic pricing incentive mechanism from the grid operator's point of view, in order to incentivize the proper demand response from multiple geographically dispersed data centers. Tran et al. <cit.> studied demand response of data centers in a multi-utilities environment, and modeled the interactions between utilities and data centers as a Stackelberg game. Different from <cit.>, we study the interaction between one utility company and one cloud provider (with multiple data centers) as a bi-level optimization problem, propose two benchmark problems to estimate the performance bounds, and propose two algorithms to solve the optimal prices and close-to-optimal prices, respectively.
§ SYSTEM MODEL
We consider a discrete time model $t \in \mathcal{T} = \{ 1,...,T \}$, where the length of a time slot matches the time-scale at which the workload allocation decisions and dynamic pricing decisions are updated, e.g. once an hour <cit.>. Let $\mathcal{N}=\{1,...,N\}$ denote the set of geographically dispersed data centers, where each data center $i \in \mathcal{N}$ has $M_{i}$ homogeneous servers, and has the same function in terms of supporting various kinds of applications (e.g., Internet services, image processing). As we will explain later, not all servers are turned on during each time slot.
Fig. <ref> illustrates the system architecture of data centers and smart grid. We assume that a group of geographically dispersed data centers are operated by a single cloud provider, and there is a traffic aggregator (e.g., a front-end portal server) responsible for distributing the total incoming computing workload $L^{t}$ within time slot $t$ to data centers in different regions <cit.>. Each data center is powered by a dedicated power substation in the power grid, and all the substations are operated by the same utility company.[Many practical examples motivates our assumption of one utility company. For example, Alibaba cloud, a Chinese cloud provider, runs five data centers at different locations in China, and three of which are served by the State Grid Corporation of China. Such scenarios also exist in deregulated electricity markets, such as in California US <cit.>.]
The architecture for data center demand response.
In each time slot $t$, we model the interactions between utility company and data centers in two stages. In Stage 1, the utility company sets a billing reference $s_{i}^{t}$, which determines the electricity tariff as we will explain later for each data center $i$ to balance the load on the power grid. In Stage 2, we assume that the data centers can predict the workload accurately at the beginning of each time slot. Then data centers cooperate with each other (as they belong to the same cloud operator) so as to minimize the total energy cost by determining the computing workload allocation $\lambda_{i}^{t}$ and the number of active servers $x_{i}^{t}$ in each data center $i$. Next, we discuss these decisions in details.
§.§ Stage 2: Data Center's Energy Cost Minimization
First, we consider the Stage-2 problem, where a cloud provider (such as Google) wants to minimize the total energy cost of multiple data centers. In practice, data centers directly negotiate with the utility company regarding the electricity rates <cit.>. In time slot $t$, the utility company charges data center $i$ with the following regional electricity price $\pi_{i}^{t}$ per unit of energy:
\begin{align}\label{unitprice}
& \pi_{i}^{t}=\alpha_{i}^{t} +\beta_{i} (e_{i}^{t}-s_{i}^{t}),
\end{align}
where $e_{i}^{t}$ is the data center's the electricity consumption, $s_{i}^{t}$ is called the billing reference, $\beta_i>0$ is a sensitivity parameter, and $\alpha_{i}^{t}>0$ denotes the base price, all at location $i$ in time slot $t$. The dynamic pricing scheme in (<ref>) is motivated by the tiered electricity pricing, which has been widely implemented in various power markets such as the United States, Japan, and China. The key idea of tiered pricing is to set several pricing tiers for the energy consumption, and the unit price per unit of energy increases with the tiers progressively <cit.>. In (<ref>), the term $\beta_{i} (e_{i}^{t}-s_{i}^{t})$ reflects the difference between electricity consumption $e_{i}^{t}$ and the billing reference $s_{i}^{t}$. The unit price $\pi_i^t$ will be higher than the base price if $e_{i}^{t} > s_{i}^{t}$.
Next, we discuss the data centers' optimization constraints.
§.§.§ Workload constraint
In each time slot $t$, users' computing requests (workload to the cloud provider) are received by a front-end portal server. Then a total of $N$ data centers should work together to complete the total workload of $L^t$, with the allocation to data center $i$ as $\lambda_i^t$:
\begin{align}\label{workload}
\sum_{i=1}^{N} \lambda_{i}^{t}=L^{t},~\lambda_{i}^{t} \geq 0, ~\forall i \in \mathcal{N}, t \in \mathcal{T}.
\end{align}
§.§.§ QoS (delay) constraint
It is important for data centers to provide QoS guarantees to the users, and QoS can be specified by the service level agreement (SLA) <cit.>. SLA usually measures the average performance for the operation of a data center during a period of time. We consider both the transmission delay (incurred before the request arrives at a data center) and the queuing delay (experienced after the request arrives at a data center). We define $d_{i}^{t}$ as the transmission delay experienced by a computing request from the aggregator to data center $i$ during time slot $t$. Notice that $d_i^t$ is usually much less than the length of a time slot. To model the queuing delay, we use queuing theory to estimate the average processing time in data center $i$ when there are $x_{i}^{t}$ active servers processing workload $\lambda_{i}^{t}$ with a service rate $\mu_{i}$ per server.[We assume that the servers in the same data center $i$ are homogeneous and have the same service rate $\mu_i$.] Applying the results from M/M/1 queuing theory <cit.>, the average waiting time is approximately $\frac{1}{\mu_i x_{i}^{t}- \lambda_{i}^{t}}$. To meet the QoS requirement, the total time delay experienced by a computing request should satisfy some delay bound $D$, which is the maximum waiting time that a request can tolerate. For simplicity, in this paper, we will assume homogeneous requests that have the same delay bound $D$. Therefore, we have the following QoS constraint
\begin{align}\label{qos}
d_{i}^{t} + \frac{1}{\mu_i x_i^t- \lambda_i^t} \leq D,~\forall i \in \mathcal{N}, t \in \mathcal{T},
\end{align}
where $\mu_i x_{i}^{t} > \lambda_i^t$.
§.§.§ Server constraint
At each data center $i$, there are tens of thousands of servers providing cloud computing services to meet users' requests. Let $M_{i}$ denote the maximum number of available servers. The cloud provider can switch on and off servers to adjust the service time. Since the number of servers is usually large, we can relax the integer constraint on the number of active servers without significantly affecting the optimal result. Therefore, we have the following server constraint[We set the minimum required number of active servers in each data center as zero. It can also be set as a non positive to reflect operational requirements for the data center, without changing the engineering insights from the analysis.]
\begin{align}\label{server}
0 \leq x_{i}^{t} \leq M_{i},~\forall i \in \mathcal{N}, t \in \mathcal{T}.
\end{align}
§.§.§ Energy consumption constraint
The energy consumption of data centers consists of IT energy consumption (e.g., CPU, memory, and storage) and ancillary energy consumption (e.g., cooling, lighting, and power facility). The quantitative relation between IT energy consumption and ancillary energy consumption is measured by the power usage efficiency (PUE) <cit.>, which is defined as the ratio of total energy consumption to IT energy consumption. The energy used by computing equipments is considered to be productive. On the contrary, the energy for ancillary infrastructure (e.g., cooling, lighting, and power facility) is auxiliary. PUE helps us understand the total energy consumption based on the IT energy consumption. Therefore, we can calculate the total energy consumption of a data center using PUE, amount of computing workload, and number of active servers. Precisely, based on the data center power model in <cit.>, we formulate the energy consumption of data center $i$ in time slot $t$ as
\begin{align*}
& e_{i}^{t} = x_{i}^{t} \left( P_{idle} + (R_{i} -1) P_{peak} \right)+ x_{i}^{t} (P_{peak}-P_{idle}) \gamma_{i}^{t} + \xi_i,
\end{align*}
where $P_{idle}$ and $P_{peak}$ represent the average idle power and average peak power of a single server, respectively. The power efficiency parameter $R_{i} > 1$ denotes PUE of data center $i$. The parameter $\xi_i$ is an empirical constant indicating the base energy consumption of data center $i$, and $\gamma_{i}^{t}$ denotes the average server utilization of data center $i$ in time slot $t$.
We substitute the average server utilization $\gamma_{i}^{t}=\lambda_{i}^{t} / (\mu_i x_{i}^{t})$, and rewrite $e_i^t$ in the following equivalent form:
\begin{equation}\label{energycon}
\begin{split}
e_{i}^{t} = \left( P_{idle} + (R_{i} -1) P_{peak} \right) x_{i}^{t}
+ \frac{P_{peak}-P_{idle}}{\mu_i} \lambda_{i}^{t} + \xi_i , \\
\forall i \in \mathcal{N}, t \in \mathcal{T},
\end{split}
\end{equation}
which is an affine function with respect to the number of active servers $x_{i}^{t}$ and the computing workload $\lambda_{i}^{t}$.
Given the operational requirements of the power substation, we limit the maximum power that can be consumed by data center $i$ in time slot $t$ as
\begin{equation}\label{maxenergy}
0 \leq e_{i}^{t} \leq Q_{i}^{t},~\forall i \in \mathcal{N}, t \in \mathcal{T}.
\end{equation}
where $Q_{i}^{t}$ denotes the available power supply to data center $i$ in time slot $t$.
With the above constraints, we can formulate the cloud provider's energy cost minimization problem in Stage 2. The objective is to minimize the data centers' total energy cost over all locations and all time slots by choosing the workload allocation $\lambda_{i}^{t}$ and the number of active servers $x_{i}^{t}$ for each data center $i\in\mathcal{N}$ and each time $t\in\mathcal{T}$. As the operational constraints (<ref>)-(<ref>) are decoupled across time slots, we formulate the energy cost minimization problem in time slot $t$ as follows:
\begin{align*}
& \leftline{\textbf{Stage-2 Problem (PS2): Total Energy Cost Minimization}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \min_{\boldsymbol{\lambda}^{t},~\boldsymbol{x}^{t}}
&& \sum_{i \in \mathcal{N}} \Big( \alpha_{i}^{t} +\beta_{i} (e_{i}^{t}-s_{i}^{t}) \Big) e_{i}^{t} \\
& \text{subject to}
&& \text{Constraints \eqref{workload}--\eqref{maxenergy}},
\end{aligned}
\end{equation*}
where $\boldsymbol{\lambda}^{t} = \{ \lambda_i^t,~\forall i\in\mathcal{N}\}$ and $\boldsymbol{x}^{t} = \{ x_i^t,~\forall i\in\mathcal{N} \}$ denote the workload allocation vector and active server number vector for each time slot $t\in\mathcal{T}$, respectively. The energy cost of data center $i$ is calculated as the product of its energy consumption $e_{i}^{t}$ and the corresponding unit price $\alpha_{i}^{t} +\beta_{i} (e_{i}^{t}-s_{i}^{t})$.
Note that, the optimal value of workload allocation $\lambda_i^t$, number of active servers $x_i^t$ and energy consumption $e_i^t$ in (<ref>) are functions of the billing references $\boldsymbol{s}^{t}=\{ s_i^t,~\forall i\in\mathcal{N} \}$ in time slot $t$. Given $\boldsymbol{s}^{t}$, we can solve Problem PS2, and will present the optimal solutions of $\lambda_i^t$, $x_i^t$ and $e_i^t$ in Section V.
§.§ Stage 1: Smart Grid's Power Load Balancing Problem
We are now ready to consider the Stage-1 power load balancing problem for the smart grid. We classify the load into two groups: data centers and others. We focus on the data centers' loads as they have geographical flexibility, and let the latter group as background loads. With the emergence of smart grid communications technologies, it is possible for the utility company to incentivize the data centers to shift loads from heavily loaded regions to lightly loaded regions. In our proposed framework, the smart grid optimizes dynamic tiered prices by setting the billing references $\boldsymbol{s}^{t}$ in each time slot $t$ to balance power load across geographical locations. To measure the power load levels in different locations, we define the electric load ratio in location $i$ and time slot $t$ as
\begin{align}\label{loadratio}
r_{i}^{t}(\boldsymbol{s}^{t}) = \frac{e_{i}^{t}(\boldsymbol{s}^{t}) + B_{i}^{t}} {C_{i}},
\end{align}
where $B_{i}^{t}$ is the background power load, and $C_{i}$ is the capacity of power substation $i$. Note that the load ratio $r_{i}^{t}$ is a function of the energy consumption $e_{i}^{t}$, and thus also depends on the billing reference $\boldsymbol{s}^{t}$ for all locations in time slot $t$. The utility company aims at balancing the load ratio $r_{i}^{t}(\boldsymbol{s}^{t})$ at all locations in each time slot.
Let $Q_{i}^{t} = C_{i} - B_{i}^{t}$ be the maximum available power supply to data center $i$ in time slot $t$. Since our study focuses on the demand response of data centers, we denote the aggregate energy usage of all the users other than data centers as the background energy load. We assume that the utility company is able to accurately forecast[We first solve the two-stage problem assuming perfect background load prediction. In section VI, we will further study the impact of prediction error.] the background energy load ahead of each time slot <cit.>.
Based on the load ratio $r_{i}^{t}$, we define the electric load index (ELI) across all locations in time slot $t$ as
\begin{align}\label{def_eli}
ELI \triangleq \sum_{i \in \mathcal{N}}
~ \Big( r_{i}^{t}(\boldsymbol{s}^{t}) \Big) ^{2} C_{i},
\end{align}
where ELI measures the overall load ratio across all locations. Note that electric load ratio $r_{i}^{t}$ is a normalized indicator, which does not reflect the importance of those locations with large capacities. Therefore, we introduce the capacities $C_{i}$ as the weighted coefficients in ELI. We can show that minimizing ELI with respect to $e_{i}^{t}$ yields an equal load ratio across all locations in the ideal case (without considering any constraints):
\begin{align*}
\frac{e_{1}^{t}+B_{1}^{t}}{C_{1}} = \cdots = \frac{e_{N}^{t}+B_{N}^{t}}{C_{N}},
\end{align*}
which indicates no overloading problem occurs in any of the locations. Therefore, the system reliability is improved at these locations.
However, such even load distribution may not be achievable in practice, because the energy consumption $e_{i}^{t}$ should also satisfy the operational constraints for workload allocation and number of active servers in (<ref>)–(<ref>). Moreover, the cloud provider and the utility company are independent entities. Data centers are operated by the cloud provider, which implies that the energy consumption of data centers cannot be directly controlled by the utility company.
In order to balance the electricity load, in this paper we focus on the scenario where the utility company charges dynamic prices to incentivize users to shift their electricity usage to less loaded locations. To encourage the participation of data centers into the demand response program and prevent the utility company from abusing its market power, constraints should be set to regulate the dynamic prices. In practice, the utility company and data centers usually negotiate with each other and enter into a contract <cit.> to specify the pricing structure. Based on related studies <cit.>, we set the following constraints for the energy price $\pi_i^t$:
\begin{align}
& \underline{\pi}_i^t \leq \alpha_{i}^{t} + \beta_{i} (e_{i}^{t}-s_i^t) \leq \overline{\pi}_i^t,~\forall i \in \mathcal{N}, t \in \mathcal{T}, \label{s_constraint1} \\
& \frac{1}{N} \sum_{i \in \mathcal{N}} \left[ \alpha_{i}^{t} + \beta_{i} (e_{i}^{t}-s_i^t) \right] \leq \pi_{\max}^{t},~t \in \mathcal{T}, \label{s_constraint2}
\end{align}
where (<ref>) ensures that the price charged to the data centers is always contained within the range $[\underline{\pi}_i^t,\overline{\pi}_i^t]$. Constraint (<ref>) enforces that the dynamic prices across all locations have an average price ceiling $\pi_{\max}^{t}$, which is specified by the contract between the utility company and data centers <cit.>. Constraint (<ref>) can prevent the utility company from charging the maximum possible price in all locations. More precisely, the utility company has to provide lower prices to other locations if it charges a higher prices at some locations, so that (<ref>) can be satisfied. This will give a guarantee to the cloud provider, such that the dynamic price will not arbitrarily increase the energy cost of the data centers.
After the contract terms (e.g., constraints (<ref>) and (<ref>)) are settled, the utility company is responsible of enforcing the price constraints (<ref>) and (<ref>).[To enforce constraints (<ref>) and (<ref>), the utility company should carefully determine the dynamic prices and consider the corresponding responses from the data centers, as the price constraints (<ref>) and (<ref>) involve both dynamic prices and energy consumption responses of data centers.] We formulate the smart grid's load balancing problem in time slot $t$ as follows:
\begin{align*}
\leftline{\textbf{Stage-1 Problem (PS1): Electric Power Load Balancing}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \underset{\boldsymbol{s}^{t}} {\min}
& & \sum_{i \in \mathcal{N}}~~ \Big( r_{i}^{t}(\boldsymbol{s}^{t}) \Big) ^{2} C_{i} \\
& \text{subject to}
& & \text{Constraints \eqref{s_constraint1} and \eqref{s_constraint2}},
\end{aligned}
\end{equation*}
where the electric load ratio $r_{i}^{t}$ depends on the energy consumption $e_{i}^{t}$, which is the optimal solution of Stage-2 Problem PS2.
§.§ Two-stage Price Optimization Problem
For Problem PS2, we can show that constraints (<ref>)–(<ref>) can be equivalently rewritten as constraints of data centers' energy consumption:
\begin{align}
& \sum_{i \in \mathcal{N}} \theta_i e_{i}^{t} = E^{t}, \label{energy_constraint1} \\
& \underline{E}_{i}^{t} \leq e_{i}^{t} \leq \overline{E}_{i}^{t},~\forall i\in\mathcal{N} \label{energy_constraint2},
\end{align}
where $\theta_i$, $E^{t}$, $\underline{E}_{i}^{t}$ and $\overline{E}_{i}^{t}$ are system parameters. Constraint (<ref>) is derived from the workload constraint in (<ref>), which specifies that the summation of $\theta_i$-weighed energy consumption of all the data centers should reach $E^{t}$ in order to process the total workload $L^{t}$. The box constraint (<ref>) sets the energy consumption upper bound $\underline{E}_{i}^{t}$ and lower bound $\overline{E}_{i}^{t}$ for each data center, to meet all the inequality constrains in (<ref>)–(<ref>). For the proof and detailed representation of the parameters, please see Appendix A.
Using constraints (<ref>) and (<ref>), we can simplify Problem PS2 into an equivalent energy consumption distribution problem, in which the cloud provider directly decides the energy consumption of data center $e_i^t$ to minimize the energy cost. The equivalent energy consumption distribution problem is presented as follows:
\begin{align*}
\leftline{\textbf{PE2: Equivalent Problem of PS2}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \underset{\boldsymbol{e}^{t}} {\min}
& & \sum_{i \in \mathcal{N}}
\Big( \alpha_{i}^{t} +\beta_{i} (e_{i}^{t}-s_{i}^{t}) \Big) e_{i}^{t} \\
& \text{subject to}
& & \text{Constraints \eqref{energy_constraint1} and \eqref{energy_constraint2}},
\end{aligned}
\end{equation*}
where $\boldsymbol{e}^{t}= \{ e_i^t,~\forall i \in \mathcal{N} \}$. Once the energy consumption $e_i^t$ is determined, we can find the corresponding workload allocation $\lambda_i^t$ and number of active servers $x_i^t$.
Two-stage optimization problem.
Fig. <ref> shows the relation between the two-stage problems PS1 and PE2, each of which is executed once in each time slot. In Stage 1, at the beginning of each time slot, the utility company sets billing references for data centers to optimize the ELI performance. This leads to the tiered price $\pi_i^t = \alpha_i^t + \beta_i (e_i^t -s_i^t)$ for each data center $i$. In Stage 2, the cloud provider optimizes the energy consumption $e_i^t$ of each data center in order to minimize the total energy consumption $\sum_{i\in\mathcal{N}} \pi_i^t e_i^t$ in time slot $t$.
The two-stage problem is a challenging optimization problem to solve, due to the coupled variables and constraints. As the utility company aims to balance the electric load across locations, it will consider the response of the cloud provider in Stage 2, when computing the optimal billing references $\boldsymbol{s}^t$ in Stage 1. Before solving the two-stage problem, we will introduce two benchmark problems to bound the optimal solution.
§ PERFORMANCE BENCHMARKS
The two-stage problem is a quadratic bilevel program with coupled constraints, which is NP-hard in general and cannot be solved effectively by standard convex optimization algorithms. Before proposing solution methods to solve the two-stage problem, we construct two benchmarks, the integrated problem and the restricted problem, to provide lower bound and upper bound of the ELI performance, which are helpful in terms of solving the two-stage problem in Section V.
§.§ The Integrated Problem
We consider the following integrated problem as a benchmark, where the utility company directly decides the optimal workload assignments and the number of active servers for each data center (without the need of dynamic pricing). This will reveal the minimum ELI that the system can achieve if the utility company and the data centers fully cooperate with each other.
The integrated problem is formulated as follows.
\begin{align*}
\leftline{\textbf{PI: Integrated Problem}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \underset{\boldsymbol{\lambda}^{t},~\boldsymbol{x}^{t}} {\min}
& & \sum_{i \in \mathcal{N}}
~~ \Big( r_{i}^{t} \Big) ^{2} C_{i} \\
& \text{subject to}
& & \text{Constraints \eqref{workload}--\eqref{maxenergy}}.
\end{aligned}
\end{equation*}
We can see that the objective is consistent with the utility company's objective of load balancing in Problem PS1. The constraints are the same in Problem PS2 for data centers' operation. Problem PI is a convex quadratic program, which can be solved by standard convex optimization techniques <cit.>.
Intuitively, compared with the scenario where the utility company incentivizes data centers through dynamic pricing, direct control of data centers' operation would be more efficient in terms of load balancing. This can lead to a lower bound of the ELI performance stated in the following proposition.
The optimal solution of the integrated problem PI provides a lower bound of the optimal ELI performance of PS1.
To prove Proposition 1, we need to the show that the feasible set of the integrated problem PI is larger than that of the original two-stage optimization problem. In the integrated problem PI, the utility company directly controls the workload allocation and the number of servers in the data centers, subject to the data center operation constraints (<ref>)-(<ref>). Whereas in the two-stage problems PS1 and PS2, the utility company aims at indirectly managing data centers' operation in PS2 through price incentives in PS1, subject to both data center operation constraints (<ref>)-(<ref>) and pricing constraints (<ref>)-(<ref>). Intuitively, when the utility company directly controls data centers' operation in PI, the decision is more flexible than incentive-based in through the two-stage problems PS1 and PS2. Hence, the performance of PI should be better, which means a lower ELI. For detailed proof, see Appendix B.
Note that the ELI performance gap between the estimated lower bound and the optimal solution is affected by constraints (<ref>) and (<ref>) in the two-stage problem. For example, enlarging the price range $[\underline{\pi}_i^t,\overline{\pi}_i^t]$ in price constraint (<ref>) can improve the optimal ELI performance to be close to the ELI lower bound, as the dynamic pricing scheme of the utility company has a larger feasible set.
§.§ The Restricted Problem
After we provide a lower bound for ELI by solving PI, we present another benchmark problem namely restricted problem RS.
In order to construct the restricted problem, first, we note that in the two-stage problem, different stages have different constraints that cannot be moved across stages. The Stage-1 problem is the upper-level problem, while the Stage-2 problem is the lower-level problem. The constraints of the Stage-1 problem PS1 also apply to the Stage-2 problem PE2, but the operational constraints of data centers in the Stage-2 problem PE2 only need to be satisfied by the data centers. Intuitively, moving constraints from the Stage-2 problem to the Stage-1 problem shrinks the utility company's action set. Thus, the way we formulate the restricted problem is to move the bounding constraint on energy consumption (<ref>) in PE2 to the Stage-1 problem PS1. Thus we formulate the restricted problem in time slot $t$ with the modified Stage-1 and Stage-2 problems as follows.
\begin{align*}
\leftline{\textbf{RS1: Stage 1 of the Restricted Problem}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \underset{\boldsymbol{s}^{t}} {\min}
& & \sum_{i \in \mathcal{N}}
~ \Big( r_{i}^{t}(\boldsymbol{s}) \Big) ^{2} C_{i} \\
& \text{subject to}
& & \text{Constraints \eqref{s_constraint1}, \eqref{s_constraint2} and \eqref{energy_constraint2}}.
\end{aligned}
\end{equation*}
\begin{align*}
\leftline{\textbf{RS2: Stage 2 of the Restricted Problem}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \underset{\boldsymbol{e}^{t}} {\min}
& & \sum_{i \in \mathcal{N}}
\Big( \alpha_{i}^{t} +\beta_{i} (e_{i}^{t}-s_{i}^{t}) \Big) e_{i}^{t} \\
& \text{subject to}
& & \text{Constraints \eqref{energy_constraint1}}.
\end{aligned}
\end{equation*}
We use backward induction to solve RS1 and RS2. We first solve Problem RS2. Since RS2 is a convex quadratic program with equality constraints, we obtain the optimal solution in the closed form as
\begin{equation}
e_{i}^{t}= \frac {s_{i}^{t}}{2} - \frac {\alpha_{i}^{t}+ \theta_i \sigma^t} {2 \beta_{i}},~\forall i \in \mathcal{N}, \label{RSsolution}
\end{equation}
where $\sigma^{t}$ is the Lagrangian multiplier corresponding to the energy equality constraint (<ref>).
Substituting the optimal solution of Problem RS2 (<ref>) into Problem RS1, we have the restricted problem as a single-level optimization problem:
\begin{align*}
\leftline{\textbf{RS: Restricted Problem}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \underset{ \{ \boldsymbol{s}^{t}, \boldsymbol{e}^{t}, \sigma^t \}} {\min}
& & \sum_{i \in \mathcal{N}}
~ \Big( r_{i}^{t} \Big) ^{2} C_{i} \\
& \text{subject to}
& & \text{Constraints \eqref{s_constraint1}--\eqref{RSsolution}},
\end{aligned}
\end{equation*}
which is a convex quadratic program, and can be solved by standard convex programming algorithms <cit.>.
Intuitively, moving constraints (<ref>) from the Stage-2 problem to the Stage-1 problem shrinks the utility company's action set. This can lead to a performance degradation in term of a higher ELI, which serves as a upper bound stated in the following proposition.
The optimal solution of the restricted problem RS provides an upper bound of the optimal ELI performance of PS1.
To prove Proposition 2, we need to the show that the feasible set of the restricted problem RS is smaller than that of the original two-stage optimization problem. Notice that the constraints (<ref>) were in PS2 of the original two-stage problem formulation, but here we move them to the Stage-1 problem RS1 of the restricted two-stage formulation. Compared with PS1, in RS1 the utility company's pricing decision in the restricted problem is more conservative, because the utility company has to satisfy the additional data center operation constraints (<ref>). Intuitively, the restricted two-stage problem has a smaller feasible set than that of the original two-stage problem. Therefore, the solution that is obtained from the restricted problem provides an upper bound for the original two-stage problem. For detailed proof, see Appendix C.
Note that the estimation of the upper bound is affected by the parameter configurations in constraints (<ref>), (<ref>) and (<ref>).
§ SOLVING THE ORIGINAL TWO-STAGE PROBLEM
After presenting ELI performance upper and lower bounds from the benchmark problems, we next solve the original two-stage problem through backward induction. We first solve the Stage-2 problem PS2, where data centers minimize the total energy cost. Then, we design a branch-and-bound algorithm for the Stage-1 problem PS1 to attain the globally optimal solution.
§.§ Solving the Stage-2 Problem
In the Stage-2 problem PS2, data centers decide the workload allocation $\lambda_{i}^{t}$ and number of active servers $x_{i}^{t}$ at all locations to minimize the total energy cost in each time slot, given the charging reference $\boldsymbol{s}^{t}$ announced by the utility company ahead of each time slot.
We have reformulated PS2 as en equivalent problem PE2 in Section III. As Problem PE2 is strictly convex, we can compute the optimal solution $e_i^{t*}$ through the Lagrangian dual method. This leads to the following result.
The unique optimal solution of Problem PE2 is
\begin{equation}\label{opt_pe2-t}
e_{i}^{t*}(\boldsymbol{s}^{t})= \min
\left\{
\max \left\{ \underline{E}_{i}^{t},
\frac {s_{i}^{t}}{2} - \frac {\alpha_{i}^{t}+ \theta_i \sigma^t} {2 \beta_{i}} \right\}
,\overline{E}_{i}^{t} \right\},~\forall i \in \mathcal{N}.
\end{equation}
where $e_{i}^{t \ast}(\boldsymbol{s}^{t})$ is called the best response of data center $i$ to the billing reference $\boldsymbol{s}^{t}$, and $\sigma^t$ is the Lagrangian multiplier corresponding to the equality constraint (<ref>).
Problem PE2 can be solved by the standard subgradient method with a constant stepsize <cit.>. For detailed proof, see Appendix D.
§.§ Solving the Stage-1 Problem
After solving the Stage-2 problem PE2, we obtain the optimal energy consumption of data centers as functions of the given charging references $\boldsymbol{s}^{t}$. We next solve the Stage-1 problem PS1. Under the assumption of complete information, the utility company knows how the data centers will respond to the dynamic prices, and can predict the energy consumptions of data centers given the dynamic prices.[We assume that the utility company can predict the energy consumptions of data centers through long-term observation, as the utility company is the electricity provider and knows the historical energy consumptions of all the data centers.] Therefore, we can replace Problem PE2 with its Karush-Kuhn-Tucker (KKT) conditions and transform the two-stage problem to a single-level optimization problem <cit.> by incorporating the KKT conditions of Problem PE2 into Problem PS1.
(Reformulation) The Stage-1 problem PS1 can be written in the following equivalent problem with quadratic objectives, linear constraints, and complementarity constraints, denoted as PE1.
\begin{align*}
\leftline{\textbf{PE1: Equivalent Problem of the Two-stage Problem}}
\end{align*}
\begin{align}
& \min_{ \{ s_i^t,e_i^t,\sigma^t,\underline{\omega}_i^t,\overline{\omega}_i^t \}, i \in \mathcal{N} }
\begin{aligned}[t]
\sum_{i \in \mathcal{N}} (r_i^t)^{2} C_i
\end{aligned} \notag \\
& \text{subject to} \notag \\
& \underline{\pi}_i^t \leq \alpha_{i}^{t} + \beta_{i} (e_{i}^{t}-s_i^t) \leq \overline{\pi}_i^t,~\forall i \in \mathcal{N}, \label{pe1-t-1} \\
& \frac{1}{N} \sum_{i \in \mathcal{N}} \left[ \alpha_{i}^{t} + \beta_{i} (e_{i}^{t}-s_i^t) \right] \leq \pi_{\max}^{t}, \label{pe1-t-2}\\
& \alpha_{i}^{t}+2 \beta_{i} e_{i}^{t} - \beta_{i} s_{i}^{t} + \theta_i \sigma^{t} - \underline{\omega}_{i}^{t} + \overline{\omega}_{i}^{t} =0,~\forall i\in\mathcal{N}, \label{pe1-t-3} \\
& \underline{\omega}_{i}^{t} (\underline{E}_{i}^{t} - e_{i}^{t})=0,~\forall i\in\mathcal{N}, \label{pe1-t-4} \\
& \overline{\omega}_{i}^{t} (e_{i}^{t} - \overline{E}_{i}^{t}) =0,~\forall i\in\mathcal{N}, \label{pe1-t-5} \\
& \sum_{i \in \mathcal{N}} \theta_i e_{i}^{t} = E^{t}, \label{pe1-t-6} \\
& \underline{E}_{i}^{t} \leq e_{i}^{t} \leq \overline{E}_{i}^{t},~\forall i\in\mathcal{N}, \label{pe1-t-7} \\
& \underline{\omega}_{i}^{t} \geq 0,~\overline{\omega}_{i}^{t} \geq 0,~~\forall i\in\mathcal{N}, \label{pe1-t-8}
\end{align}
where (<ref>)-(<ref>) are the KKT conditions of Problem PE2, and $\sigma^t$, $\underline{\omega}_i^t$, and $\overline{\omega}_i^t$ are the Lagrange multipliers associated with the equality and box constraints of PE2. Since Problem PE2 is strictly convex, the KKT conditions (<ref>)-(<ref>) are necessary and sufficient for the optimal solution of Problem PE2.
Problem PE1 is a quadratic program with nonconvex constraints, which cannot be solved efficiently by standard convex optimization techniques. However, we find that the nonconvexity only comes from the complementarity slackness conditions (<ref>) and (<ref>). We can linearize the complementarity slackness conditions (<ref>) and (<ref>) by introducing binary variables $\underline{z}_{i}^{t} \in \{ 0,1 \}$ and $\overline{z}_{i}^{t} \in \{ 0,1 \}$, and replace (<ref>) and (<ref>) by the following constraints:
\begin{align}
& e_{i}^{t} - \underline{E}_{i}^{t} \leq \underline{z}_{i}^{t} K ,~\forall i\in\mathcal {N}, \label{linearize-1} \\
& \underline{\omega}_{i}^{t} \leq (1-\underline{z}_{i}^{t}) K ,~\forall i\in\mathcal {N}, \label{linearize-2} \\
& \overline{E}_{i}^{t} - e_{i}^{t} \leq \overline{z}_{i}^{t} K ,~\forall i\in\mathcal {N}, \label{linearize-3} \\
& \overline{\omega}_{i}^{t} \leq (1-\overline{z}_{i}^{t}) K ,~\forall i\in\mathcal {N}, \label{linearize-4}
\end{align}
where $K$ is a sufficiently large constant. We can show that (<ref>) is equivalent to (<ref>) and (<ref>).
* We first show that if (<ref>) is satisfied, then (23) and (24) are also satisfied. There are three combinations to make (<ref>) be satisfied. 1) When $e_{i}^{t} = \underline{E}_{i}^{t}$ and $\underline{\omega}_{i}^{t} > 0$, we have $\underline{z}_{i}^{t} \in [0, 1-\frac{\underline{\omega}_{i}^{t}}{K}]$ from (<ref>) and (<ref>). As $\underline{z}_{i}^{t}$ is a binary variable, we obtain that $\underline{z}_{i}^{t} =0$. 2) When $e_{i}^{t} > \underline{E}_{i}^{t}$ and $\underline{\omega}_{i}^{t} = 0$, we obtain that $\underline{z}_{i}^{t} = 1$. 3) When $e_{i}^{t} = \underline{E}_{i}^{t}$ and $\underline{\omega}_{i}^{t} = 0$, we obtain that $\underline{z}_{i}^{t} \in [0,1]$, and thus either $\underline{z}_{i}^{t} = 0$ or $\underline{z}_{i}^{t} = 1$.
* We then show that if (<ref>) and (<ref>) are satisfied, then (<ref>) is also satisfied. We discuss the following two cases by exhausting the choices of the binary variable $\underline{z}_{i}^{t}$. 1) When $\underline{z}_{i}^{t} = 0$, we have $e_{i}^{t} \leq \underline{E}_{i}^{t}$ from (<ref>). Together with the constraint $e_{i}^{t} \geq \underline{E}_{i}^{t}$ as in (<ref>), we obtain $e_{i}^{t} = \underline{E}_{i}^{t}$, and thus (<ref>) is satisfied. 2) When $\underline{z}_{i}^{t} = 1$, we have $\underline{\omega}_{i}^{t} \leq 0$ from (<ref>). Together with the constraint $\underline{\omega}_{i}^{t} \geq 0$ as in (<ref>), we have $\underline{\omega}_{i}^{t} = 0$, and thus (<ref>) is also satisfied.
Following a similar reasoning, we can show that (<ref>) and (<ref>) can replace (<ref>).
To solve PE1, we design a branch-and-bound algorithm <cit.> to attain the optimal solution. We first relax the binary variables $\{ 0,1 \}$ to continuous variables within the range $[0,1]$, and define the following relaxed quadratic problem PR1.
\begin{align*}
\leftline{\textbf{PR1: Relaxed Problemof PE1}}
\end{align*}
\begin{equation*}
\begin{aligned}
& & \sum_{i \in \mathcal{N}} ~\Big( r_i^t \Big)^{2} C_i\\
& \text{subject to}
& & \text{Constraints}~ \eqref{pe1-t-1}-\eqref{pe1-t-3},~\eqref{pe1-t-6}-\eqref{linearize-4}, \\
&&& 0 \leq \underline{z}_{i}^{t} \leq 1,~\forall i\in\mathcal {N},\\
&&& 0 \leq \overline{z}_{i}^{t} \leq 1,~\forall i\in\mathcal {N},\\
& \text{Variables:}
& & \{ s_i^t,e_i^t,\sigma^t,\underline{\omega}_i^t,\overline{\omega}_i^t, \underline{z}_{i}^{t}, \overline{z}_{i}^{t} \}, i \in \mathcal{N}.
\end{aligned}
\end{equation*}
Initially, the algorithm takes the optimum of the integrated problem PI as the lower bound $\underline{F}$, and the optimum of the restricted problem RS as the upper bound $\overline{F}$. Then the algorithm start to solve the relaxed problem PR1 and builds the branch and bound tree by splitting the binary variables to enforce the binary variable constraints. Specifically, the algorithm adds the following constraints, $\underline{z}_{i}^{t}=0$ or $ \underline{z}_{i}^{t}=1$, to the relaxed problem PR1, and derives two new convex quadratic problems (e.g., two first-level children nodes in the branch-and-bound tree shown in Fig. <ref>). The algorithm continues to expand the tree by adding other constraints $\overline{z}_{i}^{t}=0$ or $ \overline{z}_{i}^{t}=1$ until all the binary variables constraints are completely enforced. Meanwhile, the algorithm updates the lower bound $\underline{F}$ after solving each relaxed problem in the children node, and updates the upper bound $\overline{F}$, when a feasible solution with lower optimum is found. The branch-and-bound algorithm terminates at a globally optimal solutions when the lower bound meets the upper bound or all the nodes in the branch and bound tree have been evaluated <cit.>. In the worst-case, the branch-and-bound algorithm will traverse $2^{2N}$ nodes.
Branch and bound tree.
§.§ Heuristic Algorithm
The branch-and-bound algorithm in general has a very high worst-case computational complexity, and hence may not be suitable for solving a large-scale load balancing problem. Therefore, we propose a heuristic algorithm to solve the two-stage problem PS1 and PS2 for suboptimal solutions. Our heuristic algorithm is designed based on the descent approach, and iteratively reduces the value of ELI in Problem PS1.
We view the solution of Problem PE2 as a function of the variables of Problem PS1. Observing the best response (<ref>) in Problem PE2, we find the following monotonic relation between the charging reference $s_i^t$, the optimal energy consumption $e_i^{t \ast}$, and the unit price $\pi_i^t = \alpha_i^t + \beta_i (e_i^{t \ast} - s_i^t)$. Specifically, increasing $s_i^t$ leads to increase in $e_i^{t \ast}$ and $\sigma^t$, and decrease in $e_j^{t \ast},~\forall j \in \mathcal{N} \backslash i$, and all the unit prices $\pi_i^t$ also decrease. On the contrary, decreasing $s_i^t$ causes decrease in $e_i^{t \ast}$ and $\sigma^t$, and increase in $e_j^{t \ast},~\forall j \in \mathcal{N} \backslash i$ and all the unit prices $\pi_i^t$. Note that minimizing the ELI performance (<ref>) yields even load ratio $r_i^t$ across all locations. Thus, we design a descent algorithm to redistribute the total energy consumptions, by decreasing $s_i^t$ in high energy-consumption locations, and increasing $s_i^t$ in low energy-consumption locations. The detailed algorithm is described in Algorithm 1. The utility company and data center iteratively compute the prices and energy consumption. In each iteration, the utility company provides a set of prices, and data centers respond to the prices and report the corresponding schedule of energy consumption (and do not reveal private information such as parameters and constraints). Algorithm 1 reduces ELI and its convergence to a feasible and possibly sub-optimal solution is guaranteed since the ELI performance is lower bounded by Problem PI. For detailed proof, see Appendix E.
Descent algorithm to solve the two-stage problem
Initialization: In each time slot $t \in \{1,...,T \}$, set the iteration count $k=1$, convergence tolerance $\epsilon>0$, and step-size $\eta(k)$. Initialize the starting point $\boldsymbol{s}^t(k) \triangleq \{ s_i^t(k),i\in \mathcal{N}\}$ by solving the restricted problem RS, and compute the average load ratio $r_{avg}^t(k) = \frac{\sum_{i \in \mathcal{N}} r_i^t(k)}{N} $.
Step1: Compute the descent direction $\boldsymbol{g}^t (k)$ for $\boldsymbol{s}^t (k)$: if $r_i^t(k) > r_{avg}^t (k)$, then set $g_{i}^t (k)= - \frac{\theta_{i}}{\beta{i}}$, $i \in \mathcal{N}$; otherwise, set $g_{j}^t (k)= \frac{\theta_{j}}{\beta{j}}$, $j \in \mathcal{N} \backslash i$.
Step2: Perform the search by using the iterations
$\boldsymbol{s}^t (k+1)=\boldsymbol{s}^t (k) + \eta(k) \boldsymbol{g}^t(k)$;
Step3: Given $\boldsymbol{s}^t(k+1)$, solve the optimal energy consumption $\boldsymbol{e}_{i}^{t}(k+1)$ according to (<ref>).
Step4: Check the feasibility based on (<ref>) and (<ref>). If yes, update $r_{avg}^t(k+1) = \frac{\sum_{i \in \mathcal{N}} r_i^t(k+1)}{N} $.
If not,
e_i^t(k+1)=e_i^t(k), s_i^t(k+1)=s_i^t(k),
r_avg^t(k+1)=r_avg^t(k), η(k+1) = 1/2 η(k).
$k \gets k+1$;
the convergence criteria $\| ELI(k) - ELI(k-1) \| \leq \epsilon$ is satisfied;
Return the sub-optimal solutions $\boldsymbol{\hat{s}}^{t}$, $\boldsymbol{\hat{e}}^{t}$.
§ IMPACT OF BACKGROUND LOAD PREDICTION ERROR
In Section V, we solved the two-stage problem based on the assumption that the utility company can forecast the background power load $B_i^t$ accurately. In practice, the prediction may have errors and the actual background load may deviate from the predicted values. We define the prediction errors for the background load in location $i$ and time slot $t$ as $\delta_{i}^{t}$. Then, we can represent the actual background load $\hat{B}_{i}^{t}$ as the summation of predicted value and the prediction error:
\begin{align*}
\hat{B}_{i}^{t} = B_{i}^{t} + \delta_{i}^{t}.
\end{align*}
Next we use the robust optimization approach <cit.> to analyze the impact of prediction errors. We assume that the prediction errors are bounded in known uncertainty sets as follows:
\begin{align}
& \Delta_{i,\min}^{t} \leq \delta_{i}^{t} \leq \Delta_{i, \max}^{t},~\forall i \in \mathcal{N}, \label{loaderror}
\end{align}
where $\Delta_{i,\min}^{t}$ and $\Delta_{i, \max}^{t}$ denote the lower bound and upper bound of the background load prediction error in location $i$ and time slot $t$, respectively.
We let $\boldsymbol{\delta}^{t} = \{ \delta_{i}^{t},~ i \in \mathcal{N} \}$ denote the prediction-error vector. Our aim is to maximize the worst-case performance of power load balancing. We formulate the worst-case performance optimization problem as:
\begin{align*}
& \leftline{\textbf{WCP: Worst-case Performance Optimization Problem}}
\end{align*}
\begin{equation*}
\begin{aligned}
& \underset{\boldsymbol{s}^{t}} {\min} ~\max_{\boldsymbol{\delta}^{t}}
& & \sum_{i \in \mathcal{N}}~~ \frac{ \Big( e_{i}^{t}(\boldsymbol{s}^{t})+B_{i}^{t} + \delta_{i}^{t} \Big)^{2} }{C_i} \\
& \text{subject to}
& & \text{Constraints \eqref{s_constraint1}, \eqref{s_constraint2}, \eqref{loaderror}},
\end{aligned}
\end{equation*}
which is a min-max optimization problem.
To solve Problem WCP, we first solve the inner ELI maximization problem of WCP (namely IWCP):
\begin{equation*}
\begin{aligned}
& \max_{\boldsymbol{\delta}^{t}}
& & \sum_{i \in \mathcal{N}}~~ \frac{ \Big( e_{i}^{t}(\boldsymbol{s}^{t})+B_{i}^{t} + \delta_{i}^{t} \Big)^{2} }{C_i} \\
& \text{subject to}
& & \text{Constraints \eqref{loaderror}},
\end{aligned}
\end{equation*}
which corresponds to the worst-case ELI performance. We can show that the objective function of Problem IWCP is convex in the prediction errors $\boldsymbol{\delta}^{t}$. Hence, the optimal solution of the IWCP problem must reach the boundary of the uncertainty set in (<ref>). Moreover, as the total actual energy consumption $e_{i}^{t}(\boldsymbol{s}^{t})+B_{i}^{t} + \delta_{i}^{t}$ is always non-negative, hence the objective function of Problem IWCP is a monotonically increasing function in $\boldsymbol{\delta}^{t}$. Thus we have the following result:
The optimal solution of Problem IWCP, i.e., the worst-case prediction error, lies at the upper bounds of the uncertainty set, i.e. $\delta_i^{t,\ast} = \Delta_{i,\max}^{t},~\forall i \in \mathcal{N} $.
Hence, we substitute the worst-case prediction error $\boldsymbol{\delta}^{t,\ast} = \{\delta_i^{t,\ast},~\forall i \in \mathcal{N}\}$ into Problem WCP, and obtain the following worst-case optimization problem:
\begin{equation*}
\begin{aligned}
& \underset{\boldsymbol{s}^{t}} {\min}
& & \sum_{i \in \mathcal{N}}~~ \frac{ \Big( e_{i}^{t}(\boldsymbol{s}^{t})+B_{i}^{t} + \delta_i^{t,\ast} \Big)^{2} }{C_i} \\
& \text{subject to}
& & \text{Constraints \eqref{s_constraint1} and \eqref{s_constraint2}},
\end{aligned}
\end{equation*}
which solves the optimal billing references $\boldsymbol{s}^{t}$ to optimize the worst-case performance of ELI. Note that the above problem shares the same structure as Problem PS1, and thus can be solved by the same methodology presented in Section V.
§ SIMULATION RESULTS
In this section, we evaluate our proposed algorithms based on realistic system parameters, and compare the corresponding electric load index and energy cost between the solutions with that of the benchmark problems.
We consider four data centers that are geographically located in four different regions in the United States: New York, Maine, Rhode Island, and Boston. In each location, there is one data center powered by a power station. The numbers of servers in the four locations are 80000, 60000, 60000, and 80000, respectively. The service rates are 4, 3, 4 and 3 requests per server, and each server consumes 200watts electricity in the peak mode and 100watts when it is idle. We set power usage effectiveness as 1.5, 1.2, 1.2 and 1.5 for four data centers, respectively. We took hourly locational marginal prices and demands of the four locations on 4th March 2013 as the base prices and background power load, according to <cit.>. The dynamic computing requests are simulated based on the normalized workload trace of Google data centers on 20th December 2013 <cit.>.
§.§ Performance of the proposed algorithms
We first evaluate the optimal solutions of the two-stage problem and benchmark problems. The upper bound and lower bound for ELI over 24 hours are shown in Fig. <ref>. The optimal solution to the integrated problem provides a lower bound for ELI, and the optimal solution to the restricted problem provides an upper bound for ELI. Input the upper bound and lower bound into the branch-and-bound algorithm, we can solve the optimal solution to the two-stage problem. The corresponding optimal ELI lies in between the upper bound and lower bound, and is very close to the lower bound. Specifically, the optimal ELI is on average 1.5% higher than the lower bound across 24 hours.
In Fig. <ref>, the solid blue curve represents the optimal ELI performance of the branch-and-bound algorithm. The dash red curve represents the sub-optimal ELI performance obtained by the heuristic algorithm, which is close to the solid blue curve. This suggests that the heuristic algorithm achieves a performance close to the optimal result.
Upper and lower bounds for ELI.
Optimal and suboptimal ELI.
§.§ Effectiveness of optimized dynamic pricing
Base pricing and optimized dynamic pricing.
We demonstrate the effectiveness of our proposed dynamic pricing by comparing to the base pricing benchmark. Fig. <ref> shows the base prices (dash red curves) and the optimized dynamic prices (solid blue curves) for four data centers, respectively. We can see that the optimized prices may significantly deviate from the base prices. Take hour 10 for example, the optimized prices in location 1 and 4 are higher than the base prices, and in location 2 and 3 are lower than the base prices. This implies that the loads in location 1 and 4 are heavier than those in location 2 and 3. The utility company optimizes the prices for the data centers to re-distribute their energy consumption for load balancing.
In Fig. <ref>, the dash red curve represents the ELI of the base price benchmark, where the data centers are charged based on the fixed base prices. The solid blue curve represents the ELI with dynamic pricing, which shows that our proposed dynamic pricing scheme reduces ELI by an average of 4$\%$ across 24 hours comparing with the base pricing benchmark.
We also evaluate data centers' total energy cost over 24 hours, shown in Fig. <ref>. The energy cost with dynamic pricing is less than the base pricing benchmark. Specifically, the data centers reduce the total energy cost by an average of 28$\%$ across 24 hours, by taking advantage of dynamic prices and reallocating the workload. Fig. <ref> and Fig. <ref> show that the dynamic interactions between smart grid and data centers bring benefits to both sides and achieve a win-win situation.
Comparison of ELI.
Comparison of energy cost.
We examine the power load distribution within one particular hour (e.g., hour 24), and plot the background power load, power load of data centers, and the total load across four locations. Fig. <ref> shows the load distribution with base pricing. The data centers' load (the white bar) is not balanced, since data centers assign workload to the location with the lowest base price as much as possible to minimize the energy cost. The consequence is that the power load is extremely high in the lowest-price location 2, bringing a risk of overloading. Fig. <ref> shows the load distribution in the two-stage model with dynamic pricing. We can see the utility company tries to drive the load more evenly across different locations. Therefore, our proposed scheme can effectively improve the reliability of smart grid through re-balancing power load across different locations.
Power load (base pricing).
Power load (dynamic pricing).
§.§ Impact of prediction errors
We conduct a case study to show the impact of prediction errors on the ELI performance. We set the bounds ($\Delta_{i,\min}^{t}$ and $\Delta_{i,\max}^{t}$) of the prediction errors as $\pm 10 \%$ of the predicted values $B_i^t$ in location $i$ and time slot $t$. Solving problem WCP in Section VI, we obtain the optimized worst-case ELI performance as dash red curve in Fig. <ref>. We also randomly generate a realization of prediction errors, and compare the ELI performance under the scenario with and without considering the prediction errors. If prediction errors are considered when optimizing the Stage-1 problem, the realized ELI performance (solid blue curve) can be guaranteed to be better than the worst-case ELI. However, if the prediction is assumed to be accurate with zero error (while in reality it is not), then the ELI performance (dash black curve) can be even worse than the worst-case benchmark (e.g. in the 20th time slot). Therefore, the results demonstrate the effectiveness of our proposed worst-case performance optimization problem, which provides a performance guarantee for ELI under prediction errors.
ELI performance with prediction error.
§.§ Impact of computing workload
We study the impact of computing workload on the performance of dynamic pricing. We consider one time slot (hour 1), and keep all the simulation parameters unchanged except the computing workload. Fig. <ref> depicts the ELI performance comparison between the baseline pricing benchmark (fixed price without incentives) and the two-stage optimal dynamic pricing (with incentives), under different computing workload ranging from $0.6$ to $1.4$ of its original value. Fig. <ref> shows that ELI increases as the workload increases for both dynamic pricing and baseline pricing, and the two-stage optimal dynamic pricing always achieves a lower ELI than the baseline pricing regardless of the computing workload. Moreover, the load balancing improvement (measured by the percentage of ELI reduction) is relatively larger under intermediate computing workload than that under light or heavy computing workload. Specifically, dynamic pricing achieves $9.7\%$ in the ELI reduction when the workload is $0.9$ of the original value. When the workload is $0.6$ and $1.4$ of the original value, the percentage of ELI reduction decreases to $4.4\%$ and $1.8\%$, respectively. The reason is as follows. When the computing workload is light, the corresponding total energy consumption is low, and is not likely to cause power overloading. Therefore, the benefit due to the dynamic pricing is relatively small. When the computing workload is heavy, the cloud provider tends to fully utilize all the data centers to provision the quality of service to all the computing requests. There is little flexibility in terms of shifting workload across different data centers, and thus the benefit of dynamic pricing becomes small as well.
Impact of computing workload.
§ CONCLUSIONS
In this paper, we studied the dynamic interactions between smart grid and data centers as a two-stage price optimization problem. To solve the two-stage optimization problem, we reformulated it as a mixed integer quadratic programming problem, and proposed a branch-and-bound algorithm to attain the globally optimal solution, and a low complexity heuristic descent algorithm to yield a close-to-optimal solution. The simulation results showed a win-win solution for both the utility company and data centers.
For future work, we would like to study the interaction between the utility company and data centers with high penetration of renewable energy and under incomplete information. Some cloud provides installed renewable energy facilities to power data centers. How to manage the renewable-powered data centers and what is the impact on the power system are worth of study. The utility company may not able to acquire private information of data-center operation, so how to incentivize data centers with asymmetric information is an interesting and practical problem for future study.
H. Wang, J. Huang, X. Lin and H. Mohsenian-Rad, “Exploring smart grid and data center interactions for electrical power load balancing," in ACM Greenmetrics, Pittsburgh, PA, June 2013.
A. Qureshi, R. Weber, H. Balakrishnan, J. Guttag, and B. Maggs, “Cutting the electric bill for Internet-scale systems," in ACM SIGCOMM, 2009.
M. Pedram, “Energy-efficient datacenters," IEEE Trans. on Computer Aided Design, 31(10) 2012.
L. Rao, X. Liu, L. Xie, and W. Liu, “Minimizing electricity cost: optimization of distributed Internet data centers in a multi-electricity-market environment," in Proc. of IEEE INFOCOM, 2010.
Z. Liu, M. Lin, A. Wierman, S.H. Low, and L.L.H. Andrew, “Greening geographical load balancing," in Proc. of ACM SIGMETRICS, 2011.
M. Lin, Z. Liu, A. Wierman, L.L.H. Andrew, “Online algorithms for geographical load balancing," in Proc. of IEEE IGCC, 2012.
M. Ghamkhari and H. Mohsenian-Rad, “Energy and performance management of green data centers: a profit maximization approach," IEEE Trans. on Smart Grid, 4(2): 1017-1025, 2013.
P. Wang, L. Rao, X. Liu and Y. Qi, “D-pro: dynamic data center operations with demand-responsive electricity prices in smart grid," IEEE Trans. on Smart Grid, 3(4): 1743-1754, 2012.
L. Zhang, S. Ren, C. Wu, and Z. Li, “A truthful incentive mechanism for emergency demand response in colocation data centers,” in Proc. of IEEE INFOCOM, 2015.
H. Mohsenian-Rad and A. Leon-Garcia, “Energy-information transmission tradeoff in green cloud computing," in Proc. of IEEE Globecom, 2010.
H. Mohsenian-Rad, and A. Leon-Garcia, “Coordination of cloud computing and smart power grids," in Proc. of IEEE SmartGridComm, 2010.
X. Fang, S. Misra, G. Xue, and D. Yang, “Smart grid-the new and improved power grid: a survey," IEEE Communications Surveys and Tutorials, 14: 944-980, 2012.
H. Mohsenian-Rad and A. Leon-Garcia, “Optimal residential load control with price prediction in real-time electricity pricing environments," IEEE Trans. on Smart Grid, 1(2): 120-133, 2010.
H.K. Nguyen, J.B. Song, and Z. Han, “Demand side management to reduce peak-to-average ratio using game theory in smart grid," in Proc. of IEEE INFOCOM Workshops, 2012.
N. Li, L. Chen, and S.H. Low, “Optimal demand response based on utility maximization in power networks," in Proc. of IEEE PES General Meeting, 2011.
C.J. Wong, S. Ha, S. Sen, and M. Chiang, “Optimized day-ahead pricing for the smart grid with device-specific scheduling flexibility," IEEE Journal on Selected Areas in Communications, 1075-1085, 2012.
N.H. Tran, D.H. Tran, S. Ren, Z. Han, and C.S. Hong, “How geo-distributed data centers do demand response: a game-theoretic approach,” IEEE Trans. on Smart Grid, to appear.
H. Xu and B. Li, “Reducing electricity demand charge for data centers with partial execution," in Proc. of ACM e-Energy, 2014.
J. Lai, et al, “Evaluation of evolving residential electricity tariffs," Ernest Orlando Lawrence Berkeley National Laboratory, 2011.
L.A. Barroso and U. Holzle, “The case for energy-proportional computing,” Computer, 40: 33–37, 2007.
A. Qureshi, “Power-demand routing in massive geo-distributed systems,” Ph.D. dissertation, Massachusetts Institute of Technology, 2010.
A. Motamedi, H. Zareipour and W.D. Rosehart, “Electricity price and demand forecasting in smart grids," IEEE Trans. on Smart Grid, 3(2): 664-674, 2012.
M. Zugno, J. Morales, P. Pinson, and H. Madsen, “A bilevel model for electricity retailers participation in a demand response market environment," Energy Economics, 36: 182-197, 2013.
S. Boyd and L. Vandenberghe, “Convex optimization," New York, NY, USA: Cambridge University Press, 2004.
Y. Yuan, Z. Li, and K. Ren, “Quantitative analysis of load redistribution attacks in power systems,” IEEE Trans. on Parallel and Distributed Systems, 23(9): 1731-1738, 2012.
J.F. Bard, “Practical bilevel optimization: algorithms and applications," Kluwer Academic Publishers, 1998.
A. Ben-tal and A. Nemirovski, “Selected topics in robust convex optimization,” Mathematical Programming, 112(1): 125-158, 2008.
Z. Sun, F. Kong, X. Liu, et. al., “Intelligent joint spatio-temporal management of electric vehicle charging and data center power consumption," in Proc. of IGCC, 2014.
|
1511.00429
|
This paper is concerned with steady, fully developed motion of a Navier-Stokes fluid with shear-dependent viscosity in a curved pipe under a given axial pressure gradient. We establish existence and uniqueness results, derive appropriate estimates and prove a characterization of the secondary flows. The approximation, with respect to the curvature ratio, of the full governing systems by some Dean-like equation is studied.
Key words. Navier-Stokes fluids, shear-dependent viscosity, shear-thinning flows, shear-thickening flows, curved pipes.
AMS Subject Classification. $35$J$65$, $76$A$05$, $76$D$03$.
§ INTRODUCTION
There is a great interest in the study of curved pipe flows due to its wide range of applications in engineering (e.g. hydraulic pipe systems related to corrosion failure) and in biofluid dynamics, such as blood flow in the vascular system. It is known since the pioneer experimental works of Eustice (<cit.>, <cit.>) that these flows are challenging and much more complex than flows in straight pipes. Among their distinguishing features is the existence of
secondary flows induced by the centrifugal force and which appear even for the slightest curvature.
Fully developed viscous flow in a curved pipe with circular cross-section was first studied theoretically by Dean (<cit.>, <cit.>) in the case of Newtonian fluids by applying regular perturbation methods, the perturbation parameter being the curvature ratio. He simplified the governing equations, by neglecting all the effects due to pipe curvature except the centrifugal forces,
and showed that for small curvature ratio the flow depends only on a single parameter, the so-called Dean number.
Following this fundamental work, the results based on perturbation solutions have been extended for a larger range of curvature ratio and Reynolds number, showing in particular the existence of additional pairs of vortices and multiple solutions (see e.g.
<cit.>, <cit.>). Different geometries including circular, elliptical and annular cross-sections have also been considered by several authors (see e.g. <cit.>, <cit.>, <cit.>, <cit.>, <cit.>).
Despite the great interest in curved pipes, a rigorous analysis of the solvability of the Dean's equations and the full Navier-Stokes equations was not available prior to the work of Galdi and Robertson <cit.>, where existence and uniqueness results for steady, fully developed flows are established. Existence of secondary motions is also studied, as well as the approximation of the Navier-Stokes equations
by the Dean's equation for small curvature ratios.
Flows of non-Newtonian fluids in curved pipes have also been studied by several authors (see e.g. <cit.>, <cit.>, <cit.>, <cit.>). Perturbation methods were used by Robertson and Muller <cit.> to study steady, fully developed flow of Oldroyd-B fluids, and to compare
the results for creeping and non-creeping flows. For a second order model, Jitchote and Robertson <cit.> obtained analytical solutions to the perturbation equations and analyze the effects of non-zero second normal stress coefficient on the behaviour of the solution. Theoretical results regarding
this problem were obtained by Coscia and Robertson in <cit.>, where existence and uniqueness of a strong solution for small non-dimensional pressure drop is established.
The aim of the present paper is to extend the analysis carried-out in <cit.> to the class of quasi-Newtonian fluids.
This class is described by partial differential equations of the quasilinear type (Navier-Stokes equations with a non constant viscosity that decreases with increasing shear rate in the case of shear-thinning flows and increases with increasing shear rate in the case of shear-thickening flows). It was first proposed and studied in bounded domains by Ladyzhenskaya in <cit.>, <cit.> and <cit.> as a modification of the Navier-Stokes system, and was similarly suggested by Lions in <cit.>. Existence of weak solutions was proved by both authors using compactness arguments and the theory of monotone operators. Much work has been done since these pioneering results and, without ambition for completeness, we cite Nečas et al. who established existence of weak solutions under less restrictive assumptions (see for example <cit.>).
Since we are dealing with fully developed flows in curved pipes, the typical issues related to the nonlinear extra stress tensor and the convective term arise and can be handled as in the case of bounded domains. However, additional difficulties related with extra terms (depending on the curvature ratio) occur and need to be managed. The splitting method, consisting in two coupled formulations respectively associated to the secondary flows and to the axial flow and used in <cit.> for the study of the Newtonian case, cannot be applied. Because of the nonlinearity of the shear stress tensor, the coercivity property of the corresponding bilinear forms in the shear-thinning case, and the monotonicity property of these forms in both shear-thinning and shear thickening cases fail to be satisfied. To overcome these difficulties, we consider a global formulation in an appropriate functional setting and adapt some standard tools, such as the Korn inequality. An existence result is established for arbitrary values of the Reynolds number, of the pressure drop and for any curvature ratio of the pipe, and a uniqueness result for small Reynolds numbers. The global formulation allows also to derive uniform estimates independent of the Dean number. Using a posteriori the splitting method, we establish other estimates for the secondary flows that highlight the connection with the Dean number. Following <cit.>, we also prove that there are no secondary flows if, and only if, the Dean number is equal to zero. Finally, we consider an approximation problem (that can be seen as a generalization of the Dean's equation), study its solvability, establish corresponding estimates and evaluate the relationship between its solutions and those of the full governing equations.
The plan of the paper is as follows. The governing equations are given in Section 2. Notation, and some preliminary results are given in Section 3. Section 4 is devoted to the statement and discussion of the main results.
In Section 5, we consider the case of the shear-thickening flows. We establish a version of the Korn inequality more appropriate for our framework and derive some estimates for the convective term and the extra stress tensor. The existence and uniqueness results for the full governing equations are then established and the approximation analysis is carried out. The shear-thinning case is treated in a similar way in Section 6.
§ GOVERNING EQUATIONS
We are concerned with steady flows of incompressible generalized Newtonian
fluids. For these fluids, the Cauchy stress tensor $\widetilde {\boldsymbol T}$ is related to the kinematic variables through
{\boldsymbol T}=-\widetilde \pi I+2\mu\left(1+|\widetilde D\widetilde { \boldsymbol u}|^2
\right)^{\frac{p-2}{2}}\widetilde D\widetilde { \boldsymbol u},$$
where $\widetilde {\boldsymbol u }$ is the velocity field,
$\widetilde D\widetilde { \boldsymbol u}=\frac{1}{2}\left(\widetilde\nabla \widetilde { \boldsymbol u}+\widetilde\nabla \widetilde { \boldsymbol u}^\top\right)$ denotes the symmetric part of the velocity gradient, $\mu >0$ is the kinematic viscosity, and $\widetilde\pi$ represents the pressure. The notation $\sim$ denotes a dimensional quantity.
The equations of conservation of momentum and mass, relative to a rectangular coordinate system, are
\begin{equation}\label{equation_dim}\left\{
\begin{array}{ll}\rho \left(\tfrac{\partial \widetilde { \boldsymbol u}}{\partial \widetilde t}+\widetilde { \boldsymbol u}\cdot \widetilde\nabla \widetilde { \boldsymbol u}\right)
+\widetilde\nabla \widetilde \pi = \widetilde\nabla\cdot\left(2\mu\left(1+|\widetilde D\widetilde { \boldsymbol u}|^2
\right)^{\frac{p-2}{2}}\widetilde D\widetilde { \boldsymbol u}\right),\vspace{3mm}\\
\widetilde\nabla\cdot \widetilde { \boldsymbol u} = 0,\end{array}
\right.\end{equation}
where $\rho>0$ is the constant density of the fluid. In this work, we consider steady flow of generalized Newtonian fluids through a curved pipe of arbitrary shaped cross-section $\Sigma$ with constant centerline radius $R$. Due to the geometric characteristics of the curved pipe, it is convenient to write system $(\ref{equation_dim})$ in the rectangular toroidal
coordinates $(\widetilde x_i)$ defined with respect to the rectangular Cartesian coordinates $(\widetilde y_i)$ through the relations
\begin{equation}\label{transf_1}\widetilde x_1=\widetilde y_3,\qquad \widetilde x_2=\sqrt{\widetilde y_1^2+\widetilde y_2^2}
-R,\qquad \widetilde x_3=
\tfrac{\widetilde y_2}{\widetilde y_1}
\end{equation}
and inverse relations
\begin{equation}\label{transf_2}\widetilde y_1=\left(R+\widetilde x_2\right)\cos\left(\tfrac{\widetilde x_3}{R}\right),\qquad
\widetilde y_2=\left(R+\widetilde x_2\right)\sin\left(\tfrac{\widetilde x_3}{R}\right),\qquad
\widetilde y_3=\widetilde x_1.\end{equation}
More details on the toroidal coordinate system, and on the corresponding formulation of the operators involved in (<ref>), can be found in Appendix <ref>.
Ł(0.85*0.55) $\widetilde{x}_{1}$
Ł(0.915*0.29) $\widetilde{x}_{2}$
Ł(0.755*0.40) $\widetilde{x}_{3}$
\strut\AffixLabels{\mbox{\psfig{figure=./coord_rect.eps,width=4.5cm}}}
Rectangular toroidal coordinates in a curved pipe
We restrict our study to curved pipes with cross section independent of $\tilde x_3$, and consider flows which are steady and fully developed, i.e. the components of the velocity vector with respect to the new basis are independent of both time and axial variable $x_3$. For such flows
\begin{equation}\label{fdev_velocity}\tfrac{\partial \widetilde u_i}{\partial \widetilde x_3}
=0 \qquad i=1,2,3\end{equation}
and the axial component of the pressure gradient
\begin{equation}\label{fdev_pressure}\tfrac{\partial \widetilde \pi}{\partial \widetilde x_3} =-\widetilde G \end{equation}
is a constant. We consider the non-dimensional form of this system by
introducing the following quantities
\begin{equation}\label{adim_var}r_0=\sup_{\widetilde
x\in\overline\Sigma}|\widetilde x|, \qquad
x_i=\tfrac{\widetilde x_i}{r_0},\qquad \boldsymbol u =\tfrac{\widetilde {\boldsymbol u }}{U_0}, \qquad
\pi=\tfrac{\widetilde \pi r_0}{\mu U_0}, \qquad G=\tfrac{\widetilde G r_0^2}{\mu U_0},\end{equation}
where $\delta\in [0,1[$ is the pipe curvature ratio, and
$U_0$ represents a characteristic velocity of the flow.
We also introduce the Reynolds number ${\cal R}e=\tfrac{\rho U_0 r_0}{\mu}$ and the pipe curvature ratio $\delta=\tfrac{r_0}{R}\in [0,1[$.
The governing equations can then be written with respect to the non-dimensional quantities as
\begin{equation}\label{equation}
\left\{ \begin{array}{ll} -\nabla^\star \cdot\left({\boldsymbol\tau}(D^\star{\boldsymbol u})
\right)+{\cal R}e \,
{\boldsymbol u}\cdot \nabla^\star {\boldsymbol u}+ \nabla^\star {\pi} =0 &\quad \mbox{in} \
\Sigma,\vspace{3mm} \\
\nabla^\star \cdot {\boldsymbol u}=0&\quad \mbox{in} \
\Sigma,\vspace{3mm} \\
\boldsymbol u=0 &\quad \mbox{on} \ \partial\Sigma\end{array}\right.
\end{equation}
$${\boldsymbol\tau}(D^\star{ \boldsymbol u})=2\left(1+ \dot{\gamma}^2|D^\star{ \boldsymbol u}|^2
\right)^{\frac{p-2}{2}}D^\star{ \boldsymbol u},$$
where $\dot{\gamma}=\tfrac{U_0}{r_0}$ is a caracteristic shear-rate. In order to simplify the redaction, we will assume without lost of generality that $\dot{\gamma}=1$. The operators involved in the definition of (<ref>), are defined by
\nabla^\star\cdot {\boldsymbol\tau}=\nabla \cdot {\boldsymbol\tau}+
\tfrac{\delta}{B} \left({\tau}_{12}\,{\boldsymbol a }_1+
\left({\tau}_{22}-{\tau}_{33}\right)\, {\boldsymbol a }_2+2{\tau}_{23}\, {\boldsymbol a }_3
\right),\vspace{3mm}\\
\nabla^\star{ \boldsymbol u}=\nabla { \boldsymbol u}+\tfrac{\delta}{B}\left(u_2\,
{\boldsymbol a }_3 \otimes{\boldsymbol a }_3-u_3\, {\boldsymbol a }_3\otimes{\boldsymbol a }_2\right),\vspace{3mm}\\
D^\star{ \boldsymbol u}=\tfrac{1}{2}
\left(\nabla^\star{ \boldsymbol u}+\nabla^\star{ \boldsymbol u}^\top\right),\vspace{3mm}\\
{ \boldsymbol u}\cdot \nabla^\star
{ \boldsymbol u}={ \boldsymbol u}\cdot \nabla
{ \boldsymbol u}+\tfrac{\delta}{B}\left(u_3u_2\, {\boldsymbol a }_3-
u_3^2\,{\boldsymbol a }_2\right),\vspace{3mm}\\
\nabla^\star {\pi}=\nabla \pi-\tfrac{G}{B}\,{\boldsymbol a }_3\vspace{3mm}\\
\nabla^\star \cdot { \boldsymbol u}=\tfrac{\partial u_1}{\partial x_1}+\tfrac{\partial u_2}{\partial x_2}+\tfrac{\delta}{B}\, u_2=\tfrac{1}{B}\, \nabla\cdot\left(B\boldsymbol u\right)\end{array}$$
$$\nabla=\big(\tfrac{\partial}{\partial x_1},\tfrac{\partial}{\partial x_2},0\big), \qquad \nabla \cdot=\tfrac{\partial}{\partial x_1}+\tfrac{\partial}{\partial x_2},\qquad B=1+\delta x_2,$$
and where $({\boldsymbol a }_1,{\boldsymbol a }_2,{\boldsymbol a }_3)$
denotes the orthonormal basis in the toroidal coordinates. (For a detailed derivation of the dimensionless equation (<ref>), see Appendix <ref>.)
§ NOTATION, ASSUMPTIONS AND PRELIMINARY RESULTS
§.§ Algebraic results
Throughout the paper, if ${ \boldsymbol u}=(u_1,u_2,u_3)$ in the rectangular toroidal coordinates, we denote by $u$ the vector with toroidal components $(u_1,u_2,0)$. Similarly, if $\boldsymbol S=(S_{ij})_{i,j=1,2,3}$ is a tensor in $\mathbb{R}^{3\times 3}$, we denote by $S$ the tensor in $\mathbb{R}^{2\times 2}$ with toroidal components $S_{ij}$, $i=1,2$.
For $\eta, \zeta\in \mathbb{R}^{d\times d}$, we define the scalar product and the corresponding norm by
$$\eta:\zeta=\sum_{i,j=1}^d \eta_{ij}\zeta_{ij} \quad
\mbox{and} \quad \left|\eta\right|=
\left(\eta:\eta\right)^{\frac{1}{2}}.$$
In the next two results, we state well known continuity, coercivity and monotonicity properties for $\boldsymbol\tau$. The corresponding proof is standard (see e.g. <cit.>). We first consider the case $p\geq 2$ corresponding to the shear-thickening flows.
Assume that $p\geq 2$ and let $\eta\in \mathbb{R}_{\mathrm{sym}}^{3\times3}$. Then the tensor $\boldsymbol\tau$ satisfies the following properties
* Continuity.
(p-1)\left(1+ |\eta|^2+|\zeta|^2\right)^{\frac{p-2}{2}}
* Coercivity.
$${\boldsymbol\tau}(\eta):\eta\geq 2|\eta|^2,\qquad \quad
{\boldsymbol\tau}(\eta):\eta\geq 2|\eta|^p, $$
* Monotonicity.
\left({\boldsymbol\tau}(\eta)-{\boldsymbol\tau}(\zeta)\right):\left(\eta-\zeta\right)\geq\left|\eta-\zeta\right|^2,\vspace{3mm}\\
\left({\boldsymbol\tau}(\eta)-{\boldsymbol\tau}(\zeta)\right):\left(\eta-\zeta\right)\geq \tfrac{1}{2^{p-1}(p-1)}
\left|\eta-\zeta\right|^p.\end{array}\right.$$
Next we consider the case $1<p<2$ corresponding to the shear-thinning flows.
Assume that $1<p<2$ and let $\eta\in \mathbb{R}_{\mathrm{sym}}^{3\times3}$. Then the tensor $\boldsymbol\tau $ satisfies the following properties
* Continuity.
$$\left|{\boldsymbol\tau }(\eta)-\boldsymbol\tau (\zeta)\right|\leq
C_p|\eta-\zeta|^{p-1} \qquad \mbox{with} \
* Coercivity.
$${\boldsymbol\tau }(\eta):\eta\geq
* Monotonicity.
$$\left({\boldsymbol\tau }(\eta)-{\boldsymbol\tau }(\zeta)\right):
\left(\eta-\zeta\right)
\geq 2(p-1)\left(1+ |\eta|^2+ |\zeta|^2
\right)^{\frac{p-2}{2}}\left|\eta-\zeta\right|^2.$$
§.§ Functional setting
Throughout the paper $\Sigma$ is a bounded domain in $\mathbb{R}^2$, with a boundary $\partial \Sigma$. Even though several of our results are valid for an arbitrary bounded domain, we will assume without loss of generality that $\Sigma$ is of class $C^2$.
By $\boldsymbol W^{k,p}(\Sigma)$ ($k\in \mathbb{N}$ and $1<p<\infty$), we denote the standard Sobolev spaces
and we denote the associated norms by $\|\cdot\|_{k,p}$. We set
$\boldsymbol W^{0,p}(\Sigma)\equiv \boldsymbol L^{p}(\Sigma)$, $\|\cdot\|_{L^{p}}\equiv \|\cdot\|_{p}$, $\boldsymbol L^{p}_0(\Sigma)=\left\{\boldsymbol u\in L^{p}(\Sigma)\mid \int_\Sigma \boldsymbol u(x)\,dx=0\right\}$, and
$$\left\|\cdot\right\|_{p,B}=\left\|B^{\frac{1}{p}}\cdot\right\|_{p}, \qquad m=\left\|\tfrac{1}{B}\right\|_\infty \qquad \mbox{and} \qquad
The dual space of $\boldsymbol W^{1,p}_0(\Sigma)$ is denoted by
$\boldsymbol W^{-1,p'}(\Sigma)$, where $p'=\tfrac{p}{p-1}$ is the dual exponent to $p$,
its norm is denoted by $\|\cdot\|_{-1,p'}$ and the duality pairing between these spaces by $\langle\cdot,\cdot\rangle$.
We will also use the following notation
$$\left(\boldsymbol u,\boldsymbol v\right)=\int_\Sigma { \boldsymbol u}(x)\cdot { \boldsymbol v}(x)\,dx, \qquad { \boldsymbol u}\in L^{p}(\Sigma), \ { \boldsymbol v}\in
\boldsymbol L^{p'}(\Sigma),$$
$$\left(\eta,\zeta\right)=\int_\Sigma \eta(x):\zeta(x)\,dx, \qquad \eta\in
\boldsymbol L^p(\Sigma,\mathbb{R}^{3\times 3}), \ \zeta\in
\boldsymbol L^{p'}(\Sigma,\mathbb{R}^{3\times 3}).$$
The space of infinitely differentiable functions with compact support in $\Sigma$ will be denoted by ${\cal D}(\Sigma)$. In order to eliminate the pressure in the weak formulation of our equation, we will work in divergence-free spaces. Consider
$$ {\cal V}=\big\{\boldsymbol\varphi\in {\cal D}(\Sigma)
\mid \nabla\cdot \boldsymbol\varphi=0\big\},$$
$${\cal V}_B=\big\{\boldsymbol\varphi\in {\cal D}(\Sigma)
\mid \nabla \cdot \left(B\boldsymbol\varphi\right)=0\big\},$$
and denote by $\boldsymbol V^{p}$ and $\boldsymbol V_B^p$ the closure of ${\cal V}$ and ${\cal V}_B$ in the $L^p$-norm of gradients, i.e.
$${\boldsymbol V}^{p}=
\left\{\boldsymbol\varphi\in \boldsymbol W_0^{1,p}(\Sigma)
\mid \nabla\cdot \boldsymbol\varphi=0\right\}, $$
$${\boldsymbol V}^{p}_B=
\left\{\boldsymbol\varphi\in \boldsymbol W_0^{1,p}(\Sigma)
\mid \nabla\cdot \left(B\boldsymbol\varphi\right)=0\right\}.$$
Next, we recall the Sobolev inequality.
Let $r$ and $q$ be such that $r\geq q$ if $q\geq 2$ and
$1<r\leq q^\ast=\tfrac{2q}{2-q}$ if $1<q<2$. Then for all
all $ \boldsymbol u\in \boldsymbol W^{1,q}_0(\Sigma)$, we have
\begin{equation}\label{sobolev0}
\left\|{ \boldsymbol u}\right\|_r\leq S_{q,r}
\left\|\nabla \boldsymbol u\right\|_q,
\end{equation}
\tfrac{\max\left(q,\frac{r}{2}\right)}{2\sqrt{2}}\, |\Sigma|^{\frac{1}{2}+\frac{1}{r}-\frac{1}{q}}& \quad \mbox{if} \ r\geq q \ \mbox{and} \
q\geq 2, \vspace{2mm}\\
\tfrac{q^\ast}{4\sqrt{2}}\,
|\Sigma|^{\frac{1}{2}+\frac{1}{r}-\frac{1}{q}}& \quad \mbox{if} \ 1<r\leq q \ \mbox{and} \ 1<q<2,\vspace{2mm}\\
\tfrac{\max\left(q,\frac{r}{2}\right)}{2\sqrt{2}}\, |\Sigma|^{\frac{1}{2}+\frac{1}{r}-\frac{1}{q}}& \quad \mbox{if} \ q<r\leq q^\ast \ \mbox{and} \ 1<q<2.\end{array}\right.
Proof. The Sobolev inequality (<ref>) is classical and follows by combining the following interpolation (see Lemma II.3.2 in <cit.>)
$$\left\|\boldsymbol u\right\|_r\leq \left(\tfrac{\max\left(q,\frac{r}{2}\right)}{2\sqrt{2}}\right)^\lambda \left\|\boldsymbol u\right\|_q^{1-\lambda}
\left\|\nabla\boldsymbol u\right\|_q^\lambda \qquad \mbox{with} \ \lambda=\tfrac{2(r-q)}{rq} $$
with the inequality $\left\|\boldsymbol u\right\|_q\leq |\Sigma|^{\frac{1}{q}-\frac{1}{r}} \left\|\boldsymbol u\right\|_r$.$
\hfill\Box$
Next, we consider a particular version of the Poincaré inequality.
For all ${ \boldsymbol u}\in { \boldsymbol W}^{1,q}_0(\Sigma)$, $1\leq q\leq +\infty$, the following estimate holds
\begin{equation} \label{poincare}\left\|{ \boldsymbol u}\right\|_q\leq
\left\|\tfrac{\partial { \boldsymbol u}}{\partial x_1}\right\|_q.\end{equation}
Proof. The result follows from Theorem II.5.1 in <cit.> by observing that
$\Sigma\subset ]-1,1[\times \mathbb{R}$.$\hfill\Box$
As well known (see <cit.>), the standard Poincaré inequality is given by
\begin{equation}
\label{poincare_opt}\left\|{ \boldsymbol u}\right\|_q\leq
S_{q,q} \left\|\nabla { \boldsymbol u}\right\|_q \end{equation}
and consequently, the Poincaré constant in $(\ref{poincare})$ is not necessarily optimal. However, since $B$ is independent of the variable $x_1$, a direct consequence of $(\ref{poincare})$ is that
\begin{equation}\label{poincare2}\left\|{ \boldsymbol u}\right\|_{q,B}\leq \left\|\tfrac{\partial { \boldsymbol u}}{\partial x_1}\right\|_{q,B}\leq
\left\|\nabla { \boldsymbol u}\right\|_{q,B}.\end{equation}
This property is particularly useful to establish the Korn inequality in the shear-thinning case.
Finally, we establish some estimates useful in the approximation analysis of the solutions with respect to the parameter $\delta$.
Let $u$, $v$ be in $W^{1,q}_0(\Sigma)$ with $q\geq 2$, and let $\sigma$ be a continous function such that
\begin{equation}\label{psi_assumption}|\sigma(\lambda)|\leq c_0|\lambda|^\alpha
\qquad \mbox{with} \ c_0>0.\end{equation}
Then for every $\alpha \geq 0$, we have
\begin{equation}
\label{est_psi_1}\left|\left(\sigma(u),v\right)\right|\leq c_0 D_{q,\alpha}
\left\|\nabla u\right\|_{q}^\alpha
\left\|\nabla v\right\|_{q},\end{equation}
\begin{equation} \label{est_psi_2}\left\|\sigma(u)\right\|_{q'}\leq c_0 E_{q,\alpha}
\left\|\nabla u\right\|_2^\alpha,\end{equation}
where $D_{q,\alpha}=\max\left(\tfrac{q}{2\sqrt{2}},\tfrac{\alpha q}{4\sqrt{2}}\right)^{\alpha+1} |\Sigma|^{\frac{1}{q'}+\frac{\alpha+1}{2}-
\frac{\alpha}{q}}$ and $E_{q,\alpha}=\max\left(\tfrac{1}{\sqrt{2}},\tfrac{\alpha}{2\sqrt{2}}\right)^{\alpha} |\Sigma|^{\frac{1}{q'}}$.
Proof. Let us first assume that $\alpha\geq 1$. Then, by using the Hölder inequality and the Sobolev inequality (<ref>) we obtain
\left|\left(\sigma(u),v\right)\right|&\leq
\left\|\sigma(u)\right\|_{q}\|v\|_{q'}\leq c_0
\left\|u\right\|_{\alpha q}^\alpha\|v\|_{q'}\leq c_0
|\Sigma|^{\frac{1}{q'}-\frac{1}{\alpha q}}\left\|u\right\|_{\alpha q}^\alpha\|v\|_{\alpha q}\vspace{2mm}\\
&\leq c_0 |\Sigma|^{\frac{1}{q'}-\frac{1}{\alpha q}}
\left( S_{q,\alpha q}\right)^{\alpha+1}
\left\|\nabla u\right\|_{q}^\alpha \|\nabla v\|_{q}\end{array}$$
which gives (<ref>). Similarly,
&\leq |\Sigma|^{\frac{1}{q'}-\frac{1}{2}}
\left\|\sigma(u)\right\|_{2}
\leq c_0 |\Sigma|^{\frac{1}{q'}-\frac{1}{2}}
\left\|u\right\|_{2\alpha}^\alpha\vspace{2mm}\\
&\leq c_0|\Sigma|^{\frac{1}{q'}-\frac{1}{2}}
\left( S_{2,2\alpha}\right)^{\alpha}
\left\|\nabla u\right\|_{2}^\alpha\end{array}$$
and we obtain (<ref>).
If $\alpha\leq 1$ then
\left|\left(\sigma(u),v\right)\right|&\leq
\left\|\sigma(u)\right\|_{\frac{q}{\alpha}}\|v\|_{\left(\frac{q}{\alpha}\right)'}\leq c_0
\left\|u\right\|^\alpha_{q}\|v\|_{\frac{q}{q-\alpha}}\leq c_0 |\Sigma|^{\frac{1}{q'}-\frac{\alpha}{q}}
\left\|u\right\|^\alpha_{q}\|v\|_{q}\vspace{2mm}\\
&\leq c_0 |\Sigma|^{\frac{1}{q'}
-\frac{\alpha}{q}}\left( S_{q,q}
\right)^{\alpha+1} \left\|\nabla u\right\|_{q}^{\alpha}
\|\nabla v\|_{q}\end{array}$$
&\leq |\Sigma|^{\frac{1}{q'}-\frac{\alpha}{2}}
\left\|\sigma(u)\right\|_{\frac{2}{\alpha}}
\leq c_0 |\Sigma|^{\frac{1}{q'}-\frac{\alpha}{2}}
\left\|u\right\|^\alpha_{2}\vspace{2mm}\\
&\leq c_0 |\Sigma|^{\frac{1}{q'}-\frac{\alpha}{2}}\left( S_{2,2}
\right)^{\alpha} \left\|\nabla u\right\|_{2}^{\alpha}\end{array}$$
and the claimed result is proved.$\hfill\Box$
Let $u,v\in W^{1,q}_0(\Sigma)$ with $\tfrac{3}{2}\leq q<2$, and let $\sigma$ be a continous function satisfying
$(\ref{psi_assumption})$. Then
\begin{equation}\label{est_psi_1_thin}\left|\left(\sigma(u),v\right)\right|\leq c_0 D_{q,\alpha}
\left\|\nabla u\right\|_{q}^\alpha
\left\|\nabla v\right\|_{q} \qquad \mbox{for every} \ \
0\leq \alpha\leq q^\ast-1,\end{equation}
\begin{equation}\label{est_psi_2_thin}
\left\|\sigma(u)\right\|_{q'}\leq c_0 E_{q,\alpha}
\left\|\nabla u\right\|_q^\alpha\qquad \mbox{for every} \ \ \tfrac{1}{q'}
< \alpha\leq \tfrac{q^\ast}{q'},\end{equation}
where $D_{q,\alpha}=\left( S_{q,q^\ast}\right)^{\alpha+1}$, $E_{q,\alpha}=\left( S_{q,\alpha q'}\right)^{\alpha}$ and $q^\ast=\tfrac{2q}{2-q}$.
Proof. Notice first that the Sobolev inequality (<ref>) is valid for $1<r\leq q^\ast$ if $1\leq q<2$. If $0<\alpha\leq q^\ast-1$, then
$\sigma(u) v$ belongs to $L^1(\Sigma)$ and by using (<ref>), we obtain
\left|\left(\sigma(u),v\right)\right|&\leq c_0
\left\|u^\alpha\right\|_{\frac{q^\ast}{\alpha}}
\|v\|_{q^\ast}\leq c_0
\left\|u\right\|^\alpha_{q^\ast}
\|v\|_{q^\ast}\vspace{2mm}\\
&\leq c_0 \left( S_{q,q^\ast}\right)^{\alpha+1}
\left\|\nabla u\right\|_{q}^\alpha
\left\|\nabla v\right\|_{q}\end{array}$$
which gives (<ref>).
To prove the last estimate, we observe that if $\tfrac{1}{q'}< \alpha\leq \tfrac{q^\ast}{q'}$ then $1<\alpha q'\leq q^\ast$. By using (<ref>) we obtain
$$\left\|\sigma(u)\right\|_{q'}\leq c_0
\left\|u\right\|_{\alpha q'}^\alpha
\leq c_0 \left( S_{q,\alpha q'}\right)^{\alpha}
\left\|\nabla u\right\|_{q}^\alpha$$
and the claimed result is proved.$\hfill\Box$
§ WEAK FORMULATION AND STATEMENT OF THE MAIN RESULTS
To give a sense to the weak solution of (<ref>), let us recall that $B=1+\delta x_2$ does not depend on $x_1$ and notice that if $\boldsymbol S$ is a symmetric tensor, then we have
\left(\nabla^\star \cdot \boldsymbol S\right)_{1}& =
\tfrac{\partial S_{11}}{\partial x_1}+
\tfrac{\partial S_{21}}{\partial x_2}
\left(\tfrac{\partial \left(B S_{11}\right)}{\partial x_1}+
\tfrac{\partial \left(B S_{21}\right)}{\partial x_2}\right)\vspace{1mm}\\
\tfrac{1}{B}\left(\nabla\cdot
\left(B\boldsymbol S\right)\right)_1,\vspace{2mm}\\
\left(\nabla^\star \cdot \boldsymbol S \right)_2 &=
\tfrac{\partial S_{12}}{\partial x_1}+
\tfrac{\partial S_{22}}{\partial x_2}+
\tfrac{\delta}{B}\left(S_{22}-S_{33}\right)
=\tfrac{1}{B}\left(\tfrac{\partial \left(B S_{12}\right)}{\partial x_1}+
\tfrac{\partial \left(BS_{22}\right)}{\partial x_2}\right)-\tfrac{\delta}{B}\,S_{33}\vspace{1mm}\\
\tfrac{1}{B}\left(\nabla\cdot
\left(B\boldsymbol S\right)\right)_2-
\tfrac{\delta}{B}\, S_{33},\vspace{2mm}\\
\left(\nabla^\star \cdot \boldsymbol S \right)_3 &=
\tfrac{\partial S_{13}}{\partial x_1}+
\tfrac{\partial S_{23}}{\partial x_2}+
\tfrac{2\delta}{B}\,S_{23}=\tfrac{1}{B}\left(\tfrac{\partial \left(B S_{13}\right)}{\partial x_1}+
\tfrac{\partial \left(B S_{23}\right)}{\partial x_2}\right)+\tfrac{\delta}{B}\,S_{23}\vspace{1mm}\\
\tfrac{1}{B}\left(\nabla\cdot
\left(B\boldsymbol S\right)\right)_3+
\tfrac{\delta}{B}\,S_{32},
\end{array}$$
that is
$$\nabla^\star \cdot \boldsymbol S=\tfrac{1}{B}
\nabla\cdot \left(B\boldsymbol S\right)
+\tfrac{\delta}{B}\left(S_{23}\mathbf a_3-S_{33}
\mathbf a_2\right).$$
Therefore, if $\boldsymbol S$ belongs to in
$\boldsymbol L^{p'}(\Sigma)$ and
${\boldsymbol\varphi}=(\varphi_1,\varphi_2,\varphi_3)\in \boldsymbol V^p_B$, then an integration by parts shows that
$$\begin{array}{ll}-\left(\nabla^\star\cdot \boldsymbol S,B{\boldsymbol\varphi}\right)
\left(B \boldsymbol S\right),B{\boldsymbol\varphi}\right)
=-\left(\nabla^\star\cdot \left(B \boldsymbol S\right),
&=-\left(\nabla \cdot \left(B\boldsymbol S\right),
\delta\left(S_{23},\varphi_3\right)+
\delta\left(S_{33},\varphi_2\right)\vspace{1mm}\\
\left(\tfrac{\partial (B S_{1i})}{\partial x_1}
+\tfrac{\partial (B S_{2i})}{\partial x_2},
\varphi_i\right)
\delta\left(S_{33},\varphi_2\right)\vspace{1mm}\\
&=\displaystyle\sum_{i=1}^3\sum_{j=1}^2\left(B S_{ij},
\tfrac{\partial \varphi_i}{\partial x_j}\right)
\delta\left(S_{33},\varphi_2\right)
=\left(\boldsymbol S,B D^\star{\boldsymbol\varphi}\right).
\end{array}$$
By taking the inner product of $(\ref{equation})_1$ and
$B\boldsymbol\varphi$ and integrating over $\Sigma$, we obtain the following weak formulation.
Assume that $p\geq \tfrac{3}{2}$. A function ${ \boldsymbol u}\in \boldsymbol V^p_B$ is a weak solution of $(\ref{equation})$ if
\begin{equation}\label{weak_formulation}\left({\boldsymbol\tau}(D^\star{ \boldsymbol u}),B D^\star{\boldsymbol\varphi}\right)+{\cal R}e
\left(B{ \boldsymbol u}\cdot \nabla^\star{ \boldsymbol u},
{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right)\qquad \mbox{for all} \ {\boldsymbol\varphi}\in {\boldsymbol V}_B^p.
\end{equation}
As in the case of bounded domains, this definition in meaningful for $p\geq \tfrac{3}{2}$ and will be used when considering both shear-thinning and shear-thickening flows. The restriction on the exponent $p$ ensures that the convective term belongs to $\boldsymbol L^1$ when considering test functions in $\boldsymbol V_B^p$.
This formulation can be splitted into two coupled formulations, respectively associated to $u=(u_1,u_2,0)$ and to $(0,0,u_3)$. Indeed, by setting $\boldsymbol{\varphi}=(\varphi_1,\varphi_2,0)$ in (<ref>), we can easily see that $u$ satisfies
\begin{equation}\label{weak_formulation_u}
\left(\tau(D^\star{ \boldsymbol u}),B D\varphi\right)+
\delta\left({\tau}_{33}(D^\star{ \boldsymbol u}),\varphi_2\right)+
{\cal R}e\, \left(Bu\cdot \nabla u,\varphi\right)
={\cal R}e\,\delta\left(u_3^2,\phi_2\right),\end{equation}
where $\tau=({\tau}_{i,j})_{i,j=1,2}$. Similarly, by setting ${\boldsymbol\varphi}=(0,0,\varphi_3)$ in (<ref>), we see that $u_3$ satisfies
\begin{equation}\label{weak_formulation_u3}
\begin{array}{ll}\left({\tau}_{13}(D^\star{ \boldsymbol u}),
B\tfrac{\partial \varphi_3}{\partial x_1}\right)
+\left({\tau}_{23}(D^\star{ \boldsymbol u}),
B\tfrac{\partial \varphi_3}{\partial x_2}
-\delta\varphi_3\right)+{\cal R}e\left((Bu\cdot \nabla u_3,
\varphi_3)+\delta\left(u_3u_2,\varphi_3\right)\right)\vspace{2mm}\\
\end{array}
\end{equation}
In the case of the Navier-Stokes equations, both formulations
(<ref>) and (<ref>)-(<ref>) can be used to establish the existence and uniqueness of a weak solution. In the present paper, the corresponding existence and uniqueness results are obtained as a particular case (by setting $p=2$) using the first formulation, while similar results were established in <cit.> by using the second formulation. One notable difference however is related with the corresponding estimates: involving $\boldsymbol u$ and $w_3$ and independent of the Reynolds number in the first case, involving $u$ and $w_3$ (and thus $\boldsymbol u$) and depending on $\delta{\cal R}e$ (at least for $u$ and $\boldsymbol u$) in the second one. This fact has no influence on the study of the solvability of our problem, but a different estimate on $u$ is missed if we do not consider (<ref>). When dealing with the generalized Navier-Stokes equations (<ref>), the
global formulation proves much more suitable than the splitting one. Because of the nonlinearity of the shear-stress, the coercivity property in the case $p<2$ and the monotonicity property in both cases $p<2$ and $p>2$ fail to be satisfied separately for $u$ and $u_3$. We will use the formulation (<ref>) a posteriori to derive an additional estimate for $u$ that highlights the dependance on the curvature ratio and the Reynolds number.
Existence and uniqueness of a weak solution in the shear-thickening case, together with associated estimates, is considered in the next result.
Assume that $p\geq 2$. Then problem
$(\ref{equation})$ admits at least a weak solution $\boldsymbol u\in \boldsymbol V^p_B$, and this solution satisfies
\begin{align}\label{main_estimates_1_thic}&
\left\|D^\star{ \boldsymbol u}\right\|_{2,B}\leq \kappa_1,\\
\label{main_estimates_2_thic}
&\left\|D^\star{ \boldsymbol u}\right\|_{p,B}
\leq\kappa_1^\frac{2}{p},\\
\label{est_u3_thic}&\left\|\nabla u_3\right\|_{2,B}\leq 2^{\frac{1}{2}}\kappa_1,\\
\label{est_u_2_thic}
&\left\|Du\right\|_{2,B}\leq \kappa_2 \delta{\cal R}e,\\
\label{est_u_p_thic}&\left\|Du\right\|_{p,B}\leq \left(\kappa_2 \delta{\cal R}e\right)^\frac{2}{p}\end{align}
$$\kappa_1=\left(\tfrac{m}{2}\right)^{\frac{1}{2}}|G||\Sigma|\qquad \mbox{and} \qquad
\kappa_2=\tfrac{m^{\frac{3}{2}}|\Sigma|}{2}\kappa_1^2.
Moreover, if the Reynolds number ${\cal R}e$ is such that
\begin{equation}\label{Re_cond}
{\cal R}e<\tfrac{2}{\kappa_1\kappa_3}
\qquad \mbox{with} \ \kappa_3=
nm^\frac{3}{2} |\Sigma|^{\frac{3}{4}}\end{equation}
then problem $(\ref{equation})$ admits a unique weak solution.
Unlike the estimates given in $(\ref{est_u_2_thic})$-$(\ref{est_u_p_thic})$, the estimates in $(\ref{main_estimates_1_thic})$-$(\ref{main_estimates_2_thic})$, and consequently in $(\ref{est_u3_thic})$, are uniformly bounded and neither depend on $\delta$ nor on ${\cal R}e$. On the other hand, the estimates in $(\ref{est_u_2_thic})$-$(\ref{est_u_p_thic})$ show that the secondary shear-thickening flows, if they exist, depend simultaneously on the pipe curvature ratio and on the Reynolds number or equivalently, and after introducing the standard adimensionalization $(\tilde u,\tilde u_3)=
(\sqrt{\delta} u,u_3)$, that they depend on the Dean number
${\cal D}e=\sqrt{\delta}\, {\cal R}e$. These results are specially useful when dealing with small Dean numbers and imply, in this situation, that the shear-thickening secondary flows are necessarily small.
Similar results are obtained in the shear-thinning case.
Assume that $\tfrac{3}{2}\leq p<2$. Then problem
$(\ref{equation})$ admits at least a weak solution $\boldsymbol u\in \boldsymbol V^p_B$, and this solution satisfies
\begin{align}\label{main_estimates_1_thin}
&\left\|D^\star{ \boldsymbol u}\right\|_{p,B}\leq \kappa_2,\\
\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^p
\leq \kappa_3,\\
&\label{est_u3_thin}\left\|\nabla u_3\right\|_{p,B}\leq \kappa_4,\\
&\label{est_u_thin}\left\|D u\right\|_{p,B}\leq \kappa_5 \,\delta {\cal R}e
\end{align}
|\Sigma|^{\frac{1}{p'}}, \qquad
\kappa_2=\kappa_1 \left(\tfrac{\|B\|_1}{p-1}+\kappa_1
\qquad \kappa_3=p'\|B\|_1+\left(2^{\frac{2-p}{2}}\kappa_1\right)^{p'}, \vspace{2mm}\\
\kappa_4=2^{2-p}\left(1+\delta m \right)\kappa_2,\end{array}$$
and where $\kappa_5$ is a positive constant only depending on $\Sigma$, $p$, $G$ and $m$.
Moreover, there exists a positive constant $\kappa_6$ only depending on $\Sigma$, $p$, $G$, $m$ and $n$ such that if
\begin{equation}\label{restriction_control}{\cal R}e<\tfrac{1}{2\kappa_1\kappa_6}\left(\tfrac{\|B\|_1}{p-1}+\kappa_1^{p'}\right)^{-\frac{2(2-p)}{p}},
\end{equation}
then problem $(\ref{equation})$ admits a unique weak solution.
As in the case of shear-thickening flows, the estimates in $(\ref{main_estimates_1_thin})$-$(\ref{est_u3_thin})$ are uniformly bounded and independent of $\delta$ and ${\cal R}e$ while estimate $(\ref{est_u_thin})$ depends simultaneously on these two parameters and shows that if the Dean number is small, then the shear-thinning secondary flows are necessarily small.
In the statement of Theorem $\ref{main2}$, the dependence of the constants $\kappa_5$ and $\kappa_6$ on the parameters $\delta$ and $m$ is explicitely known. More precisely, we have
$$\begin{array}{ll} \kappa_5=\tfrac{m^{\frac{3}{p}}}{C_{K,1}}\left( S_{p,2p'}\right)^3 \kappa_4^2\left(\tfrac{\|B\|_1}{p-1}+\kappa_1^{p'}\right)^{\frac{2-p}{p}},\vspace{2mm}\\
\kappa_6=\tfrac{8n^{\frac{p+1}{p}} m^{\frac{6}{p}}(1+\delta m)^3}{C_{K,1}^3}\left( S_{p,2p'}\right)^2,\end{array}$$
where $C_{K,1}$ is the classical Korn inequality in $\boldsymbol W^{1,p}_0(\Sigma)$. Notice also that the constants $\kappa_2$, $\kappa_3$, $\kappa_4$, $\kappa_5$ and $\kappa_6$ depend on $\kappa_1$.
Having a weak solution, the corresponding term $\nabla\pi$ can be constructed by the same way as in the linear case. The pressure is determined up to a constant and becomes unique under the additional condition $\int_\Sigma \pi\,dx=0$.
Assume that $p\geq 2$ and that $\boldsymbol u\in \boldsymbol V_B^p$ is a weak solution of
$(\ref{equation})$. Then there exists a unique $\pi \in L^{p'}_0(\Sigma)$ such that $(\ref{equation})_1$ holds in $\boldsymbol W^{-1,p'}(\Sigma)$. Moreover, we have the following estimate
$$\left\|\pi\right\|_{p'}\leq \kappa \left(\left\|Du\right\|_{p,B}+\delta^2\left\|u_2\right\|_{p,B}+{\cal R}e \left(\left\|D u\right\|_{p,B}^2+ \delta \left\|\nabla u_3\right\|_{p,B}^2\right)\right)$$
where $\kappa$ is a positive constant only depending on $\Sigma$, $p$, $m$ and $n$.
Similarly, existence of the pressure in the shear-thinning case is considered below.
Assume that $\tfrac{3}{2}\leq p<2$ and that $\boldsymbol u\in \boldsymbol V_B^p$ is a weak solution of
$(\ref{equation})$. Then there exists a unique $\pi \in L^{p'}_0(\Sigma)$ such that $(\ref{equation})_1$ holds in $\boldsymbol W^{-1,p'}(\Sigma)$. Moreover, we have the following estimate
$$\left\|\pi\right\|_{p'}\leq \kappa \left(\left\|Du\right\|_{p,B}^{p-1}+\delta^p \left\|u_2\right\|_{p,B}^{p-1}+{\cal R}e \left(\left\|D u\right\|_{p,B}^2+
\delta \left\|\nabla u_3\right\|_{p,B}^2\right)\right),$$
where $\kappa$ is a positive constant only depending on $\Sigma$, $p$, $m$ and $n$.
The next results deal with special properties of the solutions of $(\ref{equation})$. We recall that a solution $(\boldsymbol u=(u,u_3),\pi)$ is unidirectional flow if
Assume that $p\geq \tfrac{3}{2}$. If $\delta{\cal R}e=0$, then all the solutions of $(\ref{equation})$ are unidirectional flows.
Proof. Let us first assume that $p\geq 2$. Taking into account (<ref>), we deduce that
$\left\|\nabla u\right\|_p=0$ and
since $u_{\mid \partial \Sigma}=0$, by using the Poincaré inequality it follows that
$u=0$ and thus $\boldsymbol u=(0,0,u_3)$
with $u_3$ satisfying
\begin{equation}\label{u3_uni}\left\{\begin{array}{ll}-\nabla^\star\left(\left(1+\tfrac{1 }{2}|\nabla^\star u_3|^2\right)^{\frac{p-2}{2}}\nabla^\star u_3\right)=\tfrac{G}{B} & \mbox{in} \ \Sigma,\vspace{2mm}\\
u_3=0 & \mbox{in} \ \partial\Sigma.\end{array}\right.\end{equation}
The result corresponding to the shear-thinning case can be proved similarly using the estimate
(<ref>) instead of the estimate (<ref>). $\hfill\Box$
Assume that $p\geq \tfrac{3}{2}$. If $\delta{\cal R}e>0$, then the solutions of $(\ref{equation})$ are not unidirectional flows.
Proof. Let us assume that $(u_1,u_2)=(0,0)$. System (<ref>) reduces to
$$\nabla\pi={\cal R}e\,\tfrac{\delta}{B}u_3{\boldsymbol a}_2 \quad\mbox{and} \quad
u_3 \ \mbox{satisfies} \ (\ref{u3_uni}).$$
If $\delta{\cal R}e>0$, the first equation implies that $\pi$ (and thus $u_3$) does not depend on the variable $x_1$. Following the arguments developed in <cit.>, we can prove that
\begin{equation}\label{u3zero}u_3=0 \qquad \mbox{in} \ \Sigma.
\end{equation}
Indeed, let $(x_1,x_2)$ be an arbitrary point in $\Sigma$ and denote by
$(x_0,x_2)$ a point on $\partial \Sigma$ that is the intersection of the straight line with origin $(x_1,x_2)$, parallel to $\boldsymbol a_1$ and such that the segment $\mbox{seg}= \cup_{\alpha\in [0,1]}$ lies in $\overline \Sigma$. Since $u_3$ is independent of $x_1$, it follows that
$u(x_1^\alpha,x_2)$ is independent of $\alpha$ for $x_1^\alpha\in \mbox{seg}$. Therefore, by taking into account the boundary condition $u_{\mid \Sigma}=0$, we obtain
The point $(x_1,x_2)$ being arbitrary in $\Sigma$, we deduce that (<ref>) holds, and this contradicts $(\ref{u3_uni})$ and completes the proof.$\hfill\Box$
Let us now analyze the behavior of the weak solutions of $(\ref{equation})$ with respect to the parameter $\delta$. The objective would be to use these results when $\delta$ is small to approximate a solution $\boldsymbol u$ of (<ref>) by a solution of a similar but simpler system. More precisely, we consider the following problem
$$(E_\sigma) \qquad
\left\{ \begin{array}{ll} -\nabla \cdot\left({\boldsymbol\tau}(D\boldsymbol w)
\right)+{\cal R}e \,
{\boldsymbol w}\cdot \nabla {\boldsymbol w}+ \nabla {\pi} =\tfrac{G}{B}\boldsymbol a_3+\delta \sigma(w_3) \boldsymbol a_2&\quad \mbox{in} \
\Sigma,\vspace{2mm} \\
\nabla \cdot {\boldsymbol w}=0&\quad \mbox{in} \
\Sigma,\vspace{2mm} \\
\boldsymbol w=0 &\quad \mbox{on} \ \partial\Sigma,\end{array}\right.
where $\sigma$ is a non constant function and we aim to estimate the difference $\boldsymbol u-\boldsymbol w$. Obviously, the considerations concerning the monotonicity and coercivity properties of the global and the coupled formulations described above and the difficulties encountered in the treatment of the full governing equations arise in a similar way for $(E_\sigma)$. Moreover, in the derivation of the a priori estimate, the term involving $\sigma$ induces an additional difficulty that can be overcome by carrying out a careful analysis.
Begining with the shear-thickening case, we summarize the properties of the solutions of $(E_\sigma)$.
Assume that $p\geq 2$ and let
$\sigma$ be a non constant continous function satisfying
$(\ref{psi_assumption})$ for some $\alpha\geq 0$. Then problem $(E_\sigma)$ admits at least a weak solution $\boldsymbol w\in \boldsymbol V^p$, and this solution satisfies
$$\begin{array}{ll}\left\|\nabla w_3\right\|_2\leq c_1,
&\qquad \left\|\nabla w_3\right\|_p^p\leq 2^{\frac{p-2}{2}}
\left\|D w\right\|_2\leq c_0c_2\delta,&\qquad
\left\|D w\right\|_p^p\leq \left(c_0c_2\delta\right)^2,
\end{array}$$
$$c_1=\tfrac{m|\Sigma||G|}{\sqrt{2}} \quad \mbox{and} \quad
c_2= \tfrac{D_{2,\alpha}c_1^\alpha}{\sqrt{2}} \quad
\mbox{with} \ D_{2,\alpha} \ \mbox{given in Lemma} \ \ref{est_psi}.$$
Moreover, if $\delta>0$ then the solutions of $(E_\sigma)$ are not unidirectional flows.
Next we state the corresponding approximation result.
Assume that the assumptions of Proposition $\ref{w_exist_thick}$ are fulfilled and let $\boldsymbol u$, $\boldsymbol w$ be the solutions of $(\ref{equation})$ and $(E_\sigma)$. There exists ${\cal R}e_0>0$ such that if ${\cal R}e\leq {\cal R}e_0$ then
$$\left\|D\left(\boldsymbol u-\boldsymbol w\right)\right\|_{p}^p+
\left\|D\left(\boldsymbol u-\boldsymbol w\right)
\right\|_{2}^2\leq
\kappa\,\delta^{p'}$$
where $\kappa$ depends only on $p$, $\Sigma$, $m$, $n$, $c_0$ and $\alpha$.
Similarly, we consider the solvability of $(E_\sigma)$ in the shear-thinning case and the corresponding approximation result.
Assume that $\tfrac{3}{2}\leq p<2$ and let $\sigma$ is a non constant continuous function satisfying $(\ref{psi_assumption})$ for some $\alpha$ such that $\tfrac{1}{p'}<\alpha< \tfrac{p^\ast}{2p'}$.
Then problem $(E_\sigma)$ admits at least a weak solution $\boldsymbol w\in \boldsymbol V^p$, and this solution satisfies
$$\begin{array}{ll}\left\|\nabla w_3\right\|_p\leq c_1
\left(|\Sigma|+c_4^p\right)^{\frac{2-p}{p}},\vspace{2mm}\\
\left\|D w\right\|_p\leq
c_0c_1^\alpha c_2
\left(|\Sigma|+c_4^p\right)^{\frac{(2-p)(\alpha+1)}{p}}\, \delta,
\end{array}$$
$$\begin{array}{ll}c_1=m|\Sigma|^{\frac{1}{p'}}|G|, \qquad
c_2=\tfrac{D_{p,\alpha}}{2C_{K,1}} \quad
\mbox{with} \ D_{p,\alpha} \ \mbox{given in Lemma} \ \ref{est_psi_thin},
\vspace{2mm}\\
c_0c_1^{\alpha}c_2 \right)
\left(1+|\Sigma|\right)^{\frac{(2-p)(\alpha+1)}{p}},\vspace{2mm}\\
\right)^{\frac{(2-p)(\alpha+1)}{p}}.\end{array}$$
Moreover, if $\delta>0$ then the solutions of $(E_\sigma)$ are not unidirectional flows.
Assume that the assumptions of Proposition $\ref{w_exist_thin}$ are fulfilled and let $\boldsymbol u$, $\boldsymbol w$ be the solutions of $(\ref{equation})$ and $(E_\sigma)$. There exists ${\cal R}e_0>0$ such that if ${\cal R}e\leq {\cal R}e_0$ then
$$\left\|D\left(\boldsymbol u-\boldsymbol w\right)\right\|_p\leq
\kappa\,\delta^{p-1}$$
where $\kappa$ depends only on $p$, $\Sigma$, $m$, $n$, $c_0$ and $\alpha$.
Propositions <ref>, <ref>, <ref> and <ref>
show that for $\delta>0$ sufficiently small, a solution of (<ref>) can be approximated by a solution of $(E_\sigma)$, whose secondary flows exist even if they are proportionately weak. It is worth observing that this result is valid for a relatively large class of functions $\sigma$ and raises an interesting question related with the possible choices for $c_0$ and $\alpha$ that would guarantee an optimal approximation, in a sense to be correctly and adequately defined.
We finish this section by considering the case of Navier-Stokes equations obtained by setting $p=2$. Notice that the constants $\kappa_1$, $\kappa_2$ and $\kappa_3$ in the statement of Theorem $\ref{main1}$ are independent of the exponent $p$ and that the condition that guarantees the uniqueness of weak solutions only depends on
$\Sigma$, $G$, $m$ and $n$. As a consequence, the estimates and the sufficient condition on ${\cal R}e$ are identical in the particular case of Newtonian fluids.
The Navier-Stokes problem
$$\left\{ \begin{array}{ll}
-\nabla^\star \cdot\left(2D^\star{\boldsymbol u}
\right)+{\cal R}e \,
{\boldsymbol u}\cdot \nabla^\star {\boldsymbol u}+ \nabla {\pi} =\tfrac{G}{B} \, \boldsymbol a_3 &\quad \mbox{in} \
\Sigma,\vspace{2mm} \\
\nabla^\star \cdot {\boldsymbol u}=0&\quad \mbox{in} \
\Sigma,\vspace{2mm} \\
\boldsymbol u=0 &\quad \mbox{on} \ \partial\Sigma,\end{array}\right.$$
admits at least a weak solution $\boldsymbol u\in \boldsymbol V^2_B$.
This solution satisfies the estimates $(\ref{main_estimates_1_thic})$, $(\ref{est_u3_thic})$, $(\ref{est_u_2_thic})$ and if ${\cal R}e$ satisfies $(\ref{Re_cond})$, then the solution is unique. If $\delta{\cal R}e=0$, then all the solutions are unidirectional flows, otherwise they are not unidirectional flows. Finally, let $\sigma$ be a non constant continuous function satisfying $(\ref{psi_assumption})$ for some $\alpha\geq 0$. Then the following problem
$$\left\{ \begin{array}{ll}
-\Delta \boldsymbol w+{\cal R}e \,
{\boldsymbol w}\cdot \nabla {\boldsymbol w}+ \nabla {\pi} =\tfrac{G}{B}\boldsymbol a_3+\delta \sigma(w_3) \boldsymbol a_2 &\quad \mbox{in} \
\Sigma,\vspace{2mm} \\
\nabla \cdot {\boldsymbol w}=0&\quad \mbox{in} \
\Sigma,\vspace{2mm} \\
\boldsymbol w=0 &\quad \mbox{on} \ \partial\Sigma,\end{array}\right.$$
admits a weak solution in $\boldsymbol V^2$. Moreover, there exists ${\cal R}e_0>0$ such that if ${\cal R}e\leq {\cal R}e_0$ then
$$\left\|D(\boldsymbol u-\boldsymbol w)\right\|_2\leq
\kappa \delta,$$
where $\kappa$ depends on $\Sigma$, $m$, $n$, $c_0$ and $\alpha$.
The case of Navier-Stokes equations has been fully studied in <cit.>. In the previous result, we recover similar results with some differences concerning the analysis with respect to $\delta$. On the one hand, our estimate is valid for a class of problems larger than the classical Dean problem obtained by setting
\begin{equation}\label{sigma_NS}
\sigma(\lambda)={\cal R}e \lambda^2.\end{equation}
On the other hand, the estimate corresponding to the secondary flows is less accurate. Indeed, after introducing the adimensionalization
$(\tilde u,\tilde u_3)=(\sqrt{\delta} u,u_3)$, we obtain
$$\left\|D(\tilde u-\tilde w)\right\|_2\leq
\kappa \sqrt{\delta}, \qquad
\left\|\nabla(\tilde u_3-\tilde w_3)\right\|_2\leq
\kappa \delta$$
while the estimate obtained in <cit.> reads as
$$\left\|D(\tilde u-\tilde w)\right\|_2\leq
\kappa \delta, \qquad
\left\|\nabla(\tilde u_3-\tilde w_3)\right\|_2\leq
\kappa \delta.$$
This is due to some technical difficulties mainly related with the combined effect of $Du$ and $\nabla u_3$ in the shear-rate and its consequences on the monotonicity properties of the tensor $\boldsymbol\tau$. Indeed, in the case of Navier-Stokes equations with $\sigma$ given by
(<ref>), the corresponding coupled formulations allow to derive, in a first step, estimates for $\left\|D(\tilde u-\tilde w)\right\|_2$ and $\left\|\nabla(\tilde u_3-\tilde w_3)\right\|_2$ dependent on one another. The combination of these estimates in a second step gives the result. In the case of a shear-dependent viscosity, and as already observed concerning the solvability of problem (<ref>), the lack of monotonicity of the tensores $\tau=(\tau_{ij})_{i,j=1,2}$,
$\tau_{13}$ and
$\tau_{23}$ prevents from using the same arguments, and the global estimates we obtain come with a cost.
§ SHEAR-THICKENING FLOWS
The aim of this section is to study the case of shear-thickening flows (corresponding to $p\geq 2$). To achieve this goal, we first establish a Korn inequality, and then estimate the convective term as well as the extra stress tensor in an adequate setting. We finally prove the corresponding main results given above.
§.§ On the Korn inequality
The next result deals with an inequality of Korn's type in ${ \boldsymbol H}^1_0(\Sigma)$, very similar to the classical one but involving the operators $\nabla^\star$ and $D^\star$.
Let $ { \boldsymbol u}=(u_1,u_2,u_3) \in { \boldsymbol H}^1_0(\Sigma)$. Then
\begin{equation}\label{korn-l2}\left\|\nabla^\star{ \boldsymbol u}\right\|_{2,B}^2=2\left\|D^\star{ \boldsymbol u}\right\|_{2,B}^2-
\left\|\tfrac{1}{B}\nabla \cdot\left(B u \right)\right\|_{2,B}^2.\end{equation}
Proof. The definition of $D^\star$ together with standard calculations show that
\begin{align}\label{korn1}
2\left\|D^\star{ \boldsymbol u}\right\|_{2,B}^2&=2\left\|D u \right\|_{2,B}^2
+\left\|\tfrac{\partial u_3}{\partial x_1}\right\|_{2,B}^2
+\left\|\sqrt{B}\,\tfrac{\partial u_3}{\partial x_2}
-\tfrac{\delta}{\sqrt{B}} u_3\right\|_2^2\nonumber\\
&=2\left\|D u \right\|_{2,B}^2
+2\left\|\tfrac{\delta}{\sqrt{B}} u_2\right\|_2^2+
\left\|\nabla u_3\right\|_{2,B}^2
&=2\left\|Du \right\|_{2,B}^2
+2\left\|\tfrac{\delta}{B} u_2\right\|_{2,B}^2+
\left\|\nabla u_3\right\|_{2,B}^2
where $u=(u_1,u_2,0)$. Since
$$\nabla \cdot\left(2D u \right)
-\Delta u =\nabla \cdot \left((\nabla u )^T
\right)=\nabla \left(\nabla \cdot u \right),$$
we deduce that for every $ \boldsymbol\varphi\in { \boldsymbol H}^1_0(\Sigma)$ we have
$$ \left(\nabla u , \nabla \boldsymbol\varphi\right)
= 2\left(D u ,
\nabla \boldsymbol\varphi\right)-\left(\nabla \cdot u ,\nabla
\cdot\boldsymbol\varphi\right).$$
On the other hand, easy calculations show that $Du :\nabla \boldsymbol\varphi =Du :D\boldsymbol\varphi$. Combining these identities, we obtain
\left(\nabla u , \nabla \boldsymbol\varphi\right)
=2\left( D u ,D \boldsymbol\varphi\right)-
\left(\nabla \cdot u ,\nabla
\cdot\boldsymbol\varphi\right),$$
and thus
$$\begin{array}{ll}\left(\nabla u,
\nabla \left(B\boldsymbol\varphi\right)\right)&= \left(B\nabla u ,\nabla \boldsymbol\varphi\right)
+\delta\left(\tfrac{\partial u }{\partial x_2},
\boldsymbol\varphi\right)\vspace{2mm}\\
&=2\left(D u ,
D \left(B\boldsymbol\varphi\right)\right)-
\left(\nabla \cdot u ,\nabla
\cdot\left(B\boldsymbol\varphi\right)\right)\vspace{2mm}\\
&=2\left(B D u ,
D \boldsymbol\varphi\right)+\delta\left(\tfrac{\partial u }
{\partial x_2}+\nabla u_2,\boldsymbol\varphi\right)-
\left(\nabla \cdot u ,\nabla
\cdot\left(B\boldsymbol\varphi\right)\right)\vspace{2mm}\\
&=2\left(B D u ,
D \boldsymbol\varphi\right)+\delta\left(\tfrac{\partial u }
{\partial x_2},\boldsymbol\varphi\right)
-\delta\left(u_2,\nabla \cdot
\boldsymbol\varphi\right)-
\left(\nabla \cdot u ,\nabla
\cdot\left(B\boldsymbol\varphi\right)\right)\vspace{2mm}\\
&=2\left(B D u ,
D \boldsymbol\varphi\right)+\delta\left(\tfrac{\partial u }
{\partial x_2},\boldsymbol\varphi\right)+
\left(\tfrac{\delta^2}{B}u_2,\varphi_2\right)-\left(\tfrac{1}{B}\,\nabla \cdot\left(B u \right),\nabla
\cdot\left(B\boldsymbol\varphi\right)\right).\end{array}$$
The last equality implies that
$$\left(B\nabla u ,\nabla \boldsymbol\varphi\right)=
2\left(B D u , D \boldsymbol\varphi\right)-
\left(\tfrac{1}{B}\nabla \cdot\left(B u \right),\nabla
\cdot\left(B\boldsymbol\varphi\right)\right)+
\left(\tfrac{\delta^2}{B}u_2,v_2\right)$$
for all $ \boldsymbol\varphi\in { \boldsymbol H}^1_0(\Sigma)$. Setting $\boldsymbol\varphi= u $, we obtain
\begin{align}\label{korn2}\left\|\nabla u \right\|_{2,B}^2&=
2\left\| D u \right\|_{2,B}^2+
\left\|\tfrac{\delta}{\sqrt{B}}\,u_2\right\|_2^2-
\left\|\tfrac{1}{\sqrt{B}}\nabla \cdot\left(B u \right)\right\|_2^2\nonumber\\
&= 2\left\| D u \right\|_{2,B}^2+
\left\|\tfrac{\delta}{B}\,u_2\right\|_{2,B}^2-
\left\|\tfrac{1}{B}\nabla \cdot\left(B u \right)\right\|_{2,B}^2.\end{align}
On the other hand, by taking into acount the definition of $\nabla^\star$ we have
\begin{equation}\label{grad_prime}\left\|\nabla^\star{ \boldsymbol u}\right\|_{2,B}^2=
\left\|\nabla { \boldsymbol u}\right\|_{2,B}^2+\left\|\tfrac{\delta}{B}u_2\right\|_{2,B}^2
The conclusion follows from (<ref>), (<ref>) and (<ref>).$\hfill\Box$
A direct consequence of Lemma $\ref{lm-korn}$ is that
$$\left\|\nabla^\star { \boldsymbol u} \right\|_{2,B}^2\leq
2\left\|D^\star { \boldsymbol u} \right\|_{2,B}^2\qquad \mbox{for all} \ { \boldsymbol u} \in { \boldsymbol H}^1_0(\Sigma) $$
and the equality holds if $\nabla\cdot\left(B \boldsymbol u \right)=0$.
§.§ Estimates on the convective term
We point out some notable facts related with the trilinear forms $a$ and $a_\star$ defined by
a({ \boldsymbol u},{ \boldsymbol v},{ \boldsymbol w})
=\left({ \boldsymbol u}\cdot \nabla { \boldsymbol v},
{ \boldsymbol w}\right),\vspace{2mm}\\
a_\star({ \boldsymbol u},{ \boldsymbol v},{ \boldsymbol w})
=\left({ \boldsymbol u}\cdot \nabla^\star { \boldsymbol v},{ \boldsymbol w}\right)=
a({ \boldsymbol u},{ \boldsymbol v},{ \boldsymbol w})+
\left(\tfrac{\delta}{B}v_3u_2,w_3\right)
Assume that $p\geq \tfrac{3}{2}$. For every ${ \boldsymbol u}\in {\boldsymbol V}_B^p$ and every ${ \boldsymbol v}$, ${ \boldsymbol w}\in { \boldsymbol W}^{1,p}_0(\Sigma)$, we have
$$a_\star(B { \boldsymbol u},{ \boldsymbol v},{ \boldsymbol v})=0\quad \mbox{and} \quad
a_\star(B { \boldsymbol u},{ \boldsymbol v},{ \boldsymbol w})
=-a_\star(B { \boldsymbol u},{ \boldsymbol w},{ \boldsymbol v}).$$
Proof. Taking into account the definition of $a_\star$ and the fact that $\nabla\cdot \left(B { \boldsymbol u}\right)=0$, we deduce that
$$\begin{array}{ll}a_\star(B { \boldsymbol u},{ \boldsymbol v},{ \boldsymbol v})&=
a(B { \boldsymbol u},{ \boldsymbol v},{ \boldsymbol v})+
\delta\left(u_3v_2,v_3\right)-\delta\left(u_3,v_3v_2\right)=a(B { \boldsymbol u},{ \boldsymbol v},{ \boldsymbol v})=0.\end{array}$$
$$\begin{array}{ll}a_\star(B { \boldsymbol u},{ \boldsymbol v},{ \boldsymbol w})&=a(B { \boldsymbol u},{ \boldsymbol v},{ \boldsymbol w})+\delta\left(u_3v_2,w_3\right)-\delta
\left(u_3v_3,w_2\right)\vspace{2mm}\\
&=-a(B { \boldsymbol u},{ \boldsymbol w},{ \boldsymbol v})+\delta\left(u_3v_2,w_3\right)-\delta
\left(u_3v_3,w_2\right)=-a_\star(B { \boldsymbol u},{ \boldsymbol w},{ \boldsymbol v})\end{array}$$
and the proof is complete.$\hfill\Box$
Let ${ \boldsymbol u}$, ${ \boldsymbol v}$ and ${ \boldsymbol w}$ be in $\boldsymbol H^{1}_0(\Sigma)$. Then the following estimate holds
$$\left|a_\star({ \boldsymbol u},{ \boldsymbol v},B{ \boldsymbol w})\right|\leq
\kappa_3
\left\|D^\star{ \boldsymbol u}\right\|_{2,B}\left\|D^\star{ \boldsymbol v}\right\|_{2,B}
\left\|D^\star{ \boldsymbol w}\right\|_{2,B},$$
where $\kappa_3=nm^{\frac{3}{2}} |\Sigma|^{\frac{3}{4}}$.
Proof. Setting $r=4$ and $q=2$ in the Sobolev inequality (<ref>), we obtain
$$\begin{array}{ll}\left|a_\star({ \boldsymbol u},{ \boldsymbol v},B{ \boldsymbol w})\right|&\leq n
\left\|{ \boldsymbol u}\right\|_{4}\left\|\nabla^\star { \boldsymbol v}\right\|_{2}
\left\|{ \boldsymbol w}\right\|_{4}\leq n \left( S_{2,4}\right)^2
\left\|\nabla { \boldsymbol u}\right\|_{2} \left\|\nabla^\star { \boldsymbol v}\right\|_{2}
\left\|\nabla { \boldsymbol w}\right\|_{2}\vspace{2mm}\\
&\leq nm^{\frac{3}{2}}\left( S_{2,4}\right)^2
\left\|\nabla^\star { \boldsymbol u}\right\|_{2,B} \left\|\nabla^\star { \boldsymbol v}\right\|_{2,B}
\left\|\nabla^\star { \boldsymbol w}\right\|_{2,B}. \end{array}$$
The estimate follows then by taking into account Remark <ref>. $\hfill\Box$
§.§ Estimates on the extra stress tensor
Our aim now is to establish some continuity, coercivity and
monotonicity results for the stress tensor $\boldsymbol\tau$.
Assume that $p\geq 2$ and let $\boldsymbol f, \boldsymbol g\in { \boldsymbol L}^p(\Sigma,\mathbb{R}^{3\times 3})$. Then the following estimates hold
* Continuity.
\begin{equation}
\label{estim_S3_3}\Big\|\left(1+|\boldsymbol f|^2\right)^{\frac{p-2}{2}}\boldsymbol g\Big\|_{p',B}\leq {\cal F}_B\big(\left\|\boldsymbol f\right\|_{p,B}
\big)\left\|\boldsymbol g\right\|_{p,B},\end{equation}
\begin{equation}
\label{estim_S3_2}\left\|\boldsymbol\tau\left(\boldsymbol f\right)-\boldsymbol\tau\left(\boldsymbol g\right)\right\|_{p',B}\leq (p-1){\cal F}_B\big(\left\|\boldsymbol f\right\|_{p,B}
+\left\|\boldsymbol g\right\|_{p,B}\big)
\left\|\boldsymbol f-\boldsymbol g\right\|_{p,B},\end{equation}
* Coercivity.
\begin{equation}\label{estim_S2}\left({\boldsymbol\tau}(\boldsymbol f),B \boldsymbol f\right)\geq 2\left\|\boldsymbol f\right\|_{2,B}^2, \qquad \quad
\left({\boldsymbol\tau}(\boldsymbol f),B\boldsymbol f\right)\geq 2\left\|\boldsymbol f\right\|_{p,B}^p, \end{equation}
* Monotonicity.
\begin{equation}\label{estim_S3}\begin{array}{ll}\left({\boldsymbol\tau}(\boldsymbol f)-{\boldsymbol\tau}(\boldsymbol g),B\left(\boldsymbol f-\boldsymbol g\right)\right)
\geq 2\left\|\boldsymbol f-\boldsymbol g\right\|_{2,B}^2,\vspace{1mm}\\
\left({\boldsymbol\tau}(\boldsymbol f)-{\boldsymbol\tau}(\boldsymbol g),B\left(\boldsymbol f-\boldsymbol g\right)\right)\geq \tfrac{1}{2^{p-1}(p-1)}
\left\|\boldsymbol f-\boldsymbol g\right\|_{p,B}^p.\end{array}\end{equation}
where ${\cal F}_B\big(\lambda\big)=\left(\big\|B^{\frac{1}{p}}\big\|_p+\lambda \right)^{p-2}$.
Proof. Assume that $p>2$. Standard calculation show that
$$\begin{array}{ll}\Big\|\left(1+|\boldsymbol f|^2\right)^{\frac{p-2}{2}}\boldsymbol g\Big\|_{p',B}&\leq \Big\|\left(1+|\boldsymbol f|^2\right)^{\frac{p-2}{2}}
\Big\|_{\frac{p}{p-2},B}\left\|\boldsymbol g\right\|_{p,B}\leq \left(\big\|B^{\frac{1}{p}}\big\|_p+\left\|\boldsymbol f\right\|_{p,B}\right)^{p-2}\left\|\boldsymbol g\right\|_{p,B}\end{array}$$
which gives (<ref>). Estimates (<ref>) and $(\ref{estim_S3})$ are a direct consequence of the coercivity properties and the monotonicity properties in Lemma $\ref{tensor_proper2}$. Finally, observing that
$$\begin{array}{ll}\left\|{\boldsymbol\tau}\left(\boldsymbol f\right)-{\boldsymbol\tau}\left(\boldsymbol g\right)\right\|_{p',B}
&\leq \displaystyle (p-1)\left\|\left(1+|\boldsymbol f|^2
+ |\boldsymbol g|^2\right)^{\frac{p-2}{2}}|\boldsymbol f-\boldsymbol g|
\right\|_{p',B}\vspace{1mm}\\
&\leq \displaystyle (p-1)\left\|\left(1+ |\boldsymbol f|^2
+ |\boldsymbol g|^2\right)^{\frac{p-2}{2}}\right\|_{\frac{p}{p-2},B}
\|\boldsymbol f-\boldsymbol g\|_{p,B}\vspace{2mm}\\
&\leq (p-1)\left(\big\|B^{\frac{1}{p}}\big\|_p+ \|\boldsymbol f\|_{p,B}
+\|\boldsymbol g\|_{p,B}\right)^{p-2}
\left\|\boldsymbol f-\boldsymbol g\right\|_{p,B}\end{array}$$
we obtain (<ref>). The case $p=2$ is direct. $\hfill\Box$
We finish the section by a result that will be useful in the sequel.
Assume that $p\geq 2$ and let $\boldsymbol f\in { \boldsymbol L}^p(\Sigma,\mathbb{R}^{3\times 3})$. Then the following estimate holds
\begin{equation}\label{diff_div}\left\|\nabla^\star\cdot \boldsymbol\tau(\boldsymbol f)-\nabla\cdot \boldsymbol\tau(\boldsymbol f)\right\|_{p'}\leq 4\delta m\,{\cal F}_1\big(\|\boldsymbol f\|_p\big)
\left(\left\|f\right\|_{p}
where ${\cal F}_1$ is defined in Proposition $\ref{tensor_proper3}$ and
Proof. Taking into account the definition of $\nabla^\star$, we obtain
$$\begin{array}{ll}\left\|\nabla^\star\cdot \boldsymbol\tau(\boldsymbol f)-\nabla \cdot\boldsymbol\tau(\boldsymbol f)\right\|_{p'}&=\left\|\tfrac{\delta}{B}
\left({\tau}_{12}(\boldsymbol f)\,{\boldsymbol a }_1+
\left({\tau}_{22}(\boldsymbol f)-{\tau}_{33}(\boldsymbol f)\right)\, {\boldsymbol a }_2+2{\tau}_{23}(\boldsymbol f)\, {\boldsymbol a }_3
\right)\right\|_{p'}\vspace{2mm}\\
&\leq \delta m \left(\left\|{\tau}_{12}(\boldsymbol f)
\right\|_{p'}+
\left\|{\tau}_{22}(\boldsymbol f)\right\|_{p'}+
\left\|{\tau}_{33}(\boldsymbol f)\right\|_{p'}+2
\left\|{\tau}_{23}(\boldsymbol f)\right\|_{p'}
\right)\vspace{2mm}\\
&\leq 2\delta m \left(\left\|\tau(\boldsymbol f)\right\|_{p'}
+\left\|{\tau}_{33}(\boldsymbol f)\right\|_{p'}
+\left\|{\tau}_{23}(\boldsymbol f)\right\|_{p'}
\right)
\end{array}$$
and the conclusion follows from estimate (<ref>).$\hfill \Box$
§.§ Existence and uniqueness of shear-thickening flows
The aim of this section is to prove the existence and uniqueness result Theorem <ref>. As usual, we first derive some estimates that hold not only for the exact solution $\boldsymbol u$ of (<ref>), but also for corresponding standard Galerkin approximations $\boldsymbol u^k$.
Assume that $p\geq 2$ and let $\boldsymbol u$ be a weak solution of $(\ref{weak_formulation})$. Then, estimates
$(\ref{main_estimates_1_thic})$-$(\ref{est_u_p_thic})$ hold.
Proof. The proof is split into two steps.
Step 1. Global estimates.
Setting ${\boldsymbol\varphi}={ \boldsymbol u}$ in (<ref>), and using Lemma <ref> and estimate (<ref>), we deduce that
\begin{equation}\label{estimate1}2
\left\|D^\star{ \boldsymbol u}\right\|_{2,B}^2
\leq \left({\boldsymbol\tau}(D^\star{ \boldsymbol u}),
B D^\star{ \boldsymbol u}\right)=(G,u_3),\end{equation}
\begin{equation}\label{estimate1_2}2
\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^p
\leq \left({\boldsymbol\tau}(D^\star{ \boldsymbol u}),
B D^\star{ \boldsymbol u}\right)=(G,u_3).\end{equation}
Classical arguments together with (<ref>) and (<ref>) yield
\begin{align}\left(G,u_3\right)
\sqrt{B}\,u_3\right)\right|\nonumber\\
&\leq |G|
\left\|\tfrac{1}{\sqrt{B}}\right\|_2
\left\|u_3\right\|_{2,B}= |G|
\left\|B^{-1}\right\|_1^{\frac{1}{2}}
\left\|u_3\right\|_{2,B}\nonumber\\
&\leq \kappa_1\left\|\nabla
\left(\sqrt{B}\,u_3\right)\right\|_2\leq \sqrt{2}\kappa_1
\left(\left\|\nabla u_3\right\|_{2,B}^2
\right)^{\frac{1}{2}}\nonumber\\
2\kappa_1\left\|D^\star{ \boldsymbol u}\right\|_{2,B},\end{align}
where $\kappa_1=\left(\tfrac{m}{2}\right)^{\frac{1}{2}}|G||\Sigma|$.
Due to (<ref>) and (<ref>), we obtain
(<ref>) and
\begin{equation}\label{estimate4}\left(G,u_3\right)\leq 2\kappa_1^2.\end{equation}
Estimate (<ref>) follows then from (<ref>) and (<ref>).
Step 2. Estimates for $u$ and $u_3$. Let us now prove estimates (<ref>)-(<ref>). Notice first that (<ref>) is a direct consequence of $(\ref{main_estimates_1_thic})$ and (<ref>).
To derive the second estimate, we set $\varphi=u$ in the weak formulation $(\ref{weak_formulation_u})$ and get
$$\left(\tau(D^\star{ \boldsymbol u}),B D u\right)+
\delta\left({\tau}_{33}(D^\star{ \boldsymbol u}),u_2\right)=\delta{\cal R}e\,\left(u_3^2,u_2\right).$$
Therefore, by using the coercivity properties, we obtain
\begin{equation}\label{est_u_1_2}\left\|Du\right\|_{2,B}^2+
\left\|\tfrac{\delta}{B}u_2
\right\|_{2,B}^2\leq \tfrac{\delta {\cal R}e }{2}\left(u_3^2,u_2\right),
\end{equation}
\begin{equation}\label{est_u_1_p}
\left\|Du\right\|_{p,B}^p+\left\|\tfrac{\delta}{B}u_2
\right\|_{p,B}^p\leq \tfrac{\delta {\cal R}e}{2} \left(u_3^2,u_2\right).
\end{equation}
On the other hand, taking into account (<ref>) with $\alpha=q=2$, (<ref>) and (<ref>), we have
\begin{align}\label{u3_u2}
\left|\left(u_3^2,u_2\right)\right|
&\leq \left( S_{2,4}\right)^3 |\Sigma|^{\frac{1}{4}}
\left\|\nabla u_3\right\|_{2}^2
\left\|\nabla u_2\right\|_{2}\nonumber\\
&\leq \tfrac{m^{\frac{3}{2}}|\Sigma|}{2\sqrt{2}}
\left\|\nabla u_3\right\|_{2,B}^2
\left\|\nabla u_2\right\|_{2,B}\nonumber\\
&\leq \tfrac{m^{\frac{3}{2}}|\Sigma|}{2}
\left\|\nabla u_3\right\|_{2,B}^2
\left\|D u\right\|_{2,B}\nonumber\\
&\leq m^{\frac{3}{2}}|\Sigma|\,\kappa_1^2
\left\|Du\right\|_{2,B}.\end{align}
Combining (<ref>) and (<ref>), we obtain
(<ref>) and
$$\left|\left(u_3^2,u_2\right)\right|\leq 2
\left(\tfrac{m^{\frac{3}{2}}|\Sigma|\,\kappa_1^2}{2}
\right)^2\delta{\cal R}e.$$
Estimate (<ref>) follows then from (<ref>).$\hfill\Box$
Proof of Theorem <ref>. The proof, based on classical compactness and monotonicity arguments, is split into two steps.
Step 1. Let us prove the existence of a weak solution for (<ref>). Let ${ \boldsymbol u}^k$ be a classical Galerkin approximation. Arguments similar to those used in the proof of Proposition <ref> show that
$$\left\|D^\star{ \boldsymbol u}^k\right\|_{2,B}
\leq \kappa_1, \qquad\quad
\left\|D^\star{ \boldsymbol u}^k\right\|_{p,B}^p
\leq\kappa_1^2$$
and imply that the sequences $(\sqrt{B}D^\star{ \boldsymbol u}^k)_k$ and
$(B^{\frac{1}{p}}D^\star{ \boldsymbol u}^k)_k$ are bounded in $\boldsymbol L^2(\Sigma)$ and $\boldsymbol L^p(\Sigma)$, respectively. By taking into account (<ref>), we deduce that $(\sqrt{B}\nabla{ \boldsymbol u}^k)_k$ is bounded in $\boldsymbol L^2(\Sigma)$ and thus $(\nabla{ \boldsymbol u}^k)_k$ is also bounded in $\boldsymbol L^2(\Sigma)$. Moreover,
due to estimate (<ref>) we have
$$\left\|{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}^k\right)\right\|_{p',B}\leq 2{\cal F}_B\left(\left\|D^\star{ \boldsymbol u}^k\right\|_{p,B}\right)
\left\|D^\star{ \boldsymbol u}^k\right\|_{p,B}$$
and thus $\left(B^{\frac{1}{p'}}{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}^k\right)\right)_k$ is bounded in $\boldsymbol L^{p'}(\Sigma)$.
There then exist a subsequence, still indexed by $k$, ${ \boldsymbol u}\in \boldsymbol V_B^p$ and $\widetilde {\boldsymbol\tau}\in \boldsymbol L^{p'}(\Sigma)$ such that
$$\nabla { \boldsymbol u}^k\longrightarrow \nabla { \boldsymbol u} \qquad \mbox{weakly} \ in \
\boldsymbol L^2(\Sigma),$$
$$B^{\frac{1}{p'}}{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}^k\right)\longrightarrow \widetilde {\boldsymbol\tau} \qquad \mbox{weakly} \ in \
\boldsymbol L^{p'}(\Sigma).$$
By using compactness results on Sobolev spaces, we deduce that $({ \boldsymbol u}^k)_k$ strongly converges
to ${ \boldsymbol u}$ in $L^{4}(\Sigma)$ and thus
$$\sqrt{B}D^\star{ \boldsymbol u}^k\longrightarrow \sqrt{B}D^\star{ \boldsymbol u} \qquad \mbox{weakly} \ in \
\boldsymbol L^2(\Sigma).$$
Therefore, by taking into account Lemma <ref>, for all ${\boldsymbol\varphi}\in \boldsymbol V_B^p$ we have
$$\begin{array}{ll}\left|a_\star\left({ \boldsymbol u}^k,
{ \boldsymbol u}^k,B{\boldsymbol\varphi}\right)
-a_\star\left({ \boldsymbol u},
{ \boldsymbol u},B{\boldsymbol\varphi}\right)\right|
&\leq\left|a_\star\left({ \boldsymbol u}^k-{ \boldsymbol u},{ \boldsymbol u}^k,B{\boldsymbol\varphi}\right)\right|+
\left|a_\star\left({ \boldsymbol u},{ \boldsymbol u}^k-{ \boldsymbol u},
&=\left|a_\star\left({ \boldsymbol u}^k-{ \boldsymbol u},{ \boldsymbol u}^k,B{\boldsymbol\varphi}\right)\right|
+\left|a_\star\left(B{ \boldsymbol u},{\boldsymbol\varphi},{ \boldsymbol u}^k-{ \boldsymbol u}
\right)\right|\vspace{1mm}\\
& \leq \left(\left\|\nabla^\star { \boldsymbol u}^k\right\|_{2,B}
\left\|{\boldsymbol\varphi}\right\|_{4,B}+
\left\|{ \boldsymbol u}\right\|_{4,B} \left\|\nabla^\star
{\boldsymbol\varphi}\right\|_{2,B}\right) \left\|
{ \boldsymbol u}^k-{ \boldsymbol u}\right\|_{4,B}\nonumber\\
&\longrightarrow 0 \qquad \mbox{when} \ k\rightarrow +\infty.\end{array}$$
By passing to the limit in
$$\left({\boldsymbol\tau}(D^\star{ \boldsymbol u}^k),B D^\star{\boldsymbol\varphi}\right)+{\cal R}e\,a_\star\left({ \boldsymbol u}^k, { \boldsymbol u}^k,
B{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right)\qquad \mbox{for all} \ {\boldsymbol\varphi}\in \boldsymbol V_B^{p},$$
we obtain
\begin{equation}\label{state_limit_weak3}\left(\widetilde {\boldsymbol\tau},B^{\frac{1}{p}} D^\star{\boldsymbol\varphi}\right)+{\cal R}e\,
a_\star\left({ \boldsymbol u},{ \boldsymbol u},
B{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right) \qquad \mbox{for all} \ {\boldsymbol\varphi}\in \boldsymbol V_B^{p}.\end{equation}
In particular, by settin $\boldsymbol\varphi=\boldsymbol u$ and using Lemma <ref> we deduce that
\begin{equation}\label{8} \left(\widetilde{\boldsymbol\tau},B^{\frac{1}{p}} D^\star{ \boldsymbol u}\right)=
\left(G,u_3\right).
\end{equation}
On the other hand, $(\ref{estim_S3})_1$ implies
\begin{equation}\label{9}\left({\boldsymbol\tau}\left(
D^\star{ \boldsymbol u}^k\right)-{\boldsymbol\tau}\left(D^\star{\boldsymbol\varphi}\right),B D^\star\left( { \boldsymbol u}^k-{\boldsymbol\varphi}\right)\right)\geq 0
\qquad \mbox{for all} \
{\boldsymbol\varphi}\in \boldsymbol V_B^{p}.\end{equation}
D^\star{ \boldsymbol u}^k\right),B D^\star{ \boldsymbol u}^k\right)=
\left(G,u^k_3\right),$$
by substituing in (<ref>), we obtain
$$\left(G, u ^k_3\right)-\left({\boldsymbol\tau}\left(
D^\star{ \boldsymbol u}^k\right),B D^\star{\boldsymbol\varphi}\right)-\left({\boldsymbol\tau}\left(D^\star{\boldsymbol\varphi}\right),B D^\star\left( { \boldsymbol u}^k-
{\boldsymbol\varphi}\right)\right)\geq 0.$$
By passing to the limit, it follows that
$$\left(G,u_3\right)-\left(B^{\frac{1}{p}}\,\widetilde{\boldsymbol\tau},D^\star{\boldsymbol\varphi}\right)-\left(B \,{\boldsymbol\tau}\left(D^\star{\boldsymbol\varphi}\right),D^\star \left({ \boldsymbol u}-{\boldsymbol\varphi}\right)\right)\geq 0\qquad \mbox{for all} \
{\boldsymbol\varphi}\in \boldsymbol V_B^{p}.$$
This inequality together with (<ref>) implies that
-B \,{\boldsymbol\tau}\left(D^\star{\boldsymbol\varphi}\right)
,D^\star\left({ \boldsymbol u}-{\boldsymbol\varphi}\right)\right)\geq 0
\qquad \mbox{for all} \ \boldsymbol\varphi\in \boldsymbol V_B^{p}$$
and by setting ${\boldsymbol\varphi}={ \boldsymbol u}-t{\boldsymbol\psi}$ with $t>0$, we obtain
$$\left(B^{\frac{1}{p}}\,\widetilde{\boldsymbol\tau}-B \,{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}-tD{\boldsymbol\psi}\right)
\geq 0\qquad \mbox{for all} \
{\boldsymbol\psi}\in \boldsymbol V_B^{p}.$$
Letting $t$ tend to zero and using the continuity of ${\boldsymbol\tau}$, we get
B\,{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}\right),D^\star{\boldsymbol\psi}\right)\geq 0
\qquad \mbox{for all} \ {\boldsymbol\psi}
\in \boldsymbol V_B^{p}$$
and thus
\begin{equation}\label{tensor-beta}\left(B^{\frac{1}{p}}\,\widetilde{\boldsymbol\tau},D^\star{\boldsymbol\psi}\right)=\left(B \, {\boldsymbol\tau}\left(D^\star{ \boldsymbol u}\right),D^\star{\boldsymbol\psi}\right)
\qquad \mbox{for all} \ {\boldsymbol\psi}\in \boldsymbol V_B^{p}.
\end{equation}
Combining (<ref>) and (<ref>), we deduce that
$$\left(B\, {\boldsymbol\tau}\left(D^\star{ \boldsymbol u}\right),D^\star{\boldsymbol\varphi}\right)
+{\cal R}e\, a_\star\left({ \boldsymbol u},{ \boldsymbol u},
B{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right)\qquad \mbox{for all} \ {\boldsymbol\varphi}\in \boldsymbol V_B^{p}.$$
Hence ${ \boldsymbol u}$ is a solution of (<ref>)
Step 2. To prove the uniqueness result, let us assume that ${ \boldsymbol u}$ and ${ \boldsymbol v}$ are two weak solutions of $(\ref{equation})$. Substituting in the weak formulation of (<ref>), setting $\boldsymbol\varphi={ \boldsymbol u}-{ \boldsymbol v}$ and taking into account Lemma <ref>, Lemma <ref> and Proposition <ref>, we obtain
\begin{align}\label{visco_convective3}\tfrac{1}{{\cal R}e}\left({\boldsymbol\tau}(D^\star{ \boldsymbol u})-
{\boldsymbol\tau}(D^\star{ \boldsymbol v}),BD^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)&=-a_\star( { \boldsymbol u}, { \boldsymbol u},B\left({ \boldsymbol u}-{ \boldsymbol v}\right))+
a_\star\left( { \boldsymbol v}, { \boldsymbol v},B\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)\nonumber\\
&=-a_\star\left( { \boldsymbol u},{ \boldsymbol u}-{ \boldsymbol v},B\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)
-a_\star\left({ \boldsymbol u}-{ \boldsymbol v}, { \boldsymbol v},B\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)\nonumber\\
&=-a_\star\left({ \boldsymbol u}-{ \boldsymbol v}, { \boldsymbol v},B\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)\nonumber\\
&\leq \kappa_3
\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right\|_{2,B}^2
\left\|D^\star { \boldsymbol v}\right\|_{2,B}\nonumber\\
&\leq \kappa_1\kappa_3
\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right\|_{2,B}^2. \end{align}
Combining (<ref>) and $(\ref{estim_S2})_1$, we deduce that
$$\left(2-\kappa_1\kappa_3\,{\cal R}e\right)\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right\|_{2,B}^2\leq 0$$
and thus $ { \boldsymbol u}\equiv { \boldsymbol v}$ if
${\cal R}e<\tfrac{2}{\kappa_1\kappa_3}$.$\hfill\Box$
Proof of Corollary <ref>. To simplify the redaction, let us set $\boldsymbol\tau(D^\star{ \boldsymbol u})=\boldsymbol\tau$. Notice first that if ${ \boldsymbol u}$ is a weak solution of (<ref>) then
\begin{align}\label{weak_form_0}\left(\boldsymbol\tau,D \boldsymbol\varphi\right) -\left(\nabla^\star\cdot \boldsymbol\tau-\nabla\cdot \boldsymbol\tau,\boldsymbol\varphi\right)
+{\cal R}e\,a_\star\left(\boldsymbol u,
\boldsymbol u,\boldsymbol\varphi\right)=(\tfrac{G}{B}
,\varphi_3)\qquad \mbox{for all} \ \boldsymbol\varphi\in \boldsymbol V^p.\end{align}
It follows that $u=(u_1,u_2,0)$ satisfies
$$\left(\tau,D \varphi\right)-
\left(\nabla^\star\cdot \tau-\nabla\cdot \tau,\varphi\right)+{\cal R}e\left(\left(u\cdot \nabla u,\varphi\right)-\left(\tfrac{\delta}{B} u_3^2,\varphi_2\right)\right)=0$$
for all $\varphi=(\varphi_1,\varphi_2)\in \boldsymbol W^{1,p}_0(\Sigma,\mathbb R^2)$ such that $\nabla\cdot \varphi=0$, with $\tau=\left({\tau}_{ij}\right)_{i,j=1,2}$.
Taking into account (<ref>), (<ref>) and using standard arguments, we can prove that the mapping
$${\cal G}: \varphi \mapsto \left(\tau,D \varphi\right)
\left(\nabla^\star\cdot \tau-\nabla\cdot \tau,\varphi\right)+{\cal R}e\left(\left(u\cdot \nabla u,\varphi\right)
-\left(\tfrac{\delta}{B} u_3^2,\varphi_2\right)\right)$$
is a linear continuous functional on $\boldsymbol W^{1,p}_0(\Sigma,\mathbb{R}^2)$. By using a classical result (see <cit.>), we deduce that there exists $\pi\in \boldsymbol L^{p'}_0(\Sigma)$ such that
$${\cal G}(\varphi)=-\left(\nabla \pi,\varphi\right)=
\left(\pi,\nabla \cdot \varphi\right) \qquad \mbox{for all} \ \varphi\in \boldsymbol W^{1,p}_0(\Sigma,\mathbb{R}^2).$$
Moreover, there exists a positive constant $C$ depending only on $p$ and $\Sigma$ such that
\begin{equation}\label{pression1}
C\left\|\pi\right\|_{p'}\leq \left\|\nabla \pi\right\|_{-1,p'}=\sup_{\varphi\in \boldsymbol W^{1,p}_0(\Sigma,\mathbb{R}^2)}
\tfrac{\left|\left(\pi,\nabla \cdot \varphi\right)\right|}{\left\|\nabla \varphi\right\|_p}.\end{equation}
On the other hand, using (<ref>), (<ref>), (<ref>) and
(<ref>) we obtain
\begin{align}\label{pression2}\left|\left(\tau,D \varphi\right)
-\left(\nabla^\star\cdot \tau-\nabla\cdot \tau,\varphi\right)\right|
&\leq \left\|\tau
\right\|_{p'}\left\|D \varphi\right\|_p +\left\|\nabla^\star\cdot \tau-\nabla\cdot \tau\right\|_{p'}
\left\|\varphi\right\|_p\nonumber\\
&\leq \left(
\left\|\tau\right\|_{p'}+S_{p,p}\left\|\nabla^\star\cdot \tau-\nabla\cdot \tau\right\|_{p'}
\right)\left\|\nabla \varphi\right\|_p\nonumber\\
&\leq {\cal F}_1\left(\|D^\star{ \boldsymbol u}\|_p\right)\left(\left(1+4S_{p,p}\delta m\right)\left\|Du\right\|_{p}+
4S_{p,p}\delta m \, \left\|\tfrac{\delta}{B}u_2\right\|_{p}\right)\left\|\nabla \varphi\right\|_p\nonumber\\
&\leq \tilde \kappa\left(
\left\|Du\right\|_{p,B}+\delta^2
\left\|u_2\right\|_{p,B}
\right)\left\|\nabla \varphi\right\|_p,\end{align}
where $\tilde \kappa$ only depends on $p$, $\Sigma$, $m$ and $n$. Similarly, we can easily see that
\begin{align}\label{pression3}\left|\left(u\cdot \nabla u,\varphi\right)-\left(\tfrac{\delta}{B} u_3^2,\varphi_2\right)\right|&=
\left|-\left(u\otimes u,\nabla\varphi\right)-\left(\tfrac{\delta}{B} u_3^2,\varphi_2\right)\right|\nonumber\\
&\leq \left\|u\otimes u\right\|_{p'}\|\nabla\varphi\|_p+
\delta m \left\|u_3^2\right\|_{p'}
\left\|\varphi_2\right\|_p\nonumber\\
&\leq \hat\kappa \left(\left\|D u\right\|_{p,B}^2+
\delta \left\|\nabla u_3\right\|_{p,B}^2\right)\left\|\nabla\varphi\right\|_p, \end{align}
where $\hat \kappa$ only depends on $p$, $\Sigma$ and $m$.
Combining (<ref>)-(<ref>), we deduce that
$$\left\|\pi\right\|_{p'}\leq \kappa \left(\left\|Du\right\|_{p,B}+\delta^2 \left\|u_2\right\|_{p,B}+{\cal R}e \left(\left\|D u\right\|_{p,B}^2+
\delta \left\|\nabla u_3\right\|_{p,B}^2\right)\right),$$
where $\kappa$ is a positive constant only depending on $\Sigma$, $p$, $m$ and $n$.
§.§ $\delta$-approximation
Proof of Proposition <ref>. Based on a standard Galerkin approximation of the corresponding global formulation, compactness and monotonicity arguments, the existence of a weak solution for $(E_\sigma)$ can be established once suitable a priori estimates are derived. However, because of the term involving $\sigma(w_3)$, the global formulation
does not seem appropriate unless we restrict strongly the exponent $\alpha$ in (<ref>). To overcome this difficulty, we consider the coupled formulations. Arguing as in the proof of Proposition <ref>, by setting $\boldsymbol\varphi=(0,0,w_3)$ in the corresponding weak formulation we obtain
$$\int_\Sigma\left(1+|D\boldsymbol w|^2\right)^{\frac{p-2}{2}}
|\nabla w_3|^2\,dx=\left(\tfrac{G}{B},w_3\right)$$
and thus
$$\left\|\nabla w_3\right\|_2^2\leq
\left(\tfrac{G}{B},w_3\right)\leq c_1
\|\nabla w_3\|_2 \qquad \mbox{with} \ c_1=
\tfrac{m|G||\Sigma|}{\sqrt{2}} .$$
$$\left\|\nabla w_3\right\|_2\leq c_1, \qquad
\left(\tfrac{G}{B},w_3\right)\leq c_1^2$$
\left\|\nabla w_3\right\|_p^p\leq
\left(\tfrac{G}{B},w_3\right)\leq c_1^2.$$
Similarly, by setting $\boldsymbol\varphi=(w,0)=(w_1,w_2,0)$ in the corresponding weak formulation, we obtain
$$2\int_\Sigma\left(1+|D\boldsymbol w|^2\right)^{\frac{p-2}{2}}
|D w|^2\,dx=\delta \left(\sigma(w_3),w_2\right)$$
and thus
$$\|Dw\|_{2}^2\leq \tfrac{\delta}{2}\left(\sigma(w_3),w_2\right),\qquad \|Dw\|_{p}^p\leq \tfrac{\delta}{2}\left(\sigma(w_3),w_2\right).$$
Estimate (<ref>) together with the Korn inequality yield
$$\|Dw\|_{2}^2\leq \tfrac{c_0D_{2,\alpha}}{2}\, \delta
\left\|\nabla w_3\right\|_2^\alpha
\left\|\nabla w_2\right\|_2\leq\tfrac{c_0D_{2,\alpha}}{2}\,\delta
\left\|\nabla w_3\right\|_2^\alpha
\left\|\nabla w\right\|_2\leq c_0c_2 \delta
\left\|Dw\right\|_2$$
and consequently
$$\|Dw\|_{2}\leq c_0c_2 \delta, \qquad
\left(\sigma(w_3),w_2\right)\leq
$$\|Dw\|_{p}^p\leq \left(c_0c_2\delta\right)^2$$
and the a priori estimates are derived. The proof may be completed using arguments similar to those in the proof of Theorem <ref>. $\hfill\Box$
Arguing as in the proof of Corollary $\ref{pressure_1}$ and using $(\ref{est_psi_1})$, we can prove the existence of $\pi\in L^{p'}(\Sigma)$ such that $(E_\sigma)_1$ holds in $\boldsymbol W^{-1,p'}(\Sigma)$. Moreover, the following estimate holds
$$\left\|\pi\right\|_{p'}\leq \kappa\left(
\|Dw\|_{p}+
{\cal R}e\left\|Dw\right\|^2_{p}+
\delta\left\|\nabla w_3
\right\|_p^\alpha\right),$$
where $\kappa$ is a positive constant only depending on $p$, $\Sigma$, $m$, $n$ and $\alpha$.
Proof of Proposition <ref>. Let us first recall that $\boldsymbol u$ satisfies (<ref>) and that $\boldsymbol w$ satisfies
$$\left({\boldsymbol\tau}\left(D { \boldsymbol w}\right),D
+{\cal R}e\,a\left({ \boldsymbol w},{ \boldsymbol w},{\boldsymbol\varphi}\right)-\left(\pi_2,\nabla \cdot{\boldsymbol\varphi}\right)
for all ${\boldsymbol\varphi}\in \boldsymbol
W^{1,p}_0(\Sigma)$. Therefore
\begin{align}\label{difference1}\left(\boldsymbol\tau(D{ \boldsymbol u})-{\boldsymbol\tau}\left(D { \boldsymbol w}\right),D \boldsymbol\varphi\right)&=\left(\boldsymbol\tau(D{ \boldsymbol u})-{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}\right),D \boldsymbol\varphi\right)+
\left(\nabla^\star\cdot \boldsymbol\tau(D^\star{ \boldsymbol u})-\nabla\cdot \boldsymbol\tau(D^\star{ \boldsymbol u}),\boldsymbol\varphi\right)\nonumber\\
&-{\cal R}e\left(a_\star\left(\boldsymbol u,
\boldsymbol u,\boldsymbol\varphi\right)-a\left({ \boldsymbol w},{ \boldsymbol w},{\boldsymbol\varphi}\right)\right)+\left(\pi_1-\pi_2,\nabla\cdot \boldsymbol\varphi\right)-
\delta\left(\sigma(w_3),\varphi_2\right)\nonumber\\
$\circ$ Let us estimate the first term. By taking into account (<ref>), we have
$$\begin{array}{ll}\left|I_1\right|&=\left|\left({\boldsymbol\tau}\left(D^\star{ \boldsymbol u}\right)-{\boldsymbol\tau}\left(D { \boldsymbol u}\right),D\boldsymbol\varphi\right)\right|\leq \left\|{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}\right)-{\boldsymbol\tau}\left(D { \boldsymbol u}\right)\right\|_{p'}
\left\|D\boldsymbol\varphi\right\|_{p}\vspace{1mm}\\
&\leq \displaystyle (p-1) {\cal F}_1\left(\|D^\star{ \boldsymbol u}\|_{p}+\|D{ \boldsymbol u}\|_{p}\right)
\left\|D^\star{ \boldsymbol u}-D { \boldsymbol u}\right\|_{p}
\left\|D\boldsymbol\varphi\right\|_{p}.
\end{array}$$
\begin{equation}\label{w_wstar}\left\|D^\star{ \boldsymbol u}-D { \boldsymbol u}\right\|_{p}\leq \delta m \left(\|u_2\|_p+\|u_3\|_p\right),\end{equation}
we deduce that
\begin{equation}\label{difference_est3}\left|I_1\right|\leq F_ 1 \,\delta \left\|D\boldsymbol\varphi\right\|_{p},\end{equation}
where $F_1=m(p-1) {\cal F}_1\big(\|D^\star{ \boldsymbol u}\|_{p}+\|D{ \boldsymbol u}\|_{p}\big)\left(\|u_2\|_p+\|u_3\|_p\right)$.
$\circ$ Estimate (<ref>) together with the Sobolev inequality (<ref>) and the Korn inequality (<ref>) yield
\begin{align}\label{difference_est4}\left|I_2\right|&\leq
\left\|\nabla^\star\cdot \boldsymbol\tau(D^\star{ \boldsymbol u})-\nabla\cdot \boldsymbol\tau(D^\star{ \boldsymbol u})\right\|_{p'}
\left\|\boldsymbol\varphi\right\|_p\nonumber\\
&\leq 4S_{2,p}m{\cal F}_1\big(\|D^\star{ \boldsymbol u}\|_p\big)
\left(\left\|Du\right\|_{p}
+\left\|D_{33}^\star{ \boldsymbol u}\right\|_{p}+\left\|D_{23}^\star{ \boldsymbol u}\right\|_{p}\right)\, \delta
\left\|\nabla \boldsymbol\varphi\right\|_{2}\leq F_2\, \delta
\left\|D\boldsymbol\varphi\right\|_{p},\end{align}
where $F_2=4\sqrt{6}m|\Sigma|^{\frac{1}{2}-\frac{1}{p}}\, S_{2,p}\, {\cal F}_1\big(\|D^\star{ \boldsymbol u}\|_p\big)\|D^\star{ \boldsymbol u}\|_p$.
$\circ$ Similarly,
\begin{align}\label{difference_est61}\tfrac{1}{{\cal R}e}\left|I_3\right|
&=\left|a_\star({ \boldsymbol u},{ \boldsymbol u},\boldsymbol \varphi)-a({ \boldsymbol w},{ \boldsymbol w},\boldsymbol \varphi)\right|
\nonumber\\
&=\left|a_\star\left({ \boldsymbol u}-{ \boldsymbol w},{ \boldsymbol u},\boldsymbol \varphi\right)+a\left(\boldsymbol w,{ \boldsymbol u}-{ \boldsymbol w},\boldsymbol \varphi\right)+
\left({ \boldsymbol w}\cdot \nabla^\star{ \boldsymbol u}-{ \boldsymbol w}\cdot \nabla { \boldsymbol u},\boldsymbol \varphi\right)\right|\nonumber\\
&\leq\left|a_\star\left({ \boldsymbol u}-{ \boldsymbol w},{ \boldsymbol u},\boldsymbol \varphi\right)\right|
+\left|a\left(\boldsymbol w,{ \boldsymbol u}-{ \boldsymbol w},\boldsymbol \varphi\right)\right|+\left|
\left({ \boldsymbol w}\cdot \nabla^\star{ \boldsymbol u}-{ \boldsymbol w}\cdot \nabla { \boldsymbol u},\boldsymbol \varphi\right)
\right|\nonumber\\
&=\left|a_\star\left({ \boldsymbol u}-{ \boldsymbol w},{ \boldsymbol u},\boldsymbol \varphi\right)\right|+\left|a\left(\boldsymbol w,{ \boldsymbol u}-{ \boldsymbol w},\boldsymbol \varphi\right)\right|+
\left|\left(\tfrac{\delta}{B}\,w_2u_2,\varphi_3\right)-\left(\tfrac{\delta}{B}\,w_3u_3,\varphi_2\right)\right|\nonumber\\
&\leq\left|a_\star\left({ \boldsymbol u}-{ \boldsymbol w},{ \boldsymbol u},\boldsymbol \varphi\right)\right|+\left|a\left(\boldsymbol w,{ \boldsymbol u}-{ \boldsymbol w},\boldsymbol \varphi\right)\right|+
\displaystyle\left\|\tfrac{\delta}{B}|\boldsymbol w||\boldsymbol u||\boldsymbol \varphi|\right\|_1\nonumber\\
&\leq \left\|{ \boldsymbol u}-{ \boldsymbol w}\right\|_4
\left\|\nabla^\star{ \boldsymbol u}\right\|_2\left\|\boldsymbol \varphi\right\|_4 +\left\|{ \boldsymbol w}\right\|_4
\left\|\nabla\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_2\left\|\boldsymbol \varphi\right\|_4 + \delta m \left\|\boldsymbol w\right\|_4 \left\|\boldsymbol u\right\|_4
\left\|\boldsymbol \varphi\right\|_2\nonumber\\
&\leq \left(1+S_{2,2}\right)\left( S_{2,4}\right)^2
\left(\left\|\nabla\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_2\left(\left\|\nabla^\star{ \boldsymbol u}\right\|_2+\left\|\nabla{ \boldsymbol w}\right\|_2\right)+\delta m\left\|\nabla{ \boldsymbol w}\right\|_2\left\|\nabla{ \boldsymbol u}\right\|_2\right)
\left\|\nabla \boldsymbol \varphi\right\|_2\nonumber\\
F_3\left(\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{2}\left\|D\boldsymbol\varphi\right\|_2+\delta
\left\|D\boldsymbol\varphi\right\|_{p}\right)\end{align}
with $F_3=\left(1+m\right)\sqrt{8} \left(1+S_{2,2}\right)\left(S_{2,4}\right)^2\left(\left\|D^\star{ \boldsymbol u}\right\|_{2}+\left\|D{ \boldsymbol w}\right\|_{2}+\left\|D\boldsymbol u\right\|_{2}\left\|D{ \boldsymbol w}\right\|_{2}\right)$.
$\circ$ Let us now consider the term involving the pressure
$$\left|I_4\right|=\left|\left(\pi_1-\pi_2,\nabla \cdot
\boldsymbol\varphi\right)\right|\leq \left\|\pi_1-\pi_2\right\|_{p'}\left\|\nabla\cdot \boldsymbol\varphi\right\|_p.$$
Arguing as in the first part of the proof of Corollary <ref>, we can see that
&\leq \tilde \kappa \left(\left\|\tau(D^\star{ \boldsymbol u})-\tau(D{ \boldsymbol w})\right\|_{p'}+\left\|\nabla^\star\cdot \tau(D^\star{ \boldsymbol u})-\nabla\cdot \tau(D^\star{ \boldsymbol u})\right\|_{p'}\right)\vspace{2mm}\\
&+\tilde \kappa {\cal R}e
\left(\left\|u\otimes u-w\otimes w\right\|_{p'}+
\delta \left\|u_3^2\right\|_{p'}\right)+\tilde \kappa \delta
\left\|\sigma(w_3)\right\|_{p'},\end{array}$$
where $\tilde \kappa$ depends only on $\Sigma$, $p$, $m$ and $n$.
Taking into account (<ref>) and (<ref>), we have
$$\begin{array}{ll}\left\|\tau(D^\star{ \boldsymbol u})-\tau(D{ \boldsymbol w})
\right\|_{p'}&\leq
\left\|\boldsymbol\tau(D^\star{ \boldsymbol u})-\boldsymbol\tau(D{ \boldsymbol w})\right\|_{p'}\leq F_{4,1}\left\|D^\star{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}\vspace{2mm}\\
&\leq F_{4,1}\left(\left\|D^\star{ \boldsymbol u}-D{ \boldsymbol u}\right\|_{p}
+\left\|D{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}\right)\vspace{2mm}\\
&\leq F_{4,2}\left(\delta+\left\|D{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}
\right),\end{array}$$
where $F_{4,1}=(p-1){\cal F}_1\big(\|D^\star{ \boldsymbol u}\|_{p}
+\|D { \boldsymbol w}\|_{p}\big)$ and $F_{4,2}=F_{4,1}\left(1+ m\left(\|u_2\|_p+\|u_3\|_p\right)\right)$.
Moreover, by using (<ref>), we obtain
$$\left\|\nabla^\star\cdot \tau(D^\star{ \boldsymbol u})-\nabla\cdot \tau(D^\star{ \boldsymbol u})
\right\|_{p'} \leq F_{4,3} \,\delta,$$
where $F_{4,3}= 4\sqrt{3}m {\cal F}_1\big(\|D^\star{ \boldsymbol u}\|_p\big)\|D^\star{ \boldsymbol u}\|_p $.
Similarly, by using the Sobolev inequality (<ref>) and the Korn inequality (<ref>)
$$\begin{array}{ll}\left\| u\otimes u-w\otimes w\right\|_{2}+\delta \left\|u_3^2\right\|_{2}
&\leq \left\|\left(u+w\right)
\otimes \left( u-w\right)\right\|_{2}+\delta \left\|u_3\right\|^2_{4}\vspace{2mm}\\
&\leq \left\|u+w\right\|_{4}
\left\|u-w\right\|_{4}+\delta \left\|u_3\right\|^2_{4}\vspace{2mm}\\
&\leq S_{2,4}
\left\|\nabla\left(u-w\right)
\right\|_2\left\|u+w\right\|_{4}+\delta\, \left\|u_3\right\|^2_{4}\vspace{2mm}\\
&\leq S_{2,4} \,
\sqrt{2}
\left\|D\left(\boldsymbol u-\boldsymbol w\right)
\right\|_{2}\left\|u+w\right\|_{4}+\delta\, \left\|u_3\right\|^2_{4}\end{array}$$
and thus
$$\begin{array}{ll}\left\| u\otimes u-w\otimes w\right\|_{p'}+\delta \left\|u_3^2\right\|_{p'}
&\leq |\Sigma|^{\frac{1}{p'}-\frac{1}{2}}
\left(\left\| u\otimes u-w\otimes w\right\|_{2}+\delta \left\|u_3^2\right\|_{2}\right)\vspace{1mm}\\
&\leq F_{4,4} \left(
\left\|D\left(\boldsymbol u-\boldsymbol w\right)\right\|_p+\delta\right),\end{array}$$
where $F_{4,4}=|\Sigma|^{\frac{1}{p'}-\frac{1}{2}}\left(\sqrt{2}\, S_{2,4}\,
|\Sigma|^{\frac{1}{2}-\frac{1}{p}}\left\|u+w\right\|_{4}+\left\|u_3\right\|^2_{4}\right)$. On the other hand, due to (<ref>) we have
\left\|\sigma(w_3)\right\|_{p'}\leq E_{\alpha,p}
\left\|\nabla w_3\right\|_{2}^\alpha=F_{4,5}.\end{array}$$
Combining these estimates, we deduce that
\begin{align}\label{difference_est8}\left|I_4\right|&\leq \tilde\kappa\left(\left(F_{4,2}+{\cal R}e\, F_{4,4}\right)
\left\|D{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}+ \left(F_{4,2}+F_{4,3}+{\cal R}e\,F_{4,4}+F_{4,5}\right)\delta\right) \left\|\nabla\cdot \boldsymbol\varphi\right\|_p\nonumber\\
&\leq F_ 4\left(\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}+
\delta\right)\left\|\nabla\cdot \boldsymbol\varphi\right\|_p.\end{align}
$\circ$ Finally, taking into account (<ref>) we obtain
\begin{align}\label{difference_est_psi}\left|I_5\right|&=\delta \left|\left(\sigma(w_3),
\varphi_2\right)\right|\leq D_{2,\alpha} \,\delta
\left\|\nabla w_3\right\|^\alpha_2
\left\|\nabla\varphi_2\right\|_{2}\nonumber\\
&\leq D_{2,\alpha}\,\delta
\left\|\nabla w_3\right\|^\alpha_2
\left\|\nabla\boldsymbol\varphi
\right\|_{2}\leq \sqrt{2} D_{2,\alpha}\,\delta
\left\|\nabla w_3\right\|^\alpha_2
\left\|D\boldsymbol\varphi
\right\|_{2}\nonumber\\
&\leq F_5\,\delta
\left\|D\boldsymbol\varphi
\right\|_{p},\end{align}
where $F_5=\sqrt{2}\,D_{2,\alpha} |\Sigma|^{\frac{1}{2}-\frac{1}{p}}
\left\|\nabla w_3\right\|^\alpha_2$.
$\circ$ Combining (<ref>), (<ref>)-(<ref>) yields
$$\begin{array}{ll}\left|\left(\boldsymbol\tau(D{ \boldsymbol u})-{\boldsymbol\tau}\left(D { \boldsymbol w}\right),D \boldsymbol\varphi\right)\right|&\leq \left(F_1+F_2+{\cal R}e\, F_3+F_5\right)\, \delta\left\|D\boldsymbol\varphi\right\|_{p}+
{\cal R}e\,F_3 \left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{2}\left\|D\boldsymbol\varphi\right\|_2\vspace{2mm}\\
&+F_ 4\left(\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}+
\delta\right)\left\|\nabla\cdot \boldsymbol\varphi\right\|_p.\end{array}$$
Setting $\boldsymbol\varphi=\boldsymbol u-\boldsymbol w$, taking into account $(\ref{estim_S3})$ and the estimates associated to
$\boldsymbol u$ and $\boldsymbol w$, and using the Young inequality, it follows that for every $\varepsilon>0$,
\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)
\right\|_{p}^p+
\tfrac{1}{2}\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}
\right)\right\|_{2}^2
&\leq \left({\boldsymbol\tau}\left(D{ \boldsymbol u}\right)
-{\boldsymbol\tau}\left(D{ \boldsymbol w}\right),
D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right)
\vspace{2mm}\\
&\leq \left(F_1+F_2+{\cal R}e\, F_3+F_5\right)\, \delta\left\|D\left(\boldsymbol u-\boldsymbol w\right)\right\|_{p}\vspace{2mm}\\
& \ +{\cal R}e\,F_3 \left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{2}^2\vspace{2mm}\\
& \ +F_ 4 \left(\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}+
\delta\right)\left\|\tfrac{\delta}{B}u_2\right\|_p\vspace{2mm}\\
&\leq C_1\left(\delta\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}+
{\cal R}e \left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{2}^2+\delta^2\right)\vspace{2mm}\\
& \leq
\varepsilon \left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}^p+C_2(\varepsilon)\, \delta^{p'}\vspace{2mm}\\
& \ +C_1\left({\cal R}e
\left\|D\left(\boldsymbol u-\boldsymbol w\right)\right\|_{2}^2
where $C_1$ is a positive constant only depending on $\Sigma$, $p$ and $m$. Observing that $\delta^{2}<\delta^{p'}$, choosing $\varepsilon= \tfrac{1}{2^{p+1}(p-1)}$ and assuming that $C_1{\cal R}e<\tfrac{1}{4}$, we deduce that
$$ \tfrac{1}{2^{p+1}(p-1)}\left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{p}^p+
\tfrac{1}{4}\left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{2}^2\leq
and the claimed result is proved. $\hfill\Box$
§ SHEAR-THINNING FLOWS
Let us now consider the case of shear-thinning fluids (corresponding to $p<2$). As for the shear-thickening fluids, we derive a Korn inequality, establish some estimates on the convective term and on the extra stress tensor and prove existence and uniqueness results. As previously observed in Section <ref>, we will restrict the exponent $p$ in order to ensure the uniqueness of the solution and carry out the approximation analysis with respect to $\delta$.
§.§ On the Korn inequality
Let us notice that if the classical Korn inequality can be applied to
the tensor $D^\star\boldsymbol u$ with $\delta=0$ or to $Du=\left(D_{ij}^\star\boldsymbol u\right)_{i=1,2}$ with $\delta>0$, this is no more necessarily the case if we consider $D^\star\boldsymbol u$ with $\delta>0$. The difficulty, basically related with the term $D_{23}^\star \boldsymbol u=
\tfrac{\partial u_3}{\partial x_2}-\tfrac{\delta}{B} u_3$, is overcome in the case $p=2$ by using the Hilbert setting and the fact that ${u_3}_{\mid \partial \Sigma}=0$. Indeed, since
$$\left(\tfrac{\partial u_3}{\partial x_2}, u_3\right)=0,$$
we obtain
$$\left\|D_{23}^\star \boldsymbol u\right\|_{2,B}^2=
\left\|\tfrac{\partial u_3}{\partial x_2}-\tfrac{\delta}{B} u_3\right\|_{2,B}^2=\left\|\tfrac{\partial u_3}{\partial x_2}\right\|_{2,B}^2+\left\|\tfrac{\delta}{B} u_3\right\|_{2,B}^2.$$
This argument is one of the key points in the proof of the corresponding Korn inequality (see Section <ref>) but
does not apply in the $L^p$ setting. The issue is overcome by using the Poincaré inequality (<ref>) that involves only the first component $\tfrac{\partial u_3}{\partial x_1}$ of the gradient $\nabla u_3$.
Let $ { \boldsymbol u}=(u_1,u_2,u_3) \in { \boldsymbol W}^{1,p}_0(\Sigma)$ with $1<p<\infty$. Then
\begin{equation}\label{korn-inequality} C_K\left\|\nabla^\star { \boldsymbol u}\right\|_{p,B}\leq
\left\|D^\star{ \boldsymbol u}\right\|_{p,B}\end{equation}
with $C_K=\tfrac{C_{K,1}\left(n m\right)^{-\frac{1}{p}}}{2(1+\delta m)}$, where $C_{K,1}$ is the classical Korn constant
in $\boldsymbol W^{1,p}_0(\Sigma)$.
Proof. Let us first observe that due to (<ref>), we have
\begin{equation}\label{korn_in1}\left\|\tfrac{\delta}{B}u_3\right\|_{p,B}\leq
\left\|\tfrac{\delta}{B}
\tfrac{\partial u_3}{\partial x_1}\right\|_{p,B}\leq 2\delta m
\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}.\end{equation}
It follows that
\left\|D_{23}\boldsymbol u\right\|_{p,B}^p&\leq
\left(\left\|D_{23}^\star\boldsymbol u\right\|_{p,B}
\vspace{1mm}\\
\left(\left\|D_{23}^\star\boldsymbol u\right\|_{p,B}
+\delta m
\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}\right)^p
\vspace{1mm}\\
&\leq 2^{p-1}\left(\left\|D_{23}^\star\boldsymbol u
\right\|_{p,B}^p
+\left(\delta m\right)^p
\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p\right)
\end{array}$$
and thus
\begin{align}\label{korn_in2}\left\|D\boldsymbol u\right\|_{p,B}^p&=
\left\|D u\right\|_{p,B}^p+
2\left\|D_{13} \boldsymbol u\right\|_{p,B}^p+
2\left\|D_{23} \boldsymbol u\right\|_{p,B}^p\nonumber\\
&\leq \left\|D u\right\|_{p,B}^p+
2\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p
+2^{p}\left\|D_{23}^\star\boldsymbol u
\right\|_{p,B}^p
+\left(2\delta m\right)^p
\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p.\end{align}
On the other hand, taking into account the definition of $\nabla^\star$ and using the classical Korn inequality, we can easily see that
\begin{align}\label{korn_in3}
\left\|\nabla^\star{ \boldsymbol u}\right\|_{p,B}^p&=
\left\|\nabla{ \boldsymbol u}\right\|_{p,B}^p+
\left\|\tfrac{\delta}{B}u_2\right\|_{p,B}^p+
\left\|\tfrac{\delta}{B}u_3\right\|_{p,B}^p\nonumber\\
&\leq n\left\|\nabla{ \boldsymbol u}\right\|_{p}^p+
\left\|\tfrac{\delta}{B}u_2\right\|_{p,B}^p+
\left\|\tfrac{\delta}{B}u_3\right\|_{p,B}^p\nonumber\\
&\leq \tfrac{n}{C_{K,1}^p}\left\|D{ \boldsymbol u}\right\|_{p}^p+
\left\|\tfrac{\delta}{B}u_2\right\|_{p,B}^p+
\left\|\tfrac{\delta}{B}u_3\right\|_{p,B}^p\nonumber\\
&\leq \tfrac{mn}{C_{K,1}^p}
\left\|D{ \boldsymbol u}\right\|_{p,B}^p+
\left\|\tfrac{\delta}{B}u_2\right\|_{p,B}^p+
\left\|\tfrac{\delta}{B}u_3\right\|_{p,B}^p.\end{align}
Combining (<ref>)-(<ref>), we deduce that
\left\|\nabla^\star{ \boldsymbol u}\right\|_{p,B}^p\vspace{2mm}\\
&\leq \tfrac{mn}{C_{K,1}^p}
\left(\left\|D u\right\|_{p,B}^p+
2\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p
+2^{p}\left\|D_{23}^\star\boldsymbol u
\right\|_{p,B}^p
+\left(2\delta m\right)^p
\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p\right)
\vspace{2mm}\\
&+\left(2\delta m\right)^p
\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p+
\left\|D_{33}^\star \boldsymbol u\right\|_{p,B}^p\vspace{2mm}\\
&\leq \tfrac{mn}{C_{K,1}^p}\left(
\left\|Du\right\|_{p,B}^p+
\left(2+\left(2\delta m\right)^p\left(1+\tfrac{C_{K,1}^p}{mn}\right)\right)\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p+
2^p\left\|D_{23}^\star \boldsymbol u\right\|_{p,B}^p+
\left\|D_{33}^\star \boldsymbol u\right\|_{p,B}^p\right)\vspace{2mm}\\
&\leq \tfrac{\left(2+2\delta m\right)^p mn}{C_{K,1}^p}\left(
\left\|D u\right\|_{p,B}^p+
2\left\|D_{13}^\star \boldsymbol u\right\|_{p,B}^p+
2\left\|D_{23}^\star \boldsymbol u\right\|_{p,B}^p+
\left\|D_{33}^\star \boldsymbol u\right\|_{p,B}^p\right)\vspace{2mm}\\
&=\tfrac{1}{C_{K}^p} \left\|D^\star \boldsymbol u\right\|_{p,B}^p
\end{array}$$
and the claimed result is proved. $\hfill\Box$
§.§ Estimates on the convective term and extra stress tensor
We begin by a continuity property in $\boldsymbol W^{1,p}_0(\Sigma)$ of the trilinear form $a_\star$.
Let ${ \boldsymbol u}$, ${ \boldsymbol v}$ and ${ \boldsymbol w}$ be in ${ \boldsymbol W}^{1,p}_0(\Sigma)$ with $\tfrac{3}{2}\leq p<2$. Then the following estimate holds
$$\left|a_\star({ \boldsymbol u},{ \boldsymbol v},B{ \boldsymbol w})\right|\leq
\kappa_6
\left\|D^\star{ \boldsymbol u}\right\|_{p,B}\left\|D^\star{ \boldsymbol v}\right\|_{p,B}
\left\|D^\star{ \boldsymbol w}\right\|_{p,B},$$
with $\kappa_6=\tfrac{ n m^{\frac{3}{p}}}{C_K^3}\left( S_{p,2p'}\right)^2$ and where $C_K$ is the Korn constant given in $(\ref{korn-inequality})$.
Proof. Hölder's inequality and Sobolev's inequality with $r=2p'$ and $q=p$ show that
$$\begin{array}{ll}\label{a_alpha}\left|a_\star({ \boldsymbol u},{ \boldsymbol v},B{ \boldsymbol w})\right|
&\leq n\left\|{ \boldsymbol u}\right\|_{2p'}
\left\|\nabla^\star { \boldsymbol v}\right\|_{p}
\left\|{ \boldsymbol w}\right\|_{2p'}\vspace{2mm}\\
&\leq n\left( S_{p,2p'}\right)^2
\left\|\nabla \boldsymbol u\right\|_{p}
\left\|\nabla^\star { \boldsymbol v}\right\|_{p}
\left\|\nabla \boldsymbol w\right\|_{p}\vspace{2mm}\\
&\leq n m^{\frac{3}{p}}\left( S_{p,2p'}\right)^2
\left\|\nabla \boldsymbol u\right\|_{p,B}
\left\|\nabla^\star { \boldsymbol v}\right\|_{p,B}
\left\|\nabla \boldsymbol w\right\|_{p,B}.\end{array}$$
The estimate follows by using the Korn inequality (<ref>). $\hfill\Box$
In the remaining part of this section, we study some properties of the extra stress tensor and derive some associated estimates. We begin by an auxiliary result that will be useful in the sequel.
Let $1<p< 2$ and let $H_1\in L^{\frac{p}{2-p}}(\Sigma)$, $H_2\in L^1(\Sigma)$ and $H_3\in L^p(\Sigma)$ be non negative functions satisfying
$$H_3(x)^2\leq H_1(x)H_2(x)
\qquad \mbox{for a.e.} \ x\in \Sigma.$$
$$ \left\|H_3\right\|_p^2\leq \left\|H_1\right\|_{\frac{p}{2-p}}\left\|H_2\right\|_1.$$
Proof. Taking into account the condition satisfied by $H_1$, $H_2$, $H_3$, integrating and using the Hölder inequality, we obtain
\left\|H_3\right\|_p^p&=\displaystyle
\int_\Sigma \left(H_3(x)^2\right)^\frac{p}{2}dx
\leq \int_\Sigma H_1(x)^\frac{p}{2}H_2(x)^\frac{p}{2}dx
\vspace{2mm}\\
&\leq \left\|H_1^\frac{p}{2}\right\|_{\frac{2}{2-p}}
\left\|H_2^\frac{p}{2}\right\|_{\frac{2}{p}}
and the proof is complete.$\hfill\Box$
The next result deals with continuity, coercivity and monotonicity results for the extra stress tensor $\boldsymbol\tau $.
Assume that $1<p<2$ and let $\boldsymbol f, \boldsymbol g\in { \boldsymbol L}^{p}(\Sigma,\mathbb{R}^{3\times 3})$. Then the following estimates hold
* Continuity.
\begin{equation}\label{estim_S5}\left\|\left(1+|\boldsymbol f|^2\right)^{\frac{p-2}{2}}\boldsymbol g\right\|_{p',B}
\leq \left\|\boldsymbol g\right\|_{p,B}^{p-1} \qquad \mbox{if} \ |\boldsymbol g|\leq |\boldsymbol f|,\end{equation}
\begin{equation}\label{estim_S7_2}\left\|{\boldsymbol\tau }\left(\boldsymbol f\right)-{\boldsymbol\tau }\left(\boldsymbol g\right)\right\|_{p',B}\leq 2C_{p}\left\|\boldsymbol f-\boldsymbol g\right\|_{p,B}^{p-1} \qquad \mbox{with} \ C_{p}=1+2^{\frac{2-p}{2}},\end{equation}
* Coercivity.
\begin{equation}\label{estim_S6}\left({\boldsymbol\tau }\left(\boldsymbol f\right),B \boldsymbol f\right)\geq
\tfrac{2\|\boldsymbol f\|^2_{p,B}}{\left(\|B\|_1
+\|\boldsymbol f\|_{p,B}^p\right)^{\frac{2-p}{p}}}, \end{equation}
* Monotonicity.
\begin{equation}\label{estim_S7}\left({\boldsymbol\tau }\left(\boldsymbol f\right)-
{\boldsymbol\tau }\left(\boldsymbol g\right),B (\boldsymbol f-\boldsymbol g)\right)\geq
\tfrac{2(p-1)\|\boldsymbol f-\boldsymbol g\|^2_{p,B}}{\left(\|B\|_1
+\|\boldsymbol f\|_{p,B}^p+\|\boldsymbol g\|_{p,B}^p
\right)^{\frac{2-p}{p}}}.\end{equation}
Proof. Standard calculation show that if $|\boldsymbol g|\leq |\boldsymbol f|$, then
\left\|\left(1+|\boldsymbol f|^2\right)^{\frac{p-2}{2}}
\boldsymbol g
\right\|_{p',B}^{p'}&=\displaystyle\int_\Sigma B
\left((1+ |\boldsymbol f|^2)^{\frac{p-2}{2}}|\boldsymbol g|
\right)^{p'}\,dx
\vspace{2mm}\\
\displaystyle\int_\Sigma B\left((1+ |\boldsymbol g|^2)^{\frac{p-2}{2}}|\boldsymbol g|\right)^{p'}\,dx\vspace{2mm}\\
\displaystyle\int_\Sigma B\left((
|\boldsymbol g|^2)^{\frac{p-2}{2}}
|\boldsymbol g|\right)^{p'}\,dx
=\|\boldsymbol g\|_{p,B}^{p}\end{array}$$
which gives (<ref>). On the other hand, due to the monotonicity property in Lemma <ref>, we have
\begin{align}\label{monot_B}\tfrac{1}{2(p -1)}\left({\boldsymbol\tau }\left(f\right)-
{\boldsymbol\tau }\left(\boldsymbol g\right)\right):B(\boldsymbol f-\boldsymbol g)&\geq B\left(1+|\boldsymbol f|^{2}+|\boldsymbol g|^2\right)^{\frac{2-p}{2}}|\boldsymbol f-\boldsymbol g|^2\nonumber\\
&= \left(B^{\frac{2}{p}}+|B^{\frac{1}{p}}\boldsymbol f|^{2}+|B^{\frac{1}{p}}\boldsymbol g|^2\right)^{\frac{2-p}{2}}\big|B^{\frac{1}{p}}(\boldsymbol f-\boldsymbol g)\big|^2.
\end{align}
$$\begin{array}{ll}H_1=\left(B^{\frac{2}{p}}+|B^{\frac{1}{p}}\boldsymbol f|^{2}+|B^{\frac{1}{p}}\boldsymbol g|^2\right)^{\frac{2-p}{2}},\vspace{2mm}\\
H_2=\tfrac{1}{2(p -1)}\left({\boldsymbol\tau }\left(\boldsymbol f\right)-{\boldsymbol\tau }\left(\boldsymbol g\right):B(\boldsymbol f-\boldsymbol g)\right),\qquad
H_3=\big|B^{\frac{1}{p}}(\boldsymbol f-\boldsymbol g)\big|.\end{array}$$
Since $\boldsymbol f$ and $\boldsymbol g$ belong to $\boldsymbol L^{p}(\Sigma,\mathbb{R}^{3\times 3})$, it is easy to see that $H_1\in L^{\frac{p}{2-p}}(\Sigma)$, $H_2\in
L^1(\Sigma)$ and $H_3\in L^p(\Sigma)$ and that due to (<ref>), we have
$$H_3(x)^2\leq H_1(x)H_2(x) \qquad \mbox{for a.e.} \ x\in \Sigma.$$
Due to Lemma <ref>, we obtain
$$\begin{array}{ll}2(p -1) \left\|\boldsymbol f-\boldsymbol g
\right\|_{p,B}^2
&\leq\left\|\left(B^{\frac{2}{p}}+|B^{\frac{1}{p}}\boldsymbol f
|^{2}+|B^{\frac{1}{p}}\boldsymbol g|^2\right)^{\frac{2-p}{2}}
\right\|_{\frac{p}{2-p}}
\left\|\left({\boldsymbol\tau }\left(\boldsymbol f\right)-{\boldsymbol\tau }\left(\boldsymbol g\right)\right):B(\boldsymbol f-\boldsymbol g)\right\|_1\\
+\|\boldsymbol f\|_{p,B}^p+\|\boldsymbol g\|_{p,B}^p
\right)^{\frac{2-p}{p}}
\left({\boldsymbol\tau }\left(\boldsymbol f\right)-{\boldsymbol\tau }\left(\boldsymbol g\right),B(\boldsymbol f-\boldsymbol g)\right)\end{array}$$
which gives (<ref>). Estimate (<ref>) can be obtained very similarly by using the coercivity condition in Lemma <ref>. Finally, by taking into account Lemma <ref>, we have
$$\left\|{\boldsymbol\tau }\left(\boldsymbol f\right)-{\boldsymbol\tau }\left(\boldsymbol g\right)\right\|_{p',B}
\leq \displaystyle 2C_{p}\left\||\boldsymbol f-\boldsymbol g|^{p-1}\right\|_{p',B} =2C_{p}\left\|\boldsymbol f-\boldsymbol g\right\|^{p-1}_{p,B}$$
which gives estimate (<ref>). $\hfill\Box$
Assume that $\tfrac{3}{2}\leq p< 2$ and let $\boldsymbol f\in { \boldsymbol L}^p(\Sigma,\mathbb{R}^{3\times 3})$. Then the following estimate holds
\begin{equation}\label{diff_div_thin}\left\|\nabla^\star\cdot \boldsymbol\tau (\boldsymbol f)-\nabla\cdot \boldsymbol\tau (\boldsymbol f)\right\|_{p'}\leq 4\delta m
\left(\left\|f\right\|_{p}^{p-1}
\left\|f_{23}\right\|_{p}^{p-1}\right).\end{equation}
where $f=(f_{ij})_{ij=1,2}$.
Proof. Taking into account (<ref>), the estimate can be derived by following step by step the proof of Proposition <ref>.$\hfill \Box$
§.§ Existence and uniqueness of shear-thinning flows
Assume that $\tfrac{3}{2}\leq p<2$ and let $\boldsymbol u$ be a weak solution of $(\ref{weak_formulation})$. Then, estimates
$(\ref{main_estimates_1_thin})$-$(\ref{est_u_thin})$ hold.
Proof. The proof is split into three steps.
Step 1. Let us set ${\boldsymbol\varphi}={ \boldsymbol u}$ in the weak formulation (<ref>) and use Lemma <ref> and (<ref>)
to obtain
\begin{equation}\label{coerciv_B}\tfrac{2\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^2}{\left(\|B\|_1
+\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^{p}\right)^{\frac{2-p}{p}}}
\leq \left({\boldsymbol\tau }\left(D^\star{ \boldsymbol u}\right),BD^\star{ \boldsymbol u}\right)=\left(G,u_3\right).\end{equation}
On the other hand, classical arguments together with Poincaré inequality (<ref>) give
$$\left(G,u_3\right)\leq 2\kappa_1
\left\|u_3\right\|_{p,B}\leq 2\kappa_1
\left\|\tfrac{\partial u_3}{\partial x_1}\right\|_{p,B},$$
where $\kappa_1=\tfrac{m^{\frac{1}{p}}}{2}|G||\Sigma|^{\frac{1}{p'}}$. Observing that
$$\left\|D^\star{ \boldsymbol u}\right\|^p_{p,B}=
\int_\Sigma B\left(|D^\star{ \boldsymbol u}|^2\right)^{\frac{p}{2}}\,dx\geq
\left\|\tfrac{\partial u_3}{\partial x_1}\right\|^p_{p,B},$$
we deduce that
\begin{equation}\label{sec_m_B}\left(G,u_3\right)\leq 2\kappa_1
\left\|D^\star{ \boldsymbol u}\right\|_{p,B}.\end{equation}
Due to (<ref>) and (<ref>), we have
$$\left\|D^\star{ \boldsymbol u}\right\|_{p,B}\leq
\kappa_1\left(\|B\|_1
+\left\|D^\star{ \boldsymbol u}\right\|^{p}_{p,B}\right)^{\frac{2-p}{p}}$$
and thus
\begin{equation}\label{est_s5gen}\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^{\frac{p}{2-p}}\leq
\kappa_1^{\frac{p}{2-p}}\left(\|B\|_1
+\left\|D^\star{ \boldsymbol u}\right\|^{p}_{p,B}\right).
\end{equation}
The Young inequality yields
\begin{equation}\label{est_s6gen}
\kappa_1^{\frac{p}{2-p}}\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^{p}\leq (2-p)
\left\|D^\star{ \boldsymbol u}\right\|^{\frac{p}{2-p}}_{p,B}+
and by combining (<ref>) and (<ref>), we deduce that
$$(p-1)\left\|D^\star{ \boldsymbol u}\right\|^{\frac{p}{2-p}}_{p,B}\leq
\kappa_1^{\frac{p}{2-p}}\|B\|_1+(p-1)\kappa_1^{\frac{p}{(2-p)(p-1)}}.$$
$$\left\|D^\star{ \boldsymbol u}\right\|_{p,B}\leq \kappa_1 \left(\tfrac{\|B\|_1}{p-1}+\kappa_1
^{p'}\right)^{\frac{2-p}{p}} $$
and estimate $(\ref{main_estimates_1_thin})$ is proved.
Step 2. Let us now prove (<ref>). Similar arguments together with
the coercivity property and (<ref>) show that
\begin{align}\label{estima_M1}\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^p&=\displaystyle\int_ {\Sigma_{\boldsymbol u}}
B|D^\star{ \boldsymbol u}|^p\,dx+
\int_{\Sigma\setminus \Sigma_{\boldsymbol u}}B|D^\star{ \boldsymbol u}|^p\,dx\vspace{2mm}\nonumber\\
&\leq \displaystyle \int_ {\Sigma_{\boldsymbol u}}
\tfrac{|B^\frac{1}{p}D^\star{ \boldsymbol u}|^2}{|B^\frac{1}{p} D^\star{ \boldsymbol u}|^{2-p}}\,dx+
\|B\|_1\nonumber\vspace{2mm}\\
& \leq 2^{\frac{2-p}{2}} \displaystyle \int_ {\Sigma_{\boldsymbol u}}\tfrac{|B^\frac{1}{p}D^\star{ \boldsymbol u}|^2}{\left(B^\frac{2}{p}+|B^\frac{1}{p}D^\star{ \boldsymbol u}|^2\right)^{\frac{2-p}{2}}}\,dx+\|B\|_1\nonumber\vspace{2mm}\\
&\leq 2^{\frac{2-p}{2}}\int_ {\Sigma_{\boldsymbol u}}
{\boldsymbol\tau }(D^\star{ \boldsymbol u}):BD^\star{ \boldsymbol u}\,dx+ \|B\|_1\nonumber\vspace{2mm}\\
&\leq 2^{\frac{2-p}{2}} \left({\boldsymbol\tau }(D^\star{ \boldsymbol u}),BD^\star{ \boldsymbol u}\right)+ \|B\|_1= 2^{\frac{2-p}{2}}\left(G,u_3\right)+
\|B\|_1\nonumber\\
& \leq 2^{\frac{2-p}{2}}\kappa_1\left\|D^\star{ \boldsymbol u}\right\|_{p,B}+ \|B\|_1,\end{align}
where $\Sigma_{\boldsymbol u}=\left\{x\in \Sigma\mid B^\frac{1}{p}|D^\star{ \boldsymbol u}(x)|\geq 1\right\}$.
The Young inequality yields
\begin{equation}\label{estima_M4}2^{\frac{2-p}{2}}\kappa_1
\left\|D^\star{ \boldsymbol u}\right\|_{p,B}\leq
\tfrac{1}{p'}\left(2^{\frac{2-p}{2}}\kappa_1\right)^{p'}+\tfrac{1}{p}
\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^p\end{equation}
and the claimed result follows by combining (<ref>) and (<ref>).
Step 3. By taking into account (<ref>), we have
$$\begin{array}{ll}\left\|\nabla u_3\right\|_{p,B}&\leq \left\|\tfrac{\partial u_3}{\partial x_1}\right\|_{p,B}
+\left\|\tfrac{\partial u_3}{\partial x_2}-\tfrac{\delta}{B}u_3\right\|_{p,B}
&\leq 2\left(1+\delta m\right)\|D_{13}^\star \boldsymbol u\|_{p,B}+2\|D_{23}^\star \boldsymbol u\|_{p,B}\vspace{2mm}\\
2^{2-p}\left(1+\delta m \right)\|D^\star \boldsymbol u\|_{p,B}\end{array}$$
and estimate (<ref>) is then a consequence of (<ref>). Let us finally prove (<ref>). Setting $\boldsymbol\varphi=u$ in (<ref>) we obtain
\begin{equation}\label{est_u_thin1}\int_\Sigma \left(B^{\frac{2}{p}}+
|B^{\frac{1}{p}}D^\star \boldsymbol u|^2\right)^{\frac{p-2}{2}}|B^{\frac{1}{p}}D u|^2dx
\leq \tfrac{1}{2}{\cal R}e\delta \left(u_3^2,u_2\right).\end{equation}
Unlike the proof of estimates (<ref>) and
(<ref>), the coercivity property is not immediat and cannot be used. Let us then set
$$\begin{array}{ll}H_1=\left(B^{\frac{2}{p}}+ |B^{\frac{1}{p}}D^\star\boldsymbol u|^2\right)^{\frac{2-p}{2}}, \vspace{2mm}\\
H_2=\left(B^{\frac{2}{p}}+ |B^{\frac{1}{p}}D^\star\boldsymbol u|^2\right)^{\frac{p-2}{2}}|B^{\frac{1}{p}} Du|^2,
\qquad H_3=|B^{\frac{1}{p}}D u|.\end{array}$$
It is easy to see that $H_1\in \boldsymbol L^{\frac{p}{2-p}}(\Sigma)$ and that $H_3\in \boldsymbol L^p(\Sigma)$. Moreover, since
$|D^\star \boldsymbol u|\geq |Du|$, we have
$$H_2\leq \left(B^{\frac{2}{p}}+ |B^{\frac{1}{p}}D u|^2\right)^{\frac{p-2}{2}}|B^{\frac{1}{p}} Du|^2\leq
|B^{\frac{1}{p}} Du|^p\in L^1(\Sigma).$$
Using Lemma <ref>, we deduce that
\begin{align}\label{est_u_thin2}\left\|Du\right\|_{p,B}^2&\leq \left\|\big(B^{\frac{2}{p}}+ |B^{\frac{1}{p}}D^\star\boldsymbol u|^2\big)^{\frac{2-p}{2}}\right\|_{\frac{p}{2-p}}\left\|\big(B^{\frac{2}{p}}+ |B^{\frac{1}{p}}D^\star\boldsymbol u|^2\big)^{\frac{p-2}{2}}|B^{\frac{1}{p}} Du|^2\right\|_1\nonumber\\
&\leq \big(\|B\|_1+
\|D^\star\boldsymbol u\|_{p,B}^p\big)^{\frac{2-p}{p}} \left\|\big(B^{\frac{2}{p}}+ |B^{\frac{1}{p}}D^\star\boldsymbol u|^2\big)^{\frac{p-2}{2}}|B^{\frac{1}{p}} Du|^2\right\|_1.\end{align}
On the other hand, taking into account (<ref>) with $\alpha=2$ and $q=p$ and the classical Korn inequality, we have
\begin{align}\label{est_u_thin3}
\left|\left(u_3^2,u_2\right)\right|&\leq
\left(S_{p,2p'}\right)^3
\left\|\nabla u_3\right\|_{p}^2 \left\|\nabla u_2\right\|_{p}
\leq \tfrac{\left(S_{p,2p'}\right)^3}{C_{K,1}}
\left\|\nabla u_3\right\|_{p}^2 \left\|D u\right\|_{p}
\nonumber\\
&\leq \tilde\kappa \left\|\nabla u_3\right\|_{p,B}^2 \left\|D u\right\|_{p,B},\end{align}
where $\tilde\kappa=\tfrac{1}{C_{K,1}}\left(m^{\frac{1}{p}}S_{p,2p'}\right)^3$ .
Combining (<ref>)-(<ref>) and taking into account
(<ref>), we get
$$\begin{array}{ll}\|D u\|_{p,B}
&\leq \tfrac{\tilde\kappa}{2}
\left\|\nabla u_3\right\|_{p,B}^2 \left(\|B\|_1+
\|D^\star\boldsymbol u\|_{p,B}^p\right)^{\frac{2-p}{p}}\delta {\cal R}e
\vspace{2mm}\\
\left\|\nabla u_3\right\|_{p,B}^2 \left(\tfrac{2p-1}{p-1}\|B\|_1+
\left(2^{\frac{2-p}{2}}\kappa_1\right)^{p'}\right)^{\frac{2-p}{p}}\delta {\cal R}e\vspace{2mm}\\
\left\|\nabla u_3\right\|_{p,B}^2 \left(\tfrac{\|B\|_1}{p-1}+
\kappa_1^{p'}\right)^{\frac{2-p}{p}}\delta {\cal R}e\end{array}$$
and the conclusion follows from estimate (<ref>). $\hfill\Box$
Proof of Theorem <ref>. Let $\boldsymbol u^k$ be a standard Galerkin approximation. Arguments similar to those used in the proof of Proposition <ref> show that
$$\left\|D^\star \boldsymbol u^k\right\|_{p,B}^p\leq p' \|B\|_1+\left(2^{\frac{2-p}{2}}\kappa_1\right)^{p'},$$
and the sequence $\left(D^\star \boldsymbol u^k\right)_k$ is then bounded in $\boldsymbol L^p(\Sigma)$. Taking into account
the Korn inequality (<ref>), we deduce that $\left(\nabla^\star \boldsymbol u^k\right)_k$ is bounded in $\boldsymbol L^p(\Sigma)$ and thus $\left(\nabla \boldsymbol u^k\right)_k$ is bounded in $\boldsymbol L^p(\Sigma)$. Moreover,
the continuity property $(\ref{estim_S5})$ implies that
$$\left\|\boldsymbol\tau \left(D^\star\boldsymbol u^k\right)\right\|_{p',B}\leq
2\left\|D^\star\boldsymbol u^k\right\|_{p,B}^{p-1}$$
and the sequence $\left(B^{\frac{1}{p'}}\boldsymbol\tau \left(D^\star\boldsymbol u^k\right)\right)_k$ is bounded in $\boldsymbol L^{p'}(\Sigma)$. There then exist a subsequence, still indexed by $k$, $\boldsymbol u\in \boldsymbol V_B^p$ and
$\widetilde {\boldsymbol\tau }\in \boldsymbol L^{p'}(\Sigma)$ such that $\left(\nabla \boldsymbol u^k\right)_k$ converges to
$\boldsymbol u$ weakly in $\boldsymbol L^{p}(\Sigma)$ and $\left(B^{\frac{1}{p'}}\boldsymbol\tau \left(D^\star\boldsymbol u^k\right)\right)_k$ converges to $\widetilde {\boldsymbol\tau }$ weakly in $ \boldsymbol L^{p'}(\Sigma)$. Moreover, since $p >\tfrac{4}{3}$, by using compactness results on Sobolev spaces, we deduce that $\left(\boldsymbol u^k\right)_k$ strongly converges to $\boldsymbol u$ in $L^{p'}(\Sigma )$. Taking into account these convergence results, we deduce that for every $\boldsymbol\varphi \in
{\cal V}_B$, we have
$$\begin{array}{ll}\left|a_\star\left(\boldsymbol u^k,\boldsymbol u^k,B\boldsymbol\varphi\right)
-a_\star\left(\boldsymbol u,\boldsymbol u,B\boldsymbol\varphi\right)\right|&\leq\left|a\left(\boldsymbol u^k-\boldsymbol u,\boldsymbol u^k,B\boldsymbol\varphi\right)\right|
+\left|a_\star\left(\boldsymbol u,\boldsymbol u^k-\boldsymbol u,B\boldsymbol\varphi\right)\right|\vspace{2mm}\\
&=\left|a_\star\left(\boldsymbol u^k-\boldsymbol u,\boldsymbol u^k,B\boldsymbol\varphi\right)\right|
+\left|a_\star\left(B\boldsymbol u,\boldsymbol\varphi,\boldsymbol u^k-\boldsymbol u\right)\right|\vspace{2mm}\\
&\leq \left(\left\|\nabla^\star \boldsymbol u^k\right\|_{p,B}\left\|\boldsymbol\varphi\right\|_{\infty}+
\left\|\boldsymbol u\right\|_{p,B} \left\|\nabla \boldsymbol\varphi\right\|_{\infty}\right)
\left\|\boldsymbol u^k-\boldsymbol u\right\|_{p',B}\vspace{2mm}\\
& \longrightarrow 0 \qquad \mbox{when} \ k\rightarrow +\infty.\end{array}$$
Moreover, by passing to the limit in
$$\left({\boldsymbol\tau }(D^\star{ \boldsymbol u}^k),B D^\star{\boldsymbol\varphi}\right)+{\cal R}e\,a_\star\left({ \boldsymbol u}^k, { \boldsymbol u}^k,
B{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right)\qquad \mbox{for all} \ {\boldsymbol\varphi}\in {\cal V}_B,$$
we obtain
$$\left(\widetilde {\boldsymbol\tau },B^{\frac{1}{p}} D^\star{\boldsymbol\varphi}\right)+{\cal R}e\,
a_\star\left({ \boldsymbol u},{ \boldsymbol u},
B{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right) \qquad \mbox{for all} \ {\boldsymbol\varphi}\in {\cal V}_B$$
and by using the fact that ${\cal V}_B$ is dense in $\boldsymbol V_B^p$, we deduce that
$$\left(\widetilde {\boldsymbol\tau },B^{\frac{1}{p}} D^\star{\boldsymbol\varphi}\right)+{\cal R}e\,
a_\star\left({ \boldsymbol u},{ \boldsymbol u},
B{\boldsymbol\varphi}\right)=\left(G,\varphi_3\right) \qquad \mbox{for all} \ {\boldsymbol\varphi}\in \boldsymbol V_B^p.$$
The rest of the proof for the existence of a weak solution is very similar to the first step in the proof of Theorem <ref> and is omitted. To prove the uniqueness result, let us assume that ${ \boldsymbol u}$ and ${ \boldsymbol v}$ are two weak solutions of $(\ref{equation})$. Setting ${\boldsymbol\varphi}={ \boldsymbol u}-{ \boldsymbol v}$ in the corresponding weak formulation and taking into account Lemma <ref>
and (<ref>), we obtain
\begin{align}
\label{uniqueness1_1}\tfrac{\left\|D^\star({ \boldsymbol u}-{ \boldsymbol v})\right\|_{p,B}^2}{
\left(\|B\|_1+\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^{p}
+\left\|D^\star{ \boldsymbol v}\right\|_{p,B}^{p}\right)^{\frac{2-p}{p}}}
&\leq\left({\boldsymbol\tau }(D^\star{ \boldsymbol u})-
{\boldsymbol\tau }(D^\star{ \boldsymbol v}),BD^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)\nonumber\\
&= {\cal R}e\, a_\star\left({ \boldsymbol v},{ \boldsymbol v},B\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)-{\cal R}e \,a_\star\left({ \boldsymbol u},{ \boldsymbol u},B\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right)\nonumber\\
&=-{\cal R}e \,a\left({ \boldsymbol u}-{ \boldsymbol v},{ \boldsymbol v},B\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right).
\end{align}
Lemma <ref> and estimate $(\ref{main_estimates_1_thin})$ then yield
\begin{align} \label{visco_convective3_1}
\left|a_\star\left({ \boldsymbol u}-{ \boldsymbol v},{ \boldsymbol v},{ \boldsymbol u}-{ \boldsymbol v}\right)\right|&\leq \kappa_6
\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right\|_{p,B}^2
\left\|D^\star { \boldsymbol v}\right\|_{p,B}\nonumber\vspace{0mm}\\
\kappa_1\kappa_6\left(\tfrac{\|B\|_1}{p-1}+\kappa_1
\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right\|_{p,B}^2. \end{align}
On the other hand, by taking into account estimate (<ref>), we have
\begin{align}\label{tensor_suplem_1}
\left(\|B\|_1+\left\|D^\star{ \boldsymbol u}\right\|_{p,B}^{p}
+\left\|D^\star{ \boldsymbol v}\right\|_{p,B}^{p}
\right)^{\frac{2-p}{p}}
&\leq \left(\tfrac{3p-1}{p-1}\|B\|_1+\left(2^{\frac{2-p}{2}}
\kappa_1\right)^{p'}\right)^{\frac{2-p}{p}}\nonumber\\
&\leq 2\left(\tfrac{\|B\|_1}{p-1}+\kappa_1^{p'}\right)^{\frac{2-p}{p}}.\end{align}
By combining (<ref>), (<ref>) and (<ref>), we deduce that
$$\left(\tfrac{1}{2\left(\frac{\|B\|_1}{p-1}+\kappa_1^{p'}\right)^{\frac{2-p}{p}}}-{\cal R}e\,\kappa_1\kappa_6
\left(\tfrac{\|B\|_1}{p-1}+\kappa_1^{p'}\right)^{\frac{2-p}{p}}\right)
\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol v}\right)\right\|_{p,B}^2\leq 0$$
and thus ${ \boldsymbol u}= { \boldsymbol v}$ if condition (<ref>) is satisfied.$\hfill\Box$
Proof of Corollary <ref>. Arguing as in the shear-thickening case (cf. the proof of Corollary <ref>), we can prove that
$$\left\|\pi\right\|_{p'}\leq C\left(\left\|D u\right\|_{p,B}^2+
\delta \left\|\nabla u_3\right\|_{p,B}^2+
\left\|\tau \left(D^\star \boldsymbol u\right)\right\|_{p'}+
\left\|\nabla^\star\cdot\tau \left(D^\star \boldsymbol u\right)-
\nabla\cdot\tau \left(D^\star \boldsymbol u\right)\right\|_{p'}\right),$$
where $\tau=(\tau_{ij})_{i,j=1,2}$.
On the other hand, by taking into account (<ref>) and (<ref>), we obtain
$$\left\|\tau \left(D^\star \boldsymbol u\right)\right\|_{p'}= 2\left\|\left(1+|D^\star \boldsymbol u|^2\right)^{\frac{p-2}{2}}Du\right\|_{p'}\leq 2 \left\|Du\right\|_p^{p-1},$$
\begin{equation}\label{div_tau_pres}\left\|\nabla^\star\cdot\tau \left(D^\star \boldsymbol u\right)-
\nabla\cdot\tau
\left(D^\star \boldsymbol u\right)\right\|_{p'}\leq 4\delta m
\left(2\left\|Du\right\|_p^{p-1}+
\left\|\tfrac{\delta}{B}u_2\right\|_p^{p-1}\right).\end{equation}
The conclusion follows by combining the three inequalities.
§.§ $\delta$-approximation
Proof of Proposition <ref>. Even though the idea is similar to the one used in the shear-thickening case, the lack of coercivity of the stress tensor when splitting the system and considering the equations for $(0,0,w_3)$ and $(w_1,w_2,0)$ generates an additional difficulty. Setting $\phi=(0,0,w_3)$ in the corresponding weak formulation, we obtain
$$\int_\Sigma\left(1+|D\boldsymbol w|^2\right)^{\frac{p-2}{2}}
|\nabla w_3|^2\,dx=\left(\tfrac{G}{B},w_3\right).$$
$$H_1=\left(1+|D\boldsymbol w|^2\right)^{\frac{2-p}{2}},\qquad
H_2=\left(1+|D\boldsymbol w|^2\right)^{\frac{p-2}{2}}
|\nabla w_3|^2,
\qquad H_3=|\nabla w_3|.$$
Due to Lemma <ref> , we have
$$\tfrac{\left\|\nabla w_3\right\|^2_p}{\left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{\frac{2-p}{p}}}\leq
\left(\tfrac{G}{B},w_3\right).$$
On the other hand, standard arguments together with the Poincaré inequality (<ref>) and the Korn inequality give
$$\left|\left(\tfrac{G}{B},w_3\right)\right|\leq c_1
\left\|\nabla w_3\right\|_p,$$
where $c_1=m|G||\Sigma|^{\frac{1}{p'}}$.
Combining these inequalities yields
\begin{equation}\label{dw3_thin}
\tfrac{\left\|\nabla w_3\right\|_p}{\left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{\frac{2-p}{p}}}
\leq c_1.\end{equation}
Similarly, by setting $\boldsymbol\varphi=(w,0)=(w_1,w_2,0)$ in the corresponding weak formulation, we obtain
$$2\int_\Sigma\left(1+|D\boldsymbol w|^2\right)^{\frac{p-2}{2}}
|D w|^2\,dx=\delta \left(\sigma(w_3),w_2\right)$$
and thus
\left\|D\boldsymbol w\right\|_p^p\right)^{\frac{2-p}{p}}}\leq \tfrac{\delta}{2}\left(\sigma(w_3),w_2\right).$$
Estimate (<ref>) together with the Korn inequality give
\tfrac{\|Dw\|_{p}^2}{\left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{\frac{2-p}{p}}}&\leq \tfrac{D_{p,\alpha}}{2}\, c_0\delta
\left\|\nabla w_3\right\|_p^\alpha
\left\|\nabla w_2\right\|_p\vspace{2mm}\\
&\leq\tfrac{D_{p,\alpha}}{2}\, c_0\delta
\left\|\nabla w_3\right\|_p^\alpha
\left\|\nabla w\right\|_p\leq
c_2 c_0\delta
\left\|\nabla w_3\right\|_p^\alpha
\left\|Dw\right\|_p\end{array}$$
where $c_2=\tfrac{D_{p,\alpha}}{2C_{K,1}}$. Consequently, we have
\begin{equation}\label{dw_thin}\tfrac{\|Dw\|_{p}}{\left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{\frac{2-p}{p}}}\leq
c_0c_2\, \delta
\left\|\nabla w_3\right\|_p^\alpha.
\end{equation}
Combining (<ref>) and (<ref>), it follows that
\tfrac{\left\|D\boldsymbol w\right\|_p^p}{\left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{2-p}}
&\leq c_1^p+\left(c_0c_2\right)^p
\left\|\nabla w_3\right\|_p^{\alpha p}\vspace{2mm}\\
\left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{(2-p)\alpha}
\end{array}$$
yielding to
$$\begin{array}{ll}\left\|D\boldsymbol w\right\|_p^p&\leq
c_1^p \left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{2-p}+
\left(c_0c_1^{\alpha}c_2\right)^p
\left(|\Sigma|+
\left\|D\boldsymbol w\right\|_p^p\right)^{(2-p)(\alpha+1)}
\vspace{2mm}\\
&\leq c_3^p \left(1+
\left\|D\boldsymbol w\right\|_p^p\right)^{(2-p)(\alpha+1)}
\end{array}$$
\begin{equation}\label{Dw_global_1}
\left\|D\boldsymbol w\right\|_p^{\frac{p}{(2-p)(\alpha+1)}}
\leq c_3^{\frac{p}{(2-p)(\alpha+1)}}
\left(1+\left\|D\boldsymbol w\right\|_p^{p}\right),
\end{equation}
where $c_3=\left(c_1+
c_0c_1^{\alpha}c_2 \right)
\left(1+|\Sigma|\right)^{\frac{(2-p)(\alpha+1)}{p}}$. By using the Young inequality, we deduce that for $\alpha<\tfrac{p-1}{2-p}$ we have
\begin{align}\label{Dw_global_2}
c_3^{\frac{p}{(2-p)(\alpha+1)}}\left\|D\boldsymbol w\right\|_p^{p} &\leq (2-p)(\alpha+1)
\left\|D\boldsymbol u\right\|_p^{\frac{p}{(2-p)(\alpha+1)}}
\nonumber\\
c_3^{\frac{p}{\left(1-(2-p)(\alpha+1)\right)(2-p)(\alpha+1)}}. \end{align}
Combining (<ref>) and (<ref>), we obtain
\begin{equation}\label{dw3w_thin}
\left\|D\boldsymbol w\right\|_p\leq
\right)^{\frac{(2-p)(\alpha+1)}{p}}.\end{equation}
The conclusion follows from (<ref>), (<ref>) and
(<ref>). $\hfill\Box$
Proof of Proposition <ref>. The ideas of the proof are similar to the ones used in the shear-thickening case. Indeed, by arguing as in the proof of Proposition <ref>, we obtain
\begin{align}\label{difference1_thin}\left(\boldsymbol\tau(D{ \boldsymbol u})-{\boldsymbol\tau}\left(D { \boldsymbol w}\right),D \boldsymbol\varphi\right)&=\left(\boldsymbol\tau(D{ \boldsymbol u})-{\boldsymbol\tau}\left(D^\star{ \boldsymbol u}\right),D \boldsymbol\varphi\right)+
\left(\nabla^\star\cdot \boldsymbol\tau(D^\star{ \boldsymbol u})-\nabla\cdot \boldsymbol\tau(D^\star{ \boldsymbol u}),\boldsymbol\varphi\right)\nonumber\\
&-{\cal R}e\left(a_\star\left(\boldsymbol u,
\boldsymbol u,\boldsymbol\varphi\right)-a\left({ \boldsymbol w},{ \boldsymbol w},{\boldsymbol\varphi}\right)\right)+\left(\pi_1-\pi_2,\nabla\cdot \boldsymbol\varphi\right)-
\delta\left(\sigma(w_3),\varphi_2\right)\nonumber\\
$\circ$ By taking into account (<ref>), we have
\begin{align}\label{difference_est3_thin}\left|I_1\right|&\leq \left\|{\boldsymbol\tau }\left(D^\star{ \boldsymbol u}\right)-{\boldsymbol\tau }\left(D { \boldsymbol u}\right)\right\|_{p'}
\left\|D\boldsymbol\varphi\right\|_{p}\nonumber\\
&\leq 2C_{p}\left\|D^\star{ \boldsymbol u}-D { \boldsymbol u}\right\|_{p}^{p-1}
\left\|D\boldsymbol\varphi\right\|_{p}\leq F_ 1 \,\delta^{p-1} \left\|D\boldsymbol\varphi\right\|_{p},
\end{align}
where $F_1=2C_{p}m ^{p-1}\left(\|u_2\|_p+\|u_3\|_p\right)^{p-1}$.
$\circ$ Estimate (<ref>) together with the Sobolev inequality (<ref>) and the Korn inequality (<ref>) yield
\begin{align}\label{difference_est4_thin}\left|I_2\right|&\leq
\left\|\nabla^\star\cdot \boldsymbol\tau (D^\star{ \boldsymbol u})-\nabla\cdot \boldsymbol\tau (D^\star{ \boldsymbol u})\right\|_{p'}
\left\|\boldsymbol\varphi\right\|_p\nonumber\\
&\leq 4mS_{p,p}
\left(\left\|Du\right\|_{p}^{p-1}
+\left\|D_{33}^\star{ \boldsymbol u}\right\|_{p}^{p-1}+\left\|D_{23}^\star{ \boldsymbol u}\right\|_{p}^{p-1}\right)\, \delta
\left\|\nabla \boldsymbol\varphi\right\|_{p}\nonumber\\
&\leq 12mS_{p,p}
\left\|D^\star{ \boldsymbol u}\right\|_{p}^{p-1}\, \delta
\left\|\nabla \boldsymbol\varphi\right\|_{p}\leq F_2\, \delta
\left\|D\boldsymbol\varphi\right\|_{p},\end{align}
where $F_2=\tfrac{12m}{C_{K,1}} S_{p,p}\, \left\|D^\star{ \boldsymbol u}\right\|_{p}^{p-1}$.
$\circ$ The convective term is estimated similarly
\begin{align}\label{difference_est61_thin}\tfrac{1}{{\cal R}e}\left|I_3\right|
&=\left|a_\star({ \boldsymbol u},{ \boldsymbol u},\boldsymbol \varphi)-a({ \boldsymbol w},{ \boldsymbol w},\boldsymbol \varphi)\right|\nonumber\\
&\leq\left|a_\star\left({ \boldsymbol u}-{ \boldsymbol w},{ \boldsymbol u},\boldsymbol \varphi\right)\right|+\left|a\left(\boldsymbol w,{ \boldsymbol u}-{ \boldsymbol w},\boldsymbol \varphi\right)\right|+
\displaystyle\left|\left(\boldsymbol w\cdot \nabla^\star \boldsymbol u-\boldsymbol w\cdot \nabla \boldsymbol u,\boldsymbol \varphi\right)\right|\nonumber\\
&\leq\left|a_\star\left({ \boldsymbol u}-{ \boldsymbol w},{ \boldsymbol u},\boldsymbol \varphi\right)\right|+\left|a\left(\boldsymbol w,{ \boldsymbol u}-{ \boldsymbol w},\boldsymbol \varphi\right)\right|+
\displaystyle\left\|\tfrac{\delta}{B}|\boldsymbol w||\boldsymbol u||\boldsymbol \varphi|\right\|_1\nonumber\\
&\leq \left\|{ \boldsymbol u}-{ \boldsymbol w}\right\|_{2p'}
\left\|\nabla^\star{ \boldsymbol u}\right\|_p\left\|\boldsymbol \varphi\right\|_{2p'} +\left\|{ \boldsymbol w}\right\|_{2p'}
\left\|\nabla\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_p\left\|\boldsymbol \varphi\right\|_{2p'} + \delta m \left\|\boldsymbol w\right\|_{2p'} \left\|\boldsymbol u\right\|_{2p'}
\left\|\boldsymbol \varphi\right\|_p\nonumber\\
&\leq \left(1+S_{p,p}\right)\left( S_{p,2p'}\right)^2
\left(\|\nabla\left({ \boldsymbol u}-{ \boldsymbol w}\right)\|_p\left(\|\nabla^\star{ \boldsymbol u}\|_p+\|\nabla{ \boldsymbol w}\|_p\right)+\delta m\|\nabla{ \boldsymbol w}\|_p\|\nabla{ \boldsymbol u}\|_p\right)
\left\|\nabla \boldsymbol \varphi\right\|_p\nonumber\\
&\leq F_3\left(\delta
+\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}\right)\left\|D \boldsymbol\varphi\right\|_p\end{align}
with $F_3=\tfrac{1+m}{C_K^3}\left(1+S_{p,p}\right)\left( S_{p,2p'}\right)^2\left(\left\|D^\star \boldsymbol u\right\|_p+\left\|D\boldsymbol w\right\|_p+\left\|D\boldsymbol u\right\|_p\left\|D\boldsymbol w\right\|_p\right)$, and where $C_K$ is the Korn constant given in $(\ref{korn-inequality})$.
$\circ$ The estimate associated to $\left|I_4\right|$ may be obtained with slight modifications in the proof given in Corollary <ref> by observing that
&\leq \left\|\tau (D^\star{ \boldsymbol u})-\tau (D{ \boldsymbol w})\right\|_{p'}+
\left\|\nabla^\star\cdot\tau \left(D^\star \boldsymbol u\right)-
\nabla^\star\cdot\tau
\left(D^\star \boldsymbol u\right)\right\|_{p'}
\vspace{2mm}\\
&+{\cal R}e
\left(\left\|u\otimes u-w\otimes w\right\|_{p'}+
\delta \left\|u_3^2\right\|_{p'}\right)+\delta \left\|\sigma(w_3)\right\|_{p'}.\end{array}$$
Taking into account (<ref>) and (<ref>), we have
$$\left\|\nabla^\star\cdot\tau \left(D^\star \boldsymbol u\right)-
\nabla^\star\cdot\tau \left(D^\star \boldsymbol u\right)\right\|_{p'}\leq F_{4,1}\,\delta$$
$$\begin{array}{ll}\left\|\tau (D^\star{ \boldsymbol u})-\tau (D{ \boldsymbol w})\right\|_{p',B}&\leq
\left\|\tau (D^\star{ \boldsymbol u})-\tau (D{ \boldsymbol w})\right\|_{p'}
\leq 2C_{p}\left\|D^\star{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}^{p-1}\vspace{2mm}\\
&\leq 2C_{p}\left(\left\|D^\star{ \boldsymbol u}-D{ \boldsymbol u}\right\|_{p}^{p-1}
+\left\|D{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}^{p-1}\right)\vspace{2mm}\\
&\leq F_{4,2}\left(\left\|D{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}^{p-1}
where $F_{4,2}= 2C_{p}+F_1$. Moreover, by using the Sobolev inequality (<ref>) with $(q,r)=(2,4)$ and the classical Korn inequality , we obtain
$$\begin{array}{ll}\left\| u\otimes u-w\otimes w\right\|_{p'}+\delta \left\|u_3^2\right\|_{p'}
&\leq \left\|\left(u+w\right)
\otimes \left( u-w\right)\right\|_{p'}+\delta \left\|u_3\right\|^2_{2p'}\vspace{2mm}\\
&\leq \left\|u+w\right\|_{2p'}
\left\|u-w\right\|_{2p'}+\delta \left\|u_3\right\|^2_{2p'}\vspace{2mm}\\
&\leq S_{p,2p'}
\left\|\nabla\left(u-w\right)
\right\|_p\left\|u+w\right\|_{2p'}+\delta\, \left\|u_3\right\|^2_{2p'}\vspace{2mm}\\
&\leq S_{p,2p'}
\left\|\nabla\left(\boldsymbol u-\boldsymbol w\right)
\right\|_p\left\|u+w\right\|_{2p'}+\delta\, \left\|u_3\right\|^2_{2p'}\vspace{2mm}\\
&\leq \tfrac{S_{p,2p'}}{C_K}
\left\|D\left(\boldsymbol u-\boldsymbol w\right)
\right\|_p\left\|u+w\right\|_{2p'}+\delta\, \left\|u_3\right\|^2_{2p'}\vspace{2mm}\\
&\leq F_{4,3} \left(
\left\|D\left(\boldsymbol u-\boldsymbol w\right)\right\|_p+\delta\right),\end{array}$$
where $F_{4,3}=\tfrac{S_{p,2p'}}{C_{K,1}} \left\|u+w\right\|_{2p'}+ \left\|u_3\right\|^2_{2p'}$. Due to (<ref>), we have
$$\left\|\sigma(w_3)\right\|_{p'}\leq c_0 E_{\alpha,p}
\left\|\nabla w_3\right\|_{p}^\alpha=F_{4,4}.$$
Combining these estimates, we deduce that
\begin{align}\label{difference_est8_thin}\left|I_4\right|&\leq \left(F_{4,2}\left\|D{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}^{p-1}+{\cal R}e\, F_{4,3}
\left\|D{ \boldsymbol u}-D{ \boldsymbol w}\right\|_{p}\right) \left\|\nabla \cdot \boldsymbol\varphi\right\|_p\nonumber\\
&+\left( \left(F_{4,1}+{\cal R}e\,F_{4,3}\right)\delta+F_{5,2}\delta^{p-1}\right) \left\|\nabla \cdot \boldsymbol\varphi\right\|_p\nonumber\\
&\leq F_ 4\left(\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}
+\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}^{p-1}+
\delta^{p-1}\right)\left\|\nabla \cdot \boldsymbol\varphi\right\|_p.\end{align}
$\circ$ Finally, taking into account (<ref>) we obtain
\begin{align}\label{difference_est_psi_thin}\left|I_5\right|&=\delta \left|\left(\sigma(w_3),
\varphi_2\right)\right|\leq D_{p,\alpha} \,\delta
\left\|\nabla w_3\right\|^\alpha_p
\left\|\nabla\varphi_2\right\|_{p}\nonumber\\
&\leq D_{p,\alpha}\,\delta
\left\|\nabla w_3\right\|^\alpha_2
\left\|\nabla\boldsymbol\varphi
\right\|_{p}\leq F_5\,\delta
\left\|D\boldsymbol\varphi
\right\|_{p},\end{align}
where $F_5=\tfrac{D_{p,\alpha}}{C_{K,1}}
\left\|\nabla w_3\right\|^\alpha_p$.
$\circ$ Combining (<ref>)-(<ref>), and taking into the estimates associated to
$\boldsymbol u$ and $\boldsymbol w$, we deduce that
$$\begin{array}{ll}&\left|\left(\boldsymbol\tau (D{ \boldsymbol u})-{\boldsymbol\tau }\left(D { \boldsymbol w}\right),D \boldsymbol\varphi\right)\right|\vspace{2mm}\\
&\leq \left(F_1\, \delta^{p-1}+F_2\, \delta+{\cal R}e\,F_3\, \delta+F_5\, \delta\right)\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}+{\cal R}e\,F_3 \left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{p}\left\|D\boldsymbol\varphi\right\|_p\vspace{2mm}\\
& +F_ 4\left(\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}
+\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}^{p-1}+
\delta^{p-1}\right)\left\|\nabla \cdot \boldsymbol\varphi\right\|_p \vspace{2mm}\\
&\leq C_1\left(\delta^{p-1}\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p,B}+
\delta\left\|D^\star\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p,B}^{p-1}+
{\cal R}e\left\|D^\star \left(\boldsymbol u-\boldsymbol w\right)\right\|_{p,B}^2+\delta^p\right),\end{array}$$
where $C_1$ depends only on $p$, $\Sigma$, $m$, $n$, $\alpha$ and $c_0$. Setting $\boldsymbol\varphi=\boldsymbol u-\boldsymbol w$ and taking into account the estimates associated to $\boldsymbol u$ and $\boldsymbol w$, we deduce that
\left({\boldsymbol\tau }\left(D{ \boldsymbol u}\right)
-{\boldsymbol\tau }\left(D{ \boldsymbol w}\right),
D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right)
\vspace{2mm}\\
&\leq \left(F_1\, \delta^{p-1}+F_2\, \delta+{\cal R}e\,F_3\, \delta+F_5\, \delta\right)\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}+{\cal R}e\,F_3 \left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{p}^2\vspace{2mm}\\
& +F_4\left(\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}
+\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}^{p-1}+
\delta^{p-1}\right)\left\|\tfrac{\delta}{B}u_2\right\|_p \vspace{2mm}\\
&\leq C_2\left(\delta^{p-1}\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}+
\delta\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}^{p-1}+
{\cal R}e\left\|D \left(\boldsymbol u-\boldsymbol w\right)\right\|_{p}^2+\delta^p\right),\end{array}$$
where $C_2$ depends only on $p$, $\Sigma$, $m$, $n$, $\alpha$ and $c_0$. Using the Young inequality, it follows that for every $\varepsilon>0$, we have
\begin{align}\label{est_fin}&\left({\boldsymbol\tau }\left(D{ \boldsymbol u}\right)
-{\boldsymbol\tau }\left(D{ \boldsymbol w}\right),
D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right)\nonumber\\
& \leq
\left(\varepsilon+C_2\,{\cal R}e\right) \left\|D^\star{ \boldsymbol u}-D^\star{ \boldsymbol w}\right\|_{p,B}^2+C_3(\varepsilon)\left(\delta^{2(p-1)}+\delta^{\frac{2}{3-p}}\right)+C_1\,\delta^p.\end{align}
On the other hand, by taking into account $(\ref{estim_S7})$ and the estimates associated to $\boldsymbol u$ and $\boldsymbol w$, we deduce that there exists a constant $C_4$ depending on $p$, $\Sigma$, $G$ and $m$, $\alpha$ and $c_0$, but independent of $\delta$, such that
\begin{equation}\label{difference7_thin}
\left({\boldsymbol\tau }\left(D{ \boldsymbol u}\right)-{\boldsymbol\tau }\left(D{ \boldsymbol w}\right),
D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right)\geq \tfrac{2\left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}^2}{
\left(|\Sigma|+\|D{ \boldsymbol u}\|_{p}^p+\|D{ \boldsymbol w}\|_{p}\right)^{\frac{2-p}{p}}}
\geq C_4 \left\|D\left({ \boldsymbol u}-{ \boldsymbol w}\right)\right\|_{p}^2.\end{equation}
Combining (<ref>) and (<ref>), observing that $\delta^{p}<\delta^{2(p-1)}$ and $\delta^{\frac{2}{3-p}}<\delta^{2(p-1)}$, choosing $\varepsilon= \tfrac{C_4}{2}$ and assuming that ${\cal R}e<\tfrac{C_4}{2C_2}$, we deduce that
$$\left\|D\left(\boldsymbol u-\boldsymbol w\right)\right\|_{p,B}\leq
and the claimed result is proved. $\hfill\Box$
§ TOROIDAL COORDINATE SYSTEM
Let us consider the new coordinate system, in the variables $
(\widetilde x_1,\widetilde x_2 ,\widetilde x_3)$, given by the
transformations $(\widetilde x_1,\widetilde x_2,\widetilde x_3)\mapsto (\widetilde y_1,\widetilde y_2,\widetilde y_3)$ satisfying
(<ref>) and (<ref>).
Let $M$ be a generic point such that
$$M=\widetilde y_1(\widetilde x_1,\widetilde x_2 ,\widetilde x_3)\,\mathbf{e}_{1}
+\widetilde y_2(\widetilde x_1,\widetilde x_2 ,\widetilde x_3) \,\mathbf{e}_{2}+
\widetilde y_3(\widetilde x_1,\widetilde x_2 ,\widetilde x_3)
\,\mathbf{e}_{3}.$$
Defining the scale vectors
$$h_{1}=\left| \tfrac{\partial M}{\partial \widetilde x_1}
\right|, \qquad\quad h_{2}=\left|
\tfrac{\partial M}{\partial \widetilde x_2 }\right|,\quad\qquad
h_{3}=\left| \tfrac{\partial M}{\partial \widetilde x_3}
\right|$$
and the local basis
\tfrac{\partial M}{\partial \widetilde x_1}, \quad\qquad
\mathbf{a}_{2}=\tfrac{1}{h_{2}}
\tfrac{\partial M}{\partial \widetilde x_2},\quad
\qquad \mathbf{a}_{3}=\tfrac{1}{h_{3}}
\tfrac{\partial M}{\partial \widetilde x_3},$$
we obtain,
$$h_{1}=1, \quad\qquad h_{2}=1, \quad\qquad
h_{3}=1+\tfrac{1}{R }\widetilde x_2$$
\mathbf{a}_{2}=\cos \left(\tfrac{\widetilde x_3}{R }\right)
\mathbf{e}_{1}+\sin \left(\tfrac{\widetilde x_3}{R }\right)
\mathbf{e}_{2},\quad\qquad
\mathbf{a}_{3}=-\sin \left(\tfrac{\widetilde x_3}{R }\right)\mathbf{e}_{1}
+\cos \left(\tfrac{\widetilde x_3}{R }\right)\mathbf{e}_{2}.$$
The corresponding derivatives of the vector basis are given by
\frac{\partial \mathbf{a}_{1}}{\partial \widetilde x_1}=0, \quad &
\quad \frac{\partial \mathbf{a}_{1}}{\partial \widetilde x_2}=0, \quad & \quad
\frac{\partial \mathbf{a}_{1}}{\partial \widetilde x_3}=0
\medskip , \\
\frac{\partial \mathbf{a}_2}{\partial \widetilde x_1}=0,
\quad &
\quad \frac{\partial \mathbf{a}_2}{\partial \widetilde x_2 }=0, \quad &
\quad \frac{\partial \mathbf{a}_2}{\partial \widetilde x_3}
=\frac{\mathbf{a}_{3}}{R}\medskip , \\
\frac{\partial \mathbf{a}_{3}}{\partial \widetilde x_1}=0, \quad & \quad
\frac{\partial \mathbf{a}_3}{\partial \widetilde x_2}=0,
\quad & \quad
\frac{\partial \mathbf{a}_3}{\partial \widetilde x_3}=-
\frac{\mathbf a_2}{R }.
\end{array}
$\bullet$ The gradient of a scalar function $\widetilde \psi$ in the rectangular toroidal coordinates is then given by
\begin{equation}\label{grad_scalar}
\widetilde\nabla \widetilde \psi =
\tfrac{1}{h_1}
\tfrac{\partial \widetilde \psi}{\partial \widetilde x_1}\, \mathbf a_1+
\tfrac{1}{h_2}\tfrac{\partial \widetilde \psi}
{\partial \widetilde x_2}\,\mathbf a_2+\tfrac{1}{h_3}
\tfrac{\partial \widetilde \psi}{\partial \widetilde x_3}\mathbf a_3
=\tfrac{\partial \widetilde \psi}{\partial \widetilde x_1}\,\mathbf a_1+
\tfrac{\partial \widetilde \psi}{\partial \widetilde x_2}\,\mathbf a_2
+ \tfrac{1}{B}
\tfrac{\partial \widetilde \psi}{\partial \widetilde x_3}\,\mathbf a_3,
\end{equation}
where $B=1+\tfrac{1}{R}\, \widetilde x_2$.
$\bullet$ The gradient of a vector $\widetilde{\boldsymbol v}\equiv (\widetilde v_{1},\widetilde v_{2},\widetilde v_{3})$ is defined by
$$\widetilde \nabla \widetilde{\boldsymbol v}=\left(\tfrac{1}{h_1}\mathbf a_1\otimes \tfrac{\partial }{\partial \widetilde x_1}+\tfrac{1}{h_2}
\mathbf a_2\otimes \tfrac{\partial }{\partial \widetilde x_2}+
\tfrac{1}{h_3}\mathbf a_3\otimes
\tfrac{\partial }{\partial \widetilde x_3}\right)
\left(\mathbf a_1\widetilde v_1+\mathbf a_2 \widetilde v_2
+\mathbf a_3\widetilde v_3\right)$$
that is
\begin{align}\label{gradient}\widetilde \nabla \widetilde{\boldsymbol v}&=\displaystyle\sum_{j=1}^3\sum_{i=1}^2 \tfrac{\partial \widetilde v_j}{
\partial \widetilde x_i}\, \mathbf a_i\otimes \mathbf a_j
\left(\widetilde v_2\,\mathbf a_3\otimes\mathbf a_3-
\widetilde v_3\,
\mathbf a_3\otimes\mathbf a_2\right)+\tfrac{1}{B}\sum_{j=1}^3
\tfrac{\partial \widetilde v_j}{\partial \widetilde x_3} \,
\mathbf a_3\otimes\mathbf a_j\nonumber\\
&=\left( \begin{array}{lll}
\frac{\partial \widetilde v_{1}}{\partial \widetilde x_1} \ & \
\frac{\partial \widetilde v_2}{\partial \widetilde x_1} \ & \
\frac{\partial \widetilde v_3}{\partial \widetilde x_1}\vspace{3mm}\\
\frac{\partial \widetilde v_1}{\partial \widetilde x_2} \ & \
\frac{\partial \widetilde v_2}{\partial \widetilde x_2} \ & \
\frac{\partial \widetilde v_3}{\partial \widetilde x_2 }\vspace{3mm} \\
\frac{1}{B}\frac{\partial \widetilde v_{1}}{\partial \widetilde x_3} \ & \
\frac{1}{B}\left(\frac{\partial \widetilde v_{2}}{\partial \widetilde x_3}
-\frac{\widetilde v_3}{R}\right)\ & \ \frac{1}{B}\left(\frac{\partial \widetilde v_{3}}{\partial \widetilde x_3}
+\frac{\widetilde v_2}{R}\right)
\end{array}\right).\end{align}
$\bullet$ Similarly, if $\widetilde{\boldsymbol v}\equiv (\widetilde v_{1},\widetilde v_{2},\widetilde v_{3})$ and $\widetilde{\boldsymbol w}\equiv (
\widetilde w_{1},\widetilde w_{2},\widetilde w_{3})$ are two vectors, then the convective term is defined by
v·∇w= (𝐚_1 v_1+
𝐚_2 v_2+𝐚_3 v_3)·(∑_i,j𝐚_i⊗𝐚_j
=∑_i𝐚_i(∑_j v_j
that is
\begin{align}\label{convective}\widetilde {\boldsymbol v}\cdot \widetilde \nabla \widetilde{\boldsymbol w}&=\displaystyle \sum_{j=1}^3\sum_{i=1}^2 \widetilde v_i \tfrac{\partial \widetilde w_j}{\partial \widetilde x_i} \mathbf a_j+\tfrac{\widetilde v_3}{RB}\left(\widetilde w_2\mathbf a_3-
\widetilde w_3\mathbf a_2\right)+\tfrac{\widetilde v_3}{B}
\tfrac{\partial \widetilde {\boldsymbol w}}{\partial \widetilde x_3}\nonumber\\
&=\left( \begin{array}{l}
\widetilde v_1\tfrac{\partial \widetilde w_1}{\partial \widetilde x_1}+
\widetilde v_2\tfrac{\partial \widetilde w_1}{\partial \widetilde x_2} +\tfrac{\widetilde v_3}{B}
\tfrac{\partial \widetilde w_1}{\partial \widetilde x_3}
\vspace{3mm} \\
\widetilde v_1\tfrac{\partial \widetilde w_2}{\partial \widetilde x_1}+
\widetilde v_2\tfrac{\partial \widetilde w_2}{\partial \widetilde x_2} +\tfrac{\widetilde v_3}{B}
\tfrac{\partial \widetilde w_2}{\partial \widetilde x_3}
-\tfrac{1}{RB}\widetilde v_3\widetilde w_3\vspace{3mm} \\
\widetilde v_1\tfrac{\partial \widetilde w_3}{\partial \widetilde x_1}+
\widetilde v_2\tfrac{\partial \widetilde w_3}{\partial \widetilde x_2} +\tfrac{\widetilde v_3}{B}
\tfrac{\partial \widetilde w_3}{\partial \widetilde x_3}+
\tfrac{1}{RB}\widetilde v_3\widetilde w_2
\end{array}\right).\end{align}
$\bullet$ Finally, the divergence of a vector $\widetilde{\boldsymbol v}\equiv (\widetilde v_{1},\widetilde v_{2},\widetilde v_{3})$ is defined by
\begin{align}\label{div_vector}\widetilde \nabla \cdot \widetilde{\boldsymbol v}&=\tfrac{1}{h_1h_2h_3}\left( \tfrac{
\partial }{\partial \widetilde x_1}\left( h_2h_3\,\widetilde v_1\right)
+\tfrac{\partial }{\partial \widetilde x_2}
\left( h_3h_1\,\widetilde v_2\right)
+\tfrac{\partial }{\partial \widetilde x_3}
\left( h_1h_2\,\widetilde v_3 \right)
\right)\nonumber\\
&=\tfrac{\partial \widetilde v_1}{\partial \widetilde x_1}
+\tfrac{\partial \widetilde v_2}{\partial \widetilde x_2}+
\tfrac{\widetilde v_2}{RB}+
\tfrac{1}{B}\tfrac{\partial \widetilde v_3}{\partial \widetilde x_3}.\end{align}
The divergence of a tensor $\widetilde{\mathbf S}$ is defined by
$$\widetilde\nabla\cdot\widetilde{\mathbf S}=
\left(\mathbf a_1\tfrac{\partial }{\partial \widetilde x_1}+
\mathbf a_{2}
\tfrac{\partial }{\partial \widetilde x_2}
+\mathbf a_3\tfrac{1}{B}
\tfrac{\partial }{\partial \widetilde x_3}\right)\cdot
\Big(\sum_{i,j}
\mathbf a_i\otimes \mathbf a_j \widetilde S_{ij}\Big)$$
which gives
\begin{align}\label{div_tensor}
\widetilde\nabla\cdot\widetilde{\mathbf S}
&= \displaystyle \sum_{j=1}^3\sum_{i=1}^2 \tfrac{\partial \widetilde S_{ij}}{\partial \widetilde x_i}\, \mathbf a_j+
\tfrac{1}{B}\sum_{j=1}^3 \tfrac{\partial \widetilde S_{3j}}{\partial \widetilde x_3}\, \mathbf a_j+\tfrac{1}{RB}
\left(\widetilde S_{21}\mathbf a_1+
\left(\widetilde S_{22}-\widetilde S_{33}\right)
\mathbf a_2+\left(\widetilde S_{32}+\widetilde S_{23}
\right)\mathbf a_3\right)\nonumber\\
\tfrac{\partial \widetilde S_{11}}{\partial \widetilde x_1}+
\tfrac{\partial \widetilde S_{21}}{\partial \widetilde x_2}+
\tfrac{1}{B}\tfrac{\partial
\widetilde S_{31}}{\partial \widetilde x_3}
+\tfrac{\widetilde S_{21}}{RB}\vspace{3mm}\\
\tfrac{\partial \widetilde S_{12}}{\partial
\widetilde x_1}+
\tfrac{\partial \widetilde S_{22}}{\partial \widetilde x_2}+
\tfrac{1}{B}\tfrac{\partial \widetilde S_{32}}{\partial \widetilde x_3}+
\tfrac{\widetilde S_{22}-\widetilde S_{33}}{RB}\vspace{3mm}\\
\tfrac{\partial \widetilde S_{13}}{\partial \widetilde x_1}+
\tfrac{\partial \widetilde S_{23}}{\partial \widetilde x_2}+
\tfrac{1}{B}\tfrac{\partial \widetilde S_{33}}{\partial \widetilde x_3}+
\tfrac{\widetilde S_{32}+\widetilde S_{23}}{RB}\end{array}\right).
\end{align}
§ DIMENSIONLESS SYSTEM
For the confort of the reader, the dimensionless equation (<ref>) is derived hereafter. To simplify the notation, we consider the fully developed case and will assume that (<ref>) and (<ref>) are fulfilled. Taking into account (<ref>), (<ref>) and (<ref>), we can easily see that
\widetilde {\boldsymbol u}\cdot \widetilde \nabla \widetilde{\boldsymbol u}&=
&\left( \begin{array}{l}
\widetilde u_1\tfrac{\partial \widetilde u_1}{\partial \widetilde x_1}+
\widetilde u_2\tfrac{\partial \widetilde u_1}{\partial \widetilde x_2}
\vspace{3mm} \\
\widetilde u_1\tfrac{\partial \widetilde u_2}{\partial \widetilde x_1}+
\widetilde u_2\tfrac{\partial \widetilde u_2}{\partial \widetilde x_2}
-\tfrac{1}{RB}\widetilde u_3^2
\vspace{3mm} \\
\widetilde u_1\tfrac{\partial \widetilde u_3}{\partial \widetilde x_1}+
\widetilde u_2\tfrac{\partial \widetilde u_3}{\partial \widetilde x_2}+
\tfrac{1}{RB}\widetilde u_3\widetilde u_2
\end{array}\right)\vspace{4mm}\\
\underbrace{\left( \begin{array}{l}
u_1\tfrac{\partial u_1}{\partial x_1}+
u_2\tfrac{\partial u_1}{\partial x_2}
\vspace{3mm} \\
u_1\tfrac{\partial u_2}{\partial x_1}+
u_2\tfrac{\partial u_2}{\partial x_2}
\vspace{3mm} \\
u_1\tfrac{\partial u_3}{\partial x_1}+
u_2\tfrac{\partial u_3}{\partial x_2}+
\tfrac{\delta}{B}u_3u_2
\end{array}\right)}_{{\boldsymbol u}\cdot \nabla^\star {\boldsymbol u}}
%&=\tfrac{U_0^2}{r_0}\left({ \boldsymbol u}\cdot \nabla
% { \boldsymbol u}+\tfrac{\delta}{B}\left(u_3u_2\, {\boldsymbol a }_3-
% u_3^2{\boldsymbol a }_2\right)\right),
\end{array}$$
$$\widetilde\nabla \widetilde \pi =
\left(\begin{array}{ll}
\tfrac{1}{r_0}\tfrac{\partial \widetilde \pi }{\partial x_1}\vspace{3mm}\\
\tfrac{1}{r_0}\tfrac{\partial \widetilde \pi }{\partial x_2}\vspace{3mm}\\
-\tfrac{\widetilde G}{B}\end{array}\right)=
\tfrac{\mu U_0}{r_0^2}\underbrace{\left(\begin{array}{ll}
\tfrac{\partial \pi }{\partial x_1}\vspace{3mm}\\
\tfrac{\partial \pi }{\partial x_2}\vspace{3mm}\\
-\tfrac{G}{B}\end{array}\right)}_{\nabla^\star \pi}$$
with $B=1+\delta x_2$.
Similarly, by taking into account (<ref>), we obtain
$$\begin{array}{lll}\widetilde D \widetilde{\boldsymbol u}&=
&\left( \begin{array}{lll}
\frac{\partial \widetilde u_{1}}{\partial \widetilde x_1} \ & \
\tfrac{1}{2}\left(\frac{\partial \widetilde u_2}{\partial \widetilde x_1}+
\frac{\partial \widetilde u_1}{\partial \widetilde x_2}\right) \ & \
\tfrac{1}{2}\,\frac{\partial \widetilde u_3}{\partial \widetilde x_1}\vspace{3mm}\\
\tfrac{1}{2}\left(\frac{\partial \widetilde u_2}{\partial \widetilde x_1}+
\frac{\partial \widetilde u_1}{\partial \widetilde x_2}\right) \ & \
\frac{\partial \widetilde u_2}{\partial \widetilde x_2} \ & \
\tfrac{1}{2}\left(\frac{\partial \widetilde u_3}{\partial \widetilde x_2 }-\frac{\widetilde u_3}{RB}\right)
\vspace{3mm} \\
\tfrac{1}{2}\,\frac{\partial \widetilde u_3}{\partial \widetilde x_1} \ & \
\tfrac{1}{2}\left(\frac{\partial \widetilde u_3}{\partial \widetilde x_2 }-\frac{\widetilde u_3}{RB}\right)
\ & \ \frac{\widetilde u_2}{RB}
\end{array}\right)\vspace{3mm}\\
&=\tfrac{U_0}{r_0} &
\underbrace{\left( \begin{array}{lll} \frac{\partial u_{1}}{\partial x_1} \ & \
\tfrac{1}{2}\left(\frac{\partial u_2}{\partial x_1}+
\frac{\partial u_1}{\partial x_2}\right) \ & \
\tfrac{1}{2}\,\frac{\partial u_3}{\partial x_1}\vspace{3mm}\\
\tfrac{1}{2}\left(\frac{\partial u_2}{\partial x_1}+
\frac{\partial u_1}{\partial x_2}\right) \ & \
\frac{\partial u_2}{\partial x_2} \ & \
\tfrac{1}{2}\left(\frac{\partial u_3}{\partial x_2 }-\frac{\delta}{B} u_3\right)
\vspace{3mm} \\
\tfrac{1}{2}\,\frac{\partial u_3}{\partial x_1} \ & \
\tfrac{1}{2}\left(\frac{\partial u_3}{\partial x_2 }-\frac{\delta}{B}u_3\right)
\ & \ \frac{\delta}{B}u_2
\end{array}\right)}_{D^\star \boldsymbol u}.
\end{array}$$
It follows that $\big|\widetilde D \widetilde{\boldsymbol u}\big|^2=\left(\tfrac{U_0}{r_0}\right)^2\left|D^\star \boldsymbol u\right|^2$ and
$$\widetilde{\mathbf S}\big(\widetilde D \widetilde{\boldsymbol u}\big)\equiv 2\mu\left(1+\big|\widetilde D \widetilde{\boldsymbol u}\big|^2\right)^{\frac{p-2}{2}} \widetilde D \widetilde{\boldsymbol u}=
\tfrac{\mu U_0}{r_0}\underbrace{2\left(1+\left(\tfrac{U_0}{r_0}\right)^2\left|D^\star \boldsymbol u\right|^2\right)^{\frac{p-2}{2}} D^\star \boldsymbol u}_{\boldsymbol\tau\left(D^\star \boldsymbol u\right)}.$$
Due to (<ref>), to the fact that $\widetilde{\mathbf S}$ is symmetric and that
$$\tfrac{\partial \widetilde S_{3i}}{\partial \widetilde x_3}=0 \qquad i=1,2,3$$
we deduce that
\widetilde\nabla \cdot \big(\widetilde{\mathbf S}\big(\widetilde D \widetilde {\boldsymbol u}\big)\big) &= &\left(\begin{array}{ll}
\tfrac{\partial \widetilde S_{11}}{\partial \widetilde x_1}+
\tfrac{\partial \widetilde S_{12}}{\partial \widetilde x_2}+\tfrac{1}{RB}\widetilde S_{12}\vspace{3mm}\\
\tfrac{\partial \widetilde S_{12}}{\partial \widetilde x_1}+
\tfrac{\partial \widetilde S_{22}}{\partial \widetilde x_2}+
\tfrac{1}{RB}\left(\widetilde S_{22}- \widetilde\tau_{33}\right)\vspace{3mm}\\
\tfrac{\partial \widetilde S_{13}}{\partial \widetilde x_1}+
\tfrac{\partial \widetilde S_{23}}{\partial \widetilde x_2}+
\tfrac{2}{RB}\widetilde S_{23}\end{array}\right)\vspace{3mm}\\
\tfrac{\partial \widetilde S_{11}}{\partial x_1}+
\tfrac{\partial \widetilde S_{12}}{\partial x_2}+\tfrac{\delta}{B}\widetilde S_{12}\vspace{3mm}\\
\tfrac{\partial \widetilde S_{12}}{\partial x_1}+
\tfrac{\partial \widetilde S_{22}}{\partial x_2}+
\tfrac{\delta}{B}\left(\widetilde S_{22}- \widetilde S_{33}\right)\vspace{4mm}\\
\tfrac{\partial \widetilde S_{13}}{\partial x_1}+
\tfrac{\partial \widetilde S_{23}}{\partial x_2}+
\tfrac{2\delta}{B}
\widetilde S_{23}\end{array}\right)\vspace{3mm}\\
&=\tfrac{\mu U_0}{r_0^2}&\underbrace{\left(\begin{array}{ll}
\tfrac{\partial \tau_{11}}{\partial x_1}+
\tfrac{\partial \tau_{12}}{\partial x_2}+\tfrac{\delta}{B}\tau_{12}\vspace{4mm}\\
\tfrac{\partial \tau_{12}}{\partial x_1}+
\tfrac{\partial \tau_{22}}{\partial x_2}+
\tfrac{\delta}{B}\left(\tau_{22}- \tau_{33}\right)\vspace{3mm}\\
\tfrac{\partial \tau_{13}}{\partial x_1}+
\tfrac{\partial \tau_{23}}{\partial x_2}+
\tfrac{2\delta}{B} \tau_{23}\end{array}\right)}_{
\nabla^\star\cdot\left(\boldsymbol\tau\left(D^\star \boldsymbol u\right)\right)},\end{array}$$
where we dropped the dependance on $\widetilde D \widetilde{\boldsymbol u}$ and $D^\star \boldsymbol u$. Taking into account these identities and substituting in equation $(\ref{equation_dim})_1$, we obtain
$$\tfrac{\rho U_0^2}{r_0}\, \boldsymbol u\cdot \nabla^\star \boldsymbol u +\tfrac{\mu U_0}{r_0^2} \, \nabla^\star \pi=
\tfrac{\mu U_0}{r_0^2} \, \nabla^\star\cdot\left(\boldsymbol\tau\left(D^\star \boldsymbol u\right)\right)
which, by multiplying by $\tfrac{r_0^2}{\mu U_0}$, gives equation $(\ref{equation})_1$. Finally, by taking into account (<ref>), we obtain
$$\widetilde \nabla \cdot \widetilde{\boldsymbol u}=\tfrac{\partial \widetilde u_1}{\partial \widetilde x_1}
+\tfrac{\partial \widetilde u_2}{\partial \widetilde x_2}+
\tfrac{\widetilde u_2}{RB}=\tfrac{U_0}{r_0}
\underbrace{\left(\tfrac{\partial u_1}{\partial x_1}
+\tfrac{\partial u_2}{\partial x_2}+
\tfrac{\delta}{B}\, u_2\right)}_{\nabla^\star \cdot \boldsymbol u},$$
showing that $(\ref{equation_dim})_2$ implies $(\ref{equation})_2$.
amrouche C. Amrouche, V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension.
Czechoslovak Mathematical Journal, 44 (1994) 109-140.
berger A. A. Berger, L. Talbot, L.-S. Yao, Flow in curved pipes, Ann. Rev. Fluid Mech., 15 (1983) 461-512.
Rob-coscia V. Coscia, A. M. Robertson, Existence and uniqueness of steady, fully developed flows of second order fluids in curved
pipes, Math. Models Methods Appl. Sci. 6 (1) (2001) 1055-1071.
daska P. Daskopoulos, A. M. Lenhoff, Flow in curved ducts:
bifurcation structure for stationary ducts, J. Fluid Mech., 203 (1989) 125-148.
dean1 W. R. Dean, Note on the motion of fluid in curved pipe,
Philos. Mag., 20 (1927) 208-223.
dean2 W. R. Dean, The streamline motion of fluid in curved pipe,
Philos. Mag., 30 (1928) 673-695.
eustice1 J. Eustice, Flow of water in curved pipes, Proc. R. Soc. Lond. A 84 (1910) 107-118.
eustice2 J. Eustice, Experiments of streamline motion in curved pipes, Proc. R. Soc. Lond. A 85 (1911) 119-131.
fan Y. Fan, R. I. Tanner, N. Phan-Thien, Fully developed viscous and viscoelastic flows in curved pipes,
J. Fluid Mech., 440 (2001) 327-357.
galdirobertson G. P. Galdi, A. M. Robertson, On flow of a Navier-Stokes fluid in curved pipes. Part I: Steady flow, Applied Math. Letters, 18 (2005) 1116-1124.
galdi G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I and II, Springer Tracts in Natural Philosophy 38, 39, 2nd edition, Springer-Verlag, New York, 1998.
ito H. Ito, Flow in curved pipes, JSME Int. J., 30 (1987) 543-552.
Rob-jichote W. Jitchote, A. M. Robertson, Flow of second order fluids in curved pipes, J. Non-Newtonian Fluid Mech. 90, 2000, 91-116.
lady1 O. A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Stek. Inst. Math. 102 (1967) 95-118.
lady2 O. A. Ladyzhenskaya, On some modifications of the Navier-Stokes equations for large gradients of Velocity, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 7 (1968) 126-154.
lady O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Beach, New York, 1969.
lions J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.
necas J. Nečas, J. Málek, J. Rokyta, M. Ružička, Weak and measure-valued solutions to evolutionary partial differential equations, Applied Mathematics and Mathematical Computation, Vol. 13, Chapmann and Hall, London, 1996.
Rob A. M. Robertson, On viscous flow in curved pipes of non-uniform cross section, Inter. J. Numer. Meth. fluid., 22 (1996) 771-798.
Rob-mul A. M. Robertson, S. J. Muller, Flow of Oldroyd-B fluids in curved pipes of circular and annular cross-section, Int. J. Non-Linear Mechanics, 31 (1996) 1-20.
soh W. Y. Soh, S. A. Berger, Fully developed flow in a curved pipe of arbitrary curvature ratio, Int. J. Numer. Meth. Fluids, 7 (1987) 733-755.
topa H. C. Topakoglu, M. A. Ebadian, On the steady laminar flow of an incompressible viscous fluid in a curved pipe of elliptical cross-section, J. Fluid Mech., 158 (1985) 329-340.
yang Z. H. Yang, H. B. Keller, Multiple laminar flows through curved pipes, Appl. Numer. Math., 2, (1986) 257-271.
|
1511.00192
|
In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| <|\Pi_n(\tau_2)|$ for all $n>k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$.
§ INTRODUCTION
A set partition of a set $S$ is a collection of disjoint nonempty subsets $B_1,\ldots, B_m$ of $S$ whose union is $S$. We call the subsets $B_i$ blocks, and we write
$$\sigma = B_1/\cdots/ B_m \vdash S.$$
If $S$ is a set of positive integers and $\sigma\vdash S$, then the standardization of $\sigma$ is the set partition of $\{1,2,\ldots, |S|\}$ obtained by replacing the smallest element of $S$ by $1$, the second smallest element of $S$ by $2$, and so on.
If $n$ is a positive integer we let $[n] = \{1,2,\ldots,n\}$ and define
$$\Pi_n = \makeset{\sigma}{\sigma\vdash[n]}.$$
The concept of pattern avoidance for set partitions was introduced by Klazar in <cit.>. If $k\leq n$ and we have $\sigma\vdash[n]$ and $\tau\vdash[k],$ then we say $\sigma$ contains $\tau$ as a pattern if there is a subset $S$ of $[n]$ such that the standardization of the restriction of $\sigma$ to $S$ is $\tau$. If $\sigma$ does not contain $\tau$, we say that $\sigma$ avoids $\tau$. We let
$$\Pi_n(\tau) = \makeset{\sigma\in \Pi_n}{\sigma\mbox{ avoids } \tau}.$$
The present paper is a contribution to the study of Klazar's definition of pattern avoidance, but we should mention right away that there are other definitions. One of these arises from the well-known correspondence between set partitions and restricted growth functions. Recall that a restricted growth function (RGF) is a word $a_1a_2\cdots a_\ell$ of positive integers such that $a_1 = 1$ and, for $i\geq 2$, we have $a_i\leq 1+ \max\{a_1,\ldots, a_{i-1}\}.$ The integer $\ell$ is called the length of the RGF. If $\sigma\in \Pi_n$, we define a corresponding $RGF$ of length $n$, denoted by $\rgf(\sigma)$, as follows. First write
$\sigma = B_1/B_2/\cdots /B_m$, where
$$\min B_1 < \min B_2<\cdots < \min B_m.$$
(This is called the standard form of $\sigma$.) Then let $\rgf(\sigma) = a_1\cdots a_n$, where $i\in B_{a_i}$. We can obtain an alternative notion of pattern avoidance for set partitions by using a natural notion of avoidance for RGF's. If $k\leq n$ and $\textbf{a}=a_1\cdots a_n$ and $\textbf{b}=b_1\cdots b_k$ are RGFs, we say that $\textbf{a}$ contains $\textbf{b}$ if $\textbf{a}$ has a subsequence whose standardization is $\textbf{b}$, otherwise we say that $\textbf{a}$ avoids $\textbf{b}$. This notion of RGF avoidance has been studied extensively (see Mansour's comprehensive book <cit.>). It does not coincide with Klazar's notion. For if $\sigma$ avoids $\tau$ in Klazar's sense, then $RGF(\sigma)$ avoids $RGF(\tau)$; but the converse may fail. For example $145/23$ contains $12/34$, yet $\rgf(145/23) = 12211$ avoids $\rgf(12/34) = 1122$.
There is yet a third notion of pattern avoidance for set partitions that involves arc-diagrams (See <cit.>). A well studied notion in this context is that of non-nesting and non-crossing set partitions which arise from the avoidance of certain arc-configurations. We wish to point out that under Klazar's definition non-crossing set partitions are those that omit $13/24$. Interestingly, there is no single pattern $\sigma$ such that the non-nesting partitions are precisely those that omit $\sigma$, in Klazar's sense. To see this, observe that the only candidate for $\sigma$ is $14/23$ since the non-nesting set partitions of length $4$ are $\Pi_4 \setminus \{14/23\}$. Yet, the set partition $135/24$ is non-nesting but contains $14/23$.
Klazar's definition has not been as well studied as the RGF definition. The earliest paper is of course Klazar's <cit.>, and the most recent paper of which we are aware is the paper <cit.> by Dahlberg, et al. We also mention Sagan's paper <cit.>, which contains, along with many other results, an enumeration of $\Pi_n(\tau)$ for all $\tau\vdash[3]$.
In Section <ref> of our paper we enumerate $\Pi_n(\tau)$ for all $\tau \vdash[4]$, although our main purpose is to undertake a study of Wilf-equivalence in the context of Klazar's definition of avoidance. If $\tau,\pi\vdash[k]$, we say that $\tau$ and $\pi$ are Wilf-equivalent, and we write $\tau\sim \pi$, if $|\Pi_n(\tau)| = |\Pi_n(\pi)|$ for all $n>k$. The fact that $1/2/3\sim 13/2$ is established in <cit.>. In Section <ref> of our paper we establish the new Wilf-equivalences
$$12\cdots a-1,a+1\cdots k/a \sim 12\cdots b-1,b+1\cdots k/b$$
for all $k\geq 4$ and $1<a,b<k$, and based on computer evidence, we conjecture (Conjecture <ref>) that these, $1/2/3\sim 13/2$, and $\tau\sim \tau^c$ are the only Wilf-equivalences, where $\tau^c$ is called the complement of $\tau$ and is obtained from $\tau$ by subtracting each number from $k+1$. (There are similar conjectures for Wilf-equivalences in other contexts, all of which seem quite difficult to prove. See <cit.>.) We also show that
$$|\Pi_n(2\cdots k/1)| < |\Pi_n(13\cdots k/2)| \mbox{ and } |\Pi_n(1\cdots k-1/k)| < |\Pi_n(13\cdots k/2)|$$
for all $n\geq 2k-2$ but that these inequalities may become equalities when $n< 2k-2$. For example, this occurs when $k=5$ and $n= 6,7$.
Motivated by the results indicated in the last sentence, we introduce a partial ordering $\prec$ of $\Pi_k$, as follows. For $\tau, \pi\in \Pi_k$ we write $\tau\prec \pi$ if $|\Pi_n(\tau)| \leq |\Pi_n(\pi)|$ for all $n>k$, and there exists some $m\geq k$ such that $|\Pi_n(\tau)| < |\Pi_n(\pi)|$ for all $n>m$.
If $\beta_k$ denotes the partition of $[k]$ that has only one block, then computer evidence suggests that, for every $k\geq 4$ and every $\tau\neq \beta_k$ in $\Pi_k$ we have $\tau\prec \beta_k$, and in fact $|\Pi_n(\tau)|< |\Pi_n(\beta_k)|$ for all $n>k$. In Section <ref> of our paper we prove this result for all $\tau$ with exactly two blocks, and we conjecture (Conjecture <ref>) that the result holds for all $\tau\neq \beta_k$. We also consider partitions $\pi$ such that there is exactly one doubleton block and all other blocks are singletons. We prove that for every such $\pi$, if $\sigma_k$ denotes the partition all of whose blocks are singletons, then
$$|\Pi_n(\pi)|<|\Pi_n(\sigma_k)|< |\Pi_n(\beta_k)|$$
for all $n>k$. This result should be compared with Klazar's result in <cit.> that for all such $\pi$ and $\sigma_k$ the generating function of $|\Pi_n(\pi)|$ is rational (whereas the generating function for $|\Pi_n(\beta_k)|$ is not).
§ WILF-EQUIVALENCE
For a fixed $k$, we first determine all Wilf-equivalences among the patterns
$$\beta_{k,a} = 1\cdots (a-1) (a+1)\cdots k/ a,$$
where $1\leq a \leq k$.
We have
$\beta_{k,1}\sim \beta_{k,k}$. For $n> k$ and $2\leq a \leq k-1$ we have
$$|\Pi_n(\beta_{k,a+1})|\leq |\Pi_n(\beta_{k,a})|,$$
with equality if $a< k-1,$ and therefore $\beta_{k,a}\sim \beta_{k,b}$ when $2\leq a,b \leq k-1.$ Finally, we have
$$|\Pi_n(\beta_{k,k})| < |\Pi_n(\beta_{k,k-1})|$$
when $n\geq 2k-2,$ so $\beta_{k,k} \prec \beta_{k,a}$ for $2\leq a \leq k-1.$
This theorem together with computational evidence for all patterns of length $\leq 9$ suggests the following conjecture.
The only Wilf-equivalences are $1/2/3\sim 13/2$, the equivalences established by Theorem <ref>, and the equivalences resulting from complementation.
The first assertion of Theorem <ref> is a simple consequence of complementation. To prove the second assertion, we construct an injection
$$\phi_a:\Pi_n(\beta_{k,a+1}) \to \Pi_n(\beta_{k,a}),$$
for each $2\leq a \leq k-1$, and show that $\phi_a$ is a bijection if $a< k-1.$ Before we can write down this mapping we need a few lemmas and observations. We start with these immediately and delay the proof of Theorem <ref> to the end of this section.
To begin, fix $k$ and $2\leq a\leq k-1$. Observe that $\pi\vdash [n]$ contains $\beta_{k,a}$ if and only if $\pi$ contains a block $B$ such that
\begin{equation}\label{eq:blocks contain}
a-1\leq |\makeset{x\in B}{x<c}|\quad\mbox{and}\quad k-a \leq |\makeset{x\in B}{x>c}|
\end{equation}
for some $c\notin B$.
Consequently, we say a finite set $B$ of integers contains $\beta_{k,a}$ if it satisfies (<ref>), otherwise we say $B$ avoids $\beta_{k,a}$. Consequently, a set partition $\pi$ avoids $\beta_{k,a}$ if and only if all of its blocks avoid $\beta_{k,a}$.
Now, consider the anatomy of a block $B$ that avoids $\beta_{k,a+1}$ but contains $\beta_{k,a}$. Since $B$ contains $\beta_{k,a}$, there exists some $c\notin B$ so that
$$a-1 \leq |\makeset{x\in B}{x<c}|\quad\mbox{and}\quad k-a \leq |\makeset{x\in B}{x>c}|.$$
Furthermore, since $B$ avoids $\beta_{k,a+1}$, the set on the left must have size exactly $a-1$. For the same reason, the set on the right must be of the form
$$y_1-\ell,\ldots,y_1-2,y_1-1,y_1<\cdots < y_{k-a-1},$$
for some $\ell\geq 1$. Therefore the block $B$ looks like
\begin{equation}\label{eq:B decomp}
\begin{tikzpicture}[baseline={([yshift=2ex]current bounding box.center)}]
\draw (0,0) rectangle (2,1) node[pos=.5] {$<c$};
\draw[fill=gray!20] (3,0) rectangle (5,1)node[pos=.5] {};
\draw (5,0) rectangle (8,1)node[pos=.5] {};
\draw [
},decorate] (0,0) -- (2,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] {$a-1$};
\draw [
},decorate] (3,0) -- (5,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] {$\ell$};
\draw [
},decorate] (5,0) -- (8,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] {$k-(a+1)$};
\end{tikzpicture}\ \raisebox{-10pt}{,}
\end{equation}
where the elements in the shaded block are consecutive in value and the rightmost block's smallest element is $y_1$. By decrementing the values in the shaded region, we effectively slide the gray rectangle as far left as possible, obtaining the new block $B'$:
\begin{equation}\label{eq:B' decomp}
\begin{tikzpicture}[baseline={([yshift=2ex]current bounding box.center)}]
\draw (0,0) rectangle (2,1) node[pos=.5] {$<c$};
\draw[fill=gray!20] (2,0) rectangle (4,1)node[pos=.5] {};
\draw (5,0) rectangle (8,1)node[pos=.5] {};
\draw [
},decorate] (0,0) -- (2,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] {$a-1$};
\draw [
},decorate] (2,0) -- (4,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] {$\ell$};
\draw [
},decorate] (5,0) -- (8,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] {$k-(a+1)$};
\end{tikzpicture}\ \raisebox{-12pt}{.}
\end{equation}
We remark that in $B'$, the elements in the gray block are consecutive in value and the largest element in the leftmost block is one less than the smallest element in the gray block. Consequently, $B'$ avoids $\beta_{k,a}$ while containing $\beta_{k,a+1}$.
The new block $B'$ gives rise to a new set partition as summarized in our next definition.
Let $\pi\vdash [n]$ and assume the $i$th block $B$ of $\pi$ avoids $\beta_{k,a+1}$ and contains $\beta_{k,a}$. So $B$ is as depicted in (<ref>). Now let $B'$ be the block depicted in (<ref>). We define $\slide_i(\pi)\vdash [n]$ to be the set partition $\pi' = B'/ \sigma$ where $\sigma$ is the set partition of $[n]\setminus B'$ that is order-isomorphic to the set partition obtained by deleting $B$ from $\pi$.
We illustrate this definition with the following short example.
Let $k=5, a=2$ so that $\beta_{5,3} = 1245/3$ and $\beta_{5,2} = 1345/2$. If
$$\pi = 1\ 3\ / 2\ 5\ 6\ 7\ 8\ 9/ 4\ 10 \in \Pi_{10},$$
then, since its second block avoids $\beta_{5,3}$ and contains $\beta_{5,2}$, we have
$$\slide_2(\pi) = 1\ \textbf{6}\ / 2\ \textbf{3\ 4\ 5}\ 8\ 9/ \textbf{7}\ 10,$$
where the elements affected by our sliding operation are highlighted.
If $\pi$, $B$, and $B'$ are as in the above definition and $B$ is the $i$th block (in standard form) of $\pi$, then $B'$ is the $i$th block in $\slide_i(\pi)$.
First observe that as $a\geq 2$, then $\min(B) = \min(B')$. Now set
$$[n]\setminus B = \{w_1< w_2<\cdots\}\qquad\mbox{and}\qquad [n]\setminus B' = \{w_1'< w_2'<\cdots\}.$$
It follows from our construction of $B'$ that
$$w_s < \min(B)\Longrightarrow w_s = w_s',$$
so $B'$ is no earlier than the $i$th block in $\slide_i(\pi)$ and
$$w_s>\min(B)\Longrightarrow w_s\leq w_s',$$
so $B'$ is no later than the $i$th block in $\slide_i(\pi)$.
As the definition of our mapping $\phi_a$ involves repeated “slide" operations, we require a lemma guaranteeing that blocks not involved in the slide operation do not change “too much". Our next lemma spells out exactly what is meant by this.
Let $\pi=B_1/B_2/\cdots /B_m\vdash [n]$. Assume that for some fixed $i$, the block $B_i$ avoids $\beta_{k,a+1}$ and contains $\beta_{k,a}$ and set
$$\slide_i(\pi) = B_1'/B_2'/\cdots/B_m'.$$
Then for any $1\leq c\leq k$ and $j\neq i$, the block $B_j'$ avoids $\beta_{k,c}$ if and only if $B_j$ avoids $\beta_{k,c}$.
Fix $j\neq i$, $1\leq c\leq k$ and set
$$[n]\setminus B_i = \{w_1< w_2<\cdots\}\qquad\mbox{and}\qquad [n]\setminus B_i' = \{w_1'< w_2'<\cdots\}.$$
First, we claim that $w_t+1 = w_{t+1}$ if and only if $w'_t+1 = w_{t+1}'$. For the moment, let us assume this claim. Next we make the following general observation: Any block $B_j$ avoids $\beta_{k,c}$ if and only if whenever $x\in B_j$ is such that
$$c-1 \leq |\makeset{y\in B_j}{y\leq x}|\quad\mbox{and}\quad k-c \leq |\makeset{y\in B_j}{y\geq x}|,$$
then $x+1\in B_j$.
As $\pi - B_i$ is order-isomorphic to $\slide_i(\pi) - B_i'$, it now follows that $B_j$ avoids $\beta_{k,c}$ if and only if $B_j'$ does too.
It only remains to prove our claim. Since $B_i$ avoids $\beta_{k,a+1}$ and contains $\beta_{k,a}$ it is depicted in (<ref>) and $B_i'$ is depicted in (<ref>). Observe that if $I$ is the set of values in $[n]\setminus B_i$ that fall in the gap to the left of the gray block in (<ref>) and $J$ is the set of values in $[n]\setminus B_i'$ that fall in the gap to the right of the gray block in (<ref>), then $I$ and $J$ are both intervals and $|I| = |J|$. If in (<ref>) $\ell$ is the maximum of the values in the leftmost block and (for $a<k-1$) $r$ is the minimum of the values in the rightmost block, then it follows that
\begin{equation}\label{eq:left of slide}
\makeset{w_s}{w_s<\ell} = \makeset{w_s'}{w_s'< \ell}
\end{equation}
\begin{equation}\label{eq:right of slide}
\makeset{w_s}{w_s>r} = \makeset{w_s'}{w_s'> r}.
\end{equation}
Assuming $w_t$ and $w_{t+1}$ are consecutive in value it follows that they both lie in either the set (<ref>), the set (<ref>), or in the interval $I$. In the first two cases, it easily follows from our above equalities that $w_t'+1=w_{t+1}'$. In the third case, the set equality in (<ref>) along with the fact that $|I|=|J|$ implies that $w_t',w'_{t+1}\in J$. Since $J$ is an interval, we conclude that $w_t',w'_{t+1}$ are consecutive in value.
The proof that if $w_t'$ and $w_{t+1}'$ are consecutive in value, then $w_t$ and $w_{t+1}$ are too, is analogous. Its details are omitted.
Finally, we are in a position to define $\phi_a$. For any $\pi\in \Pi_n(\beta_{k,a+1})$ first set $\pi_0 = \pi$ and let $m$ be the number of blocks in $\pi$. Having defined $\pi_i$ we obtain $\pi_{i+1}$ by considering the $(i+1)$st block of $\pi_i$. If this block contains $\beta_{k,a}$, then set $\pi_{i+1}=\slide_{i+1}(\pi_i)$, otherwise we set $\pi_{i+1} =\pi_i$. Lastly we set $\phi_a(\pi) = \pi_m$.
We pause to point out that our mapping $\phi_a$ is well defined. First, we note that Lemma <ref> guarantees that the $(i+1)$st block in $\pi_i$ avoids $\beta_{k,a+1}$. (This is crucial since if this block contained $\beta_{k,a+1}$ and also contained $\beta_{k,a}$ our sliding operation would not be defined.) Additionally, Lemma <ref> together with Remark <ref> guarantees that $\phi_a(\pi)$ avoid $\beta_{k,a}$.
Before proving that $\phi_a$ is a bijection we demonstrate this mapping.
Consider $\beta_{5,3} = 1245/3$ and $\beta_{5,2} = 1345/2$ and fix
$$\pi = 1\ 10\ 11\ 12/ 2\ 4\ 5\ 8/ 3\ 6\ 7\ 9 \in \Pi_{12}(\beta_{5,3}).$$
The above algorithm for $\phi_3$ yields the following steps, where the elements affected by each slide are highlighted.
\begin{align*}
\pi_0 =&1\ 10\ 11\ 12 / 2\ 4\ 5\ 8/ 3\ 6\ 7\ 9\\
\pi_1 =& 1\ \textbf{2}\ 11\ 12 / \textbf{3\ 5\ 6\ 9}/ \textbf{4\ 7\ 8\ 10}\\
\pi_2 =& 1\ 2\ 11\ 12 / 3\ \textbf{4}\ 6\ 9/ \textbf{5}\ 7\ 8\ 10\\
\pi_3 =& 1\ 2\ 11\ 12 / 3\ 4\ \textbf{7}\ 9/ 5\ \textbf{6}\ 8\ 10
\end{align*}
As each slide in the definition of $\phi_a$ is certainly reversible, it follows that the mapping
$$\phi_a:\Pi_n(\beta_{k,a+1}) \to \Pi_n(\beta_{k,a}),$$
is injective, provided $2\leq a \leq k-1$. Under this restriction we therefore have
$$|\Pi_n(\beta_{k,a+1})|\leq |\Pi_n(\beta_{k,a})|.$$
If $a<k-1,$ then $k-a\geq 2,$ so
the composition of injective maps
$$\Pi_n(\beta_{k,a}) \xrightarrow{\comp} \Pi_n(\beta_{k,k+1-a}) \xrightarrow{\phi_{k-a}} \Pi_n(\beta_{k,k-a}) \xrightarrow{\comp} \Pi_n(\beta_{k,a+1}),$$
implies that
$$|\Pi_n(\beta_{k,a})|\leq |\Pi_n(\beta_{k,a+1})|$$
and proves that $\phi_a$ is bijective. This completes the proof of the second assertion of Theorem <ref>.
To prove the third assertion, we show that the injection $\phi_{k-1}:\Pi_n(\beta_{k,k})\rightarrow \Pi_n(\beta_{k,k-1})$ is not surjective when $n\geq 2k-2.$ To see this, it will be helpful to observe that if $\pi\in \Pi_n(\beta_{k,k})$ then at most one block of $\pi$ has $k-1$ or more elements. For if one block has elements $x_1<\cdots <x_{k-1}$ and another block has elements $y_1<\cdots <y_{k-1}$, then either $x_{k-1}< y_{k-1}$ or $y_{k-1}< x_{k-1}$, so $\pi$ contains the partition $\beta_{k,k}$, a contradiction. It now follows from the definition of $\phi_{k-1}$ that for every $\pi\in \Pi_n(\beta_{k,k})$ there is at most one block of size greater than or equal to $k-1$ in $\phi_{k-1}(\pi).$ But if $n\geq 2k-2$ then the element $1,2,\ldots,k-1/k,\ldots,2k-2/(2k-1),\ldots, n$ of $\Pi_n(\beta_{k,k-1})$ has at least two blocks of size $k-1.$
§ PARTIAL ORDERING BY $\PREC$
§.§ The pattern $\beta_k$
We recall that $\beta_k$ denotes the element of $\Pi_k$ that has only one block. If $\tau$ is any element of $\Pi_k$ other than $\beta_k$ itself, computer evidence suggests that $\tau\prec \beta_k$ and $|\Pi_n(\tau)|< |\Pi_n(\beta_k)|$ for all $n> k.$
Let $k\geq 4.$ If $\tau\in \Pi_k$ and $\tau\neq \beta_k$, then $\tau\prec \beta_k$ and $|\Pi_n(\tau)|< |\Pi_n(\beta_k)|$ for all $n> k.$
It follows from Theorems <ref>, <ref>, and <ref> that Conjecture <ref> is true for $k=4$.
Let $k\geq 4$. If $\sigma \in \Pi_k$ and $ \sigma$ has exactly two blocks, then $\sigma \prec \beta_k.$ In fact, $|\Pi_n(\sigma)| < |\Pi_n(\beta_k)|$ for all $n>k.$
Suppose $\sigma\in \Pi_k$ and $\sigma$ has exactly two blocks, so $\sigma=A/B,$ with $1\in A.$ Suppose $a_1,b_1,a_2,b_2,\ldots,a_j,b_j$ are positive integers and $a_{j+1}$ is a nonnegative integer such that the first $a_1$ elements of $[k]$ are in $A$, the next $b_1$ elements of $[k]$ are in $B$, the next $a_2$ elements of $[k]$ are in $A$, and so on, so that $a_{j+1}=0$ if $k\in B$ and $a_{j+1}> 0$ if $k\in A.$
To prove the theorem we establish, for any given $n>k,$ a nonsurjective injection
$$\varphi:\Pi_n-\Pi_n(\beta_k)\rightarrow \Pi_n-\Pi_n(\sigma).$$
To define $\varphi,$ take any $\pi\in \Pi_n-\Pi_n(\beta_k).$ If $\pi\in \Pi_n-\Pi_n(\sigma),$ let $\varphi(\pi)=\pi.$
Now suppose $\pi\in \Pi_n(\sigma)$. Since $\pi\in \Pi_n-\Pi_n(\beta_k)$, $\pi$ has at least one block with at least $k$ elements. We obtain $\varphi(\pi)$ from $\pi$ by partitioning each such block $C$ in the following way.
Recalling that $\sigma=A/B$, let $|B|=m$, so that $m=b_1+\cdots +b_j$. Write
where $q\geq 1$ and $0\leq r <m$ are integers. Then we have
$$|C|=|A|+r+q(b_1+\cdots +b_j).$$
We order the elements of $C$ from smallest to largest and define a subset $A^*$ of $C$ of cardinality $|A|+r$, as follows. Let the first $a_1$ elements of $A^*$ be the first $a_1$ elements of $C$. Skip over the next $qb_1$ elements of $C$, and let the next $a_2$ elements of $C$ be the next $a_2$ elements of $A^*.$ Then skip over the next $qb_2$ elements of $C$ and let the next $a_3$ elements of $C$ be the next $a_3$ elements of $A^*.$ Continuing in this way, define the first $a_1+\cdots +a_{j+1}=|A|$ elements of $A^*.$ Then add the last $r$ elements of $C$ to $A^*,$ so that $|A^*|=|A|+r.$
Next we define subsets $B_1,\ldots,B_q$ of $C$ such that $|B_i|=m$ for $1\leq i\leq q$. Take the first $qb_1$ elements of $C$ that were skipped over in the construction of $A^*$, and put the first $b_1$ of these elements in $B_1,$ the next $b_1$ of these elements in $B_2,$ and so on. Then take the next $qb_2$ elements of $C$ that were skipped over in the construction of $A^*$ and put the first $b_2$ of these elements in $B_1,$ the next $b_2$ of these elements in $B_2,$ and so on. Continuing in this way, complete the construction of $B_1,\ldots,B_q$.
We partition $C$ into blocks $A^*,B_1,\ldots,B_q$. By construction, we see that for each $B_i$, the partition $A^*/B_i$ contains the partition $\sigma$. We have $|B_i|=m<k$ and $|A^*|=|A|+r< |A|+m=k$.
When $\pi\in \Pi_n(\sigma)$, we obtain $\varphi(\pi)$ from $\pi$ by partitioning each block $C$ of size at least $k$ in the way just indicated. Clearly, $\varphi(\pi)\in \Pi_n-\Pi_n(\sigma)$. We also have $\varphi(\pi)\in \Pi_n(\beta_k)$, by the last sentence of the preceding paragraph. For $\pi\in \Pi_n-\Pi_n(\sigma)$ we had $\varphi(\pi)=\pi\notin\Pi_n(\beta_k)$, so to show that $\varphi$ is one-to-one it suffices to show that, for $\pi\in \Pi_n(\sigma)$, we can recover $\pi$ from $\varphi(\pi)$.
To show this, observe that each block of $\varphi(\pi)$ is a subset of a block of $\pi$, and that if $D,E$ are blocks of $\varphi(\pi)$ such that the partition $D/E$ contains $\sigma$, then since $\pi\in\Pi_n(\sigma)$, $D$ and $E$ must be subsets of the same block of $\pi$. Thus we obtain $\pi$ from $\varphi(\pi)$ by coalescing into one block the elements of any blocks $D,E$ of $\varphi(\pi)$ such that $D/E$ contains $\sigma$.
Finally, to show that $\varphi$ is not surjective, first suppose that $m+1< k$ and consider the partition $\gamma$ of $[n]$ whose blocks are $A, B\cup\{k+1\}$ and $n-k-1$ singleton blocks. Then $\gamma\in \Pi_n-\Pi_n(\sigma)$. Suppose $\pi\in \Pi_n-\Pi_n(\beta_k)$ and $\varphi(\pi)=\gamma$. Since $m+1< k$, we have $\gamma\in \Pi_n(\beta_k)$, so $\gamma \neq \pi$ and thus $\pi\in \Pi_n(\sigma)$. Since $\varphi(\pi)=\gamma,$ it follows from the definition of $\varphi$ that $A\cup B \cup \{k+1\}$ is contained in some block $F$ of $\pi,$ and $|F|> k$. In defining $\varphi(\pi)$, the block $A^*$ derived from $F$ contains the smallest element of $F$, namely 1, so $A^*=A$ (since $A$ is the block of $\gamma$ that contains 1). The other blocks derived from $F$ have $m$ elements each. This is impossible, since the block $B\cup \{k+1\}$ is derived from $F$.
Now consider the case $m+1=k$. In this case $\sigma=\beta_{k,1}$. Since $\beta_{k,1}\sim\beta_{k,k}$ by Theorem <ref> and $\varphi$ is not surjective when $\sigma=\beta_{k,k}$ by the preceding paragraph, it follows that $\varphi$ is not surjective when $\sigma=\beta_{k,1}$.
§.§ The pattern $\sigma_k$
We recall that $\sigma_k = 1/2/3/\cdots/ k$. In this subsection we prove that for all $\delta\in \Pi_k$, whose blocks consist of all singletons except for one doubleton block,
$$\delta\prec \sigma_k\prec \beta_k.$$
We begin with a lemma.
Let $\alpha\vdash [k-1]$. If
$$|\Pi_n(\alpha)| <|\Pi_n(\sigma_{k-1})|$$
for all $n>k-1$, then
$$|\Pi_n(1/\alpha')| <|\Pi_n(\sigma_{k})|\quad\mbox{and}\quad |\Pi_n(\alpha/k)| <|\Pi_n(\sigma_{k})|, $$
for all $n>k$, where $\alpha'$ is obtained by incrementing all the values in $\alpha$ by $1$.
Proving the inequality on the left is sufficient, since the inequality on the right then follows by complementation.
To do this, fix $n>k$ and assume there exists nonsurjective injections $\phi_m :\Pi_m(\alpha)\to \Pi_m(\sigma_{k-1})$ for all $m> k-1$. Observe that for any $B_1/B_2/\cdots/ B_m\in \Pi_n(1/ \alpha')$, in standard form, the set partition $B_2/\cdots /B_m$ avoids $\alpha$ as $1\in B_1$. Therefore we obtain a nonsurjective injection $\psi$ from $\Pi_n(1/\alpha')$ to $\Pi_n(\sigma_k)$ as follows: If $n-|B_1| > k-1$ we let
$$\psi(B_1/B_2/ \cdots/ B_m) = B_1/ \phi_{n-|B_1|}(B_2 / \cdots/ B_m).$$
(To be precise $\phi_{n-|B_1|}(B_2 / \cdots/ B_m)$ is obtained by first standardizing the partition $B_2/\cdots/B_m$, then applying $\phi_{n-|B_1|}$, and then incrementing the values.) If $n-|B_1| \leq k-1$ we let
$$\psi(B_1/B_2/ \cdots/ B_m) = B_1/B_2/ \cdots/ B_m,$$
unless $B_2/ \cdots/ B_m$ consists of exactly $k-1$ singleton blocks whose union is $[n]\setminus B_1$. In this exceptional case we set $\psi(B_1/B_2/ \cdots/ B_m)$ to be the partition $B_1/A$ where $A$ is the partition of $[n]\setminus B_1$ whose standardization is $\alpha$.
The existence of this injection proves our claim.
To prove our next theorem, we first recall the bijection between restricted growth functions and set partitions as described in the Introduction. Next, define $\rgf_n^{<k}$ to be the set of all $w\in \rgf$ in the letters $\{1,\ldots, k-1\}$ with length $n$. It immediately follows that the standard bijection between $\rgf$ and set partitions restricts to a bijection between $\rgf_n^{<k}$ and $\Pi_n(\sigma_k)$.
Let $\delta$ be a pattern of length $k\geq 4$ that consists of all singletons except for exactly one doubleton. Then
$$|\Pi_n(\delta)| < |\Pi_n(\sigma_k)|$$
for all $n>k\geq 3$. Consequently, $\delta\prec \sigma_k$.
If $a$ and $b$ are the elements of the doubleton block of $\delta$, we consider cases depending on the value of $|a-b|$.
If $|a-b|=1$, then since $k\geq 4$ we see by using Lemma <ref> that it suffices to show that $|\Pi_n(\alpha)| < |\Pi_n(\sigma_4)|$ where $n> 4$ and $\alpha$ is one of $12/3/4$, $1/23/4$, or $1/2/34$. By Lemma <ref> again, it then suffices to show that $|\Pi_n(\beta)|< |\Pi_n(\sigma_3)|$ where $n> 3$ and $\beta$ is one of $12/3$ or $1/23$. For either choice of $\beta$ we have $|\Pi_n(\beta)| = 1+ \binom{n}{2}$ and $|\Pi_n(\sigma_3)| = 2^{n-1}$ by Theorem 2.5 of (<cit.>), and this concludes the proof when $|a-b| =1$.
If $|a-b|=2$, then we see by using Lemma <ref> that it suffices to show that $|\Pi_n(\alpha)|< |\Pi_n(\sigma_4)|$ when $n>4$ and $\alpha$ is one of $13/2/4$ or $1/3/24$. Since $13/2/4$ is obtained from $1/3/24$ by complementation it suffices to deal with $1/3/24$. We do so in Lemma <ref>, following the enumeration of $|\Pi_n(1/3/24)|$.
To deal with the case $|a-b|\geq 3$, it suffices, by Lemma <ref>, to prove the result for $\delta = 1k/2/3/\cdots / k-1$. For such a $\delta$, we see that every set partition in $\Pi_n(\delta)$ is obtained from a unique set partition $B_1/\cdots /B_m\in \Pi_{n-1}(\delta)$ by inserting $n$ into one of the $k-2$ rightmost blocks $B_{m-(k-3)}, \ldots, B_m$ or by inserting $n$ into a new singleton block. We encode these insertion choices as follows:
1&\leftrightarrow & \mbox{ into a new singleton block}\\
2&\leftrightarrow & \mbox{ into } B_m\\
3&\leftrightarrow & \mbox{ into } B_{m-1}\\
&\vdots &\\
k-1&\leftrightarrow & \mbox{ into } B_{m-(k-3)}.
\end{array}$
Now define the set $R_n^{<k}$ to be the set of all words $w$ in the letters $1,2,\ldots, k-1$ such that $w_1 = 1$ and
$$w_s \leq 1+ \#\mbox{ of 1's in the subword } w_1\cdots w_{s-1}.$$
Using the above encoding, it is clear that we have an injective mapping from $\Pi_n(\delta)$ into the set $R_n^{<k}$. This mapping is not surjective for $n>k\geq 4$ since then the set partition
$$1(k-1)/2(k+1)/3/\cdots /k-2/k/k+2/\cdots / n\notin \Pi_{n}(\delta)$$
is mapped, under the above encoding, to the word
$$\underbrace{11\cdots 1}_{k-2} (k-1) 1 (k-1)\underbrace{11\cdots 1}_{n-k-1} \in R_{k+1}^{<k}.$$
It now suffices to prove that $|\rgf_n^{<k}|=|R_n^{<k}|$ since $|\rgf_n^{<k}| = |\Pi_n(\sigma_k)|$ as mentioned above. To see this consider a word $w\in \rgf_n^{<k}$ and decompose it according to the first occurrence, from left to right, of each letter. Doing so $w$ decomposes into the subwords
(0,0)rectangle (.5,.5) node[pos=.5] 1;
(.6,0) rectangle (3,.5) node[pos = .5] 1's;
(3.2,0)rectangle (3.7,.5) node[pos=.5] 2;
(3.9,0) rectangle (6.6,.5) node[pos = .5] 1,2's;
(6.8,0)rectangle (7.3,.5) node[pos=.5] 3;
(7.5,0) rectangle (10,.5) node[pos = .5] 1,2,3's;
at (10.7,.2) $\cdots$;
(11.5,0)rectangle (12,.5) node[pos=.5] m;
(12.2,0) rectangle (14.7,.5) node[pos = .5] $1,2,\ldots, m's$;
[thick, decoration=
raise=0.1cm, decorate] (.6,0) – (3,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] $u_1$;
[thick, decoration=
raise=0.1cm, decorate] (3.9,0) – (6.6,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] $u_2$;
[thick, decoration=
raise=0.1cm, decorate] (7.5,0) – (10,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] $u_3$;
[thick, decoration=
raise=0.1cm, decorate] (12.2,0) – (14.7,0)
node [pos=0.5,anchor=north,yshift=-0.2cm] $u_m$;
As $w\in\rgf_n^{<k}$ we know that $m\leq k-1$. In the case that $m<k-1$ we define $w'$ to be the new word given by incrementing each of the subwords $u_i$ by 1 and replacing each letter's first occurrence by a 1. So $w'$ decomposes, according to its occurrences of 1's, as
(0,0)rectangle (.5,.5) node[pos=.5] 1;
(.6,0) rectangle (3,.5) node[pos = .5] $u_1+1$;
(3.2,0)rectangle (3.7,.5) node[pos=.5] 1;
(3.9,0) rectangle (6.6,.5) node[pos = .5] $u_2+1$;
(6.8,0)rectangle (7.3,.5) node[pos=.5] 1;
(7.5,0) rectangle (10,.5) node[pos = .5] $u_3+1$;
at (10.7,.2) $\cdots$;
(11.5,0)rectangle (12,.5) node[pos=.5] 1;
(12.2,0) rectangle (14.7,.5) node[pos = .5] $u_m+1$;
It easily follows that $w'\in R_n^{<k}$ since the letters in $u_i+1$ are at most $i+1$ and are preceded by exactly $i$ occurrences of the letter 1.
In the case that $m=k-1$, we modify this mapping only slightly. Here we define $w'$ to be the word
(0,0)rectangle (.4,.5) node[pos=.5] 1;
(.6,0) rectangle (2.4,.5) node[pos = .5] $u_1+1$;
(2.6,0)rectangle (2.9,.5) node[pos=.5] 1;
(3.1,0) rectangle (5,.5) node[pos = .5] $u_2+1$;
at (7,.2) $\cdots$;
(8.5,0) rectangle (8.9,.5) node[pos=.5] 1;
(9.1,0) rectangle (11.4,.5) node[pos = .5] $u_{k-2}+1$;
(11.6,0)rectangle (11.9,.5) node[pos=.5] 1;
(12.1,0) rectangle (14,.5) node[pos = .5] $u_{k-1}$;
so that all subwords, except the last one, $u_{k-1}$, are incremented by 1. Note this second case is needed since the letter $k-1$ might occur in $u_{k-1}$ and the letter $k$ is not available. Again it is clear that $w'\in R_n^{<k}$. In this way we obtain an injective mapping from $\rgf_n^{<k}$ to $R_n^{<k}$. Furthermore, this mapping is easily seen to be surjective as we can decompose any word $v$ in $R_n^{<k}$ according to its first $k-1$ ones and then reverse the above mapping. It only remains to prove that applying the reverse mapping to any $v\in R_n^{<k}$ results in a word $w\in \rgf_n^{<k}$.
This easily follows from the observation that if $v_s$ is a letter in $v$ with $j$ ones to its left, then
$$v_s\leq j+1 = \max(w_1,\ldots, w_{s-1}) +1.$$
First assume $j<k-1$. In this case, if $v_s\neq 1$, then $w_s=v_s-1$ and if $v_s =1$, then $w_s = j+1$. Either way, such letters satisfy the condition of a restricted growth function. The case when $j\geq k-1$ is similar. Its details are left to the reader.
For all $k\geq 2$ and $n\geq 1$
$$|\Pi_n(\sigma_k)| \leq |\Pi_n(\beta_k)|.$$
Moreover, this inequality is strict provided $n>k>2$.
First, we show, by induction on $k$, that there exists a family of injections
$$\psi_{k,n}: \Pi_n(\sigma_k) \to \Pi_n(\beta_k),$$
for all $k\geq 2$ and $n\geq 1$.
To streamline this notation we drop the subscript $n$ and write $\psi_k$ instead of $\psi_{k,n}$.
To begin our induction argument note that $|\Pi_n(\sigma_2)| =1=|\Pi_n(\beta_2)|$. Now, consider $\pi = B_1/\cdots/B_m\in \Pi_n(\sigma_{k+1})$ and observe that $B_2/\cdots/B_m$ can be considered a set partition of $[n-|B_1|]$ that avoids $\sigma_k$. Therefore $\psi_k(B_2/\cdots/B_m)$ is well defined, avoids $\beta_k$, and has blocks of size $<k$. (If $B_2/\cdots/B_m = \emptyset$ we set $\psi_k(\emptyset) = \emptyset$.) With this in mind, the division algorithm yields
$$|B_1| = q\cdot k + r,$$
where $0\leq r\leq k-1$. Provided $r>0$ let $C_0/C_1/C_2/\cdots/C_q$ be the set partition of $B_1$ where $C_0$ consists of the first $r$ numbers of $B_1$ and $C_1$ consists of the next $k$ numbers, etc. Note that in the case $r=0$ we simply ignore $C_0$. Finally, we define
$$\psi_{k+1}(\pi) = C_0/C_1/ \cdots/C_q/\psi_k(B_2/\cdots/B_m),$$
so that $C_1,\ldots, C_q$ are the only blocks of size $k$ in this partition. Note that if $r>0$, then $1\in C_0$, otherwise $1\in C_1$. Consequently, $\psi_{k+1}$ is injective since $\psi_k$ is injective and $B_1$ may be recovered by taking the union of all blocks of size $k$ in $\psi_{k+1}(\pi)$ along with the block containing 1.
It only remains to show that our mappings $\psi_{n,k}$ are not surjective when $n>k>2$. To see this observe that in the construction of $\psi_{n,k}(\pi)$ the first block of $\pi$ (in standard form) is partitioned into consecutive segments as described above. As a result, no partition in the image of $\psi_{n,k}$ could contain the blocks:
$$1\ 3/ 2\ 4\ 5\ \cdots (k+1),$$
because both blocks would have to come from $B_1$, but this violates consecutiveness. Hence these mappings cannot be surjective.
Let $\delta$ be any pattern of length $k$ consisting of all singletons except for one doubleton. Then,
$$|\Pi_n(\delta)| < |\Pi_n(\sigma_k)| <|\Pi_n(\beta_k)|,$$
for all $n>k\geq 4$.
The proof of this corollary is a direct consequence of the previous two theorems.
§ ENUMERATION
In this section we concentrate on the enumeration of $|\Pi_n(\tau)|$ for patterns $\tau\vdash [4]$. As enumerations for the patterns $1/2/3/\cdots/k$ and $\beta_k=12\cdots k$ are well known using exponential generating functions and Sagan in <cit.> enumerated the general pattern $12/3/4/\cdots/k$, we concentrate on all others with one exception. As the enumeration for the pattern $1/23/4$ devolves into numerous (uninteresting) cases and Klazar in <cit.> showed that the resulting generating function is rational, we choose to omit this pattern from our discussion.
To summarize the enumerations established in the following subsections (as well as those mentioned above) we include the following table of results where $\displaystyle\exp_m =\sum_{i=0}^m\frac{x^i}{i!}$ and
$S(n,k)$ denotes the Stirling numbers of the second kind and
$\displaystyle G(z) = \frac{z-2z^2(1+z)-z\sqrt{1-4z^2}}{-2+2z(1+z)^2}$. Aside from the omitted pattern $1/23/4$, every pattern of length 4 is, up to complementation, included in the table.
Pattern Enumeration Reference
$1234$ $\displaystyle\exp(\exp_{3}(z)-1)$ -
$1/2/3/4$ $\displaystyle\exp_{3}(e^z-1)$ -
$12/3/4$ $\displaystyle 1+\sum_{k=1}^{n-1}\sum_{j=1}^{m-2}S(n-k,j)\sum_{i=1}^j \binom{j-1}{i-1}(k)_i$ <cit.>
$12/34$ $\displaystyle\sum_{k=0}^{\lfloor n/2\rfloor} k!{n \choose 2k}+\sum_{\ell=3}^{n-2k}{n\choose 2k+\ell}k!(k+1)^2$ (<ref>)
$1/234$ $\displaystyle \sum_{\ell=1}^{n} M(n-\ell) + \sum_{\ell=1}^{n-2} (n-\ell-1)M(n-\ell-1)+\sum_{\ell=1}^{n-3} {n-\ell-1 \choose 2}M(n-\ell-2)$ (<ref>)
$134/2$ $\displaystyle 1+ \sum_{k=1}^{\lfloor n/2 \rfloor}\sum_{f=0}^{n-2k} {n-f-k-1\choose k-1}{2k+f\choose f}(2k)!!$ (<ref>)
$14/23$ $\displaystyle G\left(\frac{z}{1-z}\right)\frac{1}{1-z} + \frac{1}{1-z}$ (<ref>)
$13/24$ $\displaystyle \frac{1-\sqrt{1-4z}}{2z}$ -
$14/2/3$ $\displaystyle\frac{z-3z^2+3z^3}{1-5z+8z^2-5z^3}$ (<ref>)
$1/24/3$ $\displaystyle \frac{z-4z^2+6z^3-2z^4}{(1-3z+2z^2)^2}$ (<ref>)
§.§ The pattern $14/2/3$
Let us begin by recalling the standard recursive construction for building set partitions. That is, every set partition in $\Pi_n$ is obtained by taking a set partition in $\Pi_{n-1}$ and either adding $n$ to an existing block or appending $n$ as a singleton. To enumerate $\Pi_n(14/2/3)$ we consider a refinement of this recursive construction. First, observe that if $\pi \in \Pi_n(14/2/3)$, where in standard form $\pi = B_1/\cdots /B_m$, we have either $n\in B_{m-1}$ or $n\in B_m$, as any other choice would force an occurrence of the pattern $14/2/3$. From this observation we may immediately restrict our attention to set partitions which are built by recursively placing the next largest element into either a singleton block, the last block, or the second to last block. To formalize this refinement let us define $W_n$ to be the subset of all words in the letters a, b, c, that start with an a and have the property that any occurrence of the letter c must be preceded by at least two a's. Using this we obtain an injection
$$\varphi:W_n\to \Pi_n$$
which is defined recursively as follows. First, set $\varphi(\textbf{a}) = 1$. Then, for any $w\in W_n$, define $\varphi(w)$ to be the set partition obtained from $\pi'=\varphi(w_1\cdots w_{n-1})$ by appending $n$ as a singleton if $w_n = \textbf{a}$, inserting $n$ into the (existing) rightmost block of $\pi'$ if $w_n = \textbf{b}$, or inserting $n$ into the second to last block of $\pi'$ if $w_n = \textbf{c}$. Note that our insistence that words in $W_n$ have the property that any c is preceded by at least two a's guarantees that if $w_n=\textbf{c}$, then $\pi'$ contains at least two blocks.
It is clear for our definition that $\varphi$ is injective. Additionally, it follows from our first observation in this subsection that
$$\Pi_n(14/2/3)\subseteq \varphi(W_n).$$
Before continuing we next provide an example of this construction in Example <ref>.
If $w = \textbf{aaccba}$, then
$$\varphi(w) = 134/25/6,$$
and if $v = \textbf{aacabc}$, then
$$\varphi(v) = 13/26/45$$
which is not $14/2/3$-avoiding.
An immediate consequence of Example <ref>, is that $\Pi_n(14/2/3)\subsetneq \varphi(W_n)$. Consequently, we seek a subset of $W_n$ whose image under $\varphi$ is precisely $\Pi_n(14/2/3)$. To this end consider the subset $W_n^*$ consisting of all $w\in W_n$ with the property that the letters between any two c's do not contain exactly one a. Observe that in Example <ref> $w\in W_n^*$ but $v$ is not.
The restricted mapping $\varphi:W_n^*\to \Pi_n(14/2/3)$ is a bijection.
We first show that $\Pi_n(14/2/3)\subseteq\varphi(W_n^*)$. To this end consider $w\in W_n\setminus W_n^*$ and let $i<j<k$ be such that $w_i=w_k=\textbf{c}$ and $w_j=\textbf{a}$ is the only occurrence of the letter $\textbf{a}$ between these two c's. Next, set
$$\varphi(w) = B_1/\cdots /B_m$$
so that $i\in B_t$, for some $t$. As $w_i=c$ that means that $\ell:=\min(B_{t+1})< i$. As there is no a between $w_i$ and $w_j$ we see that $j\in B_{t+2}$. Moreover, as there is no a between $w_j$ and $w_k$ we see that $k\in B_{t+1}$. This provides our desired contradiction since the integers $\ell<i<j<k$ in the blocks $B_t, B_{t+1}$, and $B_{t+2}$ create an occurrence of the forbidden pattern $14/2/3$.
To establish the other inclusion, consider $w\in W_n^*$ and assume for a contradiction that $\varphi(w)\notin\Pi_n(14/2/3)$. Again set
$$\varphi(w) = B_1/\cdots /B_m.$$
By definition of the map $\varphi$, we see that $\min(B_i) > \max(B_j)$ for all $j+1<i$. From this it follows that any occurrence of $14/2/3$ in $\varphi(w)$ must occur among three consecutive blocks $B_t, B_{t+1}, B_{t+2}$ and involve integers $\ell<i<j<k$ such that $\ell, k\in B_{t+1}$, $i\in B_t$, and $j\in B_{t+2}$. This immediately implies that $w_i = w_k = \textbf{c}$ and that $w_j= \textbf{a}$ is the only a between these two c's. This contradiction permits us to conclude that $\varphi(W_n^*) = \Pi_n(14/2/3)$ as claimed.
We have
$$F_{14/2/3}(z) = \sum_{n\geq 1} |\Pi_n(14/2/3)|z^n = \frac{z-3z^2+3z^3}{1-5z+8z^2-5z^3}.$$
By our previous lemma it suffices to find $\sum_{n\geq 1} |W_n^*|z^n$. To this end we consider three cases depending on how many c's our word contains.
The words in $W_n^*$ with no c's are easily counted by the expression $\frac{z}{1-2z}$. On the other hand, the words with exactly one c's are counted by
where the second term follows since any occurrence of a c must be preceeded by at least two a's. Lastly, we consider words containing at least two c's. It is clear from the definitions that such words decompose as
$$\textbf{a}\ \underbrace{\boxed{\textbf{b}'s\mbox{ and }\textbf{a}'s}}_{\mbox{at least one } \textbf{a}}\ \textbf{c}\cdots\textbf{c}\cdots\textbf{c}\cdots\textbf{c}\ \boxed{\textbf{b}'s\mbox{ and }\textbf{a}'s},$$
so that between any two consecutive c's we do not have exactly one a. Counting the words between any two consecutive c's we have
$$G(z) = \frac{1}{1-2z} - \sum_{i\geq 1} iz^i = \frac{1}{1-2z} - \frac{z}{(1-z)^2}$$
since such words are in the letters a and b but cannot have exactly one a. In terms of $G(z)$ we see that
counts the case where our words have at least two c's. Summing these three cases yields the desired result.
§.§ The pattern $1/24/3$
The enumeration of the pattern $1/24/3$ closely resembles that of the enumeration of $14/2/3$ found in the previous section. As a result, we begin similarly by observing that if $\pi=B_1/\cdots/B_m\in \Pi_n(1/24/3)$ then we can only have $n\in B_1$ or $n\in B_m$, as any other choice creates our forbidden pattern. Recalling the set $W_n$ defined in the previous subsection, we define (recursively) the function
$$\phi:W_n\to \Pi_n.$$
First set $\phi(\textbf{a}) = 1$. Next, for any $w\in W_n$, define $\phi(w)$ to be the set partition obtained from $\phi(w_1\cdots w_{n-1})=B_1/\cdots /B_m$ by inserting $n$ into a singleton block if $w_n = \textbf{a}$, inserting $n$ into the block $B_m$ if $w_n = \textbf{b}$, and lastly, inserting $n$ into the block $B_1$ provided $w_n=\textbf{c}$. It now follows, since the words in $W_n$ have the property that any c must be preceded by at least two a's, that $\phi$ is injective. It also follows, from our initial observation, that $\Pi_n(1/24/3)\subseteq \phi(W_n)$.
If $w = \textbf{abbacb}$, then
$$\phi(w) = 1235/46,$$
and if $v = \textbf{aacac}$, then
$$\phi(v) = 135/2/4.$$
Note that $\phi(v)\notin\Pi_5(1/24/3)$.
We see from this example that $\Pi_n(1/24/3)\subsetneq \phi(W_n)$. As in the previous section, we seek a subset of $W_n$ whose image under $\phi$ is precisely $\Pi_n(1/24/3)$. To this end, define $W_n^{**}$ to be the set of all $w\in W_n$ such that
1) no a falls between any two c's in $w$, and
2) any c in $w$ which is preceded by at least three a's cannot be immediately followed by a b.
In the next lemma we prove that $W_n^{**}$ is this desired set. To facilitate the reading of its proof we pause to highlight a couple key observations. Setting $w\in W_n$ with $\phi(w) = B_1/\cdots/B_m$, we first see that $\max(B_s)<\min(B_{s+1})$ for all $s>1$. Our second observation is that $w_i = \textbf{a}$ is the $t$th a in our word (from left to right) if and only if $\min(B_t) = i$. With these observations in mind we now state and prove our lemma.
The restricted map $\phi:W_n^{**} \to \Pi_n(1/24/3)$ is a bijection.
We first show that $\Pi_n(1/24/3)\subseteq \phi(W_n^{**})$. To do so, it suffices to show that if $w\in W_n\setminus W_n^{**}$, then $\phi(w)\notin \Pi_n(1/24/3)$. Begin by setting $\phi(w) = B_1/\cdots/ B_m$. We address the two ways in which $w$ can fail to be a member of $W_n^{**}$ separately. First, assume condition 1) fails, and let $i<j<k$ be such that $w_i=w_k=\textbf{c}$ and $w_j = \textbf{a}$. In particular, $i,k\in B_1$. Moreover, as any c must be preceded by at least two a's, then it follows that $\min(B_2)<i<k$. It also follows that if $w_j$ is the $t$th a in $w$, then $t\geq 3$. Consequently, the blocks $B_1, B_2$, and $B_t$ contain an occurrence of $1/24/3$.
Next, let us assume condition 2) fails. Here we assume that there exists some index $i$ so that $w_iw_{i+1}=\textbf{cb}$ and $w_i$ is preceded by at least three a's. It immediately follows that $\phi(w_1\cdots w_{i-1})$ has $t\geq 3$ blocks and that in $\phi(w)$, $i\in B_1$, $i+1\in B_t$ and $\min(B_2)<\min(B_t)<i<i+1$. Consequently, the blocks $B_1, B_2$, and $B_t$, contain an occurrence of our forbidden pattern. We conclude that $\Pi_n(1/24/3)\subseteq \phi(W_n^{**})$.
Next, we demonstrate the other inclusion. Fix $w\in W_n^{**}$ and, for a contradiction, assume that $\phi(w) \notin \Pi_n(1/24/3)$. Set $\phi(w) = B_1/\cdots/B_m$. As $\phi(w)$ contains an occurrence of $1/24/3$ we must have $m\geq 3$. As $\max(B_s)<\min(B_{s+1})$ for all $s>1$ we see that any occurrence of $1/24/3$ in $\phi(w)$ must involve $B_1$ and two other blocks $B_r$ and $B_s$ with $r<s$. If the integers that form this pattern are $i<j<k<\ell$ then we have exactly two cases.
Case 1: $j,\ell\in B_1$, $i\in B_r$, and $k\in B_s$
In this case, we see that as $i\in B_r$, then $w_j=w_\ell=\textbf{c}$. As the letters between $w_j$ and $w_\ell$ cannot contain an a we conclude that $w_k = \textbf{b}$. It now follows that $w_j\cdots w_\ell$ must contains a cb.
Additionally, the facts that the letters between $w_j$ and $w_\ell$ do not contain an $\textbf{a}$, and that $j<k<\ell$, and that $w_{\min(B_s)} = \textbf{a}$, imply that $\min(B_s)<j$. Furthermore, as $\min(B_r)\leq i<j$ we see that
$$w_1 = w_{\min(B_r)}=w_{\min(B_s)}=\textbf{a},$$
and $1< \min(B_r)<\min(B_s)<j$. This contradicts the fact that $w$ satisfies condition 2) in the definition of $W_n^{**}$.
Case 2: $i\in B_r, j,\ell\in B_s$ and $k\in B_1$
From the definition of $\phi$, we clearly have
$$w_1 = w_{\min(B_r)}=w_{\min(B_s)}=\textbf{a}.$$
Moreover, as $i\in B_r$ and $i<k$, then $w_k=\textbf{c}$. Further as $\ell\neq \min(B_s)$, then $w_\ell = \textbf{b}$. Lastly, it is clear that the subword $w_{k+1}\cdots w_{\ell-1}$ cannot contain the letter a. (If it did, then $\ell$ could not be in $B_s$.) This means $w$ contains a cb preceded by at least three a. Again this contradicts the fact that $w$ satisfies condition 2) in the definition of $W_n^{**}$. This completes our proof.
We have
$$F_{1/24/3}(z) = \sum_{n\geq 1} |\Pi_n(1/24/3)|z^n = \frac{z-4z^2+6z^3-2z^4}{(1-3z+2z^2)^2}.$$
By our previous lemma it suffices to enumerate $W_n^{**}$. To do so we consider three cases depending on the number of c's. Clearly, the words in $W_n^{**}$ with no c's are counted by the expression $\frac{z}{1-2z}$. Next consider words that contain at least one c and have the additional property that the first c appears after the second a but before the third a (if it exists). Such words must be of the form
$$\textbf{a}\textbf{b}\cdots\textbf{b}\textbf{a}\underbrace{\boxed{\textbf{b}'s\mbox{ and }\textbf{c}'s}}_{\mbox{at least 1 \textbf{c}}}\qquad \mbox{or}\qquad \textbf{a}\textbf{b}\cdots\textbf{b}\textbf{a}\underbrace{\boxed{\textbf{b}'s\mbox{ and }\textbf{c}'s}}_{\mbox{at least 1 \textbf{c}}}\ \textbf{a}\ \boxed{\textbf{a}'s\mbox{ and }\textbf{b}'s}.$$
These two forms are counted by
Lastly, we consider the case that our first c is preceded by at least three a's. In this case such words must be of the form
$$\textbf{ab}\cdots\textbf{bab}\cdots\textbf{ba}\ \boxed{\textbf{a}'s\mbox{ and }\textbf{b}'s}\ \textbf{c}\cdots\textbf{c}\qquad\mbox{ or } \qquad \textbf{ab}\cdots\textbf{bab}\cdots\textbf{ba}\ \boxed{\textbf{a}'s\mbox{ and }\textbf{b}'s}\ \textbf{c}\cdots\textbf{c\ a}\ \boxed{\textbf{a}'s\mbox{ and }\textbf{b}'s}.$$
Together such words are counted by the expression
Adding these three terms together and simplifying yields the desired expression.
With the recursive structure of $1/24/3$-avoiding set partitions established, we now conclude the proof the case $|a-b|=2$ in the proof of Theorem <ref>, by establishing the following lemma.
For $n\geq 5$, we have $|\Pi_n(1/24/3)| < |\Pi_n(1/2/3/4)|$.
Let $A_n$ be the set of all partitions of $[n]$ with exactly 2 blocks and let $A_n^*$ be the set of all partitions with exactly 3 blocks in which $n$ is a singleton. (Note $A_n^*$ is a subset of both $\Pi_n(1/24/3)$ and $\Pi_n(1/2/3/4)$.) Now let $C_n$ be the set of all partitions in $\Pi_n(1/24/3)$ with the property that removing $n$ gives a partition with at least 3 blocks. Similarly, let $D_n$ be the set of all partitions in $\Pi_n(1/2/3/4)$ with the property that removing $n$ results in a set partition with exactly 3 blocks.
By our above note and the fact that the patterns involved have at least 3 blocks, it follows that
$$\Pi_n(1/24/3) = \{\beta_n\} \cupdot A_n \cupdot A_n^* \cupdot C_n \qquad\textrm{and}\qquad \Pi_n(1/2/3/4) = \{\beta_n\} \cupdot A_n\cupdot A_n^* \cupdot D_n.$$
It now suffices to show that $|C_n|<|D_n|$ for $n\geq 5$. We proceed by induction on $n$. As $|\Pi_5(1/24/3)| = 39$ and $|\Pi_5(1/2/3/4)| = 41$, we must have $|C_5| < |D_5|$. From the description of $1/24/3$-avoiding permutations given in the first paragraph of this section, it follows that $|C_{n+1}|\leq 3|C_n|$. Additionally, it is clear that $|D_{n+1}| = 3|D_n|$. Together we get
$$|C_{n+1}| \leq 3|C_n| < 3|D_n| = |D_{n+1}|$$
where the second inequality follows by our inductive hypothesis. (Note the first inequality may not be an equality as in Example <ref>.)
§.§ The pattern $12/34$
We have
$$|\Pi_n(12/34)| = \sum_{k=0}^{\lfloor n/2\rfloor} k!{n \choose 2k}+\sum_{\ell=3}^{n-2k}{n\choose 2k+\ell}k!(k+1)^2.$$
Before proving this result it will be helpful to state and prove a couple of lemmas.
Any set partition which avoids $12/34$ has at most one block of size greater than $2$.
For a contradiction assume $\pi\in \Pi_n(12/34)$ contains two blocks $B=\{x_1,\ldots, x_a\}$ and $C=\{y_1,\ldots, y_b\}$ where $a,b\geq 3$. Without loss of generality we may further assume that $x_2<y_2$. On the other hand, this implies that the blocks $B$ and $C$ must contain an occurrence of $12/34$ which is impossible.
The proof of the next lemma is straightforward. The details are left to the reader.
Let $B=\{z_1<\cdots< z_a\}$, $C=\{x_1<y_1\}$, and $D=\{x_2<y_2\}$ be blocks in $\pi \in \Pi_n(12/34)$ so that $3\leq a$. Then
a) $\max(x_1,x_2)<\min(y_1,y_2)$.
b) $x_1<z_2$ and $z_{a-1}<y_1$.
It immediately follows from the previous lemma that if $x_1<\cdots<x_k$ are the minimum entries among all the blocks of size 2 and $y_1,\ldots, y_k$ are the maximum entries among all the blocks of size 2 then we must have
$$x_i < \min(y_1,\ldots, y_k),$$
for $1\leq i\leq k$.
To start, let us first count those set partitions in $\Pi_n(12/34)$ whose block sizes do not exceed 2. To count such set partitions with exactly $k$ blocks of size 2, we first choose a subset $x_1<\cdots<x_k<y_1<\cdots y_k$ of size $2k$ from $[n]$. (The $n-2k$ integers not chosen become singletons.) We then may choose to match up each of the $x_i$'s with exactly one of the $y_i$'s in any of the $k!$ ways. It follows from the above remark that all set partitions in $\Pi_n(12/34)$ whose block sizes do not exceed 2 are of this form. A simple argument further shows that any set partition built in this manner must also avoid $12/34$. Consequently, the number of such set partitions is given by
$$\sum_{k=0}^{\lfloor n/2\rfloor}k!{n \choose 2k}.$$
Now let us consider the set partitions in $\Pi_n(12/34)$ that contain $k$ blocks of size 2 and exactly one block of size $\ell\geq 3$. To build such a partition, we first choose a subset of size $2k+\ell$ from $[n]$. Let us denote the members of this subset by
$$x_1<\cdots< x_{k+1}< z_1 <\cdots < z_{\ell-2} < y_1<\cdots <y_{k+1}.$$
(Again, the $n-2k-\ell$ integers not chosen become singletons in our final set partition.)
Next, choose exactly one element from the $x_i$'s and one element from the $y_i$'s along with all the $z_i$'s to form our block of size $\ell$. Next, as in the preceding case we are free to match each of the remaining $x_i$'s to each of the remaining $y_i$'s in all $k!$ possible ways to form our $k$ blocks of size 2. Lemma <ref> guarantees that any such set partition is built in this manner. Furthermore, it is straightforward to show that any set partition built in this manner avoids $12/34$. Consequently, the number of such set partitions is given by
$$\sum_{k=0}^{\lfloor n/2\rfloor}\sum_{\ell=3}^{n-2k}{n\choose 2k+\ell}k!(k+1)^2.$$
Adding these two terms gives our final result.
§.§ The pattern $14/23$
To begin, let us concentrate on the subset $\Pi_n^*(14/23)$ consisting of all set partitions in $\Pi_n(14/23)$ that do not contain singletons. We first show that the set partitions in this subset can be constructed via a simple recursive procedure. To do this we first need the following definition.
For any $\pi\in\Pi_n^*(14/23)$ we say $k$ is a cap provided each $i\geq k$ is the maximum element in its block. Let $\iota(\pi)$ be the number of caps in $\pi$.
Next, we define the sets $\Pi_n^*$ as follows. First, set $\Pi_2^* = \{\{1,2\}\}$. For $n\geq 3$, define $\Pi_n^*$ to be the set of all partitions obtained by either of the following two operations.
For any $\pi\in \Pi_{n-1}^*$ insert $n$ into the block containing $n-1$.
For $\sigma \in \Pi_{n-2}^*$ where $n\geq 4$ doing the following. Fix $k$ to be either one of the $\iota(\pi)$ caps in $\sigma$ or set $k=n-1$. Then, increment all the values in $\sigma$ which are $\geq k$. Finally, append the doubleton block $\{k,n\}$.
We note that no partition in $\Pi_n^*$ contains a singleton.
We have $\Pi_n^* = \Pi^*_n(14/23)$.
We leave it to the reader to convince themselves that $\Pi_n^* \subseteq \Pi_n^*(14/23)$. We show the other inclusion by induction on $n$. First note that $\Pi_2^* = \{\{1,2\}\} = \Pi_2^*(14/23)$. Now take any $\pi\in \Pi_{n+1}^*(14/23)$ and let $B$ be the block whose cap is $n+1$. If $|B|\geq 3$, then we claim that $n\in B$ as well. It then follows (inductively) that $\pi\in \Pi_{n+1}^*$. To prove this claim, let $B = \{x_1<x_2<\cdots < x_k\}$ so that $k\geq 3$ and $x_k = n+1$. For a contradiction assume $x_{k-1}<n$. So there exists a block $C$, distinct from $B$, whose cap is $n$. Set $C = \{y_1<\cdots<y_\ell\}$ so that $y_\ell = n$ and $\ell\geq 2$, as $\pi$ does not contain singletons. We must have
$$y_{\ell-1} < x_{k-2}<x_{k-1} < n = y_\ell\quad\mbox{or}\qquad x_{k-2}<y_{\ell-1}<y_\ell=n<n+1 = x_k.$$
But either choice results in an occurrence of the forbidden pattern $14/23$. Hence $n\in B$ as claimed.
The other possibility is for $B=\{k,n+1\}$. As $\pi$ avoids $14/23$ it follows that every $i$ strictly between $k$ and $n+1$ must be a cap. Consequently, $\pi$ was constructed from some set partition in $\Pi_n^*(14/23)= \Pi_n^*$ via $n+1$-augmentation.
With this lemma established, we are now ready to enumerate this pattern.
We have
$$F_{14/23}(z) =\sum_{n\geq 0} |\Pi_n(14/23)|\ z^n = G\left(\frac{z}{1-z}\right)\frac{1}{1-z} + \frac{1}{1-z},$$
$$G(z) = \frac{z-2z^2(1+z)-z\sqrt{1-4z^2}}{-2+2z(1+z)^2}.$$
In light of the previous lemma we know that
$$G(z) = \sum_{n\geq 2} |\Pi^*_n(14/23)|\ z^n = \sum_{n\geq 2} |\Pi^*_n|\ z^n.$$
It is straightforward to see that an arbitrary set partition in $\Pi_n(14/23)$ is obtained by inserting $k$ singleton blocks into some set partition in $\Pi_{n-k}(14/23)$. In terms of generating functions this corresponds to
$$F_{14/23}(z) = G\left(\frac{z}{1-z}\right)\frac{1}{1-z} + \frac{1}{1-z}.$$
It only remains to prove that $G$ is given by the desired generating function. To do this, first define
$$H(z,t) = \sum_{n\geq 2}\sum_{\pi \in \Pi_n^*}z^n t^{\iota(\pi)}.$$
The recursive description of $\Pi_n^*$ translates into the functional equation
$$H(z,t) = z^2t +ztH(z,1) + \frac{z^2t}{1-t}\left(H(z,1) -tH(z,t)\right),$$
where the second term corresponds to $n$-insertion and the third term corresponds to $n$-augmentation. Solving for $G(z) = H(z,1)$ using the kernel method results in the desired expression.
§.§ The pattern $1/234$
We have
$$|\Pi_n(1/234)| = \sum_{\ell=1}^{n} M(n-\ell) + \sum_{\ell=1}^{n-2} (n-\ell-1)M(n-\ell-1)+\sum_{\ell=1}^{n-3} {n-\ell-1 \choose 2}M(n-\ell-2),$$
$$M(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} {n\choose 2k} (2k)!!\ .$$
Let $\pi = B_1/\cdots /B_m \in \Pi_n(1/234)$. First observe that $|B_i|\leq 2$ for all $i\geq 2$ since $1\in B_1$ and $\pi$ avoids $1/234$. This means that the set partition $B_2/\cdots /B_m$ is a matching with fixed points of the set $[n]\setminus B_1$ which has size $n_0 = n-|B_1|$. It is a well known result that such objects are counted by
$$M(n_0) = \sum_{k=0}^{\lfloor n_0/2 \rfloor} {n_0\choose 2k} (2k)!!\ .$$
Next, observe that $B_1$ is not free to be any subset of $[n]$. In fact $B_1$ must be of either the form
1) $\{1,\ldots, \ell\}$, or
2) $\{1,\ldots, \ell, k\}$ for some $k>\ell+1$, or
3) $\{1,\ldots, \ell, k,m\}$ for some $m>k>\ell+1$,
as any other possibility would result in an occurrence of $1/234$.
Combining these two observations we see that the first term in our formula counts the set partitions in $\Pi_n(1/234)$ whose first block is of the form in 1). Likewise, the second and third terms count those set partitions whose first block is of the form in 2) and 3) respectively.
§.§ The pattern $134/2$
To enumerate this pattern we require the well studied notion of weak integer compositions. In particular, we denote by $C_{n,k}$ the set of all weak integer compositions of $n$ with $k$ parts. Additionally we denote by $M_{k,f}$ the set of all set partitions of $[2k+f]$ with $f$ singletons and $k$ doubletons. (Observe these are just matchings with $f$ fixed points.) Lastly, we define the refined set $\Pi_{n,k,f}(134/2)$ to be the set of all set partitions in $\Pi_{n}(134/2)$ with exactly $k+f$ blocks where exactly $f$ of them are singletons.
Provided, $k\geq 1$ and $2k+f\leq n$, there exists an explicit bijection
$$\phi:\Pi_{n,k,f}(134/2)\to C_{n-f-2k,k} \times M_{k,f} .$$
Deferring the proof of this lemma to the end of this section, we continue with our enumeration.
As it is well known that $|C_{n-f-2k,k}| = {n-f-k-1\choose k-1}$
and $|M_{k,f}| = {2k+f\choose f}(2k)!!$
it now follows from Lemma <ref>, that
\begin{align*}
|\Pi_n(134/2)| &= 1+ \sum_{k=1}^{\lfloor n/2 \rfloor}\sum_{f=0}^{n-2k} |\Pi_{n,k,f}(134/2)|\\
&= 1+ \sum_{k=1}^{\lfloor n/2 \rfloor}\sum_{f=0}^{n-2k} {n-f-k-1\choose k-1}{2k+f\choose f}(2k)!!. \\
\end{align*}
We record this result in our last theorem.
We have the following formula
$$|\Pi_n(134/2)| = 1+ \sum_{k=1}^{\lfloor n/2 \rfloor}\sum_{f=0}^{n-2k} {n-f-k-1\choose k-1}{2k+f\choose f}(2k)!!.$$
We now turn our attention to the proof of Lemma <ref>. We begin with a simple characterization of $134/2$-avoiding set partitions. Its straightforward proof is omitted.
For any set partition $\pi = B_1/\cdots /B_m$, we have that $\pi$ is $134/2$-avoiding if and only if any non-singleton block is of the form
$$\{a, a+1,\ldots, a+\ell, b\},$$
where $\ell\geq 0$ and $b\geq a+\ell+1$.
Consider a set partition $\pi=B_1/\ldots/B_m\in \Pi_n(134/2)$ so that $f$ of the blocks are singletons and the remaining $k= m-f$ blocks $B_{i_1},\ldots, B_{i_k}$ are not singletons. Now let
$$\lambda= \left(|B_{i_1}|-2,|B_{i_2}|-2,\ldots, |B_{i_k}|-2\right)$$
be the resulting weak composition of $n-f-2k$ with $k$ parts. (As each of the blocks $B_{i_j}$ are of the form in Lemma <ref>, the parts of our composition are just their corresponding values for $\ell$ in this decomposition.) Furthermore, by throwing out all but the min and max of each block, and applying standardization map, we obtain a set partition $\sigma$ of $[f+2k]$ with exactly $f$ singletons and $k$ doubletons. Finally, we define $\phi(\pi)=(\lambda,\sigma)$. (We illustrate this construction in Example <ref>.) As this map is easily seen to be bijective, the proof is complete.
Consider the set partition $127/3/4568/9/10\ 11\ 12$ where $f=2$ and $k = 3$. Then
$$\phi(127/3/4568/9/10\ 11\ 12) = (\lambda, \sigma)$$
where $\lambda = (1,2,1) \in C_{4,3}$ and $\sigma = 14/2/35/6/78\in M_{3,2}$.
The authors are grateful to Bruce Sagan for recommending this area of research.
|
1511.00148
|
I. ŽliobaitėDiscrimination measures
Nowadays, many decisions are made using predictive models built on historical data.
Predictive models may systematically discriminate groups of people even if the computing process is fair and well-intentioned.
Discrimination-aware data mining studies how to make predictive models free from discrimination, when historical data, on which they are built, may be biased, incomplete, or even contain past discriminatory decisions.
Discrimination refers to disadvantageous treatment of a person based on belonging to a category rather than on individual merit.
In this survey we review and organize various discrimination measures that have been used for measuring discrimination in data, as well as in evaluating performance of discrimination-aware predictive models.
We also discuss related measures from other disciplines, which have not been used for measuring discrimination, but potentially could be suitable for this purpose. We computationally analyze properties of selected measures.
We also review and discuss measuring procedures, and present recommendations for practitioners.
The primary target audience is data mining, machine learning, pattern recognition, statistical modeling researchers developing new methods for non-discriminatory predictive modeling.
In addition, practitioners and policy makers would use the survey for diagnosing potential discrimination by predictive models.
fairness in machine learning, predictive modeling, non-discrimination, discrimination-aware data mining
§ INTRODUCTION
Nowadays, many decisions are made using predictive models built on historical data, for instance, personalized pricing and recommendations, credit scoring, automated CV screening of job applicants, profiling of potential suspects by the police, and many more. Penetration of machine learning technologies, and decisions informed by big data has raised public awareness that automated decision making may lead to discrimination <cit.>.
Predictive models may discriminate people, even if the computing process is fair and well-intentioned <cit.>.
This is because most machine learning methods are based upon assumptions that the historical data is correct, and represents the population well, which is often far from reality.
Discrimination-aware machine learning and data mining is an emerging discipline, which studies how to prevent discrimination in predictive modeling. It is assumed that non-discrimination regulations, such as which characteristics, or which groups of people are considered as protected, are externally defined by national and international legislation.
The goal is to mathematically formulate non-discrimination constraints, and develop machine learning algorithms that would be able to take into account those constraints, and still be as accurate as possible.
In the last few years researchers have developed a number of discrimination-aware machine learning algorithms, using a variety of performance measures.
Nevertheless, there is a lack of consensus how to define fairness of predictive models, and how to measure the performance in terms of discrimination.
Quite often research papers propose a new way to quantify discrimination, and a new algorithm that would optimize that measure. The variety of approaches to evaluation makes it difficult to compare the results and assess the progress in the discipline, and even more importantly, it makes it difficult to recommend computational strategies for practitioners and policy makers.
The goal of this survey is to present a unifying view towards discrimination measures in machine learning, and understand the implications of choosing to optimize one or another measure, because measuring is central in formulating optimization criteria for algorithmic discrimination discovery and prevention.
Hence, it is important to have a structured survey at an early stage of development of this research field, in order to present task settings in a systematic way for follow up research, and to enable systematic comparison of approaches.
Thus, we review and categorize measures that have been used in machine learning and data mining, and also discuss existing measures from other fields, such as feature selection, which in principle could be used for measuring discrimination.
There are several related surveys that can be viewed as complementary to this survey.
A recent review <cit.> presents a multi-disciplinary context for discrimination-aware data mining. This survey contains a brief overview of discrimination measures with does not go into analysis and comparison of the measures, since the focus is on approaches to solutions across different disciplines (law, economics, statistics, computer science).
Another recent review <cit.> discusses legal aspects of potential discrimination by machine learning, mainly focusing on American anti-discrimination law.
A matured handbook on measuring racial discrimination <cit.> focuses on surveying and collecting evidence for discrimination discovery.
The book is not considering discrimination by algorithms, only by human decision makers.
The remainder of the article is organized as follows.
Section <ref> presents legal context, terminology, and
provides an overview of research in developing non-discriminatory predictive modeling approaches. Our intention is to keep this section brief. An interested reader is referred to focused surveys <cit.> for more information.
Section <ref> reviews and organizes discrimination measures used in discrimination-aware machine learning and data mining, as well as potentially useful measures from other fields.
Section <ref> analyzes and compares a set of most popular measures, and discusses implications of using one or the other.
Finally, Section <ref> presents recommendations for researchers, and concludes the survey.
§ BACKGROUND
§.§ Discrimination and law
Discrimination translates from latin as a distinguishing.
While distinguishing is not wrong as such, discrimination has a negative connotation referring to adversary treatment of people based on belonging to some group rather than individual merits.
Public attention to discrimination prevention has been increasing in the last few years.
National and international anti-discrimination legislation are extending the scope of protection against discrimination, and expanding discrimination grounds.
Adversary discrimination is undesired from the perspective of basic human rights, and in many areas of life non-discrimination is enforced by international and national legislation,
to allow all individuals an equal prospect to access opportunities available in a society <cit.>. Enforcing non-discrimination is not only for benefiting individuals.
Considering individual merits rather than group characteristics is expected to benefit decision makers as well leading to more more informed, and thus likely more accurate decisions.
Discrimination can be characterized by three main concepts: (1) what actions (2) in which situations (3) towards whom are considered discriminatory.
Actions are forms of discrimination, situations are areas of discrimination, and grounds of discrimination describe characteristics of towards whom discrimination may occur.
For example, the main grounds for discrimination defined in European Council directives <cit.> (2000/43/EC, 2000/78/EC) are: race and ethnic origin, disability, age, religion or belief, sexual orientation, gender, nationality. Multiple discrimination occurs when a person is discriminated on a combination of several grounds.
The main areas of discrimination are: access to employment, access to education, employment and working conditions, social protection, access to supply of goods and services.
Discriminatory actions may take different forms, the two main of which are known as direct discrimination and indirect discrimination.
A direct discrimination occurs when a person is treated less favorably than another is, has been or would be treated in a comparable situation on protected grounds. For example, property owners are not renting to a minority racial tenant.
An indirect discrimination (also known as structural discrimination) occurs where an apparently neutral provision, criterion or practice would put persons of a protected ground at a particular disadvantage compared with other persons. For example, a requirement to produce an ID in a form of driver's license for entering a club may discriminate visually impaired people, who cannot have a driver's license.
A related term statistical discrimination <cit.> is often used in economic modelling. It refers to inequality between demographic groups occurring even when economic agents are rational and non-prejudiced.
Indirect discrimination applies to machine learning and data mining, since algorithms produce decision rules or decision models.
While human decision makers may make biased decisions on case by case basis, rules produced by algorithms are applied consistently, and may discriminate more systematically and at a larger scale.
Discrimination due to algorithms is sometimes referred to as digital discrimination (e.g. <cit.>) .
General population, and even many data scientists may think that algorithms are based on data, and, therefore, models produced by algorithms are always objective.
However, models are as objective as the data on which they are applied, and as long as the assumptions behind the models perfectly match the reality. In practice, this is rarely the case.
Historical data may be biased, incomplete, or record past discriminatory decisions that can easily be transferred to predictive models, and reinforced in new decision making <cit.>.
Lately, awareness of policy makers and public attention to potential discrimination has been increasing <cit.>, but there is a long way ahead before we can fully understand how such discrimination happens and how to prevent it.
§.§ Discrimination-aware machine learning and data mining
Non-discriminatory machine learning and data mining, a discipline at an intersection of computer science, law and social sciences, focuses on two main research directions: discrimination discovery, and discrimination prevention.
Discrimination discovery aims at finding discriminatory patterns in data using data mining methods.
Data mining approach for discrimination discovery typically mines association and classification rules from the data, and then assesses those rules in terms of potential discrimination <cit.>.
A more traditional statistical approach to discrimination discovery typically fits a regression model to the data including the protected features (such as race, gender), and then analyzes the magnitude and statistical significance of the regression coefficients at the protected attributes (e.g. <cit.>). If those coefficients appear to be significant, then discrimination is flagged.
Discrimination prevention develops machine learning algorithms that would produce predictive models, ensuring that those models are free from discrimination,
while, standard predictive models, induced by machine learning and data mining algorithms, may discriminate groups of people due to training data being biased, incomplete, or recording past discriminatory decisions. The goal is to have a model (decision rules) that would obey non-discrimination constraints, typically the constraints directly relate to the selected discrimination measure.
Solutions for discrimination prevention in predictive models fall into three categories: data preprocessing, model postprocessing, and model regularization.
Data preprocessing modifies the historical data such that the data no longer contains discrimination, and then uses regular machine learning algorithms for model induction.
Data preprocessing may modify the target variable <cit.>, or modify input data <cit.>.
Model postprocessing produces a regular model and then modifies it (e.g. by changing the labels of some leaves in a decision tree) <cit.>.
Model regularisation adds optimization constraints in the model learning phase (e.g. by modifying the splitting criteria in decision tree learning) <cit.>.
An interested reader is invited to consult an edited book <cit.>, a special issue in a journal <cit.>, and proceedings of three workshops in discrimination-aware data mining and machine learning <cit.> for more details.
Defining coherent discrimination measures is central for both lines of research: discrimination discovery and discrimination prevention.
Discrimination discovery needs a measure in order to judge whether there is discrimination in data. Discrimination prevention needs a measure as an optimization criteria in order to sanitize predictive models. Hence, our main focus in this survey is to review discrimination measures, and analyze their properties, and understand implications of using one or another measure.
§ MACHINE LEARNING SETTINGS, DEFINITIONS AND SCENARIOS
§.§ Definition of fairness for machine learning
In the context of machine learning non-discrimination can be defined as follows:
(1) people that are similar in terms non-protected characteristics should receive similar predictions, and
(2) differences in predictions across groups of people can only be as large as justified by non-protected characteristics.
The first condition relates to direct discrimination, and can be illustrated by so called twin test: if gender is the protected attribute and we have two identical twins that share all characteristics, but gender, they should receive identical predictions.
The first part is necessary but not sufficient condition to make sure that there is no discrimination in decision making.
The second condition ensures that there is no indirect discrimination, also referred to as redlining. For example, banks used to deny loans for residents of selected neighborhoods.
Even though race was not formally used as a decision criterion, it appeared that the excluded neighborhoods had much higher population of non-white people than average.
Even though people from the same neighborhood ("twins") are treated the same way no matter what the race is, artificial lowering of positive decision rates in the non-white-dominated neighborhoods would harm the non-white population more than white. Therefore, different decision rates across neighborhoods can only be as large as justified by non-protected characteristics, and this is what the second part of the definition controls.
More formally, let $X$ be a set of variables describing non-protected characteristics of a person, $S$ be a set of variables describing the protected characteristics, and $\hat{y}$ be the model output. A predictive model can be considered fair if:
(1) the expected value for model output does not depend on the protected characteristics $E(\hat{y}|X,S) = E(\hat{y}|X)$ for all $X$ and $S$, that is, there is no direct discrimination; and
(2) if non-protected characteristics and protected characteristics are not independent, then the expected value for model output dependence on those non-protected characteristics should be justified, that is if $E(X|S) \neq E(X)$, then $E(\hat{y}|X) = e^\star(\hat{y}|X)$, where $e^\star$ is a constraint.
Finding and justifying $e^\star$ is non-trivial and very challenging, and that is where a lot of ongoing effort in discrimination-aware machine learning concentrate.
§.§ Machine learning task settings
Machine learning settings for decision support, where discrimination may potentially occur, can take many different forms.
The variable that is to be predicted – target variable – may be binary, ordinal, or numeric, corresponding to binary classification, multiclass classification or regression tasks.
As an example of a binary classification task in the banking domain could be deciding whether to accept or decline loan application of a person.
Multiclass classification task could be to determine to which customer benefit program a person should be assigned (e.g. "golden clients", "silver clients", "bronze clients").
Regression task could be to determine the interest rate for a particular loan for a particular person.
Discrimination can occur only when target variable is polar. That is, each task setting some outcomes should be considered superior to others. For example, getting a loan is better than not getting a loan, or the "golden client" package is better than the "silver", and "silver" is better than "bronze", or assigned interest rate $3\%$ is better than $5\%$. If the target variable is not polar, there is no discrimination, because no treatment is superior or inferior to other treatment.
The protected characteristic, in machine learning settings referred to as the protected variable or sensitive attribute, may as well be binary, categorical or numeric, and it does not need to be polar.
For example, gender can be encoded with a binary protected variable, ethnicity can be encoded with a categorical variable, and age can be encoded with a numerical variable.
In principle, any combination one or more personal characteristics may be required to be protected.
Discrimination on more than one ground is known as multiple discrimination, and it may be required to ensure prevention of multiple discrimination in predictive models.
Thus, ideally, machine learning methods and discrimination measures should be able to handle any type or a combination of protected variables.
For instance, the authorities may want to enforce non-discrimination with respect to ethnicity in determining interest rate,
or non discrimination with respect to gender and age in deciding whether to accept loan applications.
In discrimination prevention it is assumed that the protected ground is externally given, for example, by law.
§.§ Principles for making machine learning non-discriminatory
A typical machine learning process is illustrated in Figure <ref>.
A machine learning algorithm is a procedure used for producing a predictive model from historical data.
A model is the resulting decision rule (or a collection of rules).
The resulting model is used for decision making for new incoming data.
The model would take personal characteristics as inputs (for example, income, credit history, employment status), and output a prediction (for example, credit risk level).
A typical machine learning setting.
Algorithms themselves do not discriminate, because they are not used for decision making. Models (decision rules) that are used for decision making may potentially discriminate people with respect to certain characteristics.
Algorithms, on the other hand, may be discrimination-aware by employing specific procedures during model construction to enforce non-discriminatory constraints into the models.
Hence, one of the main goals of discrimination-aware machine learning and data mining is to develop discrimination-aware algorithms, that would guarantee that non-discriminatory models are produced.
There is an ongoing debate in the discrimination-aware data mining and machine learning community whether models should or should not use protected characteristics as inputs. For example, a credit risk assessment model may use gender as input, or may leave the gender variable out.
Our position on the matter is as follows. Using the protected characteristic as model input may help to ensure that there is no indirect discrimination (for example, as demonstrated in the experimental section of <cit.>). However, if a model uses the protected characteristic as input, the model is not treating two persons that share identical characteristics except for the protected characteristic the same way, a direct discrimination would be propagated.
Therefore, such a model would be discriminatory discriminatory due to violation of condition #1 in the definition in Section <ref>.
Hence, the model should not use the protected characteristic for decision making.
However, we see no problem in using the protected characteristic in the model learning process, which often may help to enforce non-discrimination constraints. Thus, machine learning algorithms can use the protected characteristic in the learning phase, as long as the resulting predictive model does not require the protected characteristic when used for decision making.
Ensuring that there is no indirect discrimination is much more tricky. In order to verify to what extent non-discriminatory constraints are obeyed,
and enforce fair allocation of predictions across groups of people, machine learning algorithms must have access to the protected characteristics in the historical data. We argue that if protected information (e.g. gender or race) is not available during the model learning building process, the learning algorithm cannot be discrimination-aware, because it cannot actively control non-discrimination.
The resulting models produces without access to sensitive information may be discriminatory, may be not, but that is by chance rather than discrimination-awareness property of the algorithm.
Non-discrimination can potentially be measured on data (historical data), on predictions made by models, or on models themselves.
Different task settings and application goals may require different measurement techniques.
In order to select appropriate measures, which also typically serve as optimisation constraints in the non-discriminatory model learning process, it is important to understand underlying assumptions and basic principles behind different discrimination measures.
The next section presents a categorized survey of measures used in the discrimination-aware data mining and machine learning literature, and discusses other existing measures that could in principle be used for measuring fairness of algorithms.
The goal is to present arguments for selecting relevant measures for different learning settings.
§ DISCRIMINATION MEASURES
Discrimination measures can be categorized into (1) statistical tests, (2) absolute measures, (3) conditional measures, and (4) structural measures.
We survey measures in this order due to historical reasons, which is more or less how they came into use. First statistics tests were used which would answer yes or no, then absolute measures came into play that allow quantifying the extent of discrimination, then conditional measures appeared that take into account possible legitimate explanations of differences between different groups of people. Statistical tests, absolute measures and conditional measures are designed for indicating indirect discrimination.
Structural measures have been introduced mainly in accord to mining classification rules, aiming at discovering direct discrimination, but in principle they can also address indirect discrimination.
All these types are not intended as alternatives, but rather reflect different aspects of the problem, as summarized in Table <ref>.
Discrimination measure types
Measures Indicate what? Type of discrimination
Statistical tests presence/absence of discrimination indirect
Absolute measures magnitude of discrimination indirect
Conditional measures magnitude of discrimination indirect
Structural measures spread of discrimination direct or indirect
Statistical tests indicate presence or absence of discrimination at a dataset level, they do not measure the magnitude of discrimination, neither the spread of discrimination within the dataset.
Absolute measures capture the magnitude of discrimination over a dataset taking into account the protected characteristic, and the prediction decision; no other characteristics of individuals are considered.
It is assumed that all individuals are alike, and there should be no differences in decisions for the protected and the general group of people, disregarding any possible explanation.
Absolute measures generally are not for using stand alone on a dataset, but rather provide core principles for conditional measures, or statistical tests.
Conditional measures capture the magnitude of discrimination, which cannot be explained by any non-protected characteristics of individuals. Statistical tests, absolute and conditional measures are designed to capture indirect discrimination at a dataset level.
Structural measures do not measure the magnitude of discrimination, but the spread of discrimination, that is, a share of people in the dataset that are affected by direct discrimination.
Our survey of measures will use mathematical notation as summarized in Table <ref>.
For simplicity we will use the following short probability notation:
$p(s=1)$ will be encoded as $p(s^1)$, and $p(y=+)$ will be encoded as $p(y^{+})$.
Let $s^1$ denote the protected community, and $y^+$ denote the desired decision (e.g. positive decision to grant a loan).
Upper indices will denote values, lower indices will denote counters of variables.
Symbol Explanation
$y$ target variable, $y_i$ denotes the $i^{th}$ observation
$y^i$ a value of a binary target variable, $y \in \{y^+,y^-\}$
$s$ protected variable
$s^i$ a value of a discreet/binary protected variable, $s \in \{s^1,\ldots,s^m\}$
typically index $1$ denotes a protected group, e.g. $s^1$ - black, $s^0$ - white race
$X$ a set of input variables (predictors), $X = \{x^{(1)},\ldots,x^{(l)}\}$
$z$ explanatory variable or stratum
$z^i$ a value of explanatory variable $z \in \{z^1,\ldots,z^k\}$
$N$ number of individuals in the dataset
$n_i$ number of individuals in group $s^i$
§.§ Statistical tests
Statistical tests are the earliest measures for indirect discrimination discovery in data.
Statistical tests are formal procedures to accept or reject statistical hypotheses, which check how likely the result is to have occurred by chance. In discrimination analysis typically the null hypothesis, or the default position, is that there is no difference between the treatment of the general group and the protected group. The test checks, how likely the observed difference between groups has occurred by chance. If chance is unlikely then the null hypothesis is rejected and discrimination is declared.
Two limitations of statistical tests need to be kept in mind when using them for measuring discrimination.
* Statistical significance does not mean practical significance; statistical tests do not show the magnitude of the the differences between the groups, which can be huge, or can be minor.
* If the null hypothesis is rejected then discrimination is present, but if null hypothesis cannot be rejected, this does not prove that there is no discrimination. It maybe that the data sample is too small to declare discrimination.
Standard statistical tests are typically applied for measuring discrimination. The same tests are used in clinical trials, marketing, and scientific research.
§.§.§ Regression slope test
The test fits an ordinary least squares (OLS) regression to the data including the protected variable, and tests whether the regression coefficient of the protected variable is significantly different from zero. A basic version for discrimination discovery considers only the protected characteristic $s$ and the target variable $y$ <cit.>. In principle $s$ and $y$ can be binary or numeric, but typically in discrimination testing $s$ is binary. The regression may include only the protected variable $s$ as a predictor, but it may also include variables from $X$ that may explain some of the observed difference in decisions.
The test statistic is $t = b/\sigma$, where $b$ is the estimated regression coefficient of $s$, and $\sigma$ is the standard error, computed as
$\sigma = \frac{\sqrt{\sum_{i=1}^n (y_i - f(y_i))^2}}{\sqrt{(n-2)}\sqrt{\sum_{i=1}^n (s_i - \bar{s})^2}}$, where $n$ is the number of observations, $f(.)$ is the regression model, $\bar{.}$ indicates the mean. The t-test with $n-2$ degrees of freedom is applied.
§.§.§ Difference of means test
The null hypothesis is that the means of the two groups are equal.
The test statistic is $t = \frac{E(y|s^0) - E(y|s^1)}{\sigma \sqrt{1/n_0 + 1/n_1}}$, where $n_0$ is the number of individuals in the regular group, $n_1$ is the number of individuals in the protected group,
$\sigma = \sqrt{((n_0 - 1)\delta_0^2 + (n_1 - 1)\delta_1^2)/(n_0 + n_1 - 2)}$, where $\delta_0^2$ and $\delta_1^2$ are the sample target variances in the respective groups.
The t-test with $n_0 - n_1 -2$ degrees of freedom is applied.
The test assumes independent samples, normality and equal variances.
§.§.§ Difference in proportions for two groups
The null hypothesis is that the rates of positive outcomes within the two groups are equal.
The test statistic is
$z = \frac{p(y^+|s^0) - p(y^+|s^0)}{\sigma}$, where $\sigma = \sqrt{\frac{p(y^+|s^0)p(y^-|s^0)}{n_0} + \frac{p(y^+|s^1)p(y^-|s^1)}{n_1}}$. The z-test is used.
§.§.§ Difference in proportions for many groups
The null hypothesis is that the probabilities or proportions are equal for all the groups. This can be used for testing many groups at once. For example, equality of decisions for different ethnic groups, or age groups. If the null hypothesis is rejected that means at least one of the groups has statistically significantly different proportion.
The text statistic is
$\chi^2 = \sum_{i=1}^k \frac{(n_i- np(y^+|s^i))^2}{p(y^+|s^i)}$, where $k$ is the number of groups. The Chi-Square test is used with $k-1$ degrees of freedom.
§.§.§ Other tests and related fields
Relation to clinical trials where protected attribute is the treatment, and outcome is recovery. Prove that there is an effect (there is a discrimination). Does not prove that there is no discrimination. Neither say anything about the magnitude. For example, reduce the flue recovery by 10 min. (practically irrelevant). It may be still relevant for discrimination. Also marketing (measuring the effects of intervention).
Rank test
MannĐWhitney U test is applied for comparing two groups when the normality and equal variances assumptions are not satisfied. The null hypothesis is that the distributions of the two populations are identical.
The procedure is to rank all the observations from the largest $y$ to the smallest. The test statistic is the sum of ranks of the protected group.
§.§ Absolute measures
Absolute measures are designed to capture the magnitude of the differences between (typically two) groups of people. The groups are determined by the protected characteristic (e.g. one group is males, another group is females).
If more than one protected group is analyzed (e.g. different nationalities), typically each group is compared separately to the most favored group.
§.§.§ Mean difference
Mean difference measures the difference between the means of the targets of the protected group and the general group, $d = E(y^+|s^0) - E(y^+|s^1)$.
If there is not difference then it is considered that there is no discrimination. The measure relates to the difference of means, and difference in proportions test statistics, except that there is no correction for the standard deviation.
The mean difference for binary classification with binary protected feature, $d = p(y^+|s^0) - p(y^+|s^1)$, is also known as the discrimination score <cit.>, or sliftd <cit.>.
Mean difference has been the most popular measure in early work on non-discriminatory machine learning and data mining <cit.>.
§.§.§ Normalized difference
Normalized difference <cit.> is the mean difference for binary classification normalized by the rate of positive outcomes,
$\delta = \frac{p(y^+|s^0) - p(y^+|s^1)}{d_{\mathit{max}}}$, where $d_{\mathit{max}} = \min \left(\frac{p(y^+)}{p(s^0)},\frac{p(y^-)}{p(s^1)} \right)$.
This measure takes into account maximum possible discrimination at a given positive outcome rate, such that with maximum possible discrimination at this rate $\delta = 1$, while $\delta = 0$ indicates no discrimination.
§.§.§ Area under curve (AUC)
This measure relates to rank tests. It has been used in <cit.> for measuring discrimination between two groups when the target variable is numeric (regression task), $AUC = \frac{\sum_{(s^i,y^i) \in D^0} \sum_{(s^j,y^j) \in D^1}{\bf I}(y_i > y_j)}{n_0n_1}$, where ${\bf I}(true) = 1$ and $0$ otherwise.
For large datasets computation becomes time and memory intensive, since a quadratic number of comparisons to the number of observations is required. The authors did not mention, but there is an alternative way to compute based on ranking, which, depending on the speed ranking algorithm, may be faster.
Assign numeric ranks to all the observations, beginning with 1 for the smallest value. Let $R_0$ be the sum of the ranks for the favored group. Then $AUC = R_0 - \frac{n_0(n_0 + 1)}{2}$.
We observe that if the target variable is binary, and in case of equality half of a point is added to the sum, then AUC linearly relates to mean difference as
$AUC = p(y^+|s^0)p(y^-|s^1) + 0.5p(y^+|s^0)p(y^+|s^1) + 0.5p(y^-|s^0)p(y^-|s^0) = 0.5d + 0.5$, where $d$ denotes discrimination measured by the mean difference measure.
§.§.§ Impact ratio
Impact ratio, also known as slift <cit.>, is the ratio of positive outcomes for the protected group over the general group, $r = p(y^+|s^1)/p(y^+|s^0)$.
This measure is used in the US courts for quantifying discrimination,
the decisions are deemed to be discriminatory if the ratio of positive outcomes for the protected group is below $80\%$ of that of the general group.
Also this is the form stated in the Sex Discrimination Act of U.K. $r=1$ indicates that there is no discrimination.
§.§.§ Elift ratio
Elift ratio <cit.> is similar to impact ratio, but instead of dividing by the general group,
the denominator is the overall rate of positive outcomes $r = p(y^+|s^0)/p(y^+)$. The same measure, expressed as $p\frac{p(y,s)}{p(y)p(s)} < 1 + \eta$ for all values of $y$ and $s$, is later referred to as $\eta$-neutrality <cit.>.
§.§.§ Odds ratio
Odds ratio of two proportions is often used in natural, social and biomedical sciences to measure the association between exposure and outcome. The popularity is due to convenient relation with the logistic regression. The exponential function of the logistic regression coefficient translates one unit increase in the odds ratio.
Odds ratio has been used for measuring discrimination <cit.> as $r = \frac{p(y^+|s^0)p(y^-|s^1)}{p(y^+|s^1)p(y^-|s^0)}$.
§.§.§ Mutual information
Mutual information (MI) is popular in information theory for measuring mutual dependence between variables. In discrimination literature this measure has been referred to as normalized prejudice index <cit.>, and used for measuring the magnitude of discrimination.
Mutual information is measured in bits, but it can be normalized such that the result falls into the range between $0$ and $1$. For categorical variables $MI = \frac{I(y,s)}{\sqrt{H(y),H(s)}}$, where $I(s,y) = \sum_{(s,y)} p(s,y) \log \frac{p(s,y)}{p(s)p(y)}$, and $H(y) = - \sum_{y} p(y) \log p(y)$.
For numerical variables the summation is replaces by integral.
§.§.§ Balanced residuals
While other measures work on datasets, balanced residuals is for machine learning model outputs. This measure characterizes the difference between the actual outcomes recorded in the dataset, and the model outputs. The requirement is that underpredictions and overpredictions should be balanced within the protected and regular groups. <cit.> proposed balanced residuals as a criteria, not a measure. That is, the average residuals should be equal, but in principle the difference could be used as a measure of discrimination $d = \frac{\sum_{i \in D^1} y_i - \hat{y}_i}{n_1} - \frac{\sum_{j \in D^0} y_j - \hat{y}_j}{n_0}$, where $y$ is the true target value, $\hat{y}$ is the prediction. Positive values of $d$ would indicate discrimination towards the protected group.
One should; however, use and interpret this measure with caution. If the learning dataset is discriminatory, but the predictive model makes ideal predictions such that all the residuals are zero, this measure would show no discrimination, even though the predictions would be discriminatory, since the original data is discriminatory. Suppose, another predictive model makes a constant prediction for everybody, and the constant prediction is equal to the mean of the regular group. If the learning dataset contains discrimination, then the residuals for the regular group would be smaller than for the protected group, and the measure would indicate discrimination, however, a constant prediction to everybody means tat everybody is treated equally, and there should be no discrimination detected.
§.§.§ Other possible measures
There are many established measures in feature selection literature <cit.> for measuring the relation between two variables, which, in principle, can be used as absolute discrimination measures. The stronger the relation between the protected variable $s$ and the target variable $y$, the larger the absolute discrimination.
There are three main groups of measures for relation between variables: correlation based, information theoretic, and one-class classifiers. Correlation based measures, such as the Person correlation coefficient, are typically used for numeric variables. Information theoretic measures, such as mutual information mentioned earlier, are typically used for categorical variables. One-class classifiers present an interesting option. In discrimination the setting would be to predict the target $y$ solely on the protected variable $s$, and measure the prediction accuracy.
We are not aware of such attempts in the non-discriminatory machine learning literature, but it would be a valid option to explore.
§.§.§ Measuring for more than two groups
Most of the absolute discrimination measures are for two groups (protected group vs. regular group). Ideas, how to apply those for more than two groups, can be borrowed from multi-class classification <cit.>, multi-label classification <cit.>, and one-class classification <cit.> literature.
Basically, there are three options how to obtain sub-measures: measure pairwise for each pair of groups ($k(k-1)/2$ comparisons), measure one against the rest for each group ($k$ comparisons), measure each group against the regular group ($k-1$ comparisons). The remaining question is how to aggregate the sub-measures. Based on personal conversations with legal experts, we advocate for reporting the maximum from all the comparisons as the final discrimination score. Alternatively, all the scores could be summed weighing by the group sizes to obtain an overall discrimination score.
Even though absolute measures do not take into account any explanations of possible differences of decisions across groups, they can be considered as core building blocks for developing conditional measures. Conditional measures do take into account explanations in differences, and measure only discrimination that cannot be explained by non-protected characteristics.
Table <ref> summarizes applicability of absolute measures in different machine learning settings.
Summary of absolute measures. Checkmark () indicates that it is directly applicable in a given machine learning setting.
Tilde ($\sim$) indicates that a straightforward extension exists (for instance, measuring pairwise).
3cProtected variable 3cTarget variable
Measure Binary Categoric Numeric Binary Ordinal Numeric
Mean difference $\sim$
Normalized difference $\sim$
Area under curve $\sim$
Impact ratio $\sim$
Elift ratio $\sim$
Odds ratio $\sim$
Mutual information
Balanced residuals $\sim$ $\sim$
§.§ Conditional measures
Absolute measures take into account only the target variable $y$ and the protected variable $s$.
Absolute measures consider all the differences in treatment between the protected group and the regular group to be discriminatory.
Conditional measure, on the other hand, try to capture how much of the difference between the groups is explainable by other characteristics of individuals, recorded in $X$, and only the remaining differences are deemed to be discriminatory. For example, part of the difference in acceptance rates for natives and immigrants may be explained by the difference in education level. Only the remaining unexplained difference should be considered as discrimination. Let $z = f(X)$ be an explanatory variable. For example, if $z^i$ denotes a certain education level. Then all the individuals with the same level of education will form a strata $i$. Within each strata the acceptance rates are required to be equal.
§.§.§ Unexplained difference
Unexplained difference <cit.> is measured, as the name suggests, as the overall mean difference minus the differences that can be explained by other legitimate variable. Recall that mean difference is $d = p(y^+|s^0) - p(y^+|s^1)$.
Then the unexplained difference $d_u = d - d_e$,
$d_e = \sum_{i=1}^m p^\star(y^+|z^i)(p(z^i|s^0) - p(z^i|s^1))$, where $p^\star(y^+|z^i)$ is the desired acceptance rate within the strata $i$.
The authors recommend using
$p^\star(y^+|z^i) = \frac{p(y^+|s^0,z^i) + p(y^+|s^1,z^i)}{2}$. In the simplest case $z$ bay be equal one of the variables in $X$. The authors also use clustering on $X$ to take into account more than one explanatory variable at the same time. Then $z$ denotes a cluster, one strata is one cluster.
§.§.§ Propensity measure
Propensity models <cit.> are typically used in clinical trials or marketing for estimating the probability that an individual would receive a treatment. Given the estimated probabilities, individuals can be stratified according to similar probabilities of receiving a treatment, and the effects of treatment can be measured within each strata separately.
Propensity models have been used for measuring discrimination <cit.>, in this case a function was learned to model the protected characteristic based on input variables $X$, that is $s^1 = f(X)$. A logistic regression was used for modeling $f(.)$. Then the estimated propensity scores $\hat{s}^1$ were split into five ranges, where each range formed one strata. Discrimination was measured within each strata, treating each strata as a separate dataset, and using absolute discrimination measures discussed in the previous section. The authors did not aggregate the resulting discrimination into one measure, but in principle the results can be aggregated into one measure, for instance, using the unexplained difference formulas, reported above. In such a case each strata would correspond to one value of an explanatory variable $z$.
§.§.§ Belift ratio
Belift ratio <cit.> is similar to Elift ratio in absolute measures, but here the probabilities of positive outcome are also conditioned on input attributes, $belift = \frac{p(y^+|s^1,X^r,X^a)}{p(y^+|X^a)}$, where $X = X^r \cup X^{\not r}$ is a set of input variables, $X^{r}$ denotes so caller redlining attributes, the variables which are correlated with the protected variable $s$. The authors proposed estimating the probabilities via bayesian networks. A possible difficulty for applying this measure in practice may be that not everybody, especially non-machine learning users, are familiar enough with the Bayesian networks to an extent needed for estimating the probabilities. Moreover, construction of a Bayesian network may be different even for the same problem depending on assumptions made about interactions between the variables. Thus, different users may get different discrimination scores for the same application case.
A simplified approximation of belift could be to treat all the attributes as redlining attributes, and instead of conditioning on all the input variables, condition on a summary of input variables $z$, where $z = f(X)$. Then the measure for strata $i$ would be $\frac{p(y^+|s^1,z^i)}{p(y^+)}$.
The measure has a limitation that neither the original version, nor the simplified version allow differences to be explained by variables that are correlated with the protected variable. That is, if a university has two programmes, say medicine and computer science, and the protected group, e.g. females, are more likely to apply for a more competitive programme, then the programmes cannot have different acceptance rates. That is, if the acceptance rates are different, all the difference is considered to discriminatory.
§.§ Structural measures
Structural measures are targeted at quantifying direct discrimination. The main idea behind structural measures is for each individual in the dataset to identify whether s/he is discriminated, and then analyze how many individuals in the dataset are affected. Currently
§.§.§ Situation testing
Situation testing <cit.> measures which fraction of individuals in the protected group are considered discriminated, as
$f = \frac{\sum_{y_i \in D(y^0|s^1)} {\bf I}(\mathit{diff}(y_i) \geq t)}{|D(y^0|s^1)|}$, where $t$ is a user defined threshold, ${\bf I}$ is the indicator function that takes $1$ if true, $0$ otherwise.
The situation testing for an individual $i$ is computed as $\mathit{diff}(y_i) = \frac{\sum_{y_j \in D^0{\mathit{\kappa-nearest-neighbours} }}}{\kappa} - \frac{\sum_{y_j \in D^1{\mathit{\kappa-nearest-neighbours} }}}{\kappa}$.
Positive and negative discrimination is handled separately.
The idea is to compare each individual to the opposite group and see if the decision would be different.
In that sense, the measure relates to propensity scoring (Section <ref>), used for identifying groups of people similar according to the non-protected characteristics, and requiring for decisions within those groups to be balanced. The main difference is that propensity measures would signal indirect discrimination within a group, and situation testing aims at signalling direct discrimination for each individual in question.
§.§.§ Consistency
Consistency measure <cit.> compares the predictions for each individual with his/her nearest neighbors.
$C = 1 - \frac{1}{\kappa N}\sum_{i=1}^N \sum_{y_j \in D^{\mathit{\kappa-nearest-neighbours}}}|y_i - y_j|$.
Consistency measure is closely related to situation testing, but considers nearest neighbors from any group (not from the opposite group).
Due to this choice, consistency measure should be used with caution in situations where there is a high correlation between the protected variable and the legitimate input variables. For example, suppose we have only one predictor variable - location of an apartment, and the target variable is to grant a loan or not. Suppose all non-white people live in one neighborhood (as in the redlining example), and all the white people in the other neighborhood. Unless the number of nearest neighbors to consider is very large, this measure will show no discrimination, since all the neighbors will get the same decision, even though all black residents will be rejected, and all white will be accepted (maximum discrimination). Perfect consistency, but maximum discrimination. In their experimental evaluation the authors have used this measure in combination with the mean difference measure.
§ ANALYSIS OF CORE MEASURES
Even though absolute measures are naive in a sense that they do not take any possible explanations of different treatment into account, and due to that may show more discrimination that there actually is, these measures provide core mechanisms and a basis for measuring indirect discrimination. Conditional measures are typically built upon absolute measures. In addition, statistical tests often directly relate to absolute measures.
Thus, to provide a better understanding of properties and implications of choosing one measure over another, in this section we computationally analyze a set of absolute measures, and discuss their properties.
We analyze the following measures, introduced in Section <ref>: mean difference, normalized difference, mutual information, impact ratio, elift and odds ratio. From the measures analyzed in this section, mean difference and area under curve can be directly used in regression tasks. We focus on the classification scenario, since this scenario has been studied more extensively in the discrimination-aware data mining and machine learning literature, and there are more measures available for classification than for regression; the regression setting, except for a recent work <cit.>, remains a subject of future research, and therefore is out of the scope of a survey paper.
Table <ref> summarizes boundary conditions of the selected measures. In the difference based measures $0$ indicates no discrimination, in the ratio based measures $1$ indicates no discrimination, in AUC $0.5$ means no discrimination. The boundary conditions are reached when one group gets all the positive decisions, and the other group gets all the negative decisions.
Measure Maximum No Reverse
discrimination discrimination discrimination
Mean difference $1$ $0$ $-1$
Normalized difference $1$ $0$ $-1$
Mutual information $1$ $0$ $1$
Impact ratio $0$ $1$ $+\infty$
Elift $0$ $1$ $+\infty$
Odds ratio $0$ $1$ $+\infty$
Area under curve (AUC) $1$ $0.5$ $0$
Next we experimentally analyze the performance of the selected measures. We leave out AUC from the experiments, since in classification it is equivalent to the mean difference measure. The goal of the experiments is to demonstrate how the performance depends on variations in the overall rate of positive decisions, balance between classes and balance between the regular and protected groups of people in data.
For this analysis we use synthetically generated data which allows to represent different task settings and control the levels of underlying discrimination.
Given four parameters:
the proportion of individuals in the protected group $p(s^1)$,
the proportion of positive outputs $p(y^+)$, the underlying discrimination $d \in [-100\%,100\%]$, and the number of data points $n$, data is generated as follows.
First $n$ data points are generated assigning a score in $[0,1]$ uniformly at random, and assigning group membership at random according to the probability $p(s^1)$. This data contains no discrimination, because the scores are assigned at random. If would contain full discrimination if we ranked the observations according to the assigned scores and all the members of the regular group would appear before all the members of the protected group. Following this reasoning, half-discrimination would be if in a half of the data the members of the regular group appear before all the members of the protected group in the ranking, and the other half of the data would show a random mix of both groups in the ranking. For the experimental analysis purposes we define this as $50\%$ discrimination. It is difficult to measure discrimination in data this way, but it is easy to generate such a data. For a given level of desired discrimination $d$ we select $dn$ observations at random, sort them according to their scores, and then permute group assignments within this subsample in such a way that the highest scores get assigned to the regular group, and the lowest scores get assigned to the protected group. Finally, since the experiment is about classification, we round the scores to zero-one in such a way that the proportion of ones is as desired by $p(y^+)$. Then we apply different measures of discrimination to data generated this way, and investigate, how these measures can reconstruct the underlying discrimination. For each parameter setting we generate $n=10000$ data points, and average the results over $100$ such runs[The code for our experiments is made available at <https://github.com/zliobaite/paper-fairml-survey>.]
Figure <ref> depicts the performance of mean difference, normalized difference and mutual information. Ideally, the performance should be invariant to balance of the groups ($p(s^10)$) and the proportion of positive outputs ($p(y^+)$), and thus run along the diagonal line in as many plots, as possible. We can see that the normalized difference captures that. The mean difference captures the trends, but the indicated discrimination highly depends on the balance of the classes and balance of the groups, therefore, this measure to be interpreted with care when data is highly imbalanced. The same holds for mutual information. For instance, at $p(s^1)= 90\%$ and $p(y^+)=90\%$ the true discrimination in data may be near $100\%$, i.e. nearly the worst possible, but both measures would indicate that discrimination is nearly zero. The normalized difference would capture the situation as desired. In addition to that, we see that the mean difference and normalized difference are linear measures, while mutual information is non-linear, and would show less discrimination that actually in the medium ranges. Moreover, mutual information dos not indicate the sign of discrimination, that is, the outcome does not indicate whether discrimination is reversed or not. For these reasons, we do not recommend using mutual information for the purpose of quantifying discrimination.
Therefore, from the difference based measures we advocate normalized difference, which was designed to be robust to imbalances in data.
The normalized difference is somewhat more complex to compute than the mean difference, which may be a limitation for practical applications outside research.
Therefore, if data is closed to balanced in terms of groups and positive-negative outputs, then the mean difference can be used.
ylabel right/.style=after end axis/.append code=[rotate=90, anchor=north] at (rel axis cs:1,0.5) #1;
[name = plot1,,ylabel = measured discrim.,legend entries=,,mean difference, normalized difference, mutual information,
legend style=draw=none,font=,legend columns=3,legend to name = named,title = $p(y^+) = 10\%$]
[name = plot2, at=(plot1.right of south east), anchor=left of south west,,title = $30\%$]
[name = plot3, at=(plot2.right of south east), anchor=left of south west, ,title = $50\%$]
[name = plot4, at=(plot3.right of south east), anchor=left of south west, ,title = $70\%$]
[name = plot5, at=(plot4.right of south east), anchor=left of south west, ,title = $90\%$,ylabel right = $p(s^1) = 10\%$]
[name=plot6, at=(plot1.below south west), anchor=above north west, ,ylabel = measured discrim.]
[name=plot7, at=(plot6.right of south east), anchor=left of south west,]
[name=plot8, at=(plot7.right of south east), anchor=left of south west,]
[name=plot9, at=(plot8.right of south east), anchor=left of south west,]
[name=plot10, at=(plot9.right of south east), anchor=left of south west,,ylabel right = $30\%$]
[name=plot11, at=(plot6.below south west), anchor=above north west, ,ylabel = measured discrim.]
[name=plot12, at=(plot11.right of south east), anchor=left of south west,]
[name=plot13, at=(plot12.right of south east), anchor=left of south west,]
[name=plot14, at=(plot13.right of south east), anchor=left of south west,]
[name=plot15, at=(plot14.right of south east), anchor=left of south west,,ylabel right = $50\%$]
[name=plot16, at=(plot11.below south west), anchor=above north west, ,ylabel = measured discrim.]
[name=plot17, at=(plot16.right of south east), anchor=left of south west,]
[name=plot18, at=(plot17.right of south east), anchor=left of south west,]
[name=plot19, at=(plot18.right of south east), anchor=left of south west,]
[name=plot20, at=(plot19.right of south east), anchor=left of south west,,ylabel right = $70\%$]
[name=plot21, at=(plot16.below south west), anchor=above north west, ,ylabel = measured discrim.,xlabel = discrim. in data]
[name=plot22, at=(plot21.right of south east), anchor=left of south west,, xlabel = discrim. in data]
[name=plot23, at=(plot22.right of south east), anchor=left of south west,, xlabel = discrim. in data]
[name=plot24, at=(plot23.right of south east), anchor=left of south west,, xlabel = discrim. in data]
[name=plot24, at=(plot24.right of south east), anchor=left of south west,, xlabel = discrim. in data,ylabel right = $90\%$]
Analysis of the measures based on differences: discrimination in data vs. measured discrimination.
Figure <ref> presents similar analysis of the measures based on ratios: impact ratio, elift and odds ratio. We can see that the odds ratio, and the impact ratio are very sensitive to imbalances in groups and positive outputs. The elift is more stable in that respect, but still has some variations, particularly at high imbalance of positive outputs ($p(y^+)=90\%$ or $10\%$), when discrimination may be highly exaggerated (far from the diagonal line).
In addition, measured discrimination by all ratios grows very fast at low rates of positive outcome (e.g. see the plot $p(y^+) = 10\%$ and $p(s^1)=90\%$), while there is almost no discrimination in the data, measures indicate high discrimination.
We also can see that all the ratios are asymmetric in terms of reverse discrimination. One unit of measured discrimination is not the same as one unit of reverse discrimination. This makes ratios a bit more difficult to interpret than differences, analyzed earlier, especially at large scale explorations and comparisons of, for instance, different computational methods for prevention. Due to these reasons, we do not recommend using ratio based discrimination measures, since they are much more difficult to interpret correctly, and may easily be misleading. Instead recommend using and building upon difference based measures, discussed in Figure <ref>.
[name = plot1,,ylabel = measured discrim.,legend entries=,,impact ratio, elift, odds ratio,
legend style=draw=none,font=,legend columns=3,legend to name = named2,title = $p(y^+) = 10\%$]
[name = plot2, at=(plot1.right of south east), anchor=left of south west,,title = $30\%$]
[name = plot3, at=(plot2.right of south east), anchor=left of south west, ,title = $50\%$]
[name = plot4, at=(plot3.right of south east), anchor=left of south west, ,title = $70\%$]
[name = plot5, at=(plot4.right of south east), anchor=left of south west, ,title = $90\%$,ylabel right = $p(s^1) = 10\%$]
[name=plot6, at=(plot1.below south west), anchor=above north west, ,ylabel = measured discrim.]
[name=plot7, at=(plot6.right of south east), anchor=left of south west,]
[name=plot8, at=(plot7.right of south east), anchor=left of south west,]
[name=plot9, at=(plot8.right of south east), anchor=left of south west,]
[name=plot10, at=(plot9.right of south east), anchor=left of south west,,ylabel right = $30\%$]
[name=plot11, at=(plot6.below south west), anchor=above north west, ,ylabel = measured discrim.]
[name=plot12, at=(plot11.right of south east), anchor=left of south west,]
[name=plot13, at=(plot12.right of south east), anchor=left of south west,]
[name=plot14, at=(plot13.right of south east), anchor=left of south west,]
[name=plot15, at=(plot14.right of south east), anchor=left of south west,,ylabel right = $50\%$]
[name=plot16, at=(plot11.below south west), anchor=above north west, ,ylabel = measured discrim.]
[name=plot17, at=(plot16.right of south east), anchor=left of south west,]
[name=plot18, at=(plot17.right of south east), anchor=left of south west,]
[name=plot19, at=(plot18.right of south east), anchor=left of south west,]
[name=plot20, at=(plot19.right of south east), anchor=left of south west,,ylabel right = $70\%$]
[name=plot21, at=(plot16.below south west), anchor=above north west, ,ylabel = measured discrim.,xlabel = discrim. in data]
[name=plot22, at=(plot21.right of south east), anchor=left of south west,, xlabel = discrim. in data]
[name=plot23, at=(plot22.right of south east), anchor=left of south west,, xlabel = discrim. in data]
[name=plot24, at=(plot23.right of south east), anchor=left of south west,, xlabel = discrim. in data]
[name=plot24, at=(plot24.right of south east), anchor=left of south west,, xlabel = discrim. in data,ylabel right = $90\%$]
Analysis of the measures based on ratios: discrimination in data vs. measured discrimination.
The core measures that we have analyzed form a basis for assessing fairness of predictive models, but it is not enough to use them directly, since they do not take into account possible legitimate explanations of differences between the groups, and instead consider any differences between the groups of people undesirable. The basic principle is to try to stratify the population in such a way that in each stratum contains people that are similar in terms of their legitimate characteristics, for instance, have similar qualifications if the task is candidate selection for job interviews.
propensity score matching, reported in Section <ref>, is one possible way to stratification, but it is not the only one, and outcomes may vary depending on internal parameter choices. Thus, the principle to measuring is available, but there are still open challenges ahead to make the approach more robust to different users, and more uniform across different task setting, such that one could diagnose potential discrimination or declare fairness with more confidence.
§ RECOMMENDATIONS FOR RESEARCHERS AND PRACTITIONERS
As attention of researchers, media and general public to potential discrimination is growing, it is important to be able to measure fairness of predictive models in a systematic and accountable way. We have surveyed measures used (and potentially usable) for measuring indirect discrimination in machine learning, and experimentally analyzed the performance of the core measures in classification tasks. Based on our analysis we generally recommend using the normalized difference, and in case the classes and groups of people in the data are well balanced, it may be sufficient to use the simple (unnormalized) mean difference. We do not recommend using ratio based measures challenges associated with their interpretation in different situation.
The core measures stand alone are not enough for measuring fairness correctly. These measures can only be applied to uniform populations considering that everybody within the population is equally qualified to get a positive decision. In reality this is rarely the case, for example, different salary levels may be explained by different education levels. Therefore, the main principle of applying the core measures should be by first segmenting the population into more or less uniform segments according to their qualifications, and then applying core measures within each segment. Some of such measuring techniques have been surveyed in Section <ref> (Conditional measures), but generally there is no one easy way to approach it, and presenting sound arguments to justify the methods of allocating people into segments is very important in research and practice.
We hope that this survey can establish a basis for further research developments in this important topic. So far most of the research has concentrated on binary classification with binary protected characteristic. While this is a base scenario, relatively easy to deal with in research, many technical challenges for future research lie in addressing more complex learning scenarios with different types and multiple protected characteristics, in multi-class, multi-target classification and regression settings, with different types of legitimate variables, noisy input data, potentially missing protected characteristics, and many more.
|
1511.00117
|
This paper introduces a new notion of chaotic algorithms. These algorithms are iterative and are based on so-called chaotic iterations. Contrary to all existing studies on chaotic iterations, we are not interested in stable states of such iterations but in their possible unpredictable behaviors. By establishing a link between chaotic iterations and the notion of Devaney's topological chaos, we give conditions ensuring that these kind of algorithms produce topological chaos. This leads to algorithms that are highly unpredictable.
After presenting the theoretical foundations of our approach, we are interested in its practical aspects. We show how the theoretical algorithms give rise to computer programs that produce true topological chaos, then we propose applications in the area of information security.
§ INTRODUCTION
The use of chaos in various fields of information security such as data hiding, hash functions, or pseudo-random number generators is almost always based on the conception of algorithms that include known chaotic maps such as the logistic map. The goal is to obtain an algorithm which preserves the chaotic properties of the included chaotic functions.
For example, in <cit.> and <cit.>, a watermark $W$ is encrypted in $W_{e}$ by using the bitwise exclusive or: $W_{e}=W\otimes X$, where $X$ is a logistic map. Then, pixels of the carrier image designed to embed these bits are selected with the 2-D Arnold's cat map. A similar use of chaotic maps for watermarking can be found in e.g. <cit.>, <cit.>, <cit.>, <cit.> and <cit.>. In the domain of hash functions, the use of chaotic maps is seen in e.g. <cit.>, <cit.>, <cit.> and <cit.>.
However, without rigorous proof it is not indisputable that an algorithm that includes chaotic functions preserves chaos properties: for example, using the logistic function with other “obvious” parameters does not guarantee that the algorithm is chaotic.
Moreover, even if the algorithm obtained by the inclusion of chaotic maps is itself chaotic, the implementation of this algorithm on a machine can cause it to lose its chaotic nature. This is due to the finite nature of the machine numbers set. These issues are discussed in this document.
In this paper we don't simply integrate chaotic maps hoping that the security algorithm remains chaotic, we conceive algorithms for computer security that we mathematically prove chaotic. We raise the question of their implementation, proving in doing so that it is possible to design a chaotic algorithm and a chaotic computer program.
The chaos theory we consider is Devaney's topological chaos. In addition to being recognized as one of the best mathematical definition of chaos, this theory offers a framework with qualitative and quantitative tools to evaluate the notion of unpredictability. As an application of our fundamental results, we are interested in the area of information security. We propose in this paper a new approach of security which is based on unpredictability as it is defined by Devaney's chaos.
The paper begins by introducing the theoretical foundation of the new approach. We recall the definition of Devaney's topological chaos as well as the definition of discrete chaotic iterations. Although these definitions are distinct from each other, we establish a link between them by giving conditions under which chaotic discrete iterations generate a Devaney's topological chaos. Because chaotic iterations are very suited for computer programming, this link allows us to generate Devaney's chaos topological in the computer science field.
After having studied the theoretical aspects of our approach we focus on practical aspects. The important question is how to preserve the topological chaos properties in a set of a finite number of states. This question is answered by introducing a concept we call secure chaotic information machine. This is a Mealy machine generating chaos as defined by Devaney (Section <ref>).
We also give some applications of our approach of chaos, in the domain of information security. Algorithms intended for information security and based on this new approach are explained in Section <ref>, in the hash function domain
The rest of this paper is organized as follows. In Section <ref>, the definitions of Devaney's chaos and discrete chaotic iterations are recalled. A link between these two notions is established and sufficient conditions to obtain Devaney's topological chaos from discrete chaotic iterations are given in Section <ref>. In Section <ref>, the question on how to apply the theoretical result is raised and applications in the computer science field are given in Section <ref>. The paper ends with a conclusion in which our contribution is summarized and the planned future work is discussed.
§ BASIC RECALLS
This section is devoted to basic definitions and terminologies in the field of topological chaos and in the one of chaotic iterations.
§.§ Devaney's chaotic dynamical systems
Consider a metric space $(\mathcal{X},d)$ and a continuous function $f:\mathcal{X}\longrightarrow \mathcal{X}$.
$f$ is said to be topologically transitive if, for any pair of open sets $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq \varnothing$.
An element (a point) $x$ is a periodic element (point) for $f$ of period $n\in \mathds{N}^*,$ if $f^{n}(x)=x$. The set of periodic points of $f$ is denoted $Per(f).$
$(\mathcal{X},f)$ is said to be regular if the set of periodic points is dense in $\mathcal{X}$,
\begin{equation*}
\forall x\in \mathcal{X},\forall \varepsilon >0,\exists p\in Per(f),d(x,p)\leqslant \varepsilon .
\end{equation*}
$f$ has sensitive dependence on initial conditions
if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any neighborhood $V$ of $x$, there exists $y\in V$ and $n\geqslant 0$ such that $|f^{n}(x)-f^{n}(y)|>\delta $.
$\delta$ is called the constant of sensitivity of $f$.
Let us now recall the definition of a chaotic topological system, in the
sense of Devaney <cit.>:
$f:\mathcal{X}\longrightarrow \mathcal{X}$ is said to be chaotic on $%
\mathcal{X}$ if,
* $f$ has sensitive dependence on initial conditions,
* $f$ is topologically transitive,
* $(\mathcal{X},f)$ is regular.
Therefore, quoting Robert Devaney: “A chaotic map possesses three ingredients: unpredictability, indecomposability and an element of regularity. A chaotic system is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or decomposed into two subsystems, because of topological transitivity. And, in the midst of this random behavior, we nevertheless have an element of regularity, namely the periodic points which are dense.” Fundamentally different behaviors are thus possible and occur in an unpredictable way.
§.§ Chaotic iterations
In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$, $V_{i}$
denotes the $i^{th}$ component of a vector $V$ and $f^{k}=f\circ ...\circ f$
denotes the $k^{th}$ composition of a function $f$. Finally, the following
notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
Let us consider a system of a finite number $\mathsf{N}$ of elements (or
cells), so that each cell has a boolean state. Then a sequence of
length $\mathsf{N}$ of boolean states of the cells corresponds to a
particular state of the system. A sequence which elements belong to $ \llbracket 1;\mathsf{N} \rrbracket $ is called a strategy. The set of all strategies is denoted by $\mathbb{S}.$
The set $\mathds{B}$ denoting $\{0,1\}$, let $f:\mathds{B}^{\mathsf{N}%
}\longrightarrow \mathds{B}^{\mathsf{N}}$ be a function and $S\in \mathbb{S}
$ be a strategy. Then, the so-called chaotic iterations are defined by $x^0\in \mathds{B}^{\mathsf{N}}$ and $\forall n\in \mathds{N}^{\ast },$
\begin{equation}
\forall i\in \llbracket1;\mathsf{N}\rrbracket%
\begin{array}{ll}
x_i^{n-1} & \text{ if }S^n\neq i \\
\left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.%
\end{array}%
\right.%
\end{equation}
In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is “iterated”. Note that in a more general formulation, $S^n$ can be a subset of components and $f(x^{n-1})_{S^{n}}$ can be replaced by $f(x^{k})_{S^{n}}$ (where $k\leqslant n-1$), describing for example delays transmission (see e.g. <cit.>). For the general definition of such chaotic iterations, see e.g. <cit.>.
§ CHAOTIC ITERATIONS AS DEVANEY'S CHAOS
§.§ The new topological space
In this section we will put our study in a topological context by defining a suitable metric space where chaotic iterations are continuous.
§.§.§ Defining the iteration function and the phase space
Let $\delta $ be the discrete boolean metric, $\delta (x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
\begin{equation*}
\begin{array}{lrll}
F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} &
\longrightarrow & \mathds{B}^{\mathsf{N}} \\
& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta
(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
\end{array}%
\end{equation*}
where + and . are the boolean addition and product operations.
Consider the phase space:
\begin{equation*}
\mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times
\mathds{B}^\mathsf{N},
\end{equation*}
and the map defined on $\mathcal{X}$:
\begin{equation}
G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
\end{equation}
where $\sigma$ is the shift function defined by $\sigma (S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in \mathds{N}}\in \mathbb{S}$ and $i$ is the initial function $i:(S^{n})_{n\in \mathds{N}} \in \mathbb{S}\longrightarrow S^{0}\in \llbracket 1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in (<ref>) can be described by the following iterations:
\begin{equation*}
\left\{
\begin{array}{l}
X^0 \in \mathcal{X} \\
\end{array}%
\right.
\end{equation*}
With this formulation, a shift function appears as a component of chaotic iterations. The shift function is a famous example of a chaotic map <cit.> but its presence is not sufficient enough to claim $G_f$ as chaotic. In the rest of this section we prove rigorously that under some hypotheses, chaotic iterations generate topological chaos. Furthermore, due to the suitability of chaotic iterations for computer programming we also prove that this is true in the computer science field.
§.§.§ Cardinality of $\mathcal{X}$
By comparing $\mathbb{S}$ and $\mathds{R}$, we have the result.
The phase space $\mathcal{X}$ has, at least, the cardinality of the continuum.
Let $\varphi$ be the map which transforms a strategy into the binary representation of an element in $[0,1$[, as follows. If the $n^{th}$ term of the strategy is 0, then the $n^{th}$ associated digit is 0, or else it is equal to 1.
With this construction, $\varphi : \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \longrightarrow [0,1]$ is surjective. But $]0,1[$ is isomorphic to $\mathds{R}$ ($x \in ]0,1[\mapsto tan(\pi(x-\frac{1}{2}))$ is an isomorphism), so the cardinality of $\llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N}$ is greater or equal to the cardinality of $\mathds{R}$. As a consequence, the cardinality of the Cartesian product $\mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times \mathds{B}^\mathsf{N}$ is greater or equal to the cardinality of $\mathds{R}$.
This result is independent from the number of cells of the system.
§.§.§ A new distance
We define a new distance between two points $X = (S,E), Y = (\check{S},\check{E})\in
\mathcal{X}$ by
\begin{equation*}
\end{equation*}
\begin{equation*}
\left\{
\begin{array}{lll}
\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
}\delta (E_{k},\check{E}_{k})}, \\
\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
\sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.%
\end{array}%
\right.
\end{equation*}
If the floor value $\lfloor d(X,Y)\rfloor $ is equal to $n$,
then the systems $E, \check{E}$ differ in $n$ cells. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a measure of the differences between strategies $S$ and $\check{S}$. More precisely, this floating part is less than $10^{-k}$ if and only if the first $k$
terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the $k^{th}$ terms of the two
strategies are different.
§.§.§ Continuity of the iteration function
To prove that chaotic iterations are an example of topological chaos in the
sense of Devaney <cit.>, $G_{f}$ must be continuous in the metric
space $(\mathcal{X},d)$.
$G_f$ is a continuous function.
We use the sequential continuity.
Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
G_{f}(S,E)\right) $. Let us recall that for all $n$, $S^n$ is a strategy,
thus, we consider a sequence of strategies (i.e. a sequence of
sequences).As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
cell will change its state:
\begin{equation*}
\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Longrightarrow E^n = E.
\end{equation*}
In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
first term, which is $S^0$:
\begin{equation*}
\forall n\geqslant n_{1},S_0^n=S_0.
\end{equation*}
Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
identical and strategies $S^n$ and $S$ start with the same first term.Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.We now prove that the distance between $\left(
G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
0. Let $\varepsilon >0$.
* If $\varepsilon \geqslant 1$, we see that distance
between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
* If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
\varepsilon \geqslant 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
\begin{equation*}
\exists n_{2}\in \mathds{N},\forall n\geqslant
\end{equation*}
thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
As a consequence, the $k+1$ first entries of the strategies of $%
G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
10^{-(k+1)}\leqslant \varepsilon $.In conclusion,
\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
,\forall n\geqslant N_{0},
d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
\leqslant \varepsilon .
$G_{f}$ is consequently continuous.
In this section, we proved that chaotic iterations can be modeled as a dynamical system in a topological space. In the next section, we show that chaotic iterations are a case of topological chaos, according to Devaney.
§.§ Discrete chaotic iterations as topological chaos
To prove that we are in the framework of Devaney's topological chaos, we have to find a boolean function $f$ such that $G_f$ satisfies the regularity, transitivity and sensitivity conditions. We will prove that the vectorial logical negation
\begin{equation}
\hdots,x_{\mathsf{N}})=(\overline{x_{1}},\hdots,\overline{x_{\mathsf{N}}})
\label{f0}
\end{equation}
is a suitable function.
§.§.§ Regularity
Periodic points of $G_{f_0}$ are dense in $\mathcal{X}$.
Let $(\check{S}, \check{E})\in \mathcal{X}$ and $\varepsilon >0$. We are
looking for a periodic point $(\widetilde{S},\widetilde{E})$ satisfying $d((%
\check{S}, \check{E});(\widetilde{S},\widetilde{E}))<\varepsilon$.
As $\varepsilon$ can be strictly lesser than 1, we must choose $%
\widetilde{E} = \check{E}$. Let us define $k_0(\varepsilon) =\lfloor
log_{10}(\varepsilon )\rfloor +1$ and consider the set
\[
\mathcal{S}_{\check{S}, k_0(\varepsilon)} = \left\{ S \in \mathbb{S} / S^k =
\check{S}^k, \forall k \leqslant k_0(\varepsilon) \right\}.
\]
Then, $\forall S \in \mathcal{S}_{\check{S}, k_0(\varepsilon)}, d((S, \check{%
E});(\check{S}, \check{E})) < \varepsilon$. It remains to choose $\widetilde{%
S} \in \mathcal{S}_{\check{S}, k_0(\varepsilon)}$ such that $(\widetilde{S},%
\widetilde{E}) = (\widetilde{S},\check{E})$ is a periodic point for $%
Let $\mathcal{J} = \left\{ i \in \{1, ..., \mathsf{N}\} / E_i \neq \check{%
E}_i, \text{ where } (S, E) = G_{f_0}^{k_0} (\check{S}, \check{E}) \right\}$
, $i_0 = card(\mathcal{J})$ and $j_1 <j_2 < ... < j_{i_0}$ the elements of $%
\mathcal{J}$. Then, $\widetilde{S} \in \mathcal{S}_{\check{S},
k_0(\varepsilon)}$ defined by
* $\widetilde{S}^k = \check{S}^k$, if $k \leqslant k_0(\varepsilon)$,
* $\widetilde{S}^k = j_{k-k_0(\varepsilon)}$, if $k \in
\{k_0(\varepsilon)+1, k_0(\varepsilon)+2, ..., k_0(\varepsilon)+i_0\}$,
* and $\widetilde{S}^{k}=\widetilde{S}^{j}$, where $j\leqslant
k_{0}(\varepsilon )+i_{0}$ is satisfying $j\equiv k~(\text{mod }%
k_{0}(\varepsilon )+i_{0})$, if $k>k_{0}(\varepsilon )+i_{0}$,
is such that $%
(\widetilde{S},\widetilde{E})$ is a periodic point, of period $%
k_{0}(\varepsilon )+i_{0}$, which is $\varepsilon -$closed to $(\check{S},%
\check{E})$.As a conclusion, $(\mathcal{X},G_{f_{0}})$ is
§.§.§ Transitivity
$(\mathcal{X},G_{f_0})$ is topologically transitive.
Let us define $\mathcal{E}:\mathcal{X}\rightarrow \mathbb{B}^{\mathsf{N}},$
such that $\mathcal{E(}S,E)=E.$ Let $\mathcal{B}_{A}=\mathcal{B}(X_{A},r_{A})
$ and $\mathcal{B}_{B}=\mathcal{B}(X_{B},r_{B})$ be two open balls of $%
\mathcal{X}$, with $X_{A}=(S_{A},E_{A})$ and $X_{B}=(S_{B},E_{B})$. We are
looking for $\widetilde{X}=(\widetilde{S},\widetilde{E})$ in $\mathcal{B}_{A}
$ such that $\exists n_{0}\in \mathbb{N},G_{f_{0}}^{n_{0}}(\widetilde{X})\in
\mathcal{B}_{B}$.$\widetilde{X}$ must be in $\mathcal{B}_{A}$ and $r_{A}$ can be strictly
lesser than 1, so $\widetilde{E}=E_{A}$. Let $k_{0}=\lfloor \log
_{10}(r_{A})+1\rfloor $. Then $\forall S\in \mathbb{S}$, if $%
S^{k}=S_{A}^{k},\forall k\leqslant k_{0}$, then $(S,\widetilde{E})\in
\mathcal{B}_{A}$. Let us notice $(\check{S},\check{E}%
)=G_{f_{0}}^{k_{0}}(S_{A},E_{A})$ and $c_{1},...,c_{k_{1}}$ the elements of
the set $\{i\in \llbracket1,\mathsf{N}\rrbracket/\check{E}_{i}\neq \mathcal{E%
}(X_{B})_{i}\}.$ So any point $X$ of the set
\[
\{(S,E_{A})\in \mathcal{X}/\forall k\leqslant k_{0},S^{k}=S_{A}^{k}\text{
and }\forall k\in \llbracket1,k_{1}\rrbracket,S^{k_{0}+k}=c_{k}\}
\]
is satisfying $X\in \mathcal{B}_{A}$ and $\mathcal{E}\left(
G_{f_{0}}^{k_{0}+k_{1}}(X)\right) =E_{B}$.
Lastly, let us define $k_2 = \lfloor \log_{10}(r_B)\rfloor +1$. Then $%
\widetilde{X} = (\widetilde{S}, \widetilde{E}) \in \mathcal{X}$ defined by:
* $\widetilde{X} = E_A$,
* $\forall k \leqslant k_0, \widetilde{S}^k = S_A^k$,
* $\forall k \in \llbracket 1, k_1 \rrbracket,$ $\widetilde{S}^{k_0+k} =
* $\forall k \in \mathbb{N}^*, \widetilde{S}^{k_0+k_1+k} = S_B^k$,
is such that $\widetilde{X} \in \mathcal{B}_A$ and $G_{f_0}^{k_0+k_1}(%
\widetilde{X}) \in \mathcal{B}_B$.
§.§.§ Sensitive dependence on initial conditions
$(\mathcal{X},G_{f_0})$ has sensitive dependence on initial conditions.
Banks et al. proved in <cit.> that having sensitive dependence is a consequence of being regular and topologically transitive.
§.§.§ Devaney's Chaos
In conclusion, $(\mathcal{X},G_{f_0})$ is topologically transitive, regular and has sensitive dependence on initial conditions. Then we have the following result:
$G_{f_0}$ is a chaotic map on $(\mathcal{X},d)$ in the sense of Devaney.
We have proven that the set $\mathcal{C}$ of the iterate functions $f$ so that $(\mathcal{X}, G_f)$ is chaotic (according to the definition of Devaney), is a nonempty set. In future work, we will deepen the study of $\mathcal{C}$, among other things, by computing its cardinality and characterizing this set.
§ CHAOS IN A FINITE STATE MACHINE
§.§ The approach presented in this paper
In the section above, it has been proven that discrete chaotic iterations can be put in the field of discrete dynamical systems:
\begin{equation}
\left\{
\begin{array}{l}
x^{0}\in \mathcal{X} \\
x^{n+1} = G_f(x^{n}),
\end{array}%
\right.
\end{equation}
where $(\mathcal{X},d)$ is a metric space and $G_f$ is a continuous function. Thus, it becomes possible to study the topological behavior of those chaotic iterations.
Precisely, it has been proven that if the iterate function is based on the vectorial logical negation $f_0$, then chaotic iterations generate chaos according to Devaney. Therefore chaotic iterations, as Devaney's topological chaos, satisfy: sensitive dependence on the initial conditions, unpredictability, indecomposability, and uniform repartition.
Two major problems typically occur when trying to develop a computer program with chaotic behavior. First, computers have a finite number of states, so the computations always enter into cycles. Second, the properties of chaotic algorithms are inherited from a real chaotic sequence (like a logistic map) and this behavior is lost when computing floating-point numbers (unlike real numbers, floating-point numbers have a finite decimal part). These two problems are solved in this paper due to the two following ideas:
* Chaotic iterations are Mealy machines.
At each iteration, data corresponding to the current strategy <ref> are taken from the outside world, then computations are realized into the memory (the updates of the finite state of the system). The last state is returned after a desired number of iterations. Contrary to the existing points of view, based on a Moore machine, this machine can pass two times in a same state, without continuing the same evolution. Section <ref> explains in detail this original contribution, which allows the realization of true chaos in computers.
* As mentioned above, the strategy $S$ defined in <ref> will not depend on real numbers, but on integers taken from the outside world. We work with the set $\mathcal{X}$ defined in Subsection <ref> which has the cardinality of the continuum. Section <ref> discusses the consequences of dealing with finite strategies in practice.
§.§ A chaotic Mealy machine
The algorithms considered chaotic usually follow the principle of a Moore machine. After having received its initial states, the machine works alone with no interaction with the outside world. Its outputs only depend on the different states of the machine. The main problem is that when a machine with a finite number of states reaches a same state twice, the two following evolutions are identical. Such an algorithm always enters into a cycle. This behavior is highly predictable and cannot be set as chaotic. Attempts to define a discrete notion of chaos have been proposed, but they are not completely satisfactory and are less recognized than the notion of Devaney's topological chaos.
This problem does not occur in a Mealy machine. This finite state transducer generates an output $O$ computed from its current state $E$ and the current value of an input $S$ (Fig. <ref>).
By this accord, even if the machine reaches the same state twice, the corresponding following evolutions may be completely different depending on the values of the inputs. The method presented here is based on such a machine. Indeed, chaotic iterations are a Mealy machine: at each iteration, the computations take into account new inputs (strategies) which are obtained, for example, from the media on which our algorithm applies. Roughly speaking, as the set of all media is infinite, we obtain a finite state machine which can evolve in infinite ways, thus making it possible to obtain a true chaos in computers.
Mealy machine for chaotic algorithms.
A Mealy machine is said to be chaotic if this machine has a chaotic behavior, as expressed by Devaney.
The Mealy machine we used in this document will be the chaotic iterations with $G_{f_0}$ as iterate function. Because these chaotic iterations satisfy the Devaney's definition of chaos, as stated in Section <ref>, we can conclude that our Mealy machine is a chaotic machine.
§.§ The practical case of finite strategies
It is worthwhile to notice that even if the set of machine numbers is finite, we deal in practice with the infinite set of strategies that have finite but unbounded lengths. Indeed, as suggested before, it is not necessary to store all the terms of the strategy in the memory. Only its $n^{th}$ term (an integer less than or equal to $\mathsf{N}$) has to be stored at the $n^{th}$ step, as it is illustrated in the following example. Let us suppose that a given text is input from the outside world into the computer character by character and that the current term of the strategy is computed from the ASCII code of the current stored character. Since the set of all possible texts of the outside world is infinite and the number of their characters is unbounded, we work with an infinite set of finite but unbounded strategies. Of course, the previous example is a simplistic one. A chaotic procedure should to be introduced to generate the terms of the strategy from the stream of characters.In the computer science framework, we also have to deal with a finite set of states of the form $\mathds{B}^\mathsf{N}$ and as stated before an infinite set $\mathbb{S}$ of strategies. The sole difference with the theoretical study is that instead of being infinite the sequences of $S$ are finite with unbounded length.The proofs of continuity and transitivity are independent of the finiteness of the length of strategies (sequences of $\mathbb{S}$). Sensitivity can be proved too in this situation. So even in the case of finite machine numbers, we have the two fundamental properties of chaos: sensitivity and transitivity, which respectively implies unpredictability and indecomposability (see <cit.>, p.50). The regularity supposes that the sequences are of infinite lengths. To obtain the analogous of regularity in the context of finite sets, we define below the notion of periodic but finite sequences.
A strategy $S\in\mathbb{S}$ is said to be periodic but finite if $S$ is a finite sequence of length $n$ and if there exists a divisor $p$ of $n$, $p \neq n$, such that $\forall i \leqslant n-p, S^i = S^{i+p}$. A point $(E,S) \in \mathcal{X}$ is said to be periodic but finite, if its strategy $S$ is periodic but finite.
For example, $(1,2,1,2,1,2,1,2)$ ($p$=2) and $(2,2,2)$ ($p$=1), are periodic but finite. This definition can be interpreted as the analogous of periodicity definition on finite strategies. Following the proof of regularity (Section <ref>), it can be proven that the set of periodic but finite points is dense on $\mathcal{X}$, hence obtaining a desired element of regularity in finite sets, as quoted by Devaney (<cit.>, p.50): “two points arbitrary close to each other could have completely different behaviors, the one could have a cyclic behavior as long as the system iterates while the trajectory of the second could `visit' the whole phase space”. It should be recalled that the regularity was introduced by Devaney in order to counteract the effects of sensitivity and transitivity: two points close to each other can have fundamentally different behaviors.
§ HASH FUNCTIONS BASED ON TOPOLOGICAL CHAOS
§.§ Introduction
The use of chaotic maps to generate hash algorithms has seen several developments in recent years. In <cit.> for example, a digital signature algorithm based on an elliptic curve and chaotic mapping is proposed to strengthen the security of an elliptic curve digital signature algorithm. Other examples of the generation of a hash function using chaotic maps can be found in e.g. <cit.>, <cit.> and <cit.>. However, as for digital watermarking, the use of any chaotic map does not guarantee that the resulting hash function would behave chaotically too. To our knowledge, this point is not discussed in these referenced papers, however it should be considered as important.
We define in this section a new way to construct hash functions based on chaotic iterations. As a consequence of the theory presented before, the generated hash functions satisfy the topological chaos property. Thus, various desired properties in this domain are guaranted by our approach. For example, the avalanche criterion is closely linked to the sensitivity property.
§.§ A chaotic machine for hash functions
In this section, we explain a new way to obtain a hash value of a digital medium described by a binary sequence. It is based on chaotic iterations and satisfies the topological chaos property. The hash value will be the last state of some chaotic iterations: the initial state $X_0$, finite strategy $S$ and iterate function must then be defined.
The initial condition $X_0=\left( S,E\right) $ is composed by a $\mathsf{N} = 256$ bits sequence $E$ and a chaotic strategy $S$. In the following sequence, we describe in detail how to obtain this initial condition from the original medium.
§.§.§ How to obtain $E$
The first step of our algorithm is to transform the message in a normalized 256 bits sequence $E$. To illustrate this step we state that our original text is: “The original text”.
Each character of this string is replaced by its ASCII code (on 7 bits). Then, we add a 1 to this string.
10101001 10100011 00101010 00001101
11111100 10110100 11100111 11010011
10111011 00001110 11000100 00011101
00110010 11111000 11101001
So, the binary value (1111000) of the length of this string (120) is added, with another 1:
10101001 10100011 00101010 00001101
11111100 10110100 11100111 11010011
10111011 00001110 11000100 00011101
00110010 11111000 11101001 11110001
The whole string is copied, but in the opposite direction. This gives:
10101001 10100011 00101010 00001101
11111100 10110100 11100111 11010011
10111011 00001110 11000100 00011101
00110010 11111000 11101001 11110001
00011111 00101110 00111110 10011001
01110000 01000110 11100001 10111011
10010111 11001110 01011010 01111111
01100000 10101001 10001011 0010101
So, we obtain a multiple of 512, by duplicating this string enough and truncating at the next multiple of 512. This string in which the whole original text is contained, is denoted by $D$.
Finally, we split our obtained string into blocks of 256 bits and apply the exclusive-or function, obtaining a 256 bits sequence.
11111010 11100101 01111110 00010110
00000101 11011101 00101000 01110100
11001101 00010011 01001100 00100111
01010111 00001001 00111010 00010011
00100001 01110010 01000011 10101011
10010000 11001011 00100010 11001100
10111000 01010010 11101110 10000001
10100001 11111010 10011101 01111101
So, in the context of Subsection <ref>, $\mathsf{N}=256$ and $E$ is the above obtained sequence of 256 bits.
We now have the definitive length of our digest. Note that a lot of texts have the same string. This is not a problem because the strategy we will build depends on the whole text.
Let us now build the strategy $S$.
§.§.§ How to choose $S$
To obtain the strategy $S$, an intermediate sequence $(u^n)$ is constructed from $D$ as follows:
* $D$ is split into blocks of 8 bits. Then $u^n$ is the decimal value of the $n^{th}$ block.
* A circular rotation of one bit to the left is applied to $D$ (the first bit of $D$ is put on the end of $D$). Then the new string is split into blocks of 8 bits another time. The decimal values of those blocks are added to $(u^n)$.
* This operation is repeated again 6 times.
It is now possible to build the strategy $S$:
\begin{equation*}
S^0 = u^0,~~~
S^n=(u^n+2\times S^{n-1}+n) ~(mod ~256).
\end{equation*}
$S$ will be highly dependent to the changes of the original text, because $\theta \longmapsto 2\theta ~(mod ~1)$ is known to be chaotic as defined by Devaney <cit.>.
§.§.§ How to construct the digest
To construct the digest, chaotic iterations are done with initial state $X^0$,
\begin{equation*}
\begin{array}{rccc}
f: & \llbracket1,256\rrbracket & \longrightarrow & \llbracket1,256\rrbracket
\\
& (E_1,\hdots,E_{256}) & \longmapsto & (\overline{E_1},\hdots,\overline{%
\end{array}%
\end{equation*}
as iterate function and $S$ for the chaotic strategy.
The result of those iterations is a 256 bits vector. Its components are taken 4 per 4 bits and translated into hexadecimal numbers, to obtain the hash value:
To compare, if instead of using the text “The original text” we took “the original text”, the hash function returns:
In this paper, the generation of hash value is done with the vectorial boolean negation $f_{0} $ defined in eq. (<ref>). Nevertheless, the procedure remains general and can be applied with any function $f$ such that $G_f$ is chaotic. In the following subsection, a complete example of the procedure is given.
§.§ Application example
Let us consider the two black and white images of size $64 \times 64$ in Fig. <ref>, in which the pixel in position (40,40) has been changed.
[Original image.]
[Modified image.]
Hash of some black and white images.
In this case, our hash function returns:
for the Fig. <ref> and
for the Fig. <ref>.
Let us consider now the two 256 graylevel images of Lena ($256 \times 256$ pixels) in figure <ref>, in which the grayscale level of the pixel in position (50,50) has been transformed from 93 (fig. <ref>) to 94 (fig. <ref>).
[Original lena.]
[Modified lena.]
Hash of some grayscale level images.
In this case, our hash function returns:
for the left Lena and
for the right Lena.
These examples give an illustration of the avalanche effect obtained by this algorithm. A more complete study of the properties possessed by our hash functions and resistance under collisions will be studied in future work.
§ CONCLUSION
In this paper, a new approach to generate algorithms with chaotic behaviors is proposed. This approach is based on the well-known Devaney's topological chaos. The algorithms which are of iterative nature are based on the so-called chaotic iterations.
This is achieved by establishing a link between the notions of topological chaos and chaotic iterations. This is the first time that such an approach is considered for chaotic iterations. Indeed, we are not interested in stable states of such iterations as it has always been the case in the literature, but in their unpredictable behavior.
After a solid theoretical study, we consider the practical implementation of the proposed algorithms by evaluating the case of finite sets. We study the behavior of the induced computer programs proving that it is possible to design true chaotic computer programs.
A simple application is proposed in the area of hash functions. The security in this case is defined by the unpredictability of the behavior of the proposed algorithm.
The algorithm derived from our approach satisfies important properties of topological chaos such as sensitivity to initial conditions, uniform repartition (as a result of the transitivity), and unpredictability.
The results expected in our study have been experimentally checked. The choices made in this first study are simple: the aim was not to find the best hash function, but to give simple illustrated examples to prove the feasibility in using the new kind of chaotic algorithms in computer science.
In future work, we will investigate other choices of iteration functions and chaotic strategies. We will try to characterize transitive functions. Other properties induced by topological chaos, such as entropy, will be explored and their interest in the information security framework will be shown.
|
1511.00239
|
A. Saleev & N. Nikolaev & F. Rathmann
Systematics limitations in the EDM searches.
Institut für Kernphysik, Forschungszentrum Jülich
Jülich, 52425, Germany,
and Samara State Aerospace University
Samara, 443086, Russia
L. D. Landau Institute for Theoretical Physics
Moscow region, Chernogolovka, 142432, Russia
Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich
Jülich, 52425, Germany
Day Month Year
Day Month Year
Day Month Year
Searches of the electric dipole moment (EDM) at a pure magnetic ring, like COSY, encounter strong background coming from magnetic dipole moment (MDM). The most troubling issue is the MDM spin rotation in the so-called imperfection, radial and longitudinal, B-fields.
To study the systematic effects of the imperfection fields at COSY we proposed the original method which makes use of the two static solenoids acting as artificial imperfections. Perturbation of the spin tune caused by the spin kicks in the solenoids probes the systematic effect of cumulative spin rotation in the imperfection fields all over the ring. The spin tune is one of the most precise quantities measured presently at COSY at $10^{-10}$ level. The method has been successfully tested in September 2014 run at COSY, unravelling strength of spin kicks in the ring's imperfection fields at the level of $10^{-3} rad$.
PACS numbers: 13.40.Em, 11.30.Er, 29.20.Dh, 29.27.Hj
§ EDM SEARCHES AT ALL-MAGNETIC STORAGE RINGS
Spin motion in the pure magnetic ring with a vertical guiding field $\vec{B}$ is governed by T-BMT equation<cit.>
\begin{eqnarray}
\frac{d\vec{S}}{dt}&=&\vec{\Omega}\times\vec{S} \nonumber\\
\vec{\Omega}&=& -\frac{e}{m}\left\{G\vec{B} + \eta\vec{\beta}\times\vec{B}\right\}
\end{eqnarray}
where $G$ is a magnetic anomaly and $\eta=\frac{dm}{e}$ has a meaning of the ratio of the EDM, $d$, to the MDM. The generic signal of the EDM is a rotation of the spin in an electric field. The EDM interacts with the motional electric field, which tilts the stable spin axis,
\begin{equation}
\vec{\Omega}= 2\pi f_R \frac{G\gamma}{\cos\xi}\left\{\vec{e}_x\sin\xi + \vec{e}_y\cos\xi\right\}\, ,
\end{equation}
where $\tan\xi=\frac{\eta\beta}{G}$ and $f_R$ is cyclotron frequency.
In a realistic ring, the MDM spin rotation in the imperfection fields all over the ring adds to the EDM rotation in non-trivial way. For the particle on the closed orbit, the imperfection fields induce the spin kicks which repeat after each turn. The product of consecutive spin rotations all over the ring for one turn gives a rotation around spin closed orbit $\vec{c}$ with spin tune $Q_S$. As a result, spin rotation, given at some point of the ring after $k=f_R t$ turns is defined as:
\begin{equation}\label{eq:otm}
\Psi(k)=\left(e^{-i\vec{\sigma}\cdot\vec{c}\pi Q_S}\right)^k\Psi(0)
\end{equation}
where $\Psi$ is a two-component spinor and the $i$-th component of the spin vector is given by
\begin{equation}
\end{equation}
$\vec{\sigma}$ is a vector of Pauli matricies,
\begin{equation}
\vec{\sigma}\cdot\vec{c}=c_1\sigma_1+c_2\sigma_2+c_3\sigma_3
\end{equation}
where $c_{1,2,3}$ are directional cosines for the vector of spin closed orbit. In case of non-zero EDM and no imperfections present in the ring, $c_1=\sin\xi$, $c_2=\cos\xi$, $c_3=0$, and spin tune $Q_S= \frac{G\gamma}{\cos\xi}$.
To study how the systematic effects from the imperfection fields at COSY induce changes in the spin closed orbit we proposed an original method in which an artificial imperfection fields are manipulated and then specific changes in the value of spin tune are observed.
§.§ JEDI September 2014 run
At September 2014 run, conducted by JEDI (Jülich Electric Dipole moment Investigations) collaboration at COSY, two solenoids from the compensation magnets of the electron coolers have been used as static spin rotators which produce artificial imperfection spin kicks.
Let us consider the case when the solenoids are switched off. They are located at straight sections of the ring opposite to each other (see fig. $\ref{rsketch}$) and effectively split the ring in two halves. Each half is described by the corresponding spin rotation matrix, $\hat{R}_1$ and $\hat{R}_2$:
\begin{eqnarray}
\label{eq:otm1}\hat{R}_1&=&e^{-\frac{i}{2}\vec{\sigma}\cdot\vec{m}~\pi Q_1}\\
\label{eq:otm2}\hat{R}_2&=&e^{-\frac{i}{2}\vec{\sigma}\cdot\vec{n}~\pi Q_2}
\end{eqnarray}
Scheme of the ring cofiguration. The solenoids are located at points 1 and 2. Vectors $\vec{c}$ and $\vec{c}^{\,*}$ define direction of spin closed orbit when solenoids are turned off.
Then the total spin rotation matrix at the point downstream the first solenoid:
\begin{equation}\label{eq:rax}
\hat{R}=\hat{R}_1\hat{R}_2=\exp\left\{-\frac{i}{2}\vec{\sigma}\cdot\vec{c}~2\pi Q_S\right\} ,
\end{equation}
whereas downstream the second solenoid it is different:
\begin{equation}\label{eq:raxx}
\hat{R}=\hat{R}_2\hat{R}_1=\exp\left\{-\frac{i}{2}\vec{\sigma}\cdot\vec{c}^{\,*}2\pi Q_S\right\}.
\end{equation}
The rotation axis, $\vec{c}^{\,*}=c^*_1\vec{e}_x + c^*_2\vec{e}_y + c^*_3\vec{e}_z$, is different from that one in “($\ref{eq:rax}$)", $\vec{c}^{\,*}\neq\vec{c}$ due to non-commuting property of the rotations. The spin tune, $\nu_s=Q_S$, does not depend on the order of rotations and it is the same in both cases.
The biggest contribution to the spin rotation in each half of the ring comes from the vertical guiding field of the main dipoles in the arcs. Vertical shifts of the quadrupoles, inclinations of the dipole fields and other misalignments of magnets produce small horizontal imperfection fields. Spin kicks in the imperfection fields slightly perturb spin rotation in the vertical guiding field, and give rise to the $m_{1,3}$ and $n_{1,3}$ in Eqs. $\ref{eq:otm1},\ref{eq:otm2}$. But misalignments differ in each half of the ring, so the corresponding spin rotations for the two halves are slightly different as well, $\hat{R}_1\approx\hat{R}_2$. In the first order,
\begin{eqnarray}
Q_1&\cong& Q_2\approx G\gamma\\
m_2&\cong& n_2\approx 1.
\end{eqnarray}
When the e-cooler's solenoids are switched on, they produce spin kicks around the longitudinal axis,
\begin{equation}\label{eq:skick}
\chi_i= (1+G)\frac{\int B_i dl}{B\rho}
\end{equation}
where $B\rho$ represents magnetic rigidity of the ring. The magnitude of $\int B dl$ has linear dependence with respect to the applied current $J_1$ or $J_2$ in each solenoid. It is calculated by using corresponding calibration factors $f_1$ or $f_2$:
\begin{equation}
\int B_{i}dl=f_{i}J_{i}
\end{equation}
The spin kicks in the solenoids produce perturbation of the spin tune in the ring, $\nu_s=Q_S+\Delta\nu_s$. The total spin rotation matrix, defined downstream the first solenoid, becomes
\begin{equation}
\hat{R}=\hat{R}_2 \hat{R}_z(\chi_2) \hat{R}_1 \hat{R}_z(\chi_1)
\end{equation}
Using the relation $\cos(\pi\nu_s)=Tr[\hat{R}]/2$, perturbation of the spin tune is given by
\begin{eqnarray}\label{eq:stsh}
\cos(\pi Q_S)&-&\cos(\pi (Q_S+\Delta\nu_s))=\nonumber\\
&&(E+\cos(\pi Q_S))\sin^2\left(\frac{y_+}{2}\right)-\frac{1}{2}(c_3+c^*_3)\sin(\pi Q_S)\sin y_+ \nonumber\\
&-&(E-\cos(\pi Q_S))\sin^2\left(\frac{y_-}{2}\right)+\frac{1}{2}(c_3-c^*_3)\sin(\pi Q_S)\sin y_-
\end{eqnarray}
where the solenoid's spin kicks are defined as
\begin{equation}
\end{equation}
and paramaeter $E$ is
\begin{equation}
E\approx\cos\frac{\pi(Q_1-Q_2)}{2}\approx 1
\end{equation}
a) Expected spin tune behaviour during each measurement. b) Typical output of the spin tune analysis software, here the spin tune shift $\Delta\nu_s\simeq10^{-5}$ has been observed between 20 and 45 seconds.
The experiment consists of multiple spin tune measurements<cit.>. At first, the polarized deuteron beam is prepared for the spin tune measurement: after injection and acceleration to $T=270~MeV$, the beam is cooled, then bunched. Then RF-solenoid flips the initial vertical polarization into the horizontal plane by partial Froissart-Stora scan and time clock starts. All particle spins starts precessing with the spin tune $\nu_s=Q_S$ in the horizontal plane. After 20 seconds the static solenoids are switched on at specified currents $J_1$ and $J_2$. The spin tune becomes $\nu_s=Q_S+\Delta\nu_s(\chi_1,\chi_2)$. To suppress systematic effects related to the drift of the spin tune during the cycle, at 45 seconds the solenoids are switched off and $\nu_s=Q_S$ again (see fig. $\ref{fstep}$). Special data analysis software determines the spin tune shift $\Delta\nu_s$ and base spin tune $Q_S$ with the precision of $\sim10^{-10}$. To maintain a long spin coherence time of horizontal polarization, the sextupole magnets in the ring have been set up at specified settings which compensate decoherence effects from emittance and momentum spread of the beam<cit.>.
Spin tune map. Each white dot represent single measurement of $\Delta\nu_s(y_+,y_-)$, the surface is a fit to the dataset. Error bars are smaller than the size of the symbols. Note that weak parabolic dependence in $y_-$ has negative curvature.
The applied currents for the solenoids have been chosen such that corresponding spin kicks $\chi^i_1$ and $\chi^j_2$ form a rectangular mesh of $i\times j$ points.
According to "Eq.$\ref{eq:stsh}$", the functional form of $\Delta\nu_s(y_+,y_-)$ is a sum of two, concave and convex, parabolas:
\begin{equation}
\Delta\nu_s(y_+,y_-)\sim\pm(y_\pm-a_\pm)^2
\end{equation}
and the extremum, the saddle point, is located at
\begin{eqnarray}
a_+&=&\sin(\pi Q_S)\frac{c_3+c^*_3}{1+\cos(\pi Q_S)} \\
a_-&=&\sin(\pi Q_S)\frac{c_3-c^*_3}{1-\cos(\pi Q_S)}
\end{eqnarray}
where $c_3$ and $c^*_3$ are the longitudinal projections of spin closed orbit at the points 1 and 2 of the ring, as defined in "Eq. $\ref{eq:rax}$" and "Eq. $\ref{eq:raxx}$".
The procedure of the spin tune measurements with different strength of the spin kicks in static spin rotators can be referred as “spin tune mapping". As a matter of fact, it provides a partial determination of the stable spin axis at the point of the applied spin kick, which in turn, gives a hint on the magnitude of the imperfection spin kicks in the ring.
To recover a saddle point in the spin tune map with two solenoids, the function “($\ref{eq:stsh}$)" was fit to data points $\Delta\nu_s(y_+,y_-)$. Fig. $\ref{map}$ shows one of the maps that has been measured during September 2014 run. The saddle point is located at $a_+=-0.00111077\pm 6.1*10^{-8}~rad$ and $a_-=-0.00244326\pm 2.05*10^{-7}~rad$, which defines $c_3=-0.00299124\pm 1.8*10^{-7}$ and $c^*_3=-0.00163653\pm 7.1*10^{-8}$. The results for $a_\pm$ are the preliminary ones, more scrutiny of the impact of possible misalignment of the solenoid axes is in order.
The method also enables to measure $c_1$ and $c^*_1$ if the static Wien filters operating as the MDM rotators with the radial rotation axis are used alongside the solenoids<cit.>. Part of the September 2014 run has also applied this method to studies of the systematic effects coming from the steering magnets<cit.>.
§ SUMMARY AND OUTLOOK
The spin tune mapping emerges as a tool to determine the spin closed orbit with an unprecedented accuracy. In the present experiment only the longitudinal projection of the stable spin axis has been measured, but adding a static Wien filter next to the solenoid would enable to determine the radial projection of the stable spin axis as well. We anticipate that the spin tune mapping technique would prove most useful in a calibration of various devices to be employed in the high precision EDM searches at all magnetic rings.
jpap F. Rathmann et al, J. Phys.: Conf. Ser. 012011, 447 (2013).
llmnfA. Lehrach, B. Lorentz, W. Morse, N. Nikolaev, and F. Rathmann, arXiv:1201.5773.
autbk S. Y. Lee, Spin Dynamics and Snakes in Synchrotrons. (World Scientific, 1997).
publ D. Eversmann et al, Precise Spin Tune Measurement in a Storage Ring, in print.
proed E. Stephenson, Polarimetry for Stored Polarized Hadron Beams, Proc. of SPIN'14, (2015).
seb S. Mey, A Novel E$\times$B Spin Manipulator at COSY, Proc. of SPIN'14, (2015).
stas S. Checkmenev, Estimation of Systematic Errors for Deuteron Electric Dipole Moment at COSY, Proc. of SPIN'14, (2015).
|
1511.00462
|
How does a star cluster of more than few 10,000 solar masses form? We present the case of the cluster NGC 346 in the Small
Magellanic Cloud, still embedded in its natal star-forming region N66, and we propose a scenario for its formation, based on observations of
the rich stellar populations in the region. Young massive clusters host a high fraction of early-type stars, indicating an extremely high star formation efficiency.
The Milky Way galaxy hosts several young massive clusters that fill the gap between
young low-mass open clusters and old massive globular clusters. Only a handful, though, are young enough to study their formation.
Moreover, the investigation of their gaseous natal environments suffers from contamination by the Galactic disk. Young massive clusters are very abundant in distant starburst and interacting galaxies, but the
distance of their hosting galaxies do not also allow a detailed analysis of their formation. The Magellanic Clouds, on the other hand, host young massive clusters in a wide
range of ages with the youngest being still embedded in their giant HII regions. Hubble Space Telescope imaging of such star-forming
complexes provide a stellar sampling with a high dynamic range in stellar masses, allowing the detailed study of star formation at scales typical for
molecular clouds. Our cluster analysis on the distribution of newly-born stars in N66 shows that star formation in the region proceeds in a clumpy hierarchical fashion, leading to the formation of both a dominant young massive cluster, hosting about half of the observed pre–main-sequence population, and a self-similar dispersed distribution of the
remaining stars. We investigate the correlation between stellar surface density (and star formation rate derived from star-counts) and molecular gas surface density (derived
from dust column density) in order to unravel the physical conditions that gave birth to NGC 346.
A power law fit to the data yields a steep correlation between these two parameters with a considerable scatter. The fraction of stellar over the total (gas plus young stars) mass is found to be
systematically higher within the central 15 pc (where the young massive cluster is located) than outside, which suggests variations in the star formation
efficiency within the same star-forming complex. This trend possibly reflects a change of star formation efficiency in N66 between clustered and non-clustered star formation.
Our findings suggest that the formation of NGC 346 is the combined result of star formation regulated by turbulence and of early
dynamical evolution induced by the gravitational potential of the dense interstellar medium.
§ INTRODUCTION
It is generally accepted that most stars form in a clustered mode, i.e., in gravitationally bound concentrations.
Star-forming clusters vary in size, mass, and stellar content, from small compact groups of protostars, still embedded in their natal
star-forming regions (<cit.>), to large Young Massive Clusters (YMCs) that cover more than about $10^4$ M$_\odot$
(<cit.>). The latter, being fundamental contributors to the stellar mass budget of galaxies,
are valuable laboratories for the investigation of clustered star formation and subsequent evolution. Star clusters themselves are generally not
formed in isolation. They are the densest assemblings of larger stellar aggregates and complexes <cit.>,
within a hierarchy of stellar structures, which extends up to galactic scales (e.g., <cit.>).
There is only a handful of YMCs close enough to be resolved into their stellar populations. The study of those located in the Milky Way is limited by
the line-of-site contamination by the Galactic disk, while in the Magellanic Clouds deep multi-band photometric surveys allow the study of star formation
across the full extent of YMCs natal environments <cit.>. In this study we explore the formation of
the most massive young stellar cluster in the Small Magellanic Cloud, NGC 346, located in the star-forming complex LHA 115-N66 (in short
N66), the brightest HII region in this galaxy. We exploit the exceptionally rich sample of young stars collected with the Hubble Space Telescope
(HST) in N66 to address the formation of a YMC from two aspects: (1) The clustering of stars across the length-scale of a giant molecular cloud
where the YMC forms (<cit.>), and (2) the relation between the star formation rate and the environmental
conditions of the star-forming complex (<cit.>).
The observed ACF of the young stellar population in N66 (solid black line), and the set of
twelve modeled ACFs (light-grey area), which correspond to the best-representative
simulated mixed distributions that assume a central compact stellar component and an extended fractal
one. The ACF of the best-matching model is plotted with a short dashed line. The vertical dashed line corresponds
to the scale beyond which edge-effects hamper the analysis. Plot from
§ THE CLUSTERING OF YOUNG STARS ACROSS N66
The young population in NGC 346 consists of low-mass pre–main-sequence stars, identified from their positions in
the color-magnitude diagram (<cit.>), and high-mass upper–main-sequence
stars (with $m_{\rm 555}-m_{\rm 814} \leq$ 0.0 mag; 12 $\lesssim m_{\rm 555} \lesssim$ 17 mag),
compiling a total sample of 5,150 stars. An age of $\sim$ 5 Myr has been established for these stars by
<cit.>. We quantify the degree of clustering of the stars by using the autocorrelation function
(ACF), based on <cit.>. For a two-dimensional self-similar
distribution this function, $1+\xi$, has a power-law dependency to separation $r$ of the form $1+\xi (r) \propto r^{\eta}$, with the
exponent $\eta$ being related to the 2D fractal dimension as $D_{2} = \eta +2$. The ACF of the young
stellar populations of N66 does not follow a single power-law but it has two distinct parts (Fig. <ref>), and therefore
the stellar distribution is not entirely hierarchical.
The broken power-law suggests a complex distribution of stars in N66, which is influenced by the
star cluster NGC 346. In order to interpret this behavior we applied numerical simulations of centrally-concentrated stellar clusters, following
an <cit.> surface density profile, and of three-dimensional fractal stellar distributions, constructed using a
box-counting technique. From the comparison of the ACFs of synthetic distributions and their combinations we find that the ACF of N66 is best
reproduced as the composite of two distinct spatial distributions; a centrally-condensed cluster with a core radius of $\sim$ 2.5 pc and a profile
index $\gamma \simeq$ 2.27, and a self-similar stellar distribution with a 3D fractal dimension $D_3\approx$ 2.3 (Fig. <ref>). About 40%
of the total young stellar population belongs to the cluster, while the remaining is spread across the whole extent of the observed star-forming
complex. The separation of $\sim$ 6 pc where the ACF power-law breaks indicates the length-scale where the clustering of stars
changes from one pattern to the other.
Our findings confirm the appearance of substructure in the region, established in a previous study of ours <cit.>,
as the result of hierarchical stellar distribution. The fractal dimension $D_3 \simeq 2.3$, derived for the self-similar stellar component, fits very well
to the value established from numerical experiments of supersonic isothermal turbulence (<cit.>), implying that the
observed hierarchy is inherited from the turbulent interstellar gas. On the other hand, numerical simulations find a higher
degree of clumping for sink particles formed from turbulent molecular clouds ($D_3 \sim$ 1.6; <cit.>).
The differences in the derived fractal dimensions may reflect discrepancies between the methods used for their measurement,
but they may as well demonstrate that stars have a different spatial distribution than the gas from which they formed. Even if stars were
originally distributed according to a turbulent-driven hierarchy, N-body simulations suggest that a moderate amount of dynamical interactions will partly
erase substructure, preventing a star-forming region from retaining a strong signature of the primordial ISM distribution (<cit.>). The substructure in N66 is unlikely to have formed after the onset of star formation, and it is therefore
the observed fractal dimension is an upper limit to the primordial value. Dynamics may have facilitated the formation of the
YMC within the original self-similar distribution of stars. Kinematic studies in the region will certainly shed light to the importance of dynamical
evolution in the formation of NGC 346.
In conclusion, our findings suggest that star formation in N66 takes place in a clustered, as well as a dispersed mode, revealed by the bimodal – centrally-condensed and self-similar –
clustering of newly-born stars in the region <cit.>.
§ THE RELATION BETWEEN GAS AND YOUNG STARS IN N66
To understand the formation of NGC 346 in relation to the surrounding interstellar medium (ISM), we examine the ISM properties
as a function of the surface density of young stars across N66 (<cit.>). The young stellar sample used
for the cluster analysis discussed in the previous section is also used for the derivation of the surface density of the star formation rate
($\Sigma_{\rm SFR}$) through star counts. We employ the “dust method” to determine the ISM column density, i.e. using the infrared (IR) to submillimetre (submm) dust continuum emission to derive via radiative transfer modeling of its spectral energy
distribution (SED) the dust surface density, which we transform into gas surface density ($\Sigma_{\rm gas}$). This method has the advantage of not being sensitive to the state of the gas (ionized, neutral or molecular) as long as dust and gas are well mixed. This is important because at the age of the young stars in N66 the parental molecular cloud may have been subject to significant photodissociation and using a molecular gas tracer (such as CO) may lead to an incomplete view of the total gas surface densities.
In our analysis, we use IR-to-submm photometric maps from Spitzer (<cit.>) and Herschel (<cit.>), which we convolve to the lowest available resolution of 20$^{\prime\prime}$, corresponding to the SPIRE 250 $\mu$m beam. We
have also convolved the stellar density map, which has an exceedingly good intrinsic angular resolution, to this effective spatial resolution. After convolution, all maps
have been re-projected on the same pixels scheme with pixel size of 20$\times$20 arcsec$^{2}$ in order to have spatially independent measurements and to allow pixel-by-pixel
SED extraction and comparison between the stellar densities and derived dust surface densities. We thus study the small-scale correlation between newly formed stars and the gas reservoir in 115 pixels (of $\sim$ 6 pc size), while covering the entire star-forming complex. The conversion of dust to gas surface density was made with the use of a
gas-to-dust ratio ($r_{\rm gd} =$ 1250), calibrated using gas tracers on large scales. The SFR map is derived from the observed number of young stars in each pixel of the stellar density map using a mass per detected star of 4.3 M$_{\odot}$, derived
using the stellar Initial Mass Function of the region (<cit.>), and a star formation duration of 5 Myr <cit.>. This SFR from star-counts is found to be more reliable than indirect SFR tracers (e.g., H$\alpha$, 24 $\mu$m emission) on parsec scales. We show that these tracers break down on small scales ($<$ 100 pc) by up to a factor of 10.
Scatter plot of surface density of young stars versus surface density of dust (and the corresponding SFR and gas surface density)
per pixel of 20$\times$20 arcsec$^{2}$ (6$\times$6 pc$^{2}$) in the star-forming complex N66. The blue points have a strong warm dust
component in their SED ($\Sigma_{24\,{\mu}{\rm m}}$/$\Sigma_{250\,{\mu}{\rm m}}$ [F$_\nu$/F$_\nu$] $>$ 0.3). The lines-of-sight selected
using this color criterion clearly occupy a separate region of the diagram. The dashed and dotted lines are the power-law fits to the data
with or without the warm pixels, respectively. The blue areas correspond to the 2$\sigma$ uncertainties of the best-fitting exponents, which are
given in parentheses. There is a clear correlation between $\Sigma_{\rm SFR}$ and $\Sigma_{\rm gas}$ with some considerable scatter,
suggesting a dependence of the SFR to the density of the gas reservoir. Plot from <cit.>.
A correlation between the stellar surface density, $\Sigma_{\star}$, and the dust surface density, $\Sigma_{\rm dust}$, and between the corresponding
parameters, $\Sigma_{\rm SFR}$ and $\Sigma_{\rm gas}$, is found with considerable scatter (Fig. <ref>). A power-law fit to the data yields a
steep relation with an exponent of 2.6 $\pm$ 0.2. We find that sight-lines towards the central $\lesssim$ 15 pc, where NGC 346 is located, exhibit
systematically high values of $\Sigma_{\star}$/$\Sigma_{\rm dust}$ by a factor of $\sim$5 compared to the rest of the complex <cit.>.
We investigate the variations in terms of the mass fraction of young stars, $f_{\star} = \Sigma_{\star}/\left(\Sigma_{\star}+\Sigma_{\rm gas}\right)$,
across the star-forming complex. The spatial distribution of the observed stellar mass fraction
shows that it is systematically high (with a maximum of 15 per cent) within the central 15 pc of the complex, where the YMC is located (Fig. <ref>). These very high values
correspond to very high stellar surface densities, but not to low gas column densities. The stellar mass fraction becomes lower outside the central 15 pc,
across the remaining surveyed area (66 per cent of the pixels), where it has values systematically below 2 per cent. There is a clear monotonic decrease in
$f_{\star}$ with radial distance from the center, which can be approximated by a power law with exponent $-0.7$. These variations show also a clear dependence
of $f_{\star}$ on other measurable quantities, such as the surface density of hot dust emission at 24 $\mu$m. These findings indicate clearly a higher SFE in the
inner 15 pc area, which is dominated by the young stars belonging to the YMC, while the outer parts, characterized by lower SFE, are those dominated
by the dispersed self-similar distribution of young stars (Sect. 2).
Map of N66 showing the spatial distribution of the derived stellar mass fraction.
The size of each point is dependent on $f_{\star}$ and its color depends on $\Sigma_{24\,{\mu}{\rm m}}$.
The mass fraction is indicated in each point. Stellar mass fraction clearly peaks on the stellar cluster where
the dust is heated to higher temperatures due to the intense radiation field. The warm pixels identified in
Fig. <ref> are encircled. Plot from <cit.>.
Our analysis allow us to place these measurements on a Schmidt-Kennicutt (SK) diagram. We find that individual pixels fall systematically above the
fiducial SK relation for $\Sigma_{\rm SFR}$ versus $\Sigma_{{\rm HI}+{\rm H}_{2}}$ for integrated disk-galaxies <cit.>
by on average a factor of $\sim$7.
This behavior is consistent with the results by <cit.>, who found that the measured SFR lies above the galaxy
averages when `zooming-in' to parsec-scale Galactic star-forming clumps. This is probably caused by less dense gas, which is inefficient in
forming stars, and which is included in the galaxy-scale averages but not measured when `zooming-in' on individual clumps.
In our analysis, on the other hand, even though we analyze 6$\times$6 pc$^{2}$ sized regions, we cover the complete star-forming
complex and find that the entire region (of $\sim$50 pc), being quite active, lies consistently above the SK relation. For N66 one should `zoom-out'
beyond 50 pc to probe this less dense, non-star-forming gas. Indeed, averaging over a larger area (90 pc in radius) we derive a measurement
for N66 that lies closer to the SK-relation, but which still remains high by a factor of $\sim$3.
In conclusion, we find an above-average star formation activity in N66, and that the observed correlations between stellar and ISM properties reflect a change in star formation efficiency between clustered and non-clustered star-formation
within the same star-forming complex <cit.>.
Based on observations made with NASA/ESA Hubble Space Telescope,
obtained from the data archive at the Space Telescope Science Institute (STScI),
Spitzer Space Telescope, Herschel and Atacama Pathfinder Experiment
(APEX). STScI is operated by the Association of Universities for Research in
Astronomy, Inc. under NASA contract NAS 5-26555. Herschel is an ESA
space observatory with science instruments provided by European-led Principal
Investigator consortia and with important participation from NASA. Spitzer Space
Telescope is operated by the Jet Propulsion Laboratory, California Institute of
Technology under a contract with NASA. APEX is a collaboration between the
Max-Planck-Institut für Radioastronomie, the European Southern Observatory,
and the Onsala Space Observatory. The authors kindly acknowledge support from the
German Research Foundation (DFG) through the individual grants GO 1659/3-1
and GO 1659/3-2,
and the collaborative research project SFB881 “The Milky Way System” (subprojects B1,
B2, and B5) respectively.
Elmegreen B. G., 2011, in: C. Charbonnel & T. Montmerle (eds.), Star Formation in the Local Universe,
EAS Publications Series Vol. 51 (Cambridge: Cambridge Univ. Press), p. 31
[Elson, Fall & Freeman (1987)]eff87
Elson, R. A. W., Fall, S. M., & Freeman, K. C. 1987, ApJ, 323, 54
[Federrath et al.(2009)]federrath09
Federrath, C., Klessen, R. S., & Schmidt, W. 2009, ApJ, 692, 364
[Girichidis et al.(2012)]girichidis12
Girichidis, P., Federrath, C., Allison, R., et al. 2012, MNRAS, 420, 3264
[Goodwin & Whitworth(2004)]goodwin04
Goodwin, S. P., & Whitworth, A. P. 2004, A&A, 413, 929
[Gordon et al.(2011)]gordon11
Gordon, K. D., Meixner, M., Meade, M. R., et al. 2011, AJ, 142, 102
Gouliermis, D. A. 2012, Space Sci. Revs, 169, 1
[Gouliermis et al.(2006)]gouliermis06
Gouliermis, D. A., Dolphin A. E., Brandner W., Henning T., 2006, ApJS, 166, 549
[Gouliermis et al.(2010)]gouliermis10 Gouliermis, D. A.,
Schmeja, S., Klessen, R. S., et al. 2010, ApJ, 725, 1717
[Gouliermis et al.(2014)]gouliermis14 Gouliermis, D. A.,
Hony, S., & Klessen, R. S. 2014, MNRAS, 439, 3775
[Gouliermis et al.(2015)]gouliermis15 Gouliermis, D. A.,
Thilker, D., Elmegreen, B. G., et al. 2015, MNRAS, 452, 3508
[Heiderman et al.(2010)]heiderman10
Heiderman, A., Evans, N. J., II, Allen, L. E., Huard, T., & Heyer, M. 2010, ApJ, 723, 1019
[Hony et al.(2010)]hony2010
Hony, S., Galliano, F., Madden, S. C., et al. 2010, A&A, 518, L76
[Hony et al.(2015)]hony2015 Hony, S., Gouliermis,
D. A., Galliano, F., et al. 2015, MNRAS, 448, 1847
[Kennicutt & Evans(2012)]kennicutt12
Kennicutt, R. C., & Evans, N. J. 2012, ARAA, 50, 531
[Lada & Lada(2003)]ladalada03
Lada, C. J., & Lada, E. A. 2003, ARAA, 41, 57
[Meixner et al.(2013)]2013AJ....146...62M Meixner, M., Panuzzo,
P., Roman-Duval, J., et al. 2013, AJ, 146, 62
[Mokiem et al.(2006)]mokiem06
Mokiem, M. R., de Koter, A., Evans, C. J., et al. 2006, A&A, 456, 1131
[Parker et al.(2014)]parker14 Parker, R. J., Wright,
N. J., Goodwin, S. P., & Meyer, M. R. 2014, MNRAS, 438, 620
Peebles, P. J. E. 1980, The large-scale structure of the universe (Princeton Univ. Press)
[Portegies Zwart et al.(2010)]portegieszwart2010
Portegies Zwart, S. F., McMillan, S. L. W., & Gieles, M. 2010, ARAA, 48, 431
[Sabbi et al.(2008)]sabbi08
Sabbi, E., Sirianni, M., Nota, A., et al. 2008, AJ, 135, 173
[Schmeja et al.(2009)]schmeja09
Schmeja, S., Gouliermis, D. A., & Klessen, R. S. 2009, ApJ, 694, 367
|
1511.00547
|
*ld_thm:mixTheorem <ref>
Università degli Studi di Roma Tor Vergata, Dipartimento di
Matematica, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Complex Fourth Moment Theorems]Fourth Moment Theorems for complex
Gaussian approximation
Research was supported by ERC grant 277742 Pascal
We prove a bound
for the Wasserstein distance between vectors of smooth complex random
variables and complex Gaussians in the framework of complex Markov diffusion
generators. For the special case of chaotic
eigenfunctions, this bound can be expressed in terms of certain fourth
moments of the vector, yielding a quantitative Fourth Moment Theorem for
complex Gaussian approximation on complex Markov diffusion chaos. This extends
the results of <cit.>
and <cit.> for the real case. Our main ingredients
are a complex version of the so called $\Gamma$-calculus and Stein's method
for the multivariate complex Gaussian distribution.
[2000]60F05, 60J35, 60J99
§ INTRODUCTION
Let $F_n$ be a sequence of random variables and $\gamma$ be a probability
measure. A Fourth Moment Theorem for $F_n$ holds, if there exists a
polynomial $P=P(m_1(F_n),\dots,m_4(F_n))$ in the first four moments of $F_n$
such that $P(m_1(F_n),\dots,m_4(F_n)) \to 0$ characterizes, or at least implies
the convergence of the laws of $F_n$ towards $\gamma$. The first discovery of
such a Fourth Moment Theorem dates back to 2005, when Nualart and Peccati, in
their seminal paper <cit.>, characterized
convergence in distribution of a sequence $F_n$ of multiple Wiener-Itô
integrals towards a Gaussian distribution by the convergence of the moment
sequence $m_4(F_n)-3m_2(F_n)^2$. The next developments were an extension to the
multidimensional case by Peccati and Tudor in <cit.>, the
introduction of Malliavin calculus by Nualart and Ortiz-Latorre
in <cit.> and, starting in <cit.> by
Nourdin and Peccati and then followed by numerous other contributions of the
same two authors and their collaborators, the combination of Malliavin calculus
and Stein's method to obtain quantitative central limit theorems in strong
distances. This Malliavin-Stein method has found widespread applications, for
example in statistics, mathematical physics, stochastic geometry or free
probability (see the webpage <cit.> for an
overview). For a textbook
introduction to the method, we refer to <cit.>.
Recently, initiated by Ledoux in <cit.> and then further
refined by Azmoodeh, Campese and Poly in <cit.>, it was
shown that Fourth Moment Theorems hold in the very general framework of Markov
diffusion generators (see <cit.> for an
exhaustive treatment
of this framework), in which the aforementioned setting of multiple
Wiener-Itó integrals appears as the special case of the Ornstein-Uhlenbeck
generator. This abstract point of view not only allows for a drastically
simplified proof of Nualart and Peccati's classical Fourth Moment Theorem, but
also provides
new Fourth Moment Theorems for previously uncovered structures such as Laguerre
and Jacobi chaos and new target laws (such as the Beta
distribution). We refer
to <cit.> for details. Stein's method continues
to work in this framework as well and can be used to associate
quantitative estimates to these results. The
abstract framework also led to new results for the Ornstein-Uhlenbeck setting,
such as an “even-moment theorem” (see <cit.>) and
advances regarding the Gaussian product conjecture
(see <cit.>).
In this paper, we extend this abstract diffusion generator framework to the
complex case, thus covering complex valued random variables.
Our main result is a quantitative bound between the Wasserstein distance of a
multivariate complex Gaussian random vector and vectors of square integrable
complex random variables in the domain of a carré du champ operator associated
to a diffusive Markov generator (all unexplained terminology will be
below). For the case of chaotic
eigenfunctions, this bound can be expressed purely in terms of the second and
fourth absolute moments of the vector and thus yields a quantitative complex
Fourth Moment Theorem. To obtain these results, we extend Stein's method to
cover the complex Gaussian distribution and develop a complex version of the so
called $\Gamma$-calculus, in which a central role will be played by the Wirtinger
derivatives $\partial_z$ and $\partial_{\overline{z}}$.
Of course, from a purely algebraic point of view, our approach is
equivalent to the multidimensional real case, which has been treated
in <cit.>, and indeed one could handle
sequences of complex valued random variables by separating real and
imaginary part and considering them as two-dimensional real vectors. For the
complex Ornstein-Uhlenbeck generator, such an ad-hoc strategy has been followed
in <cit.> and <cit.>.
However, by staying completely inside the complex domain, our approach has the
advantage of making available many powerful tools connected to concepts such as
holomorphy from complex analysis which have no equivalents in the real
case. Although not needed for an abstract derivation of our main results, these
tools might be useful in the future, also when taking the reverse route and
trying to prove results for the real case by translating them to the complex
domain. Even for the special case of the complex Ornstein-Uhlenbeck
generator, there is much structure present in the complex domain, such as a
fine decomposition of the eigenspaces as shown in Example <ref> or the
unitary group from complex White noise analysis
(see <cit.>), which has no real counterparts.
Complex random variables are encountered naturally in many applications and
indeed this paper was motivated by the study of spin random fields arising in
cosmology (see for
example <cit.>). A
followup paper with an application of our results to this case is in
The rest of this paper is organized as follows. In
Section <ref>, we introduce the notation used throughout the
rest of this paper and provide some necessary background material. The abstract
complex diffusion generator framework is presented in
Section <ref>, while our main results are stated
and proved in
Section <ref>.
§ PRELIMINARIES
§.§ Wirtinger calculus
Let $x,y\in \R$ be two real variables and define the complex variables $z=x +
\mi y$ and $\overline{z} = x-\mi y$, where $\mi^2=-1$. Then, every function
$\widetilde{f}=\widetilde{f}(x,y) \colon \R^{2} \to \C$ can be considered as a
function $f=f(z) \colon \C \to \C$ via the identity
\begin{equation}
\label{eq:7}
\widetilde{f}(x,y) = \widetilde{f} \left( \frac{z+\overline{z}}{2},
\frac{z-\overline{z}}{2} \right) = f(z).
\end{equation}
Conversely, every function $f=f(z) \colon \C \to \C$ can be considered
as a function $\widetilde{f}=\widetilde{f}(x,y) \colon \R^{2} \to \C$ by writing
\begin{equation}
\label{eq:10}
f(z) = f(x+ \mi y) = \widetilde{f}(x,y).
\end{equation}
With this notation, the Wirtinger derivatives are defined as
\begin{equation*}
\partial_z = \frac{1}{2} \left( \partial_x - \mi \partial_y \right)
\qquad \text{and} \qquad
\partial_{\overline{z}} = \frac{1}{2} \left( \partial_x + \mi \partial_y
\right),
\end{equation*}
where, here and in the following, we use the shorthand
$\partial_{a} = \frac{\partial}{\partial a}$, and, more generally,
$\partial_{a_1 a_2 \dots a_d} =
\frac{\partial^d}{\partial_{a_{1}} \partial_{a_2} \dots \partial_{a_d}}$ to
denote derivatives with respect to the variables $a_1,\dots,a_d$ (the concrete
interpretation as a partial or Wirtinger derivative will always be
clear from the context).
Starting from the Wirtinger derivatives $\partial_z$ and $\partial_{\overline{z}}$,
one can get back the partial derivatives $\partial_x$ and $\partial_y$ by the
\begin{equation*}
\partial_x = \partial_z + \partial_{\overline{z}}
\qquad \text{and} \qquad
\partial_y = \mi \left( \partial_z - \partial_{\overline{z}} \right).
\end{equation*}
It is straightfoward to see that both Wirtinger derivatives are
linear and merit their names by satisfying the derivation properties (product rules)
\begin{equation*}
\partial_z (fg) = (\partial_z f) g + f \partial_z g
\qquad \text{and} \qquad
\partial_{\overline{z}} (fg) = (\partial_{\overline{z}} f) g + f \partial_{\overline{z}} g,
\end{equation*}
Moreover, the conjugation identities
\begin{equation*}
\overline{\partial_z f} = \partial_{\overline{z}} \overline{f}
\qquad \text{and} \qquad
\overline{\partial_{\overline{z}} f } = \partial_z \overline{f}.
\end{equation*}
hold. The chain rules take the form
\begin{align*}
\partial_z (f \circ g) &= \left((\partial_z f) \circ g \right) \partial_z g +
\left( \left( \partial_{\overline{z}} f \right) \circ g \right) \partial_z
\overline{g}, \\
\partial_{\overline{z}} (f \circ g)
\left( \left( \partial_z f \right) \circ g
\right) \partial_{\overline{z}} g
\left( \left( \partial_{\overline{z}} f \right)
\circ g
\right) \partial_{\overline{z}} \overline{g}.
\end{align*}
In particular, we see that $\partial_z z
= 1$, $\partial_z \overline{z}=0$, $\partial_{\overline{z}} \overline{z}=1$ and
$\partial_{\overline{z}} z = 0$, which, in view of the chain and product rule,
allows to formally treat $z$ and $\overline{z}$ as if they were independent
variables when differentiating. Heuristically, when applying the
Wirtinger derivatives to a function $f \colon \C \to \C$, one does not need to
consider $f(z)$ as a function
$\widetilde{f}(x,y)$ and then compute the partial derivatives with respect to
$x$ and $y$, but can instead directly apply the formal rules of differentiation
to the complex variables $z$ and $\overline{z}$. For example, we have
for $p,q \neq -1$ that
$\partial_z (z^p \overline{z}^q)=pz^{p-1}\overline{z}^q$ and
$\partial_{\overline{z}} (z^p\overline{z}^q) = qz^p\overline{z}^{q-1}$.
In the sequel, we will work in general dimension $d \geq 1$, considering
functions $f \colon \C^d \to \C$ and $\widetilde{f} \colon \R^{2d} \to \C$ which
continue to be related through (<ref>) and (<ref>), where now
$x=(x_1,\dots,x_d)$, $y=(x_1,\dots,x_d)$ and $z=(z_1,\dots,z_d)$ are vectors of
variables. For these functions, we define the gradients $\nabla f$ and
$\overline{\nabla}f$ by
\begin{equation*}
\nabla f = \left( \partial_{z_1} f, \partial_{z_2} f, \dots, \partial_{z_d} f
\right)^{T}
\end{equation*}
\begin{equation*}
\overline{\nabla} f = \left( \partial_{\overline{z}_1}
f, \partial_{\overline{z}_2} f, \dots, \partial_{\overline{z}_d} f \right)^T,
\end{equation*}
and the complex Hessians $\nabla\nabla f$,
$\overline{\nabla\nabla} f$, $\overline{\nabla}
\nabla f$ and $\nabla \overline{\nabla} f$ by
\begin{align*}
\nabla\nabla f &= \left( \partial_{z_jz_k} f
\right)_{1 \leq j,k \leq d},
\overline{\nabla\nabla} f &= \left( \partial_{\overline{z}_j \overline{z}_k} f
\right)_{1 \leq j,k \leq d},
\\
\overline{\nabla} \nabla f &= \left( \partial_{\overline{z}_j z_k} f \right)_{1
\leq j,k \leq d}
&\text{and \qquad}
\nabla \overline{\nabla} f &= \left( \partial_{z_j \overline{z}_k} f \right)_{1
\leq j,k \leq d}.
\end{align*}
With some abuse of notation, we say that $f$ is an element of $\mathcal{C}^{m}(\C^d)$, if
the associated function $\widetilde{f}$ belongs to
$\mathcal{C}^m(\R^{2d})$. Similarly, a function $f$ has bounded Wirtinger
derivatives up to some order $m$, if the associated function $\widetilde{f}$ has
bounded partial derivatives up to this order.
§.§ The complex normal distribution
Given a probability space $\mathcal{P}=(\Omega,\mathcal{F},P)$ and two
real-valued random variables $X$ and $Y$, the quantity $Z=X+\mi Y$ is called a
complex valued random variable. The characteristic function, law and, if it
exists, density of $Z$ are defined as being the corresponding quantities of
the two-dimensional real random vector $(X,Y)$. With the notation of the
previous subsection, we clearly have that if $f(z)$ is the density of $Z$, then
$\widetilde{f}(x,y)$ is the density of $(X,Y)$, and the analogous statement is
true for the law. From the characteristic function $\widetilde{\rho}(\xi,\upsilon)$
of $(X,Y)$, we readily calculate the characteristic function
$\rho(\zeta)=\rho(\xi+ \mi \upsilon)$ of $Z$ to be
\begin{equation*}
\widetilde{\rho}(\zeta) = \Ex{\me^{\mi \mathfrak{Re} (\left\langle \zeta,Z \right\rangle_{\C^d})}},
\end{equation*}
where, here and in the following, $\on{E}$ denotes mathematical expectation and
$\mathfrak{Re}(z)$ and $\mathfrak{Im}(z)$ stand for the real and imaginary parts
of a complex number $z$.
For $d\geq 1$, $\mu \in \C^d$ and a positive definite $d \times d$ Hermitian
$\Sigma$, a complex random vector $Z \in \C^d$ is said to follow a multivariate
complex normal distribution with mean $\mu$ and covariance $\Sigma$,
short: $Z \sim C\mathcal{N}_d(\mu,\Sigma)$, if it has the density function
\begin{equation}
\label{eq:34}
f(z) =
\frac{1}{\pi^d \abs{\on{det} \Sigma}} \on{exp} \left( - \overline{\left( z -
\mu\right)}^{T} \Sigma^{-1} \left( z - \mu \right) \right),
\end{equation}
where $A^{T}$ denotes the transpose of $A$.
Let $Z \sim C\mathcal{N}_d(\mu,\Sigma)$.
* Straightforward calculations show that
\begin{equation*}
\Ex{ (Z-\mu) \overline{(Z_k-\mu)}^{T} } = \Sigma
\end{equation*}
\begin{equation}
\label{eq:35}
\Ex{ (Z-\mu) (Z - \mu)^T} = 0.
\end{equation}
Furthermore, it can be shown that $Z$ is circularly symmetric: For
any $\alpha \in \R$, the rotated vector $\me^{\mi \alpha} Z$ has the same
distribution as $Z$. Each circularly symmetric complex
Gaussian vector can be obtained via a linear transformation of a standard
complex Gaussian vector $Z \sim C\mathcal{N}_d(0,\Id_d)$ whose real and
imaginary part are independent real-valued standard Gaussian
vectors. Some authors drop the independency assumption and thus obtain
more general complex Gaussian vectors for which the matrix on the left
hand side of (<ref>), sometimes called the relation matrix, is no
longer zero. However, when we speak of a complex Gaussian vector, we
always mean the circularly symmetric case of Definition <ref>.
* The characteristic function $\rho$ of $Z$ is given by
\begin{equation*}
\rho(\zeta)
\exp \left( \mi \mathfrak{Re}( \left\langle \mu,\zeta \right\rangle_{\C^d}) -
\frac{1}{4} \left\langle \Sigma \zeta,\zeta \right\rangle_{\C^d} \right),
\end{equation*}
which shows that $Z$ is determined by its moments, i.e. (assuming $\mu=0$ for
notational convenience) any $d$-dimensional
complex random vector $W$ satisfying
\begin{equation*}
\Ex{
\prod_{j=1}^d (W_j-\mu_j)^{p_j} (\overline{W}_j-\overline{\mu}_j)^{q_j}
\Ex{
\prod_{j=1}^d (Z_j-\mu_j)^{p_j} (\overline{Z}_j-\overline{\mu}_j)^{q_j}
\end{equation*}
for all $p_j,q_j \in \N_0$, $j=1,\dots,d$, has the same law as $Z$.
We will need the following complex version of the Gaussian integration by parts
formula, which for convenience will be stated for the centered case.
Let $Z \sim CN_d(0,\Sigma)$ and $\varphi \colon \mathbb{C}^d \to \mathbb{C}$
such that, for $1 \leq i \leq d$, the Wirtinger derivatives
$\partial_{z_i} \varphi$ and $\partial_{\overline{z}_i} \varphi$
exist and have at most polynomial growth in $z_i$ and
$\overline{z}_i$ respectively. Then it holds that
\begin{equation}
\label{eq:15}
\Ex{Z_i \varphi(Z_1,\dots,Z_d)} = \sum_{j=1}^d \Ex{ Z_i \overline{Z}_j}
\Ex{ \partial_{\overline{z}_j} \varphi(Z_1,\dots,Z_d)}
\end{equation}
\begin{equation}
\label{eq:16}
\Ex{\overline{Z}_i \varphi(Z_1,\dots,Z_d)} = \sum_{j=1}^d \Ex{ Z_j
\overline{Z}_i}
\Ex{ \partial_{z_j} \varphi(Z_1,\dots,Z_d)}
\end{equation}
The proof is standard and straightforward but is included nevertheless to
demonstrate the use of Wirtinger calculus. We will
prove formula (<ref>) and note that (<ref>) then follows by conjugation.
As $\Sigma$ is positive definite Hermitian, it admits a normal square root
$A$ such that $\Sigma=A^{\ast}A$, where $A^{\ast}$ denotes the conjugate
transpose of $A$. Clearly, both $A$ and $A^{\ast}$ are
invertible and we have that $\Sigma^{-1} = (A^{\ast})^{-1} A^{-1}$. The linear
\begin{equation*}
\xi = A^{-1} z
\end{equation*}
induces a linear transformation on $\R^{2d}$ whose constant volume element
will be
denoted by $v$. Writing $z=x+iy$ and $\zeta=\xi+\mi \upsilon$ (with
$z_i=x_i+\mi y_i$ and $\zeta_i=\xi_i + \mi \upsilon_i$), we get that
\begin{align*}
\Ex{Z_i \varphi(Z)}
\frac{1}{\pi^d \abs{\on{det}\Sigma}}
\int_{\R^{2d}}^{}
z_i \varphi(z)
\,
\me^{- z^{\ast} \Sigma^{-1} z}
\diff{(x,y)}
\\ &=
\frac{v}{\pi^d \abs{\on{det}\Sigma}}
\sum_{j=1}^d A_{ij}
\int_{\R^{2d}}^{}
\zeta_j \varphi(A \zeta)
\,
\me^{- \zeta^{\ast} \zeta}
\diff{(\xi,\upsilon)}
\\ &=
\frac{- v}{\pi^d \abs{\on{det}\Sigma}}
\sum_{j=1}^d A_{ij}
\int_{\R^{2d}}^{}
\varphi(A \zeta)
\left(
\partial_{\overline{\zeta}_j} \me^{- \zeta^{\ast} \zeta} \right)
\diff{(\xi,\upsilon)}.
\end{align*}
By the product rule, it holds that
\begin{equation*}
\varphi(A \zeta)
\left(
\partial_{\overline{\zeta}_j} \me^{- \zeta^{\ast} \zeta} \right)
\partial_{\overline{\zeta}_j} \left( \varphi(A \zeta)
\me^{- \zeta^{\ast} \zeta} \right)
\left( \partial_{\overline{\zeta}_j} \varphi(A \zeta)
\right)
\me^{- \zeta^{\ast} \zeta}.
\end{equation*}
Now, by a Fubini argument,
\begin{equation*}
\int_{\R^{2d}}^{} \partial_{\overline{\zeta}_j} \left( \varphi(A \zeta)
\me^{- \zeta^{\ast} \zeta} \right) \diff{(\xi,\upsilon)} = 0.
\end{equation*}
Furthermore, by the chain rule,
\begin{equation*}
\partial_{\overline{\zeta}_j} \varphi(A \zeta)
\sum_{k=1}^d
\overline{A}_{k,j}
(\partial_{\overline{\zeta}_k} \varphi) (A \zeta).
\end{equation*}
\begin{align*}
\Ex{Z_i \varphi(Z)}
\frac{v}{\pi^d \abs{\on{det}\Sigma}}
\sum_{j,k=1}^d A_{ij} \overline{A}_{k,j}
\int_{\R^{2d}}^{}
(\partial_{\overline{\zeta}_k} \varphi)
(A \zeta)
\me^{- \zeta^{\ast} \zeta}
\diff{(\xi,\upsilon)}
\\ &=
\frac{1}{\pi^d \abs{\on{det}\Sigma}}
\sum_{j,k=1}^d A_{ij} \overline{A}_{k,j}
\int_{\R^{2d}}^{}
\partial_{\overline{z}_k} \varphi
\me^{- z^{\ast} z}
\diff{(x,y)}.
\end{align*}
Noting that $\sum_{j=1}^d A_{i,j} \overline{A}_{k,j} = \sum_{j=1}^d A_{i,j}
A^{\ast}_{j,k} = \Sigma_{i,k} = \Ex{Z_i \overline{Z}_k}$
proves (<ref>).
As an immediate consequence of Lemma <ref>, we see that for $Z\sim
C\mathcal{N}_d(0,\Sigma)$ and all multi-indices $p=(p_1,\dots,p_d) \in \N_0^d$ of order
at least one it holds that
\begin{equation*}
\Ex{ \prod_{j=1}^d Z_j^{p_j}} = \Ex{ \prod_{j=1}^d \overline{Z}_j^{p_j}} = 0.
\end{equation*}
Furthermore, for the case $Z \sim C\mathcal{N}_1(0,\sigma^2)$, Lemma <ref>
yields the well-known moment-formula
\begin{equation*}
\begin{cases}
p! \, \sigma^{2p} &\qquad \text{if $p=q$} \\
0 &\qquad \text{if $p \neq q$},
\end{cases}
\end{equation*}
valid for $p,q \in \N_0$ (with the usual convention that $0!=1$).
§.§ Stein's method for the complex normal distribution
For a quadratic matrix $A$, the Hilbert-Schmidt norm $\norm{A}_{\text{HS}}$ is
defined via the inner product
$\left\langle A,B \right\rangle_{\text{HS}} = \on{tr}(A \overline{B}^T)$.
The next lemma is an adaptation of the Stein characterization for the
multivariate real normal distribution (see <cit.>, <cit.> and also <cit.>) to the complex case. Recall the definitions of
complex gradients and Hessians given in
Section <ref>.
For $d\geq 1$, let $\Sigma$ be a positive definite, Hermitian matrix and $Z
\sim C\mathcal{N}_d(0,\Sigma)$.
* A $d$-dimensional complex random vector $Y$ has the complex normal
distribution $C\mathcal{N}_d(0,\Sigma)$, if, and only if, the identity
\begin{multline*}
\Ex{ \left\langle \overline{\nabla} \nabla f(Y), \overline{\Sigma}
\right\rangle_{\text{HS}}}
\Ex{ \left\langle \nabla \overline{\nabla} f(Y), \Sigma
\right\rangle_{\text{HS}}}
\\ -
\Ex{
\left\langle \nabla f(Y), \overline{Y} \right\rangle_{\C^d}}
\Ex{
\left\langle \overline{\nabla} f(Y), Y \right\rangle_{\C^d}}
= 0
\end{multline*}
holds for any $f \in \mathcal{C}^2(\C^d)$ which satisfies
\begin{multline*}
\Ex{\abs{\left\langle \overline{\nabla} \nabla f(Y),
\overline{\Sigma}
\right\rangle_{\text{HS}}}}
\Ex{ \abs{\left\langle \nabla \overline{\nabla} f(Y), \Sigma
\right\rangle_{\text{HS}}}}
\\ +
\Ex{
\abs{
\left\langle \nabla f(Y), \overline{Y} \right\rangle_{\C^d}}}
\Ex{\abs{
\left\langle \overline{\nabla} f(Y), Y \right\rangle_{\C^d}}}
< \infty.
\end{multline*}
* Given $h \in \mathcal{C}^2(\C^d)$ with bounded derivatives up to
order two, the function
\begin{equation}
\label{eq:4}
U_h(z) = \int_0^1 \frac{1}{2t}
\left(
\Ex{ h(Z_{z,t})}
\Ex{h(Z)}
\right)
\diff{t},
\end{equation}
where $Z_{z,t}= \sqrt{t}z + \sqrt{1-t}Z$, is a solution to the complex Stein
\begin{multline}
\label{eq:22}
\left\langle \overline{\nabla} \nabla f(z), \overline{\Sigma}
\right\rangle_{\text{HS}}
\left\langle \nabla \overline{\nabla} f(z), \Sigma
\right\rangle_{\text{HS}}
\\ -
\left\langle \nabla f(z), \overline{z} \right\rangle_{\C^d}
\left\langle \overline{\nabla} f(z), z
\right\rangle_{\C^d}
h(z) - \Ex{ h(Z)}.
\end{multline}
The real counterpart of this lemma has been proven by direct calculations
using Gaussian integration by parts in <cit.>, and based on the generator approach
of <cit.> in <cit.>. Both of these proofs can be straightforwardly
adapted to the complex case (either using complex Gaussian by parts or the
complex Ornstein-Uhlenbeck generator) which is why we omit the details.
We also need the following technical lemma, which can be deduced by adapting the
proof of inequality (3.4) in <cit.>.
In the setting and with the notation of Lemma <ref>, let $c(\Sigma) =
\norm{\Sigma^{-1}}_{\text{op}} \norm{\Sigma}_{\text{op}}^{1/2}$. Then, for any
$\alpha \in [0,1]$, the complex
Hessians of the Stein solution (<ref>) satisfy the bounds
\begin{align*}
\norm{\nabla \overline{\nabla} U_h(z)}_{\text{HS}}
\left( \alpha \max_{1 \leq k \leq d} \norm{\partial_{z_j}h}_{\infty} +
(1-\alpha) \max_{1 \leq k \leq d} \norm{\partial_{\overline{z}_j}
h}_{\infty} \right),
\\
\norm{\overline{\nabla} \nabla U_h(z)}_{\text{HS}}
\left( \alpha \max_{1 \leq k \leq d} \norm{\partial_{z_j}h}_{\infty} +
(1-\alpha) \max_{1 \leq k \leq d} \norm{\partial_{\overline{z}_j}
h}_{\infty} \right),
\\
\norm{\nabla \nabla U_h(z)}_{\text{HS}}
\max_{1 \leq k \leq d} \norm{\partial_{z_j}h}_{\infty}
\\
\intertext{and}
\norm{\overline{\nabla}\overline{\nabla} U_h(z)}_{\text{HS}}
\max_{1 \leq k \leq d} \norm{\partial_{\overline{z}_j}h}_{\infty}.
\end{align*}
* The proofs of Lemmas <ref> and <ref>
depend on the characterizing property that any complex Gaussian vector $Z
\sim
C\mathcal{N}_d(0,\Sigma)$ can be obtained via a linear transformation of a
standard complex Gaussian vector $\widetilde{Z}\sim
C\mathcal{N}_d(0,\Id_d)$. An adaptation of these proofs to the larger class
of not necessarily circularly symmetric complex Gaussian vectors hinted at
in Remark <ref> is thus not possible.
* As $\Sigma$ and its inverse are positive definite and Hermitian, the
operator norms of these two matrixes coincide with their spectral
radii. Thus, the constant $c(\Sigma)=\norm{\Sigma^1}_{op}
\norm{\Sigma}_{op}^{1/2}$ appearing in the bounds for $U_h$ in
Lemma <ref> coincides with
$\sqrt{\lambda_{max}}/\lambda_{min}$ where $\lambda_{max}$ and
$\lambda_{min}$ denote the largest and smallest eigenvalues of $\Sigma$,
respectively. In particular, in the case $d=1$, where $\Sigma=\lambda > 0$, this
constant becomes $1/\sqrt{\lambda}$.
For the case $d=1$, it is actually possible to derive a simpler characterizing
differential equation.
A complex valued random variable $Z$ has the standard complex normal
distribution if, and only if
\begin{equation}
\label{eq:8}
\Ex{\partial_z f(Z)} - \Ex{\overline{Z} f(Z)} = 0
\end{equation}
for all Wirtinger differentiable functions $f \colon \C \to \C$
such that $\partial_zf$ has at most polynomial growth.
Necessity of condition (<ref>) is implied by Gaussian integration by parts
(Lemma <ref>). Sufficiency follows by
inserting the polynomials $f(z)=\overline{z}^pz^q$, which
immediately yields the moment recursion for the standard complex
Gaussian, and noting that the complex Gaussian distribution is
determined by its moments.
From identity (<ref>), one is led to the “Stein equation”
\begin{equation}
\label{eq:9}
\partial_z f(z) - \overline{z} f(z) = h(z) - \Ex{h(Z)},
\end{equation}
where $h \colon \C \to \C$ is some given function and $Z$ has the standard
complex Gaussian distribution. Note that if we formally replace the complex
variable $z$ with a real variable $x$ and the Wirtinger derivative
$\partial_z$ with the partial derivative $\partial_x$, we obtain the
classical Stein equation of
the one-dimensional Gaussian distribution. However, as one sees after writing
$z=x+\mi y$ and separating real and imaginary parts, this equation
is not solvable
in general. This was to be expected, as otherwise, using Stein's method for the
real case, one could obtain bounds in total variation and Kolmogorov distance
for two-dimensional real Gaussian approximation, which is not possible using
this approach (see for
example <cit.>. Lemma <ref> can thus
not be quantified.
§ COMPLEX MARKOV DIFFUSION GENERATORS
As in the real case, we start with a good measurable space $E$ in the sense
of <cit.> (for example, take $E$ to be a Polish space),
equipped with a probability measure $\mu$. On $L^2(E,\R,\mu)$, let $\LL$ be a
symmetric Markov diffusion generator $\LL$ with discrete spectrum $S =
\Set{ - \lambda_{k}}$, where the eigenvalues $-\lambda_k$ are ordered by magnitude,
i.e. $0=\lambda_0 < \lambda_1 < \dots$. In the language of functional analysis,
$-\LL$ is a positive, self-adjoint linear operator vanishing on the
constants. The associated bilinear carré du champ operator $\Gamma$, acting on
a set $\mathcal{A}_0$ which we assume to be dense in $L^p(E,\R,\mu)$ for all $p
\geq 1$, is defined in the usual way as
\begin{equation}
\label{eq:18}
2\Gamma(U_1,U_2) = \LL (U_1U_2)- U_{1} \LL U_2 - U_2 \LL U_1
\end{equation}
and for any
smooth function $\varphi \colon \R^d \to \R$ the diffusion property
\begin{multline}
\label{eq:14}
\LL \varphi (U) = \LL \varphi(U_1,\dots,U_d) \\ = \sum_{k=1}^d \partial_{x_k}
\varphi(U_1,\dots,U_d) \LL U_k + \sum_{j,k=1}^d \partial_{x_j x_k} \varphi(U_1,\dots,U_d)
\Gamma(U_j,U_k)
\end{multline}
holds. As is well known, this diffusion property is equivalent to the chain rule
\begin{equation}
\label{eq:30}
\Gamma(\varphi(U_1,\dots,U_d),V) = \sum_{j=1}^d \partial_j \varphi(U_1,\dots,U_d) \Gamma(U_k,V).
\end{equation}
Through straightfoward complexification, we can extend $\LL$ and $\Gamma$ to act
on the space $L^2(E,\C,\mu)$. Writing $F=U+\mi V$ for a generic element of
$L^2(E,\C,\mu)$, this extension of $\LL$, which we denote for
the moment by $\widehat{\LL}$, is simply defined as
\begin{equation}
\label{eq:20}
\widehat{\LL} =
\widehat{\LL}(U + \mi V) = \LL U + \mi \LL V.
\end{equation}
We immediately see that $-\widehat{\LL}$ remains a positive, self-adjoint
operator vanishing on constants and that its spectrum coincides with the one
of $\LL$. Indeed, assuming that $U+\mi V$ is an eigenfunction of $\widehat{\LL}$
with some eigenvalue $\lambda$, we have that
\begin{equation*}
\lambda (U + \mi V) = \widehat{\LL} (U + \mi V) = \LL U + \mi \LL V.
\end{equation*}
Comparing imaginary and real parts, this gives that $\lambda$ lies in the
spectrum of $\LL$ and that both, $U$ and $V$ are eigenfunctions of $\LL$ with eigenvalue $\lambda$.
Similarly, for $U_1,U_2,V_1,V_2 \in \mathcal{A}_0$, the extended carré du
champ operator $\widehat{\Gamma}$, defined on $\widehat{\mathcal{A}}_0 =
\mathcal{A}_0 \oplus \mi \mathcal{A}_0$, is given by
\begin{equation}
\label{eq:26}
\widehat{\Gamma}(U_1+\mi V_1,U_2 + \mi V_2) = \Gamma(U_1,U_2) +
\Gamma(V_1,V_2) + \mi \left( \Gamma(V_1,U_2) - \Gamma(U_1,V_2) \right).
\end{equation}
It follows that $\widehat{\Gamma}$ is sesquilinear, positive ($\widehat{\Gamma}(F,F) \geq
0$) and Hermitian
($\widehat{\Gamma}(F,G)= \overline{\widehat{\Gamma}(G,F)}$). Furthermore, using
identity (<ref>), we get that
\begin{equation}
\label{eq:19}
2\widehat{\Gamma}(F,G) = \widehat{\LL}(F\overline{G}) - F \widehat{\LL}\overline{G} - \overline{G}
\widehat{\LL} F,
\end{equation}
which yields the integration by parts formula
\begin{equation*}
\int_E^{} \widehat{\Gamma}(F,G) \diff{\mu} = - \int_E^{} F \widehat{\LL} \,
\overline{G} \diff{\mu}.
\end{equation*}
As $-\widehat{\LL}$ is
positive, it holds that $\overline{\widehat{\LL} F} = \widehat{\LL}
\overline{F}$. In particular, all eigenspaces of $\widehat{\LL}$ are closed under complex
conjugation and, if $\pi_j$ denotes the orthogonal projection onto
$\ker \left(\widehat{L}+ \lambda_k \Id \right)$, we have for all $F \in
L^2(E,\C,\mu)$ that $\overline{\pi_j(F)} = \pi_j(\overline{F})$. Furthermore, by
the defining equation (<ref>) of $\Gamma$, we also see that
Using (<ref>) and (<ref>), it is straightforward to verify the
diffusion property
\begin{align}
\label{eq:28}
\widehat{\LL} \varphi(F) &= \widehat{\LL} \varphi(F_1,\dots,F_d)
\\ &= \notag
\sum_{j=1}^d \left( \partial_{z_j} \varphi(F) \widehat{\LL} F_j
+ \partial_{\overline{z}_j} \varphi(F) \widehat{\LL} \overline{F}_j \right)
\\ \notag &\qquad \qquad+ \sum_{j,k=1}^d \left( \partial_{z_j z_k} \varphi(F)
\widehat{\Gamma}(F_j,
\overline{F}_k) + \partial_{\overline{z}_j \overline{z}_k} \varphi(F)
\widehat{\Gamma}(\overline{F}_j,F_k)
\right)
\\ \notag &\qquad \qquad+ \sum_{j,k=1}^d \left( \partial_{z_j \overline{z}_k}
\varphi(F)
\widehat{\Gamma}(F_j,F_k)
+ \partial_{\overline{z}_j z_k} \varphi(F)
\widehat{\Gamma}(\overline{F}_j,\overline{F}_k)
\right),
\end{align}
and the chain rule
\begin{multline}
\label{eq:29}
\widehat{\Gamma}(\varphi(F),G) = \widehat{\Gamma}(\varphi(F_1,\dots,F_d))
\\ = \sum_{j=1}^d \left( \partial_{z_j} \varphi(F) \widehat{\Gamma}(F_j,G)
+ \partial_{\overline{z}_j} \varphi(F) \widehat{\Gamma}(\overline{F}_j,G)
\right),
\end{multline}
both valid for smooth functions $\varphi \colon \C^d \to
\C$ and $F=(F_1,\dots,F_d) \in \widehat{\mathcal{A}}_0^d$. Here,
$\partial_z$, $\partial_{\overline{z}}$, $\partial_{zz}$ etc. denote the
(iterated) Wirtinger derivatives introduced in
Section <ref>. To be more clear,
the diffusion property (<ref>) and the chain rule (<ref>) can be
translated into versions of their real counterparts (i.e. identities (<ref>)
and (<ref>)), by simply writing
$\varphi(z_1,\dots,z_d)=u(x_1,\dots,x_d,y_1,\dots,y_d)+ \mi
v(x_1,\dots,x_d,y_1,\dots,y_d)$, where $z_j=x_j + \mi y_j$ and the functions $u$
and $v$ are real valued, decomposing the vector $F$ into real and imaginary
and writing the Wirtinger derivatives in terms of derivatives with respect to
the real variables $x_j$ and $y_j$.
Because of this, we also see that, as in the real case, the diffusion property of
$\widehat{\LL}$ is equivalent to the chain rule of $\widehat{\Gamma}$.
Of course, using the fact that $\widehat{\Gamma}$ is Hermitian, we can
derive a chain rule for the second argument:
\begin{multline*}
\widehat{\Gamma}(F,\varphi(G))
= \widehat{\Gamma}(F,\varphi(G_1,\dots,G_d))
\\ = \sum_{j=1}^d \left( \partial_{\overline{z}_j} \overline{\varphi}(G)
\widehat{\Gamma}(F,G_j)
+ \partial_{z_j} \overline{\varphi}(G) \widehat{\Gamma}(F,\overline{G}_j)
\right).
\end{multline*}
We are now ready to define a complex Markov diffusion generator.
Given a good measurable space $(E,\mathcal{F})$, equipped with a probability
measure $\mu$, a self-adjoint, linear operator $\widehat{\LL}$ acting on
$L^2(E,\C,\mu)$ is called a complex symmetric Markov diffusion
generator with invariant measure $\mu$, if $-\LL$ is positive, $\LL 1 = 0$
and the diffusion property (<ref>) holds.
Note that the construction above also works in reverse: Given a complex
symmetric Markov diffusion generator $\widehat{\LL}$, we obtain a corresponding
generator on $L^2(E,\R,\mu)$. From this abstract point of view, the real and
complex approaches are thus completely equivalent. We state this as a short
$\widehat{\LL}$ is a complex Markov diffusion generator, if, and only if,
there exists a real Markov diffusion generator $\LL$ with (the same)
spectrum such that $\widehat{\LL}F = \LL \mathfrak{Re}(F) + \mi \LL
\mathfrak{Im}(F)$ for all $F \in \on{dom}\widehat{\LL}$.
Note that $\widehat{\LL}$ and $\widehat{\Gamma}$ coincide with $\LL$ and
$\Gamma$ when restricted to real valued arguments. Because of this and
Proposition <ref> we will from now on no longer notationally distinguish
$\LL$ and $\Gamma$ from their complexified versions $\widehat{\LL}$ and
$\widehat{\Gamma}$ and instead denote both versions by $\LL$ and $\Gamma$,
As already mentioned in the introduction, it should be noted that although
everything is equivalent from an abstract point of view, the complex case
often introduces features absent from the real case when turning to
special cases such as the complex Ornstein-Uhlenbeck generator (see the next
example). Furthermore, in many applications it is often much more natural
to use complex Gamma-calculus, taking the direct route through the complex
domain instead of making a detour through $\R^2$.
[The complex Ornstein-Uhlenbeck generator]
Starting from a standard isonormal Gaussian process framework (see for
example <cit.>
or <cit.>) for the real-valued,
infinite-dimensional Ornstein-Uhlenbeck generator, we obtain the complex
Ornstein-Uhlenbeck generator $\LL_{\text{OU}}$ by carrying out the
construction outlined above (i.e. via
Proposition <ref>). Equivalently, one can directly start from an
isonormal complex Gaussian process (see <cit.>).
The generator $\LL_{\text{OU}}$ can then be decomposed in the form
$\LL_{\text{OU}}= - \delta D$, where $\delta$ and $D$ are the complex
Malliavin divergence and Malliavin derivative operators
(see <cit.> for more details on
complex Malliavin calculus). The carré du champ operator
$\Gamma_{\text{OU}}$ takes the form
$\Gamma_{\text{OU}}= \left\langle DF,DG \right\rangle_H$, where the inner
product is now taken in a complex Hilbert space $H$ (coming from the
underlying complex isonormal Gaussian process) and $D$ denotes the complex
Malliavin derivative. The spectrum of $\LL_{\text{OU}}$ is $-\N_0$ and by our
above we know that all eigenfunctions $F_\lambda \in \ker \left(
\LL_{\text{OU}}+ \lambda \Id \right)$, $\lambda \in \N_0$, are of the form
$F_\lambda=U_\lambda +
\mi V_{\lambda}$, where $U_\lambda$ and $V_\lambda$ are eigenfunctions of the
real-valued Ornstein-Uhlenbeck generator and thus multiple Wiener-Itô
integrals. In contrast to the real case, however, one has a finer
decomposition of the eigenspaces with a rich structure: For each $\lambda \in
\N_0$, it holds that
\begin{equation*}
\ker \left( \LL_{\text{OU}} + \lambda \Id \right) = \bigoplus_{\substack{p,q
\in \N_0\\ p+q=\lambda}} \mathcal{H}_{p,q},
\end{equation*}
where the sum on the right is orthogonal and the spaces $\mathcal{H}_{p,q}$
consist of complex Wiener-Itô integrals of the form $I_{p,q}(f)$
(see <cit.>). Furthermore, one can show that
$\overline{\mathcal{H}_{p,q}}=\mathcal{H}_{q,p}$ and that only the eigenspaces
of even eigenvalues $\lambda=2p$ contain real-valued eigenfunctions, belonging
to $\mathcal{H}_{p,p}$.
Let us briefly outline the construction of an
orthonormal basis for $\mathcal{H}_{p,q}$ (see again <cit.>
for details). For integers $p,q \geq 0$, the complex Hermite polynomials
$H_{p,q}$ are given by
\begin{align*}
H_{p,q}(z) &= (-1)^{p+q} \me^{\abs{z}^2} \left( \partial_z \right)^p
\left( \partial_{\overline{z}} \right)^q \me^{- \abs{z}^2}
\\ &=
\sum_{j=1}^{p \land q} \binom{p}{j} \binom{q}{j} j! (-1)^j z^{p-j}
\overline{z}^{q-j},
\end{align*}
where summation ends at the smaller of the two parameters $p$ and $q$.
Now let $\left\{ e_j \colon j \geq 1 \right\}$ be an orthonormal basis of the
underlying complex Hilbert space $H$ and $\left\{ Z(h) \colon h \in H
\right\}$ denote the complex isonormal Gaussian process.
Furthermore, for $n \in \N_0$, denote by $M_{n}$ the set of all multi-indices
of order $n$ (sequences with a finite number of positive non-zero entries
which sum up to $n$) and, for $(m_p,m_q) \in M_p \times M_q$, define
\begin{equation*}
\Phi_{m_p,m_q} = \prod_{j=1}^{\infty} H_{m_p(j),m_q(j)}(Z(e_j)).
\end{equation*}
Then, the family $\left\{ \Phi_{m_p,m_q} \colon (m_p,m_q) \in M_p \times M_q
\right\}$ is an orthonormal basis of $\mathcal{H}_{p,q}$.
In particular, as $H_{p,0}=z^p$, we see that for any multi-index $m \in
M_{p}$, the monomial $\prod_{j=1}^{\infty} Z(e_j)^{m(j)}$ is an element of
$\mathcal{H}_{p,0}$ and remains an eigenfunction when taking powers. In other
words, for these eigenfunctions the Wick product coincides with the ordinary
product. In the real case, there exist no non-constant eigenfunctions with
this property.
§ FOURTH MOMENT THEOREMS FOR COMPLEX GAUSSIAN APPROXIMATION
Throughout the whole section, $\LL$ denotes a complex symmetric Markov diffusion
generator with invariant measure $\mu$ and discrete spectrum $S = \Set{ -
\lambda_k}$, acting on $L^2(E,\C,\mu)$, where $E$ is a good measurable
space. The associated carré du champ operator, acting on $\mathcal{A}_0$ which
we assume to be dense in $L^p(E,\C,\mu)$ for all $p \geq 1$, is
denoted by $\Gamma$.
We start by introducing the notion of chaos, which for the real case was given
in <cit.> and extended to the multidimensional case
in <cit.>. These definitions can be
generalized as follows to the complex setting.
* Two eigenfunctions $F_1 \in \ker \left(\LL +\lambda_{p_1} \on{Id}\right)$ and $F_2 \in
\ker \left(\LL +\lambda_{p_2} \on{Id}\right)$ are called jointly
chaotic, if $F_1F_2$, $F_1 \overline{F}_2$ (and thus, as the eigenspaces
are closed under conjugation, also $\overline{F_1F_2}$ and
$\overline{F}_1F_2$) have an expansion over the first $p_1+p_2+1$
eigenspaces. In formulas, we require that
\begin{equation*}
F_1 F_2 \in \bigoplus_{k=0}^{p_1+p_2} \ker \left(\LL +\lambda_{k} \on{Id}
\right)
\qquad \text{and} \qquad
F_1 \overline{F}_2 \in \bigoplus_{k=0}^{p_1+p_2} \ker \left(\LL +\lambda_{k}
\on{Id} \right).
\end{equation*}
* A single eigenfunction is called chaotic, if it is jointly chaotic
with itself.
* A vector $F=(F_1,\dots,F_d)$ of eigenfunctions $F_j \in \ker \left(\LL +
\lambda_{p_j} \on{Id}\right)$ is called chaotic, if any two of its
components are jointly chaotic (in particular, each component is
chaotic in the sense of part (ii)).
Note that indeed, by taking all involved eigenfunctions to be real
valued, we obtain the corresponding notions of real Markov chaos
(namely <cit.>)
and <cit.>) as special cases of
Definition <ref>.
As in the real case, a crucial ingredient for our main results will be the
following general principle. The proof for the real case
(see <cit.>) can be straightforwardly generalized
to the complex case and is therefore omitted.
Let $F \in \bigoplus_{k=0}^p \on{ker} \left( \LL + \lambda_k \Id
\right)$. Then, for any $\eta \geq \lambda_p$ it holds that
\begin{equation*}
\int_E^{} F \left( \LL + \eta \Id \right)^2 \overline{F} \diff{\mu}
\leq
\eta
\int_E^{} F \left( \LL + \eta \Id \right) \overline{F} \diff{\mu}
\leq
\int_E^{} F \left( \LL + \eta \Id \right)^2 \overline{F} \diff{\mu},
\end{equation*}
where $1/c$ is the minimum of the set $\Set{ \eta - \lambda_k \mid 0 \leq k
\leq p} \setminus \Set{ 0}$.
Again, we note that by specializing to real valued eigenfunctions,
we obtain <cit.> as a special case. From
Theorem <ref>, we immediately deduce the following corollary.
For two jointly chaotic eigenfunctions $F_{1} \in
\ker \left( \LL + \lambda_{p_1} \Id \right)$ and $F_{2} \in
\ker \left( \LL + \lambda_{p_2} \Id \right)$ it holds that
\begin{equation}
\label{eq:33}
\int_E^{} \abs{\Gamma(F_1,F_2)}^2 \diff{\mu} \leq \frac{p_1+p_2}{2} \int_E^{}
\overline{F}_1 F_2 \Gamma(F_1,F_2) \diff{\mu}.
\end{equation}
By definition, $2 \Gamma(F_1,F_2) = \left( \LL + (p_1+p_2)\Id
\right) (F_1\overline{F}_2)$ and thus, using the fact that $\LL$ and the
identity are both self-adjoint and then Theorem <ref>, it follows that
\begin{align*}
\int_E^{} \abs{\Gamma(F_1,F_2)}^2 \diff{\mu}
&= \frac{1}{4} \int_E^{} \overline{F}_1 F_2
\left( \LL +
(p_1+p_2) \Id \right)^2 (F_1 \overline{F}_2) \diff{\mu}
\\ &\leq
\frac{p_1+p_2}{4} \int_E^{} \overline{F}_1F_2
\left( \LL +
(p_1+p_2) \Id \right) (F_1 \overline{F}_2) \diff{\mu}
\\ &=
\frac{p_1+p_2}{2} \int_E^{} \overline{F}_1F_2
\Gamma(F_1,F_2) \diff{\mu}.
\end{align*}
Before continuing, we need to introduce some notation.
For $d\geq 1$, let $F=(F_1,\dots,F_d)$ and $G=(G_1,\dots,G_d)$ be two complex
random vectors. The Wasserstein distance $d_W(F,G)$ between $F$ and $G$ is then
defined as
\begin{equation}
d_W(F,G) = \sup_{h \in \mathcal{H}} \abs{ \Ex{h(F)} - \Ex{h(G)}},
\end{equation}
where $\mathcal{H} = \Set{ h \colon \C^d \to \C \mid \norm{h}_{\text{Lip}} \leq
1 }$ and $\norm{h}_{\text{Lip}}$ denotes the Lipschitz norm, defined as
\begin{equation*}
\norm{h}_{Lip} = \sup_{w,z \in \C^d} \frac{\abs{h(w) - h(z)}}{\norm{w-z}_2} =
\sup_{w,z \in \C^d} \frac{\abs{h(w) -h(z)}}{\sqrt{\sum_{j=1}^d \abs{w_j-z_j}^{2}}}.
\end{equation*}
Furthermore, we will use the shorthand $\Gamma(F,\LL^{-1}G)$ to denote the
matrix $\left( \Gamma(F_j,\LL^{-1}G_k) \right)_{1 \leq j,k \leq d}$.
The following Theorem provides a
quantitative bound on the Wasserstein distance between a complex random vector
and a multivariate complex Gaussian.
For $d\geq 1$, let $Z \sim C\mathcal{N}_d(0,\Sigma)$ and denote by
$F=(F_1,\dots,F_d)$ a complex random vector whose components are elements of
$\mathcal{A}_0$. Then it holds that
\begin{multline}
\label{eq:5}
\leq 2 \norm{\Sigma^{-1}}_{\text{op}} \norm{\Sigma}_{\text{op}}^{1/2}
\left(
\int_E^{}
\norm{\Gamma(\overline{F},-\LL^{-1} F)}_{\text{HS}}^2 \diff{\mu}
\right.
\\ +\left. \int_E^{}
\norm{\Gamma(F,-\LL^{-1} F) - \Sigma}_{\text{HS}}^2 \diff{\mu}
\right)^{1/2}.
\end{multline}
In the case of the real Ornstein-Uhlenbeck generator, a similar bound was
obtained in <cit.> through the use of
Malliavin calculus. Formulated in the language of Markov diffusion generators,
this bound reads
\begin{equation*}
\leq \norm{C^{-1}}_{\text{op}} \norm{C}_{\text{op}}^{1/2}
\left(
\int_E^{}
\norm{\Gamma(U,-\LL^{-1} U) - C}_{\text{HS}}^2 \diff{\mu}
\right)^{1/2},
\end{equation*}
where $U$ is a real-valued, smooth random vector and $W \sim
\mathcal{N}_d(0,C)$ a $d$-dimensional centered real Gaussian vector. Compared
to the bound (<ref>), we see that in the complex setting a second
$\Gamma$-term appears.
Let $h \in C^2(\C^d)$ with bounded first and second derivatives and denote
by $U_h$ the solution (<ref>) to the Stein equation of Lemma <ref>.
By the integration by parts formula and the chain rule for $\Gamma$, it holds
\begin{align*}
\int_E^{} \left\langle \nabla U_h(F),\overline{F} \right\rangle_{\C^d} \diff{\mu}
\sum_{k=1}^d \int_E^{} \left(\partial_{z_k} U_h(F)\right) F_k \diff{\mu}
\\ &=
\sum_{k=1}^d \int_E^{} \left(\partial_{z_k} U_h(F)\right)
\LL \LL^{-1}F_k \diff{\mu}
\\ &=
\sum_{k=1}^d \int_E^{} \Gamma \left(\partial_{z_k} U_h(F),
-\LL^{-1}\overline{F}_k\right) \diff{\mu}
\\ &=
\sum_{j,k=1}^d \int_E^{} \Big( \partial_{z_jz_k} U_h(F) \Gamma \left( F_j,
\\ & \qquad \qquad \qquad \qquad +
\partial_{\overline{z}_jz_k} U_h(F) \Gamma \left( \overline{F}_j,
\Big) \diff{\mu}
\\ &=
\int_E^{}
\left\langle \nabla\nabla U_h(F), \Gamma(\overline{F},-\LL^{-1} F)
\right\rangle_{\text{HS}} \diff{\mu}
\\ &\qquad \qquad \qquad \qquad+
\int_E^{}
\left\langle \overline{\nabla}\nabla U_h(F), \Gamma(F,-\LL^{-1} F)
\right\rangle_{\text{HS}}
\diff{\mu}.
\end{align*}
\begin{align*}
\int_E^{} \left\langle \overline{\nabla} f(F),F
\right\rangle_{\C^d} \diff{\mu}
\int_E^{}
\left\langle \overline{\nabla\nabla} f(F), \Gamma(F,-\LL^{-1} \overline{F})
\right\rangle_{\text{HS}} \diff{\mu}
\\ &\qquad \qquad \qquad \qquad+
\int_E^{}
\left\langle \nabla \overline{\nabla} f(F), \Gamma(\overline{F},-\LL^{-1} \overline{F})
\right\rangle_{\text{HS}}
\diff{\mu}.
\end{align*}
Plugging these two identities into the complex Stein equation yields
\begin{align*}
\notag
\int_E^{} h(F) \diff{\mu} - \Ex{h(Z)} &=
\int_E^{}
\left\langle \nabla\nabla U_h(F), \Gamma(\overline{F},-\LL^{-1} F)
\right\rangle_{\text{HS}} \diff{\mu}
\\ &\qquad+ \notag
\int_E^{}
\left\langle \overline{\nabla\nabla} U_h(F), \Gamma(F,-\LL^{-1} \overline{F})
\right\rangle_{\text{HS}} \diff{\mu}
\\ & \qquad+ \notag
\int_E^{}
\left\langle \nabla\overline{\nabla} U_h(F), \Gamma(F,-\LL^{-1} F) - \Sigma
\right\rangle_{\text{HS}}
\diff{\mu}
\\ &\qquad+ \notag
\int_E^{}
\left\langle \overline{\nabla} \nabla U_h(F),
\Gamma(\overline{F},-\LL^{-1} \overline{F} - \overline{\Sigma})
\right\rangle_{\text{HS}}
\diff{\mu}
\\ &= I_1 + I_2 + I_3 +I_4,
\end{align*}
so that
\begin{align}
\notag
\abs{ \int_E^{} h(F) \diff{\mu} - \Ex{h(Z)}}
&= \sqrt{\abs{I_1+I_2+I_3+I_4}^2}
\\ \label{eq:11} &\leq 2
\sqrt{\abs{I_1}^2 + \abs{I_2}^2 + \abs{I_3}^2 + \abs{I_4}^2}.
\end{align}
Using Lemma <ref>, we obtain
\begin{align*}
\abs{I_1}^2+\abs{I_3}^2
\int_E^{} \left(
\abs{ \left\langle \nabla\nabla U_h(F),
\Gamma(\overline{F},-\LL^{-1} F) \right\rangle_{\text{HS}}}^2 \right.
\\ &\qquad \qquad \qquad \qquad+
\left.
\abs{\left\langle \overline{\nabla\nabla} U_h(F), \Gamma(F,-\LL^{-1}
\overline{F})
\right\rangle_{\text{HS}}}^2
\right) \diff{\mu}
\\ &\leq
\norm{\Sigma^{-1}}_{\text{op}}^2 \norm{\Sigma}_{\text{op}}
\norm{h}_{\text{Lip}}^2
\int_E^{}
\norm{\Gamma(\overline{F},-\LL^{-1} F)}_{\text{HS}}^2
\diff{\mu}
\\
\intertext{and similarly}
\abs{I_2}^2+\abs{I_4}^2 &\leq
\norm{\Sigma^{-1}}_{\text{op}}^2
\norm{\Sigma}_{\text{op}} \norm{h}_{\text{Lip}}^2
\int_E^{} \norm{\Gamma(F,-\LL^{-1} F) - \Sigma}_{\text{HS}}^2 \diff{\mu}.
\end{align*}
Plugged back into (<ref>), this gives
\begin{multline*}
\abs{ \int_E^{}h(F) \diff{\mu} - \Ex{h(Z)}}
\\ \leq 2
\norm{\Sigma^{-1}}_{\text{op}} \norm{\Sigma}_{\text{op}}^{1/2}
\norm{h}_{\text{Lip}}
\Big(
\int_E^{}
\norm{\Gamma(\overline{F},-\LL^{-1} F)}_{\text{HS}}^2
\diff{\mu}
\\ +
\int_E^{} \norm{\Gamma(F,-\LL^{-1} F) - \Sigma}_{\text{HS}}^2 \diff{\mu}
\Big)^{1/2},
\end{multline*}
where $h \in \mathcal{C}^2(\C^d,\C)$ with bounded first and second
derivatives. The proof is finished by noting that any Lipschitz function $g$
can be uniformly approximated by functions of this type (take for example
$g_{\varepsilon}(z) = \Ex{g(z + \sqrt{\varepsilon} Z)}$, where $Z \sim
C\mathcal{N}_d(0,\Id_d)$; see <cit.>).
In the framework of real Markov diffusion generators, one can obtain bounds
for the stronger Kolmogorov and total variation distances when specializing to
dimension one. As a complex random variable corresponds to a two-dimensional
real random vector, this strengthening is of course no longer possible using
this approach.
For chaotic complex random vectors, the integrals appearing in the
bound (<ref>) of Theorem <ref> can be expressed purely in terms of
moments as follows.
For $d\geq 1$, let $Z \sim C\mathcal{N}_d(0,\Sigma)$, where
$\Sigma=(\sigma_{j,k})_{1 \leq j,k \leq d}$, and
be a chaotic complex random vector. Then it holds that
\begin{equation}
\label{eq:21}
d_{W}(F,Z) \leq
\norm{\Sigma^{-1}}_{\text{op}}
\norm{\Sigma}_{\text{op}}^{1/2}
\sqrt{\Psi_1(F) + \Psi_2(F) + \Psi_3(F)},
\end{equation}
\begin{align*}
\Psi_1(F) &= \sum_{j,k=1}^d \abs{
\int_E^{} F_j \overline{F}_k \diff{\mu} - \sigma_{j,k}}^2
\\
\Psi_2(F) &= \sum_{j,k=1}^d
\sqrt{
\int_E^{} \abs{F_j}^4 \diff{\mu}
\left( \frac{1}{2} \int_E^{} \abs{F_k}^4 \diff{\mu} -
\left( \int_E^{} \abs{F_k}^2 \diff{\mu} \right) \right)
\intertext{and}
\Psi_3(F) &= \sum_{j,k=1}^d
\int_E^{} \abs{F_jF_k}^2 \diff{\mu} - \int_E^{}
\abs{F_j}^2 \diff{\mu} \int_E^{} \abs{F_k}^2
\diff{\mu}
\abs{
\int_E^{} F_j \overline{F}_k \diff{\mu} }^2.
\end{align*}
In view of Theorem <ref>, we have to show that
\begin{multline*}
\int_E^{}
\left(
\norm{\Gamma(\overline{F},-\LL^{-1} F)}_{\text{HS}}^2
\norm{\Gamma(F,-\LL^{-1} F) - \Sigma}_{\text{HS}}^2 \right) \diff{\mu}
\\ \leq \Psi_1(F) + \Psi_2(F) + \Psi_3(F).
\end{multline*}
When expanding the two Hilbert-Schmidt norms, the integral on the left hand
side becomes
\begin{equation}
\label{eq:43}
\sum_{j,k=1}^d
\int_E^{} \left(
\abs{\Gamma(\overline{F}_j,-\LL^{-1}F_k)}^2
\abs{\Gamma(F_j,-\LL^{-1}F_k) - \sigma_{j,k}}^2
\right)
\diff{\mu}
\end{equation}
Now note that by integration by parts and Corollary <ref>,
\begin{align}
\int_E^{}\abs{\Gamma(F_j,-\LL^{-1}F_k) - \sigma_{j,k}}^2 \diff{\mu} \notag
\int_{E} \abs{\Gamma(F_j,-\LL^{-1}F_k)}^2 \diff{\mu} + \abs{\sigma_{j,k}}^2
\\ &\qquad - \notag
\mathfrak{Re}
\left( \int_E^{}\Gamma(F_j,-\LL^{-1}F_k) \diff{\mu} \,
\overline{\sigma}_{j,k} \right)
\\ &= \notag
\int_{E} \abs{\Gamma(F_j,-\LL^{-1}F_k)}^2 \diff{\mu} +
\abs{\sigma_{j,k}}^2
\\ &\qquad \notag - 2
\mathfrak{Re}
\left( \int_E^{}F_j \overline{F}_k \diff{\mu} \,
\overline{\sigma}_{j,k} \right)
\\ &= \notag
\int_{E} \abs{\Gamma(F_j,-\LL^{-1}F_k)}^2 \diff{\mu}
- \abs{\int_E^{} F_j \overline{F}_k \diff{\mu}}^2
\\ &\qquad \notag
+ \abs{ \int_E^{} F_j \overline{F_k} \diff{\mu} - \sigma_{j,k}}^2
\\ &\leq \label{eq:40}
\int_{E} \overline{F}_j F_k\Gamma(F_j,-\LL^{-1}F_k) \diff{\mu}
- \abs{\int_E^{} F_j \overline{F}_k \diff{\mu}}^2
\\ &\qquad \notag
+ \abs{ \int_E^{} F_j \overline{F_k} \diff{\mu} - \sigma_{j,k}}^2.
\end{align}
On the other hand, by Corollary <ref>, the chain rule and integration by parts,
\begin{align}
\int_E^{} \abs{\Gamma(\overline{F}_j,-\LL^{-1}F_k)}^2 \diff{\mu}
& \leq \notag
\int_E^{} F_j F_k \Gamma(\overline{F}_j,-\LL^{-1}F_k) \diff{\mu}
\\ &= \label{eq:42}
\int_E^{} \Gamma(F_j\overline{F}_jF_{k},-\LL^{-1}F_k) \diff{\mu}
\\ & \qquad \qquad - \notag
\int_E^{} \abs{F_j}^{2} \Gamma(F_{k},-\LL^{-1}F_k)
\diff{\mu}
\\ & \qquad \qquad - \notag
\int_E^{} \overline{F}_jF_{k} \Gamma(F_j,-\LL^{-1}F_k)
\diff{\mu}
\end{align}
Plugging (<ref>) and (<ref>) into (<ref>) yields that
\begin{multline*}
\sum_{j,k=1}^d
\int_E^{} \left(
\abs{\Gamma(\overline{F}_j,-\LL^{-1}F_k)}^2
\abs{\Gamma(F_j,-\LL^{-1}F_k) - \sigma_{jk}}^2
\right)
\diff{\mu}
\\ \leq
\Psi_{1} - \sum_{j,k=1}^d \int_E^{} \abs{F_j}^2 \left( \Gamma(F_k,-\LL^{-1}F_k) -
\int_E^{} \abs{F_k}^2 \right) \diff{\mu} + \Psi_3.
\end{multline*}
To see that the sum in the middle is bounded by $\Psi_2(F)$, we apply
Cauchy-Schwarz to each summand and then make use of the complex Gamma calculus
once more to transform the remaining $\Gamma$-expression into a moment:
\begin{align*}
\int_E^{} \Gamma(F_k,-\LL^{-1}F_k)^2 \diff{\mu}
\int_E^{} F_k \overline{F}_k \Gamma(F_k,-\LL^{-1}F_k) \diff{\mu}
\\ &=
\frac{1}{2}
\int_E^{} \Gamma(F_k^2\overline{F_k},-\LL^{-1}F_k) \diff{\mu}
\\ &\qquad \qquad -
\int_E^{} F_k^2 \Gamma(\overline{F_k},-\LL^{-1}F_k) \diff{\mu}
\\ &\leq
\frac{1}{2}
\int_E^{} \Gamma(F_k^2\overline{F_k},-\LL^{-1}F_k) \diff{\mu}
\\ &=
\frac{1}{2} \int_E^{} \abs{F_k}^4 \diff{\mu},
\end{align*}
where the last inequality follows from Corollary <ref>, which implies
that $\int_E^{}F_k^2 \Gamma(\overline{F}_k,-\LL^{-1}F_k)\diff{\mu} \geq 0$.
For eigenfunctions of the real Ornstein-Uhlenbeck generator, a bound of a
similar type has been proven in <cit.>
using Malliavin calculus, with the notable difference that only
non-mixed fourth moments appear. The same strategy could be followed to
prove a refined version of the bound (<ref>) for eigenfunctions of the
complex Ornstein-Uhlenbeck generator (i.e. complex multiple Wiener-Itô
integrals; see Example <ref>), exclusively involving the second moments
$\int_E^{} F_j \overline{F}_k \diff{\mu}$, $1 \leq j,k \leq d$ and the
non-mixed fourth moments $\int_E^{} \abs{F_j}^4 \diff{\mu}$, $1 \leq j \leq
Applying the Gaussian integration by parts formula, one sees for $Z \sim
C\mathcal{N}_d(0,\Sigma)$ and $j=1,2,3$ that indeed $\Psi_j(Z)=0$. Therefore, we
have the following corollary.
For $d \geq 1$, let $Z \sim C\mathcal{N}_d(0,\Sigma)$ and
be a sequence of centered chaotic complex random vectors.
Then, $(F_n)$ converges in distribution towards $Z$, if, and only if,
\begin{equation}
\label{eq:44}
\int_E^{} F_{j,n} \overline{F}_{k,n} \diff{\mu} \to \Ex{ Z_j \overline{Z}_k}
\end{equation}
\begin{equation}
\label{eq:6}
\int_E^{} \abs{F_{j,n} F_{k,n}}^2 \diff{\mu} \to \Ex{ \abs{Z_j Z_k}^2}
\end{equation}
for $1 \leq j,k \leq d$.
* For $d=1$ and $\Sigma=\sigma^2>0$, Corollary <ref> says that a
sequence of centered chaotic eigenfunctions converges in distribution
towards a one-dimensional centered complex Gaussian random variable with
variance $\sigma^2$, if, and only if, its second and fourth absolute
moments converge towards $\sigma^2$ and
$2\sigma^4$, respectively. This is the complex counterpart
of the abstract Fourth Moment Theorem for Gaussian approximation
(<cit.>). If, in addition, we take $\LL$
to be the complex Ornstein-Uhlenbeck generator (see Example <ref>), we
obtain <cit.>).
* For $d \geq 2$, Corollary <ref> is the complex counterpart
of <cit.>.
If $\LL$ is the real Ornstein-Uhlenbeck generator, the Peccati-Tudor Theorem
(<cit.>) says that a centered sequence $(F_{n})$ of
vectors of eigenfunctions of $\LL$ (i.e. multiple integrals) converges jointly
in distribution towards a centered Gaussian random vector with covariance
$\Sigma$, if, and only if, $\on{Var}(F_n) \to 0$ and each component sequence
converges separately towards a (one-dimensional) Gaussian. This result has been
generalized in <cit.> to the abstract
diffusion generator framework. A straightforward adaptation of the latter
finding yields the following complex Peccati-Tudor Theorem:
For $d \geq 2$, let $Z\sim C\mathcal{N}_d(0,\Sigma)$, where $\Sigma$ is
positive definite and Hermitan, and let $F_n=(F_{1,n},\dots,F_{d,n})$ be a
centered chaotic vector whose covariance converges towards $\Sigma$ as $n \to
\infty$. Furthermore, assume that
* The underlying generator $\LL$ is ergodic, in the sense that its kernel
only consists of constants
* If, for $1 \leq j < k \leq d$ and $j \neq k$, the pair
$(F_{j,n},F_{k,n})_n$ has a subsequence $(F_{j,n_{l}},F_{k,n_l})_l$ such
that $F_{j,n_l}$ and $F_{k,n_l}$ are elements of the same eigenspace with
eigenvalue $\lambda_{l}$ for all $l$, it holds that
\begin{equation*}
\int_E^{} \pi_{2\lambda_l}(F_{j,n_l}^2)
\pi_{2\lambda_l}(\overline{F}_{j,n_l}^2) \diff{\mu}
2 \left( \int_E^{} F_{j,n_l} \overline{F}_{j,n_l} \diff{\mu} \right)^2 \to 0,
\end{equation*}
where $\pi_{\lambda}$ denotes the orthogonal projection onto $\ker
\left(L+\lambda\Id\right)$.
Then, the following two assertions are equivalent.
* $F_n \xrightarrow{d} Z$.
* For $1 \leq j \leq d$ it holds that $F_{j,n} \xrightarrow{d} Z_j$.
In <cit.>, Fourth Moment Theorems for the Gamma and
Beta distribution were derived. We would like to mention that our techniques
could be readily applied to extend these results and cover
target random variables whose real and imaginary parts are independent real
Gamma or Beta random variables (in the Ornstein-Uhlenbeck case, this has been
done for the Gamma case in <cit.> by treating real and
imaginary part separately).
§ ACKNOWLEDGEMENTS
The author thanks Domenico Marinucci for many useful remarks and a careful
reading of an earlier version of this paper.
|
1511.00437
|
In this paper, we proved that if the solution to damped focusing Klein-Gordon equations is global forward in time, then it will decouple into a finite number of equilibrium points with different shifts from the origin. The core ingredient of our proof is the existence of the “concentration-compact attractor" which yields a finite number of profiles. Using damping effect, we can prove all the profiles are equilibrium points.
§ INTRODUCTION
In this paper, we consider the following damped focusing Klein-Gordon equation:
\begin{align}\label{1}
\left\{ \begin{array}{l}
{u_{tt}} - \Delta u + u +2\alpha {u_t} - {\left| u \right|^{p - 1}}u = 0, \\
u(0) = {u_0},\mbox{ }{\partial _t}u(0) = {u_1}\in \mathcal{H}, \\
\end{array} \right.
\end{align}
where $\mathcal{H}=H^1(\Bbb R^d)\times L^2(\Bbb R^d)$, $\alpha\ge0$. The energy is given by
$$E(f,g) = \int_{{\Bbb R^d}} {\left( {\frac{1}{2}{{\left| {\nabla f} \right|}^2} + \frac{1}{2}{{\left| f \right|}^2} + \frac{1}{2}{{\left| g \right|}^2} - \frac{1}{{p + 1}}{{\left| f \right|}^{p + 1}}} \right)} dx.$$
Dispersive equations such as Klein-Gordon equations, wave equations, Schrödinger equations have been intensively studied for decades. For $\alpha=0$, namely nonlinear Klein-Gordon equation, T. Cazenave <cit.> gave the following dichotomy: solutions either blow up at finite time or are global forward in time and bounded in $\mathcal{H}$, provided $1<p<\infty$, when $d=1,2$ and $1<p<\frac{d}{d-2}$ if $d\ge3$. For $\alpha>0$, E. Feireisl <cit.> gave an independent proof of the boundedness of the trajectory to global solutions, for $1<p<1+\min(\frac{d}{d-2},\frac{4}{d})$ when $d\ge3$, and in his paper <cit.>, the case $d=1$ is considered. N. Burq, G. Raugel, W. Schlag <cit.> studied the long time behaviors of solutions to nonlinear damped Klein-Gordon equations in radial case. They proved that radial global solutions will converge to equilibrium points as time goes infinity. A natural problem is what happens for non-radial solutions? It is widely conjectured that the solutions will decouple into the superposition of equilibrium points. A positive result given by E. Feireisl <cit.> implied there exists a global solution which decouples into a finite number of equilibrium points with different shifts from origin. Indeed, this problem is closely related to the soliton resolution conjecture in dispersive equations. The ( imprecise sense ) soliton resolution conjecture states that for “generic" large global solutions, the evolution asymptotically decouples into the superposition of divergent solitons, a free radiation term, and an error term tending to zero as time goes to infinity. For more expression and history, see A. Soffer <cit.>.
There are a lot of works devoted to the verification of the soliton resolution conjecture. T. Duyckaerts, C. Kenig, and F. Merle <cit.> first make a breakthrough on this topic. For radial data to three dimensional focusing energy-critical wave equations, they proved the solution with bounded trajectory is in fact a superposition of a finite number of rescalings of the ground state plus a radiation term which is asymptotically a free wave. One of the key ingredient of their arguments is the novel tool, called “channels of energy" introduced by <cit.> <cit.>. The method developed by them has been applied to many other situations, such as <cit.> <cit.> <cit.> <cit.> <cit.> for wave maps, <cit.> <cit.> <cit.> <cit.> for semilinear wave equations. By a weak version of outer energy inequality, the soliton resolution along a sequence of times was proved by R. Cote, C. Kenig, A. Lawrie and W. Schlag <cit.> for four dimensional critical wave equations in radial case, by R. Cote <cit.> for equivariant wave maps, and by H. Jia, and C. Kenig <cit.> for semilinear wave equations, wave maps.
It is known that (<ref>) admits a radial positive stationary solution with the minimized energy among all the non-zero stationary solutions.
Besides the ground state, (<ref>) also has an infinite number of nodal solutions which owns zero points. (see for instance H. Berestycki, and P.L. Lions <cit.>). Hence it seems that subcritical problems need different techniques. The dynamics of solutions below and slightly above the ground state is known. If $\alpha=0$, for initial data with energy below the ground state, I.E. Payne, and D.H. Sattinger <cit.> proved that the solution either blows up in finite time or scatters to zero. K. Nakanishi, and W. Schlag <cit.> described the asymptotics of the solutions with energy slightly larger than the ground state. In fact they proved the trichotomy forward in time: the solution (1) either blows up at finite time (2) or globally exists and scatters to zero (3) or globally exists and scatters to the ground states. In radial setting, the above trichotomy was obtained in K. Nakanishi, and W. Schlag <cit.>, followed by K. Nakanishi, and W. Schlag <cit.> in non-radial case. The main technical ingredient of their papers is the “one pass" theorem which excludes the existence of (almost) homoclinic orbits between the ground state and (almost) heteroclinic orbits connecting ground state $Q$ with $-Q$. N. Burq, G. Raugel, and W. Schlag <cit.> studied the longtime dynamics for damped Klein-Gordon equations in radial case. By developing some dynamical methods especially invariant manifolds, they proved the $\omega$-limit set of the trajectory is just one single point, hence they showed the dichotomy in forward time (1) the solution either blows up at finite time, (2) or converges to some equilibrium point.
In this paper, we aim to study the long time behaviors of damped Klein-Gordon equations without radial assumptions.
$$\lambda (d) = \left\{ \begin{array}{l}
\infty ,\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }d = 1,2 \\
1 + \frac{4}{{d - 2}},\mbox{ }d = 3,4.
\end{array} \right.
Then, we have the main theorem as follows:
Let $\alpha>0$, $1\le d\le 4$, $1<p<\lambda(d)$. For any data $(u_0,u_1)\in\mathcal{H}$. Then
$(i)$ either the solution of (<ref>) blows up at finite positive time,
$(ii)$ or it is global forward in time with unbounded trajectory;
$(iii)$ or for any time sequence $t_n\to\infty$, up to a subsequence, there exist $0\le J<\infty$, $x_{j,n}\in \Bbb R^d$ for $j=1,2,...,J$ and equilibrium points $\{Q^j\}$ such that
$$u(t_n) = \sum\limits_{j = 1}^J {Q^j}(x - x_{j,n}) + o_{H^1}(1),$$
and $\mathop {\lim }\limits_{t \to \infty } {\partial _t}u(t) = 0$, in $L^2$, where $\{x_{j,n}\}$ satisfies the separation property:
\mathop {\lim }\limits_{n \to \infty } \left| {{x_{j,n}} - {x_{i,n}}} \right| = 0, \mbox{ }{\rm{for}}\mbox{ }i\neq j.
An adaptation of arguments in T. Cazenave <cit.> shows for $1<p<\infty$, when d=1,2, $1<p<\frac{d}{d-2}$, when $d\ge 3$, every global solution has bounded trajectory. Therefore, for this range of $p$, we have the following dichotomy:
For $1<p<\infty$, when d=1,2, $1<p<\frac{d}{d-2}$, when $d\ge 3$, either the solution to (<ref>) blows up at finite time $((i)$ in Theorem 1.1$)$ or decouples into the superposition of equilibriums $((iii)$ in Theorem 1.1$)$.
For $3\le d\le 6$, $1<p<\frac{d}{d-2}$, the proof in E. Feireisl <cit.> is sufficient to give Corollary <ref>. Thus, in this paper, when $d\ge 3$, we always assume $p\ge \frac{d}{d-2}$.
In order to describe our proof, the following notions are needed:
Given any $h\in \Bbb R^d$, let $\tau_h:\mathcal{H}\to\mathcal{H}$ be the shift operator $\tau_h f(x)=f(x-h)$, and we denote the translation group by $G=\{\tau_h:h\in\Bbb R^d\}$. Given any $K\subseteq\mathcal{H},$ we denote the orbit of $K$ by $GK=\{gf:g\in G, f\in K\}$. If $GK=K$, then we call $K$ G-invariant. Suppose that $J\ge0$ is an integer, we let
$$JK \equiv \left\{ {{f_1} + ... + {f_J}:{f_1},{f_2},...,{f_J} \in K} \right\}.$$
We say $E\subseteq\mathcal{H}$ is G-precompact with $J$ components if $E\subseteq J(GK)$ for some compact $K\subseteq\mathcal{H}$ and $J\ge1$.
Our proof is divided into three parts. In the first step, we prove the trajectory of $u(t)$ is attracted by a G-precompact set with $J$ components, namely the existence of concentration-compact attractor. The key ingredient in this step is frequency localization and spatial localisation. The idea of “concentration compact" attractor was introduced by T. Tao <cit.>. In the second step, for any sequence going to infinity, we prove up to a subsequence there exist a finite number profiles. Then by applying perturbation theorem, we obtain a nonlinear profile decomposition. Using damping effect of (<ref>), we can show all the profiles are exactly equilibriums. Finally we prove the convergence for all time.
Our paper is organized as follows: In Section 2, we recall some preliminaries, such as Strichartz estimates, local wellposedness, perturbation theorem. In Section 3, we prove the frequency localization and spatial localization. In Section 4, we prove the existence of concentration-compact attractor. In Section 5, we extract the profiles and finish our proof by using damping.
Notation and Preliminaries
We will use the notation $X\lesssim Y$ whenever there exists some positive constant $C$ so that $X\le C Y$. Similarly, we will use $X\sim Y$ if $X\lesssim Y \lesssim X$.
We define the Fourier transform on $\mathbb{R}^d$ to be
$$F(f)(\xi) =\tfrac1{(2\pi)^d} \int_{\mathbb{R}^d} e^{- ix\cdot\xi} f(x)\,\mathrm{d}x,$$
$P_N$ is the usual Littlewood-Paley decomposition operator with frequency truncated in $N$. Similarly, we use $P_{\le N}$ and $P_{\ge N}$. Sometimes, we denote $P_{<\mu}u$ by $u_{<\mu}$. $\|u(t)\|_\mathcal{H}$ means $\|(u(t),\partial_t u(t))\|_{\mathcal{H}}$.
All the constants are denoted by $C$ and they can change from line to line.
§ PRELIMINARIES
As explained in Remark <ref>, we only need to consider
$$\left\{ \begin{array}{l}
1 < p < \infty ,\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }d = 1,2 \\
\frac{d}{{d - 2}} \le p < 1 + \frac{4}{{d - 2}},d = 3,4.
\end{array} \right.
In this section we give the Strichartz estimates, local wellposedness and perturbation theorem, we closely follow notations in <cit.>.
Consider the linear equation,
\begin{align}\label{2.1}
u_{tt}+2\alpha u_t-\Delta u+u=G, \mbox{ }\big(u(0,x),u_t(0,x)\big)=(u_0,u_1)\in \mathcal{H},
\end{align}
then by Duhamel principle,
\begin{align*}
u(t) &= {e^{ - \alpha t}}\left[ {\cos (t\sqrt { - \Delta + 1 - {\alpha ^2}} ) + \alpha \frac{{\sin (t\sqrt { - \Delta + 1 - {\alpha ^2}} )}}{{\sqrt { - \Delta + 1 - {\alpha ^2}} }}} \right]{u_0} \\
&+ {e^{ - \alpha t}}\frac{{\sin (t\sqrt { - \Delta + 1 - {\alpha ^2}} )}}{{\sqrt { - \Delta + 1 - {\alpha ^2}} }}{u_1} + \int_0^t {\frac{{\sin ((t - s)\sqrt { - \Delta + 1 - {\alpha ^2}} )}}{{\sqrt { - \Delta + 1 - {\alpha ^2}} }}} {e^{ - \alpha (t - s)}}G(s)ds \\
&\triangleq {S_{1,\alpha}}(t){u_0} + {S_{2,\alpha}}(t){u_1} + \int_0^t {{S_{2,\alpha}}} (t - s)G(s)ds.
\end{align*}
Define $S_L(t)(u_0,u_1)=S_{1,\alpha}u_0+S_{2,\alpha}u_1$.
The Strichartz estimates are given by the following lemma. We emphasize that since we only need local Strichartz estimates in this paper, it is possible to get $L^p_t(I;L^q_x)$ estimates for non-admissible pair by Hölder inequality (see (<ref>) below).
Let $t_0>T>0$, $u$ be a solution to (<ref>) on $[t_0-T,t_0+T]\times\Bbb R^d$. For $d\ge3$, $\theta^*=\frac{d+2}{d-2}$, we have
\mathop {\sup }\limits_{t \in [-T,T]} {\left\| {(u(t),\partial_tu(t))} \right\|_{\mathcal{H}}} + {\left\| u \right\|_{L_t^{{\theta ^ * }}([-T,T];L_x^{2{\theta ^ * }})}} \lesssim e^{T\alpha}\left[ {{{\left\| {({u_0},{u_1})} \right\|}_\mathcal{H}} + \int_{-T}^T {{{\left\| {G(s)} \right\|}_2}ds} } \right].
If $d=3,4$, then
$${\left\| u \right\|_{L_t^{\frac{4}{{d - 2}}}([ - T,T];L_x^{\frac{{4d}}{{d - 2}}})}} \lesssim {e^{T\alpha }}\left[ {{{\left\| {({u_0},{u_1})} \right\|}_{\cal H}} + \int_{ - T}^T {{{\left\| {G(s)} \right\|}_2}ds} } \right].
If $d=1,2$, it holds that
$$\mathop {\sup }\limits_{t \in [-T,T]} {\left\| {(u(t),\partial_tu(t))} \right\|_{\mathcal{H}}} \lesssim e^{T\alpha}\left[ {{{\left\| {({u_0},{u_1})} \right\|}_\mathcal{H}} + \int_{-T}^T {{{\left\| {G(s)} \right\|}_2}ds} } \right].
Define $Q=p+1$, when $d=1,2$, $p<Q<\frac{2p}{p(d-2)-d}$, when $3\le d\le 4$, $\frac{d}{d-2}\le p<1+\frac{4}{d-2}$, we have
\begin{align}\label{wang}
{\left\| u \right\|_{L_t^Q([ - T,T];L_x^{2p})}}\lesssim C(T)\left[ {{{\left\| {({u_0},{u_1})} \right\|}_{\cal H}} + \int_{ - T}^T {{{\left\| {G(s)} \right\|}_2}ds} } \right].
\end{align}
As a corollary of Strichartz estimates, we have the perturbation theorem.
Let $M>0$. there exists $\varepsilon_0=\varepsilon_0(M)$ satisfying the following: Let $I\subset \Bbb R^+$ is a finite interval containing $t_0$, $\tilde{u}$ is defined on $I\times\Bbb R^d$, and satisfies
\mathop {\sup }\limits_{t \in I} {\left\| {(\tilde{u},{\partial _t}\tilde{u})(t)} \right\|_\mathcal{H}}\le M.
Suppose that $v$ is a solution to (<ref>) with initial data $(v(t_0),\partial_tv(t_0))$ at time $t_0$. Let $\varepsilon\in (0,\varepsilon_0)$, suppose that
\begin{align*}
\partial_{tt}\tilde{u}+2\alpha \partial_t \tilde{u}-\Delta \tilde{u} +\tilde{u}-|\tilde u|^{p-1}\tilde{u}&=e,\\
\|e\|_{L^1_t(I;L^2_x)}+\|S_{1,\alpha}(t-t_0)(\tilde{u}-v)(t_0)\|_{L^{Q}_t(I;L^{2p}_x)}+\|S_{2,\alpha}(t-t_0)
(\partial_t\tilde{u}-\partial_tv)(t_0)\|_{L^{Q}_t(I;L^{2p}_x)}&\le \varepsilon.
\end{align*}
$$\|\tilde u-v-S_L(t-t_0)(\tilde{u}-v)(t_0)\|_{L^{\infty}_t(I;\mathcal{H})}+\|\tilde u-v\|_{L^{Q}_t(I;L^{2p}_x)}\le C(M)\varepsilon.
Combining T. Cazenave <cit.> and N. Burq, G. Raugel, W. Schlag <cit.>, we obtain the local wellposedness theorem as follows:
For $(u_0,u_1)\in \mathcal{H}$, there exists $T>0$ such that (<ref>) is well-defined in $[0,T)$, with $T$ depending on $\|(u_0,u_1)\|_{\mathcal{H}}$. Furthermore, if $\|(u_0,u_1)\|_{ \mathcal{H}}<\epsilon$ with $\epsilon$ sufficiently small, then there exits $\gamma>0$ such that
$$\|(u(t),\partial_tu(t))\|_{\mathcal{H}}\le Ce^{-\gamma t}\|(u_0,u_1)\|_{\mathcal{H}}.$$
Moreover, if the solution $u(t)$ is globally defined, then we have
\int^{\infty}_0\|\partial_tu(s)\|_2^2ds<\infty.
§ FREQUENCY LOCALIZATION AND SPATIAL LOCALIZATION
Since we focus on bounded solution throughout the paper, we assume
$$\mathop {\sup }\limits_{t \in [0,\infty)} {\left\| {(u,{\partial _t}u)(t)} \right\|_\mathcal{H}} \le E.$$
In the first step, we prove the localization of frequency, namely
For any $\mu_0>0$ there exists $c(\mu_0)>0$ depending on $E$ such that
\begin{align}
&\mathop {\lim \sup }\limits_{t \to \infty } \|P_{\ge \frac{1}{c(\mu_0)}}u(t)\|_{H^1}\le \mu_0, \label{345}\\
&\mathop {\lim \sup }\limits_{t \to \infty } \|P_{\ge \frac{1}{c(\mu_0)}}\partial_tu(t)\|_{L^2}\le \mu_0. \label{kj}
\end{align}
From Duhamel principle,
\begin{align*}
{P_{ \ge \frac{1}{{{\mu}}}}}u(t) = {S_{1,\alpha }}{P_{ \ge \frac{1}{{{\mu}}}}}{u_0} + {S_{2,\alpha }}{P_{ \ge \frac{1}{{{\mu}}}}}{u_1} + \int_0^t {{S_{2,\alpha }}} (t - s){P_{ \ge \frac{1}{{{\mu}}}}}\left( {{{\left| u \right|}^{p - 1}}u} \right)(s)ds.
\end{align*}
\begin{align*}
{\left\| {{S_{1,\alpha }}{P_{ \ge \frac{1}{{{\mu}}}}}{u_0}} \right\|_{{H^1}}}{\rm{ }} \le {e^{ - \alpha t}}{\left\| {{u_0}} \right\|_{{H^1}}},\mbox{ }\mbox{ }\mbox{ }{\left\| {{S_{2,\alpha }}{P_{ \ge {\mu}^{ - 1}}}{u_1}} \right\|_{{H^1}}}{\rm{ }} \le {e^{ - \alpha t}}{\left\| {{u_1}} \right\|_{{L^2}}},
\end{align*}
for $\mu$ sufficiently small, we have
\begin{align*}
&{\left\| {{P_{ \ge \mu^{ - 1}}}u(t)} \right\|_{{H^1}}} \le C{e^{ - \alpha t}}{\left\| {\left( {{u_0},{u_1}} \right)} \right\|_\mathcal{H}} + \int_0^t {{e^{ - \alpha (t - s)}}} {\left\| {{P_{ \ge \mu^{ - 1}}}\left( {{{\left| u \right|}^{p - 1}}u} \right)(s)} \right\|_2}ds.
\end{align*}
Let $h(u) = {\left| u \right|^{p - 1}}u$, split $u$ into $u = {P_{ \le {\mu ^{ - 1}}}}u + {P_{ \ge {\mu ^{ - 1}}}}u $, then
$$h(u) = h({P_{ \le {\mu ^{ - 1}}}}u) + {P_{ \ge {\mu ^{ - 1}}}}uO({\left| u \right|^{p - 1}}).$$
Case 1.$\mbox{ }1<p<\frac{d}{d-2}$ for $d\ge3$
Bernstein's inequality and Hölder's inequality imply
\begin{align*}
{\left\| {{P_{ \ge {\mu ^{ - 1}}}}h(u)} \right\|_2} &\le {\left\| {{P_{ \ge {\mu ^{ - 1}}}}h({P_{ \le {\mu ^{ - 1}}}}u)} \right\|_2} + {\left\| {{P_{ \ge {\mu ^{ - 1}}}}\left( {{P_{ \ge {\mu ^{ - 1}}}}uO({{\left| u \right|}^{p - 1}})} \right)} \right\|_2} \\
&\le {\mu }{\left\| {\nabla h({P_{ \le {\mu ^{ - 1}}}}u)} \right\|_2} + {\left\| {{P_{ \ge {\mu ^{ - 1}}}}uO({{\left| u \right|}^{p - 1}})} \right\|_2} \\
&\le {\mu }{\left\| {\nabla {P_{ \le {\mu ^{ - 1}}}}u{{\left| {{P_{ \le {\mu ^{ - 1}}}}u} \right|}^{p - 1}}} \right\|_2} + {\left\| {{P_{ \ge {\mu ^{ - 1}}}}u} \right\|_m}{\left\| {{{\left| u \right|}^{p - 1}}} \right\|_{\frac{{{2^*}}}{{p - 1}}}},
\end{align*}
where $\frac{1}{m}+\frac{p-1}{2^*}=\frac{1}{2}$.
By Bernstein's inequality, we have
{\mu }{\left\| {\nabla {P_{ \le {\mu ^{ - 1}}}}u{{\left| {{P_{ \le {\mu ^{ - 1}}}}u} \right|}^{p - 1}}} \right\|_2} \le {\mu }{\left\| {\nabla u} \right\|_2}\left\| {{P_{ \le {\mu ^{ - 1}}}}u} \right\|_\infty ^{p - 1} \le {\mu ^{-\frac{{d(p - 1)}}{{{2^*}}} + 1}}{\left\| {\nabla u} \right\|_2}\left\| {{P_{ \le {\mu ^{ - 1}}}}u} \right\|_{{2^*}}^{p - 1}.
Since $1<p<\frac{d}{d-2}$, we conclude for some $\kappa>0$,
\begin{align}\label{u7}
{\left\| {{P_{ \ge {\mu ^{ - 1}}}}h({P_{ \le {\mu ^{ - 1}}}}u)} \right\|_2} \le {\mu ^{ \kappa }}{\left\| u \right\|_{{H^1}}}.
\end{align}
Applying Bernstein's inequality, we have
\begin{align*}
{\left\| {{P_{ \ge {\mu ^{ - 1}}}}u} \right\|_m} &\le {\left( {\sum\limits_{N \ge {\mu ^{ - 1}}}^\infty {\left\| {{P_N}u} \right\|_m^2} } \right)^{1/2}} \le {\left( {\sum\limits_{N \ge {\mu ^{ - 1}}}^\infty {{N^{2d\left( {\frac{1}{2} - \frac{1}{m}} \right) - 2}}} {N^2}\left\| {{P_N}u} \right\|_2^2} \right)^{1/2}} \\
&\le {\mu ^{ - d\left( {\frac{1}{2} - \frac{1}{m}} \right) + 1}}{\left( {\sum\limits_{N \ge {\mu ^{ - 1}}}^\infty {{N^2}\left\| {{P_N}u} \right\|_2^2} } \right)^{1/2}}.
\end{align*}
which combined with (<ref>) gives (<ref>) by $1<p<\frac{d}{d-2}$.
Next, we bound $\partial_tu$. From Duhamel principle, we have
\begin{align*}
{\partial _t}u(t) &= - \alpha u(t) + {e^{ - \alpha \delta }}\left[ { - \sqrt { - \Delta + 1 - {\alpha ^2}} \sin \left( {t\sqrt { - \Delta + 1 - {\alpha ^2}} } \right) + \alpha \cos \left( {t\sqrt { - \Delta + 1 - {\alpha ^2}} } \right)} \right]u(t - \delta ) \\
&\mbox{ }+ {e^{ - \alpha \delta }}\cos \left( {t\sqrt { - \Delta + 1 - {\alpha ^2}} } \right){\partial _t}u(t - \delta ) + \int_{t - \delta }^{t} {\cos \left( {(t-s)\sqrt { - \Delta + 1 - {\alpha ^2}} } \right){e^{ - \alpha (t - s)}}} \left( {{{\left| u \right|}^{p - 1}}u} \right)(s)ds.
\end{align*}
For $\mu_1\ll\mu_0$, (<ref>) implies that there exist $\eta>0$ and $T_0>0$ such that
\|P_{\ge\eta^{-1}}u(t)\|_{H^1}<\mu_1,
for $t>T_0$. Taking $\delta$ large such that $e^{-\alpha\delta}<\mu_1$, then for $t>T_0+\delta$, it suffices to prove
$${\left\| {{P_{ \ge {\eta ^{ - 1}}}}h(u(t))} \right\|_2} \le {\eta ^{\lambda }},$$
for some $\lambda>0$.
The rest of the proof of (<ref>) is the same as (<ref>).
Case 2.$\mbox{ }1<p<\infty$ for $d=1$.
By Bernstein's inequality, Hölder's inequality, Sobolev embedding theorem,
$${\left\| {{P_{ \ge {\mu ^{ - 1}}}}h(u)} \right\|_2} \le \mu {\left\| {\nabla {P_{ \le {\mu ^{ - 1}}}}u} \right\|_2}{\left\| {{{\left| {{P_{ \le {\mu ^{ - 1}}}}u} \right|}^{p - 1}}} \right\|_\infty } + {\left\| {{P_{ \ge {\mu ^{ - 1}}}}u} \right\|_2}{\left\| {{{\left| u \right|}^{p - 1}}} \right\|_\infty } \le \mu.
The remaining proof is the same as Case 1.
Case 3.$\mbox{ }1<p<\infty$ for $d=2$.
The proof is also similar to Case 1, we omit it.
Case 4.$\mbox{ }\frac{d}{d-2}\le p<1+\frac{4}{d-2}$ for $d=3,4$.
Choosing $\delta$ sufficiently large, such that $e^{-\alpha\delta}\ll \mu_0$. Fix $t_0>\delta$, consider the interval $I\equiv[t_0-\delta,t_0]$.
By Duhamel principle, for $t\in I$, we have
Fix $\varepsilon$ sufficiently small, divide $I$ into subintervals $I_1, I_2,...,I_n$, such that $|I_j|\sim \varepsilon$, then $n\sim \frac{\delta}{\varepsilon}$.
Taking $p-1<R<\frac{4}{d-2}$, Hölder's inequality and Strichartz estimates give
\begin{align*}
\|u(t)\|_{L^R(I_j;L^{\frac{4d}{d-2}})}&\le |I_j|^{\frac{1}{R}-\frac{d-2}{4}}\|u(t)\|_{L^{\frac{4}{d-2}}_t(I_j;L^{\frac{4d}{d-2}}_x)}\\
\end{align*}
By Hölder's inequality, Sobolev embedding theorem,
$${\left\| {h(u(s))} \right\|_{L_t^1({I_j};{L^2})}} \le \int_{{I_j}} {{{\left\| {{{\left| u \right|}^{p - 1}}} \right\|}_{L_x^{\frac{{4d}}{{(p - 1)(d - 2)}}}}}{{\left\| u \right\|}_{L_x^\theta }}dt} \le C(E){\left| {{I_j}} \right|^{ - \frac{{p - 1}}{R} + 1}}\left\| u \right\|_{L_t^R({I_j};{L^{4d/(d - 2)}})}^{p - 1},
where $\frac{1}{\theta}+\frac{{(p - 1)(d - 2)}}{{4d}}=\frac{1}{2}$.
\|u(t)\|_{L^{R}_t(I_j;L^{\frac{4d}{d-2}}_x)}\le C(E)\varepsilon^{\frac{1}{R}-\frac{d-2}{4}}\big(1+{\left| {{\varepsilon}} \right|^{ - \frac{{p - 1}}{R} + 1}}\left\| u \right\|_{L_t^R({I_j};{L^{4d/(d - 2)}})}^{p - 1}\big).
Since $p\ge 2$ in Case 3, by continuity method, we have
$$\|u(t)\|_{L^{R}_t(I;L^{\frac{4d}{d-2}}_x)}\le 8C(E)\varepsilon^{\frac{1}{R}-\frac{d-2}{4}}.
Summing up all the intervals, we get
\begin{align}\label{kjna}
\|u(t)\|_{L^{R}_t(I;L^{\frac{4d}{d-2}}_x)}\lesssim C(\delta,E),
\end{align}
where $C(E,\delta)$ is independent of $t$.
Again by Duhamel principle,
$${\left\| {{P_{ \ge {\mu ^{ - 1}}}}u(t_0)} \right\|_{{H^1}}} \le C(E){e^{ - \alpha \delta }} + \int_{t_0 - \delta }^{t_0} {{e^{ - \alpha (t - s)}}{{\left\| {{P_{ \ge {\mu ^{ - 1}}}}h(u(s))} \right\|}_2}} ds.$$
Hence it suffices to bound ${\left\| {{P_{ \ge {\mu ^{ - 1}}}}h(u)} \right\|_2}.$
Using similar arguments in Case 1, by Bernstein's inequality and Hölder's inequality, we have
$${\left\| {{P_{ \ge {\mu ^{ - 1}}}}h(u)} \right\|_2} \le \mu {\left\| {\nabla {P_{ \le {\mu ^{ - 1}}}}u} \right\|_2}{\left\| {{{\left| {{P_{ \le {\mu ^{ - 1}}}}u} \right|}^{p - 1}}} \right\|_\infty } + {\left\| {{P_{ \ge {\mu ^{ - 1}}}}u} \right\|_\theta }{\left\| {{{\left| u \right|}^{p - 1}}} \right\|_{\frac{{4d}}{{(p - 1)(d - 2)}}}},
where $\frac{1}{\theta}+\frac{{(p - 1)(d - 2)}}{{4d}}=\frac{1}{2}$.
Applying Bernstein's inequality, we get
$${\left\| {{{\left| {{P_{ \le {\mu ^{ - 1}}}}u} \right|}^{p - 1}}} \right\|_\infty } \le {\mu ^{ - (p - 1)\frac{{d - 2}}{4}}}{\left\| u \right\|^{p-1}_{\frac{{4d}}{{d - 2}}}},$$
which combined with Hölder's inequality and (<ref>) give
\begin{align*}
&\int_{{t_0} - \delta }^{{t_0}} {{e^{ - \alpha (t - s)}}} \mu {\left\| {\nabla {P_{ \le {\mu ^{ - 1}}}}u(s)} \right\|_2}{\left\| {{{\left| {{P_{ \le {\mu ^{ - 1}}}}u(s)} \right|}^{p - 1}}} \right\|_\infty }ds \\
&\le C(E){\mu ^{1 - (p - 1)\frac{{d - 2}}{4}}}\int_{{t_0} - \delta }^{{t_0}} {{e^{ - \alpha (t - s)}}} \left\| u \right\|_{4d/(d - 2)}^{p - 1}ds \\
&\le C(E){\mu ^{1 - (p - 1)\frac{{d - 2}}{4}}}\left\| u \right\|_{L_t^R(I;{L_x^{4d/(d - 2)}})}^{p - 1} \\
&\le C(E,\delta ){\mu ^{1 - (p - 1)\frac{{d - 2}}{4}}}.
\end{align*}
This bound is acceptable since $1<p<1+\frac{4}{d-2}$.
The same arguments as Case 1 yield the desired bound for
${\left\| {{P_{ \ge {\mu ^{ - 1}}}}u} \right\|_\theta }$ by $1<p<1+\frac{4}{d-2}$.
Therefore, we finish our proof.
Now, we prove the spatial localization, namely the following proposition:
Let $u$ be a global solution to (<ref>) with $\mathcal{H}$ norm at most $E>0$. Then there exit $J=J(E)$ depending only on $E$, and functions $x_1(t),...,x_J(t):\Bbb R^+\to \Bbb R^d$, such that for any $\mu>0$ there exits $\eta=\eta(E,\mu)>0$ such that
$$\mathop {\lim \sup }\limits_{t \to \infty } {\int_{dist(x,\{ {x_1}(t),...,{x_J}(t)\} ) > {\eta ^{ - 1}}} {\left| {\nabla u} \right|} ^2} + {\left| u \right|^2} + {\left| {{\partial _t}u} \right|^2} \le \mu.$$
Before proving Proposition <ref>, we first prove a weaker proposition:
Let $u$ be a global solution to (<ref>) with $\mathcal{H}$ norm at most $E>0$. Then for $\mu_0>0$, there exits $J=J(E,\mu_0)$ and functions $\tilde{x}_1(t),...,\tilde{x}_J(t):\Bbb R^+\to \Bbb R^d$, and $\eta=\eta(E,\mu_0)>0$ such that
$$\mathop {\lim \sup }\limits_{t \to \infty } \int_{dist(x,\{ {\tilde{x}_1}(t),...,{\tilde{x}_J}(t)\} ) > {\eta ^{ - 1}}} {\left| u \right|^2}\le \mu_0.$$
The whole proof is divided into five parts. Fix $E>0$ and $\mu_0$,
choose parameters $\mu_0\gg\mu_1\gg\mu_2\gg\mu_3\gg\mu_4>0$.
Step One. Selecting a “good" time sequence
For any $t_0>T_0$, consider the time interval $[t_0-\mu_1^{-1}, t_0+\mu_1^{-1}]$. Since
$$\mathop {\lim }\limits_{{t_0} \to \infty } \int_{{t_0} - \mu _1^{ - 1}}^{{t_0} + \mu _1^{ - 1}} {\left\| {{\partial _t}u(s)} \right\|_2^2} ds = 0,$$
there exists $T_1$ sufficiently large such that for $t_0>T_1$,
$$\int_{{t_0} - \mu _1^{ - 1}}^{{t_0} + \mu _1^{ - 1}} {\left\| {{\partial _t}u(s)} \right\|_2^2}ds\le \mu_2^2.$$
Thus there exits good time $t_*\in [t_0-\mu_1^{-1}, t_0+\mu_1^{-1}]$, such that
\begin{align}\label{pk}
\|\partial_tu(t_*)\|_2\le \mu_2^2.
\end{align}
Step Two. $L^{\infty}_x$ spatial localization at fixed time.
From Lemma 3.1, for any $\mu_2>0$ there exists $c(\mu_2)>0$, such that for $T>T_0$,
\begin{align}\label{s1}
\|u_{>c(\mu_2)^{-1}}\|_{H^1}\le \mu_2^2.
\end{align}
As step one, we fix time $t>T_1$. Now we claim there exist $J(E,\mu_2,\mu_3)$ and $x_1(t),...,x_J(t):\Bbb R^+ \to \Bbb R^d$, such that
\begin{align}\label{45}
|u_{<c(\mu_2)^{-1}}(t,x)|<\mu_3, \mbox{ }{\rm{whenever}}\mbox{ }dist(x,\{x_1(t),...,x_J(t)\}) \ge 2\mu_3^{-1}.
\end{align}
Indeed, let $x_1(t),...,x_J(t)$ be a maximal $2\mu_3^{-1}$-separated set of points in $R^d$ such that
\begin{align*}
|u_{<c(\mu_2)^{-1}}(t,x_j(t))|\ge \mu_3 \mbox{ }\mbox{ }{\rm{for}}\mbox{ }\mbox{ }{\rm{all}} \mbox{ }\mbox{ } 1\le j\le J(t).
\end{align*}
It is easy to verify
|u_{<c(\mu_2)^{-1}}(t,x_j(t))|\lesssim c(\mu_2)^{d/2}\int_{|x-x_j(t)|\le \mu_3^{-1}}|u|^2dx+\mu_3^d\|u\|_2.
Then we have
\mu_3\le |u_{<c(\mu_2)^{-1}}(t,x_j(t))|\lesssim c(\mu_2)^{d/2}\int_{|x-x_j(t)|\le \mu_3^{-1}}|u|^2dx.
Since $x_j(t)$ are $2\mu_3^{-1}$-separated, thus $J$ is finite depending on $\mu_2,\mu_3$. By the maximal property of the set $\{x_1,...,x_J\}$,
we conclude
|u_{<c(\mu_2)^{-1}}(t,x_j(t))|<\mu_3, \mbox{ }\mbox{ } {\rm{whenever}} \mbox{ }\mbox{ }dist(x,\{x_1,...,x_J\})\ge 2\mu^{-1}_3.
Step Three. $L^{\infty}_x$ spatial localization on an interval centered at good time.
For $t>T_1$, consider good time $t_*$ in $[t-\mu_1^{-1},t+\mu_1^{-1}]$. Then $[t-\mu_1^{-1},t+\mu_1^{-1}]\subset[t_*-4\mu_1^{-1},t_*+4\mu_1^{-1}]\equiv I$. Define the distance function $D(x)=dist(x,\{x_1(t_*),x_2(t_*),...,x_J(t_*)\})$. Let $\chi:\Bbb R^d\to R^+$ be a smooth cutoff function which equals 1 for $D(x)\le 2\mu_3^{-1}$, vanishes for $D(x)\ge 3\mu_3^{-1}$, and $\nabla^k\chi=O_k(\mu_3^k)$ for $k\ge0$. Then we have
Claim 1. $\|S_{1,\alpha}[(1-\chi)u(t_*)]\|_{L^{Q}_t(I;L^{2p}_x)}+\|S_{2,\alpha}[(1-\chi)u(t_*)]\|_{L^{\theta*}_t(I;L^{2\theta^*}_x)}\lesssim_{\mu_1} \mu_2^2.$
By Strichartz estimates,
$$\|S_{2,\alpha}[(1-\chi)u(t_*)]\|_{L^{Q}_t(I:L^{2p}_x)} \le Ce^{C\mu_1^{-1}}\|\partial_tu(t_*)\|_2,$$
which combined with (<ref>) yields the desired bounds for $S_{2,\alpha}$. Since high frequency is small by (<ref>), it suffices to prove
$${\left\| {{S_{1,\alpha}}[(1 - \chi ){P_{ \le C(\mu _2)^{-1}}}u({t_*})]} \right\|_{L_t^{Q}(I;L_x^{2{p}})}}\lesssim_{\mu_1}{\mu _2}^2.$$
From the rapid decay of the convolution kernel of $P_{<c(\mu_2)^{-1}}$ and the support of $1-\chi$, we see that $(1-\chi)P_{<c(\mu_2)^{-1}}(1_{D<\mu_3^{-1}}P_{<c(\mu_2)^{-1}}u)$ can be absorbed by $\mu_2^2$, it suffices to prove
$${\left\| {{S_{1,\alpha }}\left[ {(1 - \chi ){P_{ < C{{({\mu _2})}^{ - 1}}}}\left( {{1_{D > \mu _3^{ - 1}}}{P_{ < C{{({\mu _2})}^{ - 1}}}}u({t_*})} \right)} \right]} \right\|_{L_t^{{Q}}(I;L_x^{2p})}} \le \mu _2^2.
Indeed, stationary phase shows that the operator ${{S_{1,\alpha }}(1 - \chi ){P_{ < C{{({\mu _2})}^{ - 1}}}}}$ have an operator norm of $C_1(\mu_2)$ on $L^{2\theta^*}$, then thanks to (<ref>), for some $\delta>0$, we have
\begin{align*}
&{\left\| {{S_{1,\alpha }}\left[ {(1 - \chi ){P_{ < C{{({\mu _2})}^{ - 1}}}}\left( {{1_{D > \mu _3^{ - 1}}}{P_{ < C{{({\mu _2})}^{ - 1}}}}u({t_*})} \right)} \right]} \right\|_{L_t^{{Q}}(I;L_x^{2{p}})}} \\
&\le {C_1}({\mu _2}){\left\| {{1_{D > \mu _3^{ - 1}}}{P_{ < C{{({\mu _2})}^{ - 1}}}}u({t_*})} \right\|_{L_t^{{Q}}(I;L_x^{2{p}})}} \\
&\le {C_1}({\mu _2})\left\| {{1_{D > \mu _3^{ - 1}}}{P_{ < C{{({\mu _2})}^{ - 1}}}}u({t_*})} \right\|_{L_t^{{Q}}(I;L_x^\infty )}^\delta \left\| {{1_{D > \mu _3^{ - 1}}}{P_{ < C{{({\mu _2})}^{ - 1}}}}u({t_*})} \right\|_{L_t^{{Q}}(I;L_x^{2})}^{1 - \delta } \\
&\le {C_1}({\mu _2},{\mu _1})\mu _3^\delta \lesssim {\mu_2 ^2}.
\end{align*}
Claim 2. $\|1_{D>\mu_4^{-2}}u\|_{L^{Q}_t(I:L^{2p}_x)}\lesssim_{\mu_1}{\mu _2}.$
This claim can be proved by perturbation theorem and Strichartz estimates. Indeed, let $v$ be a solution to (<ref>) on $I$ with initial data $v(t_*)=\chi u(t_*)$, $\partial_tv(t_*)=\partial_t u(t_*)$. Then by perturbation theorem, Claim 1, (<ref>), we have
\|u-v\|_{L_t^{{Q}}(I;L_x^{2p})}\lesssim_{\mu_1} \mu_2^2.
It suffices to prove
\begin{align}\label{dfr}
\|1_{D>\mu_4^{-2}}v\|_{L_t^{{Q}}(I;L_x^{2p})}\lesssim_{\mu_1} \mu_2^2.
\end{align}
Choosing another weight function $W:\Bbb R^d\to \Bbb R^+$ comparable to $1+\mu_4D$ which obeys the bounds $\nabla W, \nabla^2W=O(\mu_4)$. Since $W\chi=O(1)$, we have
\|Wv(t_0)\|_2\lesssim1.
Since $v$ solves (1), we have
{\partial _{tt}}(Wv) + 2\alpha {\partial _t}(Wv) - \Delta (Wv) + Wv = W{\left| v \right|^{p - 1}}v + O({\mu _4}\left| v \right|) + O({\mu _4}\left| {\nabla v} \right|).
Strichartz estimates imply
$${\left\| {Wv} \right\|_{L_t^{{\theta ^*}}(I;L_x^{2{\theta ^*}})}} + {\left\| {\big(Wv,\partial_t(Wv)\big)} \right\|_{L_t^\infty (I;\mathcal{H})}} \lesssim {\left\| {(Wv(t'),{\partial _t}Wv(t'))} \right\|_\mathcal{H}} + {\left\| {W{{\left| u \right|}^p}} \right\|_{L_t^1(I;L_x^2)}} + {\mu _4},
for any subinterval $I'$ of $I$ and any $t'\in I'$. Denote the left side by $X(I')$, Hölder's inequality and Sobolev embedding theorem reveal that
{\left\| {W{{\left| u \right|}^p}} \right\|_{L_t^1(I';L_x^2)}} \le C{\left| {I'} \right|^{\frac{4}{{d + 2}}}}X(I').
Chopping $I$ up to sufficiently small intervals, we have
$$X(I')\le C(\mu_1);$$
particularly, we conclude
\|1_{D>\mu^{-2}_4}v\|_{L_t^{{\theta ^*}}(I;L_x^{2\theta^*})}\le C(\mu_1)\mu_4,
which yields (<ref>) by interpolation inequality thus finishing the proof of Claim 2.
Claim 3. $\|1_{D>\mu_4^{-3}}\int_IS_{2,\alpha}(t-s)\big(|u|^{p-1}u\big)(s)ds\|_{L^2(\Bbb R^d}\lesssim_{\mu_1}\mu _2.$
From finite speed of propagation, $S_{2,\alpha}(t-s)\big(|u|^{p-1}u1_{D\le \mu_4^{-2}}\big)$ is supported in $D\le \mu_4^{-2}+4\mu_1^{-1}$.
Therefore $1_{D>\mu_4^{-3}}\int_IS_{2,\alpha}(t-s)\big(|u|^{p-1}u(s)1_{D\le \mu_4^{-2}}\big)ds=0$. Thus it suffices to prove
\begin{align}\label{ko}
{\left\| {\int_I {{S_{2,\alpha}}} (t - s)|u{|^{p - 1}}u(s){1_{D \ge \mu _4^{ - 2}}}ds} \right\|_{{L^2}}}{ \mathbin{\lower.3ex\hbox{$\buildrel<\over
{\smash{\scriptstyle\sim}\vphantom{_x}}$}} _{{\mu _1}}}{\mu _2}.
\end{align}
By Strichartz estimates, the left is bounded by
$$\||u|^p1_{D\ge \mu_4^{-2}}\|_{L^1_tL^2_x}.$$
Then (<ref>) follows from Hölder's inequality and Claim 2.
Step Four. $L^2$ localization of good times.
In this step, we prove the $L^2$ localization of $u(t_*)$, namely for $T_1$ sufficiently large, $t>T_1$,
\begin{align}\label{ol}
\|1_{D>\mu_4^{-3}}u(t_*)\|_2=O_{L^2}(\mu_1).
\end{align}
The proof is based on the decay of linear part and Claim 3 in step three. Indeed, from Duhamel principle and Claim 3,
Then since $S_{1,\alpha}$ and $S_{2,\alpha}$ have a exponential decay, we obtain
\begin{align*}
\left\| {{1_{D > \mu _4^{ - 3}}}u({t_*})} \right\|_2^2 &= \left\langle {{1_{D > \mu _4^{ - 3}}}u({t_*}),{S_{1,\alpha}}(\mu _1^{ - 1})u({t_*} - \mu _1^{ - 1})} \right\rangle + \left\langle {{1_{D > \mu _4^{ - 3}}}u({t_*}),{S_{2,\alpha}}(\mu _1^{ - 1}){\partial _t}u({t_*} - \mu _1^{ - 1})} \right\rangle + O({\mu _2}) \\
&\le {e^{ - \mu _1^{ - 1}\alpha }}\left\| {u({t_*})} \right\|_2^2 + O({\mu _2}) \lesssim {\mu _1}.
\end{align*}
Thus (<ref>) follows.
Step Five. $L^2$ localization of all time.
First from Duhamel principle and similar arguments as step four, it is easy to verify,
$$\|1_{D\ge \mu_4^{-3}}u(t)\|_2\le \mu_1,$$
for $t\in(t_*, t_*+4\mu_1^{-1})$.
Indeed, from Duhamel principle, finite speed of propagation, Claim 2 and Claim 3, we have
\begin{align*}
{\left\| {{1_{D > \mu _4^{ - 4}}}u(t)} \right\|_2} &\le {\left\| {{1_{D > \mu _4^{ - 4}}}{S_{1,\alpha }}(t - {t_*})u({t_*})} \right\|_2} + {\left\| {{1_{D > \mu _4^{ - 4}}}{S_{2,\alpha }}(t - {t_*})u({t_*})} \right\|_2} \\
&+ {\left\| {{1_{D > \mu _4^{ - 4}}}\int_{{t_*}}^t {{e^{ - \alpha (t - s)}}{S_{2,\alpha }}(t - s)\left( {{{\left| u \right|}^{p - 1}}u} \right)(s)} } \right\|_2} \\
&\lesssim {\left\| {{1_{D > \mu _4^{ - 3}}}u({t_*})} \right\|_2} + {\left\| {{1_{D > \mu _4^{ - 3}}}u({t_*})} \right\|_2} + {\mu _2} \lesssim {\mu _1}.
\end{align*}
Splitting the whole interval $[T_1,\infty)$ into subintervals with length $2\mu_1^{-1}$, denote these subintervals as $I_1, I_2, I_3,...$. Denote $t_*\in I_j$ by $t^j_*$. It is obvious that $I_{j+1}$ is covered by $(t^j_*,t^j_* + 4\mu _1^{ - 1})$, for $j=1,2,...$. Now let's define $\tilde{x}_j(t)$ for each $t$ by the following rule:
For $t\in I_{j+1}$, take $\tilde{x}_j(t)=x_j(t^j_*)$.
It is direct to see $\tilde{x}_j(t)$ defined above satisfies Proposition <ref> for all $t>T_1+2\mu_1^{-1}$.
Let $u$ be a global solution to (<ref>) with $\mathcal{H}$ norm at most $E>0$. Then for $\mu_0>0$, there exits $J=J(E,\mu_0)$ and functions $\tilde{x}_1(t),...,\tilde{x}_J(t):\Bbb R^+\to \Bbb R^d$, and $\eta=\eta(E,\mu_0)>0$ such that
$$\mathop {\lim \sup }\limits_{t \to \infty } {\int_{dist(x,\{ {\tilde{x}_1}(t),...,{\tilde{x}_J}(t)\} ) > {\eta ^{ - 1}}} {\left| {\nabla u} \right|} ^2} + {\left| u \right|^2} + {\left| {{\partial _t}u} \right|^2} \le \mu_0.$$
Choose $\mu_4\ll\mu_3\ll\mu_2\ll\mu_1\ll\mu_0$ as Proposition <ref>. Suppose that $t\in I_{j+1}$, then $0<t-t^j_*<4\mu_1^{-1}$.
Define $D(t)={dist(x,\{ {\tilde{x}_1}(t),...,{\tilde{x}_J}(t)\})}$. Then the proof of Proposition <ref> implies that for all $t\in I_{j+1}$,
\begin{align}\label{xcv}
\end{align}
Let $\chi_1$ be a cutoff function supported in $D(t^j_*)>\mu_4^{-4}$, which equals one in $D>\mu_4^{-5}$, with bound $|\nabla\chi_1|\lesssim \mu_4.$ Therefore we have
$$ \int_{D(t_*^j)> \mu _4^{ - 5}} {{{\left| {\nabla u(t)} \right|}^2} dx\le } {\left\| {u(t){\chi _1}} \right\|_{{H^1}}} + {\mu _4}.$$
Duhamel principle and finite speed of propagation give
\begin{align*}
u(t)\chi_1 &= {S_{1,\alpha }}(\mu _1^{ - 1}){1_{D(t_*^j) > \mu _4^{ - 3}}}u(t - \mu _1^{ - 1}) + {S_{2,\alpha }}(\mu _1^{ - 1}){1_{D(t_*^j) > \mu _4^{ - 3}}}{\partial _t}u(t - \mu _1^{ - 1}) \\
&+ \int_{t - \mu _1^{ - 1}}^t {{S_{2,\alpha }}(t - s) {{{\left| u \right|}^{p - 1}}u} } {1_{D(t_*^j) > \mu _4^{ - 3}}}(s)ds.
\end{align*}
By Strichartz estimates and exponential decay of ${S_{1,\alpha }},S_{2,\alpha}$, we get
\begin{align*}
{\left\| {u(t){\chi _1}} \right\|_{{H^1}}} \le C(E){e^{ - \alpha \mu _1^{ - 1}}}
+ {\left\| {{{\left| u \right|}^{p - 1}}u{1_{D(t_*^j) > \mu _4^{ - 3}}}} \right\|_{L_t^{{1}}((t - \mu _1^{ - 1},t);L_x^{2})}}.
\end{align*}
Then Claim 2, $(t-\mu^{-1}_{1},t)\subset (t^j_*-4\mu_1^{-1},t^j_*+4\mu_1^{-1})$, and (<ref>) imply,
\begin{align}\label{bounds}
\int_{D(t) > \mu _4^{ - 5}} {{{\left| {\nabla u(t)} \right|}^2} dx\le } {\mu _1},
\end{align}
which gives us the desired bound for $\nabla u(t)$.
Next, we prove the desired bound for $\partial_tu$.
By Duhamel principle, and finite speed of propagation, we obtain
\begin{align*}
{1_{D(t_*^j) > \mu _4^{ - 4}}}u(t) &= {S_{1,\alpha }}(t - t_*^j){1_{D(t_*^j) > \mu _4^{ - 3}}}u(t_*^j) + {S_{2,\alpha }}(t - t_*^j){1_{D(t_*^j) > \mu _4^{ - 3}}}{\partial _t}u(t_*^j)\\
&+ \int_{t_*^j}^t {{e^{ - \alpha (t - s)}}}S_{2,\alpha}(t-s) {1_{D(t_*^j) > \mu _4^{ - 3}}}{\left| {u(s)} \right|^{p - 1}}u(s)ds.
\end{align*}
Direct calculations yield
\begin{align*}
{\left\| {{\partial _t}\left[ {{1_{D(t_*^j) > \mu _4^{ - 4}}}u(t)} \right]} \right\|_2} \le&{\left\| {{1_{D(t_*^j) > \mu _4^{ - 3}}}u(t_*^j)} \right\|_{{H^1}}} + {\left\| {{1_{D(t_*^j) > \mu _4^{ - 3}}}{\partial _t}u(t_*^j)} \right\|_2} \\
&+ {\left\| {{1_{D(t_*^j) > \mu _4^{ - 3}}}{{\left| u \right|}^{p - 1}}u} \right\|_{L_t^{{1}}(I;L_x^{2})}},
\end{align*}
where $I=[t^j_*-\mu_1^{-1},t^j_*+\mu_1^{-1}]$. Then Claim 2, (<ref>) and (<ref>) imply
$${\left\| {{\partial _t}\left[ {{1_{D(t^j_*) > \mu _4^{ - 4}}}u(t)} \right]} \right\|_2} \lesssim {\mu _1}.$$
Then the desire bound follows from (<ref>).
From the proof of Proposition 10.1 in T. Tao <cit.>, Proposition <ref> is a corollary of Proposition <ref> and the following lemma.
Let $u$ be a global solution with $\mathcal{H}$ norm at most $E$. Suppose that we have the energy concentration bound
\int_{|x-x_0|<R}|u(t_0,x)|^2+|\nabla u(t_0,x)|^2+|\partial_t u(t_0,x)|^2dx\ge \eta_1^2
for some $x_0\in \Bbb R^d$, $t_0\in \Bbb R^+$, $R>0$, and sufficiently small $\eta_1>0$. Then, if $t_0$ is sufficiently large depending on $u,E,x_0,R.\eta_1$, we have the improved energy concentration
\int_{|x-x_0|<R'}|u(t_0,x)|^2+|\nabla u(t_0,x)|^2+|\partial_t u(t_0,x)|^2dx\ge \beta(E),
for some $\beta(E)>0$ independent of $\eta_1$ and some $R'$ depending on $E,R,\eta_1$.
The proof of Lemma <ref> can be reduced to the following lemma.
Given $E>0$, $\eta_1>0$ sufficiently small, there exists $\beta>0$ with the following property:
Suppose that we have the energy concentration bound
\eta_1^2\le \int_{|x-x_0|<R}{\left| {\nabla u}(t_0,x) \right|} ^2 + {\left| u(t_0,x) \right|^2} + \left| {{\partial _t}u(t_0,x)} \right|^2\le \beta
for some $x_0\in \Bbb R^d$, $t_0\in \Bbb R^+$, $R>0$, and some global solution $u$ with $\mathcal{H}$ norm at most $E$. Then, if $t_0$ is sufficiently large depending on $E,x_0,R, \eta_1$, we have
\int_{|x-x_0|<R'}{\left| {\nabla u(t_0,x)} \right|} ^2 + {\left| u (t_0,x)\right|^2} + \left| {{\partial _t}u(t_0,x)} \right|^2\ge \int_{|x-x_0|<R}{\left| {\nabla u(t_0,x)} \right|} ^2 + {\left| u (t_0,x)\right|^2} + \left| {{\partial _t}u(t_0,x)} \right|^2+\eta_4^2,
for some $\eta_4(E,\eta_1)>0$ and $R'(E,R,\eta_1,\eta_4)$.
For simplicity, define $e={\left| {\nabla u(t_0,x)} \right|} ^2 + {\left| u (t_0,x)\right|^2} + \left| {{\partial _t}u(t_0,x)} \right|^2$.
Fix $E>0$, let $\beta>0$ be a sufficiently small quantity to be determined. Choose parameters
$\eta_1\gg\eta_2\gg\eta_3\gg\eta_4>0$. Let $R_0>max(32R, \frac{32}{\eta_3})$. Suppose by contradiction that there exists $R'>R_0$ such that our claim fails, then
$$\int_{R<|x-x_0|<R'}e(t_0,x)dx\lesssim \eta_4^2,$$
especially, we have
\int_{|x|<R'}e(t_0,x)\lesssim \beta.
Choose $\beta<\epsilon$, where $\epsilon$ is the constant in Proposition <ref>. Denote the solution of (1.1) with initial data $1_{|x|<R'}u(t_0,x)$ at $t_0$. Proposition <ref> implies
\begin{align}\label{fg}
\|\tilde{u}\|_{\mathcal{H}}\le e^{-\gamma (t-t_0)}\|\tilde{u}(t_0)\|_{\mathcal{H}}\lesssim \beta .
\end{align}
Finite speed of propagation implies
$$u(x,t)=\tilde{u}(x,t) \mbox{ }{\rm{in}}\mbox{ } \{(x,t):|x|<R'-|t-t_0|\}.
Consider a time interval $I=[t_0,t_0+\eta_3^{-1}]$. Then for $t\in I$, we have
Combining with (<ref>), we have verified
\int_{|x|<R'/2}e(t+\eta_3^{-1},x)dx\lesssim e^{-\eta_3^{-1}}\beta\lesssim \eta^3.
If we have obtained
\begin{align}\label{cv}
\inf_{t\in I}\int_{|x|<R'/2}e(t,x)\ge \eta_1^2,
\end{align}
then contradiction follows. Hence, it suffices to prove (<ref>). By Lemma 3.1, there exists $\mu>0$, $T_0>0$, such that for any $t>T_0$,
\|P_{>\mu^{-1}}u\|_{H^1}+\|P>{\mu^{-1}}\partial_tu\|_{L^2}\le \eta_1^6.
Hence it suffices to prove
\mathop {\inf }\limits_{t \in I} {\int_{\left| x \right| < R'/2} {\left| {{P_{ \le {\mu ^{ - 1}}}}u} \right|} ^2} + {\left| {\nabla {P_{ \le {\mu ^{ - 1}}}}u} \right|^2} + {\left| {{\partial _t}{P_{ \le {\mu ^{ - 1}}}}u} \right|^2}dx \gtrsim \eta _1^2.
Let $\psi(x)$ be a smooth cutoff function which equals 1 in $\{|x|<R'/4\}$, vanishes when $|x|>R'/2$, with bound $|\nabla \psi(x)|=O(R'^{-1})$, then
\begin{align*}
&\frac{d}{{dt}}\int_{{\Bbb R^d}} {\psi (x)} \left[ {{{\left| {{P_{ \le {\mu ^{ - 1}}}}u} \right|}^2} + {{\left| {\nabla {P_{ \le {\mu ^{ - 1}}}}u} \right|}^2} + {{\left| {{P_{ \le {\mu ^{ - 1}}}}{u_t}} \right|}^2}} \right]dx \\
&= 2\int_{{\Bbb R^d}} {\psi (x)} \left( {{P_{ \le {\mu ^{ - 1}}}}{u_t}} \right){P_{ \le {\mu ^{ - 1}}}}h(u) - 4\alpha \int_{{\Bbb R^d}} {\psi (x)} {\left| {{P_{ \le {\mu ^{ - 1}}}}{u_t}} \right|^2}dx - 2\int_{{\Bbb R^d}} {\nabla \psi (x)} \left( {\nabla {P_{ \le {\mu ^{ - 1}}}}u} \right)\left( {{P_{ \le {\mu ^{ - 1}}}}{u_t}} \right)dx.
\end{align*}
$${e_\mu }(t) = \left| {{P_{ \le {\mu ^{ - 1}}}}u} \right| + {\left| {\nabla {P_{ \le {\mu ^{ - 1}}}}u} \right|^2} + {\left| {{\partial _t}{P_{ \le {\mu ^{ - 1}}}}u} \right|^2}.$$
Hölder's inequality yield,
\begin{align*}
&\left| {\int_{{\Bbb R^d}} {\psi (x)} {e_\mu }(t)dx - \int_{{\Bbb R^d}} {\psi (x)} {e_\mu }({t_0})dx} \right| \\
&\le \int_{{t_0}}^{{t_0} + 1/{\eta _3}} {\left| {\frac{d}{{dt}}\int_{{\Bbb R^d}} {\psi (x)} {e_\mu }dx} \right|dt} \\
&\le C\int_{{t_0}}^{{t_0} + 1/{\eta _3}} {{{\left\| {{u_t}} \right\|}_2}{{\left\| {{P_{ \le {\mu ^{ - 1}}}}h(u)} \right\|}_2} + \left\| {{u_t}} \right\|_2^2 + {{\left\| {{u_t}} \right\|}_2}{{\left\| {\nabla u} \right\|}_2}dtdt}.
\end{align*}
For $d\ge3$, $1<p\le \frac{d}{d-2}$, Sobolev embedding theorem implies ${{{\left\| {{P_{ \le {\mu ^{ - 1}}}}h(u)} \right\|}_2}}\le C(E)$, thus
\begin{align}\label{guf}
\left| {\int_{{\Bbb R^d}} {\psi (x)} {e_\mu }(t)dx - \int_{{\Bbb R^d}} {\psi (x)} {e_\mu }({t_0})dx} \right| \le C(E,\mu)\int_{{t_0}}^{{t_0} + 1/{\eta _3}} {\left( {\left\| {{u_t}} \right\|_2^2 + {{\left\| {{u_t}} \right\|}_2}} \right)} dt.
\end{align}
For $d\ge3$, $\frac{d}{d-2}<p<1+\frac{4}{d-2}$, by Bernstein's inequality,
$${\left\| {{P_{ \le {\mu ^{ - 1}}}}h(u)} \right\|_2} \le {\mu ^{ - d\left( {\frac{p}{{{2^*}}} - \frac{1}{2}} \right)}}{\left\| {{P_{ \le {\mu ^{ - 1}}}}h(u)} \right\|_{\frac{{{2^*}}}{p}}} \le {\mu ^{ - d\left( {\frac{p}{{{2^*}}} - \frac{1}{2}} \right)}}C(E),$$
which yields (<ref>) again.
For $d=1,2$, (<ref>) can be obtained directly by Sobolev embedding theorem.
Since ${\int_0^\infty {\left\| {{u_t}} \right\|_2^2dt} < \infty }$,
choose $t_0$ sufficiently large such that
$$\int_{{t_0}}^{{t_0} + 1/{\eta _3}} {\left\| {{u_t}} \right\|_2^2} dt \le \sqrt {{\eta _3}} {\rm{ }}{\mu ^{d\left( {\frac{p}{{{2^*}}} - \frac{1}{2}} \right)}}\eta _1^6,
$$\int_{|x| < R'/2} {e_{\mu}(t)} dx \ge \int_{{\Bbb R^d}} {\psi (x)e_{\mu}(t)} dx \ge \int_{{\Bbb R^d}} {\psi (x)e_{\mu}({t_0})} dx -\sqrt {{\eta _3}} \eta _1^3\gtrsim \eta _1^2,$$
thus proving (<ref>), from which our lemma follows.
§ CONCENTRATION COMPACT ATTRACTOR
In this section, we first derive the global attractor, then we prove Theorem 1.1 immediately.
§.§ Concentration-compactness attractor
We recall the following criterion for compact attractors proved by Proposition B.2 in Tao <cit.>.
Let $\mathcal{U}$ be a collection of trajectories $u:\Bbb R^+\to \mathcal{H}$. If $\mathcal{U}$ is bounded in $\mathcal{H}$, and for any $\mu_0>0$ there exists $\mu_1>0$ such that
\begin{align*}
&\mathop {\lim \sup }\limits_{t \to \infty } {\left\| {{P_{ > 1/{\mu _1}}}u(t)} \right\|_\mathcal{H}} \le {\mu _0}, \\
&\mathop {\lim \sup }\limits_{t \to \infty } {\int_{|x| > 1/{\mu _1}} {\left| {u(x,t)} \right|} ^2} + {\left| {\nabla u(x,t)} \right|^2} + {\left| {{\partial _t}u(x,t)} \right|^2}dx \le {\mu _0}.
\end{align*}
Then there exists a compact set $K\subset\mathcal{H}$ such that $\mathop {\lim }\limits_{t \to \infty }dist_{\mathcal{H}}(u(t),K)=0$.
Let $\mathcal{U}$ be a collection of trajectories $u:\Bbb R^+\to \mathcal{H}$, and let $J\ge1$. If $\mathcal{U}$ is bounded in $\mathcal{H}$, and for any $\mu_0>0$ there exists $\mu_1>0$ such that for every $u\in \mathcal{U}$ we have $x_1,...,x_J:\Bbb R^+\to\Bbb R^d$ for which
\begin{align*}
&\mathop {\lim \sup }\limits_{t \to \infty } {\left\| {{P_{ > 1/{\mu _1}}}u(t)} \right\|_\mathcal{H}} \le {\mu _0}, \\
&\mathop {\lim \sup }\limits_{t \to \infty } {\int_{dist\left( {x,\left\{ {{x_1}(t),{x_2}(t),...,{x_J}(t)} \right\}} \right) > 1/{\mu _1}} {\left| {u(x,t)} \right|} ^2} + {\left| {\nabla u(x,t)} \right|^2} + {\left| {{\partial _t}u(x,t)} \right|^2}dx \le {\mu _0}.
\end{align*}
Then there exists a G-precompact set $K\subset\mathcal{H}$ with $J$ components such that $\mathop {\lim }\limits_{t \to \infty }dist_{\mathcal{H}}(u(t),K)=0$.
Although the proof is almost the same as proposition B.3 of T. Tao <cit.>, for reader's convenience, we give a sketch here.
We use the partition of unity
1 = \sum\limits_{j = 1}^J {{\psi _{j,t}}(x)},
{\psi _{j,t}}(x) \equiv \frac{{{{\left\langle {x - {x_j}(t)} \right\rangle }^{ - 1}}}}{{\sum\limits_{l = 1}^J {{{\left\langle {x - {x_l}(t)} \right\rangle }^{ - 1}}} }}.
Split $(u(x,t),\partial_tu(x,t))$ as
\begin{align}\label{oi}
u(t) = \sum\limits_{j = 1}^J {{\tau _{{x_j}(t)}}} {w_j}(t),{\rm{ }}{\partial _t}u(t) = \sum\limits_{j = 1}^J {{\tau _{{x_j}(t)}}} {v_j}(t),
\end{align}
$${w_j}(t) = {\tau _{ - {x_j}(t)}}{\psi _{j,t}}(x)u(t),\mbox{ }{v_j}(t) = {\tau _{ - {x_j}(t)}}{\psi _{j,t}}(x){\partial _t}u(t).
The localization of $u$ and $\partial_tu$ implies for any $\mu_0>0$, there exists $\eta>0$ such that
\begin{align*}
\mathop {\lim \sup }\limits_{t \to \infty } {\left\| {{P_{ > \eta }}{w_j}} \right\|_{{H^1}}} + {\left\| {{P_{ > \eta }}{v_j}} \right\|_{{L^2}}} \le {\mu _0},\mbox{ }\mbox{ }\mathop {\lim \sup }\limits_{t \to \infty } \int_{\left| x \right| > \eta } {{{\left| w_j \right|}^2} + {{\left| {\nabla w_j} \right|}^2} + {{\left| {v_j} \right|}^2}} \le {\mu _0}.
\end{align*}
From Proposition <ref>, there exist a compact set $K_1\subset H^1$ and a compact set $K_2\subset L^2$, such that
\mathop {\lim }\limits_{t \to \infty } dist({w_j}(t),{K_1}) = 0,\mbox{ }\mbox{ }\mathop {\lim }\limits_{t \to \infty } dist({v_j}(t),{K_2}) = 0,
for all $j=1,2,...,J$. Combining with (<ref>), we obtain
where $K=K_1\times K_2$.
As a corollary of Proposition <ref>, Proposition <ref>, Lemma <ref>, we have
There exists a compact set $K\subset \mathcal{H}$ and $0\le J<\infty$, such that
$$\mathop {\lim }\limits_{t \to \infty } dist_{\mathcal{H}}(u(t),J(GK))=0.$$
§ PROOF OF THEOREM 1.1
Step one.
Combining Corollary <ref> with Lemma B.7 in Tao <cit.>, we have for any $t_n\to \infty$, up to a subsequence there exits $J_1,J_2,...,J_M$ and $w_m\in J_m(GK)$ such that
\begin{align*}
u({t_n}) &= \sum\limits_{m = 1}^M {{\tau _{{x_{m,n}}}}} {w_m} + {o_{H^1}}(1) \\
{\partial _t}u({t_n}) &= \sum\limits_{m = 1}^M {{\tau _{{x_{m,n}}}}} {v_m} + {o_{L^2}}(1),\\
\end{align*}
where $x_{m,n}\in \Bbb R^d$ and they satisfies
$\mathop {\lim }\limits_{n \to \infty } \left| {{x_{m,n}} - {x_{k,n}}} \right| = \infty$, for $k\neq m$.
Step two.
By linear energy decoupling property, we have $\mathop {\sup }\limits_m {\left\| {\left( {{w_m},{v_m}} \right)} \right\|_\mathcal{H}} < C$, by the local theory, there exists $T>0$ such that the solution $W_j$ to (<ref>) with initial data $(w_j,v_j)$ is wellposed on $[0,T]$. From perturbation theorem and separation of $x_{m,n}$,
we obtain
\partial_tu({t_n} + t) = \sum\limits_{j = 1}^M {\partial_t{W_j}(x - {x_{j.n}}} ,t) + {o_{L^2}}(1).
Since $\mathop {\lim }\limits_{n \to \infty } \int_0^T {\left\| {{\partial _t}u({t_n} + t)} \right\|_2^2} dt = 0$, by the separation of linear energy, we conclude
\int_0^T {\left\| {{\partial _t}{W_j}(t)} \right\|_2^2} dt = 0.
Therefore, $W_j$ is an equilibrium, the same holds for $w_j$, thus we have proved
there exists a finite number of equilibrium points $Q_m$ such that for any sequence $t_n\to\infty$, there exists $x_{m,n}$ for which
u(t_n)=\sum\limits_{m = 1}^M Q_m(x-x_{m,n})+ {o_{H^1}}(1), \mbox{ }\mbox{ }\partial_tu(t_n)={o_{L^2}}(1).
By contradiction arguments, we can prove our theorem.
BL H. Berestycki, P.L. Lions, Nonlinear scalar field equations. II. Existence of infinite many solutions, Arch. Rational Mech. Anal. 82 (1983), 347-375.
BRS N. Burq, G. Raugel, W. Schlag, Long time dynamics for damped Klein-Gordon equations, arXiv:1505.05981.
Ca T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Functional Analysis, 60 (1985), 36-55.
C R. Cote, Soliton resolution for equivariant wave maps to the sphere, arXiv:1305.5325, to appear in Comm.
Pure. Appl. Math.
CKLS3 R.Cote, C. Kenig, A. Lawrie, W, Schlag, Profiles for the radial focusing 4d energy-critical wave equation, Preprint arXiv: 1402.2307.
CKLS1 R. Cote, C. Kenig, A. Lawrie, W. Schlag, Characterization of large energy solutions of the equivariant
wave map problem: II , Amer. J. Math. 137 (2015), no. 1, 209-250, see also arXiv:1209.3684v2.
CKLS2 R. Cote, C. Kenig, A. Lawrie, W. Schlag, Characterization of large energy solutions of the equivariant
wave map problem: I, American Journal of Mathematics, 137 (2015), no.1, 139-207, see also
DKM T. Duyckaerts, C. Kenig, F. Merle, Classification of radial solutions of the focusing, energy-critical wave
equation, Cambridge Journal of Mathematics 1 (2013), no. 1, 75-144.
DKM2 T. Duyckaerts, C. Kenig, F. Merle, Universality of blow-up profile for small radial type II blow-up
solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 533-599.
DKM4 T. Duyckaerts, C. Kenig, F. Merle, Universality of blow-up profile for small radial type II blow-up
solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS) 14 (2011), 1389-1454.
DKM3 T., Duyckaerts, C., Kenig, F., Merle, Scattering for radial, bounded solutions of focusing supercritical
wave equations, Int. Math. Res. Not. IMRN 2014, no. 1, 224-258.
F2 E. Feireisl, Convergence to an equilibrium for semilinear wave equations on unbounded intervals, Dynam. Syst. Appl. 3 (1994),
F E. Feireisl, Finite energy travelling waves for nonlinear damped wave equations, Quarterly Journal of Applied mathematics LVI(1998),
JLX H., Jia, B.P., Liu, G.X., Xu, Long time dynamics of defocusing energy critical 3 + 1 dimensional wave
equation with potential in the radial case, to appear in in Communications in Mathematical Physics, see
also arXiv:1403.5696.
JK H. Jia, C. Kenig, Asymptotic decomposition for semilinear wave and equivariant wave map equations, arXiv. 1503.06715.
KLSC., Kenig, A., Lawrie, W., Schlag, Relaxation of wave maps exterior to a ball to harmonic maps for all
data, arXiv:1301.0817.
KLLS2C., Kenig, A., Lawrie, B.P., Liu, W., Schlag, Stable soliton resolution for exterior wave maps in all
equivariance classes, arXiv:1409.3644.
KLLS3 C., Kenig, A., Lawrie, B.P., Liu, W., Schlag, Channels of energy for the linear radial wave equation,
NS K. Nakanishi, W. Schlag, Invariant manifolds and dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced mathematicas. European Mathematical Society (EMS), Zürich, 2011.
NS1 K. Nakanishi, W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differetnial Equations, 250 (2011), 2299-2333.
NS2 K. Nakanishi, W. Schlag, Global dynamics above the ground state energy for the nonlinear Klein-Gordon equation without a radial assumption, Arch. Rational Mech. Anal. 203 (2012) 809-851.
PS I. E. Payne, D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. math. 22(1975), 273-303.
R C. Rodriguez, Profiles for the radial focusing energy-critical wave equation in odd dimensions, arXiv:1412.1388.
S A. Soffer, Soliton dynamics and scattering, http://www.icm2006.org/proceedings/VolIII/contents/ICM Vol3, 24.pdf
TT T. Tao, A (concentration-) compact attractor for highdimensional nonlinear Schrödinger equations, Dynamics of Partial Differential Equations, 4 (2007) 1-53.
|
1511.00586
|
Strong local–global phenomena for representations]Strong local–global phenomena for Galois and automorphic representations
Department of Mathematics, University of Oklahoma, Norman, OK 73019 USA
Many results are known regarding how much local information is required to determine
a global object, such as a modular form, or a Galois or automorphic representation.
We begin by surveying some things that are known and expected, and then explain recent
joint work with Dinakar Ramakrishnan about comparing degree 2 Artin and automorphic
representations which a priori may not correspond at certain infinite sets of places.
These notes are based on a talk I gave at the RIMS workshop, “Modular forms and
automorphic representations,” from Feb 2–6, 2015, which was in turn based
on the joint work <cit.> with Ramakrishnan. I am grateful to the organizers
for the opportunity to present this exposition.
These notes were written while I was visiting
Osaka City University under a JSPS Invitation Fellowship. I was also supported in part by
a Simons Collaboration Grant. I am happy to thank all of these organizations for their kind support.
I would also like to thank Christina Durfee for enlightening discussions about characters of finite
groups and Nahid Walji for helpful feedback.
§ INTRODUCTION
A local–global principle, or phenomenon, is a situation where certain
local conditions are sufficient to imply a corresponding global condition.
Examples both of local–global principles (e.g., zeroes of quadratic forms, norms in cyclic
extensions, Grunwald–Wang, splitting of central simple algebras)
as well as
examples of failures of local–global principles (e.g., unique factorization, Grunwald–Wang,
points on varieties, zeros or poles of $L$-functions,
vanishing of periods) abound in number theory and are of consummate interest.
See, for example, Mazur's (8th out of 11 so far!) Bulletin article <cit.> for
local–global principles and obstructions for varieties.
On the other hand, for certain objects like idele class characters,
modular (new) forms, Galois representations or
automorphic representations, we have much more rigid local–global phenomena.
Here the usual local–global principle is more-or-less tantamount to the existence of an Euler product
for the associated $L$-function. We will discuss stronger versions of this, where
knowing local $L$-factors at a sufficiently large set of places determines the global
$L$-function (and hence, often, the global object up to isomorphism).
Specifically, consider the following 3 results. Let $F$ be a number field, $\Sigma_F$
the set of places of $F$ and $\Gamma_F$ the absolute Galois group of $F$.
Denote by $\rho, \rho'$ irreducible $n$-dimensional complex representations
of $\Gamma_F$ (i.e., irreducible Artin representations)
and by $\pi, \pi'$ irreducible cuspidal automorphic representations of $\GL_n(\A_F)$.
* If $L(s, \rho_v) = L(s, \rho'_v)$ for almost all $v$, then $L(s, \rho) = L(s, \rho')$, and in fact
$\rho \simeq \rho'$.
* If $L(s, \pi_v) = L(s, \pi'_v)$ for almost all $v$, then $L(s, \pi) = L(s, \pi')$,
and in fact $\pi \simeq \pi'$.
* If $L(s, \rho_v) = L(s, \pi_v)$ for almost all $v$, then $L(s, \rho) = L(s, \pi)$.
To be more precise, by the notation $L(s, \rho)$, $L(s, \pi)$, etc., for global $L$-functions we will
mean the incomplete $L$-function (the product over all finite places of local factors). When we
want to denote completed $L$-functions, we will write $L^*(s, \rho)$, $L^*(s, \pi)$, etc.
By equality of two global $L$-functions, we mean as Euler products over the base field, i.e., not
just equality of meromorphic functions but all local factors are equal as well.
Result (1) is an elementary consequence of Chebotarev density, and
(2) is the strong multiplicity one (SMO) theorem for $\GL(n)$ due to Jacquet and Shalika <cit.>.
Result (3) follows from an argument of Deligne and Serre <cit.> (see
Appendix A of my thesis <cit.>).
While the first two statements are usually stated just
with the conclusion of the two representations being isomorphic, stating the conclusion in terms of a
global $L$-function equality puts all three results on the same footing. In addition, if one wants to
think about representations of other groups, this seems to be the right point of view.
E.g., cuspidal representations of $\mathrm{SO}_n(\A)$ will not satisfy SMO in the usual sense, but equality
at almost all places should give an equality of global $L$-functions (in fact, $L$-packets).
Now one can ask a more general type of question.
Suppose two global $L$-functions over $F$ agree at all primes outside
of some set $S \subset \Sigma_F$. Under what conditions can we conclude that the $L$-factors
are equal everywhere?
The above 3 results are about when $S$ is a finite set, but some results and conjectures exist
generalizing (1) and (2) if $S$ is “not too big”, or of a certain form. We will discuss each of
these situations, and conclude by explaining recent joint work with
Ramakrishnan <cit.>, where we generalized (3) to certain kinds of infinite sets for $n=2$.
Of course it is interesting to consider when $\rho$ and $\rho'$ are $\ell$-adic Galois representations
as well. We will make some remarks about $\ell$-adic representations, but for simplicity focus
on Artin representations.
§ GALOIS REPRESENTATIONS
Suppose $\rho$ and $\rho'$ are irreducible $n$-dimensional Artin representations of $\Gamma_F$.
They both factor through Galois groups of finite extensions of $F$, so we can choose a single
finite Galois extension $K/F$ such that $\rho$ and $\rho'$ may be considered as representations of
$G = \Gal(K/F)$.
Our basic problem is to determine if knowing $L^S(s, \rho) = L^S(s, \rho')$ for some fixed $S \subset
\Sigma_F$ implies $L(s, \rho) = L(s, \rho')$.
Since a representation of a finite group is determined by its character, for Artin representations
it suffices to consider a weaker hypothesis.
Namely, let $S \subset \Sigma_F$ and suppose
$\tr \rho(Fr_v) = \tr \rho'(Fr_v)$ for $v \not \in S$. (We may assume $S$ contains all places where
$\rho$ and $\rho'$ are ramified, so that this makes sense.) Note this is weaker than the condition
on $L$-factors because, at unramified places,
$L_v(s, \rho) = (\det(I-\rho(\Fr_v)q_v^{-s}))^{-1}$ determines $\tr \rho(\Fr_v)$ but not conversely.
Recall we define the (natural) density of $S$ to be
\[ \den(S) := \lim_{x \to \infty} \frac{ \# \{ v \in S : q_v < x \} } { \# \{ v \in \Sigma_F : q_v < x \} }, \]
if this limit exists. Now Chebotarev density
says that if $\den(S) < \frac 1{|G|}$, then
$\{ \Fr_v : v \not \in S \}$ hits all conjugacy classes in $G$. So if $\den(S) < \frac 1{|G|}$, then
$\tr \rho(g) = \tr \rho'(g)$ for all $g \in G$, whence $\rho \simeq \rho'$. Often it is easier
to work with Dirichlet density, which is defined by
\[ \delta(S) := \lim_{s\to 1^+} \frac{ \sum q_v^{-s} }{\log \frac 1{s-1}} \]
If $\den(S)$ exists, so does $\delta(S)$ and they are equal.
Suppose $\rho$ and $\rho'$ are $n$-dimensional Artin representations of
$\Gal(K/F)$. If $\tr \rho(\Fr_v) = \tr \rho'(\Fr_v)$ for $v$ outside of a set $S$ of places with
$\delta(S) < \frac 1{2n^2}$, then $\rho \simeq \rho'$.
This follows from combining the above Chebotarev density argument and the following result about finite group characters.
If $\chi$ and $\chi'$ are irreducible characters of degree $n$ of a finite group $G$
and $X = \{ g \in G : \chi(g) = \chi'(g) \}$ has size $> |G|(1-1/2n^2)$, then $\chi = \chi'$.
Put $Y = G - X$. Since $\chi, \chi'$ have maximum absolute value $n$, we see
\[ \sum_{g \in Y} |\chi(g) \bar \chi(g)|, \sum_{g \in Y} |\chi(g) \bar \chi'(g)| \le |Y|n^2 < \frac{|G|}2. \]
Since $\sum_{g \in G} \chi(g) \bar \chi(g) = |G|$, this means $\sum_{g \in X} \chi(g) \bar \chi(g) \ge
\frac{G}2$. Then
\[ \sum_{g \in G} \chi(g) \bar \chi(g) = \sum_{g \in X} \chi(g) \bar \chi(g) + \sum_{g \in Y} \chi(g) \bar \chi'(g) \ne 0, \]
which implies $\chi = \chi'$ by orthogonality relations and irreducibility.
It is known that one cannot do better than this, cf. <cit.>.
Namely, if $n=2^m$ then Buzzard, Edixhoven
and Taylor constructed distinct $n$-dimensional irreducibles $\rho$ and $\rho'$ such that
$\den(S) = |G|(1-1/2n^2)$ where $G$ is a central quotient of $Q_8^m$ ($Q_8$ is the quaternion
group of order 8) times $\{ \pm 1 \}$. Serre showed the existence of similar examples for arbitrary
We remark that Rajan <cit.> proved an analogue for (semisimple, finitely ramified)
$\ell$-adic Galois representations of $\Gal(\bar F/F)$.
§ AUTOMORPHIC REPRESENTATIONS
Let $\pi$ and $\pi'$ be irreducible automorphic cuspidal unitary representations of $\GL_n(\A_F)$.
For a finite place $v$, we can write
\[ L(s, \pi_v) = \prod_{i=1}^k (1-\alpha_{v,i} q_v^{-s})^{-1} \]
for some $0 \le k \le n$ and nonzero complex numbers $\alpha_{v,i}$.
Note that $L(s, \pi_v)$ is a nowhere vanishing meromorphic function whose set of poles are
precisely the values of $s$ such that $q_v^{s} = \alpha_{v,i}$ for some $1 \le i \le k$. The latter
condition implies $q_v^{\Ree(s)} = |\alpha_{v,i}|$, and conversely for each real $x$ with
$q_v^x = |\alpha_{v,i}|$ for some $i$, there exists an $s$ such that $\Ree(s) = x$ and there
is a pole at $s$. (If $k=0$, then $L(s, \pi_v) = 1$ and there are no poles.)
Similarly, write
\[ L(s, \pi_v') = \prod_{i=1}^{k'} (1-\alpha'_{v,i} q_v^{-s})^{-1}. \]
The following observation, while simple, will be key for us in several places, so I will set
it off to highlight it.
Fix a finite place $v$ with $k, k' \ge 1$. If the first (rightmost) pole for $L(s, \pi_v)$
occurs on the vertical line $\Ree(s) = x_0$, then $x_0 = \max \{ \frac{\log |\alpha_{v, i} |}{\log q_v} : 1 \le i \le k \}$.
Similarly, if the first pole for $L(s, \pi_v \times \bar \pi'_v)$
occurs on the vertical line $\Ree(s) = x_0$, then $x_0 = \max \{ \frac{\log |\alpha_{v, i} \alpha'_{v,j} |}{\log q_v} : 1 \le i \le k,
1 \le j \le k' \}$.
Suppose $L(s, \pi_v) = L(s, \pi'_v)$ for all $v$ outside of a
finite set $S$. Then $\pi \simeq \pi'$.
This generalizes earlier results of Miyake <cit.> for $n=2$ and Piatetski–Shapiro <cit.>,
who needed to also assume the archimedean components match.
For $n=1$, this follows from strong approximation.
There are two ingredients, both proved in <cit.>.
(i) We have $\pi \simeq \pi'$ if and only if
$L(s, \pi \times \bar \pi')$ has a pole at $s=1$
(use the integral representation and orthogonality of cusp forms).
(ii) For finite $v$, we have the bound $|\alpha_{v,i}| < q_v^{1/2}$. (The ramified case reduces to the unramified case.)
Now to prove strong multiplicity one, consider the ratio
\begin{equation} \label{eq:1}
\frac{L(s, \pi \times \bar \pi)}{L(s, \pi \times \bar \pi')} =
\frac{L_S(s, \pi \times \bar \pi)}{L_S(s, \pi \times \bar \pi')}.
\end{equation}
By (ii), we see that
$L_S(s, \pi \times \bar \pi)$ and $L_S(s, \pi \times \bar \pi')$ both have no poles on $\Ree(s) \ge 1$.
Since these functions are also never zero, the right hand side has no pole in
$\Ree(s) \ge 1$.
Since $L(s, \pi \times \bar \pi)$ has a pole at $s=1$ by (i), it must be canceled out
by a pole of $L(s, \pi \times \bar \pi')$ at $s=1$ in order for the ratio on the left to not have a pole
there. Thus, again by (i), we get $\pi \simeq \pi'$.
We remark that Moreno <cit.> proved an “analytic” SMO:
if $\pi$ and $\pi'$ have bounded conductors and archimedean
parameters, there is an effective (but exponential) constant $X$
(depending on the bounds on conductors and archimedean
parameters, $F$ and $n$) such that if $\pi_v \simeq \pi'_v$
for all $v$ with $q_v < X$, then $\pi \simeq \pi'$. Note that for $n=2$, such a result gives you a
bound on the number of Fourier coefficients needed to distinguish modular forms of bounded
level and weight with the same nebentypus. (Of course there will be some finite bound
because the space of such forms is finite dimensional.)
Further work has been done along these lines (for $n=2$ and general $n$),
but this is not our focus now and we will not discuss it further. We are interested in results where
one does not impose a priori bounds on ramification or infinity types.
Coming back to the usual SMO, note that in the above proof it was
crucial $S$ be finite to conclude the RHS of (<ref>)
has no pole in $\Ree(s) \ge 1$.
To refine this, we need a couple more ingredients.
First is an improvement on (ii). Recall the Generalized Ramanujan
Conjecture (GRC) asserts that each $\pi_v$ is tempered, i.e., each $|\alpha_{v,i}| = 1$.
For general $n$, the best that is known is the Luo–Rudnick–Sarnak bound from <cit.>,
which says $|\alpha_{v,i}| < q_v^{1/2-1/(n^2+1)}$. For $n=2$ we can do better. Using $\Sym^2$,
Gelbart–Jacquet <cit.> got a bound of $q_v^{1/4}$. With $\Sym^3$ this was improved to
exponent $\frac 19$ by Kim–Shahidi <cit.>, then further improved to $\frac 7{64}$ by
Kim–Sarnak <cit.> and Blomer–Brumley <cit.> using $\Sym^4$.
In fact, for us, a bound of the form $q_v^{\delta}$ for some $\delta < \frac 14$ is sufficient.
The second ingredient we need is Landau's lemma, which we explain now.
Let us say a Dirichlet series $L(s)$
is of positive type if it has an Euler product (on some right half plane) and $\log L(s)$ is a Dirichlet
series with positive ($\ge 0$) coefficients. Note
\[ \log \frac 1{1-\alpha q^{-s}} = \sum_n \frac{\alpha^n/n}{q^{ns}} \]
is a Dirichlet series with positive coefficients if $\alpha \ge 0$. Since the sum of Dirichlet series
with positive coefficients is again a Dirichlet series with positive coefficients (admitting convergence)
we see an $L$-series of the form
\[ L(s) = \prod_i \frac 1{1-\alpha_i q_i^{-s}} \]
with each $\alpha_i \ge 0$ is of positive type (admitting convergence), e.g., a Dedekind zeta
function. More important for us will be examples like $L(s, \pi \times \bar \pi)$,
which are also of positive type.
Suppose $L(s)$ is a Dirichlet series of positive type. Then no zero of $L(s)$
occurs to the right of the first (rightmost) pole, and the first pole occurs on the real axis.
This will be extremely useful because we can now control the locations of not just poles of
$L$-functions, but also zeroes.
Suppose $n=2$ and $L(s, \pi_v) = L(s, \pi'_v)$ for all $v$
outside a set $S$ with $\delta(S) < \frac 18$. Then $\pi \simeq \pi'$.
This was used by Taylor <cit.> for constructing families of $\ell$-adic Galois representations to
modular forms over imaginary quadratic field.
Suppose $\pi \neq \pi'$ and assume $S$ contains all places of ramification. Put
\[ Z(s) = \frac{L(s, \pi \times \bar \pi)L(s, \pi' \times \bar \pi')}{L(s, \pi \times \bar \pi') L(s, \pi' \times \bar \pi)}. \]
Then, by (i), the numerator has a double pole at $s=1$ while the denominator has no pole there.
Hence $Z(s)$ has a pole of order 2 at $s=1$. By definition, $Z_v(s) = 1$ for any $v \not \in S$,
so we also have
\[ Z(s) = Z_S(s) = \frac{L_S(s, \pi \times \bar \pi)L_S(s, \pi' \times \bar \pi')}{L_S(s, \pi \times \bar \pi') L_S(s, \pi' \times \bar \pi)}. \]
\[ D_S(s) = L_S(s, \pi \times \bar \pi)L_S(s, \pi' \times \bar \pi')L_S(s, \pi \times \bar \pi')
L_S(s, \pi' \times \bar \pi) \]
\[ Z_S(s) = \frac{L_S(s, \pi \times \bar \pi)^2 L_S(\pi' \times \bar \pi')^2}{D_S(s)}. \]
This is convenient because $D_S(s)$ is a Dirichlet series of positive type,
and one can check it is nonvanishing for $s \ge 1$, so it has no zero at $s=1$ by Landau's lemma.
We would like to get a contradiction by saying that $L_S(s, \pi \times \bar \pi)$ and
$L_S(\pi' \times \bar \pi')$ can't have poles at $s=1$ for $S$ of sufficiently small density, but there is no reason they even need to be meromorphic at $s=1$.
Instead, we observe that $Z_S(s)$ having a pole of order 2 at $s=1$ means
\[ \lim_{s \to 1^+} \frac{\log Z_S(s)}{\log \frac 1{s-1}} = 2. \]
Hence to obtain a contradiction, it will suffice to show
\begin{equation} \label{eq:desired}
\lim_{s\to 1^+} \frac{\log L_S(s, \pi \times \bar \pi)}{\log \frac 1{s-1}} < \frac 12,
\end{equation}
as the same argument will apply to $L_S(s,\pi \times \bar \pi')$.
For simplicity, assume $F=\Q$. Say $\pi_p$ has Satake parameters
$\{ \alpha_{1,p}, \alpha_{2,p} \}$.
\begin{align*}
\log L(s, \pi_p \times \bar \pi_p) &= \sum_{1 \le i, j \le 2 } \log \frac 1{1-\alpha_{i,p}\alpha_{j,p}p^{s}}
= \sum_{1 \le i, j \le 2 } \sum_{n \ge 1} \frac{(\alpha_{i,p}\alpha_{j,p})^n}{np^{ns}} \\
&= \frac{c_p} {p^{s}} + O(p^{-2s}),
\end{align*}
\[ c_p = \sum_{1 \le i, j \le 2 } \alpha_{i,p}\alpha_{j,p}. \]
It is well known that the “prime zeta function” satisfies
\[ \sum_p \frac 1{p^s} = \log \frac 1{s-1} + O(1), \quad s \to 1^{+}. \]
So if $\pi$ is tempered at $p$, and then $|c_p| \le 4$ and one deduces
\begin{equation} \label{eq:Lpipi-tb}
\log L_S(s, \pi \times \bar \pi) \le 4\delta(S) \log \frac 1{s-1} + o(\log \frac 1{s-1}),
\end{equation}
and we are done as $\delta(S) < \frac 18$.
So the difficulty is when $\pi$ is not tempered. Here one needs the above-mentioned bound
towards GRC: $|\alpha_{i,v}| \le q_v^{\delta}$ with $\delta < \frac 14$.
Then Ramakrishnan does a careful analysis involving $L(s, \Ad(\pi))$ and
$L(s, \Ad(\pi) \times \Ad(\pi))$ to treat the case when $\Ad(\pi)$ is cuspidal.
(This is the technical crux of the proof, but it will not come up later for us, so we will not explain
this analysis.)
If $\Ad(\pi)$ is not
cuspidal, then $\pi$ is induced from a character of a quadratic extension, and therefore tempered
In fact, Rajan <cit.> observed this is also true if one just assumes equality of
coefficients of Dirichlet series (i.e., sums of Satake parameters—or, for modular forms, Fourier
coefficients) at primes $v \not \in S$. This is analogous to only requiring equalities of traces
$\tr \rho(\Fr_v) = \tr \rho'(\Fr_v)$ for Galois representations.
Note that Ramakrishnan's result is sharp, which one can deduce from $n=2$ examples which show
Proposition <ref> is sharp. Nevertheless, Walji <cit.> was able to
prove some refinements, such as the following:
if $n=2$ and $\pi$ and $\pi'$ are not dihedral (induced from quadratic extensions),
then a refined SMO is true with the stronger bound $\delta(S) < \frac 14$.
Now let's go back to considering arbitrary $n$.
A refined SMO is true with $\delta(S) < \frac 1{2n^2}$.
For $n=1$ this is true by class field theory and Proposition <ref>. For $n=2$, this
is precisely the content of Theorem <ref>.
Let's think back to the proofs of Theorems <ref> and <ref> to see what
is needed to prove a refined SMO result. In the proof of the usual SMO (Theorem <ref>)
we wanted to show $L_S(s, \pi \times \bar \pi)$ has no pole at $s=1$ and $L_S(s, \pi \times \bar \pi')$
has no zero at $s=1$. To prove refined SMO for $\GL(2)$ (Theorem <ref>), Ramakrishnan
considered a ratio $Z(s)$ and used Landau's lemma to essentially translate the problem
into showing both $L_S(s, \pi \times \bar \pi)$ and $L_S(s, \pi \times \bar \pi')$ have no poles in
$\Ree(s) \ge 1$. Let's just consider $L_S(s, \pi \times \bar \pi)$ since the idea for
$L_S(s, \pi \times \bar \pi')$ is similar.
Suppose we have a bound towards GRC which says
each $L(s, \pi_v \times \bar \pi_v)$ has no pole in $\Ree(s) > 2\delta < 1$. Then, morally,
if $S$ is not too dense the bound for the first pole of $L_S(s, \pi \times \bar \pi)$ should not
be pushed too far to the right of $2\delta$. (If $S$ has density 1, then $L_S(s, \pi \times \bar \pi)$
can have a pole up to 1 unit to the right of $2\delta$.) The actual argument is more subtle than
this, but we will return to this moral shortly.
Looking at the argument for the tempered case of
Theorem <ref>, we see
the fact that $n=2$ was not really crucial. For general $n$, the $4\delta(S)$ in (<ref>)
becomes $n^2\delta(S)$, and this is less than the $\frac 12$
required in (<ref>) precisely when $\delta(S) < \frac 1{2n^2}$.
In other words, this conjecture should follow from GRC and. Moreover,
the bound $\delta(S) < \frac 1{2n^2}$ must be sharp by the existence of examples of
Galois representations showing Proposition <ref> is sharp—here one can take
these examples to be of finite nilpotent Galois groups, where one knows modularity by
Arthur–Clozel <cit.>.
Unfortunately, the Luo–Rudnick–Sarnak bounds toward GRC only tell us
each $L(s, \pi_v \times \bar \pi_v)$ has no pole in $\Ree(s) \ge 1-\frac{2}{n^2+1}$, which
does not seem to be enough to force $L_S(s, \pi \times \bar \pi)$ to have no pole in $\Ree(s) \ge 1$
for any $S$ of positive density. So for not-necessarily tempered representations of $\GL(n)$
we don't know any refined SMO for $n > 2$ and $S$ of positive density at present, but
we can treat certain infinite sets $S$ of density 0.
(In fact, at the time of his conjecture, Ramakrishnan announced he had a weak result for $n > 2$ (<cit.>, <cit.>),
but did not publish a result of this type until recently—see below.)
We remark that there have been spectacular results on proving GRC for certain classes of
representations for $\GL(n)$ to which one often knows how to associate Galois representations,
e.g., cohomological self-dual representations over a totally real field. For instance,
see Clozel's aphoristically titled article <cit.>.
In <cit.>, Rajan showed that a refined SMO is true for arbitrary $n$ if
\[ \sum_{v \in S} q_v^{- \frac 2{n^2+1} } < \infty. \]
This is not difficult—this condition implies that the first pole of
$L_S(s, \pi \times \bar \pi)$ is not more than $\frac 2{n^2+1}$ to the right of the first pole of a
local factor (the argument is the same as for the Key Observation below).
So by the Luo–Rudnick–Sarnak bound, this is precisely what one needs to conclude
$L_S(s, \pi \times \bar \pi)$ has no pole in $\Ree(s) \ge 1$.
However this condition only holds for very sparse sets of primes.
It is much stronger than $\sum q_v^{-1} < \infty$, which is in turn stronger
than requiring $\delta(S) = 0$, so one cannot handle $S$ of positive density.
An example of where this applies is: let $F/\Q$ be cyclic of prime degree
$p > \frac{n^2+1}2$ and let $S \subset \Sigma_F$ consist of inert primes in $F/\Q$.
(In <cit.>, Rajan says $S$ has positive density in this example,
but presumably he means the corresponding primes of $\Q$, rather than $F$, have
positive density: by Chebotarev, the density of the underlying primes of $S$ in
$\Sigma_\Q$ has density $\frac{p-1}p$ in $\Sigma_\Q$.)
Recently, Ramakrishnan proved the following result.
Suppose $F$ is a cyclic extension of prime degree $p$ of some number field $k$.
A refined SMO is true when
$S \subset \Sigma_F$ contains only finitely many primes which are split over $k$.
This is still density 0, and satisfies Rajan's criterion when $p$ is large, so the main content is for
$p$ small. In fact, $p=2$ is the hardest case, and this case was used in a crucial
way in trace formula comparisons of Wei Zhang <cit.> and Feigon–Martin–Whitehouse
More recently, Ramakrishnan <cit.>
has extended this to the arbitrary Galois case, where a quite different
approach was required.
As explained above, the key point is to show that $L_S(s, \pi \times \bar \pi)$ has no pole in
$\Ree(s) \ge 1$. There are two ingredients to the proof. First is the following elementary but key
fact, which we want to highlight because we will use it again in the next section.
Let $S_j$ be the set of primes of degree $j$. Suppose for each $v \in S_j$
we have numbers $\alpha_v$ such that $|\alpha_v| < q_v^\delta$. Then $\prod_{S_j} \frac 1{1-\alpha_v q_v^{-s}}$ converges absolutely in $\Re(s) > \delta + \frac 1j$.
This is a special case where we can make our above-mentioned “moral” precise. It says that
a product of local factors over primes of degree $j$ will not have in a pole which is more than
$\frac 1j$ to the right of a pole of any local factor.
\[ \log \prod_{v \in S_j} \frac 1{1-\alpha_v q_v^{-s}} = \sum_v \sum_m \frac{\alpha_v^m}{mq_v^{sm_v}}
\le \sum_v \sum_m \frac 1{q_v^{(s-\delta)m}} \]
If we denote by $p_v$ the rational prime below $q_v$, then $q_v \ge p_v^j$ so the above is bounded
(absolutely) by
\[ \sum_v \sum_m \frac 1{p_v^{(s-\delta)jm}} \le \sum_m \frac 1{m^{(s-\delta)j}}, \]
which converges if $(\Ree(s)-\delta)j > 1$, i.e., if $\Ree(s) > \delta + \frac 1j$.
Now by Luo–Rudnick–Sarnak, we know each local factor of $L(s, \pi \times \bar \pi)$ has no pole
in $\Ree(s) \ge 1- \frac 2{n^2+1}$, so the above observation tells us $L_S(s,\pi \times \bar \pi)$
has no pole in $\Ree(s) > 1 - \frac 2{n^2+1} + \frac 1p$, since all but a finite number (which do not
matter) of primes in $S$ have degree $p$. Hence if $p \ge \frac{n^2+1}2$ we are done.
To deal with smaller $p$, Ramakrishnan uses Kummer theory to prove the following.
Suppose $K/F$ is a degree $p^{m-1}$ extension such that $K/k$ is a nested chain of cyclic $p^2$-extensions. If $v$ is a prime of $F$ of degree $p$ over $k$, and $w$ is an unramified
prime of $K$ over $v$, then $w$ has degree $p^m$ over $k$.
Consequently, given such an extension $K/F$, the Key Observation tells us that
$L_S(s, \pi_{K} \times \bar \pi_{K})$ has no poles in $\Ree(s) > 1- \frac 2{n^2+1} + \frac 1{p^m}$.
Taking $m$ large enough we can conclude $L_S(s, \pi_{K} \times \bar \pi_{K})$ has no poles
in $\Ree(s) \ge 1$, and thus that $\pi_K \simeq \pi'_K$.
To finish the proof, one must carefully vary the field $K$ to get $\pi_K \simeq \pi'_K$ over sufficiently
many extension $K/F$ to deduce the isomorphism $\pi \simeq \pi'$ over $F$.
We will a use similar idea for our result in the next section.
§ MODULARITY
Let $\rho$ be an irreducible $n$-dimensional Artin representation of $\Gamma_F = \Gal(\bar F/F)$.
Let $\pi$ be a cuspidal automorphic representation of $\GL_n(\A_F)$. Here we are
interested in comparing $\rho$ and $\pi$.
Recall we say $\rho$ is modular if $L(s,\rho)$ agrees with $L(s,\pi(\rho))$ at almost all places, for
some cuspidal automorphic representation $\pi(\rho)$ of $\GL_n(\A_F)$. (By SMO, $\pi(\rho)$
is unique up to isomorphism.) The strong Artin, or modularity, conjecture asserts that
every $\rho$ is modular. The following well-known result tells us an equivalent definition of
modularity is $L(s,\rho) = L(s,\pi)$ (in the sense of equality of Euler products).
If $L(s,\rho_v) = L(s,\pi_v)$ for almost all $v$, then $L(s, \rho) = L(s,\pi)$. Further
we have an identity of total archimedean factors $L_\infty(s,\rho) = L_\infty(s,\pi)$.
The proof follows from an argument due to Deligne and Serre <cit.>, and the details are
given in <cit.>. The idea is to twist by a highly ramified character $\chi$
at bad places, which makes the $L$-factors 1 at these places so we get a global
equality $L(s, \rho \otimes \chi) = L(s, \pi \otimes \chi)$. We may take $\chi$ to be trivial at each
archimedean place, and then comparing poles in functional equations allows us to deduce
$L_\infty(s,\rho) = L_\infty(s, \pi)$. Now we repeat the argument with $\chi$ which is highly ramified
at all but one bad place $v$, where $\chi$ is trivial. This gives $L(s, \rho_v) = L(s,\pi_v)$.
In fact, in <cit.>, when $n=2$ we show the stronger statement that $\rho$ and $\pi$ correspond
via local Langlands at all (finite and infinite) places. However, this argument relies on the fact that
the local Langlands correspondence is characterized by twists of $L$- and $\epsilon$- factors
by characters, which is not true for $n \ge 4$ (see
<cit.> for an example with $n=4$), so this argument does not generalize
to arbitrary $n$.
Now we can ask, to compare $\rho$ and $\pi$, how large of a set of places do we need to deduce
$L(s,\rho) = L(s, \pi)$?
The above proposition says it suffices to compare them at almost all places.
But since $\rho$ and $\pi$ should be determined by their local $L$-factors outside any set $S$
of places of density less then $\frac 1{2n^2}$, to show they correspond it should suffice to
check matching of local $L$-factors outside such a set $S$. Precisely, we have
If $L(s,\rho_v) = L(s,\pi_v)$ for all $v$ outside of some set $S$ of places of
density less than $\frac 1{2n^2}$, then $L(s, \rho) = L(s, \pi)$.
This is true for $n=1$ by class field theory. In general, this is a consequence of the
strong Artin conjecture together with the refined SMO Conjecture (Conjecture <ref>).
Namely, if $\rho$ is modular and its $L$-function agrees with that of $\pi$ outside of $S$, then
$L(s,\pi_v) = L(s, \pi(\rho)_v)$ for $v \not \in S$. By Conjecture <ref>, if
$\delta(S) < \frac 1{n^2}$, then $\pi \simeq \pi(\rho)$.
Consequently, by Theorem <ref>, we know this conjecture is true whenever $n=2$
and $\rho$ is modular. This is the case if $\rho$ has solvable image by the
Langlands–Tunnell theorem, or if $\rho$ is odd and $F= \Q$ by Khare and Winterberger's work
on Serre's conjecture (see <cit.>).
Arguing similarly in the reverse direction, Conjecture <ref> is true whenever $\pi$
corresponds to some Artin representation $\rho(\pi)$ by Proposition <ref>.
This is known if $\pi$ corresponds to a weight 1 Hilbert modular form by Wiles <cit.>.
However, even for $n=2$, this is not solved completely, and the case where $\rho$ is even (so
$\pi$ should correspond to a Maass form, say, if $F=\Q$) with nonsolvable image seems particularly difficult.
In any case, one might hope that if one could prove this conjecture independent of modularity,
then this may help establish new cases of modularity. Recently, Ramakrishnan and I proved
the following mild result towards this conjecture.
Suppose $n=2$ and $F$ is cyclic extension of prime degree $p$ of some number field $k$. Let $S \subset \Sigma_F$ be a set of primes such that almost all $v \in S$
are inert over $k$. Then $L(s, \rho_v) = L(s, \pi_v)$ for $v \not \in S$ implies $L(s, \rho) = L(s, \pi)$,
and in fact $\rho_v \leftrightarrow \pi_v$ in the sense of local Langlands at all $v$.
By the generalization of the Deligne–Serre argument we mentioned above,
it suffices to show $L(s, \rho_v) = L(s, \pi_v)$ for almost all $v$.
We show this in 4 steps.
Step 1: Show $\pi$ is tempered (at each place).
It is immediate from the equality $L(s, \rho_v) = L(s, \pi_v)$ that $\pi_v$ is tempered for any
$v \not \in S$, so we just need to show temperedness at $v \in S$.
Note, from the Fact before Theorem <ref>,
$\pi_v$ is tempered if and only if $L(s, \pi_v \times \bar \pi_v)$
has no pole in $\Ree(s) > 0$.
Now consider the ratio
\[ \Lambda(s) = \Lambda_F(s) = \frac{L^*(s, \pi \times \bar \pi)}{L^*(s, \rho \times \bar \rho)} =
\frac{L_S(s, \pi \times \bar \pi)}{L_S(s, \rho \times \bar \rho)}. \]
Then $\Lambda(s)$ satisfies a functional equation with $\Lambda(1-s)$, and if we can show
$\Lambda$ has no poles in $\Ree(s) \ge \frac 12$,
this will mean it is entire by the functional equation.
Take $\delta < \frac 14$ such that the bound of $q_v^{\delta}$ towards GRC is satisfied. The Key Observation implies that $L_S(\pi \times \bar \pi)$ has no poles in $\Ree(s) > 2\delta + \frac 1p$
and $L_S(\rho \times \bar \rho)$ has no poles (and thus no zeroes by Landau's lemma) in
$\Ree(s) > \frac 1p$. For $p$ sufficiently large, this means $\Lambda$ has no poles in
$\Ree(s) \ge \frac 12$, so $\Lambda$ is entire. For small $p$, we use Lemma <ref>
to pass to an extension $K$ to push our bounds on the poles of the numerator and denominator
to the left of $\Ree(s) = \frac 12$, and get that an analogous ratio $\Lambda_K$ is entire.
In either case, write the ratio as $\Lambda_K$, where we take $K=F$ if $p$ is sufficiently large.
If $\pi_v$ is not tempered for some $v \in S$, then Landau's lemma implies
$L_{T}(s, \pi_K \times \bar \pi_K)$ has a pole
at some $s_0 > 0$, where $T$ is the set of primes of $K$ above $S$.
But for $\Lambda_K$ to be entire, we also need a pole at $s_0$ for
$L(s, \rho_K \times \bar \rho_K)$. By taking $K$ larger if needed, we can make it so that
$L(s, \rho_K \times \bar \rho_K)$ has no poles in $\Ree(s) > \frac 1{p^m} < s_0$, a contradiction.
Step 2: Show $L(s, \rho_K)$, for some finite solvable
extension $K/F$, is entire.
Here we consider the ratio
\[ \Lambda_K(s) = \frac{L^*(s, \rho_K)}{L^*(s, \pi_K)} = \frac{L_T(s, \rho_K)}{L_T(s, \pi_K)}, \]
where as before $T$ is the set of places of $K$ above $S$.
Again, by looking at a functional equation, it suffices to show $L_T(s, \rho_K)$ has no pole in
$\Ree(s) \ge \frac 12$. The bound from the Key Observation is that $L_S(s, \rho)$ has no
pole in $\Ree(s) > \frac 1p$, so can take $K=F$ unless $p=2$. In this case we need to pass
to an extension $K$, so the bound becomes $\Ree(s) > \frac 1{p^2}$. By a refinement of
Lemma <ref>, we can do this with $K/F$ quadratic or biquadratic, according to whether
$\sqrt{-1} \in F$ or not.
Step 3: Deduce $L(s, \rho_K) = L(s, \pi_K)$.
The point is, up until now, everything we did is valid for twists, and the choice of $K$ in the
previous step only depends on $F/k$. So we get that $L(s, \rho_K \otimes \chi)$ is entire for
any finite order idele class character of $K$. For Artin representations, it is known that this is
sufficient to use the $\GL(2)$ converse theorem, namely one gets boundedness in vertical strips
for free. Thus $\rho_K$ corresponds to an automorphic representation $\Pi$ of $\GL_2(\A_K)$,
which must have the same $L$-factors as $\pi_K$ at all places not above a place in $S$, i.e.,
at all places outside a density 0 set. By refined SMO (Theorem <ref>), this means
$\pi_K \simeq \Pi$, so $L(s, \rho_K) = L(s, \pi_K)$.
Step 4: Descend the previous step to $F$, i.e., show $L(s, \rho) = L(s, \pi)$.
If $p>2$, then $K=F$ so there is nothing to do. Assume $p=2$. I will just discuss the proof
in the simpler case that $K/F$ is quadratic (i.e., when $\sqrt{-1} \in F$),
and refer to <cit.> for the biquadratic case.
Remember, it suffices to show for almost all $v \in S$ that
$L(s, \rho_v) = L(s, \pi_v)$. Fix any place $v \in S$ such that $\rho_v$ and $\pi_v$ are
unramified, and let $w$ be a place above $v$, which will be inert. By the previous step, we know
$\rho_{K,w} \leftrightarrow \pi_{K,w}$ (in the sense of local Langlands, since unramified
representations are determined by their local $L$-factors).
This means that $\rho_v$ must correspond to either $\pi_v$ or $\pi_v \otimes \mu$,
where $\mu$ is the quadratic character associated $K_w/F_v$. Now the point is that we have
sufficient flexibility in our choice of $K$ so that we can get the correspondence
$\rho_{K,w} \leftrightarrow \pi_{K,w}$ both when $K_w/F_v$ is ramified.
But it is impossible for an unramified $\rho_v$ to correspond to a ramified twist of the
unramified $\pi_v$, so we must have $\rho_v \leftrightarrow \pi_v$, and we are done.
Finally we remark that a similar conjecture should be true for (families of compatible)
$\ell$-adic Galois representations. In order for the proof of the above theorem to go through for
$\ell$-adic Galois representations, first we would need to know a purity result (which is known
in some cases), such as $L(s,\rho_v)$
has no poles in $\Ree(s) > 0$, to conclude temperedness of $\pi$. Then we would need to know
that entirety of the twists $L(s, \rho \otimes \chi)$ also implies boundedness in vertical strips, so we
can use the converse theorem in Step 3. (For Artin representations, this follows from a theorem of Brauer
which tells us Artin $L$-functions are quotients of products of degree 1 $L$-functions.)
author=Arthur, James,
author=Clozel, Laurent,
title=Simple algebras, base change, and the advanced theory of the trace
series=Annals of Mathematics Studies,
publisher=Princeton University Press, Princeton, NJ,
review=1007299 (90m:22041),
author=Blomer, Valentin,
author=Brumley, Farrell,
title=On the Ramanujan conjecture over number fields,
journal=Ann. of Math. (2),
author=Clozel, Laurent,
title=Purity reigns supreme,
journal=Int. Math. Res. Not. IMRN,
author=Deligne, Pierre,
author=Serre, Jean-Pierre,
title=Formes modulaires de poids $1$,
journal=Ann. Sci. École Norm. Sup. (4),
pages=507–530 (1975),
review=0379379 (52 #284),
author=Feigon, Brooke,
author=Martin, Kimball,
author=Whitehouse, David,
title=Periods and nonvanishing of central $L$-values for $\mathrm{GL}(2n)$,
author=Gelbart, Stephen,
author=Jacquet, Hervé,
title=A relation between automorphic representations of ${\rm GL}(2)$
and ${\rm GL}(3)$,
journal=Ann. Sci. École Norm. Sup. (4),
review=533066 (81e:10025),
author = Jacquet, H,
author = Langlands, R. P.,
title = Automorphic forms on ${\rm GL}(2)$,
series = Lecture Notes in Mathematics,
publisher = Springer-Verlag, Berlin-New York,
date = 1970,
pages = vii+548,
review=0401654 (70),
author=Jacquet, Hervé,
author=Piatetski-Shapiro, Ilja Iosifovitch,
author=Shalika, Joseph,
title=Automorphic forms on ${\rm GL}(3)$. I,
journal=Ann. of Math. (2),
review=519356 (80i:10034a),
author=Jacquet, H.,
author=Shalika, J. A.,
title=On Euler products and the classification of automorphic
representations. I,
journal=Amer. J. Math.,
review=618323 (82m:10050a),
author=Khare, Chandrashekhar,
title=Serre's conjecture and its consequences,
journal=Jpn. J. Math.,
review=2609324 (2011d:11121),
author=Kim, Henry H.,
title=Functoriality for the exterior square of ${\rm GL}_4$ and the
symmetric fourth of ${\rm GL}_2$,
note=With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and
Peter Sarnak,
journal=J. Amer. Math. Soc.,
review=1937203 (2003k:11083),
author=Kim, Henry H.,
author=Shahidi, Freydoon,
title=Cuspidality of symmetric powers with applications,
journal=Duke Math. J.,
review=1890650 (2003a:11057),
author=Luo, Wenzhi,
author=Rudnick, Zeév,
author=Sarnak, Peter,
title=On the generalized Ramanujan conjecture for ${\rm GL}(n)$,
title=Automorphic forms, automorphic representations, and arithmetic,
address=Fort Worth, TX,
series=Proc. Sympos. Pure Math.,
publisher=Amer. Math. Soc., Providence, RI,
review=1703764 (2000e:11072),
author=Martin, Kimball,
title=Four-dimensional Galois representations of solvable type and
automorphic forms,
note=Thesis (Ph.D.)–California Institute of Technology,
publisher=ProQuest LLC, Ann Arbor, MI,
author=Martin, Kimball,
author=Ramakrishnan, Dinakar,
title=A comparison of automorphic and Artin $L$-series of GL(2)-type agreeing at degree one primes,
status=Contemp. Math., to appear,
author=Mazur, B.,
title=On the passage from local to global in number theory,
journal=Bull. Amer. Math. Soc. (N.S.),
review=1202293 (93m:11052),
author=Miyake, Toshitsune,
title=On automorphic forms on ${\rm GL}_{2}$ and Hecke operators,
journal=Ann. of Math. (2),
review=0299559 (45 #8607),
author=Moreno, Carlos J.,
title=Analytic proof of the strong multiplicity one theorem,
journal=Amer. J. Math.,
review=778093 (86m:22027),
author=Piatetski-Shapiro, I. I.,
title=Multiplicity one theorems,
title=Automorphic forms, representations and $L$-functions,
address=Proc. Sympos. Pure Math., Oregon State Univ., Corvallis,
series=Proc. Sympos. Pure Math., XXXIII,
publisher=Amer. Math. Soc., Providence, R.I.,
review=546599 (81m:22027),
author=Rajan, C. S.,
title=On strong multiplicity one for $l$-adic representations,
journal=Internat. Math. Res. Notices,
review=1606395 (99c:11064),
author=Rajan, C. S.,
title=On strong multiplicity one for automorphic representations,
journal=J. Number Theory,
review=1994478 (2004f:11050),
author=Ramakrishnan, Dinakar,
title=A refinement of the strong multiplicity one theorem for ${\rm
GL}(2)$. Appendix to: “$l$-adic representations associated to modular
forms over imaginary quadratic fields. II” [Invent. Math. 116
(1994), no. 1-3, 619–643; MR1253207 (95h:11050a)] by R. Taylor,
journal=Invent. Math.,
review=1253208 (95h:11050b),
author=Ramakrishnan, Dinakar,
title=Pure motives and automorphic forms,
address=Seattle, WA,
series=Proc. Sympos. Pure Math.,
publisher=Amer. Math. Soc., Providence, RI,
review=1265561 (94m:11134),
author=Ramakrishnan, Dinakar,
title=A mild Tchebotarev theorem for ${\rm GL}(n)$,
journal=J. Number Theory,
author=Ramakrishnan, Dinakar,
title=A theorem à là Tchebotarev for ${\rm GL}(n)$,
author=Taylor, Richard,
title=$l$-adic representations associated to modular forms over
imaginary quadratic fields. II,
journal=Invent. Math.,
review=1253207 (95h:11050a),
author=Walji, Nahid,
title=Further refinement of strong multiplicity one for $\rm GL(2)$,
journal=Trans. Amer. Math. Soc.,
author=Wiles, A.,
title=On ordinary $\lambda$-adic representations associated to modular
journal=Invent. Math.,
review=969243 (89j:11051),
author=Zhang, Wei,
title=Fourier transform and the global Gan-Gross-Prasad conjecture for
unitary groups,
journal=Ann. of Math. (2),
|
1511.00214
|
§ MOTIVATION
Lattice QCD (LQCD) is a non-perturbative
approach that provides a powerful tool for ab initio evaluation
of hadron observables. These include both quantities that are well
determined experimentally, but also those that are not easily
accessible in experiment. Thus, LQCD provides input to
phenomenology and to searches for beyond the Standard Model
Physics. Recent progress in the simulation of LQCD has been
impressive, mainly due to the improvements in the algorithms,
development of new techniques, and increase in computational
power. This enabled simulations to be carried out at parameters very
close to their physical values.
Understanding nucleon structure from first principles is considered a
milestone of hadron physics and numerous experiments have been
devoted to its study, starting with the measurements of the
electromagnetic form factors initiated more than 50 years
ago. Reproducing these key observables within the lattice QCD
formulation is a prerequisite to obtaining reliable predictions on
observables that explore Physics beyond the Standard Model. There is a
rich experimental program in major facilities (CERN, JLab, MAMI, MESA, PSI,
JPARC, etc) investigating hadron structure, such as the proton radius,
electric dipole moments and scalar and tensor interactions.
The 12 GeV upgrade of the Continuous Electron Beam
Accelerator Facility at JLab will allow to employ new
methods for studying the basic properties of hadrons. Hadron structure has
been an essential part of the physics program, which involves new and
interesting high precision experiments, such as nucleon resonance
studies with CLAS12, the longitudinal spin structure of the nucleon,
meson spectroscopy with low momentum transfer electron scattering,
high precision measurement of the proton charge radius, and many more.
The experiments on the proton radius have attracted a lot of interest
since accurate measurements of the root mean square charge radius from muonic
hydrogen <cit.> ($\langle r^2_p \rangle_{\mu H} = 0.84\,{\rm fm}^2$)
is 7.7$\sigma$ yielded a value smaller that the radius determined from
elastic e-p scattering and hydrogen spectroscopy ($\langle r^2_p
\rangle_{e p} =0.88\,{\rm fm}^2$) <cit.> (see
Ref. <cit.> for a review). The 4$\%$
difference in the two measurements is currently not explained. We note
that the measurements in the muonic hydrogen experiments are ten times
more accurate than other measurements and they are very sensitive to
the proton size. In particular, the radius is measured from the energy
difference between the 2P and 2S states of the muonic
hydrogen <cit.> and more accurate experiments are planned at
The above few examples illustrate that hadron structure is a very rich
field of research relevant to new physics searches. Thus, lattice QCD
does not only provide input to on-going experiments, but also gives
guidance to new experiments within a robust theoretical framework.
Being one of the building-blocks in the universe, the nucleon provides
an extremely valuable laboratory for studying strong dynamics
providing important input that can also shed light in new physics searches.
Although it is the only stable hadron in the Standard Model,
its structure is not fully understood yet. There have been several
recent lattice QCD results on nucleon observables. In these
proceedings we discuss representative observables probing hadron
structure, as well as, challenges involved in their computations. Topics to
be covered include the nucleon axial and tensor charges, the nucleon
spin, including disconnected contributions, neutron electric dipole
moment, the first gluon moment of parton distribution functions (PDFs), and a direct method for
computing quasi-distribution functions on the lattice. The systematic
uncertainties related to nucleon matrix elements are also investigated.
§ NUCLEON MATRIX ELEMENTS
In the evaluation of nucleon matrix elements in LQCD there are two type of diagrams
entering, shown in Fig. <ref>. The disconnected diagram has been neglected
in most of the studies because it is very noisy and expensive to
compute. However, in the last few years a number of groups are
studying various techniques for its computation using dynamical simulations.
Connected (left) and disconnected (right) contributions to
the nucleon three-point function.
For the computation of nucleon matrix elements one constructs
two- and three-point correlation functions defined as
G^2pt(q⃗, t_f) = ∑_x⃗_f e^-ix⃗_f ·q⃗
Γ^0_βα ⟨J_α(x⃗_f,t_f)J_β(0) ⟩ ,
G^3pt_O(Γ^μ,q⃗, t_f) = ∑_x⃗_f, x⃗ e^ix⃗ ·q⃗ e^-ix⃗_f ·p⃗'
Γ^μ_βα ⟨ J_α(x⃗_f,t_f) O(x⃗,t) J_β(0) ⟩ ,
appropriately projected in order to compute the quantities of
interest. For instance, the projectors $\Gamma^\mu$ are usually defined as
$\Gamma^0 \equiv \frac{1}{4}(1+\gamma_0),\,\, \Gamma^k \equiv \Gamma^0\cdot\gamma_5\cdot\gamma_k\,$.
The lattice data are extracted from dimensionless ratio of the two-
and three-point correlation functions
R_O(Γ,q⃗, t, t_f) = G^3pt_O(Γ,q⃗,t)/G^2pt(0⃗, t_f)
×√(G^2pt(-q⃗, t_f-t)G^2pt(0⃗, t)G^2pt(0⃗, t_f)/G^2pt(0⃗ , t_f-t)G^2pt(-q⃗,t)G^2pt(-q⃗,t_f))
→t_f-t→∞ t-t_i→∞
Π(Γ,q⃗) .
The above ratio is considered optimized since it does not contain
potentially noisy two-point functions at large separations and also
correlations between its different factors reduce the statistical
noise. The most common method to extract the desired matrix element is to
look for a plateau with respect to the current insertion time, $t$ (or,
alternatively, the sink time, $t_f$), which should be located at a
time well separated from the creation and annihilation times in order
to ensure single state dominance. To establish proper connection to
experiments we apply renormalization which, for most of the quantities
discussed in this review, is multiplicative
Π^R (Γ,q⃗) = Z_O Π(Γ,q⃗) .
The renormalized matrix elements can be parameterized in terms of
Generalized Form Factors (GFFs), and the decomposition follows the
symmetry properties of QCD. As an example we take the axial current
insertion, which decomposes into two Lorentz invariant Form Factors
(FFs), the axial ($G_A$) and induced pseudoscalar ($G_p$)
⟨N(p',s')|ψ̅(x) γ_μ γ_5 ψ(x)|N(p,s)⟩= i (
u̅_N(p',s') [
]u_N(p,s) ,
where $q^2$ is the momentum transfer in Minkowski space (hereafter, $Q^2=-q^2$).
Here, I will mostly consider the flavor isovector
combination for which the disconnected contribution cancels out;
strictly speaking, this happens for actions with exact isospin
symmetry. Another advantage of the isovector combination is that
the renormalization simplifies considerably.
§.§ Systematic uncertainties
The systematic uncertainties are important aspects of lattice
computations that need to be addressed carefully. In a nutshell, such
systematics are:
$\bullet$ cut-off effects due to the introduction of a finite lattice
spacing. For a proper continuum extrapolation one requires
simulations for, at least, three values of the lattice spacing, which is
computationally very costly, especially as we approach the physical
point. To minimize this systematic, gauge configurations are generated
employing improved actions with a value of the lattice that ensures small or negligible cut-off effects compared to the statistical accuracy.
$\bullet$ finite volume effects due to the finite extent of the space-time box. These in general depend on the quantity under
study. Ideally, simulations should be performed at multiple volumes, so
that the infinite volume limit can be taken. This requires significant
computer resources. As a rule of thumb one needs $L\,m_\pi$ larger than $3.5$
to suppress finite volume effects.
$\bullet$ contamination from other hadron states due to the fact that
the interpolating field used to create a hadron of given quantum
numbers couple in addition to states higher in energy. While
for two-point functions identification of the lowest energy state is straight forward,
for three-point functions it is more saddle, and there are various
methods to extract information from lattice data. The most common
approach is the so called plateau method in which one probes the large
Euclidean time evolution of the ratio in Eq. (<ref>)
R_O(Γ,t_i,t,t_f) →(t_f-t) Δ>>1 (t-t_i) Δ>>1
M[ 1 + α e^-(t_f-t) Δ(p') +
β e^-(t-t_i) Δ(p) +⋯] .
In the above equation the excited states contributions
fall exponentially with the sink-insertion ($t_f-t$) and
insertion-source ($t-t_i$) time separation. So, it is possible to
reduce the unwanted excited states contamination by increasing the
source-sink separation, but this comes with a cost of increased
statistical noise.
Another method is the so-called summation method in which we sum the
ratio from the source to the sink, and thus, the excited state
contaminations are suppressed by exponentials decaying with
$(t_f-t_i)$ rather than $(t_f - t)$ and $(t - t_i)$.
However, one needs the slope of the summed ratio
∑_t=t_i^t_f R(t_i,t,t_f) = const. +
M (t_f-t_i) +
O( e^-((t_f-t_i) Δ(p'))) +
O( e^-((t_f-t_i) Δ(p)) ) .
$\bullet$ simulations at unphysically large values of the pion mass
due to limitations on the computational resources and optimization
techniques. Then one typically uses chiral perturbation theory
($\chi$PT) to carry the extrapolation to the physical point, with low
energy constants determined by over-constraining the fits using
experimental, as well as, lattice data. Over the last years
simulations at physical parameters have become feasible, which can be
compared directly to experimental and phenomenological data. This is a
substantial step forward since the chiral extrapolation to the
physical point is avoided, which is often difficult and can lead to
rather large systematic uncertainties, in particular in the baryon sector.
$\bullet$ renormalization, which might involve mixing with other
observables. In addition the data should be converted to the $\msbar$
scheme in order to be compared to experimental and phenomenological
data. This conversion is performed using perturbative expressions to
finite order in the coupling constant and this might bring in
systematic uncertainties; using higher-loop expressions (typically
${\cal O}(g^6)$) exhibit very small systematics. More importantly,
renormalization functions computed non-perturbatively may carry
lattice artifacts, which can be removed by subtracting them utilizing
perturbation theory <cit.>.
§ NUCLEON CHARGES
§.§ Axial Charge
One of the fundamental nucleon observables is the axial charge,
$g_A \equiv G_A(0)$, which is determined from the forward matrix
element of the axial current, and gives the intrinsic quark spin
in the nucleon. It governs the rate of $\beta$-decay and has been
measured precisely. In LQCD the axial charge can be determined directly
from the evaluation of the matrix element and thus, there is no
ambiguity associated to fits. For this reasons, $g_A$ is an optimal
benchmark quantity for hadron structure computations, and it is
essential for LQCD to reproduce its experimental value or if a
deviation is observed to understand its origin.
Collection of lattice results for $g_A$ corresponding to: $N_f{=}2{+}1$ DWF
(RBC/UKQCD <cit.>,
RBC/UKQCD <cit.>, $\chi$QCD <cit.>),
$N_f{=}2{+}1$ DWF on asqtad sea (LHPC <cit.>), $N_f{=}2$
TMF (ETMC <cit.>), $N_f{=}2$ Clover
(QCDSF/UKQCD <cit.>, CLS/MAINZ <cit.>,
QCDSF <cit.>, RQCD <cit.>),
$N_f{=}2{+}1$ Clover (LHPC <cit.>,
CSSM <cit.>), $N_f{=}2{+}1{+}1$ TMF
(ETMC <cit.>), $N_f{=}2{+}1{+}1$ HISQ
(PNDME <cit.>), $N_f{=}2$ TMF with Clover
(ETMC <cit.>). The asterisk is the experimental value <cit.>.
There are numerous computations of $g_A$ from many collaborations
and selected results are shown in Fig. <ref> as a function
of $m_\pi^2$. These results have been obtained using dynamical
gauge field configurations with ${\cal O}(a)$-improved lattice QCD
actions, namely Domain Wall Fermions (DWF), Hybrid, Clover,
Twisted Mass Fermions (TMF) and HISQ fermions (see caption of
Fig. <ref> for references). For a meaningful comparison we
include only results obtained from the plateau method without any
volume corrections. The latest achievement of the Lattice Community
are the results at the physical point for which there is no necessity
of chiral extrapolation eliminating an up to now uncontrolled
extrapolation. The ones at the two lowest values of the pion
mass correspond to PNDME (128 MeV) <cit.> and ETMC
(133 MeV) <cit.>, and are in agreement with the
experimental value: $g_A^{\rm exp}=0.2701(25)$ <cit.>.
Of course the statistical errors are still large and it is necessary
to increase the statistics and study the volume and lattice spacing
dependence before finalizing these results. In addition, the results
shown in Fig. <ref> are at a given lattice spacing and volume
and, thus, systematic effects should be investigated.
In summary, based on current results on the axial charge <cit.>,
we conclude that cut-off effects are small, at least for $a
\le 0.1$ fm, and no indication of significant excited state
contamination has been observed indicating that sink-source time
separation of about 1 fm is sufficient. No clear conclusion can be
extracted regarding finite volume effects that need further
investigation. It is worth stressing, however, that the value of $g_A$
determined close to the physical point by ETMC with $L\,m_\pi \sim 3$
($a<0.1$ fm), and by PNDME with $L\,m_\pi \sim 3.75$ ($a<0.1$ fm)
are in agreement with the experimental data.
We note that all high statistics studies of systematic uncertainties
have been performed at relatively large values of the pion mass. It is
thus essential to also perform similar investigations at values of the
pion mass closer to the physical one. Given that the signal to noise
error decreases exponentially as the pion mass decreases
one needs to increase considerably the number of independent
measurements leading to increase computational cost, and thus, noise
reduction methods are highly valuable.
§.§ Tensor Charge
The nucleon scalar and tensor charges have not been studied in LQCD as extensively as
since the contributions of effective
scalar and tensor interactions in the Standard Model are very small
(per-mil level). These interactions correspond to the non $V-A$
structure of weak interactions and serve as a test for new physics.
Ongoing experiments using ultra-cold
neutrons <cit.>,
as well as planned ones <cit.> will reach the
necessary precision to investigate such interactions.
In order to study the scalar and tensor interactions we add a term in
the effective Hamiltonian corresponding to new BSM physics,
H_eff = G_F ( J_V_A^l×J_V_A^q + ∑_i ϵ_i O_i^l ×O_i^q )
where the sum includes operators with novel structure, such as the
scalar and tensor, which come with low-energy couplings that are
related to masses of new TeV-scale particles.
Experimentally, bounds on the tensor coupling constant arise
in the radiative pion decay $\pi \to e \nu\gamma$, while new
experiments at Jefferson lab are planned using polarized 3He/Proton
aim at increasing its accuracy by an order of magnitude <cit.>.
Also, experiments at LHC are expected to increase the limits to
contributions from tensor and scalar interactions by an order of
magnitude, making these observables interesting probes of new
physics. Computations of the scalar charge will also provide input for
dark matter searches, since experiments aiming at a direct detection
of dark matter, are based on measuring the recoil energy of a nucleon
hit by a dark matter candidate. In several supersymmetric scenarios
<cit.> the dark matter nucleon interaction is mediated
through a Higgs boson. In this case the theoretical expression of the
spin independent scattering amplitude at zero momentum transfer
involves the quark content of the nucleon or the nucleon
$\sigma$-term, which is closely related to the scalar charge. Thus,
computations of the nucleon scalar and tensor charges within LQCD will
provide useful input for the ongoing experimental searches for BSM
In these proceedings we focus on the tensor charge, which is the zeroth
moment of the transversity distribution functions, the last part among
the three quark distributions at leading twist
⟨N(p',s')| O_T^α^μν |N(p,s)⟩ , O_T^α^μν = ψ̅ σ^μν τ^α/2 ψ .
Results on the isovector tensor charge are compared in Fig <ref>
for several discretizations, lattice spacings, and volumes. For all
results, the plateau method has been chosen for a meaningful
comparison. Overall, there is a very good agreement among lattice data, which
also exhibit very mild pion mass dependence. We would like to highlight
the data at the physical point <cit.>
that provide a prediction for this quantity, free of uncontrolled
systematics due to chiral extrapolations.
Lattice results for $g_T$ as a function of the $m_\pi^2$, corresponding to:
$N_f{=}2$ Clover (QCDSF/UKQCD <cit.>),
$N_f{=}2$ DWF (RBC <cit.>),
$N_f{=}2{+}1$ DWF (RBC/UKQCD <cit.>, LHPC <cit.>),
$N_f{=}2{+}1$ DWF on asqtad sea (LHPC <cit.>),
$N_f{=}2{+}1$ Clover (LHPC <cit.>),
$N_f{=}2{+}1{+}1$ TMF (ETMC <cit.>),
$N_f{=}2{+}1{+}1$ HISQ (PNDME <cit.>),
$N_f{=}2$ TMF with Clover (ETMC <cit.>).
The excited states contamination for $g_T$ have been also investigated,
revealing a weak dependence on the source-sink time separation, $T_{sink}$
the values from the plateau method do not vary as a function of
$T_{sink}$ and are in agreement with the value extracted from the
summation method, within statistical uncertainties.
On the experimental side there are available data for $g_T$ obtained
from combined global analysis of the measured azimuthal asymmetries in
SIDIS and in $e^+ e^- \to h_1 h_2 X$ (see, e.g. <cit.>).
There are also results from model predictions, for instance the
predictions by a covariant quark-diquark model. However, direct
comparison of the tensor charges from different models and scales is
not always meaningful, since the tensor charges are strongly scale-dependent
quantities, in particular at low values of the renormalization scale.
Therefore, ab initio calculations of $g_T$ from Lattice QCD are
extremely useful in providing reliable and model-independent predictions.
§ PDFS ON THE LATTICE
Measurements of parton distribution functions in high-energy
processes such as deep-inelastic lepton scattering and Drell-Yan in
hadron-hadron collisions are very interesting since they provide
information on the quark and gluon structure of a hadron. To leading
twist, these quantities give the probability of finding a specific
parton in the hadron carrying certain momentum and spin, in the
infinite momentum frame. Due to the fact that PDFs are light-cone
correlation functions (quark and gluons fields are separated along the
light-cone, defined in the real Minkowski time), what is
calculated in LQCD are Mellin moments expressed in terms of hadron matrix elements
of local operators. Although there is intense activity on the
computation of such moments in lattice QCD, it is highly desirable to
have information on the PDFs themselves. The Mellin moments are
related to the original PDFs through the operator product expansion
(OPE). However, the reconstruction of the PDFs appears to be
unfeasible since the signal-to-noise ratio becomes very small for
higher moments. Also for moments with more than 3 derivatives there is
unavoidable mixing with lower dimension operators, which complicates
the renormalization procedure. In addition, there is limited progress
in calculations of gluon moments (see Section <ref>),
which require a disconnected insertion, has low signal quality and
operator mixing.
Recently, a novel direct approach has been proposed by
Ji <cit.>, suggesting that one can compute a parton
quasi-distribution function, $\tilde{q}(x, \Lambda, P, \Gamma)$, where
$x=k/|\vec{P}|$, $\Lambda$ is an ultraviolet cutoff scale, $\vec{P}$
is the momentum of the nucleon, and $\Gamma$ is the Dirac structure of
the operator under study. $\tilde{q}(x, \Lambda, P)$ is accessible on
the lattice, and for large momenta, one can establish connection with
the PDFs through a matching procedure. Such a matching appears in
one-loop perturbation theory <cit.> and the computation
of quasi-distribution functions has been carried out in
Ref. <cit.> using $N_f{=}2{+}1{+}1$ HISQ gauge ensembles
with clover valence quarks, and more recently in
Ref <cit.> for $N_f{=}2{+}1{+}1$ twisted mass
The momentum-dependent non-local static correlation is written as
q̃(x,Λ,P_z,Γ) = ∫_-∞^infty dz/4π e^-izk ×⟨P |ψ̅(z)Γe^i g∫_0^z
A_z(z^') dz^' ψ(0) |P⟩ ,
where $x$ is the momentum distribution and $e^{ig\int_0^z A_z(z^\prime)
dz^\prime}$ is the Wilson line introduced to ensure gauge invariance
in the quark distribution. Also, for simplicity, the momentum
$\vec{P}$ is taken in the $z$-direction. One of the characteristics of
$\tilde{q}$ is that, unlike the case of the physical PDFs, is non=zero for $|x|>1$. When the momentum approached infinity one
recovers the physical distribution functions, $q(x,\mu)$, with the
infrared region remaining the same. For finite
but large enough momenta, $\tilde q$ and $q$ are related via
q(x,μ) = q_bare(x){1 + α_s/2π Z_F(μ)
} + α_s/2π∫_x^1 q^(1) (x/y,μ)
q_bare(y) dy/y + 𝒪(α_s^2) ,
q̃ (x,Λ,P_z) = q_bare(x){1 +
α_s/2π Z̃_̃F̃(Λ,P_z) } +
α_s/2π∫_x/x_c^1 q̃^(1) (x/y,Λ,P_z )
q_bare(y) dy/y + 𝒪(α_s^2) ,
where $q_{bare}$ is the bare distribution, $Z_F$ and $\tilde Z_F$ are
the wave function corrections, $q^{(1)}$ and $\tilde{q}^{(1)}$ are the
vertex corrections. Also, $\mu$ is the renormalization scale and
$x_c=\Lambda/P_x$ is the largest possible value of the momentum
distribution, $x$. The leading UV divergences to the
quasi-distribution functions are computed in perturbation theory by
having $P_z$ fixed, while sending $\Lambda \to \infty$. The UV
regulator $\Lambda$ will be set to the renormalization scale $\mu$
when relating $\tilde{q}$ at finite momentum to $\tilde{q}$ at
infinite momentum.
On the lattice one computes $\tilde{q}(x,\Lambda,P_z,\Gamma)$ as defined
in Eq. (<ref>), which is then used in the lhs of Eq. (<ref>)
to calculate the rhs of Eqs. (<ref>) - (<ref>) in order to
extract the quark distribution. This makes use of perturbation theory
and to date results exist for the non-singlet case and to ${\cal O}(a_s)$
for the vertex corrections and the self-energy. Thus, a combination of
Eqs. (<ref>) - (<ref>) leads to
q̃ (x,Λ,P_z) = q(x, μ) + α_s/2π
q(x,μ) {Z̃_̃F̃(Λ,P_z)-Z_F(μ) }
+ α_s/2π∫_x/x_c^1 (q̃^(1)
(x/y,Λ,P_z )-q^(1) (x/y,μ)) q(y, μ) dy/y +
𝒪(α_s^2) ,
From the above expression one can isolate $q(x,\mu)$, and by including at
the same time anti-quarks ($\bar{q}(x)=-q(-x)$), Eq. (<ref>)
can be rewritten as
q(x, μ) = q̃ (x,Λ,P_z) - α_s/2π
q̃ (x,Λ,P_z) δ(Z̃_̃F̃(Λ,P_3)-Z_F(μ))
- α_s/2π∫_-1^1
(q̃^(1) (ξ,Λ,P_3 )- q^(1) (ξ,μ) ) q̃ (y,Λ,P_z) dy/|y| +
𝒪(α_s^2) .
A nucleon mass correction in $M_N/P_z$ can be also made to an
arbitrary order. More details are given in Refs. <cit.>.
A first study appeared in Ref <cit.> for the unpolarized
and polarized quasi-distribution functions using clover valence
fermions on an $N_f{=}2{+}1{+}1$ ensemble of HISQ quarks corresponding
to $m_\pi\sim$310 MeV. The authors apply HYP smearing to the gauge
links, which appears to minimize the discretization effects. In
addition, since the multiplicative renormalization of $\tilde{q}$ has
not been computed yet, the smearing is important because it shift the
renormalization functions close to unity.
For the unpolarized case, the matrix element calculated on the lattice is
h(z,Λ,P_z) = ⟨P⃗|ψ̅(z) γ_3
( ∏_n U_z(nẑ)) ψ(0) |P⃗⟩ .
Note that, in order to study the polarized operator one should simply
substitute the Dirac structure $\gamma_3$ with $\gamma_5\,\gamma_3$.
The matrix elements are computed with the nucleon boosted with
momentum $P_z=\frac{2\pi}{L},\,\frac{4\pi}{L},\,\frac{6\pi}{L}$. In
the left panel of Fig. <ref> we show results for the
isovector $\tilde{q}$ which is the Fourier transformation of
$h(z,\Lambda,P_z)$ in the $z$ coordinate
q̃_lat(x,Λ,P_z) = ∫dz/4π e^-izkh(z,Λ,P_z) .
The values for the momentum $P_z=\frac{2\pi}{L},\,\frac{4\pi}{L},\,\frac{6\pi}{L}$
are shown with a red, green and cyan color, respectively. One observation
is that at momentum $\frac{2\pi}{L}$ the peak of the distribution is
centered around $x=1$, where the physical distribution is, in fact,
zero. For larger momenta the peak of the band moves towards zero and
centered around $x=0.5,\,0.4$ for $P_z=\frac{4\pi}{L},\,\frac{6\pi}{L}$,
respectively. Also, the value of $\tilde{q}$ at $x=1$ reduces as the
momentum increases, which is expected, since for asymptotically large
momenta the quasi-distribution functions approach the physical ones.
In lattice calculations the maximum momenta are limited by statistical
accuracy, and thus, a large-momentum effective field theory should be
utilized to relate the finite-momentum $\tilde{q}$ to the physical ones.
To do so, one can use perturbative corrections, available to 1-loop
level <cit.>. Due to the fact that the momenta are not
infinitely large, the nucleon-mass corrections -expressed in terms of
$M_N^2/(4P_z^2)$- are also important and should taken into account as
explained in Ref. <cit.>. The lattice data for $\tilde{q}$
upon the mass and 1-loop corrections have a milder momentum dependence,
and one should finally extrapolate to the infinite momentum limit,
using the fit $a + b/P_z^2$. The extrapolated unpolarized isovector
$\tilde{q}$ is shown in the right panel of Fig. <ref>,
where one observes that outside the region $|x| \ge 1$ the curve
drops significantly, as expected. The lattice data are plotted with
results from global analysis by MSTW <cit.> and
CTEQ-JLab <cit.>, although no attempt for comparison is
made since the available lattice data cannot be extrapolated to the
chiral limit nor the continuum, and no source of systematics has been
Left: Lattice results for the isovector $\tilde{q}(x)$ using
momenta $P_z$: $\frac{2\pi}{L}$ (red), $\frac{4\pi}{L}$
(green),$\frac{6\pi}{L}$ (cyan). Right: Unpolarized isovector
$\tilde{q}(x)$ upon extrapolation in $P_z$ with 68$\%$ C.L (orange
band). Brown (green) dotted line corresponds to global analysis of
MSTW <cit.> (CTEQ-JLab <cit.>).
Another recent study of the unpolarized quasi-distribution function
was performed by ETMC, using an ensemble of $N_f{=}2{+}1{+}1$ twisted
mass fermions at $m_\pi\sim 373$MeV. The matrix elements are computed
for the 3 lowest momenta:
$P_z=\frac{2\pi}{L},\,\frac{4\pi}{L},\,\frac{6\pi}{L}$, since the
statistical noise does not allow to explore higher momenta. The
dependence on the source-sink separation is also studied, showing
compatibility within error bars. Two to five HYP smearing steps are
applied to the gauge links of the operator, which as mentioned above,
it is expected to bring the renormalization function closer to unity.
Since the renormalization function for this quantity is not yet
available (a perturbative calculation is in progress <cit.>),
a comparison between the unsmeared results with the data from various
HYP smearing steps, reveals the influence of the renormalization; it
appears that the smearing affect is stronger in the imaginary
part. However, the authors of Ref. <cit.> use the
renormalization function of the ultra-local vector current, since for
$z=0$, the operator reduces to the local vector current. As a test,
one can check the value of the renormalized $h(z=0,\Lambda,P_z)$,
defined in Eq. <ref>, which is expected to be equal to 1. The
authors find for instance that $h^R(z=0,\Lambda,4\pi/L)=0.99(3)$.
In the left panel of Fig. <ref> we show the isovector
$\tilde{q}$ for $P_z=\frac{4\pi}{L}$ and 0, 2, 5 HYP smearing
steps. As can be seen, the difference between the 0 and 2 steps is
more pronounced than the difference of 2 to 5 steps, indicating a
saturation of the smearing effect. These data correspond to the 1-loop
and nucleon mass corrected results, and both the real and imaginary
parts are taken into account. However, the difference between the
various HYP steps indicated that the proper renormalization will play
an important role, and the renormalized results are expected to agree
within statistical errors.
The physical quark distribution function $q$ can be extracted from
$\tilde{q}$ and then the mass corrections may be applied. This is
shown in the right panel of Fig. <ref> for momentum
$P_z=\frac{6\pi}{L}$. The authors find that while increasing the
momentum from $\frac{2\pi}{L}$ to $\frac{6\pi}{L}$, the peak of the
$u(x)-d(x)$ moves to smaller values of $x$, for the negative region,
the $\bar{d}(x)-\bar{u}(x)$ becomes very small for most of the $x<0$
region. The latter is in qualitative agreement with the behavior of
the antiquark distributions as extracted from phenomenological
analyses <cit.>. The nucleon
mass corrections lead to a desirable decrease of the distributions in
the large $x$ region. Also, the mild oscillatory behavior in the large
$x$ region is due to the fact that the Fourier transformation was
performed over a finite interval, that is $−L/2 \le z \le +L/2$.
Left: Results for the isovector $\tilde{q}$ upon one-loop and
mass corrections for the momentum $P_z=4\pi/4$ and for 0, 2 and 5
HYP smearing steps. Right: The quasi-distribution function
$\tilde{q}$, the PDF without subtracting the mass correction $q$,
and the final PDF, $q(0)$, shown for momentum
$P_z=\frac{6\pi}{L}$. Various black lines show phenomenological
results at 6.25 MeV$^2$ from MSTW <cit.>
(CJ12 <cit.>), ABM11 <cit.>.
§ NUCLEON SPIN
In 1989 DIS experiments at CERN showed that only a small amount of the
proton spin was actually carried by the valence quarks. This was
called “the proton spin crisis”, but since then our understanding
on the proton spin has evolved. We now know that both the gluons and
sea quarks are polarized and, thus, their contribution to the spin is
essential. It is also understood that a complete description of the
spin requires to take into account the non-perturbative structure of
the proton. Using the lattice QCD formalism one can provide
significant input towards understanding this open issue. The total
nucleon spin is generated by the sum of the quark orbital angular momentum
($L^q$), the quark spin ($\Sigma^q$) and the gluon angular momentum
($J^g$). The quark components are related to $g^q_A$ and the GFFs of
the one-derivative vector at $Q^2=0$
1/2 = ∑_q (L^q + 1/2ΔΣ^q ) + J^g ,
J^q = 1/2 ( A_20^q + B_20^q ) , L^q=J^q-Σ^q , Σ^q = g_A^q ,
where $L^q,\,\Delta\Sigma^q$ and $J^g$ are gauge invariant.
Since we are interested in the individual quark contributions to the
various components of the spin, one needs to consider the
disconnected contributions.
The computation of disconnected diagrams using improved actions with
dynamical fermions became feasible over the last years and for the
proper renormalization of the individual quark and isoscalar
contributions one should take into account the singlet operator and
its proper renormalization (see Ref. <cit.> for
further discussion).
A number of results have appeared recently where the disconnected loop
contributions to $g_A$ are evaluated as shown in Fig. <ref>. We
observe a nice agreement among results using a number of methods both
for the light <cit.>, as well as for the strange
quark contributions <cit.>.
For $g_A^{light}$ we find $\sim10\%$ contributions compared to the
connected part that must be taken into account in the discussion of
the spin carried by quarks in the proton. These contributions are
negative and thus reduce the value of $g_A^q$.
Disconnected contributions for $g_A^q$ for the light (left) and
strange (right) quark contributions.
The total spin, $J^q$, and the quark spin, $\Sigma^q$,
carried by the up and down quarks. The lattice data correspond to:
$N_f{=}2{+}1$ DWF and DWF on asqtad (LHPC <cit.>),
$N_f{=}2$ Clover (QCDSF/UKQCD <cit.>),
$N_f{=}2$ TMF (ETMC <cit.>)
$N_f{=}2{+}1{+}1$ TMF (ETMC <cit.>)
$N_f{=}2$ TMF with Clover (ETMC <cit.>).
In Fig. <ref> we show results for the u and d contributions to
the total spin, $J^q$. It is found that the u-quark exclusively
carries the spin attributed to the quarks in the nucleon since $J^d$
is consistent with zero for all pion masses and lattice discretization
schemes. The quark distribution to the intrinsic spin in also shown in
Fig. <ref>. There is a nice agreement between the results at
the physical pion mass using TM fermions <cit.> and the
experimental values for both the u- and the d-quarks. The disconnected
contributions have been neglected from most data except for one TMF
ensemble at $m_\pi=375$ MeV. The effect is shown by the shift of
the filled blue square -which ignores disconnected contributions- to
the violet triangle -which includes them. Although the effect is small,
it is larger than the statistical error and thus one needs to take
them into account. The lattice results thus corroborate the missing
spin contribution arising from the quarks.
§ GLUON MOMENTS OF THE NUCLEON
To go further in our understanding for the structure of the nucleon
and the missing contributions to its spin we need to consider
contribution from the gluonic degrees of freedom. In this section,
we will discuss lattice results that predict a sizeable
contribution of the gluons to the nucleon spin.
Investigation of the gluon distribution functions has also become of
high importance in major experimental facilities such as COMPASS
and STAR. Experimentally the gluon distributions can be determined from
the QCD evolution of the DIS and DIS measurements and a number of groups are
carrying out extended analyses.
While the quark moments have been studied
extensively <cit.> there are only a few
computations for the gluon moments, mainly due to the bad
signal-to-noise ratio, as well as the fact that there is a mixing with
the corresponding quark singlet operator. Here we discuss recent results for
the unpolarized gluon moment using different methods. This observable
can be evaluated by employing the following gluon operator
O_μν^g = -Tr[ G_μρ G_νρ ],
from which one may extract the gluon moment by constructing
appropriate combinations of Dirac indices
⟨N(p) |O^L_b | N(p) ⟩=
( m_N + 2/3 E_Np⃗^2 ) ⟨x ⟩_g , O^L_b ≡O^L_44 - 1/3 ∑_j=1^3 O^L_jj.
The lattice discretization of
the gluon operator is denoted by ${\cal O}^L_b $ and can be expressed
in terms of plaquettes
O^L_b = 4/9β/a∑_x(∑_i_c[U_i4(x,t)]-∑_i<j_c[U_ij(x,t)]) .
The advantage of this operator is that the gluon moment, $\langle x
\rangle_g$, can be extracted directly from lattice data at zero
momentum transfer, as can be seen from the rhs of
Eq. (<ref>). However, the fact that terms of similar magnitude
are subtracted leads potentially to a noisy signal.
A direct computation of the gluon moment is related to the following
ratio at zero momentum transfer
⟨[N(t)N(0)]_p=0 ℬ(τ)⟩/⟨N(t)N(0)_p=0 ⟩
0≪τ≪t=m_N ⟨x ⟩_g ,
which comprises of disconnected diagrams only.
The three-point function can, thus, be written as a
product of nucleon two-point functions and the gluon operator.
Although disconnected contributions are notoriously difficult and
noisy, applying smearing to the gauge links
in the gluon operator improves the quality of the
signal. This was demonstrated in Ref. <cit.> using
$N_f{=}2+1+1$ twisted mass fermions at $m_\pi\sim 373$ MeV. The
authors test both HYP and stout smearings and find a significant
reduction of the noise-to-signal ratios after five steps for the HYP
smearing, and ten steps for the stout smearing.
The ratio for $\langle x \rangle_g$ for $N_f{=}2+1+1$ twisted
mass fermions at $m_\pi\sim 375$ MeV after 10 iterations of stout
smearing steps.
An example is shown in
Fig. <ref> for the ratio leading to $\langle x\rangle_g$ after
10 iterations of stout smearing steps.
A challenge with such a computation is that to obtain physical results
for $\langle x \rangle_g$, the lattice matrix element needs to be
renormalized. Since the gluon operator is singlet it mixes
with the quark momentum fraction $\langle x \rangle_q$, as well as
with other operators that are: (a) gauge invariant, or (b)
BRS-variations, or (c) vanish by the equations of motion. However, in
physical matrix elements the mixing with the operators (a)-(c)
vanishes and the mixing reduces to a $2\times2$ matrix, that is
⟨x ⟩_g^(μ)
∑_q ⟨x ⟩_q^(μ)
= (
) (
⟨x ⟩_g
∑_q ⟨x ⟩_q
) ,
where $\mu$ is the renormalization scale, usually set to 2 GeV.
Note that, in the quenched approximation the mixing matrix simplifies
considerably since both $Z_{gq}$ and $Z_{qg}$ become $1-Z_{qq}$
and $1-Z_{gg}$, respectively. For the renormalization of
$ \langle x \rangle_g$ the relevant matrix elements are
$Z_{gg}$ and $Z_{qq}$ and the relation to the bare quark and gluon
moments is
⟨x ⟩_g^ =
Z_gg^⟨x ⟩_g +
Z_gq^∑_q ⟨x ⟩_q .
Due to the mixing and the involvement of disconnected contributions,
an appropriate renormalization scheme
to extract the multiplicative renormalization functions and the mixing
coefficients non-perturbatively is a difficult task. As a first step we thus use perturbation theory to compute the
elements of the mixing matrix. One of the advantages of the perturbative
calculation <cit.> is that the results can be computed
directly in the $\msbar$ scheme without an intermediate RI-type step.
Since the gauge links of the operator are smeared for signal
improvement, an equivalent procedure is also followed in the
perturbative calculation in order to match the non-perturbative
calculation of $ \langle x \rangle_g$. In the calculation of
Ref. <cit.> the preferred smearing is stout since it is
analytically defined in both the perturbative and non-perturbative
evaluations. Note, however, that in the perturbative computation of
the renormalization functions introduction of smearing increases
the number of algebraic expression, which explodes as the stout
smearing steps increase. Currently, the computation is performed to 2
smearing steps, which already involved millions of terms. The smearing
parameter is chosen to be small, and thus, more levels of smearing is
expected to bring in a very small effect, due to the polynomial
dependence on the smearing parameter. For the work of Ref. <cit.>
the renormalized matrix element at $m_\pi=373$ MeV in the $\msbar$ at 2
GeV is found to be: $ \langle x \rangle_g=0.309(25)$.
An alternative approach to extract the matrix elements of the gluon
operator utilizes the Euclidean form of the Feynman-Hellman
theorem. In this methodology an
operator $\lambda {\cal O}$ is introduced into the total QCD action,
and the matrix element of the operator can be extracted from the
derivative of the energy of the state with respect to $\lambda$
∂E_N(λ)/∂λ = ( :∂Ŝ(λ)/∂λ:)_N(p)N(p),λ ,
where $:...:$ denotes the subtraction of the vacuum expectation value
of the operator. By combining the above equations with the
continuum decomposition expression, one can extract the gluon moment
at zero momentum transfer
⟨x ⟩_g = 2/3m_N∂m_N/∂λ|_λ=0 .
Note that the Feynman-Hellman methodology requires production of new
gauge ensembles for each value of the $\lambda$ parameter (and each
operator), which is computationally costly, especially for $m_\pi$
close to the physical value. This methodology was applied in
Ref. <cit.> in the quenched approximation for
clover fermions at ensembles corresponding to several values of $m_\pi$ so
that the extrapolation to the chiral limit can be taken, as shown in
Fig. <ref>. The extrapolated value is $\langle x \rangle_g
= 0.43(7)(5)$, which, despite being quenched, it is close to the value
of ETMC using dynamical twisted mass fermions <cit.>.
$\langle x \rangle_g$ for $N_f{=}0$ clover
fermions <cit.> as a functions of the pion mass squared,
$(a\,m_\pi)^2$. The open circle corresponds to the chirally
extrapolated result.
A different direction for the computation of not only the gluon, but
also the quark moments relies on a complete gauge-invariant
decomposition of the nucleon spin (see Eq. (<ref>)) in terms of the
quark spin, the quark orbital angular momentum, and the glue angular
momentum operators, as defined from the symmetric energy-momentum
tensor. Thus, instead of computing directly $J_q$ and $J_g$ from the
explicit definitions
J⃗_q = 1/2 Σ⃗_q + L⃗_q = ∫d^3x [ 1/2 ψ γ⃗ γ^5 ψ+ ψ^† { x⃗ ×(i D⃗) } ψ] , J⃗_g = ∫d^3x [ x⃗ ×( E⃗ ×B⃗ )] ,
one can calculate them from the energy-momentum tensor
J_q,g^i = 1/2 ϵ^ijk ∫ d^3x (𝒯_q,g^0k x^j - 𝒯_q,g^0j x^k) ,
which in Euclidean space the quark and gluon operators are
= (-1) i/4∑_f ψ_f
γ_4 → D_i
+ γ_i → D_4
- γ_4 ← D_i
- γ_i ← D_4
] ψ_f ,
= (+i) [-1/2 ∑_k=1^3 2
^color [G_4k G_ki + G_ik
G_k4 ]] .
The complete calculation of the quark and glue momenta and angular
momenta on a quenched lattice for Wilson fermions has been presented
in Ref. <cit.>, including both connected and disconnected
insertions for the quark contributions. Three ensembles have been
employed with $m_\pi=$478, 538, 650 MeV. The overlap operator is used
for the gauge field tensor, which leads to less noisy results
than that from usual gauge links. Details on the computation can be
found in Ref. <cit.>. Regarding the renormalization of
the operators, the authors use sum rules to define renormalization
conditions on the lattice. This results from the fact that although
the momentum and angular momentum fractions of the quark and glue are
renormalization scale and scheme dependent individually, their sums are
not because the nucleon total momentum and angular momentum are
conserved. One thus obtains
J_q,g + 1/2 Z_q,g^L [T_1(0) + T_2(0)]_q,g , ⟨x⟩_q,g = Z_q,g^L T_1(0)_q,g ,
⟨x⟩_q + ⟨x⟩_g = Z_q^L T_1 (0)_q
+ Z_g^L T_1 (0)_g = 1 ,
J_q+ J_g = 1/2 {
Z_q^L [ T_1 (0) + T_2 (0) ]_q + Z_g^L [ T_1 (0) +
T_2 (0) ]_g
} = 1/2 ,
which also leads to
Z_q^L T_2 (0)_q + Z_g ^L T_2 (0)_g =0 ,
providing sufficient conditions and cross-checks to obtain the
renormalization functions.
$T_1$, $T_2$ for gluons and angular momentum $J_g$ using
quenched Wilson fermions <cit.>.
In Fig. <ref> we show results for the gluon contributions
corresponding to 478 MeV. The extrapolated value for the gluon
unpolarized moment is $\langle x \rangle_g=0.313(56)$, which is
compatible with the results presented above for both quenched
<cit.> and dynamical fermions <cit.>.
§ NEUTRON ELECTRIC DIPOLE MOMENT
There is recently a major activity in LQCD
computation of the neutron electric dipole moment (EDM),
$\vec{d}_N$, which we review in this section.
A non-zero EDM indicates the violation of parity
($P$) and time ($T$) symmetries, and consequently of $CP$, probing
physics BSM <cit.>. So far, no finite neutron EDM (nEDM) has been
reported and current bounds are still several orders of magnitude
below what one expects from $CP$-violation induced by weak
interactions. Several experiments are under way to improve the upper
bound on the nEDM value, with the best experimental upper limit
being <cit.>
|d⃗_N |< 2.9 ×10^-13 e ·fm (90% CL) .
To investigate theoretically a finite nEDM, we add to the
$CP$-conserving QCD Lagrangian density a $CP$-violating
interaction term, proportional to the topological charge, $q$
L_QCD ( x ) = 1/2 g^2
Tr [ G_μν ( x ) G_μν ( x ) ] +
∑_f ψ_f ( x ) (γ_μ D_μ +m_f) ψ_f( x )
- i θq ( x ) ,
q ( x ) = 1/ 32 π^2 ϵ_μνρσ
Tr [ G_μν ( x ) G_ρσ ( x ) ] ,
where $\psi_f$ denotes a fermion field of flavor $f$ with bare mass
$m_f$ and $G_{\mu \nu}$ is the gluon field tensor. The so called
$\theta$-parameter controls the strength of the $CP$-breaking, and the
addition of the $CP$-violating term leads to a non-zero value for
nEDM. The $\theta$-parameter can be taken as a small continuous
parameter allowing a perturbative expansion and only keep first
order contributions in $\theta$. This is in accordance to effective
field theory calculations (see references within <cit.>)
that give a bound of the order
$\theta \lesssim {\cal O} \left(10^{-10} - 10^{-11} \right)$.
The nEDM versus $m_\pi^2$ for: a) $N_f{=}2{+}1{+}1$ twisted
mass fermions <cit.> (blue square) corresponding to a
weighted average using different methods for extracting $F_3(0)$),
b) $N_f{=}0$ DWF <cit.> (magenta upward triangles),
$N_f{=}2{+}1$ DWF <cit.> (red circles) and
$N_f{=}0$ clover fermions <cit.> (green downward
triangles) by extracting the $CP$-odd $F_3(Q^2)$ and fitting its
$Q^2$-dependence, c) $N_f{=}2$ Clover fermions <cit.>
(turquoise left triangles) obtained using a background electric field,
d) $N_f{=}2{+}1$ clover fermions <cit.> (orange diamonds)
by implementing an imaginary $\theta$.
The quantity, which is of interest is the nEDM,
$\vec{d}_N$, which at leading order of $\theta$ and in momentum space
is given by <cit.>
|d⃗_N |= θlim_Q^2 →0 |F_3(Q^2) |/2 m_N ,
where $m_N$ denotes the mass of the neutron, $Q^2{=}-q^2$ the
four-momentum transfer in Euclidean space ($q{=}p_f-p_i$) and
$F_3(Q^2)$ is the $CP$-odd form factor. In a theory
with $CP$ violation we can, therefore, calculate the electric dipole
moment by evaluating the zero momentum transfer limit of the
$CP$-odd form factor. However, the $CP$-violating matrix element gives
access to $Q_k F_3(Q^2)$ and not to $F_3(Q^2)$ alone, hindering a
direct evaluation of $F_3(0)$. Details on different methods for the
extraction of $F_3(Q^2)$ can be found in Ref. <cit.>.
Besides extracting the nEDM from the $CP$-odd form factor
there are alternative methods to compute the nEDM, such are the
implementation of an imaginary $\theta$ <cit.>, or with an
application of an external electric field and measuring the associated
energy shifts <cit.>. A collection of lattice results
are displayed in Fig. <ref>, using different methods for
obtaining $d_N$, as well as, different definition of the topological
charge. We note that the results of Ref. <cit.> have
been extrapolated to the continuum limit.
§ SUMMARY AND CHALLENGES
Lattice QCD has entered a new era in terms of simulations with the
light quark masses fixed to their physical value. This is due to major
improvements in algorithm and techniques coupled with increase in the
computational power. However, many challenges lie ahead:
development of appropriate algorithms to reduce the statistical errors
at reduced cost and addressing systematic uncertainties in order to
compute accurately observables that reproduce experimental data or can
probe beyond the standard model physics.
For hadron structure, simulations at different lattice spacings and
larger volumes are crucial for a proper study of lattice artifacts in
order to provide reliable results at the continuum limit. Such studies
require an accuracy, which is difficult to achieve with standard methods.
Noise reduction techniques are, thus, essential in order to settle
some of the long-standing discrepancies reviewed in this talk.
Similarly techniques developed for the computation of disconnected quark
loop diagrams, such as the truncated solver method <cit.>
need to be improved since they become inefficient at the physical
point <cit.>. Thus, new ideas will be needed to compute
disconnected contributions to hadron structure to an accuracy of a few
percent. Utilization of new computer architectures such as GPUs has
proved essential for the evaluation of disconnected diagrams and this
is a direction that we will further pursue in the future.
Other open issues such as the nucleon spin need the evaluation
of quantities that are challenging to compute, such as gluonic contributions. As
discussed in this review, there are several challenges in the
computation of the gluon moments, including increased gauge noise and
mixing with other operators. Perturbation theory has been utilized in
order to successfully compute the multiplicative renormalization and
disentangle the operator mixing.
Evaluation of the nucleon matrix elements for the electromagnetic
current in a theory with a $CP$-violating term in the Lagrangian,
yields the value of the neutron electric dipole moment from
first principles. Despite its difficulties, such a calculation can
guide planned experiments.
Despite the challenges of LQCD calculations, simulations at the
physical point have eliminated one of the systematic uncertainties
that was inherent in all lattice calculations in the past, that is the
difficulty to quantify systematic error due to the chiral extrapolation.
Calculating observables directly at the physical point holds the promise of resolving
discrepancies on benchmark quantities like $g_A$ and reliably compute
quantities relevant for revealing possible new physics BSM.
Acknowledgments: I would like to thank the organizers for
their invitation to participate and for providing partial financial
support. I also want to thank my collaborators for fruitful
A. Antognini, et al. [CREMA Collaboration],
Science 339 (2013) 417.
C. Carlson,
Prog. Part. Nucl. Phys. 82, 59 (2015),
P. Mohr, B. Taylor, D. Newell,
Rev. Mod. Phys. 84, 1527 (2012),
R. Pohl et al., Nature 466 (2010) 213.
M. Constantinou et al. [ETM Collaboration],
JHEP 1008, 068 (2010),
M. Constantinou, R. Horsley, H. Panagopoulos, H. Perlt,
P. Rakow, G. Schierholz, A. Schiller, J. Zanotti,
Phys. Rev. D 91, no. 1, 014502 (2015),
C. Alexandrou, M. Constantinou, H. Panagopoulos [ETM Collaboration],
R. Babich, R. Brower, M. Clark, G. Fleming, J. Osborn, C. Rebbi, D. Schaich,
Phys. Rev. D 85, 054510 (2012),
G. Bali et al. [QCDSF Collaboration],
Phys. Rev. Lett. 108, 222001 (2012),
M. Engelhardt,
Phys. Rev. D 86, 114510 (2012),
A. Abdel-Rehim, C. Alexandrou, M. Constantinou, V. Drach,
K. Hadjiyiannakou, K. Jansen, G. Koutsou, A. Vaquero,
Phys. Rev. D 89, 034501 (2014),
T. Yamazaki, Y. Aoki, T. Blum, H.W. Lin, M.F. Lin, S. Ohta, S. Sasaki,
R. Tweedie, J. Zanotti [RBC/UKQCD Collaboration],
Phys. Rev. Lett. 100, 171602 (2008),
T. Yamazaki, Y. Aoki, T. Blum, H.W. Lin, S. Ohta, S. Sasaki, R. Tweedie, J. Zanotti,
Phys. Rev. D 79, 114505 (2009),
S. Ohta [RBC/UKQCD Collaboration], PoS LATTICE 2013 274 (2013),
Y.B. Yang, Y. Chen, T. Draper, M. Gong, K.F Liu,
Z. Liu, J.P. Ma [$\chi$QCD Collaboration], PoS(LATTICE2014)137,
J. Bratt et al. [LHP Collaboration],
Phys. Rev. D 82, 094502 (2010),
C. Alexandrou, M. Brinet, J. Carbonell, M. Constantinou, P. Harraud,
P. Guichon, K. Jansen, T. Korzec, M. Papinutto [ETM Collaboration],
Phys. Rev. D 83, 045010 (2011),
D. Pleiter et al. [QCDSF/UKQCD Collaboration],
PoS(LATT2010), 153 (2010),
S. Capitani, M. Della Morte, G. von Hippel, B. Jager, A. Juttner,
B. Knippschild, H. Meyer, H. Wittig,
Phys. Rev. D 86, 074502 (2012),
R. Horsley, Y. Nakamura, A. Nobile, P. Rakow, G. Schierholz
, J. Zanotti,
Phys. Lett. B 732, 41 (2014),
G. Bali, S. Collins, B. Glässle, M. Göckeler, J. Najjar,
R. Rödl, A. Schäfer, R. Schiel, W. Söldner, A. Sternbeck ,
Phys. Rev. D 91, no. 5, 054501 (2015),
J. Green, M. Engelhardt, S. Krieg, J. Negele, A. Pochinsky, S. Syritsyn,
PoS LATTICE 2012, 170 (2012),
B. J. Owen, J. Dragos, W. Kamleh, D. Leinweber, M. Mahbub, B. Menadue, J. Zanotti,
Phys. Lett. B 723, 217 (2013),
C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, K. Jansen, C. Kallidonis, G. Koutsou,
Phys. Rev. D 88, 014509 (2013),
T. Bhattacharya, S. Cohen, R. Gupta, A. Joseph, H. W. Lin, B. Yoon,
Phys. Rev. D 89, 094502 (2014),
R. Gupta, T. Bhattacharya, A. Joseph, H.W. Lin, B. Yoon [PNDME
Collaboration], PoS(LATTICE2014)152, [arXiv:1501.07639].
A. Abdel-Rehim et al.,
J. Beringer et al., Particle Data Group,
Phys. Rev. D 86, 010001 (2012).
M. Constantinou,
PoS(LATTICE2014)001 (2015),
H. Abele, M. Astruc Hoffmann, S. Baessler, D. Dubbers, F. Gluck,
U. Muller, V. Nesvizhevsky, J. Reich, O. Zimmer,
Phys. Rev. Lett. 88, 211801 (2002),
J. Nico,
J. Phys. G 36, 104001 (2009).
A. Young et al.,
J. Phys. G 41, no. 11, 114007 (2014).
S. Baeßler, J. Bowman, S. Penttilä, D. Počanić,
J. Phys. G 41, 114003 (2014),
T. Bhattacharya, V. Cirigliano, S. Cohen, A. Filipuzzi,
M. Gonzalez-Alonso, M. Graesser, R. Gupta, H. W. Lin,
Phys. Rev. D 85, 054512 (2012),
H. Gao et al.,
Eur. Phys. J. Plus 126, 2 (2011),
J. Ellis, K. Olive,
M. Gockeler, P. Hagler, R. Horsley, D. Pleiter, P. Rakow, A. Schafer,
G. Schierholz, J. Zanotti [QCDSF and UKQCD Collaborations],
Phys. Lett. B 627, 113 (2005),
H. W. Lin, T. Blum, S. Ohta, S. Sasaki, T. Yamazaki,
Phys. Rev. D 78, 014505 (2008),
S. Ohta, T. Yamazaki [RBC and UKQCD Collaborations],
Y. Aoki, T. Blum, H. W. Lin, S. Ohta, S. Sasaki, R. Tweedie, J. Zanotti, T. Yamazaki,
Phys. Rev. D 82, 014501 (2010),
J. R. Green, J. W. Negele, A. V. Pochinsky, S. N. Syritsyn, M. Engelhardt, S. Krieg,
Phys. Rev. D 86, 114509 (2012),
T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, A. Joseph, H. W. Lin, B. Yoon,
C. Alexandrou, M. Constantinou, K. Jansen, G. Koutsou, H. Panagopoulos,
PoS LATTICE 2013, 294 (2014),
M. Anselmino, M. Boglione, U. D'Alesio, S. Melis, F. Murgia
A. Prokudin,
Phys. Rev. D 87, 094019 (2013),
X. Ji,
Phys. Rev. Lett. 110, 262002 (2013),
X. Xiong, X. Ji, J. Zhang, Y. Zhao,
Phys. Rev. D 90, no. 1, 014051 (2014),
H. W. Lin, J. Chen, S. Cohen, X. Ji,
Phys. Rev. D 91, 054510 (2015),
C. Alexandrou, K. Cichy, V. Drach, E. Garcia-Ramos, K. Hadjiyiannakou, K. Jansen, F. Steffens, C. Wiese,
Phys. Rev. D 92, no. 1, 014502 (2015),
A. Martin, W. Stirling, R. Thorne, G. Watt,
Eur. Phys. J. C 63, 189 (2009),
J. Owens, A. Accardi, W. Melnitchouk,
Phys. Rev. D 87, no. 9, 094012 (2013),
M. Constantinou, H. Panagopoulos,
Perturbative renormalization of quasi-distribution functions in Lattice QCD.
S. Alekhin, J. Blumlein, S. Moch,
Phys. Rev. D 86, 054009 (2012),
Y.B. Yang, M. Gong, K.F. Liu, M. Sun [$\chi$QCD Collaboration],
PoS(LATTICE2014)138, [arXiv:1504.04052].
S. Meinel et al. [LHP Collaboration], PoS(LATTICE2014)139.
A. Chambers et al. [CSSM/QCDSF/UKQCD Collaboration],
PoS(LATTICE2014)165, [arXiv:1412.6569].
T. Bhattacharya, R. Gupta, B. Yoon,
PoS LATTICE 2014, 141 (2014),
C. Alexandrou, M. Constantinou, V. Drach, K. Hadjiyiannakou, K. Jansen
G. Koutsou, A. Vaquero [ETM Collaboration], PoS(LATTICE2014)140,
S. Syritsyn, J. Green, J. Negele, A. Pochinsky, M. Engelhardt, P. Hagler, B. Musch, W. Schroers,
PoS LATTICE 2011, 178 (2011),
A. Sternbeck et al.,
PoS LATTICE 2011, 177 (2011),
C. Alexandrou, J. Carbonell, M. Constantinou, P. Harraud, P. Guichon, K. Jansen, C. Kallidonis, T. Korzec, M. Papinutto,
Phys. Rev. D 83, 114513 (2011),
C. Alexandrou, V. Drach, K. Hadjiyiannakou, K. Jansen, B. Kostrzewa, C. Wiese,
PoS LATTICE 2013, 289 (2014),
M. Constantinou, H. Panagopoulos, in preparation.
C. Alexandrou, M. Constantinou, V. Drach, K. Hadjiyiannakou, K. Jansen, B. Kostrzewa, H. Panagopoulos, C. Wiese,
in preparation.
R. Horsley, R. Millo, Y. Nakamura, H. Perlt, D. Pleiter, P. Rakow,
G. Schierholz, A. Schiller, F. Winter, J. Zanotti [QCDSF/UKQCD Collaborations],
Phys. Lett. B 714, 312 (2012),
M. Deka et al.,
Phys. Rev. D 91, no. 1, 014505 (2015),
M. Pospelov, A. Ritz,
Annals Phys. 318, 119 (2005), [hep-ph/0504231].
V. Helaine [nEDM Collaboration],
EPJ Web Conf. 73 (2014) 07003.
C. Baker et al.,
Phys. Rev. Lett. 97, 131801 (2006), [hep-ex/0602020].
C. Baker et al.,
Phys. Rev. Lett. 98, 149102 (2007), [arXiv:0704.1354].
S. Barr, W. Marciano in “CP Violation”, ed. C Jarlskog (World Scientific, Singapore, 1988).
C. Alexandrou, A. Athenodorou, M. Constantinou, K. Hadjiyiannakou,
K. Jansen, G. Koutsou, K. Ottnad, M. Petschlies, arXiv:1510.05823.
E. Shintani, S. Aoki, N. Ishizuka, K. Kanaya, Y. Kikukawa,
Y. Kuramashi, M. Okawa, Y. Tanigchi, A. Ukawa, T. Yoshie,
Phys. Rev. D 72, 014504 (2005), [hep-lat/0505022].
E. Shintani S. Aoki, N. Ishizuka, K. Kanaya, Y. Kikukawa,
Y. Kuramashi, M. Okawa, A. Ukawa, T. Yoshie,
Phys. Rev. D 75, 034507 (2007), [hep-lat/0611032].
F. Guo, R. Horsley, U. Meissner, Y. Nakamura, H. Perlt, P. Rakow,
G. Schierholz, A. Schiller, J. Zanotti,
Phys. Rev. Lett. 115, no. 6, 062001 (2015), [arXiv:1502.02295].
E. Shintani, S. Aoki, Y. Kuramashi,
Phys. Rev. D 78, 014503 (2008), [arXiv:0803.0797].
E. Shintani, T. Blum, A. Soni, T. Izubuchi,
PoS LATTICE 2013, 298 (2014).
A. Shindler, T. Luu, J. de Vries,
G. Bali, S. Collins, A. Schäfer,
Comput. Phys. Commun. 181, 1570 (2010),
|
1511.00084
|
Newton polygons]Newton polygons of $L$-functions of polynomials $x^d+ax^{d-1}$ with $p\equiv-1\bmod d$
Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China
[email protected], [email protected]
[2010]Primary 11; Secondary 14
Corresponding author: S. Zhang. Email: [email protected]
For prime $p\equiv-1\bmod d$ and $q$ a power of $p$, we obtain the slopes of the $q$-adic Newton polygons of $L$-functions of $x^d+ax^{d-1}\in \F_q[x]$ with respect to finite characters $\chi$ when $p$ is larger than an explicit bound depending only on $d$ and $\log_p q$. The main tools are Dwork's trace formula and Zhu's rigid transform theorem.
§ MAIN RESULTS
Let $q=p^h$ be a power of the rational prime number $p$. Let $v$ be the normalized valuation on $\overline{\Q}_p$ with $v(p)=1$. For a polynomial $f(x)\in\F_q[x]$, let $\hat f\in\Z_q[x]$ be its Teichmüller lifting. For a finite character $\chi:\Z_p\ra\C_p^\times$ of order $p^{m_\chi}$, define the
\begin{equation} L^*(f,\chi,t)=\exp\left(\sum_{m=1}^\infty S_m^*(f,\chi)\frac{t^m}{m}\right), \end{equation}
where $S^*_m(f,\chi)$ is the exponential sum
\begin{equation} S^*_m(f,\chi)=\sum_{x\in\mu_{q^m-1}}\chi(\Tr_{\Q_{q^m}/\Q_p}\hat f(x)) \end{equation}
and $\mu_n$ is the group of $n$-th roots of unity. Then $L^*(f,\chi,t)$ is a polynomial of degree $p^{m_\chi-1}d$ by Adolphson-Sperber <cit.> and Liu-Wei <cit.>. We denote $\NP_q(f,\chi,t)$ the $q$-adic Newton polygon of $L^*(f,\chi,t)$.
We fix a character $\Psi_1:\Z_p\ra\C_p^\times$ of order $p$, and denote $L^*(f,t)=L^*(f,\Psi_1,t)$ and $\NP_q(f,t)=\NP_q(f,\Psi_1, t)$. When $p\equiv 1\bmod d$, it is well-known that $\NP_q(f,t)$ coincides the Hodge polygon with slopes $\{i/d:0\le i\le d-1\}$.
Let $a$ be a nonzero element in $\F_q$. For $f(x)=x^d+ax^s (s<d)$, Liu-Niu and Zhu obtained the slopes of $\NP_q(f,t)$ for $p$ large enough under certain conditions in <cit.> and <cit.>, but these conditions are not so easy to check. For $f(x)=x^d+ax$, Zhu, Liu-Niu and Ouyang-J. Yang obtained the slopes in <cit.>, <cit.> and <cit.>, see also R. Yang <cit.> for earlier results.
In <cit.>, Davis–Wan–Xiao gave a result on the behavior of the slopes of $\NP_q(f,\chi,t)$ when the order of $\chi$ is large enough. In this way for $p$ sufficiently large, they can obtain the slopes of $\NP_q(f,\chi, t)$ based on the slopes of $\NP_q(f,\chi_0, t)$ with $\chi_0$ a character of order $p^2$. In <cit.>, Niu gave a lower bound of the Newton polygon $\NP_q(f,\chi, t)$. In <cit.>, Ouyang–Yang showed that if the Newton polygon of $L^*(f,t)$ is sufficiently close to its Hodge polygon, the slopes of $\NP_q(f,\chi,t)$ for $\chi$ in general follow from the slopes of $\NP_q(f,t)$. As a consequence they obtained the slopes of $\NP_q(x^d+ax,\chi,t)$ when $p$ is bigger than an explicit bound depending only on $d$ and $h$.
Our main results are the following two theorems.
Let $f(x)=x^d+ax^{d-1}$ be a polynomial in $\F_q[x]$ with $a\neq 0$. Let $N(d)=\frac{d^2+3}{4}$ for $q=p$ and $\frac{d^2}{2}$ for general $q$. If $p\equiv-1\bmod d$ and $p> N(d)$, the $q$-adic Newton polygon of $L^*(f,t)$ has slopes
\[\{w_0,w_1,\ldots,w_{d-1}\},\]
\[ w_i=\begin{cases}
\frac{(p+1)i}{d(p-1)},&\ \text{if}\ i<\frac{d}{2};\\
\frac{(p+1)i-d}{d(p-1)}=\frac{1}{2},&\ \text{if}\ i=\frac{d}{2};\\
\frac{(p+1)i-2d}{d(p-1)},&\ \text{if}\ i>\frac{d}{2}.
\end{cases} \]
(1) For general $p$, write $pi=dk_i+r_i$ with $1\le i,r_i\le d-1$. If $r_i>s$ for any $1\le i\le s$, then one can decide that the first $s+1$ slopes of $\NP_q(f,t)$ are $\{0,\frac{k_1+1}{p-1},\ldots,\frac{k_s+1}{p-1}\}$ by our method for sufficiently large $p$. For the rest of slopes, one needs to calculate the determinants of submatrices of “Vandermonde style” matrices.
(2) The slopes in our case coincide Zhu's result in <cit.>.
Assume $f(x)$ and $N(d)$ as above. For any non-trivial finite character $\chi$, if $p\equiv-1\bmod d$ and $p>\max\{N(d),\frac{h(d^2-1)}{4d}+1\}$, the $q$-adic Newton polygon of $L^*(f,\chi,t)$ has slopes
\[\{p^{1-m_\chi}(i+w_j):0\le i\le p^{m_\chi-1}-1,0\le j\le d-1\}.\]
§ PRELIMINARIES
§.§ Dwork's trace formula
We will recall Dwork's work for $f(x) =x^d+ax^{d-1}$. For general $f$, one can see <cit.>.
Let $\g\in \Q_p(\mu_p)$ be a root of the Artin-Hasse exponential series
\[E(t)=\exp(\sum_{m=0}^\infty p^{-m}t^{p^m})\]
such that $v(\g)=\frac{1}{p-1}$. Fix a $\g^{1/d}\in\bar\Q_p$. Let
\[ \theta(t)=E(\g t)=\sum_{m=0}^\infty \g_m t^m \]
be Dwork's splitting function. Then $v(\g_m)\ge m/(p-1)$, and $\g_m=\g^m/m!$ for $0\le m\le p-1$. Let
\[ F(x)=\theta(x^d)\theta(ax^{d-1})=\sum_{i=0}^\infty F_i x^i, \]
\[ F_i=\sum_{dm+(d-1)n=i} \g_m\g_n a^n.\]
One can see $m+n\ge i/d$ and $v(F_i)\ge \frac{i}{d(p-1)}$.
Set $A_1=(F_{pi-j}\g^{(j-i)/d})_{i,j\ge 0}$. This is a nuclear matrix over $\Q_q(\g^{1/d})$ with
\[ v(F_{pi-j}\g^{(j-i)/d})\ge \frac{pi-j}{d(p-1)}+\frac{j-i}{d(p-1)}=\frac{i}{d}. \]
We extend the Frobenius $\varphi$ to $\Q_q(\g^{1/d})$ with $\varphi(\g^{1/d})=\g^{1/d}$.
Let $A_h=A_1\varphi(A_1)\cdots\varphi^{h-1}(A_1)$. Then
\[ L^*(f,t)=\frac{\det^{\varphi^{-1}}(I-tA_h)}{\detphi(I-tqA_h)}. \]
§.§ Zhu's rigid transformation theorem
Let $U_1=(u_{ij})_{i,j\ge0}$ be a nuclear matrix over $\Q_q(\g^{1/d})$. Then the Fredholm determinant $\det(I-tU_1)$ is well defined and $p$-adic entire (see <cit.>). Write
\[ \det(I-tU_1)=c_0+c_1t+c_2t^2+\cdots.\]
For $0\le t_1<t_2<\cdots<t_s$, denote by $U_1(t_1,\ldots,t_s)$ the principal sub-matrix consisting of $(t_i,t_j)$-entries of $U_1$ for $1\le i,j\le s$. In particular, denote $U_1[s]=U_1(0,1,\ldots,s-1)$. Then we have $c_0=1$ and for $s\ge 1$,
\[ c_s=(-1)^s\sum_{0\le t_1<t_2<\cdots<t_s} \det U_1(t_1,t_2,\ldots,t_s).\]
Let $U_h=U_1\varphi(U_1)\cdots\varphi^{h-1}(U_1)$. Write
\[ \det(I-tU_h)=C_0+C_1t+C_2t^2+\cdots.\]
(See <cit.>.)
Suppose $(\beta_s)_{s\ge 0}$ is a strictly increasing sequence such that
\[ \beta_i\le v(a_{ij})\ \text{and}\ \lim_{s\ra +\infty} \beta_s=+\infty. \]
\[ \sum_{s<i} \beta_s\le v(\det U_1[i])\le \frac{\beta_i-\beta_{i-1}}{2}+\sum_{s<i}\beta_s \]
holds for every $1\le i\le k$, then $v(C_i)=hv(\det U_1[i])$ for $1\le i\le k$ and
\[\NP_q(\det(I-tA_h[k]))=\NP_p(\det(I-tA_1[k])).\]
§ SLOPES OF THE NEWTON POLYGON OF $L^*(F,\CHI,T)$
From now on, we assume $p\equiv-1\bmod d$ and write $p=dk-1$.
§.§ The case $\chi=\Psi_1$
Let $M(s)=(a_{ij})_{1\le i,j\le s}$ be an $s\times s$ matrix with entries
\[ a_{i,j}=\frac{a^{i+j}}{(ki-i-j)!(i+j)!}.\]
Then $v(\det M(s))=0$ for $1\le s\le d-1$.
Denote $x[0]=1$ and $x[n]:=x(x-1)\cdots(x-n+1)$ for $n\geq 1$. Then $x[n]$ is a polynomial of $x$ of degree $n$ and $\{(x+j)[t]:0\le t\le j-1\}$ is a basis of the space of polynomials of degree $\le j-1$. Thus we can write
\[((k-1)x-1)[j-1]=c_0(j)+\sum_{t=1}^{j-1}c_t(j)\cdot (x+j)[t].\]
Let $x=-j$, we get
\[c_0(j)=((k-1)(-j)-1)[j-1]=((1-k)j-1)[j-1].\]
For any $1\le u\le j-1$,
\[ 1\le(k-1)j+u<kj\le k(d-1)\le p. \]
Hence $p\nmid (1-k)j-u$ and $v(c_0(j))=0$.
Let $D=\diag\{a,a^2,\ldots,a^s\}$ and $M'=(a'_{ij})_{1\leq i,j\leq s}$ with $a'_{ij}=a_{ij}a^{-i-j}$, then
\begin{equation}\label{mateq0} M(s)=D M' D. \end{equation}
Let $a''_{ij}:=(ki-i-1)!(i+s)!a'_{ij}$. Then
\[\begin{split}
&=\sum_{t=0}^{j-1}c_t(j)\cdot (i+j)[t]\cdot (i+s)[s-j],\\
&=\sum_{t=0}^{j-1}c_t(j)\cdot (i+s)[s-j+t]\\
&=\sum_{t=1}^{j} (i+s)[s-t] \cdot c_{j-t}(j).
\end{split}\]
Define $c_{j-t}(j):=0$ for $j<t$. Write $M''=(a''_{ij})_{1\le i,j\le s}$, $M_1=((i+s)[s-t])_{1\leq i,t\leq s}$ and $M_2=(c_{j-t}(j))_{1\leq t,j\leq s}$. Then
\begin{equation}\label{mateq1} M''=M_1 M_2. \end{equation}
\[ x[n]=\sum_{t=0}^{n}c'_t(n)x^t, \]
then $c'_n(n)=1$ and
\[ (i+s)[s-j]=\sum_{t=0}^{s-j}c'_t(s-j)(i+s)^t. \]
Define $c'_t(n):=0$ for $t>n$. Write $M_{11}=((i+s)^{t-1})_{1\leq i,t\leq s}$ and $M_{12}= (c'_{t-1}(s-j))_{1\leq t,j\leq s}$. Then
\begin{equation}\label{mateq2} M_1=M_{11} M_{12}. \end{equation}
Notice that $M_{11}$ is a Vandermonde matrix with determinant $\det M_{11}=\prod_{t=1}^s t^{s-t}$. One can also easily find
\begin{equation}\label{mateq3} \det M_{12}=(-1)^{[s/2]} \quad\text{and}\quad \det M_2=\prod_{i=1}^sc_0(i). \end{equation}
Now by (<ref>), (<ref>), (<ref>) and (<ref>),
\[ \det M(s)=a^{s(s+1)} (-1)^{[s/2]}\prod_{i=1}^s \frac{i^{s-i}c_0(i)}{(ki-i-1)!(i+s)!}. \]
Hence $v(\det M(s))=0$.
Denote $O(x)$ a number in $\overline{\Q}_p$ with valuation $\ge v(x)$ for $x\in\overline{\Q}_p$.
$(i)$ For $i+j<d$, $F_{pi-j}=\g^{ki}(a_{ij}+O(\g))$.
$(ii)$ For $i+j\ge d$, $v(F_{pi-j})=ki-1$ and
\[ F_{pi-(d-i)}=\frac{\g^{ki-1}(1+O(\g))}{(ki-1)!}.\]
\[ m=\begin{cases}
ki-i-j,&\ \text{if}\ j<d-i;\\
ki-i-j+d-1, &\ \text{if}\ j\ge d-i,
\end{cases}\]
\[ n=\begin{cases}
i+j,&\ \text{if}\ j<d-i;\\
i+j-d, &\ \text{if}\ j\ge d-i.
\end{cases}\]
Then $pi-j=dm+(d-1)n$ and $0\le n\le d-1$. This lemma follows from
\[F_{pi-j}=\sum_{l\ge 0} \g_{m-(d-1)l}\g_{n+dl}a^{n+dl}=\g_m\g_na^n(1+O(\g))=\frac{\g^{m+n}a^n}{m!n!}(1+O(\g)). \qedhere \]
For $1\leq s\leq d-1$, the valuation of $\det A_1[s+1]$ is $w_0+w_1+\cdots+w_s$.
Note that the first row of $A_1$ is $(1,0,0,\ldots)$. Let $A$ be the matrix by deleting the first row and column of $A_1[s+1]$. Then $\det A_1[s+1]=\det A$.
Let $D_1=\diag\{\g^{0/d},\g^{1/d},\ldots,\g^{s/d}\}$, $D_2=\diag\{\g^{k-1},\g^{2k-1},\ldots,\g^{(d-1)k-1}\}$ and $B[s]=(\g^{1-ki}F_{pi-j})_{1\le i,j\le s}$. Then $A=D_1^{-1}D_2B[s]D_1$. It suffices to compute $v(\det B[s])$.
Note that for $s=d-1$,
\[ B[d-1]=\begin{pmatrix}
\g a_{11}+O(\g^2)&\cdots&\g a_{1,d-2}+O(\g^2) &\frac{1+O(\g)}{(k-1)!}\\
\vdots &\udots&\frac{1+O(\g)}{(2k-1)!} & b_{2,d-1} \\
\g a_{d-2,1}+O(\g^2)&\udots&\udots&\vdots\\
\frac{1+O(\g)}{((d-1)k-1)!}&b_{d-1,2}&\cdots&b_{d-1,d-1}
\end{pmatrix} \]
with $v(b_{ij})=0$. If $1\le s\le \frac{d-1}{2}$, then
\[B[s]=\begin{pmatrix}
\g a_{11}+O(\g^2)&\cdots&\g a_{s1}+O(\g^2)\\
\vdots&\ddots&\vdots\\
\g a_{s1}+O(\g^2)&\cdots&\g a_{ss}+O(\g^2)\\
\end{pmatrix}\]
has determinant
\[ \det B[s]=\g^s(\det M(s)+O(\g)). \]
The valuation of $\det B[s]$ is $sv(\g)$.
If $\frac{d}{2}\le s\le d-1$, then
\[B[s]=\begin{pmatrix}
B[d-1-s] &P_1\\
\end{pmatrix}.\]
The valuation of any entry of $B[d-1-s],P_1,P_2$ is $v(\g)$ and
\[Q\equiv \begin{pmatrix}
\multirow{2}{*}{{\Huge0}}& &\frac{1}{(k-1)!}\\
\frac{1}{((d-1)k-1)!} & &\multirow{2}*{{\Huge*}}
\end{pmatrix}\bmod\g.\]
Thus $Q$ is invertible over the ring of integers of $\Q_p(\g)$. The determinant
\[\det B[s]=\det Q\det(B[d-1-s]-P_1Q^{-1}P_2)=\det Q \det B[d-1-s](1+O(\g))\]
has valuation $(d-1-s)v(\g)$.
Finally, $A=D_1^{-1}D_2B[s]D_1$ has valuation
\[(\sum_{i=1}^{s}(ki-1) +\min\{s,d-1-s\})v(\g)=w_0+w_1+\cdots+w_s. \qedhere \]
For $1\leq s\leq d-1$, we have
\[\begin{split}
v(\det A_1[s+1])&=\sum_{i\le s} w_i\\
\frac{s(s+1)}{2d}+\frac{s(s+1)}{d(p-1)},&\ \text{if}\ s\le (d-1)/2;\\
\frac{s(s+1)}{2d}+\frac{(d-s)(d-s-1)}{d(p-1)},&\ \text{if}\ s\ge d/2;
\end{cases}\\
&\le \frac{s(s+1)}{2d}+\frac{d^2-1}{4d(p-1)}.
\end{split}\]
If $p> \frac{d^2+3}{4}$, then $\frac{d^2-1}{4d(p-1)}<1/d$. For $0\le t_0<t_1<\cdots<t_s$, assume $t_s\neq s$. Since
\[v(F_{pi-j}\g^{(j-i)/d})\ge i/d,\]
we have
\[ v(\det A_1[t_0,\ldots,t_s])\ge \frac{s^2+s+2}{2d}>v(\det A_1[s+1]). \]
Thus $v(c_{s+1})=v(\det A_1[s+1])=\sum_{i\le s}w_s$ and $\{w_0,w_1,\ldots,w_{d-1}\}$ are slopes of $\NP_p(\det(I-tA_1))$.
If moreover $p>\frac{d^2}{2}$, then $p\ge \frac{d^2+1}{2}$ and $\frac{d^2-1}{4d(p-1)}\le\frac{1}{2d}$. Choose $\beta_i=i/d$ in Theorem <ref>, we have
\[v(C_{s+1})=h(w_0+w_1+\cdots+w_s)\]
\[\NP_q(\det(I-tA_h[d]))=\NP_p(\det(I-tA_1[d])).\]
Thus $w_0,w_1,\dots,w_{d-1}$ are $q$-adic slopes of $\NP_q(\detphi(I-tA_h))$.
By Theorem <ref>,
\[ \detphi(I-tA_h)=L^*(f,t)\detphi(I-tqA_h). \]
Since the valuation of any entry of $A_h$ is $\ge0$, the $q$-adic slopes of $\detphi(I-tA_h)$ are $\ge0$ and the $q$-adic slopes of $\detphi(I-tqA_h)$ are $\ge1$. Thus any $q$-adic slope of $\detphi(I-tA_h)$ less than $1$ must be a $q$-adic slope of $L^*(f,t)$. But $L^*(f,t)$ has degree $d$, hence $w_0,\ldots,w_{d-1}$ are all slopes of $L^*(f,t)$.
§.§ The case for general $\chi$
Let $f(x)\in\F_q[x]$ be a polynomial with degree $d$. Assume $p\nmid d$. Let $\NP(f,x)$ be the piecewise linear function whose graph is the $q$-adic Newton polygon of $\det(I-tA_h)$. Let $\HP(f,x)$ be the piecewise linear function whose graph is the polygon with vertices
\[ (k,\frac{k(k-1)}{2d}),\quad k=0,1,2,\ldots. \]
Then $\NP(f,x)\ge \HP(f,x)$ (cf. <cit.>). Set
\[ \gap(f)=\max_{x\ge 0}\{\NP(f,x)-\HP(f,x)\}. \]
(See <cit.>.)
Let $0=\alpha_0<\alpha_1<\cdots<\alpha_{d-1}<1$ denote the slopes of the $q$-adic Newton polygon of $L^*(f,t)$. If $\gap(f)<1/h$, then the $q$-adic Newton polygon of $L^*(f,\chi,t)$ has slopes
\[\{p^{1-m_\chi}(i+\alpha_j):0\le i\le p^{m_\chi-1}-1,0\le j\le d-1\}\]
for any non-trivial finite character $\chi$.
The slopes of $\NP(f,x)$ are
\[\{i+w_j:i\ge0,0\le j\le d-1\}.\]
Notice that
\[\sum_{i=0}^{d-1} w_i=\sum_{i=0}^{d-1}\frac{i}{d},\]
$\NP(f,x)-\HP(f,x)$ is a periodic function with period $d$. For $0\le k<d$,
\[\NP(f,x)-\HP(f,x)\le\sum_{i\le(d-1)/2}\frac{2i}{d(p-1)}\le\frac{d^2-1}{4d(p-1)}.\]
If $p>\frac{h(d^2-1)}{4d}+1$, then $\gap(f)<1/h$ and this concludes the proof.
Acknowledgements. This paper was prepared when the authors were visiting the Academy of Mathematics and Systems Science and the Morningside Center of Mathematics of Chinese Academy of Sciences. We would like to thank Professor Ye Tian for his hospitality. We also would like to thank Jinbang Yang for many helpful discussions. This work was partially supported by NKBRPC(2013CB834202) and NSFC(11171317 and 11571328).
A. Adolphson, S. Sperber,
Newton polyhedra and the degree of the $L$-function associated to an exponential sum,
Invent. Math., 1987, 88(3): 555–569.
C. Davis, D. Wan, L. Xiao,
Newton slopes for Artin-Schreier-Witt towers,
Math. Ann., DOI: 10.1007/s00208-015-1262-4.
C. Liu, C. Niu,
Generic twisted $T$-adic exponential sums of binomials,
Sci. China. Math., 2011, 54(5): 865–875.
C. Liu, C. Niu,
Generic twisted $T$-adic exponential sums of polynomials,
J. Number Theory, 2014, 140(7): 38-¨C59.
C. Liu, D. Wan,
$T$-adic exponential sums over finite fields,
Algebra Number Theory, 2009, 3(5): 489–509.
C. Liu, D. Wei,
The $L$-function of Witt coverings,
Math. Z., 2007, 255(1): 95–115.
C. Niu,
The $p$-adic Riemann hypothesis of exponential sums associated to certain binomials over finite fields,
(in Chinese) Journal of Liaocheng University: Natural Science Edition, 2011, 24(4): 89–94.
Y. Ouyang, J. Yang,
Newton polygons of $L$-functions of polynomials $x^d+ax$,
J. Number Theory.
J. P. Serre, Endomorphismes complétement continus des espaces de Banach $p$-adiques,
(in French) Inst. Hautes Études Sci. Publ. Math. No., 1962, 12: 69–85.
R. Yang,
Newton polygons of $L$-functions of polynomials of the form $x^d+\lambda x$,
Finite Fields Appl., 2003, 9(1): 59–88.
H. J. Zhu,
Asymptotic variations of L-functions of exponential sums,
H. J. Zhu,
Generic $A$-family of exponential sums,
J. Number Theory, 2014, 143: 82–101.
|
1511.00164
|
Recent results on charmless hadronic $B$ decays at Belle
Bilas Pal[[email protected]], University of Cincinnati
On behalf of the Belle collaboration
Two-body charmless hadronic decays of $B$ mesons are
important for determining Standard Model parameters
and for detecting the presence of new physics. We present recent results from the Belle experiment on the charmless
hadronic decays $B\rightarrow \eta \pi^0$ and $B\rightarrow \pi^0 \pi^0$.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
Two-body charmless hadronic decays of $B$ mesons are
important for determining Standard Model parameters
and for detecting the presence of new physics. We present recent results from the Belle experiment on the charmless
hadronic decays $B\rightarrow \eta \pi^0$ and $B\rightarrow \pi^0 \pi^0$.
§ EVIDENCE FOR THE DECAY $B\RIGHTARROW \ETA \PI^0$
The decay $B\rightarrow \eta \pi^0$ proceeds mainly via a $b\to u$ Cabibbo- and
color-suppressed “tree” diagram, and via a $b\to d$
“penguin" diagram, as shown
in Fig. <ref>.
The branching fraction can
be used to constrain isospin-breaking effects on the value of
$\sin2\phi_2~(\sin 2\alpha)$ measured in $B\to\pi\pi$
decays <cit.>.
It can also be used
to constrain $CP$-violating parameters ($C^{}_{\eta' K}$
and $S^{}_{\eta' K}$) governing the time dependence of
$B^0\to\eta' K^0$ decays <cit.>.
The branching fraction is estimated
using QCD factorization <cit.>, soft collinear effective
field theory <cit.>, and flavor SU(3) symmetry <cit.>
and is found to be in the range $(2 - 12)\times10^{-7}$.
(a) Tree and (b) penguin diagram
contributions to $B\rightarrow \eta \pi^0$ .
Several experiments <cit.>, including Belle, have searched for this decay mode.
The current most stringent limit on the branching fraction
is $\mathcal{B}(B^0\to\eta\pi^0)<1.5\times10^{-6}$ at
90% confidence level (C.L.) <cit.>. The analysis presented here uses the full data set of the Belle experiment running on
the $\Upsilon(4S)$ resonance at the KEKB asymmetric-energy
$e^+e^-$ collider. This data set corresponds to
$753\times10^{6}$ $B\overline{B}$ pairs, which is a factor of 5
larger than that used previously. Improved tracking, photon reconstruction,
and continuum suppression algorithms are also used in this analysis.
We find the evidence of the decay $B\rightarrow \eta \pi^0$ <cit.>, where the candidate $\eta$ mesons
are reconstructed via $\eta\rightarrow\gamma\gamma~(\eta_{\gamma\gamma})$ and $\eta\rightarrow\pi^+\pi^-\pi^0~(\eta_{3\pi})$ decays and
$\pi^0$ via $\pi^0\rightarrow\gamma\gamma$. Results of the fit to the variables, beam-energy-constrained mass $M_{\rm bc}=\sqrt{E^2_{\rm beam}-|\vec{p_B}|^2c^2}/c^2$, energy difference $\Delta E=E_B-E_{\rm beam}$ and continuum suppression variable $C'_{NB}=\ln(\frac{C_{NB}-C^{\rm min}_{NB}}{C^{\rm max}_{NB}-C_{NB}})$, are given in Table. <ref>.
Fitted signal yield $Y_{\rm sig}$,
reconstruction efficiency $\epsilon$,
$\eta$ decay branching fraction $\mathcal{B}_{\eta}$,
signal significance, and $B^0$ branching fraction
$\mathcal{B}$ for the decay $B^0\rightarrow\eta\pi^0$. The errors listed are statistical only.
The significance includes both statistical and
systematic uncertainties.
Mode $Y_{\rm sig}$ $\epsilon(\%)$ $\mathcal{B}_{\eta}(\%)$
Significance $\mathcal{B}(10^{-7})$
$B^0\to\eta_{\gamma\gamma}\pi^0$ $30.6^{+12.2}_{-10.8}$
18.4 39.41 3.1 $5.6^{+2.2}_{-2.0}$
$B^0\to\eta_{3\pi}\pi^0$ $0.5^{+6.6}_{-5.4}$
14.2 22.92 0.1 $0.2^{+2.8}_{-2.3}$
Combined 3.0 $4.1^{+1.7}_{-1.5}$
The combined branching fraction
is determined by simultaneously fitting both $B^0\to\eta_{\gamma\gamma}\pi^0$
and $B^0\to\eta_{3\pi}\pi^0$ samples for a common $\mathcal{B}(B^0\to\eta\pi^0)$. Signal enhanced projections of the simultaneous fit are
shown in Fig. <ref>.
Signal enhanced projections of the simultaneous fit for the decay $B^0\rightarrow\eta\pi^0$:
(a), (b) $M_{\rm bc}$; (c), (d) $\Delta E$;
(e), (f) $C'_{NB}$.
The top (bottom) row corresponds to $\eta\to\gamma\gamma$
($\eta\to\pi^+ \pi^-\pi^0$) decays.
Points with error bars are data; the (green) dashed, (red) dotted and (magenta) dot-dashed curves represent the signal, continuum and charmless rare backgrounds, respectively, and the (blue) solid curves represent
the total PDF.
The branching
fraction for $B\rightarrow \eta \pi^0$ decays is measured to be
\begin{eqnarray}
\mathcal{B}(B^0\to\eta\pi^0) & = &
\left( 4.1^{+1.7+0.5}_{-1.5-0.7}\right) \times 10^{-7}\nonumber,
\end{eqnarray}
where the first uncertainty is statistical and the second
is systematic. This corresponds to a 90% C.L. upper limit of $\mathcal{B}(B^0\to\eta\pi^0)<6.5\times 10^{-7}$. The significance of this result is $3.0$ standard deviations.
The measured branching fraction is in good agreement with theoretical
expectations <cit.>. Inserting our measured value
into Eq. (19) of Ref. <cit.> gives the result that the
isospin-breaking correction to the weak phase $\phi_2$ measured in $B\to\pi\pi$ decays due to $\pi^0$–$\eta$–$\eta'$ mixing is less than $0.97^{\circ}$ at 90% C.L.
§ THE DECAY $B^0\RIGHTARROW\PI^0\PI^0$ (PRELIMINARY RESULTS)
This decay is an important input for the isospin analysis in the $B\rightarrow\pi\pi$ system.
A fit to the variables $\Delta E$, $M_{\rm bc}$ and a fisher discriminant $T_C$ is performed. We measure a preliminary branching fraction of
$\mathcal{B}(B^0\to\pi^0\pi^0)=(0.9\pm0.12( {\rm stat.}) \pm0.10({\rm sys.}))\times10^{-6}$ , with a significance of 6.7 standard deviations
and the direct $CP$ asymmetry of $A_{CP}=-0.054\pm0.086$. Signal enhanced projections are shown in Fig. <ref>. With this result, the constraint to the $\phi_2$ using the isospin relation in the $B\rightarrow\pi\pi$ system will be re-evaluated.
Signal enhanced projections of the fit for the decay $B^0\rightarrow \pi^0\pi^0$:
(left) $\Delta E$, (middle) $M_{\rm bc}$ and (right) $T_C$.
Contributions from signal, continuum, $\rho\pi^+$ and other $B$ decays are shown by blue, green, red and cyan curves respectively.
§ SUMMARY
Using the full set of Belle data, recent and preliminary measurements of charmless hadronic $B$ decays are presented. Our measurement of $B^0\rightarrow\eta\pi^0$ branching fraction constitutes the first evidence of the decay.
§ ACKNOWLEDGEMENTS
The author thanks the organizers of DPF 2015 for excellent hospitality and for assembling a nice scientific program.
The author would also like to thank Alan Schwartz for reviewing and providing valuable feedback to the final manuscript. This work is supported by the U.S. Department of Energy.
M. Gronau and J. Zupan,
Phys. Rev. D 71, 074017 (2005).
S. Gardner,
Phys. Rev. D 72, 034015 (2005).
M. Gronau, J. L. Rosner and J. Zupan,
Phys. Lett. B 596, 107 (2004);
Phys. Rev. D 74, 093003 (2006).
M. Z. Yang and Y. D. Yang,
Nucl. Phys. B609, 469 (2001);
M. Beneke and M. Neubert,
Nucl. Phys. B675, 333 (2003);
J. f. Sun, G. h. Zhu and D. s. Du,
Phys. Rev. D 68, 054003 (2003);
Z. j. Xiao and W. j. Zou,
Phys. Rev. D 70, 094008 (2004);
H. s. Wang, X. Liu, Z. j. Xiao, L. b. Guo and C. D. Lu,
Nucl. Phys. B738, 243 (2006);
H. Y. Cheng and C. K. Chua,
Phys. Rev. D 80, 114008 (2009);
H. Y. Cheng and J. G. Smith,
Annu. Rev. Nucl. Part. Sci. 59, 215 (2009).
A. R. Williamson and J. Zupan,
Phys. Rev. D 74, 014003 (2006);
Phys. Rev. D 74, 039901 (2006).
C. W. Chiang, M. Gronau and J. L. Rosner,
Phys. Rev. D 68, 074012 (2003);
H. K. Fu, X. G. He and Y. K. Hsiao,
Phys. Rev. D 69, 074002 (2004);
C. W. Chiang, M. Gronau, J. L. Rosner and D. A. Suprun,
Phys. Rev. D 70, 034020 (2004);
C. W. Chiang and Y. F. Zhou,
J. High Energy Phys. 12 (2006) 027;
H. Y. Cheng, C. W. Chiang and A. L. Kuo,
Phys. Rev. D 91, 014011 (2015).
H. Albrecht et al. (ARGUS Collaboration),
Phys. Lett. B 241, 278 (1990).
M. Acciarri et al. (L3 Collaboration),
Phys. Lett. B 363, 127 (1995).
S. J. Richichi et al. (CLEO Collaboration),
Phys. Rev. Lett. 85, 520 (2000).
P. Chang et al. (Belle Collaboration),
Phys. Rev. D 71, 091106 (2005).
B. Aubert et al. (BaBar Collaboration),
Phys. Rev. D 78, 011107 (2008).
B. Pal et al. (Belle Collaboration),
Phys. Rev. D 92, 011101 (2015).
|
1511.00419
|
Two particularly simple ideal clocks exhibiting intrinsic circular motion with the speed of light and opposite spin alignment are described. The clocks are singled out by singularities of an inverse Legendre
transformation for relativistic rotators of which mass and spin are fixed parameters. Such clocks work always the same way, no matter how they move. When subject to high accelerations or falling in strong gravitational fields of black holes, the clocks could be used to test the clock hypothesis.
An ideal clock is a mathematical abstraction of
a nearly perfect material clocking mechanism.
The clock hypothesis asserts that an ideal clock measures its proper time. This means that the
number of consecutive cycles registered by the
clock increases steadily with the affine parameter of the worldline of the clock's center of mass ().
On the dimensional grounds, we may expect that the hypothesis could be violated for extreme accelerations of order $c\omega$ (e.g. $\tfrac{mc^3}{\hbar}\!\sim \!10^{^{_{29}}}\!\,\!\tfrac{\mathrm{m}}{\mathrm{s}^2}$ for the electron).
A recent result <cit.> suggests that quantum field-theoretical realizations of extended clocks (experiencing the Unruh effect) do not measure their proper times. But the clock hypothesis refers to classical concepts of the relativity theory (e.g. a single worldline), and as such should be first of all tested within the same conceptual framework. A candidate clock should be a relatively simple mathematical device so as to minimize the influence of external disturbances on its structure. If some fundamental limitation were to concern such a clocking standard, the more it would concern more complicated clocks.
A mathematical clock can be devised by an analogy with a quantum particle such as Dirac electron. The intrinsic clock of such particle cannot be impaired – the phase of its wave function oscillates in the rest frame with a fixed frequency determined by only
the fundamental constants of nature. But quantum phase is not observable, it would be useless as a clock. Something similar happens with the basic classical analogue of a quantum particle – a structureless material point. The action functional of the material point (to some extent related to a quantum phase) increases linearly with the affine parameter of its worldline. But classical observables are reparameterization invariant, and do not distinguish any particular time variable. In order to play the role of an ideal clock, the material point must be endowed with an additional structure repeatedly changing with the proper time, e.g. connected with some sort of intrinsic rotation. Additionally, for the clock to resemble a quantum particle with its invariable structure as much as possible, it may be required that the clock's mass and magnitude of its spin should have fixed numerical values.
§ A RELATIVISTIC CLOCKING MECHANISM
In the relativity theory, a rotation can be described as a continuous action of an elliptic homography mapping points on a complex plane at one instant to those at another instant and leaving fixed
a pair of points: $\kappa_+$ and $\kappa_-$. It is natural to identify $\kappa_{\pm}$
with stereographic images $\mcal{Z}(k_{\pm})$ of a pair of null vectors preserved in free motion of any isolated massive system with spin:
Here, $p$ is the momentum vector
and $w$ is the Mathisson pseudovector
(customarily ascribed to Pauli and Lubański, see <cit.>).
Given a pair $k_{\pm}$, the homography is set by specifying the motion
of a single point $\kappa\equiv\mcal{Z}(k)$ – a stereographic image
of a null vector $k$ – about an invariant circle of that homography.
The $\kappa$ is invariant under a local scaling $\delta{k} =\epsilon k$, so should be the Lagrangian.
It is thus necessary that
$\SP{k}{\Kpi}= 0$ identically (with $\Kpi$ being the momentum conjugate to $k$), since otherwise the variation
$\delta{L} = \epsilon({ k\partial_k{L} +
\SP{\dk}{\Kpi}})+ \SP{k}{\Kpi}\dot{\epsilon}$ would not be vanishing for arbitrary $\epsilon$. Accordingly, there are two structure constraints:
\begin{equation}\label{eq:fundcond2}
\Psi_3:\quad\SP{k}{\Kpi}\weq0,
\qquad \Psi_4:\quad\SP{k}{k}\weq0.
\end{equation}
Now, it can be deduced what the invariant circle must be.
$w\propto \ast (p\wedge k \wedge \Kpi)$ it follows
that $\SP{k}{w}=0$. This means that
the image point of $k$ moves about the image circle of $w$. As so, $k$ may be thought
of as representing the clock's pointer and $w$ as representing the clock's dial (see figure).
In free motion in Minkowski spacetime,
the vectors $k_{\pm}$
are parallel transported. Then a Lorentz invariant phase can be assigned to $\kappa$ between instants $\tau_o$ and $\tau$ through:
\begin{equation}\label{eq:phase}
\phi
\br{\frac{\kappa(\tau)-\kappa_{+}}{\kappa(\tau)-\kappa_{-}}\cdot
\frac{\kappa(\tau_o)-\kappa_{-}}{\kappa(\tau_o)-\kappa_{+}} }}\,.\end{equation}
The phase $\phi$ is a real number. In free motion of the clock, a rotation through $\phi=2\pi$
represents a single full clocking cycle.[A massless system ($\SP{p}{p}=0$) would be structurally
something different, because for a
parabolic homography (preserving a null direction $p$ and an orthogonal to it spatial direction $w$) the analogous phase is not a Lorentz scalar.]
§ DYNAMICAL REQUIREMENTS AND THE HAMILTONIAN
The intuition derived from the theory of Eulerian rigid bodies suggests that the the above clock will be insensitive to external influences if both its mass and size is fixed. This requirement can be fulfilled in an invariant way by imposing constraints on the Casimir invariants of the Poincaré group:
\begin{equation}\label{eq:fundcond}\Psi_1:\quad\SP{p}{p}-\elm^2\weq0,
\qquad\Psi_2:\quad\SP{w}{w}+\frac{1}{4}\elm^4\ell^2\weq0.\end{equation}
constants $\elm$ and $\ell$ are fixed parameters with the dimension of mass and that of length.
These constraints should be regarded as primary, i.e., implied by the form of the Lagrangian.
Motivated by devising an ideal
clock, Staruszkiewicz observed <cit.> that (unlike unitarity) the irreducibility of quantum systems has a
classical counterpart realized in postulating (<ref>) as a means to singling out physically appealing Lagrangians. This postulate
is in essence equivalent to the earlier, strong conservation idea due to Kuzenko, Lyakhovich and Segal <cit.>, introduced
as a basic dynamical principle for devising Lagrangians suitable for geometric models of particles with spin.
As established in sec:mech,
the phase space of the simplest clock
can be parameterized using components of the position fourvector $x$ and three tangent fourvectors $p,k,\Kpi$ bound to satisfy constraints eq:fundcond2eq:fundcond, where
\begin{array}{@{}c@{\;}c@{\;}c@{}}
\SP{p}{p}&\SP{p}{k}&\SP{p}{\Kpi}\\
\SP{k}{p}&\SP{k}{k}&\SP{k}{\Kpi}\\
\SP{\Kpi}{p}&\SP{\Kpi}{k}&\SP{\Kpi}{\Kpi}
\end{array}\right]\weq \SP{p}{k}^2\SP{\Kpi}{\Kpi}.$$
Between these dynamical variables we assume a Poisson bracket $\PB{U}{V}\equiv\SP{\partial_{x}{U}}{\partial_{p}{V}}-\SP{\partial_{p}{U}}{\partial_{x}{V}}+\SP{\partial_{k}{U}}{\partial_{\Kpi}{V}}
eq:fundcond2eq:fundcond form a system of independent first class constraints with respect to this bracket.
In line with Dirac method <cit.>, the most general Hamiltonian is
a linear combination of all first class constraints with arbitrary functions $u$'s as coefficients.
It is convenient that the combination be taken as:[
The original KLS Hamiltonian
<cit.> involved a complex
variable $\zeta$, ($\zeta\equiv \mcal{Z}(k)$), inherited from a primary Lagrangian.
Starting with a related Lagrangian expressed in terms of $k$,
a Hamiltonian analogous in form to (<ref>)
was arrived at in <cit.> (upon earlier reducing an extended phase space).
Our approach goes in the opposite direction. We start with a Hamiltonian deduced from first principles. In <cit.> we generalized this method onto systems described by a collection of fourvectors.]
\begin{multline}\label{eq:hamiltonian} \hspace{-0.05\linewidth}
H\seq\frac{u_1}{2\elm}\br{\SP{p}{p}-{\elm}^2} + \frac{u_2}{2\elm}
\br{\!\SP{p}{p}+\frac{4}{{\ell}^2{\elm}^2}\SP{k}{p}^2\SP{\Kpi}{\Kpi}}
\\
The equations $\partial_{u_i}H\weq0$ form a system of first class constraints equivalent to eq:fundcond2eq:fundcond. Next, we introduce velocities $\dot{\wq}\weq\partial_{\wp}{H}$ <cit.>:
\begin{equation}\label{eq:DefVel}
\begin{split}
\hspace{-.08\linewidth} &\dx\weq\br{{u_1} + {u_2}}
\frac{p}{\elm} +{u_2}\frac{4\,\SP{k}{p}\,\SP{{\Kpi}}{{\Kpi}}}
{{{\ell}}^2\,{{\elm}}^3}\,k \weq {u_1} \frac{p}{\elm}+ {u_2}n,&\\
\hspace{-.08\linewidth} & \dk\weq u_2\,\frac{4{\SP{k}{p}}^2}
{{\ell}^2{\elm}^3}{\,\Kpi}+ u_3\,k,\hspace{.175\linewidth} n:={\frac{p}{\sqrt{\SP{p}{p}}}
- \frac{\sqrt{\SP{p}{p}}\,k}{\SP{k}{p}}}.&
\end{split}
\end{equation}
By taking projective derivatives, defined recursively by
$\udnp{n+1}{\slot}\eqdef\udp{(\udnp{n}{\slot})}$, where $\udp{\slot}\eqdef\br{\dot{\slot}}_{\perpp}$ and $\ts{(\slot)_{\perpp}\eqdef \slot-
\frac{\SP{p}{\slot}}{\SP{p}{p}}p}$,
it can be shown that a curvature $\varkappa$ (defined by analogy with Frenet-Serret formulas)
is fixed: $\varkappa\equiv- \SP{\udp{x}}{\udp{x}}^{-3}\gramm{\udp{x}}{\uddp{x}}\weq\sfrac{4}{\ell^2}$,
and that torsion vanishes
on account of $\udp{x}$, $\uddp{x}$, and $\udddp{x}$ being coplanar as $p$-orthogonal linear combinations of
$p,k,\Kpi$. Hence, the trajectory perceived in the frame
is a circle of a fixed radius $\sfrac{\ell}{2}$ (without constraints (<ref>), the radius would vary with the actual state <cit.>). Correspondingly, the worldline's path is winding up
around a fixed
space-time cylinder, the main axis of which represents the inertial motion.
To measure the rate of
change of the unit spatial vector $n$ in the frame (see (<ref>) for the definition of $n$), a frequency scalar $\Omega$ can be introduced:
\begin{equation}\label{eq:Omega}\Omega^2:={-\frac{\SP{\dot{n}}{\dot{n}}\SP{p}{p}}{\SP{p}{\dx}^2}};
\quad \Rightarrow\quad
\Omega= \frac{\SP{p}{p}\sqrt{-\SP{\dk}{\dk}}}{\abs{
\SP{k}{p}\SP{p}{\dx}}}\;\; \mathrm{if}\;\; \dot{p}=0.\end{equation}
For solutions, it reduces to a ratio:
$\Omega\weq\br{\sfrac{2}{\ell}}\abs{\sfrac{u_2}{u_1}}$, and if $\abs{u_1}>\abs{u_2}$ it is related
to a hyperbolic angle $\Lambda$ between $p$ and $\dx$: $\Omega\weq\br{\sfrac{2}{\ell}}\tanh{\Lambda}$.
Both $\Omega$ and $\Lambda$ are reparametrization-invariant scalars with
obvious physical meaning. On the other hand, $\Omega$ and $\Lambda$ are functions of the arbitrary ratio
$\sfrac{u_2}{u_1}$. Thus the motion is indeterminate.
To solve this paradox, this ratio needs to be set based on a sound guiding principle, so as not to
introduce arbitrary features into the dynamics.
§ SINGULARITIES IN THE INVERSE LEGENDRE TRANSFORMATION
The form of a Lagrangian $L\equiv \SP{\dx}{p}+\SP{\dk}{\Kpi}-H$
corresponding to the Hamiltonian
is subject to invertibility of the
map (<ref>)
restricted to a submanifold
determined by the constraints eq:fundcondeq:fundcond2.
For the purpose of the invertibility analysis, it must suffice to focus upon Lorentz scalars only.
On the submanifold of interest, we may consider a map between two sets of variables:
$u_1$, $u_2$, $u_3$, $
\SP{k}{p}$, $\SP{p}{\Kpi}$ and
$\SP{\dk}{\dk}$, $\SP{\dk}{\dx}$, $\SP{\dx}{\dx}$, $\SP{k}{\dx}$,
\begin{equation}
\label{eq:map}
\begin{split}
&\SP{\dx}{\dx}\weq u_1^2-u_2^2,\qquad\qquad \SP{k}{\dx}\weq \frac{\SP{k}{p}}{\elm}\left(u_1+u_2\right),\\
&{{\SP{\dk}{\dx}}\weq \ {
\frac{\SP{k}{p}}{\elm} \left(
\frac{4\,\SP{k}{p}\,\SP{p}{\Kpi}}{
{\elm}^3{\ell}^2}\,{u_2} + {u_3} \right) }}\,\left( {u_1} + {u_2}
\right),\\
&\SP{\dk}{\dk}\weq -
\frac{4{\SP{k}{p}}^2}{{\ell}^2{\elm}^2}\,u_2^2,\quad\quad
\SP{\dk}{k}\weq0.
\end{split}
\end{equation}
The number of new constraints for velocities depends on the rank of the Jacobian matrix of this map.
Non-zero minors of maximal rank $4$
for this Jacobian are: $j_1=\tfrac{16{\SP{k}{p}}^3}{{\ell}^2{\elm}^4}\, u_2
\br{u_1 + u_2}(u_2^2-u_1^2)$ and $ j_2={\tfrac{4\,\SP{k}{p}}{{{\ell}}^2\,{{\elm}}^3}}{u_2}\,j_1$.
Since $\SP{k}{p}\neq0$ (for a timelike $p$ and a null $k$), this implies that the Jacobian rank, $\mathrm{Rk}$, is dependent on $u_{1,2}$. Full analysis distinguishes the following 4 regimes:
Rk velocity constraints
i) $4$
$u_1^2\neq u_2^2\ne0$ $\SP{k}{\dk}\weq0$
ii) $3$ ${u_1}={u_2}\neq0$ $\SP{k}{\dk}\weq0$, $\SP{\dx}{\dx}\weq0$, $ \ell^2\SP{\dk}{\dk}
iii) $2$ ${u_1}=-{u_2}\neq0$ $\SP{k}{\dk}\weq0$, $\dx\propto k$ $\quad \Rightarrow\quad \SP{\dx}{\dx}\weq0$
ii') 3 $u_2=0$, $u_1\ne0$ $\SP{k}{\dk}\weq0$, $\SP{\dk}{\dk}\weq0$
The ii' case will not be of concern here, and $u_2\neq0$ is
assumed from now on. To find explicit expressions for momenta, two cases are to be considered: $ u_1+u_2\neq0$ /i, ii/ or $ u_1+u_2=0$ /iii/.
$\bullet$ For $u_2(u_1+u_2)\ne0$
the ansatz $p=\alpha_1 \dx+\alpha_2 k$ and $\Kpi=\beta_1 \dk +\beta_2 k$ allows to express momenta in terms of velocities and $u$'s. On substituting to (<ref>) and solving for $\alpha_{1,2},\beta_{1,2}$, one gets:
\begin{align}\label{eq:momenta_gen}
\begin{split}
&p=\frac{{\elm}}{{u_1} + {u_2}}\,{\dot{x}} -
\frac{{{\ell}}^2\,{\elm}\,{\left( {u_1} + {u_2} \right) }^2\,
\left( \SP{{\dot{k}}}{{\dot{k}}} - 2\,\SP{k}{\dk}\,{u_3} \right) }{4\,
&{\Kpi}= \frac{{{\ell}}^2\,{\elm}\,{\left( {u_1} + {u_2} \right) }^2}
{4\,{\SP{k}{\dx}}^2\,{u_2}}\,\left( {\dot{k}} - k\,{u_3} \right).
\end{split}
\end{align}
The $\Psi_3$ constraint leads to $\SP{k}{\dk}\weq0$ (consistently with $\Psi_4$), while
the $\Psi_{1,2}$ constraints give conditions for $u_{1,2}$:
\tfrac{1}{{\br{u_1+u_2}}^2} \SP{\dx}{\dx}+\tfrac{u_1+u_2}{2u_2}\,\xi=1
\quad\wedge\quad
\tfrac{\br{u_1 + u_2}^2}{4\,{u_2}^2}\,\xi
=1. \quad
\boxed{\xi:=-\ell^2\tfrac{\SP{\dk}{\dk}}{{\SP{k}{\dx}}^2}}
The resulting $u_{1,2}$ can be expressed as independent functions of velocities, provided that the Jacobian determinant
${\scriptsize \tfrac{\partial(\Psi_1,\Psi_2)}{\partial(u_1,u_2)}}$, equal to $
{\scriptsize {\frac{-\elm^6\ell^2\,\xi
}{4\,u_2^3\,\br{u_1 + u_2}}\, \SP{\dx}{\dx}}}$,
is nonzero, which leads to
a Lagrangian of the first kind.
Otherwise, if $\SP{\dx}{\dx}=0$,
then one gets $u_1=u_2$ and a frequency constraint $\SP{k}{\dx}^2+\ell^2\SP{\dk}{\dk}\weq0$.
This leads to a Lagrangian of the second kind.
$\bullet$ For $u_2\neq0$ and $u_1+u_2=0$, one is led to a Lagrangian of the third kind
with $\dx\propto k$ (when $\SP{\dx}{\dx}\weq0$, $\SP{k}{\dx}\weq0$ and $\SP{\dk}{\dx}\weq0$).
§.§ Null worldlines principle
Above, the rank of the inverse Legendre transformation, qualitatively discriminated between two separate regimes: $\SP{\dx}{\dx}\neq0$ (maximal rank) and $\SP{\dx}{\dx}=0$ (lower ranks).
Now, two other premises can be brought to the attention, as to why specifically the condition $\SP{\dx}{\dx}=0$
is so particular.
In the maximal rank case, assuming any constraints
such that $\SP{\dx}{\dx}\neq0$ would be a matter of
arbitrary decision. For $\SP{\dx}{\dx}>0$, choosing a given function for $\Omega$ is equivalent to setting the hyperbolic angle $\Lambda$. But there is no privileged
hyperbolic angle in the
(homogeneous) Lobachevsky space of fourvelocities (a similar argument on de Sitter hyperboloid would apply to the 'tachionic' sector $u_2^2>u_1^2$). On the contrary, null worldlines are
distinguished by the lightcone structure of the spacetime.
$\SP{\dx}{\dx}=0$, the velocity can be fixed in a manifestly relativistically invariant manner independently of parameterization.
We stress this important circumstance, since outside the light cone, a more general condition $\SP{\dx}{\dx}=\sigma$ with a given nonzero function $\sigma$, neither would set a velocity nor be reparameterization invariant. This qualitative difference should find its reflection also in the structure of the respective Lagrangians.[
This difference is already seen for a material point
described by a general Lagrangian $L=\frac{1}{2}({w^{-1}{\SP{\dx}{\dx}}+m^2 w})$. The equation $\partial_wL=0$ implies two qualitatively distinct regimes: 1) in which $w$ is a function of $\dx$, then $w=m^{-1}\sqrt{\SP{\dx}{\dx}}$,
and 2) in which $w$ is independent of $\dx$, requiring $m=0$ and
a constraint $\SP{\dx}{\dx}=0$. The resulting Lagrangians are:
1) that of a massive particle $L=m\sqrt{\SP{\dx}{\dx}}$ with $p=m\frac{\dx}{\sqrt{\SP{\dx}{\dx}}}$ and 2) that of a massless particle $L=\frac{1}{2}w^{-1}\SP{\dx}{\dx}$ with $p=w^{-1}\dx$ and an arbitrary $w$ transforming as $\delta{w}=w\delta \epsilon$ under a reparameterization $\delta{\dx}=\dx \delta\epsilon$. The analytic form of $L=m\sqrt{\SP{\dx}{\dx}}$ would not be suitable in a region containing the surface $\SP{\dx}{\dx}=0$, where the corresponding $p$ would be divergent. ]
Yet, there is an insightful remark due to Dirac, showing the distinguished role of the condition $\SP{\dx}{\dx}=0$. It is a
[counterintuitive] consequence of Dirac equation, that a measurement of the electron's
instantaneous motion
is certain to give the speed of light,
which Dirac mentions in his Principles
and asserts this result to
be generally true
in a relativistic theory.
The Dirac observation in conjunction with previous findings tempts one
to conjecture
worldlines of classical analogs of quantum elementary particles should be null.
LAGRANGIANS OF THE FIRST KIND
In the sub-luminal sector ($u_1^2>u_2^2$) let
$u_2=\rho\,\sinh{\psi}$, $\rho>0$, $\abs{\psi}<\infty$. Then from
${\scriptstyle\tanh{\psi}=\mp\, \frac{\sqrt{\xi }}{2 {\pm}{\sqrt{\xi }}}}$.
With the resulting $u_{1,2}$ substituted in (<ref>), two Lagrangians follow:
\sqrt{\SP{\dx}{\dx}} \sqrt{ 1\pm\sqrt{\xi}
}+\lambda_1\SP{k}{k}+\lambda_2\SP{k}{\dk}$ ($\lambda_{1,2}$ involve arbitrary $u_{3,4}$).
In the super-luminal sector ($u_1^2<u_2^2$) – which may be considered on account of $x$ not being assigned to a motion –
a similar analysis (with $u_1=-\etasign\hat{\rho}\,\sinh{\hat{\psi}}$ and
$u_2=-\etasign\hat{\rho}\,\cosh{\hat{\psi}}$, $\hat{\rho}\neq0$)
leads to a single Lagrangian
\sqrt{-\SP{\dx}{\dx}} \sqrt{\sqrt{\xi}-1
In both cases, the last term in $\tilde{L}$ (whose only effect is an additive gauge-like term
in the canonical momentum
$\partial_{\dk}\tilde{L}\to\partial_{\dk}\tilde{L}+\alpha k$)
can be integrated
off by parts. On denoting the
remaining term $(\lambda_1-\br{\sfrac{1}{2}}\dot{\lambda}_2)\SP{k}{k}$ by $ \lambda\SP{k}{k}$,
one finally ends up with two Lagrangians (equivalent to those arrived at in <cit.>):
\begin{equation}\label{eq:LagrFRR}L_{\pm}=
\etasign\elm \sqrt{\SP{\dx}{\dx}\br{ 1\pm\sqrt{- {\ell}^2\frac{
\SP{\dk}{\dk} }{ {\SP{k}{\dx}}^2 } }}\; }+\lambda\,
\SP{k}{k},\end{equation}
with their respective
Lagrange multipliers $\lambda$.
The sub-luminal Lagrangian $L_{+}$ is that
of the Fundamental Relativistic Rotator <cit.>. With the Lagrangian
$L_{-}$ we could consider both sub- or super-luminal motions.
In the clock context, it is appropriate to recall an earlier result <cit.> published in <cit.> that the Lagrangians (<ref>) can be alternatively arrived at by adopting a physically dubious condition
that the Hessian matrix ${\partial}_{\wdq\wdq}L$ for a general Lagrangian $L=f(\xi)\sqrt{\SP{\dx}{\dx}}$ expressed in terms of only the $5$ degrees of freedom
characteristic of a rotator – Cartesian $\vec{x}(t)$ and spherical $\vartheta(t),\varphi(t)$ (considered as functions of $x^0\equiv t$) – must be zero.
As shown therein, this leads to a differential equation for $f$: $\dc f+2\xi({\dc}^2+\ddc f)=0$.
As a direct consequence of this, the clocking frequency becomes indeterminate. This conforms with what has been concluded in sec:hamiltonian.
For reasons described in sec:nwp, with the Lagrangian (<ref>), there would be no privileged velocity constraint suitable to set this frequency so as to make the motion determinate, while conditions involving $\SP{\dx}{\dx}=0$ (such as ii or iii) would not be compatible with the analytic structure of these Lagrangians (the canonical momenta $\partial_{\wdq}L$ would involve indeterminate forms $0/0$). For these reasons we must come to the conclusion that (<ref>) does not describe a clock at all.
It seems that neither considering more complicated systems <cit.> nor
introducing interactions <cit.> would help to remove this indeterminacy of motion. For example, in the electromagnetic field, the consistency requirements $\PB{\Psi_{1,2}}{H}\weq0$ (with $p-eA$ substituted for $p$ by the minimal coupling principle) lead to a secondary constraint $F_{\mu\nu}p^{\mu}k^{\nu}\weq0$, which for rotators reduces to a condition $F_{\mu\nu}\dx^{\mu}k^{\nu}=0$ strictly connected with the Hessian singularity
alluded to above. Although this condition might lead to a unique motion in some situations (e.g. with appropriate initial data in a uniform magnetic field <cit.>) this may not to be so in general (see, a toy model <cit.>).
§ IDEAL CLOCKS
§.§ Second kind Lagrangian
The new velocity constraints arranged to forms of the first degree in the velocities read:
\begin{equation}\label{eq:cnstrI}
\frac{\SP{\dx}{\dx}}{\SP{k}{\dx}}\weq0,\quad
\ell^2 \frac{\SP{\dk}{\dk}}
{\SP{k}{\dx}} + \SP{k}{\dx}\weq0.
\end{equation}
By eliminating these constraints from (<ref>), one finds
$u_1=\chi$, $u_2=\chi$, $u_3=\upsilon$,
$\scriptstyle {\SP{k}{p}=
\frac{\elm\SP{\dot{x}}{k}}{2\,\chi}}$ and
$\scriptstyle \SP{p}{{\Kpi}} =
\frac{{{\ell}}^2\,{{\elm}}^2}{2\,\SP{k}{\dx}}\,
\left( \frac{\SP{{\dot{k}}}{{\dot{x}}}}{\SP{k}{\dx}} - \upsilon \right) $, where
$\chi$ and $\upsilon$ are arbitrary functions.
After discarding a total derivative involving $\SP{k}{\dk}$ and
the higher order terms in the velocity constraints (irrelevant for the Dirac variational procedure <cit.>), the resulting Lagrangian
can be arranged in a form with a new independent variable $\scriptstyle \kappa(\chi)\equiv \frac{\SP{k}{p}}{\elm}$ and a Lagrange multiplier $\lambda$:
\begin{equation}\label{eq:LagI}
\boxed{L=\frac{\elm\kappa}{2} \frac{\SP{\dx}{\dx}}{\SP{k}{\dx}} +
\frac{\elm}{4 \kappa}\left( \ell^2\frac{
\SP{\dk}{\dk}}{\SP{k}{\dx}} + \SP{k}{\dx} \right) + \lambda\, \SP{k}{k}.}
\end{equation}
As expected, this Lagrangian is linear in the velocity constraints, with functions of momenta as coefficients.
In view of the equation $\partial_{\kappa}L=0$, the conditions
(<ref>) can be regarded as consequences of one another, and hence, only $\SP{\dx}{\dx}=0$ may be imposed as a subsidiary condition. Then $\kappa$ becomes arbitrary. Conversely, if $\partial_{\kappa}L=0$ is to be satisfied for arbitrary $\kappa$, then both conditions in (<ref>) follow.
The Casimir invariants
${\scriptstyle \SP{p}{p}=\frac{\elm^2}{2}\br{1+\xi}}$
${\scriptstyle \SP{w}{w}=-\frac{\ell^2\elm^4}{4}\xi}$
are bound to satisfy only a single constraint
${\scriptstyle \SP{p}{p}\weq\frac{\elm^2}{2}-\frac{2}{\ell^2\elm^2}\SP{w}{w}}$
and off the surface ${\scriptstyle \frac{{{\ell}}^2\,\SP{\dk}{\dk}}
{\SP{k}{\dx}} + \SP{k}{\dx}\weq0}$ they
would be functions of the velocities. But for (<ref>) the principal conditions are satisfied on the basis of Hamilton's principle, either supplemented with the null worldlines principle or with the condition that $\kappa$ be independent of the velocities.[Because $\SP{k}{\partial_{\dx}L}\equiv m\kappa$,
the freedom in choosing $\kappa$ at an instant (with $k$ being set) involves the freedom in choosing a combination of momentum variables in $p$. This dependence of a Lagrangian on momentum variables is characteristic of systems with velocity constraints <cit.>.]
The latter requirement is crucial, since otherwise, by solving $\partial_{\kappa}L=0$ for $\kappa$, one would end up with a qualitatively different Lagrangian ${\scriptstyle \elm\sqrt{\frac{\SP{\dx}{\dx}} {2}\br{1+\ell^2\frac{\SP{\dk}{\dk}}{\SP{k}{\dx}^2}}}\,+\lambda\SP{k}{k}}$ whose analytic form is not admissible
on the surface (<ref>) (the momenta $\partial_{\wdq}L$ would involve indeterminate forms $\frac{0}{0}$).
§.§.§ Connection with a family of Relativistic Rotators.
To extend the construction in <cit.> so as to include also the case of bib:secLag, let
a class of projection invariant Lagrangians of the first degree in the velocities be considered, whose form would be admissible also on the surface $\SP{\dx}{\dx}=0$ and compatible with the condition $\SP{k}{\dx}\neq0$:
\begin{equation}\label{eq:rotators}L_{\mcal{F}}=\frac{\elm{\kappa}}{2}\,\frac{\SP{{\dx}}{{\dx}}}{\SP{k}{\dx}}
+\frac{\elm}{2{\kappa}}\SP{k}{\dx} \mcal{F}(\xi) + \lambda\, \SP{k}{k}.\end{equation}
The ${\kappa}$ must have appeared in this precise way for the dimensional grounds and it must transform as $\kappa\to \alpha \kappa$ when $k\to\alpha k$ on account of the assumed projection invariance. Here, $\mcal{F}$ is any function.
If $\partial_{{\kappa}}{L_{\mcal{F}}}=0$, the principal constraints reduce to $\mcal{F}(\xi ) - 2\,\xi \,\mcal{F}'(\xi )=1$
and $4\,\xi \,{\mcal{F}'(\xi )}^2=1$ for any $\kappa$.
If $\kappa$ is not a function of velocities, then $\partial_{{\kappa}}{L_{\mcal{F}}}=0$ implies $\mcal{F}=0$ (and $\SP{\dx}{\dx}=0$),
then the principal conditions give $\mcal{F}'=-\frac{1}{2}$ and $\xi=1$. Hence, to a linear order, $\mcal{F}(\xi)=
\br{1-\xi}/2+o(1-\xi)$ in the vicinity of $\xi=1$. And this is another way of arriving at (<ref>).
In contrast, for ${\kappa}$ not independent of the velocities, one would conclude from $\partial_{{\kappa}}L_{\mcal{F}}=0$ that ${\kappa}={\SP{k}{\dx}}{}\sqrt{\mcal{F}(\xi)/\SP{\dx}{\dx}}$ and end up with a class of Lagrangians $\elm
describing relativistic rotators considered in <cit.> (which includes
Lagrangians $L_{\pm}$ of sec:maxreg as a special case with $\mcal{F}(\xi)=1\pm\sqrt{\xi}$).
§.§ Third kind Lagrangian
Putting $u_1=-u_2$, consider for a while a restricted Legendre transformation with
$p$ left unaltered. Taking ${\scriptstyle
\Kpi=\mp\frac{\ell\elm^2}{2}
\frac{\dk-k u_3}{\SP{k}{p}\sqrt{-\SP{\dk}{\dk}}}}$ and ${\scriptstyle u_2=\mp
\frac{\ell\elm}{2 \SP{k}{p}}\sqrt{-\SP{\dk}{\dk}}}$ implied by eq:DefVeleq:map into account (where ${\scriptstyle \sgn{\frac{\SP{p}{\dx}}{\SP{k}{p}}}=\pm1}$) and integrating off the term linear in $\SP{k}{\dk}$, one arrives at
a Lagrangian:
\begin{equation}\label{eq:lagB1}
L= \SP{\dx}{p} \pm\,
\frac{\ell \elm^2}{2} \frac{
\sqrt{-\SP{\dk}{\dk}}}{ \SP{k}{p}}
By making arbitrary variations w.r.t. $p$ ($\delta L$ must be independent of $\delta p$ <cit.>),
the result ${\scriptstyle
\dx = \pm\frac{\ell\elm^2}{2}
\frac{\sqrt{-\SP{\dk}{\dk}}}{\SP{k}{p}^2}k}$ following from eq:DefVeleq:map can be re-obtained.
It implies $\scriptstyle{\SP{{\dot{x}}}{e}} =\pm
\frac{{\ell}\,{{\elm}}^2\,{\sqrt{-\SP{{\dot{k}}}{{\dot{k}}}}}}
{2\,{\SP{k}{p}}^2}\SP{e}{k}$ for any
vector $e$, and this fact can be used to eliminate $p$ from $L$.
the alternative Lagrangian takes
on a form involving arbitrary $e$ such that $\SP{k}{e}\neq0$:
\begin{equation}\label{eq:lagB2}
+\lambda \SP{k}{k}.\end{equation}
$\SP{e}{k}$ to be nonzero, it would
suffice that $e$ be obtained from any timelike
vector by a two-parameter transformation group $e\to \alpha
(e+\beta k)$, with $\alpha$, $\beta$ being arbitrary functions.
This freedom in choosing
$e$ must be
physically irrelevant, and this will be so if $\partial_e L=0$.
implies $\dx=\frac{\SP{\dx}{e}}{\SP{k}{e}}k$.
Furthermore, $p:= \partial_{\dx}L$ is collinear with $e$ and is independent of the scale of $e$.
As so, $p$ may be substituted in place of $e$ in the expression for $\partial_{\dx}L$, hence
\begin{equation}\label{eq:freq}\frac{2}{\ell} =
\frac{{{\elm}}^2\,{\sqrt{-\SP{{\dot{k}}}{{\dot{k}}}}}}{\abs{\SP{{\dot{x}}}{p}\SP{k}{p}}} \quad\Rightarrow\quad \Omega=\frac{2}{\ell}\frac{\SP{p}{p}}{\elm^2} \quad \mathrm{(from\, \eqref{eq:Omega})}.\end{equation}
constraint $\SP{p}{p}-\elm^2\weq0$ does not follow from the Lagrangians eq:lagB1eq:lagB2, nevertheless it is essential for consistency with the map (<ref>). It must be regarded as a secondary first class constraint (whose purpose is to set $\Omega$ to
$\sfrac{2}{\ell}$ and the orbital radius to $\sfrac{\ell}{2}$, consistently with the equations of motion).
§.§ Comparison of the clocks
It is convenient to write down the Hamiltonian equations in the gauge: $\SP{p}{\Kpi}=0$,
$\SP{k}{p}=\elm$, $\SP{p}{\dx}=\elm$ and to consider a unit space-like direction $n$ defined in (<ref>), which is collinear with the projection of $k$ onto a subspace orthogonal to $p$. Together with the consistency requirements
$0=\PB{\SP{p}{\Kpi}}{H}$, this
implies for $u_1=\pm u_2$ that $u_1=1$, $u_2=\pm 1$, $u_3=0$ and $u_4=
\pm\frac{\elm }{2}$.
This way the
Hamiltonian equations reduce to
$$ \dx=\frac{p}{\elm}\mp n , \qquad
\dot{p}=0,\qquad \dot{n}=\pm\frac{4}{\elm\,\ell^2}\,\Kpi,\qquad
\dot{\Kpi}=\mp\elm\,n,$$
with $\SP{n}{n}=-1$, $\SP{n}{p}=0$ (then $k=\frac{p}{\elm}+n$).
The equations for $\dot{n}$ and $\dot{\Kpi}$ imply a
uniform motion with frequency $\Omega=\frac{2}{\ell}$ about a great circle on the unit sphere: $\ddot{n}+\frac{4}{\ell^2}\,n =0$.
The null directions of the clocks' velocity vectors $\dx$ are conjugate to one another by the reflection $\frac{\elm\dx}{\SP{p}{\dx}}\to\frac{2p}{\elm}-\frac{\elm\dx}{\SP{p}{\dx}}$.
Interestingly, the two clocks
have opposite spin alignment:
$$p\wedge k\wedge\Kpi=\pm\frac{\elm\ell^2}{4}p\wedge n\wedge{\dot{n}}=\pm\elm\,p\wedge x\wedge{\dot{x}},$$
where $x=\frac{p}{\elm}t+\frac{\ell}{2}r(\varphi)$, $\SP{r}{r}=-1$, $\SP{p}{r}=0$, $\varphi=\frac{2}{\ell}t$ (then $n=\pm r'(\varphi)$).
In a sense, the two clocks can be regarded as a limiting case of the Lagrangian (<ref>) when
$\SP{\dx}{\dx}\to0$ with: a) $\frac{\ell^2\SP{\dk}{\dk}}{\SP{k}{\dx}^2}\to -1$ for the clock (<ref>) or b) $\SP{k}{\dx}\to0$ for the clock (<ref>).
§ SUMMARY AND FUTURE APPLICATIONS
In this paper were described two mathematical clocks which are relativistic rotators exhibiting intrinsic circular motion with the speed of light and opposite spin alignment. The Lagrangians of the clocks were distinguished by a singularity of an inverse Legendre map for rotators of which Casimir scalars are fixed parameters. Such clocks are perfect, they work always the same way, no matter how they move.
In future works, the two ideal clocks can be used to test the clock hypothesis.
In free motion, the phase associated with the intrinsic circular motion of these clocks increases steadily with the affine parameter of the center of mass (). But it is not a priori obvious (even in the limit $\ell\to0$) if this property will survive for accelerated motions of the , e.g. for a constrained motion along a strongly curved worldline.
For such motions, the chronometric curve – that is, properly parameterized helical null worldline of an ideal clock – would undergo additional distortions and this could affect the steady clocking rate.
Testing the clock hypothesis requires introducing interactions. However, usual coupling with external fields may lead to inconsistencies, see <cit.>. In this context, it would be instructive to see the implications of the secondary constraint $F_{\mu\nu}p^{\mu}k^{\nu}\weq0$ appearing when ideal clocks are minimally coupled with the electromagnetic field. Furthermore,
it would be interesting to study the motion in strong gravitational fields of black holes.
In curved spacetimes, there might arise problems even with defining the rotation phase: when
global teleparallelism is lost, the local reference frames, used to measure the infinitesimal phase increments at various instants, cannot be unambiguously connected; in addition, some disturbances in the phase could appear due to rotation of local inertial frames.
|
1511.00398
|
$^{1}$ Department of Physics, College of William and Mary, Williamsburg, VA 23187, USA
$^{2}$ Department of Physical Sciences, Kutztown University of Pennsylvania, Kutztown, PA 19530, USA
$^{3}$ School of Natural Sciences, University of California, Merced, CA 95344, USA
A ballistic atom pump is a system containing two reservoirs of neutral atoms or molecules and a junction connecting them containing a localized time-varying potential. Atoms move through the pump as independent particles. Under certain conditions, these pumps can create net transport of atoms from one reservoir to the other. While such systems are sometimes called “quantum pumps,” they are also models of classical chaotic transport, and their quantum behavior cannot be understood without study of the corresponding classical behavior. Here we examine classically such a pump's effect on energy and temperature in the reservoirs, in addition to net particle transport. We show that the changes in particle number, of energy in each reservoir, and of temperature in each reservoir vary in unexpected ways as the incident particle energy is varied.
67.85.Hj, 05.60.Gg, 03.65.Sq, 37.10.Vz
Particle transport is an ongoing topic of interest in a variety of systems including solid state electronics, microfluidic devices, and atomtronics components. In electronic solid-state systems, the transport of electrons through mesojunctions having time-dependent potential barriers, a phenomenon often called “quantum pumping,” has been theorized for decades <cit.>. It has been shown that such a system can pump electrons from one reservoir to another with no bias (such as a potential difference). More recently, Das and Aubin have proposed simulating such electron pumps using a system of neutral cold atoms with optical potentials as the driving forces <cit.>. Neutral atom transport is becoming increasingly important in its own right due to the ongoing development of atomtronics, which seeks to replicate properties of electronics using neutral atoms, and in the field of quantum computing <cit.>. Analogues of batteries, diodes, transistors, and recently hysteresis <cit.> have been explored in ultracold neutral atom systems.
Previous studies of these pumps have generally been within the quantum regime, and largely focus on charge or spin transport associated with fermionic carriers. In this paper we study the classical analogues of such pumps, and focus on the differential transfer of particles, energy, and heat. This broadens the study of quantum pumps into a new and largely unexplored regime. These classical analogues of quantum pumps are also interesting because they provide models of chaotic transport, which occurs in a great variety of systems on scales from nuclei to galaxies <cit.>.
The pumps we consider are effectively one-dimensional, so the Hamiltonian is
\begin{eqnarray}
H(p,x,t) = p^2/2m + V(x,t).\label{ham}
\end{eqnarray}
We choose $V(x,t)$ to consist of two repulsive barriers, one or both of which oscillates. When both barriers oscillate they have the
same frequency $\omega$, but not the same phase. We examine the classical scattering of equal numbers of particles which approach such pumps from each reservoir with equal and fixed incident energy.
It has been shown that flows may be zero or negligible in pumps with idealized limits such as delta-function barriers or uniform phase-space density <cit.>. Here we show that under more realistic conditions, such pumps can generate significant net transfer of both matter and energy. Understanding heat flow is also essential for any transport mechanism, and is of fundamental importance for thermoelectric devices <cit.>. Studies that have been done in the context of mesoscopic pumps <cit.> used a strictly quantum picture involving exchange of quasi-particles, and heat flow was shown to be outwards from the pump towards the reservoirs. The classical model discussed here is more appropriate for higher temperatures, and we show that the pump can heat or cool one or both reservoirs.
Summary of results: In previous papers <cit.> we have shown that two-barrier pumps have the following properties when at least one barrier oscillates. (1) These so-called “quantum pumps” provide nice models of classical chaotic scattering, and their behavior is governed by a heteroclinic tangle. (2) Quantum theory shows that monoenergetic particles incident on periodically oscillating barriers have final energies equal to $E_n=E_i+n\hbar\omega$, where $E_i$ is their initial energy and $\omega$ is the frequency of the pump; classical and semiclassical theories are needed to understand the range of $n$ and the heights of the peaks. (3) Net pumping of particles from one reservoir to another can occur if monoenergetic particles approach the pump from both sides. (4) Pumping can go in either direction, depending on the incident energy and the pump parameters. (5) The amount of pumping is very sensitive to incident energy and to pump parameters, and cannot be predicted without detailed calculation. (6) It is possible to design a “particle diode” which only allows net particle transport in one direction for low-energy incident particles, and in the opposite direction for high-energy incident particles.
In this paper we show that for monoenergetic incident particles: (A) Such pumps can transfer energy from one reservoir to the other, and energy can be transferred from pump to particles or vice versa. (B) A net change of energy in each reservoir can occur even if there is no net particle transport. The direction of energy change is distinct from the direction of particle transport. (C) Such pumps can heat or cool one or both reservoirs, and the heating or cooling is distinct from the existence or direction of net particle transport and distinct from energy flow. (D) At some incident energies, such pumps can generate net particle transport while at the same time particles give energy to the pump.
System We will establish properties (A)-(D) by examining one specific pump: a particle diode consisting of two Gaussian-shaped potential barriers, only one of which oscillates. We choose this pump as our example because the dynamics of the system become much more complicated when both barriers oscillate <cit.>. However, allowing the second barrier to oscillate (or changing the barrier parameters) only affects the conclusions discussed below quantitatively. Therefore properties (A)-(D) apply to general ballistic atom pumps.
In the chosen diode, the distance between the barriers is substantially larger than their widths, so their overlap is negligible. The right-hand barrier has a fixed height, while the left-hand barrier oscillates between zero and the height of the right-hand barrier. The pump is described by
\begin{align}
V(x,t)&=\hat{U}_L \left(1+\alpha_L \cos(\omega t)\right)\exp\left(\frac{-(x+\hat{x})^2}{2\sigma^2}\right) \label{gaussleft}\nonumber \\
&+\hat{U}_R \exp\left(\frac{-(x-\hat{x})^2}{2\sigma^2}\right),
\end{align}
where $\hat{U}_{L,R}$ is the average height of each barrier, $\alpha_{L}$ is the amplitude of
oscillation of the left barrier, $\omega=2\pi/T$ is the frequency and $T$ is the period, and $\sigma$ is the standard deviation of each Gaussian. The left and right barriers are centered at $x=-\hat{x}$ and $x=\hat{x}=4.5$, respectively. In our calculations, we set $\hat{U}_L=1$, $\hat{U}_R=2\hat{U}_L=2$, $\alpha_L=1$, $\omega=1$, $\sigma=1.5$, and $m=1$. These are scaled units <cit.>.
The effects of the pump can be understood qualitatively as follows. For incident energies less than the height of the static barrier, all particles from the right reflect from the static barrier, but particles incident from the left may gain enough energy from the oscillating barrier to scatter past both barriers. Consequently, the only possible direction of net particle transport is from left-to-right. For incident energies greater than the height of the static barrier, computations show that, in this case, all particles incident from the right transmit past both barriers, but particles incident from the left may lose energy to the oscillating barrier, reflect from the static barrier, and ultimately scatter to the left reservoir. Thus the only possible direction of net particle transport reverses to right-to-left.
Our computational algorithm can be summarized as follows: 1) For each
initial energy, launch particles toward the barriers from the left and right. Particles begin with a range of positions $\Delta x=|p_i|2\pi/\omega$ where $p_i$ is the initial momentum, which ensures that all barrier phases are encountered. 2) Record the reservoir to which each particle is scattered, and sum the results to obtain the net fractional transport (defined below) of particles scattered to the right (which may be negative if more particles are scattered to the left).
3) Compute the total energy gain of the two reservoirs after scattering, which may be negative if the system loses energy to the pump. 4) Compute the net gain (or loss) in the total energy of each reservoir. Energy being an extensive quantity, a reservoir gains total energy by gain in the number of particles as well as by gain of energy of individual particles passing though the pump.
5) Compute the change of energy of each particle scattered into each reservoir, and compute the average of these changes for all particles scattered into each reservoir. The average change of energy per scattered particle may be regarded as corresponding to a change of temperature of the reservoir. Temperature being an intensive property, the direction of temperature change need not be the same as the direction of energy change in each reservoir. Formulas for computation of these quantities are given below.
The fractional transport of particles through the pump is defined as
\begin{align}
\end{align}
where $R(|p_i|)$ is the number of particles scattered to the right for each $|p_i|$, and $L(|p_i|)$ is the number of particles scattered to the left. The sum $R(|p_i|)+L(|p_i|)$ represents all particles incident on the pump for a given $|p_i|$. $C_P(|p_i|)$ is positive when more particles are scattered to the right (net particle transport to the right reservoir), negative when more particles are scattered to the left (net particle transport to the left reservoir) and zero when equal numbers of particles scatter to the right and left reservoirs.
(Color online) (a) Net particle transport, $C_P(|p_i|)$ [thickest curve], and the change in average energy per particle in the left reservoir, $(k_B/2)\detl$ [thin (green) curve] and right reservoir, $(k_B/2)\detr$ [medium (purple) curve]. When $C_P(|p_i|)$ is positive (negative) there is net particle transport from left-to-right (right-to-left). When $(k_B/2)\Delta T^{L,R}(\p)$ is positive (negative), the pump increases (decreases) the average energy of particles scattered into the respective reservoir, and the temperature in that reservoir increases (decreases). (b) Total energy change in both reservoirs, $\detot$ [thickest curve], in the left reservoir, $\del$ [thin (green) curve], and in the right reservoir, $\der$ [medium (purple) curve]. When $\detot$ is positive, the pumps adds net energy to the reservoirs; when negative, the reservoirs lose net energy to the pump. When $\Delta E^{L,R}(\p)$ is positive (negative), the pump increases (decreases) the total energy in the respective reservoir. (c) Summary of (a) and (b). Dark gray (red) indicates an increase, and light gray (blue) represents a decrease. No color is plotted if the quantity does not change.
For each initial particle energy, the total energy change of the system and each reservoir are defined as
\begin{equation}
\Delta E^\alpha(|p_i|) = E_f^\alpha(|p_i|)-E_i^\alpha(|p_i|),
\end{equation}
where $\alpha=\{T,L,R\}$. When $\alpha=T$, $E_f^T(|p_i|)$ and $E_i^T(|p_i|)$ represent the total final and initial energies, respectively, of all particles incident upon one cycle of the pump. When $\Delta E^T>0$, the pump has added energy to the reservoirs; when $\Delta E^T<0$, the reservoirs have lost energy to the pump. When $\alpha=L$ or $R$, $E_f^{\alpha}(|p_i|)$ represents the total final energy of all particles which scatter to the left or right reservoirs, and $E_i^{\alpha}(|p_i|)$ represents the corresponding total initial energy of all particles beginning in the left or right reservoir.
The last quantities examined in this paper are the changes in average energy per particle scattered into each reservoir. These quantities are defined as
\begin{align}
\overline{\Delta E^\beta(\p)}=\frac{E_f^\beta(|p_i|)}{M^\beta}-\frac{E_i^\beta(|p_i|)}{N^\beta}
=\frac{k_B}{2}{\Delta T^\beta(|p_i|)},\label{temp}
\end{align}
where $\beta=\{L,R\}$ and corresponds to the left and right reservoirs, respectively. $N^\beta$ is the number of particles incident on the pump from the $\beta$ reservoir in one cycle, and $M^\beta$ is the number of particles scattered to the $\beta$ reservoir. A total of $2N^\beta$ particles approach the pump for each incident energy ($N^\beta$ from each reservoir); consequently $M^\beta>N^\beta$ corresponds to an increase in particle number for the $\beta$ reservoir. This change of average energy per particle can be regarded as a change of temperature of those scattered particles. Then a positive (negative) $\Delta T^{L,R}$ produces an increase (decrease) in the temperature of the corresponding reservoir after thermalization.
Results In Fig. <ref> we show the results of calculations for net particle transport, energy changes in the total system, and temperature and energy changes in each reservoir for the selected pump. We discuss all properties in relation to the net particle transport, which is the thick curve in Fig. <ref>(a). There are four distinct regions of particle transport direction, and we discuss them in order of increasing complexity. This complexity arises for two reasons. (1) Depending on the initial energy and the frequency of the barrier, a particle can ride repeatedly up and down the oscillating barrier. (2) A particle can undergo multiple reflections between the two barriers; this is the source of chaos in the system.
Region I: No particle transport; left reservoir heated ($0<|p_i|\lesssim 1.176$) At these low energies, no particle gets past the static barrier, so there is no net particle transport, and $C_P(|p_i|)=0$ [thickest curve in Fig. <ref>(a)]. Particles incident from the right reflect from the static barrier into the right reservoir without a change in energy. Therefore the number of particles, their average energy, and the total energy in the right reservoir do not change, i.e. $\detr=0$ [medium curve (purple online) in Fig. <ref>(a)] and $\Delta E^R(|p_i|)=0$ [medium (purple online) curve in Fig. <ref>(b)].
All particles incident from the left are scattered into the left reservoir, but the oscillating barrier changes their energy. They may gain or lose energy to the pump, depending on their time of arrival. On average, they gain energy. Accordingly, the temperature (average energy per particle) and total energy both rise in the left reservoir, i.e., $\detl>0$ and $\del>0$ [thin (green online) curves in Fig. <ref>(a) and <ref>(b)]. Considering both reservoirs together, there has been net addition of energy from the pump to the reservoirs ($\detot>0$) [thickest curve in Fig. <ref>(b)], and this energy is entirely added to the left reservoir.
These results are summarized in Fig <ref>(c), in which the light gray (blue online) represents a loss, dark gray (red online) represents an increase, and white represents no change.
Region IV: No particle transport; both reservoirs cooled ($\p\gtrsim2.63$) At high incident momentum, all particles incident from both sides transmit past both barriers, and there is no net particle transport ($C_P(\p)=0$). Particles incident from both sides lose energy (on average) to the pump, which causes a decrease in the total energy of each reservoir ($\Delta E^{L,R}(\p)<0$) and total energy of the system ($\detot<0$). The average energy changes of particles scattered into each reservoir are equal ($\detl=\detr<0$) and each reservoir is cooled. Fig. <ref>(c) summarizes these results. Calculations show that the loss of energy to the pump decreases exponentially with $|p_i|$, a result that calls for a general proof.
Region II: Net left-to-right particle transport ($1.176\lesssim |p_i| \leq 2$) This region is defined by the fact that all particles from the right are reflected by the static barrier, but some particles incident from the left gain enough energy from the pump to scatter into the right reservoir. Accordingly, the right-hand reservoir gains particles ($C_P(|p_i|)>0$) and average evergy per particle ($\detr>0$), and the reservoir is heated. The total energy of the reservoir increases ($\der>0$).
Some particles which begin on the left scatter to the left, and the pump can change their energy. Over most of region II ($1.176\lesssim|p_i|\lesssim 1.95$), the left-to-left scatterers gain energy from the pump (on average) ( $\detl>0$), and temperature of the left reservoir increases. However at the high end of this region ($1.95 \lesssim \p <2$), the left-to-left scatterers lose energy (on average) to the pump ( $\detl<0$), and the left reservoir is cooled.
The total energy change of this reservoir depends on the average energy change of left-scattered particles, and on the loss of particles to the right-hand reservoir. Over most of region II ($1.243\lesssim \p <2$), there is a net loss of energy in the left-hand reservoir ($\del<0$). However at the lower end of this region ($1.176\lesssim|p_i|\lesssim 1.243$), the gain of energy of left-to-left scatterers exceeds the loss of energy associated with particle transport to the right, and the total energy in the left-hand reservoir rises ($\del>0$)
Combining the energy changes of both reservoirs, the pump has added energy to the reservoirs for the entirety of region II ($\detot>0$). These results are summarized in Fig <ref>(c).
Region III: Net right-to-left particle transport ($2<|p_i|\lesssim 2.63$) This region is the most complex. For $|p_i|>2$, for this pump, all particles incident from the right have enough energy to transmit past both barriers. Particles incident from the left initially have enough energy to get over the static barrier, but they may lose energy to the oscillating barrier, be reflected from the static barrier, and scatter into the left reservoir. Therefore the only possible direction of net particle transport is from right-to-left. Fig. <ref>(a) shows right-to-left particle transport ($C_P(|p_i|)<0$) in the range $2<|p_i|\lesssim 2.63$.
Particles which scatter to the right reservoir begin in the left reservoir. In the majority of this region ($2<|p_i|\lesssim 2.616$) they (on average) gain energy from the pump ($\detr>0$), and the temperature in the right reservoir rises. Combining the gain of energy per particle with the loss of particles, the result is a loss of total energy in the right reservoir ($\der<0$). In the remainder of region III, ($2.616<|p_i|\lesssim 2.63$), the left-to-right scatterers lose energy to the pump (on average) ($\detr<0$), the right reservoir is cooled, and its total energy decreases ($\der<0$) because of loss of particles and loss of average particle energy.
Particles which scatter to the left reservoir can begin in either reservoir. These particles on average lose energy to the pump ($\detl<0$), so the left reservoir is cooled. However its total energy rises ($\del>0$) because scattering increases particle number in the reservoir. Examining both reservoirs together, over most of the lower portion of region III ($2<|p_i|\lesssim 2.267$), the pump adds energy to the reservoirs, while over the remainder of the region ($2.267\lesssim|p_i|\lesssim 2.63$), it removes energy from the reservoirs. Fig. <ref>(c) summarizes these results.
Averaging over energies Thermodynamics (and physical intuition) tell us that if a pump is connected to a single reservoir (or two reservoirs with the same temperature, pressure, and chemical potential) then the net energy transfer can only go from the pump to the reservoirs. Accordingly if we average the energy input $\detot$ over a Maxwellian distribution at any temperature, that result must be nonnegative ($\int{\detot e^{-p_i^2/2mk_BT}dp}\geq 0$). Scrutiny of $\detot$ in Fig. <ref>(b) shows that this is satisfied in the example pump. Also the observation that at low incident particle energies (Region I), the net energy flow is from pump to particles must hold for any pump. This is another point that calls for a dynamical proof.
We have therefore by example established the properties (A)-(D) stated in the “Summary of results.”
Acknowledgements KKD acknowledges support of the NSF under Grant No. PHY-1313871, and of a PASSHE
-FPDC grant and a research grant from Kutztown University.. KAM acknowledges NSF support from Grant No. PHY-0748828. JBD and TAB acknowledge NSF support via Grant No. PHY-1068344. Calculations were performed on the SciClone computing complex at The College of William and Mary, which were provided with the assistance of the National Science Foundation, the Virginia Port Authority, Sun Microsystems, and Virginia's Commonwealth Technology Research Fund.
authorD. J. Thouless,
journalPhys. Rev. B volume27,
pages6083 (year1983).
authorP. W. Brouwer,
journalPhys. Rev. B volume58,
pagesR10135 (year1998).
[Moskalets and
authorM. Moskalets and
authorM. Büttiker,
journalPhys. Rev. B volume66,
pages205320 (year2002a).
[Das and Opatrný(2010)]das-opatrny
authorK. K. Das and
authorT. Opatrný,
journalPhys. Lett. A volume374,
pages485 (year2010).
[Mucciolo et al.(2002)Mucciolo, Chamon,
and Marcus]chamon-spin
authorE. R. Mucciolo,
authorC. Chamon, and
authorC. M. Marcus,
journalPhys. Rev. Lett. volume89,
pages146802 (year2002).
[M. Garttner, F. Lenz, C. Petri, F. K. Diakonos and P.
authorM. Garttner, F. Lenz, C. Petri, F. K. Diakonos
and P. Schmelcher, journalPhys. Rev. E
volume81, pages051136
[Ferry et al.(2009)Ferry, Goodnick, and
authorD. Ferry,
authorS. M. Goodnick,
and authorJ. Bird,
titleTransport in Nanostructures
(publisherCambridge, addressUSA,
year2009), edition2nd ed.
[K. K. Das and S. Aubin(2009)]DasAubin_PRL2009
authorK. K. Das and S. Aubin,
journalPhys. Rev. Lett. volume103,
pages123007 (year2009).
authorK. K. Das,
journalPhys. Rev. A volume84,
pages031601 (year2011).
[K. K. Das, M. R. Meehan, and A. J.
authorK. K. Das, M. R. Meehan, and A. J. Pyle,
journalPhys. Rev. A volume89,
pages063626 (year2014).
[B. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel, C.-S.
Chuu, X.-M. Jin, J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W.
authorB. Zhao, Y.-A. Chen, X.-H. Bao, T. Strassel,
C.-S. Chuu, X.-M. Jin, J. Schmiedmayer, Z.-S. Yuan, S. Chen, and J.-W.
Pan, journalNature Physics volume5,
pages95 (year2009).
[A. A. Zozulya and D. Z. Anderson(2013)]zoz
authorA. A. Zozulya and D. Z. Anderson,
journalPhys. Rev. A volume88,
pages043641 (year2013).
[R. A. Pepino, J. Cooper, D. Z. Anderson, and M. J.
Holland (2009)]pepinocooper
authorR. A. Pepino, J. Cooper, D. Z. Anderson, and M.
J. Holland , journalPhys. Rev. Lett.
volume103, pages140405
[ S. Eckel, J. G. Lee, F. Jendrzejewski, N. Murray, C.
W. Clark, C. J. Lobb, W. D. Phillips, M. Edwards, and G. K. Campbell
author S. Eckel, J. G. Lee, F. Jendrzejewski, N.
Murray, C. W. Clark, C. J. Lobb, W. D. Phillips, M. Edwards, and G. K.
Campbell , journalNature volume506,
pages200 (year2014).
[C. C. Rankin and W. H. Miller(1971)]53
authorC. C. Rankin and W. H. Miller,
journalJ. Chem. Phys. volume55,
pages3150 (year1971).
[T. Uzer, C. Jaffé, J. Palacian, P. Yanguas, S.
authorT. Uzer, C. Jaffé, J. Palacian, P. Yanguas,
S. Wiggins, journalNonlinearity volume15,
pages957 (year2002).
[P. Gaspard(1998)]59
authorP. Gaspard, titleChaos,
scattering, and statistical mechanics (publisherCambridge
University Press, addressCambridge, MA, year1998).
[J. U. Nöckel and A. D. Stone(1996)]63
authorJ. U. Nöckel and A. D. Stone,
titleOptical processes in microcavities
(publisherWorld Scientific Publishers,
addressSingapore, year1996).
[S. Sridhar, W. Lu, W. Sinai(2002)]76
authorS. Sridhar, W. Lu, W. Sinai,
journalJ. Stat. Phys. volume108,
pages755 (year2002).
[K. Burke and K. A. Mitchell(2009)]78
authorK. Burke and K. A. Mitchell,
journalPhys. Rev. A volume80,
pages033416 (year2009).
[K. Mitchell, J. Handley, B. Tighe, A. Flower, and J.
authorK. Mitchell, J. Handley, B. Tighe, A. Flower,
and J. Delos, journalPhys. Rev. Lett.
volume92, pages73001 (year2004).
[K. A. Mitchell and D. A. Steck(2007)]79
authorK. A. Mitchell and D. A. Steck,
journalPhys. Rev. A volume76,
pages031403 (year2007).
[J. M. Ottino, A. Souvaliotis, and G.
authorJ. M. Ottino, A. Souvaliotis, and G.
Metcalfe, journalChaos, Solitons, and Fractals
volume6, pages425 (year1995).
[R. Samelson(1996)]105
authorR. Samelson, titleStochastic
Modelling in Physical Oceanography (publisherBirkhauser,
addressBoston, year1996).
[C. Jaffé and T. Uzer(2004)]111
authorC. Jaffé and T. Uzer,
journalAnn. N Y Acad. Sci. volume1017,
pages39 (year2004).
[M. K. Ivory, T. A. Byrd, A. J. Pyle, K. K. Das, K. A.
Mitchell, S. Aubin, and J. B. Delos(2014)]double
authorM. K. Ivory, T. A. Byrd, A. J. Pyle, K. K. Das,
K. A. Mitchell, S. Aubin, and J. B. Delos, journalPhys. Rev. A
volume90, pages023602
[G. Casati, C.Mejía-Monasterio, and T.
authorG. Casati, C.Mejía-Monasterio, and T.
Prosen, journalPhys. Rev. Lett.
volume101, pages016601
[Moskalets and
authorM. Moskalets and
authorM. Büttiker,
journalPhys. Rev. B volume66,
pages035306 (year2002b).
[Wang and Wang(2002)]Wang-heat
authorB. Wang and
authorJ. Wang,
journalPhys. Rev. B volume66,
pages125310 (year2002).
[T. A. Byrd and J. B. Delos(2014)]ByrdDelos
authorT. A. Byrd and J. B. Delos,
journalPhys. Rev. E volume89,
pages022907 (year2014).
[T. A. Byrd, M. K. Ivory, A. J. Pyle, S. Aubin K. A.
Mitchell, J. B. Delos, and K. K. Das(2012)]single
authorT. A. Byrd, M. K. Ivory, A. J. Pyle, S. Aubin K.
A. Mitchell, J. B. Delos, and K. K. Das, journalPhys. Rev. A
volume86, pages013622
|
1511.00146
|
Several recent works have explored stochastic gradient methods for variational inference that exploit the geometry of the variational-parameter space. However, the theoretical properties of these methods are not well-understood and these methods typically only apply to conditionally-conjugate models. We present a new stochastic method for variational inference which exploits the geometry of the variational-parameter space and also yields simple closed-form updates even for non-conjugate models. We also give a convergence-rate analysis of our method and many other previous methods which exploit the geometry of the space. Our analysis generalizes existing convergence results for stochastic mirror-descent on non-convex objectives by using a more general class of divergence functions. Beyond giving a theoretical justification for a variety of recent methods, our experiments show that new algorithms derived in this framework lead to state of the art results on a variety of problems. Further, due to its generality, we expect that our theoretical analysis could also apply to other applications.
§ INTRODUCTION
Variational inference methods are one of the most widely-used computational tools to deal with the intractability of Bayesian inference, while stochastic gradient (SG) methods are one of the most widely-used tools for solving optimization problems on huge datasets. The last three years have seen an explosion of work exploring SG methods for variational inference <cit.>. In many settings, these methods can yield simple updates and scale to huge datasets.
A challenge that has been addressed in many of the recent works on this topic is that the “black-box" SG method ignores the geometry of the variational-parameter space. This has lead to methods like the stochastic variational inference (SVI) method of <cit.>, that uses natural gradients to exploit the geometry. This leads to better performance in practice, but this approach only applies to conditionally-conjugate models. In addition, it is not clear how using natural gradients for variational inference affects the theoretical convergence rate of SG methods.
In this work we consider a general framework that (i) can be stochastic to allow huge datasets, (ii) can exploit the geometry of the variational-parameter space to improve performance, and (iii) can yield a closed-form update even for non-conjugate models. The new framework can be viewed as a stochastic generalization of the proximal-gradient method of <cit.>, which splits the objective into conjugate and non-conjugate terms. By linearizing the non-conjugate terms, this previous method as well as our new method yield simple closed-form proximal-gradient updates even for non-conjugate models.
While proximal-gradient methods have been well-studied in the optimization community <cit.>, like SVI there is nothing known about the convergence rate of the method of <cit.> because it uses “divergence" functions which do not satisfy standard assumptions.
Our second contribution is to analyze the convergence rate of the proposed method. In particular, we generalize an existing result on the convergence rate of stochastic mirror descent in non-convex settings <cit.> to allow a general class of divergence functions that includes the cases above (in both deterministic and stochastic settings). While it has been observed empirically that including an appropriate divergence function enables larger steps than basic SG methods, this work gives the first theoretical result justifying the use of these more-general divergence functions. It in particular reveals how different factors affect the convergence rate such as the Lipschitz-continuity of the lower bound, the information geometry of the divergence functions, and the variance of the stochastic approximation. Our results also suggest conditions under which the proximal-gradient steps of <cit.> can make more progress than (non-split) gradient steps, and sheds light on the choice of step-size for these methods.
Our experimental results indicate that the new method leads to improvements in performance on a variety of problems, and we note that the algorithm and theory might be useful beyond the variational inference scenarios we have considered in this work.
§ VARIATIONAL INFERENCE
Consider a general latent variable model where we have a data vector $\vy$ of length $N$ and a latent vector $\vz$ of length $D$.
In Bayesian inference, we are interested in computing the marginal likelihood $p(\vy)$, which can be written as the integral of the joint distribution $p(\vy,\vz)$ over all values of $\vz$.
This integral is often intractable, and in variational inference we typically approximate it with the evidence lower-bound optimization (ELBO) approximation $\elbofinal$. This approximation introduces a distribution $q(\vz|\vlambda)$ and chooses the variational parameters $\vlambda$ to maximize the following lower bound on the marginal likelihood:
\begin{equation}
\begin{aligned}
& \log p(\vy) = \log\int q(\vz|\vlambda)\frac{p(\vy,\vz)}{q(\vz|\vlambda)}\, d\vz, \\
& \ge \max_{\boldsymbol{\lambda} \in \mathcal{S}} \elbofinal(\vlambda) := \myexpect_{q(\mathbf{z}|\boldsymbol{\lambda})} \sqr{ \log \frac{p(\vy,\vz)}{q(\vz|\vlambda)} } .
\end{aligned}
\label{eq:LB}
\end{equation}
The inequality follows from concavity of the logarithm function. The set $\mathcal{S}$ is the set of valid parameters $\vlambda$.
To optimize $\vlambda$, one of the seemingly-simplest approaches is gradient descent:
$\vlambda_{k+1} = \vlambda_k + \beta_k \nabla \elbofinal(\vlambda_k)$,
which can be viewed as optimizing a quadratic approximation of $\elbofinal$,
\begin{align}
\vlambda_{k+1} &= \argmin{\boldsymbol{\lambda}\in\mathcal{S}} \sqr{ -\vlambda^T \nabla \elbofinal(\vlambda_k) + \frac{1}{2\beta_k} \|\vlambda - \vlambda_k \|^2_2}. \label{eq:equivalent_grad_descent}
%\vlambda_{k+1} &= \vlambda_k + \delta_k \bigtriangledown \elbofinal(\vlambda_k)
\end{align}
While we can often choose the family $q$ so that it has convenient computational properties, it might be impractical to apply gradient descent in this context when we have a very large dataset or when some terms in the lower bound are intractable.
Recently, SG methods have been proposed to deal with these issues <cit.>: they allow large datasets by using random subsets (mini-batches) and can approximate intractable integrals using Monte Carlo methods that draw samples from $q(\vz|\vlambda)$.
A second drawback of applying gradient descent to variational inference is that it uses the Euclidean distance and thus ignores the geometry of the variational-parameter space, which often results in slow convergence. Intuitively, (<ref>) implies that we should move in the direction of the gradient, but not move $\vlambda_{k+1}$ too far away from $\vlambda_k$ in terms of the Euclidean distance. However, the Euclidean distance is not appropriate for variational inference because $\vlambda$ is the parameter vector of a distribution; the Euclidean distance is often a poor measure of dissimilarity between distributions. The following example from <cit.> illustrates this point: the two normal distributions $\gauss(0,10000)$ and $\gauss(10,10000)$ are almost indistinguishable, yet the Euclidean distance between their parameter vectors is 10, whereas the distributions $\gauss(0,0.01)$ and $\gauss(0.1,0.01)$ barely overlap, but their Euclidean distance between parameters is only $0.1$.
Natural-Gradient Methods:
The canonical way to address the problem above is by replacing the Euclidean distance in (<ref>) with another divergence function. For example, the natural gradient method defines the iteration by using the symmetric Kullback-Leibler (KL) divergence <cit.>,
\begin{equation}
\label{eq:naturalgradient}
\begin{aligned}
&\vlambda_{k+1} =\\
&\argmin{\boldsymbol{\lambda}\in\mathcal{S}} \sqr{ -\vlambda^T \nabla \elbofinal(\vlambda_k) + \frac{1}{\beta_k}\dkls{sym}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)}}.
%\vlambda_{k+1} &= \vlambda_k + \delta_k \vI_k^{-1} \bigtriangledown \elbofinal(\vlambda_k), &
\end{aligned}
\end{equation}
This leads to the update
\begin{align}
\vlambda_{k+1} = \vlambda_{k} + \beta_k \sqr{\nabla^2 \vG(\vlambda_k)}^{-1} \nabla \elbofinal(\vlambda_{k}), \label{eq:natural_grad}
\end{align}
where $\vG(\vlambda)$ is the Fisher information-matrix,
\begin{align*}
\vG(\vlambda) := \myexpect_{q(\mathbf{z}|\boldsymbol{\lambda})} \crl{\sqr{\nabla \log q(\vz|\vlambda)}\sqr{\nabla \log q(\vz|\vlambda)}^T} . \nonumber
\end{align*}
<cit.> show that the natural-gradient update can be computationally simpler than gradient descent for conditionally-conjugate exponential family models. In this family, we assume that the distribution of $\vz$ factorizes as $\prod_i p(\vz^i|\pa^i)$ where $\vz^i$ are disjoint subsets of $\vz$ and $\pa^i$ are the parents of the $\vz^i$ in a directed acyclic graph. This family also assumes that each conditional distribution is in the exponential family,
\[p(\vz^i|\pa^i) := h^i(\vz^i) \exp\sqr{ [\veta^i(\pa^i)]^T \vT^i(\vz^i) - A^i(\veta^i)},\]
where $\veta^i$ are the natural parameters, $\vT^i(\vz^i)$ are the sufficient statistics, $A^i(\veta^i)$ is the partition function, and $h^i(\vz^i)$ is the base measure.
<cit.> consider a mean-field approximation $q(\vz|\vlambda) = \prod_i q^i(\vz^i|\vlambda^i)$ where each $q^i$ belongs to the same exponential-family distribution as the joint distribution,
\begin{align*}
q^i(\vz^i) := h^i(\vz^i) \exp\sqr{ (\vlambda^i)^T \vT^i(\vz^i) - A^i(\vlambda^i)} . %\label{eq:approxq1}
\end{align*}
The parameters of this distribution are denoted by $\vlambda^i$ to differentiate them from the joint-distribution parameters $\veta^i$.
As shown by <cit.>, the Fisher matrix for this problem is equal to $\nabla^2 A^i(\vlambda^i)$ and the gradient of the lower bound with respect to $\vlambda^i$ is equal to $\nabla^2 A^i(\vlambda^i) (\vlambda^i - \vlambda^i_*)$ where $\vlambda^i_*$ are the mean-field parameters <cit.>. Therefore, when computing the natural-gradient, the $\nabla^2 A^i(\vlambda^i)$ terms cancel out and the natural-gradient is simply $\vlambda^i - \vlambda^i_*$ which is much easier to compute than the actual gradient. Unfortunately, for non-conjugate models this cancellation does not happen and the simplicity of the update is lost. The Riemannian conjugate-gradient method of <cit.> has similar issues, in that computing $\nabla^2 A(\vlambda)$ is typically very costly.
KL-Divergence Based Methods:
Rather than using the symmetric-KL, <cit.> consider using the KL divergence $\dkls{}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)}$ within a stochastic proximal-point method:
\begin{equation}
\begin{aligned}
\vlambda_{k+1} &= \argmin{\boldsymbol{\lambda}\in\mathcal{S}} \sqr{ - \elbofinal(\vlambda) + \frac{1}{\beta_k}\dkls{}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)} }. \label{eq:theis}
%\vlambda_{k+1} &= \vlambda_k + \delta_k \vI_k^{-1} \bigtriangledown \elbofinal(\vlambda_k), &
\end{aligned}
\end{equation}
This method yields better convergence properties, but requires numerical optimization to implement the update even for conditionally-conjugate models. <cit.> considers a deterministic proximal-gradient variant of this method by splitting the lower bound into $-\elbofinal := f + h$, where $f$ contains all the “easy" terms and $h$ contains all the “difficult" terms. By linearizing the “difficult" terms, this leads to a closed-form update even for non-conjugate models. The update is given by:
\begin{equation}
\begin{aligned}
\vlambda_{k+1} &= \argmin{\boldsymbol{\lambda}\in\mathcal{S}}\left[ \vlambda^T[ \nabla f(\vlambda_k)] + h(\vlambda) \right. \\
&\quad\quad\quad\quad\quad\quad \left. + \frac{1}{\beta_k}\dkls{}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)}\right].\label{eq:khan2015}
%\vlambda_{k+1} &= \vlambda_k + \delta_k \vI_k^{-1} \bigtriangledown \elbofinal(\vlambda_k), &
\end{aligned}
\end{equation}
However, this method requires the exact gradients which is usually not feasible for large dataset and/or complex models.
Mirror Descent Methods:
In the optimization literature, mirror descent (and stochastic mirror descent) algorithms are a generalization of (<ref>) where the squared-Euclidean distance can be replaced by any Bregman divergence $\mathbb{D}_F(\vlambda \| \vlambda_k)$ generated from a strongly-convex function $F(\vlambda)$ <cit.>,
\begin{align}
\vlambda_{k+1} &= \argmin{\boldsymbol{\lambda}\in\mathcal{S}} \left\{-\vlambda^T \nabla \elbofinal(\vlambda_k) + \frac{1}{\beta_k} \mathbb{D}_{F}(\vlambda\|\vlambda_k) \right\}. \label{eq:mirror_descent}
%\vlambda_{k+1} &= \vlambda_k + \delta_k \bigtriangledown \elbofinal(\vlambda_k)
\end{align}
The convergence rate of mirror descent algorithm has been analyzed in convex <cit.> and more recently in non-convex <cit.> settings. However, mirror descent does not cover the cases described above in (<ref>) and (<ref>) when a KL divergence between two exponential-family distributions is used with $\vlambda$ as the natural-parameter. For such cases, the Bregman divergence corresponds to a KL divergence with swapped parameters <cit.>,
\begin{align}
\mathbb{D}_{A}(\vlambda\|\vlambda_k) &:= A(\vlambda) - A(\vlambda_k) - [\bigtriangledown A(\vlambda_k)]^T (\vlambda - \vlambda_k) \nonumber \\
&= \dkls{}{q(\vz|\vlambda_k)}{q(\vz|\vlambda)}. \label{eq:breg}
\end{align}
where $A(\vlambda)$ is the partition function of $q$. Because (<ref>) and (<ref>) both use a KL divergence where the second argument is fixed to $\vlambda_k$, instead of the first argument, they are not covered under the mirror-descent framework.
In addition, even though mirror-descent has been used for variational inference <cit.>,
Bregman divergences do not yield an efficient update in many scenarios.
§ PROXIMAL-GRADIENT SVI
Our proximal-gradient stochastic variational inference (PG-SVI) method extends (<ref>) to allow stochastic gradients $\widehat{\nabla} f(\vlambda_k)$ and general divergence functions $\mathbb{D}(\vlambda\|\vlambda_k)$
by using the iteration
\begin{align}
\vlambda_{k+1} &= \argmin{\boldsymbol{\lambda} \in \mathcal{S}} \crl{ \vlambda^T \sqr{ \widehat{\bigtriangledown} f(\vlambda_k)} + h(\vlambda) + \frac{1}{\beta_k} \mathbb{D}(\vlambda\, \|\, \vlambda_k) }. \label{eq:subproblem}
\end{align}
This unifies a variety of existing approaches since it allows:
* Splitting of $\elbofinal$ into a difficult term $f$ and a simple term $h$, similar to the method of <cit.>.
* A stochastic approximation $\widehat{\nabla} f$ of the gradient of the difficult term, similar to SG methods.
* Divergence functions $\mathbb{D}$ that incorporate the geometry of the parameter space, similar to methods discussed in Section <ref> (see (<ref>), (<ref>), (<ref>), and (<ref>)).
Below, we describe each feature in detail, along with the precise assumptions used in our analysis.
§.§ SPLITTING
Following <cit.>, we split the lower bound into a sum of a “difficult" term $f$ and an “easy" term $h$, enabling a closed-form solution for (<ref>).
Specifically, we split using $p(\vy,\vz)/q(\vz|\vlambda) = c\, \tp_d(\vz|\vlambda) \tp_e(\vz|\vlambda)$, where $\tp_d$ contains all factors that make the optimization difficult, and $\tp_e$ contains the rest (while $c$ is a constant).
By substituting in (<ref>), we get the following split of the lower bound:
\begin{align*}
&\elbofinal(\vlambda) = \underbrace{\myexpect_q[ \log \tilde{p}_d(\vz|\vlambda)]}_{-f(\boldsymbol{\lambda})} + \underbrace{\myexpect_q[ \log \tilde{p}_e(\vz|\vlambda)]}_{-h(\boldsymbol{\lambda})} + \log c.
% \label{eq:split}
\end{align*}
Note that $\tp_d$ and $\tp_e$ need not be probability distributions.
We make the following assumptions about $f$ and $h$:
(A1) The function $f$ is differentiable and its gradient is $L-$Lipschitz-continuous, i.e. $\forall \vlambda$ and $\vlambda' \in \mathcal{S}$ we have
\[
\norm{\nabla f(\vlambda) - \nabla f(\vlambda')} \le L\norm{\vlambda -\vlambda'}.
\]
(A2) The function $h$ can be a general convex function.
These assumptions are very weak. The function $f$ can be non-convex and the Lipschitz-continuity assumption is typically satisfied in practice (and indeed the analysis can be generalized to only require this assumption on a smaller set containing the iterations). The assumption that $h$ is convex seems strong, but note that we can always take $h = 0$ in the split if the function has no “nice" convex part.
Below, we give several illustrative examples of such splits for variational-Gaussian inference with $q(\vz|\vlambda) := \gauss(\vz|\vm,\vV)$, so that $\vlambda = \{\vm,\vV\}$ with $\vm$ being the mean and $\vV$ being the covariance matrix.
Gaussian Process (GP) Models: Consider GP models <cit.> for $N$ input-output pairs $\{y_n,\vx_n\}$ indexed by $n$. Let $z_n := f(\vx_n)$ be the latent function drawn from a GP with mean 0 and covariance $\vK$. We use a non-Gaussian likelihood $p(y_n|z_n)$ to model the output. We can then use the following split, where the non-Gaussian terms are in $\tp_d$ and the Gaussian terms are in $\tp_e$:
\begin{align}
\frac{p(\vy,\vz)}{q(\vz|\vlambda)} = \underbrace{\prod_{n=1}^N p(y_n|z_n)}_{\tp_d(\mathbf{z}|\boldsymbol{\lambda})} \underbrace{\frac{\gauss(\vz|0,\vK)}{\gauss(\vz|\vm,\vV)}}_{\tp_e(\mathbf{z}|\boldsymbol{\lambda})}. \label{eq:glm_joint}
\end{align}
The detailed derivation is in the appendix. By substituting in (<ref>), we obtain the lower bound $\elbofinal(\vlambda)$ shown below along with its split:
\begin{align}
%\underbrace{\myexpect_q \sqr{\sum_{n=1}^N \log p(y_n|z_n)}}_{\tp_d(\mathbf{z}|\boldsymbol{\lambda})} \underbrace{\myexpect_q \sqr{\log \frac{\gauss(\vz|0,\vK)}{\gauss(\vz|\vm,\vV)}}}_{\tp_e(\mathbf{z}|\boldsymbol{\lambda})}. \\
\underbrace{ \sum_n \mathbb{E}_{q}[\log p(y_n|z_n)] }_{-f(\boldsymbol{\lambda})} - \underbrace{ \dkls{}{\gauss(\vz|\vm,\vV)}{\gauss(\vz|0,\vK)}}_{h(\boldsymbol{\lambda})}. \label{eq:gp_lb}%\nonumber
\end{align}
A1 is satisfied for common likelihoods, while it is easy to establish that $h$ is convex. We show in Section <ref> that this split leads to a closed-form update for iteration (<ref>).
Generalized Linear Models (GLMs): A similar split can be obtained for GLMs <cit.>, where the non-conjugate terms are in $\tp_d$ and the rest are in $\tp_e$. Denoting the weights by $\vz$ and assuming a standard Gaussian prior over it, we can use the following split:
\begin{align*}
\frac{p(\vy,\vz)}{q(\vz|\vlambda)} = \underbrace{\prod_{n=1}^N p(y_n|\vx_n^T\vz)}_{\tp_d(\mathbf{z}|\boldsymbol{\lambda})} \underbrace{\frac{\gauss(\vz|0,\vI)}{\gauss(\vz|\vm,\vV)}}_{\tp_e(\mathbf{z}|\boldsymbol{\lambda})}.% \label{eq:glm_joint}
\end{align*}
We give further details about the bound for this case in the appendix.
Correlated Topic Model (CTM): Given a text document with a vocabulary of $N$ words, denote its word-count vector by $\vy$. Let $K$ be the number of topics and $\vz$ be the vector of topic-proportions. We can then use the following split:
\begin{align*}
\frac{p(\vy,\vz)}{q(\vz|\vlambda)} = \underbrace{\prod_{n=1}^N \sqr{\sum_{k=1}^K \beta_{n,k} \frac{e^{z_k}}{\sum_j e^{z_j}}}^{y_{n}} }_{\tp_d(\mathbf{z}|\boldsymbol{\lambda})} \underbrace{\frac{\gauss(\vz|\vmu,\vSigma)}{\gauss(\vz|\vm,\vV)}}_{\tp_e(\mathbf{z}|\boldsymbol{\lambda})},% \label{eq:glm_joint}
\end{align*}
where $\vmu,\vSigma$ are parameters of the Gaussian prior and $\beta_{n,k}$ are parameters of $K$ multinomials.
We give further details about the bound in the appendix.
§.§ STOCHASTIC-APPROXIMATION
The approach of <cit.> considers (<ref>) in the special case of (<ref>) where we use the exact gradient $\nabla f(\vlambda_k)$ in the first term. But in practice this gradient is often difficult to compute. In our framework, we allow a stochastic approximation of $\nabla f(\vlambda)$ which we denote by $\widehat{\nabla} f(\vlambda_k)$.
As shown in the previous section, $f$ might take a form $f(\vlambda) := \Sigma_{n=1}^N \myexpect_{q} [\tilde{f}_n(\vz)]$ for a set of functions $\tilde{f}_n$ as in the GP model (<ref>).
In some situations, $\myexpect_q[\tilde{f}_n(\vz)]$ is computationally expensive or intractable. For example, in GP models the expectation is equal to $\myexpect_q[\log p(y_n|z_n)]$, which is intractable for most non-Gaussian likelihoods. In such cases, we can form a stochastic approximation by using a few samples $\vz^{(s)}$ from $q(\vz|\vlambda)$, as shown below:
\[
\nabla \myexpect_{q} [\tilde{f}_n(\vz)] \approx
\widehat{\vg}(\vlambda, \vxi_n) := \frac 1 S \sum_{s=1}^S \tilde{f}_n(\vz^{(s)})\nabla [\log q(\vz^{(s)} | \vlambda)]
\]
where $\vxi_n$ represents the noise in the stochastic approximation $\widehat{\vg}$ and we use the identity $\nabla q(\vz|\vlambda) = q(\vz|\vlambda) \nabla [\log q(\vz|\vlambda)]$ to derive the expression <cit.>.
We can then form a stochastic-gradient by randomly selecting a mini-batch of $M$ functions $\tilde{f}_{n_i}(\vz)$ and employing the estimate
\begin{align}
\widehat{\nabla} f(\vlambda) = \frac N M \sum_{i=1}^{M} \widehat{\vg}(\vlambda,\vxi_{n_i}). \label{eq:gradient_approx}
\end{align}
In our analysis we make the following two assumptions regarding the stochastic approximation of the gradient:
(A3) The estimate is unbiased: $\myexpect[ \widehat{\vg}(\vlambda,\vxi_{n}) ] = \bigtriangledown f(\vlambda)$.
(A4) Its variance is upper bounded: $\textrm{Var}[\widehat{\vg}(\vlambda,\vxi_{n})] \le \sigma^2$.
In both the assumptions, the expectation is taken with respect to the noise $\vxi_{n}$. The first assumption is true for the stochastic approximations of (<ref>). The second assumption is stronger, but only needs to hold for all $\vlambda_k$ so is almost always satisfied in practice.
§.§ DIVERGENCE FUNCTIONS
To incorporate the geometry of $q$ we incorporate a divergence function $\mathbb{D}$ between $\vlambda$ and $\vlambda_k$. The set of divergence functions need to satisfy two assumptions:
(A5) $\mathbb{D}(\vlambda\, \|\, \vlambda') > 0$, for all $\vlambda \ne \vlambda'$.
(A6) There exist an $\alpha>0$ such that for all $\vlambda, \vlambda'$ generated by (<ref>) we have:
\begin{align}
(\vlambda - \vlambda')^T \nabla_{\boldsymbol{\lambda}} \mathbb{D}(\vlambda\, \|\, \vlambda') \ge \alpha \|\vlambda - \vlambda'\|^2. \label{eq:A4}
\end{align}
The first assumption is reasonable and is satisfied by typical divergence functions like the squared Euclidean distance and variants of the KL divergence. In the next section we show that, whenever the iteration (<ref>) is defined and all $\vlambda_k$ stay within a compact set, the second assumption is satisfied for all divergence functions considered in Section <ref>.
§ SPECIAL CASES
Most methods discussed in Section <ref> are special cases of the proposed iteration (<ref>).
We obtain gradient descent if $h = 0$, $f = -\elbofinal$ , $\widehat{\nabla} f = \nabla f$, and $\mathbb{D}(\vlambda \| \vlambda_k) = (1/2)\|\vlambda - \vlambda_k\|^2$ (in this case A6 is satisfied with $\alpha = 1$). From here, there are three standard generalizations in the optimization literature: SG methods do not require that $\widehat{\nabla} f = \nabla f$, proximal-gradient methods do not require that $h = 0$, and mirror descent allows $\mathbb{D}$ to be a different Bregman divergence generated by a strongly-convex function. Our analysis applies to all these variations on existing optimization algorithms because A1 to A5 are standard assumptions <cit.> and, as we now show, A6 is satisfied for this class of Bregman divergences. In particular, consider the generic Bregman divergence shown in the left side of (<ref>)
for some strongly-convex function $A(\vlambda)$.
By taking the gradient with respect to $\vlambda$ and substituting in (<ref>), we obtain that A6 is equivalent to
\[
(\vlambda -\vlambda_k)^T [ \bigtriangledown A(\vlambda) - \bigtriangledown A(\vlambda_k) ] \ge \alpha \|\vlambda - \vlambda_k\|^2,
\]
which is equivalent to strong-convexity of the function $A(\vlambda)$ <cit.>.
The method of <cit.> corresponds to choosing $h = -\elbofinal$, $f = 0$, and $\mathbb{D}(\vlambda || \vlambda_k) := \dkls{}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)}$ where $q$ is an exponential family distribution with natural parameters $\vlambda$. Since we assume $h$ to be convex, only limited cases of their approach are covered under our framework. The method of <cit.> also uses the KL divergence and focuses on the deterministic case where $\widehat{\nabla} f(\vlambda) = \nabla f(\vlambda)$, but uses the split $-\elbofinal = f + h$ to allow for non-conjugate models.
In both of these models, A6 is satisfied when the Fisher matrix $\bigtriangledown^2 A(\vlambda)$ is positive-definite. This can be shown by using the definition of the KL divergence for exponential families <cit.>:
\begin{equation}
\begin{aligned}
&\dkls{}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)} \\
&\quad := A(\vlambda_k) - A(\vlambda) - [\bigtriangledown A(\vlambda)]^T (\vlambda_k - \vlambda). \label{eq:kl_exp}
\end{aligned}
\end{equation}
Taking the derivative with respect to $\vlambda$ and substituting in (<ref>) with $\vlambda' = \vlambda_k$, we get the condition
\begin{align*}
(\vlambda - \vlambda_k)^T [\bigtriangledown^2 A(\vlambda)] (\vlambda - \vlambda_k) \ge \alpha \|\vlambda - \vlambda_k\|^2,
\end{align*}
which is satisfied when $\bigtriangledown^2 A(\vlambda)$ is positive-definite over a compact set for $\alpha$ equal to its lowest eigenvalue on the set.
Methods based on natural-gradient using iteration (<ref>) (like SVI) correspond to using $h=0$, $f=-\elbofinal$, and the symmetric KL divergence. Assumption A1 to A5 are usually assumed for these methods and, as we show next, A6 is also satisfied. In particular, when $q$ is an exponential family distribution the symmetric KL divergence can be written as the sum of the Bregman divergence shown in (<ref>) and the KL divergence shown in (<ref>),
\begin{align*}
&\dkls{sym}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)} \nonumber\\
&:= \dkls{}{q(\vz|\vlambda_k)}{q(\vz|\vlambda)} + \dkls{}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)} \nonumber\\
&= \mathbb{D}_A(\vlambda\| \vlambda_k) + \dkls{}{q(\vz|\vlambda)}{q(\vz|\vlambda_k)}
\end{align*}
where the first equality follows from the definition of the symmetric KL divergence and the second one follows from (<ref>). Since the two divergences in the sum satisfy A6, the symmetric KL divergence also satisfies the assumption.
§ CONVERGENCE OF PG-SVI
We first analyze the convergence rate of deterministic methods where the gradient is exact, $\widehat{\nabla} f(\vlambda) = \nabla f(\vlambda)$. This yields a simplified result that applies to a wide variety of existing variational methods. Subsequently, we consider the more general case where a stochastic approximation of the gradient is used.
§.§ DETERMINISTIC METHODS
We first establish the convergence under a fixed step-size when using the exact gradient. We use $C_0 = \elbofinal^* - \elbofinal(\vlambda_0)$ as the initial (constant) sub-optimality, and express our result in terms of the quantity
\[
G_k := \frac{1}{\beta} (\vlambda_k -\vlambda_{k+1}),
\]
where $\vlambda_{k+1}$ is computed using (<ref>).
Let A1, A2, A5, and A6 be satisfied. If we run $t$ iterations of (<ref>) with a fixed step-size $\beta_k = \beta = \alpha/L$ for all $k$ and an exact gradient $\nabla f(\vlambda)$, then we have
\begin{equation}
\label{eqtm21}
\begin{split}
& \min_{k \in \{0,1,\dots,t-1\}} \, \norm{G_k}^2 \le \frac{2LC_0}{\alpha^2t}
\end{split}
\end{equation}
We give a proof in the appendix.
Stating the result in terms of $G_k$ may appear to be unconventional, but this quantity is natural measure of first-order optimality. For example, consider the special case of gradient descent where $h = 0$ and $\mathbb{D}(\vlambda,\vlambda_k) = \frac{1}{2}\norm{\vlambda-\vlambda_k}^2$. In this case, $\alpha=1$ and $\beta_k = 1/L$, therefore we have $\norm{G_k} = \norm{\nabla f(\vlambda_k)}$ and Proposition <ref> implies that $\min_{k}\norm{\nabla f(\vlambda_k)}^2$ has a convergence rate of $O(1/t)$. This means that the method converges at a sublinear rate to an approximate stationary point, which would be a global minimum in the special case where $f$ is convex.
In more general settings, the quantity $G_k$ provides a generalized notion of first-order optimality for problems that may be non-smooth or use a non-Euclidean geometry.
Further, if the objective is bounded below ($C_0$ is finite), this result implies that the algorithm converges to such a stationary point and also gives a rate of convergence of $O(1/t)$.
If we use a divergence with $\alpha>1$ then we can use a step-size larger than $1/L$ and the error will decrease faster than gradient-descent.
To our knowledge, this is the first result that formally shows that natural-gradient methods can achieve faster convergence rates. The splitting of the objective into $f$ and $h$ functions is also likely to improve the step-size. Since $L$ only depends on $f$, sometimes it might be possible to reduce the Lipschitz constant by choosing an appropriate split.
We next give a more general result that allows a per-iteration step size.
If we choose the step-sizes $\beta_k$ to be such that $0<\beta_k\le 2\alpha/L$ with $\beta_k< 2\alpha/L$ for at least one $k$, then,
\begin{equation}
\label{eqtm2}
\begin{split}
& \min_{k \in \{0,1\dots t-1\}} \, \norm{G_k}^2 \le \frac{C_0} {\sum_{k=0}^{t-1} \rnd{ \alpha \beta_k - L\beta_k^2/2}}
\end{split}
\end{equation}
We give a proof in the appendix. For gradient-descent, the above result implies that we can use any step-size less than $2/L$, which agrees with the classical step-size choices for gradient and proximal-gradient methods.
§.§ STOCHASTIC METHODS
We now give a bound for the more general case where we use a stochastic approximation of the gradient.
Let A1-A6 be satisfied. If we run $t$ iterations of (<ref>) for a fixed step-size $\beta_k = \alpha_*/L$ (where $0<\gamma<2$ is a scalar) and fixed batch-size $M_k = M$ for all $k$ with a stochastic gradient $\widehat{\nabla} f(\vlambda)$, then we have
\[
\myexpect_{R,\boldsymbol \xi} (\norm{G_R}^2) \leq \sqr{\frac{2LC_0}{\alpha_*^2 t} + \frac{ c\sigma^2}{M\alpha^*}}.
\]
where $c$ is a constant such that $c > 1/(2\alpha)$ and $\alpha_* := \alpha - 1/(2c)$. The expectation is taken with respect to the noise $\vxi := \{\vxi_0,\vxi_1,\ldots,\vxi_{t-1}\}$, and a random variable $R$ which follows the uniform distribution $Prob(R=k) = 1/t, \forall k\in \{0,1,2,\ldots,t-1\}$.
Unlike the bound of Proposition <ref>, this bound depends on the noise variance $\sigma^2$ as well the mini-batch size $M$. In particular, as we would expect, the bound gets tighter as the variance gets smaller and as the size of our mini-batch grows. Notice that the dependence on the variance $\sigma^2$ is also improved if we have a favourable geometry that increases $\alpha^*$. Thus, we can achieve a higher accuracy by either increasing the mini-batch size or improving the geometry.
In the appendix we give a more general result that allows non-constant sequences of step sizes,
although we found that constant step-sizes work better empirically.
Note that while stating the result in terms of a randomized iteration might seem strange, in practice we typically just take the last iteration as the approximate minimizer.
The following theorem generalizes the previous result to the case where the step-size is not necessarily a constant.
If we choose the step-size $\beta_k$ such that $0<\beta_k\le 2\alpha/L$ with $\beta_k< 2\alpha/L$ for at least one $k$, then,
\begin{equation}
\label{eqtm22}
\begin{split}
&\myexpect_{P_R} \rnd{\frac{\| \vlambda_R - \vlambda_{R-1}\|^2}{\beta_R}}
\le \frac{C_0+ \frac{c\sigma^2}{2} \sum_{k=1}^t \frac{\beta_k}{M_k}} {\sum_{k=1}^t \rnd{ \alpha_* \beta_k - L\beta_k^2/2}}.
\end{split}
\end{equation}
where $R\in \{1,2,\ldots,t\}$ is a discrete random variable drawn from the probability mass function
\begin{align}
P_R(k) := Prob(R=k) = \frac{\alpha_*\beta_k - L\beta_k^2/2}{ \sum_{k=1}^t \rnd{\alpha_*\beta_k- L\beta_k^2/2} }, \nonumber
\end{align}
§ CLOSED-FORM UPDATES FOR NON-CONJUGATE MODELS
We now give an example where iteration (<ref>) attains a closed-form solution. We expect such closed-form solution to exist for a large class of problems, including models where $q$ is an exponential-family distribution, but here we focus on the GP model discussed in Section <ref>.
For the GP model, we rewrite the lower bound (<ref>) as
\begin{align}
-\elbofinal(\vm,\vV) := \underbrace{\sum_{n=1}^N f_n(m_n,v_n)}_{f(\boldsymbol{m},\boldsymbol{V})} + \underbrace{\dkls{}{q}{p}}_{h(\boldsymbol{m},\boldsymbol{V})} \label{eq:gp_lb_1}
\end{align}
where we've used $q:=\gauss(\vz|\vm,\vV)$, $p:=\gauss(\vz|0,\vK)$, and $f_n(m_n,v_{n}):= -\mathbb{E}_{q}[\log p(y_n|z_n)]$ with $m_n$ being the entry $n$ of $\vm$ and $v_n$ being the diagonal entry $n$ of $\vV$. We can compute a stochastic approximation of $f$ using (<ref>) by randomly selecting an example $n_k$ (choosing $M=1$) and using a Monte Carlo gradient approximation of $f_{n_k}$. Using this approximation, the linearized term in (<ref>) can be simplified to the following:
\begin{align}
\vlambda^T \sqr{ \widehat{\bigtriangledown} f(\vlambda_k)} \nonumber
%&= \vm^T [ N \nabla_{\mathbf{m}} f_{n_k}(m_{n_k,k}, v_{n_k,k}) ] + \trace [ \vV \{N \nabla_{\mathbf{V}} f_{n_k}(m_{n_k,k}, v_{n_k,k}) \} ] \nonumber \\
&= m_n \underbrace{N [ \nabla_{m_n} f_{n_k}(m_{n_k,k}, v_{n_k,k}) ]}_{:= \alpha_{n_k,k}} \nonumber \\
&\quad + v_{n} \underbrace{N [\nabla_{v_{n}} f_{n_k}(m_{n_k,k}, v_{n_k,k}) ]}_{ := 2\,\gamma_{n_k,k}} \nonumber\\
&= m_n \alpha_{n_k,k} + \half v_{n} \gamma_{n_k,k}
\end{align}
where $m_{n_k,k}$ and $v_{n_k,k}$ denote the value of $m_n$ and $v_n$ in the $k$'th iteration for $n=n_k$.
By using the KL divergence as our divergence function in iteration (<ref>), and by denoting $\gauss(\vz|\vm_k,\vV_k)$ by $q_k$, we can express the two last two terms in (<ref>) as a single KL divergence function as shown below:
\begin{equation*}
\begin{split}
&\vlambda^T \sqr{ \widehat{\bigtriangledown} f(\vlambda_k)} + h(\vlambda) + \frac{1}{\beta_k} \mathbb{D}(\vlambda\|\vlambda_k),\\
&= (m_n \alpha_{n,k} + \half v_{n} \gamma_{n,k}) + \dkls{}{q}{p} + \frac{1}{\beta_k} \dkls{}{q}{q_k}, \\
&= (m_n \alpha_{n,k} + \half v_{n} \gamma_{n,k}) + \frac{1}{1-r_k} \dkls{}{q}{p^{1-r_k} q_k^{r_k}},
\end{split}
\end{equation*}
where $r_k := 1/(1+\beta_k)$.
Comparing this to (<ref>), we see that this objective is similar to that of a GP model with a Gaussian prior[Since $p$ and $q$ are Gaussian, the product is a Gaussian.] $p^{1-r_k} q_k^{r_k}$ and a linear Gaussian-like log-likelihood. Therefore, we can obtain closed-form updates for its minimization.
The updates are shown below and a detailed derivation is given in the appendix.
\begin{align}
&\tvgamma_k = r_k \tvgamma_{k-1} + (1-r_k) \gamma_{n_k,k} \vone_{n_k} , \nonumber \\
&\vm_{k+1} = \vm_k - (1-r_k) (\vI - \vK\vA_k^{-1}) (\vm_k + \alpha_{n_k,k}\boldsymbol{\kappa}_{n_k}) , \nonumber\\
&v_{n_{k+1}, k+1} = \kappa_{n_{k+1},n_{k+1}} - \boldsymbol{\kappa}_{n_{k+1}}^T \vA_k^{-1} \boldsymbol{\kappa}_{n_{k+1}}, \label{eq:effupdate1}
\end{align}
where $\tvgamma_0$ is initialized to a small positive constant to avoid numerical issues, $\vone_{n_k}$ is a vector with all zero entries except $n_k$'th entry which is equal to 1, $\boldsymbol{\kappa}_k$ is $n_k$'th column of $\vK$, and $\vA_k := \vK + [\diag(\tvgamma_k)]^{-1}$.
For iteration $k+1$, we use $m_{n_{k+1},k+1}$ and $v_{n_{k+1}, k+1}$ to compute the gradients $\alpha_{n_{k+1},k+1}$ and $\gamma_{n_{k+1},k+1}$, and run the above updates again. We continue until a convergence criteria is reached.
There are numerous advantages of these updates. First, We do not need to store the full covariance matrix $\vV$.
The updates avoid forming the matrix and only update $\vm$. This works because we only need one diagonal element in each iteration to compute the stochastic gradient $\gamma_{n_k,k}$.
For large $N$ this is a clear advantage since the memory cost is $O(N)$ rather than $O(N^2)$.
Second, computation of the mean vector $\vm$ and a diagonal entry of $\vV$ only require solving two linear equations, as shown in the second and third line of (<ref>). In general, for a mini-batch of size $M$, we need a total of $2 M$ linear equations, which is a lot cheaper than an explicit inversion.
Finally, the linear equations at iteration $k+1$ are very similar to those at iteration $k$, since $\vA_k$ differ only at one entry from $\vA_{k+1}$. Therefore, we can reuse computations from the previous iteration to improve the computational efficiency of the updates.
§ EXPERIMENTAL RESULTS
In this section, we compare our method to many existing approaches such as SGD and four adaptive gradient-methods (ADAGRAD, ADADELTA, RMSprop, ADAM), as well as two variational inference methods for non-conjugate models (the delta method and Laplace method). We show results on Gaussian process classification <cit.> and correlated topic models <cit.>. The code to reproduce these experiments can be found on GitHub.[<https://github.com/emtiyaz/prox-grad-svi>]
§.§ GAUSSIAN PROCESS CLASSIFICATION
The number of examples required for convergence versus the step-size for binary GP classification for differnet methods.
The vertical lines show the step-size above which a method diverges.
Comparison of different stochastic gradient methods for binary classification using GPs.
Each column shows results for a dataset. The top row shows the negative of the lower bound while the bottom row shows the test log-loss. In each plot, the x-axis shows the number of passes made through the data. Markers are shown at 0, 1, 2, 4, 7, and 9 passes through the data.
We first consider binary classification by using a GP model with a Bernoulli-logit likelihood on three datasets: Sonar, Ionosphere, and USPS-3vs5. These datasets can be found at the UCI data repository[<https://archive.ics.uci.edu/ml/datasets.html>] and their details are discussed in <cit.>. For the GP prior, we use the zero mean-function and a squared-exponential covariance function with hyperparameters $\sigma$ and $l$ as defined in <cit.> (see Eq. 33). We set the values of the hyperparameters using cross-validation. For the three datasets, the hyperparameters $(\log l, \log\sigma)$ are set to $(-1,6)$, $(1,2.5)$, and $(2.5,5)$, respectively.
§.§.§ Performance Under a Fixed Step-Size
In our first experiment, we compare the performance under a fixed step-size.
The results also demonstrate the faster convergence of our method compared to gradient-descent methods. We compare the following four algorithms on the Ionosphere dataset: (1) batch gradient-descent (referred to as `GD'), (2) batch proximal-gradient algorithm (referred to as `PG'), (3) batch proximal-gradient algorithm with gradients approximated by using Monte Carlo (referred to as `PG-MC' and using $S=500$ samples), and (4) the proposed proximal-gradient stochastic variational-inference (referred to as `PG-SVI') method where stochastic gradients are obtained using (<ref>) with $M=5$.
Figure <ref> shows the number of examples required for convergence versus the step-size. A lower number implies faster convergence. Convergence is assessed by monitoring the lower bound, and when the change in consecutive iterations do not exceed a certain threshold, we stop the algorithm.
We clearly see that GD requires many more passes through the data, and proximal-gradient methods converge faster than GD. In addition, the upper bound on the step-size for PG is much larger than GD. This implies that PG can potentially take larger steps than the GD method. PG-SVI is surprisingly as fast as PG which shows the advantage of our approach over the approach of <cit.>.
§.§.§ Comparison with Adaptive Gradient Methods
We also compare PG-SVI to SGD and four adaptive methods, namely ADADELTA <cit.>, RMSprop <cit.>, ADAGRAD <cit.>, and ADAM <cit.>. The implementation details of these algorithms are given in the appendix. We compare the value of the lower bound versus number of passes through the data. We also compare the average log-loss on the test data,$- \sum_n \log \hat{p}_n/N_*$, where $\hat{p}_n = p(y_n|\sigma,l,\data_t)$ is the predictive probabilities of the test point $y_n$ given training data $\data_t$ and $N_*$ is the total number of test-pairs. A lower value is better for the log-loss, and a value of 1 is equal to the performance of random coin-flipping.
Figure <ref> summarizes the results. In these plots, lower is better for both objectives and one “pass" means the number of randomly selected examples is equal to the total number of examples.
Our method is much faster to converge than other methods. It always converged within 10 passes through the data while other methods required more than 100 passes.
§.§ CORRELATED TOPIC MODEL
We next show results for correlated topic model on two collections of documents, namely the NIPS and Associated Press (AP) datasets. The NIPS[<https://archive.ics.uci.edu/>] dataset contains 1500 documents from the NIPS conferences held between 1987 and 1999 (a vocabulary-size of 12,419 words and a total of around 1.9M words). The AP[<http://www.cs.columbia.edu/ blei/lda-c/index.html>] collection contains 2,246 documents from the Associated Press (a vocabulary-size of 10,473 words and a total of 436K observed words). We use 50% of the documents for training and 50% for testing.
We compare to the delta method and the Laplace method discussed in <cit.>, and also to the original mean-field (MF) method of <cit.>. For these methods, we use an available implementation.[<https://www.cs.princeton.edu/ chongw/resource.html>] All of these methods approximate the lower bound by using approximations to the expectation of log-sum-exp functions (see the appendix for details). We compare these methods to the two versions of our algorithm which do not use such approximations, but instead use a stochastic gradient as explain in Section <ref>. Specifically, we use the following two versions: one with full covariance (referred to as PG-SVI), and the other with a diagonal covariance (referred to as PG-SVI-MF). For both of these algorithms, we use a fixed step-size of 0.001, and a mini-batch size of 2 documents.
Following <cit.> we compare the held-out log-likelihood which is computed as follows: a new test document $\vy$ is split into two halves $(\vy^1,\vy^2)$, then we compute the approximate posterior $q(\vz)$ to the posterior $p(\vz|\vy^1)$ and use this to compute the held-out log-likelihood for each $y_n \in \vy^2$ using
\begin{align}
\log p(y_n) \approx \log \int_z \sqr{\sum_{k=1}^K \beta_{n,k} \frac{e^{z_k}}{\sum_j e^{z_j}}}^{y_{n}} q(\vz) d\vz
\end{align}
We use a Monte Carlo to this quantity by using a large number of samples from $q$ (unlike <cit.> who approximate it by using the Delta method). We report the average of this quantity over all words in $\vy^2$.
Results on NIPS (left) and AP (right) datasets using correlated topic model with 10 topics. We plot the negative of the average held-out log-likelihood versus time. Markers are shown at iterations after second and fifth passes through the data.
Figure <ref> shows the negative of the average held-out log-likelihood versus time for 10 topics (lower values are better).
We see that methods based on proximal-gradient algorithm converge a little bit faster than the existing methods. More importantly, they achieves better performance. This could be due to the fact that we do not approximate the expectation of the log-sum-exp function, unlike the delta and Laplace method. We obtained similar results when we used a different number of topics.
§ DISCUSSION
This work has made two contributions. First, we proposed a new variational inference method that combines variable splitting, stochastic gradients, and general divergence functions. This method is well-suited for a huge variety of the variational inference problems that arise in practice, and we anticipate that it may improve over state of the art methods in a variety of settings.
Our second contribution is a theoretical analysis of the convergence rate of this general method. Our analysis generalizes existing results for the mirror descent algorithm in optimization, and establishes convergences rates of a variety of existing variational inference methods. Due to its generality we expect that this analysis could be useful to establish convergence rates of other algorithms that we have not thought of, perhaps beyond the variational inference settings we consider in this work. However, an open problem that is also discussed by <cit.> it to esbatlish convergence to an arbitrary accuracy with a fixed batch size.
One issue that we have not satisfactorily resolved is giving a theoretically-justified way to set the step-size in practice; our analysis only indicates that it must be sufficiently small. However, this problem is common in many methods in the literature and our analysis at least suggests the factors that should be taken into account.
Another open issue is the applicability our method to many other latent variable models; in this paper we have shown applications to variational-Gaussian inference, but we expect that our method should result in simple updates for a larger class of latent variable models such as non-conjugate exponential family distribution models. Additional work on these issues will improve usability of our method.
#1𝔼 [#1]
§ EXAMPLES OF SPLITTING FOR VARIATIONAL-GAUSSIAN INFERENCE
We give detailed derivations for the splitting-examples shown in Section 3.1 in the main paper.
As in the main paper, we denote the Gaussian posterior distribution by $q(\vz|\vlambda) := \gauss(\vz|\vm,\vV)$, so that $\vlambda = \{\vm,\vV\}$ with $\vm$ being the mean and $\vV$ being the covariance matrix.
§.§ Gaussian Process (GP) Models
Consider GP models for $N$ input-output pairs $\{y_n,\vx_n\}$ indexed by $n$.
Let $z_n := f(\vx_n)$ be the latent function drawn from a GP with a zero-mean function and a covariance function $\kappa(\vx,\vx')$. We denote the Kernel matrix obtained on the data $\vx_n$ for all $n$ by $\vK$.
We use a non-Gaussian likelihood $p(y_n|z_n)$ to model the output, and assume that each $y_n$ is independently sampled from this likelihood given $\vz$. The joint-distribution over $\vy$ and $\vz$ is shown below:
\begin{align}
p(\vy,\vz) = \prod_{n=1}^N p(y_n|z_n) \gauss(\vz|0,\vK)
\end{align}
The ratio required for the lower bound is shown below, along with the split, where non-Gaussian terms are in $\tp_d$ and Gaussian terms are in $\tp_e$:
\begin{align}
\frac{p(\vy,\vz)}{q(\vz|\vm,\vV)} = \underbrace{\prod_{n=1}^N p(y_n|z_n)}_{\tp_d(\mathbf{z}|\boldsymbol{\lambda})} \underbrace{\frac{\gauss(\vz|0,\vK)}{\gauss(\vz|\vm,\vV)}}_{\tp_e(\mathbf{z}|\boldsymbol{\lambda})}.
\end{align}
By substituting in Eq. 1 of the main paper, we can obtain the lower bound $\elbofinal$ after a few simplifications, as shown below:
\begin{align}
\elbofinal(\vm,\vV) &:= \myexpect_{q(\mathbf{z})} \sqr{ \log \frac{p(\vy,\vz)}{q(\vz|\vm,\vV)} } , \\
&= \myexpect_{q(\mathbf{z})} \sqr{ \sum_{n=1}^N \log p(y_n|z_n)} + \myexpect_{q(\mathbf{z})} \sqr{\log \frac{\gauss(\vz|0,\vK)}{\gauss(\vz|\vm,\vV)} } , \\
&= \underbrace{ \sum_{n=1}^N \mathbb{E}_{q}[\log p(y_n|z_n)] }_{-f(\boldsymbol{\lambda})} - \underbrace{ \dkls{}{\gauss(\vz|\vm,\vV)}{\gauss(\vz|0,\vK)}}_{h(\boldsymbol{\lambda})}. %\nonumber
\end{align}
The assumption A2 is satisfied since the KL divergence is convex in both $\vm$ and $\vV$. This is clear from the expression of the KL divergence:
\begin{align}
D_{KL}\sqr{\gauss(\vz|\vm,\vV) || \gauss(\vz|0,\vK) } &= \half[-\log|\vV\vK^{-1}| + \trace(\vV\vK^{-1}) + \vm^T \vK^{-1} \vm - D]
\end{align}
where $D$ is the dimensionality of $\vz$.
Convexity w.r.t. $\vm$ follows from the fact that the above is quadratic in $\vm$ and $\vK$ is positive semi-definite. Convexity w.r.t. $\vV$ follows due to concavity of $\log|\vV|$ (trace is linear, so does not matter).
Assumption A1 depends on the choice of the likelihood $p(y_n|z_n)$, but is usually satisfied. The simplest example is a Gaussian likelihood for which the function $f$ takes the following form:
\begin{align}
f(\vm,\vV) &= \sum_{n=1}^N \myexpect_q[-\log p(y_n|z_n)] = \sum_{n=1}^N \myexpect_q[-\log \gauss(y_n|z_n,\sigma^2)] \\
&= \sum_{n=1}^N \half \log (2\pi\sigma^2) + \frac{1}{2\sigma^2} \sqr{ (y_n-m_n)^2 + v_n}
\end{align}
where $m_n$ is the $n$'th element of $\vm$ and $v_n$ is the $n$'th diagonal entry of $\vV$. This clearly satisfies A1, since the objective is quadratic in $\vm$ and linear in $\vV$.
Here is an example where A1 is not satisfied: for Poisson likelihood $\log p(y_n|z_n) = \exp[y_nz_n-e^{z_n}] / y_n!$ with rate parameter equal to $e^{z_n}$, the function $f$ takes the following form:
\begin{align}
f(\vm,\vV) &= \sum_{n=1}^N \myexpect_q[-\log p(y_n|z_n)] = \sum_{n=1}^N [- y_nm_n + e^{m_n+v_n/2} + \log (y_n!) ]
\end{align}
whose derivative is not Lipschitz continuous since exponential functions are not Lipschitz.
§.§ Generalized Linear Models (GLMs)
We now describe a split for generalized linear models. We model the output $y_n$ by using an exponential family distribution whose natural-parameter is equal to $\eta_n := \vx_n^T\vz$. Assuming a standard Gaussian prior over $\vz$, the joint distribution can be written as follows:
\begin{align}
p(\vy,\vz) := \prod_{n=1}^N p(y_n|\vx_n^T\vz) \gauss(\vz|0,\vI)
\end{align}
A similar split can be obtained by putting non-conjugate terms $p(y_n|\vx_n^T\vz)$ in $\tp_d$ and the rest in $\tp_e$:
\begin{align*}
\frac{p(\vy,\vz)}{q(\vz|\vlambda)} = \underbrace{\prod_{n=1}^N p(y_n|\vx_n^T\vz)}_{\tp_d(\mathbf{z}|\boldsymbol{\lambda})} \underbrace{\frac{\gauss(\vz|0,\vI)}{\gauss(\vz|\vm,\vV)}}_{\tp_e(\mathbf{z}|\boldsymbol{\lambda})}.% \label{eq:glm_joint}
\end{align*}
The lower bound can be shown to be the following:
\begin{align}
\elbofinal(\vm,\vV) := \underbrace{ \sum_{n=1}^N \mathbb{E}_{q}[\log p(y_n|\vx_n^T \vz)] }_{-f(\boldsymbol{\lambda})} - \underbrace{ \dkls{}{\gauss(\vz|\vm,\vV)}{\gauss(\vz|0,\vI)}}_{h(\boldsymbol{\lambda})}.
\end{align}
which is very similar to the GP case. Therefore, Assumptions A1 and A2 will follow with similar arguments.
§.§ Correlated Topic Model (CTM)
We consider text documents with a vocabulary size $N$.
Let $\vz$ be a length $K$ real-valued vector which follows a Gaussian distribution shown in (<ref>).
Given $\vz$, a topic $t_{n}$ is sampled for the $n$'th word using a multinomial distribution shown in (<ref>).
Probability of observing a word in the vocabulary is then given by (<ref>).
\begin{align}
p(\vz|\vtheta) &= \gauss(\vz|\vmu,\vSigma), \label{eq:ctm1}\\
p(t_{n}=k|\vz) &= \frac{\exp(z_{k})}{\sum_{j=1}^K\exp(z_{j})}, \label{eq:ctm2}\\
p(\textrm{Observing a word v}|t_{n},\vtheta) &= \beta_{v,t_{n}} . \label{eq:ctm3}
\end{align}
Here $\vbeta$ is a $N\times K$ real-valued matrix with non-negative entries and columns that sum to 1.
The parameter set for this model is given by $\vtheta = \{\vmu, \vSigma, \vbeta\}$.
We can marginalize out $t_n$ and obtain the data-likelihood given $\vz$,
\begin{align}
p(\textrm{Observing a word v}|\vz,\vtheta) &= \sum_{k=1}^K p(\textrm{Observing a word v}|t_{n}=k,\vtheta) p(t_{n}=k|\vz) , \\
&= \sum_{k=1}^K \beta_{vk} \frac{e^{z_{k}}}{\sum_{j=1}^K e^{z_{j}}} . \label{eq:CTMlik}
\end{align}
Given that we observe $n$'th word $y_n$ times, we can write the following joint distribution:
\begin{align}
p(\vy,\vz) := \prod_{n=1}^N \sqr{\sum_{k=1}^K \beta_{n,k} \frac{e^{z_k}}{\sum_j e^{z_j}}}^{y_{n}} \gauss(\vz|\vmu,\vSigma)
\end{align}
We can then use the following split:
\begin{align*}
\frac{p(\vy,\vz)}{q(\vz|\vlambda)} = \underbrace{\prod_{n=1}^N \sqr{\sum_{k=1}^K \beta_{n,k} \frac{e^{z_k}}{\sum_j e^{z_j}}}^{y_{n}} }_{\tp_d(\mathbf{z}|\boldsymbol{\lambda})} \underbrace{\frac{\gauss(\vz|\vmu,\vSigma)}{\gauss(\vz|\vm,\vV)}}_{\tp_e(\mathbf{z}|\boldsymbol{\lambda})},% \label{eq:glm_joint}
\end{align*}
where $\vmu,\vSigma$ are parameters of the Gaussian prior and $\beta_{n,k}$ are parameters of $K$ multinomials.
The lower bound is shown below:
\begin{align}
\elbofinal(\vm,\vV) &:= \sum_{n=1}^N y_n \crl{ \mathbb{E}_{q}\sqr{ \log \rnd{\sum_{k=1}^K \beta_{n,k} e^{z_k}} }} - W \myexpect_q \crl{\log \sqr{ \sum_{j=1}^K e^{z_j} }} \nonumber\\
&\quad\quad - \dkls{}{\gauss(\vz|\vm,\vV)}{\gauss(\vz|0,\vI)}.
\end{align}
where $W = \sum_n y_n$ is the total number of words. The top line is the function $[-f(\vlambda)]$ while the bottom line is $[-h(\vlambda)]$.
There are two intractable expectations in $f$, each involving expectation of a log-sum-exp function. Wang and Blei (2013) use the Delta method and Laplace method to approximate these expectations. In contrast, in PG-SVI algorithm, we use Monte Carlo to approximate the gradient of these functions.
§ PROOF OF PROPOSITION 1 AND 2
We first prove the Proposition 2. Proposition 1 is obtained as a special case of it. Our proof technique is borrowed from Ghadimi et. al. (2014). We extend their results to general divergence functions.
We denote the proximal projection at $\vlambda_k$ with gradient $\vg$ and step-size $\beta$ by,
\begin{align}
&\mathcal{P}(\vlambda_k,\vg,\beta) := \frac{1}{\beta} (\vlambda_k -\vlambda_{k+1}), \label{eq:def_prox}\\
&\quad\quad \textrm{ where }
\vlambda_{k+1} = \arg\min_{\boldsymbol{\lambda}\in\mathcal{S}} \,\, \vlambda^T \vg + h(\vlambda) + \frac{1}{\beta} \ddd{}{\vlambda}{\vlambda_k} \label{eq:prox_proj_l1}.
\end{align}
The following lemma gives a bound on the norm of $\mathcal{P}(\vlambda_k,\vg,\beta)$.
The following holds for any $\vlambda_k\in\mathcal{S}$, any real-valued vector $\vg$ and $\beta>0$.
\begin{align}
&\vg^T \mathcal{P}(\vlambda_k, \vg,\beta) \ge \alpha ||\mathcal{P}(\vlambda_k, \vg,\beta)||^2 + \frac{1}{\beta} [h(\vlambda_{k+1}) - h(\vlambda_k)]
\end{align}
By taking the gradient of $\vlambda^T \vg + \frac{1}{\beta} \ddd{}{\vlambda}{\vlambda_k}$ and picking any sub-gradient $\nabla h$ of $h$ at $\vlambda_{k+1}$, the corresponding sub-gradient of the right hand side of (<ref>) is given as follows:
\begin{align}
\vg + \nabla h(\vlambda_{k+1}) + \frac{1}{\beta} \nabla_{\lambda} \ddd{}{\vlambda_{k+1}}{\vlambda_k} .
\end{align}
We use this to derive the optimality condition of (<ref>). For any $\vlambda$, the following holds from the optimality condition:
\begin{align}
(\vlambda - \vlambda_{k+1})^T \Bigg[&\vg + \nabla h(\vlambda_{k+1}) + \frac{1}{\beta} \nabla_{\lambda} \ddd{}{\vlambda_{k+1}}{\vlambda_k} \Bigg] \ge 0 .
\end{align}
Letting $\vlambda = \vlambda_k$,
\begin{align}
(\vlambda_k - \vlambda_{k+1})^T \Bigg[&\vg + \nabla h(\vlambda_{k+1}) + \frac{1}{\beta} \nabla_{\lambda} \ddd{}{\vlambda_{k+1}}{\vlambda_k} \Bigg] \ge 0 ,
\end{align}
which implies,
\begin{align}
\vg^T(\vlambda_k -\vlambda_{k+1}) &\ge \frac{1}{\beta} (\vlambda_{k+1} - \vlambda_k)^T \nabla_{\lambda} \ddd{}{\vlambda_{k+1}}{\vlambda_k} + h(\vlambda_{k+1}) - h(\vlambda_k) , \\
&\ge \frac{\alpha}{\beta} ||\vlambda_{k+1} - \vlambda_k||^2 + h(\vlambda_{k+1}) - h(\vlambda_k) .
\end{align}
The first line follows from Assumption A2 (convexity of $h$), and the second line follows from Assumption A6.
Now, we are ready to prove Proposition 2:
Let $\gt:=\mathcal{P}(\vlambda_{k},\nabla f(\vlambda_k),\beta_{k})$.
Since $\nabla f$ is L-smooth (Assumption A1), for any $k=0,1,\ldots, t-1$ we have,
\begin{align*}
f(\vlambda_{k+1}) &\leq f(\vlambda_{k}) + \left< \nabla f(\vlambda_k), \vlambda_{k+1}-\vlambda_k\right>+ \frac L 2 \norms{\vlambda_{k+1}-\vlambda_{k}}, \\
& = f(\vlambda_{k}) - \beta_k\left< \nabla f(\vlambda_k), \gt \right>+ \frac L 2 \beta_k^2\norms{\gt}, \\
&\leq f(\vlambda_{k}) - \beta_k\alpha\norms{\gt}-[h(\vlambda_{k+1}) - h(\vlambda_k)]+ \frac L 2 \beta_k^2\norms{\gt}.
\end{align*}
The second line follows from the definition of $\mathcal{P}$ and the last line is due to Lemma 1. Rearranging the terms and using $-\elbofinal = f + h$ we get:
\begin{align*}
&-\elbofinal(\vlambda_{k+1}) + \elbofinal(\vlambda_{k}) \leq -[\beta_k\alpha - \frac L 2 \beta_k^2]\norms{\gt} ,\\
\Rightarrow\quad\quad & \elbofinal(\vlambda_{k+1}) -\elbofinal(\vlambda_{k}) \geq [\beta_k\alpha - \frac L 2 \beta_k^2]\norms{\gt} .
\end{align*}
Summing these term for all $k=0,1,\dots t-1$, we get the following:
\begin{align*}
\elbofinal(\vlambda_{t-1}) -\elbofinal(\vlambda_{0}) \geq \sum_{k=0}^{t-1} [\beta_k\alpha - \frac L 2 \beta_k^2]\norms{\gt} .
\end{align*}
By noting that the global maximum of the lower bound always upper bounds any other value, we get $\elbofinal(\vlambda_{*}) -\elbofinal(\vlambda_{0}) \geq \elbofinal(\vlambda_{t-1}) -\elbofinal(\vlambda_{0})$. Using this,
\begin{align*}
&\elbofinal(\vlambda_{*}) -\elbofinal(\vlambda_{0}) \geq \sum_{k=0}^{t-1}[\beta_k\alpha - \frac L 2 \beta_k^2]\norms{\gt} , \\
\Rightarrow\quad\quad &\min_{k=0,1,\dots, t-1 }\norms{\gt} [{\sum_{k=0}^{t-1}[\beta_k\alpha - \frac L 2 \beta_k^2]}] \leq \elbofinal(\vlambda_{*}) -\elbofinal(\vlambda_{0}) .
\end{align*}
Since we assume at least one of $\beta_k < 2\alpha/L$, we can divide by the summation term, to get the following:
\[
\min_{k=0,1,\dots,t-1 }\norms{\gt} \leq \frac {\elbofinal(\vlambda_{*}) -\elbofinal(\vlambda_{1})}{{\sum_{k=0}^{t-1} [\beta_k\alpha - \frac L 2 \beta_k^2]} },\]
which proves Proposition 2.
Proposition 1 can be obtained by simply plugging in $\beta_k = \alpha/L$,
\[
\min_{k=0,1,\dots,t-1 }\norms{\gt} \leq \frac {C_0}{{\sum_{k=0}^{t-1}[\frac{\alpha^2}{L} - \frac{\alpha^2}{2L} ]} } = \frac{2C_0L}{\alpha^2 t} .
\]
§ PROOF OF PROPOSITION 3
We will first prove the following theorem, which gives a similar result to Proposition 2 but for a stochastic gradient $\widehat{\nabla} f$.
If we choose the step-size $\beta_k$ such that $0<\beta_k\le 2\alpha_*/L$ with $\beta_k< 2\alpha_*/L$ for at least one $k$, then,
\begin{equation}
\label{eqtm22}
\begin{split}
&\myexpect_{R,\boldsymbol{\xi}} [\norm{G_R}^2]
\le \frac{C_0+ \frac{c\sigma^2}{2} \sum_{k=0}^{t-1} \frac{\beta_k}{M_k}} {\sum_{k=0}^{t-1} \rnd{ \alpha_* \beta_k - L\beta_k^2/2}}.
\end{split}
\end{equation}
where the expectation is taken over $R\in \{0,1,2,\ldots,t-1\}$ which is a discrete random variable drawn from the probability mass function
\begin{align}
Prob(R=k) = \frac{\alpha_*\beta_k - L\beta_k^2/2}{ \sum_{k=0}^{t-1} \rnd{\alpha_*\beta_k- L\beta_k^2/2} }, \nonumber
\end{align}
and over $\vxi := \{\vxi_1,\vxi_2,\ldots,\vxi_{t-1}\}$ with $\vxi_k$ is the noise in the stochastic approximation $\widehat{\nabla} f$.
Let $\gt:=\mathcal{P}(\vlambda_{k},\widehat{\nabla} f(\lambda_{k}),\beta_{k}), \, \delta_k:=\widehat{\nabla} f(\lambda_{k})- \nabla f(\vlambda_k)$. Since $\nabla f$ is L-smooth, for any $k=0,1,\dots, t$ we have,
\begin{align}
f(\vlambda_{k+1}) &\leq f(\vlambda_{k}) + \left< \nabla f(\vlambda_k), \vlambda_{k+1}-\vlambda_k\right>+ \frac L 2 \norms{\lambda_{k+1}-\vlambda_{k}}\\
& = f(\vlambda_{k}) - \beta_k\left< \nabla f(\vlambda_k), \gt \right>+ \frac L 2 \beta_k^2\norms{\gt}\\
& = f(\vlambda_{k}) - \beta_k\left<\widehat{\nabla} f(\lambda_{k}), \gt \right>+ \frac L 2 \beta_k^2\norms{\gt}+\beta_k\left<\delta_k, \gt \right>
\end{align}
where we have used the definition of $\gt$ and $\delta_k$.
Now using Lemma <ref> on the second term and Cauchy-Schwarz for the last term, we get the following:
\begin{align}
f(\vlambda_{k+1}) &\leq f(\vlambda_{k}) -\left[ \alpha \beta_k\norms{\gt}+h(\vlambda_{k+1})-h(\vlambda_{k}) \right] + \frac L 2 \beta_k^2\norms{\gt}+\beta_k\norm{\delta_k}\norm{\gt}
\end{align}
After rearranging and using Young's inequality $\norm{\delta_k}\norm{\gt} \leq (c/2)\norm{\delta_k}^2+1/(2c) \norm{\gt}^2$ given a constant $c>0$, we get
\begin{align}
-\elbofinal(\vlambda_{k+1}) &\leq -\elbofinal(\vlambda_{k}) - \alpha \beta_k\norms{\gt} + \frac L 2 \beta_k^2\norms{\gt}+ \frac {\beta_k}{ 2c} \norms{\gt}+\frac {\beta_k c} 2 \norms{\delta_k}\\
&= -\elbofinal(\vlambda_{k}) - \left( (\alpha-1/(2c))\beta_k-\frac L 2 \beta_k^2\right)\norms{\gt}+ \frac {c\beta_k} 2 \norms{\delta_k}
\end{align}
Now considering $c > 1/(2\alpha)$, $\alpha_* = \alpha - 1/(2c)$ and $\beta_k \leq \frac {2\alpha_*} {L}$, and summing up both sides for iteration $k=0,1\dots, t-1$, we obtain
\begin{align}
\sum_{k=0}^{t-1} &\left( \alpha_*\beta_k-\frac L 2 \beta_k^2\right)\norms{\gt} \leq \elbofinal^* - \elbofinal(\vlambda_0)+ \sum_{k=0}^{t-1} \frac {c\beta_k} 2 \norms{\delta_k}
\end{align}
Now by taking expectation w.r.t. $\vxi$ on both sides and using the fact that $\myexpect_{\boldsymbol{\xi}}{\norms{\delta_k}} \leq \frac {\sigma^2}{M_k}$ by assumption $A3$ and $A4$, we get
\begin{align}
\sum_{k=0}^{t-1} &\left( \alpha_*\beta_k-\frac L 2 \beta_k^2\right)\myexpect_{\boldsymbol \xi}{\norms{\gt}} \leq C_0 + \frac {c\sigma^2}{2}\sum_{k=0}^{t-1} \frac {\beta_k} {M_k} \label{eq:ex111}
\end{align}
Writing the expectation with respect to $R$ and $\vx$ we get
\begin{align}
\label{eq:36}
\mathbb E_{R,\xi}[\norms{\tilde{g}_{\lambda_k,R}}]= \frac {\sum_{k=0}^{t-1} \left(\alpha_*\beta_k-\frac L 2 \beta_k^2\right)\myexpect_{\xi} {\norms{\gt}}}{\sum_{k=0}^{t-1} \left( \alpha_*\beta_k-\frac L 2 \beta_k^2\right)},
\end{align}
whose numerator is the left side of (<ref>). Dividing (<ref>) by $\sum_{k=0}^t \left(\alpha_*\beta_k-\frac L 2 \beta_k^2\right)$ and using this in (<ref>) we get the result.
By substituting $\beta_k = \alpha_*/L$ and $M_k = M$ in (<ref>),
\begin{align}
\myexpect_{R,\boldsymbol{\vxi}} [\norm{G_R}^2]
&\le \frac{C_0+ \frac{c\sigma^2}{2} \sum_{k=0}^{t-1} \frac{\beta_k}{M_k}} {\sum_{k=0}^{t-1} \rnd{ \alpha_* \beta_k - L\beta_k^2/2}} \\
&= \frac{C_0+ \frac{c\sigma^2\alpha_* t}{2L M}} {\frac{\alpha_*^2 t}{2L} } = \rnd{\frac{2LC_0}{\alpha_*^2 t} + \frac{c\sigma^2}{M\alpha^*}}
\end{align}
The probability distribution for $R$ reduces to a uniform distribution in this case, with the probability of each iteration being $1/t$. This proves Proposition 3.
§ DERIVATION OF CLOSED-FORM UPDATES FOR THE GP MODEL
The PG-SVI iterations $\vlambda_{k+1} = \min_{\lambda\in \mathcal{S}} \vlambda^T \sqr{ \widehat{\nabla} f(\vlambda_k)} + h(\vlambda) + \frac{1}{\beta_k} \mathbb{D}(\vlambda\|\vlambda_k)$ takes the following form for the GP model, as discussed in Section 6 of the main paper:
(_k+1,_k+1) = min_𝐦,𝐕≻0
(m_n α_n_k,k + v_n γ_n_k,k)
+ D_KL(|,) || (|0,)
+ 1/β_k D_KL(|,) || (|_k,_k) .
where $n_k$ is the example selected in $k$'th iteration.
We will now show that its solution can be obtained in closed-form.
§.§ Full Update of $\vV_{k+1}$
We first derive the full update of $\vV_{k+1}$.
The KL divergence between two Gaussian distributions is given as follows:
\begin{align}
D_{KL}\sqr{\gauss(\vz|\vm,\vV) || \gauss(\vz|0,\vK) } &= -\half[\log|\vV\vK^{-1}| - \trace(\vV\vK^{-1}) - \vm^T \vK^{-1} \vm + D]
\end{align}
Using this, we expand the last two terms of (<ref>) to get the following,
\begin{align}
\label{eq:proximal_objective_function}
&- \half \sqr{\log|\vV\vK^{-1}| - \trace(\vV\vK^{-1}) - \vm^T \vK^{-1} \vm + D} \nonumber\\
& - \half\frac{1}{\beta_k}\sqr{\log|\vV\vK^{-1}| - \trace\{\vV\vV_k^{-1}\} - (\vm -\vm_k)^T \vV_k^{-1} (\vm -\vm_k) + D} \nonumber \\
&= -\half \left[ \rnd{1+\frac{1}{\beta_k}} \log|\vV| - \trace\{ \vV(\vK^{-1} + \frac{1}{\beta_k} \vV_k^{-1}) \} - \vm^T \vK^{-1}\vm \nonumber \right. \\
&\quad\quad\quad \left. - \frac{1}{\beta_k} (\vm -\vm_k)^T \vV_k^{-1} (\vm -\vm_k) + \rnd {1+\frac{1}{\beta_k}} \rnd{D - \log|\vK| } \right]
\end{align}
Taking derivative of (<ref>) with respect to $\vV$ at $\vV=\vV_{k+1}$ and setting it to zero, we get the following (here $\vI_n$ is a matrix with all zeros, except the $n$'th diagonal element which is set to 1):
\begin{align}
\Rightarrow\quad& - \rnd{1+ \frac{1}{\beta_k}} \vV_{k+1}^{-1} + \rnd{\vK^{-1} + \frac{1}{\beta_k}\vV_{k}^{-1}} + \gamma_{n_k,k} \vI_{n_k} = 0 \\
\Rightarrow\quad& \vV_{k+1}^{-1} = \frac{1}{1+\beta_k} \vV_k^{-1} + \frac{\beta_k}{1+\beta_k} \rnd{ \vK^{-1} + \gamma_{n_k,k}\vI_{n_k} } \\
\Rightarrow\quad& \vV_{k+1}^{-1} = r_k \vV_k^{-1} + (1-r_k) \rnd{ \vK^{-1} + \gamma_{n_k,k}\vI_{n_k} } \label{eq:full_update_V}
\end{align}
which gives us the update of $\vV_{k+1}$ for $r_k := 1/(1+\beta_k)$.
§.§ Avoiding a full update of $\vV_{k+1}$
A full update will require storing the matrix $\vV_{k+1}$. Fortunately, we can avoid storing the full matrix and still do an exact update. The key point here is to notice that to compute the stochastic gradient in the next iteration we only need one diagonal element of $\vV_{k+1}$ rather than the whole matrix. Specifically, if we sample $n_{k+1}$'th example at the iteration $k+1$, then we need to compute $v_{n_{k+1},k+1}$ which is the $n_{k+1}$'th diagonal element of $\vV_{k+1}$. This can be done by solving one linear equation, as we show in this section.
Specifically, we show that the following updates can be used to compute $v_{n_{k+1},k+1}$:
\begin{align}
v_{n_{k+1},k+1} &= \kappa_{n_{k+1},n_{k+1}} - \boldsymbol{\kappa}_{n_{k+1}}^T \rnd{\vK + [\diag(\tvgamma_k)]^{-1}}^{-1} \boldsymbol{\kappa}_{n_{k+1}}, \label{eq:updateV11}
\end{align}
where $\tvgamma_{k} = r_k \tvgamma_{k-1} + (1-r_k) \gamma_{n_k,k}\vone_{n_k}$ ($\vone_n$ is a vector of all zeros except its $n$'th entry which is equal to 1).
We start the recursion with $\tvgamma_0 = \epsilon$ where $\epsilon$ is a small positive number.
We will now show that $\vV_k$ can be reparameterized in terms of a vector $\tvgamma_k$ which contains accumulated weighted sum of the gradient $\gamma_{n_j,j}$, for all $j\le k$. To show this, we recursively substitute the update of $\vV_{j}$ for $j<k+1$, as shown below (recall that $n_k$ is the example selected at the $k$'th iteration).
The second line is obtained by substituting the full update of $\vV_k$ by using (<ref>). The third line is obtained after a few simplifications. The fourth line is obtained by substituting the update of $\vV_{k-1}$ and a few simplifications.
\begin{align}
& \vV_{k+1}^{-1} = r_k \vV_k^{-1} + (1-r_k) \sqr{ \vK^{-1} + \gamma_{n_k,k}\vI_{n_k} } \\
&= r_k \sqr{r_{k-1} \vV_{k-1}^{-1} + (1-r_{k-1}) \rnd{ \vK^{-1} + \gamma_{n_{k-1},k-1}\vI_{n_{k-1}}} } + (1-r_k) \sqr{ \vK^{-1} + \gamma_{n_k,k}\vI_{n_k} }\\
&= r_k r_{k-1} \vV_{k-1}^{-1} + (1-r_kr_{k-1}) \vK^{-1} + \sqr{ r_k(1-r_{k-1}) \gamma_{n_{k-1},k-1} \vI_{n_{k-1}} + (1-r_k)\gamma_{n_k,k}\vI_{n_k} } \nonumber\\
%&= r_k r_{k-1} \sqr{r_{k-2} \vV_{k-2}^{-1} + (1-r_{k-2}) \rnd{ \vSigma^{-1} + \vX^T \diag(\vgamma_{k-2}) \vX }} \nonumber \\
%&\quad\quad\quad + (1-r_kr_{k-1}) \vSigma^{-1} + \vX^T \sqr{ r_k(1-r_{k-1}) \diag(\vgamma_{k-1}) + (1-r_k) \diag(\vgamma_k)} \vX\\
%&= r_k r_{k-1} r_{k-2} \vV_{k-2}^{-1} + (r_kr_{k-1}-r_k r_{k-1} r_{k-2}) \vSigma^{-1} + (1-r_kr_{k-1}) \vSigma^{-1} \nonumber\\
%& + \vX^T \sqr{ r_k r_{k-1} (1-r_{k-2})\diag(\vgamma_{k-2}) + r_k(1-r_{k-1}) \diag(\vgamma_{k-1}) + (1-r_k) \diag(\vgamma_k)} \vX \\
&= r_k r_{k-1} r_{k-2} \vV_{k-2}^{-1} + (1-r_k r_{k-1} r_{k-2}) \vK^{-1} \nonumber\\
& + \sqr{ r_k r_{k-1} (1-r_{k-2})\gamma_{n_{k-2},k-2}\vI_{n_{k-2}} + r_k(1-r_{k-1}) \gamma_{n_{k-1},k-1}\vI_{n_{k-1}} + (1-r_k) \gamma_{n_{k},k}\vI_{n_{k}}}
\end{align}
This update expresses $\vV_{k+1}$ in terms of $\vV_{k-2}$, $\vK$, and gradients of the data example selected at $k,k-1,$ and $k-2$.
Continuing in this fashion until $k=0$, we can write the update as follows:
\begin{align}
&\vV_{k+1}^{-1} = t_k \vV_0^{-1} + (1 - t_k) \vK^{-1} + [ r_k r_{k-1} \ldots r_3 r_2 (1-r_1) \gamma_{n_1,1} \vI_{n_1} \nonumber \\
&+ r_k r_{k-1} \ldots r_4 r_3 (1-r_2) \gamma_{n_2,2} \vI_{n_2} + r_k r_{k-1} \ldots r_5 r_4 (1-r_3) \gamma_{n_3,3} \vI_{n_2} + \ldots \nonumber\\
& + r_k r_{k-1} (1-r_{k-2})\gamma_{n_{k-2},k-2}\vI_{n_{k-2}} + r_k(1-r_{k-1}) \gamma_{n_{k-1},k-1}\vI_{n_{k-1}} + (1-r_k) \gamma_{n_{k},k}\vI_{n_{k}} ] \label{eq:temp345}
\end{align}
where $t_k$ is the product of $r_k,r_{k-1},\ldots, r_0$. We can write the updates more compactly by defining the accumulation of the gradients $\gamma_{n_j,j}$ for all $j\le k$ by a vector $\tvgamma_k$,
\begin{align}
&\vV_{k+1}^{-1} = t_k \vV_0^{-1} + (1 - t_k) \vK^{-1} + \diag(\tvgamma_k)
\end{align}
The vector $\tvgamma_k$ can be obtained by using a recursion. We illustrate this below, where we have grouped the terms in (<ref>) to show the recursion for $\tvgamma_k$ (here $\vone_{n}$ is a vector with all zero entries except $n$'th entry which is set to 1):
Therefore, $\tvgamma_k$ can be recursively updated as follows:
\begin{align}
\tvgamma_k = r_k \tvgamma_{k-1} + (1-r_k) \gamma_{n_k,k} \vone_{n_k}
\end{align}
with an initialization $\tvgamma_0 = \epsilon$ where $\epsilon$ is a small constant to avoid numerical issues.
If we set $\vV_0 = \vK$, then the formula simplifies to the following:
\begin{align}
\vV_{k+1}^{-1} &= \vK^{-1} + \diag(\tvgamma_k) \label{eq:updateVk}
\end{align}
which is completely specified by $\tvgamma_k$, eliminating the need to compute and store $\vV_{k+1}$.
The $n_{k+1}$'th diagonal element can be obtained by using Matrix Inversion Lemma, which gives us the update (<ref>).
§.§ Update of $\vm$
Taking derivative of (<ref>) with respect to $\vm$ at $\vm=\vm_{k+1}$ and setting it to zero, we get the following (here $\vone_{n}$ is a vector with all zero entries except $n$'th entry which is set to 1):
\begin{align}
\Rightarrow\quad& - \vK^{-1}\vm_{k+1} - \frac{1}{\beta_k} \vV_k^{-1} (\vm_{k+1} - \vm_k) -\alpha_{n_k,k} \vone_{n_k} = 0 \\
\Rightarrow\quad& - \sqr{\vK^{-1} + \frac{1}{\beta_k} \vV_k^{-1}}\vm_{k+1} + \frac{1}{\beta_k} \vV_k^{-1} \vm_k -\alpha_{n_k,k} \vone_{n_k} = 0 \\
\Rightarrow\quad& \vm_{k+1} = \sqr{\vK^{-1} + \frac{1}{\beta_k} \vV_k^{-1}}^{-1} \sqr{ \frac{1}{\beta_k} \vV_k^{-1} \vm_k - \alpha_{n_k,k} \vone_{n_k}} \\
\Rightarrow\quad& \vm_{k+1} = \sqr{(1-r_k)\vK^{-1} + r_k \vV_k^{-1}}^{-1} \sqr{-(1-r_k)\alpha_{n_k,k} \vone_{n_k} + r_k \vV_k^{-1} \vm_k }
\end{align}
where the last step is obtained using the fact that $1/\beta_k = r_k/(1-r_k)$.
We simplify as shown below. The second line is obtained by adding and subtracting $(1-r_k)\vK^{-1}\vm_k$ in the square bracket at the right. In the the third line, we take $\vm_{k}$ out. The fourth line is obtained by plugging in the updates of $\vV_k^{-1} = \vK^{-1} +\diag(\tvgamma_k)$. The fifth line is obtained by using Matrix-Inversion lemma, and the sixth line is obtained by taking $\vK^{-1}$ out of the right-most term.
\begin{align}
&\vm_{k+1} = \sqr{(1-r_k) \vK^{-1} + r_k \vV_k^{-1}}^{-1} \sqr{-(1-r_k)\alpha_{n_k,k} \vone_{n_k} + r_k \vV_k^{-1} \vm_k}\\
&= \sqr{(1-r_k) \vK^{-1} + r_k \vV_k^{-1}}^{-1} \sqr{(1-r_k) \{- \vK^{-1}\vm_k - \alpha_{n_k,k} \vone_{n_k} \} + \{ (1-r_k)\vK^{-1} + r_k \vV_k^{-1}\} \vm_k} \nonumber\\
&= \vm_k + (1-r_k) \sqr{(1-r_k) \vK^{-1} + r_k \vV_k^{-1}}^{-1} \rnd{-\vK^{-1}\vm_k - \alpha_{n_k,k} \vone_{n_k}} \\
&= \vm_k - (1-r_k) \sqr{\vK^{-1} + r_k \diag(\tvgamma_{k-1})}^{-1} \rnd{ \vK^{-1}\vm_k + \alpha_{n_k,k} \vone_{n_k}} \\
&= \vm_k - (1-r_k) \sqr{\vK -\vK \rnd{\vK + \diag(r_k \tvgamma_{k-1})^{-1}}^{-1} \vK} \rnd{ \vK^{-1}\vm_k + \alpha_{n_k,k} \vone_{n_k}} \\
&= \vm_k - (1-r_k) \sqr{\vI - \vK \rnd{\vK + \diag(r_k \tvgamma_{k-1})^{-1}}^{-1}} \rnd{ \vm_k + \alpha_{n_k,k}\boldsymbol{\kappa}_{n_k}} \\
&= \vm_k - (1-r_k) (\vI - \vK\vB_k^{-1}) (\vm_k + \alpha_{n_k,k}\boldsymbol{\kappa}_{n_k})
\end{align}
where $\vB_k := \vK + [\diag(r_k \tvgamma_{k-1})]^{-1}$.
Since $r_k\tvgamma_{k-1}$ and $\tvgamma_k$ differ only slightly (by the new example gradient $\gamma_{n_k}$, we can instead use the following approximate update:
\begin{align}
\vm_{k+1} &= \vm_k - (1-r_k) (\vI - \vK\vA_k^{-1}) (\vm_k + \alpha_{n_k,k}\boldsymbol{\kappa}_{n_k})
\end{align}
where $\vA_k := \vK + [\diag(\tvgamma_{k})]^{-1}$.
§ CLOSED-FORM UPDATES FOR GLMS
We rewrite the lower bound as
\begin{align}
-\elbofinal(\vm,\vV) := \underbrace{\sum_{n=1}^N f_n(\tm_n,\tv_n)}_{f(\boldsymbol{m},\boldsymbol{V})} + \underbrace{\dkls{}{\gauss(\vz|\vm,\vV)}{\gauss(\vz|0,\vI)}}_{h(\vm,\vV)} \label{eq:glm_lb_1}
\end{align}
where $f_n(\tm_n,\tv_{n}):= -\mathbb{E}_{q}[\log p(y_n|\vx_n^T\vz)]$ with $\tm_n := \vx_n^T$ and $\tv_n := \vx_n^T\vV\vx_n$. We can compute a stochastic approximation to the gradient of $f$ by randomly selecting an example $n_k$ (choosing $M=1$) and using a Monte Carlo gradient approximation to the gradient of $f_{n_k}$.
Similar to GP, we define the following as our gradients of function $f_n$:
\begin{align}
\alpha_{n_k,k} := N \nabla_{\tm_{n_k}} f_{n_k} (\tm_{n_k},\tv_{n_k}), \quad
\gamma_{n_k,k} := 2N \nabla_{\tv_{n_k}} f_{n_k} (\tm_{n_k},\tv_{n_k})
\end{align}
The PG-SVI iteration can be written as follows:
(_k+1,_k+1) = min_𝐦,𝐕≻0
(_n α_n_k,k + _n γ_n_k,k)
+ D_KL(|,) || (|0,)
+ 1/β_k D_KL(|,) || (|_k,_k) .
Using a similar derivation to the GP model, we can show that the following updates will give us the solution:
\begin{align}
&\tvgamma_k = r_k \tvgamma_{k-1} + (1-r_k) \gamma_{n_k,k} \vone_{n_k} , \nonumber \\
&\tvm_{k+1} = \tvm_k - (1-r_k) (\vI - \vK\vA_k^{-1}) (\vm_k + \alpha_{n_k,k}\boldsymbol{\kappa}_{n_k}) , \nonumber\\
&\tv_{n_{k+1}, k+1} = \kappa_{n_{k+1},n_{k+1}} - \boldsymbol{\kappa}_{n_{k+1}}^T \vA_k^{-1} \boldsymbol{\kappa}_{n_{k+1}},
\end{align}
where $\vK = \vX\vX^T$ and $\tvm_k := \vX^T\vm$.
§ DESCRIPTION OF THE DATASET FOR BINARY GP CLASSIFICATION
Sonar Ionosphere USPS
# of data points 208 351 1,781
# of features 60 34 256
# of training data points 165 280 884
§ DESCRIPTION OF ALGORITHMS FOR BINARY GP CLASSIFICATION
We give implementation details of all the algorithms used for binary GP- classification experiment.
For all methods, we compute a stochastic estimate of the gradient by using a mini-batch size of 5, 5, and 20 for the three datasets: Sonar, Ionosphere, and USPS-3vs5 respectively.
Similarly, the number of MC samples used are 2000, 500, and 2000.
For GD, SGD, and all the adaptive methods, $\vlambda := \{\vm,\vL\}$ where $\vL$ is the Cholesky factor of $\vV$.
The algorithmic parameters of these methods is given in Table <ref>. Below, we give details of their updates.
For the GD method, we use the following update:
\begin{equation}
\vlambda_{k+1} = \vlambda_k + \alpha \nabla \elbofinal(\vlambda_k),
\end{equation}
where $\alpha$ is a fixed step-size.
For the SGD method, we use a stochastic gradient, instead of the exact gradient:
\begin{equation}
\vlambda_{k+1} = \vlambda_k- \alpha_k \vg_k,
\end{equation}
where $\alpha_k = (k+1)^{-\kappa}$ is the step-size and $\vg_k := -\widehat{\nabla} \elbofinal(\vlambda_k)$.
We use the following updates for ADAGRAD:
\begin{align}
\vs_k &= \vs_{k-1} + \rnd{\vg_k \odot \vg_k}, \\
\vlambda_{k+1} &= \vlambda_k - \alpha_0 \sqr{\frac{1}{\sqrt{\vs_k + \epsilon}}} \odot \vg_k .
\end{align}
where $\alpha_0$ is a fixed step-size and $\epsilon$ is a small constant used to avoid numerical errors.
We use the following update for RMSprop:
\begin{align}
\vs_k &= \rho \vs_{k-1} + (1-\rho) \rnd{\vg_k \odot \vg_k}, \\
\vlambda_{k+1} &= \vlambda_k - \alpha_0 \sqr{\frac{1}{\sqrt{\vs_k + \epsilon}}} \odot \vg_k,
\end{align}
where $\alpha_0$ is a fixed step-size and $\rho$ is the decay factor.
We use the following updates for ADADELTA:
\begin{align}
\vs_k &= \rho \vs_{k-1} + (1-\rho) \rnd{\vg_k \odot \vg_k}, \\
\vlambda_{k+1} &= \vlambda_k - \vg_k^{AD}, \quad \textrm{ where } \vg_k^{AD} = \alpha_0 \rnd{\frac{\sqrt{\vdelta_k + \epsilon}}{\sqrt{\vs_k + \epsilon}}} \odot \vg_k, \\
\vdelta_{k+1} &= \rho \vdelta_k + (1-\rho) \rnd{\vg_k^{AD} \odot \vg_k^{AD}}.
\end{align}
where again $\alpha_0$ is a fixed step-size, and $\rho$ is the decay factor.
Finally, the updates for ADAM are shown below:
\begin{align}
\vmu_k &= \rho_\mu \vmu_{k-1} + (1-\rho_\mu) \vg_k, \\
\vs_k &= \rho_s \vs_{k-1} + (1-\rho_s) \rnd{\vg_k \odot \vg_k}, \\
\vg_{s,k} &= \sqrt{\frac{\vs_k}{1-\rho_s^k}}, \\
\vlambda_{k+1} &= \vlambda_k - \alpha_0 \sqr{\frac{1}{\vg_{s,k} + \epsilon}} \odot \sqr{\frac{\vmu_k}{1-\rho_\mu^k}}.
\end{align}
where $\alpha_0$ is a fixed step-size and $\rho_\mu, \rho_s$ are decay factors.
Algorithmic parameters for Binary GP classification experiment (Figure 2 in the main paper). $N$ is the number of training examples.
Parameter Sonar Ionosphere USPS
$\kappa$ 0.8 0.51 0.6
$\alpha_0 \times N$ 1200 25 800
$\alpha_0$ 4.5 4 8
$\alpha_0$ 0.1 0.04 0.1
$\rho$ 0.9 0.9999 0.9
$\alpha_0$ 1.0 0.1 1.0
$1 - \rho$ $5 \times 10^{-10} $ $10^{-11}$ $10^{-12}$
$\alpha_0$ 0.04 0.25 2.5
$\rho_{\mu}$ 0.9 0.9 0.9
$\rho_{s}$ 0.999 0.999 0.999
$\beta_k \times N$ 0.2 2.0 2.5
|
1511.00181
|
Extrapolation Techniques and Systematic Uncertainties in the NO$\nu$A Muon Neutrino Disappearance Analysis
Louise Suter, for the Collaboration
The NOvA long-baseline neutrino experiment consists of two highly active, finely segmented, liquid scintillator detectors located 14 mrad off Fermilab's NuMI beam axis, with a Near Detector located at Fermilab, and a Far Detector located 810 km from the target at Ash River, MI.
NO$\nu$A released it first preliminary results of the muon neutrino disappearance parameters, measuring $\sin^2(\theta_{23}) =~0.51~\pm~0.10$ and or the normal hierarchy $\Delta~m^2_{32}~=~2.37^{+0.16}_{-0.15}~\times~10^{-3}$ eV$^2$ and for the inverted hierarchy
$\Delta m^2_{32}~ =~-2.40^{+0.14}_{-0.17}~\times~10^{-3}$ eV$^2$.
This talk will present a discussion of the systematic uncertainties and extrapolation methods used
for this first analysis which uses $2.74\times10^{20}$ POT-equivalent collected between July 2013 and March 2015.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
The experiment, a long baseline neutrino oscillation experiment,
consists of two almost completely active, segmented, liquid scintillator detectors. The 0.3 kton Near Detector (ND) is located on site at Fermilab, 105 m underground and 1 km away from Fermilab's NuMI beam production target. The 14 kton Far Detector (FD) is located at Ash River, Minnesota, 810 km away from Fermilab's NuMI neutrino source.
The FD building is covered by a 3 m equivalent mound of barite rock. This provides an overburden of more than ten radiation lengths to reduce background from cosmic rays.
The relative sizes of the detectors are shown diagrammatically in Figure <ref>.
The relative sizes of the Far and Near Detectors. The structure of the detector layers is also shown.
The simulated ratio of the Far to the Near Detector flux for charged current $\nu_\mu$ events as a function of true energy.
The two detectors are located 14.6 mrad off the NuMI beam axis, resulting in a relatively narrow neutrino energy band centered at 2 GeV, where
the $\nu_\mu$($\bar{\nu}_{\mu}) \rightarrow \nu_e(\bar{\nu}_{e})$ oscillation maximum occurs (see Figure <ref>). This narrow band beam results not only in a increased flux of events at 2 GeV events but also in a suppression of the neutral current background which is very important for $\nu_\mu$ to $\nu_e$ measurements.
The neutrino energy relies on the angle between pion decay and neutrino interaction inside the detector. As one goes to an off-axis location the dependence on pion energy becomes flat.
Both detectors are highly segmented tracking calorimeters built in their entirety from low Z (0.18 radiation lengths per layer) and highly reflective (15% TiO2) PVC cells <cit.>. The PVC cells are filled with liquid scintillator consisting of mineral oil infused with 5% pseudocumene.
The detectors are constructed from extrusions consisting of planes of 6 cm $\times$ 4 cm cells, where each cell extends the full width or height of the detector. These PVC extrusions are assembled in alternating layers either vertically or horizontally, as can be seen in Figure <ref>. This orientation of the cells allows for 3D event reconstruction.
In total there are 344,054 cells in the FD and 21,192 cells in the ND.
The scintillation light is collected in every cell by a loop of wavelength shifting fiber and each cell is read out individually, using 32-pixel avalanche photo-diodes (APDs).
The NO$\nu$A FD has been taking data since July 2013, taking advantage of the modular nature of NO$\nu$A. The first data was recorded with the first 1 kton block of the detector, with additional kton blocks being added once they were fully commissioned. This allowed for the detector to take data as it was constructed.
Both NO$\nu$A detectors have been fully constructed and commissioned since August 2014. As the detector volume was changing size as the early data was recorded this information is encoded in POT quoted, hence exposure is giving the POT-14-ton-equivalent.
The first results from the NO$\nu$A experiment were presented in August 2015
and these proceedings will present a discussion of the systematic uncertainties associated
with the analysis of first $2.74\times10^{20}$ POT-equivalent collected for the muon-neutrino disappearance analysis. This data was collected between July 2013 and March 2015.
The methods developed to predict the FD spectrum as extrapolated from the observed ND data will also be presented.
§ NEAR TO FAR DETECTOR EXTRAPOLATION METHOD
The neutrino energy spectrum at the ND is measured close to the neutrino source before neutrino oscillations have occurred. This large statistics data sample is used to validate the Monte Carlo (MC) prediction of the expected beam flux and the simulation of the detector response. All beam intrinsic backgrounds can be measured at the ND to a high precision.
uses the ND energy spectrum to make a prediction of the energy spectrum that will be seen at the FD. This prediction technique is known as extrapolation and reduces the dependence on systematics uncertainties which apply to both detectors.
employs a direct extrapolation technique where the ratio of the FD to ND flux, as determined from MC, was used to predict the expected FD energy spectrum from the measured ND energy spectrum. This extrapolation is performed bin-by-bin in reconstructed energy. To extend the extrapolation technique beyond the prediction of the null oscillation spectrum,
migration matrixes are used to apply corrections to the measured ND reconstructed energy spectra to obtain the MC true energy spectra. This allows for oscillation probability predictions to be applied.
To fully qualify the oscillation parameters that describe the observed FD spectrum we minimize $\chi^2$ between observed FD data best-fit, fitted using the full systematic suite (described in Section <ref>), and the predicted FD spectrum under different oscillation predictions.
The full three-flavor parameterization of neutrino oscillations is used, with the other oscillation parameters and their uncertainties marginalized over. The oscillation parameters included in fit are; $\Delta m^2_{21} = 7.53 \pm 0.18\times 10^5 $eV$^2$; $\sin^2(2\theta_{13}) = 0.086 \pm 0.005$; $\sin^2(2\theta_{12}) = 0.846 \pm 0.021 $; and $\delta_{CP} $ is unconstrained.
As the two detectors are functionally identical this ratio based method allows for reductions in detector-response, object-identification, and energy reconstruction based uncertainties. Slight differences in acceptance, due to the size of the detector, and in the flux lead to not complete cancellation of systematics uncertainties.
The MC flux prediction results in one of the largest single detector systematic uncertainties.
The two detectors see slightly different fluxes so this uncertainty does not cancel completely.
This arrises as the ND sees a line source around 14.6 mrad where as the FD a point source at exactly 14.6 mrad, see Figure <ref>.
The experiment performs a blinded analysis technique where are all tools and algorithms are constructed using simulations or side band regions, only once an analysis is classified, by the collaboration, to be complete is the analysis run over the FD data.
The energy spectrum for $\nu_{\mu}$ charged current events both on-axis (open histogram) and 14.6 mrad off-axis (red histogram), in the NuMI beam. The left spectrum is for the FD and the right is for the ND.
§ SYSTEMATIC UNCERTAINTIES
I will discuss the non-negligible systematic uncertainties associated with the NO$\nu$A muon neutrino disappearance analysis. Multiple other effects where considered, for example the detector response modeling and the attenuation calibration corrections, but will not be discussed here as they were determined to be negligible. A summary of all the non-negligible systematic uncertainties is given in Table <ref>.
a) Uncertainty of Background Rates
The only non-negligible contaminations in the in the selected muon-neutrino sample (backgrounds) for the muon-neutrino disappearance analysis are the neutral current and tau neutrino backgrounds. These contamination rates are estimated from simulation and a 100% uncertainty is taken on them.
The largest background at the NO$\nu$A FD is the
rate of cosmic muons. This rate is determined from minimum-bias data taking outside of the neutrino beam spill.
The statistical uncertainty of this minimum-bias sample is negligible, as along with each beam spill a much larger (35x) minimum-bias sample is recorded. This sample is recorded using the same detector conditions so they can be directly matched.
b) Calibration uncertainty: Absolute Hadronic Energy Scale
The NO$\nu$A experiment uses muons which stop in the detector to provide a standard candle for setting the absolute energy scale. The
uncertainty on this is estimated from maximum difference between the multiple probes of calibration which are available at NO$\nu$A. The observed difference is propagated through the full analysis framework, including the extrapolation and oscillation parameter minimization. The probes available at NO$\nu$A include the Michele electrum spectrum, the $\pi^0$ mass peak and the $dE/dx$ of the muon and the proton. Using this method a 5% percent absolute and a 5% relative calibration uncertainty are determined.
Comparing the the off-track energy measured in NO$\nu$A ND charged current muon-neutrino interactions to the simulation a discrepancy is seen. We define off-track energy to be the sum of all energy associated with the neutrino interaction that is not part of the muon track. This is referred to as Hadronic Energy.
As the NO$\nu$A detectors are located off-axis the location of the neutrino energy peak at this location is known to a high precision. Therefore we use this knowledge to tune the hadronic energy such at the neutrino energy is peaked, as expected, at 2 GeV.
Using the ND data a 21% hadronic energy correction is determined. This correction translates into a 6% correction to the neutrino energy.
We conservatively take a 100% absolute uncertainty on this correction. This is our largest systematic uncertainty.
Combining this correction with the absolute hadronic energy scale we get a 22% total absolute hadronic energy uncertainty.
The distribution of the off-track energy (left) and the reconstructed neutrino energy (right) shown for both the simulated ND events (red) and the recalibrated ND data (blue) after the 21% correction factor is applied.
c) Calibration uncertainty: Relative Hadronic Energy Scale
In addition, we calculate the relative hadronic energy uncertainty due to the different detector acceptances.
As the 21% correction factor is calculated using ND data it may be optimized only for the ND. Due to the smaller size of the ND the acceptance is sculpted as compared to the FD and a higher percentage of the events that pass the selection are quasi-elastic.
This effect is
investigated by allowing the normalization and the energy scale of deep inelastic scattering, resonant and quasi-elastic events (as defined by GENIE <cit.>) to float. A
three parameter simultaneous fit of the muon energy, the off-track energy and normalization is done. The difference between the
one-parameter 21% scaling and this interaction-dependent scaling is used to determine the relative uncertainty.
A 2% relative uncertainty and 1% relative normalization uncertainty are determined. The relative uncertainty is combined with the uncertainty discussed in b) to give a 5% total relative hadronic energy uncertainty. The distribution of the off-track energy and the reconstructed neutrino energy at the NO$\nu$A ND are shown in Figure <ref>.
d) Flux Uncertainties
The NO$\nu$A flux is modeled using FLUKA/FLUGG <cit.>.
For each individual detector the flux uncertainty is large (20% at the 2 GeV peak) and dominated by the hadron production uncertainties. The hadron production uncertainties are
estimated by comparing the NuMI target MC predictions to the the thin-target data from NA49 <cit.>.
The hadron transport uncertainties were also investigated. Uncertainties due to the
NuMI target and horn positions, the horn current and the magnetic field, and the beam spot size and position were
determined to be small compared to hadron production uncertainties and are considered negligible.
The flux uncertainties are highly correlated between the two detectors. For each individual detector the flux uncertainty is large but due to the use of the extrapolation method it is the ratio of the uncertainties that is relevant. As the fluxes are very highly correlated between the two detectors the flux uncertainty is reduced to the percent level. The fraction uncertainty on the NO$\nu$A ND and FD and the ratio is shown in Figure <ref>.
The NO$\nu$A flux uncertainty for the far detector (left), near detector (middle) and the ratio (right).
e) Absolute Normalization
There are two sources of absolute normalization uncertainty on NO$\nu$A. The first arrises from the
an uncertainty in the detector mass which leads to uncertainty in the exposure. The NO$\nu$A detectors are
constructed from PVC cells which are filled with a liquid scintillator. Each cell contains a loop of wavelength shifting fiber. These cells are extruded in sets of 16 which are glued together to make a plane of the detector. A
0.7% normalization uncertainty is taken on the amount of plastic, glue, scintillator and wavelength shifting fiber. This is determined from the uncertainties on these components as built and as compared to what is in the simulation.
The second source of absolute normalization is due to a potential proton-on-target, POT, skew between the two detectors.
As data taking at the ND and FD was over different periods if there had been a POT mis-measurement this could result in a normalization skew. The NuMI beam has been shown to be very stable and a conservative 0.5% proton-on-target normalization uncertainty is taken.
Combining this with the mass uncertainty gives an overall 0.9% normalization uncertainty
f) Neutrino Interaction Modeling
NO$\nu$A uses GENIE to study the uncertainty on cross sections and final state particles exiting the nucleus. The effect of 1 and 2 $\sigma$ variations of the 67 parameters provided in GENIE on the muon-neutrino charged-current energy spectrum was studied.
Of these 67 parameters only 6 were seen to have a noticeable effect. There are;
the axial mass of the charged current quasi-elastic cross section; the axial mass of the neutral current quasi-elastic cross section; the axial mass of the charged current resonant cross section; the axial mass of the neutral current resonant cross section; the
vector mass for the charged current resonant cross section; and the vector mass for the neutral current resonant cross section.
As well as these 6 largest, and an effective parameter that includes the effect of the other 61 parameters added in quadrature, was added as penalty terms in the fit. From this a 10 - 25% uncertainty on neutrino interaction dynamics was determined but again this uncertainty mostly cancels out due to the use of a ratio method.
Systematic Value (1$\sigma$) Best fit ($\sigma$)
Bkg. (neutral current and $\nu_\tau$) 100% 0.06
Absolute Normalization 1.3% 0.0008
Absolute Hadronic energy scale 22% -0.67
Absolute energy scale 1% 0.06
Beam Energy dependent -0.02
(20% at 2 GeV)
Relative Normalization 1.4% -0.03
Relative Hadronic energy scale 5.4% 0.05
GENIE $M_a$ 15-25% -0.18
GENIE $M_v$ 10% -0.06
Summary table of the non-negligible systematics in NO$\nu$A muon neutrino oscillation measurement.
§ RESULTS
NO$\nu$A predicted a event rate of 201 $\nu_\mu$ charge current events at its FD extrapolated from the ND data, in range 0 – 5 GeV. This included a predicted background of $1.4 \pm 0.2$ comic muons determined from minimum-bias data and $2.0 \pm 2.0$ neutral current and $\nu_\tau$ events determined from simulation. An observed FD $\nu_\mu$ charge current rate of 33 events was seen, giving a clear signature of neutrino oscillations.
Using these results the atmospheric neutrino oscillation parameters were measured to be $\sin^2(\theta_{23}) = 0.51 \pm 0.10$ and $\Delta m^2_{32} = 2.37^{+0.16}_{-0.15} \times 10^{-3}$ eV$^2$ for the normal hierarchy and
$\Delta m^2_{32} = -2.40^{+0.14}_{-0.17} \times 10^{-3}$ eV$^2$ for the inverted hierarchy. The energy spectrum of the observed events, along with the best-fit distribution to these events (with and without systematics) is shown in Figure <ref>. The best-fit point along with the 68% and 90% contours in $\sin^2(\theta_{23})-\Delta m^2_{32}$ space for the normal hierarchy is also shown.
The energy spectrum of the observed events, along with the best fit distribution to these events (with and without systematics) (right). The best fit point along with the 68% and 90% contours in $\sin^2(\theta_{23})-\Delta m^2_{32}$ space for the normal hierarchy (left).
§ CONCLUSION
In conclusion, the extrapolation methods and systematic uncertainties associated with the analysis of the first data for the muon-neutrino disappearance analysis at NO$\nu$A have been described. This corresponded to $2.74\times10^{20}$ POT-equivalent collected between July 2013 and March 2015.
This analysis is statistically limited and all systematic uncertainties are dominated by the absolute hadronic energy scale uncertainty.
Fully quantifying the hadronic response will be essential for the next generation of results.
With these results NO$\nu$A has showcased its ability to produce world class physics and to be a leader in precision atmospheric neutrino oscillations measurements. With only 7.6% of the nominal final statistics NO$\nu$A is already competitive with the world limits.
D. S. Ayres et al. [NOvA Collaboration], (2005)
genie C. Andreopoulos et al (2010), Nucl. Instrum. Meth. A614, 87-104.
fluka A. Ferrari, P. R. Sala, A. Fasso, and J. Ranft, (2005), CERN-2005-010.
C. Alt et al. [NA49 Collaboration],
Eur. Phys. J. C 49 897 (2007),
T. Anticic et al. [NA49 Collaboration],
Eur. Phys. J. C 68 1 (2010).
|
1511.00576
|
Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs.
With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs).
Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies.
We prove the following fundamental structural and algorithmic results on GIRGs. (1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor $O(\sqrt{n})$. (2) We establish that GIRGs have clustering coefficients in $\Omega(1)$, (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits.
§ INTRODUCTION
Real-world networks, like social networks or the internet infrastructure, have structural properties that can best be described using geometry.
For instance, in social networks two people are more likely to know each other if they live in the same region and share hobbies, both of which can be encoded as spatial information.
This geometric structure may be responsible for some of the key properties of real-world networks, e.g., an underlying geometry naturally induces a large number of triangles, or large clustering coefficient: Two of one's friends are likely to live in one's region and have similar hobbies, so they are themselves similar and thus likely to know each other.
Almost always, large real-world networks are
scale-free, i.e., their degree distribution follows a power law. Such networks have been studied in detail since the 60s. One of the key findings is the small-world phenomenon, which is the observation that two nodes in a network typically have very small graph-theoretic distance. Therefore, classic mathematical models for real-world networks reproduce these two key findings. But since they have no underlying geometry their clustering coefficient is as small as $n^{-\Omega(1)}$; this holds in particular for preferential attachment graphs <cit.> and Chung-Lu random graphs <cit.> (and their variants
In order to close this gap between the empirically observed clustering coefficient and theoretical models, much of the recent work on models for real-world networks focussed on scale-free random graph models that are equipped with an underlying geometry, such as hyperbolic random graphs <cit.>, spatial preferred attachment <cit.>, and many others <cit.>. The basic properties – scale-freeness, small-world, and large clustering coefficient – have been rigorously established for most of these models.
Beyond the basics, experiments suggest that these models have some very desirable properties.
In particular, hyperbolic random graphs are a promising model, as Boguñá et al. <cit.> computed a (heuristic) maximum likelihood fit of the internet graph into the hyperbolic random graph model and demonstrated its quality by showing that greedy routing in the underlying geometry of the fit finds near-optimal shortest paths.
Further properties that have been studied on hyperbolic random graphs, mostly agreeing with empirical findings on real-world networks, are scale-freeness and clustering coefficient <cit.>, existence of a giant component <cit.>, diameter <cit.>, average distance <cit.>, separators and treewidth <cit.>, bootstrap percolation <cit.>, and clique number <cit.>. Algorithmic aspects include sampling algorithms <cit.> and compression schemes <cit.>.
Our goal is to improve algorithmic and structural results on the promising model of hyperbolic random graphs. However, it turns out to be beneficial to work with a more general model, which we introduce with this paper:
In a geometric inhomogeneous random graph (GIRG), every vertex $v$ comes with a weight $\w v$ (which we assume to follow a power law in this paper) and picks a uniformly random position $\x v$ in the $d$-dimensional torus[We choose a toroidal ground space for the technical simplicity that comes with its symmetry and in order to obtain hyperbolic random graphs as a special case. The results of this paper stay true if $\Space$ is replaced, say, by the $d$-dimensional unitcube $[0,1]^d$.] $\Space$. Two vertices $u,v$ then form an edge independently with probability $p_{uv}$, which is proportional to $\w u \w v$ and inversely proportional to some power of their distance $\|\x u - \x v\|$, see Section <ref> for details[A major difference between hyperbolic random graphs and our generalisation is that we ignore constant factors in the edge probabilities $p_{uv}$. This allows to greatly simplify the edge probability expressions, thus reducing the technical overhead.]. The model can be interpreted as a geometric variant of the classic Chung-Lu random graphs. Recently, with scale-free percolation a related model has been introduced <cit.> where the vertex set is given by the grid $\Z^d$. This model is similar with respect to component structure, clustering, and small-world properties <cit.>, but none of the algorithmic aspects studied in the present paper (sampling, compression, also separators) has been regarded thereon.
The basic connectivity properties of GIRGs follow from more general considerations in <cit.>, where a model of generic augmented Chung-Lu graphs is studied. In particular, with high probability[We say that an event holds with high probability (whp) if it holds with probability $1 - n^{-\omega(1)}$.] GIRGs have a giant component and polylogarithmic diameter, and almost surely doubly-logarithmic average distance. However, general studies such as <cit.> are limited to properties that do not depend on the specific underlying geometry. Very recently, GIRGs turned out to be accesible for studying processes such as bootstrap percolation <cit.> and greedy routing <cit.>.
*Our contribution: Models of complex networks play an important role in algorithm development and network analysis as typically real data is scarce. Regarding applications and simulations, it is crucial that a model can be generated fast in order to produce sufficiently large samples. As our main result, we present a sampling algorithm that generates a random graph from our model in expected linear time. This improves the trivial sampling algorithm by a factor $O(n)$ and the best-known algorithm for hyperbolic random graphs by a substantial factor $O(\sqrt{n})$ <cit.>.
We also prove that the underlying geometry indeed causes GIRGs to have a clustering coefficient in $\Omega(1)$. Moreover, we show that GIRGs have small separators of expected size $n^{1-\Omega(1)}$; this is in agreement with empirical findings on real-world networks <cit.>. We then use the small separators to prove that GIRGs can be efficiently compressed (i.e., they have low entropy); specifically, we show how to store a GIRG using $O(n)$ bits in expectation.
Finally, we show that hyperbolic random graphs are indeed a special case of GIRGs, so that all aforementioned results also hold for hyperbolic random graphs.
*Organization of the paper: We present the details of the model and our results in Section <ref>. In Section <ref> we introduce notation and a geometric ordering of the vertices, and we present some basic properties of the GIRG model. Afterwards, we prove our main result on sampling algorithms in Section <ref>. We analyze the clustering coefficient in Section <ref>, and determine the separator size and the entropy in Section <ref>. Finally in Section <ref> we establish that hyperbolic random graphs are a special case of GIRGs, and in Section <ref> we make some concluding remarks.
§ MODEL AND RESULTS
§.§ Definition of the Model
We prove algorithmic and structural results in a new random graph model which we call geometric inhomogeneous random graphs. In this model, each vertex $v$ comes with a weight $\w v$ and with a random position $\x v$ in a geometric space, and the set of edges $E$ is also random.
We start by defining the by-now classical Chung-Lu model and then describe the
changes that yield our variant with underlying geometry.
*Chung-Lu random graph:
For $n \in \N$ let $\w{} = (\w 1, \ldots, \w n)$ be a sequence of positive weights.
We call $\W := \sum_{v=1}^n \w v$ the total weight. The Chung-Lu random graph $G(n,\w{})$ has
vertex set $V = [n] = \{1,\ldots,n\}$, and two vertices $u \ne v$ are connected by
an edge independently with probability $p_{uv} = \Theta\big(\min\big\{1,
\frac{\w u \w v}{\W}\big\}\big)$ <cit.>. Note that the term
$\min\{1,.\}$ is necessary, as the product $\w u \w v$ may be larger than $\W$.
Classically, the $\Theta$ simply hides a factor 1, but by introducing the
$\Theta$ the model also captures similar random graphs, like the
Norros-Reittu model <cit.>, while important properties stay
asymptotically invariant.
*Geometric inhomogeneous random graph (GIRG):
Note that we obtain a circle by identifying the endpoints of the interval $[0,1]$. Then the distance of $x,y \in [0,1]$ along the circle is $|x-y|_C := \min\{|x-y|,1-|x-y|\}$.
We fix a dimension $d \ge 1$ and use as our ground space the $d$-dimensional torus $\Space=\R^d / \Z^d$, which can be described as the $d$-dimensional cube $[0,1]^d$ where opposite boundaries are identified. As distance function we use the $\infty$-norm on $\Space$, i.e., for $x,y \in \Space$ we define $\|x-y\| := \max_{1 \le i \le d} | x_i - y_i|_C$.
As for Chung-Lu graphs, we consider the vertex set $V = [n]$ and a weight sequence $\w{}$ (in this paper we require the weights to follow a power law with exponent $\beta > 2$, see next paragraph). Additionally, for any vertex $v$ we draw a point $\x v \in \Space$ uniformly and independently at random.
Again we connect vertices $u \ne v$ independently with probability $p_{uv} = p_{uv}(r)$, which now depends not only on the weights $\w u,\w v$ but also on the positions $\x u,\x v$, more precisely, on the distance $r=\|\x u - \x v\|$. We require for some constant $\alpha > 1$ the following edge probability condition:
\begin{equation}\label{eq:puv} p_{uv} = \Theta\bigg( \min\bigg\{ \frac{1}{\|\x u - \x v\|^{\alpha d}} \cdot \Big( \frac{\w u \w v}{\W}\Big)^{\alpha} ,1 \bigg\} \bigg). \tag{EP1}
\end{equation}
We also allow $\alpha = \infty$ and in this case require that
\begin{equation}\label{eq:puv2}
p_{uv} = \begin{cases} \Theta(1), & \text{if } \|\x u - \x v\| \le O\big(\big(\tfrac{\w u \w v}\W\big)^{1/d}\big), \\ 0, & \text{if } \|\x u - \x v\| \ge \Omega\big(\big(\tfrac{\w u \w v}\W\big)^{1/d}\big), \end{cases} \tag{EP2}
\end{equation}
where the constants hidden by $O$ and $\Omega$ do not have to match, i.e., there can be an interval $[c_1 (\tfrac{\w u \w v}\W)^{1/d}, c_2 (\tfrac{\w u \w v}\W)^{1/d}]$ for $\|\x u - \x v\|$ where the behaviour of $p_{uv}$ is arbitrary.
This finishes the definition of GIRGs. The free parameters of the model are $\alpha \in (1,\infty]$, $d \in \N$, the concrete weights $\w{}$ with power-law exponent $\beta > 2$ and average weight $\W/n$, the concrete function $f_{uv}(\x u, \x v)$ replacing the $\Theta$ in $p_{uv}$, and for $\alpha = \infty$ the constants hidden by $O,\Omega$ in the requirement for $p_{uv}$. We will typically hide the constants $\alpha,d,\beta,\W/n$ by $O$-notation.
*Power-law weights:
As is often done for Chung-Lu graphs, we assume throughout this paper that the weights follow a power law: the fraction of vertices with weight at least $w$ is proportional to $w^{1-\beta}$ for some $\beta>2$ (the power-law exponent of $\w{}$). More precisely, we assume that for some $\barw = \barw(n)$ with $n^{\omega(1/\log\log n)}\leq\barw \leq n^{(1-\Omega(1))/(\beta-1)}$, the sequence $\w{}$ satisfies the following conditions:
* the minimum weight is constant, i.e., $\wmin := \min\{\w{v} \mid 1 \le v \le n\} = \Omega(1)$,
* for all $\eta >0$ there exist constants $c_1,c_2>0$ such that
\[
c_1\frac{n}{w^{\beta-1+\eta}} \leq \#\{1 \le v \le n \mid \w{v} \geq w\} \leq c_2\frac{n}{w^{\beta-1-\eta}},
\]
where the first inequality holds for all $\wmin \leq w \leq \barw$ and the second for all $w \geq \wmin$.
We remark that these are standard assumptions for power-law graphs with average degree
In particular, (PL2) implies that the average weight $\W / n$ is $\Theta(1)$.
An example is the widely used weight function $\w v := \delta\cdot
(n/v)^{1/(\beta-1)}$ with parameter $\delta= \Theta(1)$.
*Discussion of the model:
The choice of the ground space $\Space$ is in the spirit of the classic random geometric graphs <cit.>. We prefer the torus to the hyper-cube for technical simplicity, as it yields symmetry. However, one could replace $\Space$ by $[0,1]^d$ or other manifolds like the $d$-dimensional sphere; our results will still hold verbatim.
Moreover, since in fixed dimension all $L_p$-norms on $\Space$ are equivalent and since the edge probabilities $p_{uv}$ have a constant factor slack, our choice of the $L_\infty$-norm is without loss of generality (among all norms).
The model is already motivated since it generalizes the celebrated hyperbolic random graphs (see Section <ref>).
Let us nevertheless discuss why our choice of edge probabilities is natural: The term $\min\{.,1\}$ is necessary, as in the Chung-Lu model, because $p_{uv}$ is a probability. To obtain a geometric model, where adjacent vertices are likely to have small distance, $p_{uv}$ should decrease with increasing distance $\|\x u - \x v\|$, and an inverse polynomial relation seems reasonable. The constraint $\alpha > 1$ is necessary to cancel the growth of the volume of the ball of radius $r$ proportional to $r^d$, so that we expect most neighbors of a vertex to lie close to it. Finally, the factor $\big( \frac{\w u \w v}{\W}\big)^{\alpha}$ ensures that the marginal probability of vertices $u,v$ with weights $\w u,\w v$ forming an edge is
$\Pr[u \sim v] = \Theta\left(\min\left\{\frac{\w u \w v}{\W},1 \right\} \right)$,
as in the Chung-Lu model, and this probability does not change by more than a constant factor if we fix either $\x{u}$ or $\x{v}$. This is why we see our model as a geometric variant of Chung-Lu random graphs.
For a fixed vertex $u \in V$ we can sum up $\Pr[u \sim v \mid \x u]$ over all vertices $v \in V \setminus \{u\}$, and it follows
The main reason why GIRGs are also technically easy is that for any vertex $u$ with fixed position $\x u$ the incident edges $\{u,v\}$ are independent. Details of these basic properties can be found in Section <ref>.
*Sampling the weights: In the definition we assume that the weight sequence $\w{}$ is fixed. However, if we sample the weights independently according to an appropriate power-law distribution with minimum weight $\wmin$ and density
$f(w) \sim w^{-\beta}$,
then for a given $\eta>0$ the sampled weight sequence will follow a power law and fulfils (PL1) and (PL2) with probability $1-n^{-\Omega(1)}$. Hence, a model with sampled weights is almost surely included in our model. For the precise statement, see Lemma <ref>.
§.§ Structural Properties of GIRGs
As discussed in the introduction, reasonable random graph models for real-world networks should reproduce a power-law degree distribution and small graph-theoretical distances between nodes. Before giving a detailed list of our results in Section <ref>, we first want to ensure that GIRGs have these desired structural properties. Indeed, they follow from a more general class of generic augmented Chung-Lu random graphs that have been studied in <cit.>. This framework has weaker assumptions on the underlying geometry than GIRGs. A short comparison reveals that GIRGs are a special case of this general class of random graph models. In the following we list the results of <cit.> transferred to GIRGs. As we are using power-law weights and $\Ex[\deg(v)]=\Theta(\w{v})$ holds for all $v \in V$, it is not surprising that the degree sequence follows a power-law.
Whp the degree sequence of a GIRG follows a power law with exponent $\beta$ and average degree $\Theta(1)$.
The next result determines basic connectivity properties. Note that for $\beta > 3$, they are not well-behaved, in particular since in this case even threshold hyperbolic random graphs do not possess a giant component of linear size <cit.>. Therefore, $2<\beta<3$ is assumed, which is the typical regime of empirical data. For the following theorem, we require the additional assumption $\barw = \omega(n^{1/2})$ in the limit case $\alpha=\infty$.
Let $2<\beta<3$. Then whp the largest component of a GIRG has linear size and diameter $\log^{O(1)} n$, while all other components have size $\log^{O(1)} n$. Moreover, the average distance of vertices in the largest component is $(2 \pm o(1))\frac{\log \log n}{|\log(\beta-2)|}$ in expectation and with probability $1-o(1)$.
We remark that most results of this paper crucially depend on an underlying geometry, and thus do not hold in the general model from <cit.>.
§.§ Results
Sampling algorithms that generate a random graph from a fixed distribution are known for Chung-Lu random graphs and others, running in expected linear time <cit.>.
As our main result, we present such an algorithm for GIRGs. This greatly improves the trivial $O(n^2)$ sampling algorithm (throwing a biased coin for each possible edge), as well as the best previous algorithm for threshold hyperbolic random graphs with expected time $O(n^{3/2})$ <cit.>. It allows to run experiments on much larger graphs than the ones with $\approx 10^4$ vertices in <cit.>.
In addition to our model assumptions, here we assume that the $\Theta$ in our requirement on $p_{uv}$ is sufficiently explicit, i.e., we can compute $p_{uv}$ exactly and we know a constant $c>0$ such that replacing $\Theta$ by $c$ yields an upper bound on $p_{uv}$, see Section <ref> for details.
Sampling a GIRG can be done in expected time $O(n)$.
In social networks, two friends of the same person are likely to also be friends with each other. This property of having many triangles is captured by the clustering coefficient, defined as the probability when choosing a random vertex $v$ and two random neighbors $v_1 \ne v_2$ of $v$ that $v_1$ and $v_2$ are adjacent (if $v$ does not have two neighbors then its contribution to the clustering coefficient is 0). While Chung-Lu random graphs have a very small clustering coefficient of $n^{-\Omega(1)}$, it is easy to show that the clustering coefficient of GIRGs is $\Theta(1)$. This is consistent with empirical data of real-world networks <cit.> and the constant clustering coefficient of hyperbolic random graphs determined in <cit.>.
Whp the clustering coefficient of a GIRG is $\Theta(1)$.
For real-world networks, a key property to analyze is their stability under attacks. It has been empirically observed that many real-world networks have small separators of size $n^{c}$, $c<1$ <cit.>. In contrast, Chung-Lu random graphs are unrealistically stable, since any deletion of $o(n)$ nodes or edges reduces the size of the giant component by at most $o(n)$ <cit.>.
We show that GIRGs agree with the empirical results much better. Specifically, if we cut the ground space $\Space$ into two halves along one of the axes then we roughly split the giant component into two halves, but the number of edges passing this cut is quite small, namely $n^{1-\Omega(1)}$.
Thus, GIRGs are prone to (quite strong) adversarial attacks, just as many real-world networks. Furthermore, their small separators are useful for many algorithms, e.g., the compression scheme of the next paragraph.
Let $2<\beta<3$. Then almost surely it suffices to delete
$$O\left(n^{\max\{2-\alpha, 3-\beta, 1-1/d \}+o(1)}\right)$$
edges of a GIRG to split its giant component into two parts of linear size each.
Since we assume $\alpha > 1$, $\beta > 2$, and $d = \Theta(1)$, the number of deleted edges is indeed $n^{1-\Omega(1)}$. Recently, Bläsius et al. <cit.> proved a better bound of $O(n^{(3-\beta)/2})$ for threshold hyperbolic random graphs which correspond to GIRGs with parameters $d=1$ and $\alpha=\infty$.
The internet graph has empirically been shown to be well compressible, using only 2-3 bits per edge <cit.>. This is not the case for the Chung-Lu model, as its entropy is $\Theta(n \log n)$ <cit.>. We show that GIRGs have linear entropy, as is known for threshold hyperbolic random graphs <cit.>.
We can store a GIRG using $O(n)$ bits in expectation. The resulting data structure allows to query the degree of any vertex and its $i$-th neighbor in time $O(1)$. The compression algorithm runs in time $O(n)$.
*Hyperbolic random graphs:
We establish that hyperbolic random graphs are an example of one-dimensional GIRGs, and that the often studied special case of threshold hyperbolic graphs is obtained by our limit case $\alpha = \infty$. Specifically, we obtain hyperbolic random graphs from GIRGs by setting the dimension $d=1$, the weights to a specific power law, and the $\Theta$ in the edge probability $p_{uv}$ to a specific, complicated function.
For every choice of parameters in the hyperbolic random graph model, there is a choice of parameters in the GIRG model such that the two resulting distributions of graphs coincide.
In particular, all our results on GIRGs hold for hyperbolic random graphs, too.
Moreover, as our proofs are much less technical than typical proofs for hyperbolic random graphs, we suggest to switch from hyperbolic random graphs to GIRGs in future studies.
§ PRELIMINARIES
§.§ Notation
For $w\in \R_{\geq 0}$, we use the notation
$ V_{\geq w} := \{v\in V\; |\; \w{v}\geq w\}$ and $V_{\leq w} := \{v\in V\; |\; \w{v}\leq w\}$,
as well as
\W_{\geq w}:=\sum_{v\in V_{\geq w}} \w{v}$ and $\W_{\leq w}:=\sum_{v\in V_{\geq w}}\w{v}.
For $u,v\in V$ we write $u\sim v$ if $u$ and $v$ are adjacent, and for
$A,B\subseteq V$ we write $A\sim v$ if there exists $u\in A$ such that $u\sim v$, and we write $A \sim B$ if there exists $v \in B$ such that $A\sim v$. For a vertex
$v\in V$, we denote its neighborhood by $\Gamma(v)$, i.e. $\Gamma(v):=\{u\in
V\mid u\sim v\}$. We say that an event holds with high probability (whp) if it holds with probability $1-n^{-\omega(1)}$.
§.§ Cells
We introduce a geometric ordering of the vertices, which we will use both for the sampling and for the compression algorithms. Consider the ground space $\Space$, split it into $2^d$ equal cubes, and repeat this process with each created cube; we call the resulting cubes cells.
Cells are cubes of the form $C = [x_1 2^{-\ell}, (x_1+1) 2^{-\ell}) \times \ldots \times [x_d 2^{-\ell}, (x_d+1)2^{-\ell})$ with $\ell \ge 0$ and $0 \le x_i < 2^\ell$. We represent cell $C$ by the tuple $(\ell,x_1,\ldots,x_d)$.
The volume of $C$ is $\Vol(C) = 2^{-\ell \cdot d}$.
For $0 < x \le 1$ we let $\ceilpowtwo{x}$ be the smallest number larger or equal to $x$ that is realized as the volume of a cell, or in other words $x$ rounded up to a power of $2^d$, $\ceilpowtwo{x} = \min\{2^{- \ell\cdot d} \mid \ell \in \mathbb N_0 \colon 2^{- \ell\cdot d} \ge x \}$.
Note that the cells of a fixed level $\ell$ partition the ground space. We obtain a geometric ordering of these cells by following the recursive construction of cells in a breadth-first-search manner, yielding the following lemma.
There is an enumeration of the cells $C_1,\ldots,C_{2^{\ell d}}$ of level $\ell$ such that for every cell $C$ of level $\ell' < \ell$ the cells of level $\ell$ contained in $C$ form a consecutive block $C_i,\ldots,C_j$ in the enumeration.
We construct the geometric ordering by induction on the level $\ell$. For $\ell=0$ there is only one cell to enumerate, so let $\ell >0$. Given an enumeration $C_1,\ldots,C_{2^{(\ell-1)d}}$ of the cells of level $\ell-1$, we first enumerate all cells of level $\ell$ contained in $C_1$, starting with the cell which is smallest in all $d$ coordinates, and ending with the cell which is largest in all $d$ coordinates. Then we enumerate all cells of level $\ell$ contained in $C_2$ (starting with smallest coordinates, and ending with largest coordinates), and so on. Evidently this gives us a geometric ordering of the cells of level $\ell$.
§.§ Basic Properties of GIRGs
In this section, we list some basic properties about GIRGs which we mentioned already in Section <ref> and which repeatedly occur in our proofs. In particular we consider the expected degree of a vertex and the marginal probability that an edge between two vertices with given weights is present. The proofs of all statements follow from more general considerations and can be found in <cit.>. Let us start by calculating the partial weight sums $\W_{\le w}$ and $\W_{\ge w}$. The values of these sums follow from the assumptions on power-law weights in Section <ref>.
The total weight satisfies $\W=\Theta(n)$. Moreover, for all sufficiently small $\eta > 0$,
* $\W_{\ge w} = O( n w^{2-\beta+\eta})$ for all $w \ge \wmin$,
* $\W_{\ge w} = \Omega( n w^{2-\beta-\eta})$ for all $\wmin \le w \le \barw$,
* $\W_{\le w} = O(n)$ for all $w$, and
* $\W_{\le w} = \Omega(n)$ for all $w=\omega(1)$.
Next we consider the marginal edge probability of two vertices $u$, $v$ with weights $\w{u}$, $\w{v}$. In GIRGs, this probability is essentially the same as in Chung-Lu random graphs. Furthermore, the marginal probability does not change by more than a constant factor if we fix the position $\x{u}$ or $\x{v}$ (but not both!). Moreover, conditioned on a fixed position $\x{v} \in \Space$, all edges $\{u,v\}$ are independently present. This is a central feature of our model.
Fix $u \in [n]$ and $\x u \in \Space$. All edges $\{u,v\}$, $u \ne v$, are independently present with probability
\begin{equation*}
\Pr[u \sim v \mid \x{u}] = \Theta(\Pr[u \sim v]) = \Theta\left(\min\left\{1,\frac{\w{u} \w{v}}{\W} \right\}\right).
\end{equation*}
The following statement shows that the expected degree of a vertex is of the same order as the weight of the vertex, thus we can interpret a given weight sequence $\w{}$ as a sequence of expected degrees.
For any $v \in [n]$ we have $\Ex[\deg(v)]=\Theta(\w{v})$.
As the expected degree of a vertex is roughly the same as its weight, it is no surprise that whp the degree of all vertices with weight sufficiently large is concentrated around the expected value. The following lemma gives a precise statement.
The following properties hold whp for all $v \in [n]$.
* $\deg(v) = O(\w{v} + \log^2 n)$.
* If $\w{v} = \omega(\log^2 n)$, then $\deg(v)= (1+o(1))\Ex[\deg(v)]= \Theta(\w{v})$.
* $\sum_{v \in V_{\ge w}} \deg(v) = \Theta(\W_{\ge w})$ for all $w=\omega(\log^2 n)$.
We conclude this section by proving that if we sample the weights randomly from an appropriate distribution, then almost surely the resulting weights satisfy our conditions on power-law weights. In particular, the following lemma implies that all results of this paper for weights satisfying (PL1) and (PL2) also hold almost surely in a model of sampled weights.
Let $\wmin=\Theta(1)$, let $\varepsilon>0$, and let $F=F_n: \R \rightarrow [0,1]$ be non-decreasing such that $F(z)=0$ for all $z \le \wmin$, and $F(z)=1-\Theta(z^{1-\beta})$ for all $z \in [\wmin,n^{1/(\beta-1-\varepsilon)}]$. Suppose that for every vertex $v \in [n]$, we choose the weight $\w{v}$ independently according to the cumulative probability distribution $F(.)$. Then for all $\eta=\eta(n)=\omega(\log\log n / \log n)$ , with probability $1-n^{-\Omega(\eta)}$, the resulting weight vector $\w{}$ satisfies the power-law conditions (PL1) and (PL2) with $\barw = (n/\log^2 n)^{1/(\beta-1)}$.
§ SAMPLING ALGORITHM
In this section we show that GIRGs can be sampled in expected time $O(n)$. The running time depends exponentially on the fixed dimension $d$. In addition to our model assumptions, in this section we require that (1) edge probabilities $p_{uv}$ can be computed in constant time (given any vertices $u,v$ and positions $\x u, \x v$) and (2) we know an explicit constant $c > 0$ such that if $\alpha < \infty$ we have
$$ p_{uv} \le \min\bigg\{c \frac 1{\|\x u - \x v\|^{\alpha d}} \cdot \Big(\frac{\w u \w v}\W \Big)^\alpha, 1\bigg\}, $$
and if $\alpha = \infty$ we have
$$ p_{uv} \le \begin{cases} 1, & \text{if } \|\x u - \x v\| < c \big(\tfrac{\w u \w v}\W\big)^{1/d}, \\ 0, & \text{otherwise.} \end{cases} $$
Note that existence of $c$ follows from our model assumptions.
In the remainder of this section we introduce building blocks of our algorithm (Section <ref>) and present our algorithm (Section <ref>) and its analysis (Section <ref>). Thereby, we always assume $\alpha<\infty$. In the last part of this chapter (Section <ref>), we show how the sampling algorithm can be adapted to the case $\alpha=\infty$.
§.§ Building Blocks
*Data structures:
Recall the definition of cells from Section <ref>. We first build a basic data structure on a set of points $P$ that allows to access the points in a given cell $C$ (of volume at least $\nu$) in constant time.
Given a set of points $P$ and $0 < \nu \le 1$, in time $O(|P|+1/\nu)$ we can construct a data structure $\mathcal D_\nu(P)$ supporting the following queries in time $O(1)$:
* given a cell $C$ of volume at least $\nu$, return $|C \cap P|$,
* given a cell $C$ of volume at least $\nu$ and a number $k$, return the $k$-th point in $C \cap P$ (in a fixed ordering of $C \cap P$ depending only on $P$ and $\nu$).
Let $\mu = \ceilpowtwo{\nu} = 2^{-\ell \cdot d}$, so that $\nu \le \mu \le O(\nu)$. Following the recursive construction of cells, we can determine a geometric ordering of the cells of volume $\mu$ as in Lemma <ref> in time $O(1/\mu) = O(1/\nu)$; say $C_1,\ldots,C_{1/\mu}$ are the cells of volume $\mu$ in the geometric ordering. We store this ordering by storing a pointer from each cell $C_i = (\ell,x_1,\ldots,x_d)$ to its successor $C_{i+1} = (\ell,x_1',\ldots,x_d')$, which allows to scan the cells $C_1,\ldots,C_{1/\mu}$ in linear time.
For any point $x \in P$, using the floor function we can determine in time $O(1)$ the cell $(\ell,x_1,\ldots,x_d)$ of volume $\mu$ that $x$ belongs to. This allows to determine the numbers $|C_i \cap P|$ for all $i$ in time $O(|P| + 1/\nu)$. We also compute each prefix sum $s_i := \sum_{j < i} |C_j \cap P|$ and store it at cell $C_i = (\ell,x_1,\ldots,x_d)$. Using an array $A[.]$ of size $|P|$, we store (a pointer to) the $k$-th point in $C_i \cap P$ at position $A[s_i + k]$. Note that this preprocessing can be performed in time $O(|P|+1/\nu)$.
A given cell $C$ of volume at least $\nu$ may consist of several cells of volume $\mu$. By Lemma <ref>, these cells form a contiguous subsequence $C_i,C_{i+1},\ldots,C_{j-1},C_j$ of $C_1,\ldots,C_{1/\mu}$, so that the points $C \cap P$ form a contiguous subsequence of $A$. For constant access time, we store for each cell $C$ of volume at least $\nu$ the indices $s_C,e_C$ of the first and last point of $C \cap P$ in $A$. Then $|C \cap P| = e_C - s_C + 1$ and the $k$-th point in $C \cap P$ is stored at $A[s_C+k]$. Thus, both queries can be answered in constant time. Note that the ordering $A[.]$ of the points in $C \cap P$ is a mix of the geometric ordering of cells of volume $\mu$ and the given ordering of $P$ within a cell of volume $\mu$, in particular this ordering indeed only depends on $P$ and $\nu$.
Next we construct a partitioning of $\Space \times \Space$ into products of cells $A_i \times B_i$. This partitioning allows to split the problem of sampling the edges of a GIRG into one problem for each $A_i \times B_i$, which is beneficial, since each product $A_i \times B_i$ has one of two easy types.
For any $A,B \subseteq \Space$ we denote the distance of $A$ and $B$ by $d(A,B) = \inf_{a \in A, b \in B} \|a-b\|$.
Let $0 < \nu \le 1$. In time $O(1/\nu)$ we can construct a set
$\mathcal P_\nu = \{(A_1,B_1),\ldots,(A_s,B_s)\}$ such that
* $A_i,B_i$ are cells with $\Vol(A_i) = \Vol(B_i) \ge \nu$,
* for all $i$, either $d(A_i,B_i) = 0$ and $\Vol(A_i) = \lceil \nu \rceil_{2^d}$ (type I) or $d(A_i,B_i) \ge \Vol(A_i)^{1/d}$ (type II),
* the sets $A_i \times B_i$ partition $\Space \times \Space$,
* $s = O(1/\nu)$.
Note that for cells $A,B$ of equal volume we have $d(A,B) = 0$ if and only if either $A=B$ or (the boundaries of) $A$ and $B$ touch. For a cell $C$ of level $\ell$ we let $\parent(C)$ be its parent, i.e., the unique cell of level $\ell-1$ that $C$ is contained in.
Let $\mu = \ceilpowtwo{\nu}$. We define $\mathcal P_\nu$ as follows. For any pair of cells $(A,B)$ with $\Vol(A) = \Vol(B) \ge \nu$, we add $(A,B)$ to $\mathcal P_\nu$ if either (i) $\Vol(A) = \Vol(B) = \mu$ and $d(A,B) = 0$, or (ii) $d(A,B) > 0$ and $d(\parent(A),\parent(B)) = 0$.
Property (1) follows by definition.
Regarding property (2), the pairs $(A,B)$ added in case (i) are clearly of type I. Observe that two cells $A,B$ of equal volume that are not equal or touching have distance at least the sidelength of $A$, which is $\Vol(A)^{1/d}$. Thus, in case (ii) the lower bound $d(A,B) > 0$ implies $d(A,B) \ge \Vol(A)^{1/d}$, so that $(A,B)$ is of type II.
For property (3), consider $(x,y) \in \Space \times \Space$ and let $A,B$ be the cells of volume $\mu$ containing $x,y$.
Let $A^{(0)} := A$ and $A^{(i)} := \parent(A^{(i-1)})$ for any $i\ge 1$, until $A^{(k)} = \Space$. Similarly, define $B = B^{(0)} \subset \ldots \subset B^{(k)} = \Space$ and note that $\Vol(A^{(i)}) = \Vol(B^{(i)})$. Observe that each set $A^{(i)} \times B^{(i)}$ contains $(x,y)$. Moreover, any set $A' \times B'$, where $A',B'$ are cells with $\Vol(A') = \Vol(B')$ and $(x,y) \in A' \times B'$, is of the form $A^{(i)} \times B^{(i)}$. Thus, to show that $\mathcal P_\nu$ partitions $\Space \times \Space$ we need to show that it contains exactly one of the pairs $(A^{(i)},B^{(i)})$ (for any $x,y$). To show this, we use the monotonicity $d(A^{(i)},B^{(i)}) \ge d(A^{(i+1)},B^{(i+1)})$ and consider two cases. If $d(A,B) = 0$ then we add $(A,B)$ to $\mathcal P_\nu$ in case (i), and we add no further $(A^{(i)},B^{(i)})$, since $d(A^{(i)},B^{(i)}) = 0$ for all $i$. If $d(A,B) > 0$ then since $d(A^{(k)},B^{(k)}) = d(\Space,\Space) = 0$ there is a unique index $0 \le i < k$ with $d(A^{(i)},B^{(i)}) > 0$ and $d(A^{(i+1)},B^{(i+1)}) = 0$. Then we add $(A^{(i)},B^{(i)})$ in case (ii) and no further $(A^{(j)},B^{(j)})$. This proves property (3).
Property (4) follows from the running time bound of $O(1/\nu)$, which we show in the following. Note that we can enumerate all $1/\mu = O(1/\nu)$ cells of volume $\mu$, and all of the at most $3^d = O(1)$ touching cells of the same volume, in time $O(1/\nu)$, proving the running time bound for case (i). Moreover, we can enumerate all $2^{\ell \cdot d}$ cells $C$ in level $\ell$, together with all of the at most $3^d = O(1)$ touching cells $C'$ in the same level. Then we can enumerate all $2^d = O(1)$ cells $A$ that have $C$ as parent as well as all $O(1)$ cells $B$ that have $C'$ as parent. This enumerates (a superset of) all possibilities of case (ii). Summing the running time $O(2^{\ell\cdot d})$ over all levels $\ell$ with volume $2^{-\ell \cdot d} \ge \nu$ yields a total running time of $O(1/\nu)$.
*Weight layers:
We set $w_0 := \wmin$ and $w_i := 2 w_{i-1}$ for $i \ge 1$. This splits the vertex set $V = [n]$ into weight layers $V_i := \{v \in V \mid w_{i-1} \le v < w_i\}$ for $1 \le i \le L$ with $L = O(\log n)$.
We write $V_i^C$ for the restriction of weight layer $V_i$ to cell $C$, $V_i^C := \{v \in V_i \mid \x v \in C\}$.
*Geometric random variates:
For $0 < p \le 1$ we write $\Geo(p)$ for a geometric random variable, taking value $i \ge 1$ with probability $p (1-p)^{i-1}$. $\Geo(p)$ can be sampled in constant time using the simple formula $\big\lceil \frac{\log(R)}{\log(1-p)} \big\rceil$, where $R$ is chosen uniformly at random in $(0,1)$, see <cit.>[To evaluate this formula exactly in time $O(1)$ we need to assume the RealRAM model of computation. However, also on a bounded precision machine like the WordRAM $\Geo(p)$ can be sampled in expected time $O(1)$ <cit.>.].
§.§ The Algorithm
Sampling algorithm for GIRGs in expected time $O(n)$
$E := \emptyset$
sample the positions $\x v$, $v \in V$, and determine the weight layers $V_i$
$1 \le i \le L$
build data structure $\mathcal D_{\nu(i)}(\{\x v \mid v \in V_i\})$ with $\nu(i) := \frac{w_i w_0}\W$
$1 \le i \le j \le L$
construct partitioning $\mathcal P_{\nu(i,j)}$ with $\nu(i,j) := \frac{w_i w_j}\W$
$(A,B) \in \mathcal P_{\nu(i,j)}$ of type I
$u \in V_i^A$ and $v \in V_j^B$ with probability $p_{uv}$ add edge $\{u,v\}$ to $E$
$(A,B) \in \mathcal P_{\nu(i,j)}$ of type II
$\bar p := \min\big\{c \cdot \frac 1{d(A,B)^{\alpha d}} \cdot \big(\frac{w_i w_j}\W\big)^\alpha, 1\big\}$
$r := \Geo(\bar p)$
$r \le |V_i^A| \cdot |V_j^B|$
determine the $r$-th pair $(u,v)$ in $V_i^A\times V_j^B$
with probability $p_{uv} / \bar p$ add edge $\{u,v\}$ to $E$
$r := r + \Geo(\bar p)$
remove all edges with $u>v$ sampled in this iteration
Given the model parameters, our Algorithm <ref> samples the edge set $E$ of a GIRG. To this end, we first sample all vertex positions $\x v$ uniformly at random in $\Space$. Given weights $w_1,\ldots,w_n$ we can determine the weight layers $V_i$ in linear time (we may use counting sort or bucket sort since there are only $L = O(\log n)$ layers). Then we build the data structure from Lemma <ref> for the points in $V_i$ setting $\nu = \nu(i) = \frac{w_i w_0}\W$, i.e., we build $\mathcal D_{\nu(i)}(\{\x v \mid v \in V_i\})$ for each $i$. In the following, for each pair of weight layers $V_i,V_j$ we sample the edges between $V_i$ and $V_j$. To this end, we construct the partitioning $\mathcal P_{\nu(i,j)}$ from Lemma <ref> with $\nu(i,j) = \frac{w_i w_j}\W$. Since $\mathcal P_{\nu(i,j)}$ partitions $\Space \times \Space$, every pair of vertices $u \in V_i, v \in V_j$ satisfies $\x u \in A, \x v \in B$ for exactly one $(A,B) \in \mathcal P_{\nu(i,j)}$. Thus, we can iterate over all $(A,B) \in \mathcal P_{\nu(i,j)}$ and sample the edges between $V_i^A$ and $V_j^B$.
If $(A,B)$ is of type I, then we simply iterate over all vertices $u \in V_i^A$ and $v \in V_j^B$ and add the edge $\{u,v\}$ with probability $p_{uv}$; this is the trivial sampling algorithm. Note that we can efficiently enumerate $V_i^A$ and $V_j^B$ using the data structure $\mathcal D_{\nu(i)}(\{\x v \mid v \in V_i\})$ that we constructed above.
If $(A,B)$ is of type II, then the distance $\|x-y\|$ of any two points $x \in A, y \in B$ satisfies $d(A,B) \le \|x-y\| \le d(A,B) + \Vol(A)^{1/d} + \Vol(B)^{1/d} \le 3 d(A,B)$, by the definition of type II. Thus, $\bar p = \min\big\{c \cdot \frac 1{d(A,B)^{\alpha d}} \cdot \big(\frac{w_i w_j}\W\big)^\alpha, 1\big\}$ is an upper bound on the edge probability $p_{uv}$ for any $u \in V_i^A, v \in V_j^B$, and it is a good upper bound since $d(A,B)$ is within a constant factor of $\|\x u - \x v\|$ and $w_i,w_j$ are within constant factors of $\w u, \w v$. Now we first sample the set of edges $\bar E$ between $V_i^A$ and $V_j^B$ that we would obtain if all edge probabilities were equal to $\bar p$, i.e., any $(u,v) \in V_i^A \times V_j^B$ is in $\bar E$ independently with probability $\bar p$. From this set $\bar E$, we can then generate the set of edges with respect to the true edge probabilities $p_{uv}$ by throwing a coin for each $\{u,v\} \in \bar E$ and letting it survive with probability $p_{uv} / \bar p$. Then in total we choose a pair $(u,v)$ as an edge in $E$ with probability $\bar p \cdot (p_{uv}/\bar p) = p_{uv}$, proving that we sample from the correct distribution. Note that here we used $p_{uv} \le \bar p$. It is left to show how to sample the “approximate” edge set $\bar E$. First note that the data structure $\mathcal D_\nu(\{\x v \mid v \in V_i\})$ defines an ordering on $V_i^A$, and we can determine the $\ell$-th element in this ordering in constant time, similarly for $V_j^B$. Using the lexicographic ordering, we obtain an ordering on $V_i^A\times V_j^B$ for which we can again determine the $\ell$-th element in constant time. In this ordering, the first pair $(u,v) \in V_i^A\times V_j^B$ that is in $\bar E$ is geometrically distributed, according to $\Geo(\bar p)$. Since geometric random variates can be generated in constant time, we can efficiently generate $\bar E$, specifically in time $O(1 + |\bar E|)$.
Finally, the case $i=j$ is special. With the algorithm described above, for any $u,v \in V_i$ we sample whether they form an edge twice, once for $\x u \in A, \x v \in B$ (for some $(A,B) \in \mathcal P_{\nu(i,j)}$) and once for $\x v \in A', \x u \in B'$ (for some $(A',B') \in \mathcal P_{\nu(i,j)}$). To fix this issue, in the case $i=j$ we only accept a sampled edge $(u,v) \in V_i^A \times V_j^B$ if $u < v$; then only one way of sampling edge $\{u,v\}$ remains. This changes the expected running time only by a constant factor.
§.§ Analysis
Correctness of our algorithm follows immediately from the above explanations. In the following we show that Algorithm <ref> runs in expected linear time. This is clear for lines 1-2. For line 3, since building the data structure from Lemma <ref> takes time $O(|P| + 1/\nu)$, it takes total time $\sum_{i=1}^L O\big(|V_i| + \W/(w_i w_0)\big)$. Clearly, the first summand $|V_i|$ sums up to $n$. Using $w_0 = \wmin = \Omega(1)$, $\W = O(n)$, and that $w_i$ grows exponentially with $i$, implying $\sum_i 1/w_i = O(1)$, also the second summand sums up to $O(n)$. For line 5, all invocations in total take time $O\big( \sum_{i,j} \W/(w_i w_j)\big)$, which is $O(n)$, since again $\W = O(n)$ and $\sum_i 1/w_i = O(1)$.
We claim that for any weight layers $V_i,V_j$ the expected running time we spend on any $(A,B) \in \mathcal P_{\nu(i,j)}$ is $O(1 + \Ex[|E^{A,B}_{i,j}|])$, where $E^{A,B}_{i,j}$ is the set of edges in $V_i^A \times V_j^B$. Summing up the first summand $O(1)$ over all $(A,B) \in \mathcal P_{\nu(i,j)}$ sums up to $1/\nu(i,j) = \W/(w_i w_j)$. As we have seen above, this sums up to $O(n)$ over all $i,j$. Summing up the second summand $O(\Ex[|E^{A,B}_{i,j}|])$ over all $(A,B) \in \mathcal P_{\nu(i,j)}$ and weight layers $V_i,V_j$ yields the total expected number of edges $O(\Ex[|E|])$, which is $O(n)$, since the average weight $\W / n = O(1)$ and thus the expected average degree is constant.
It is left to prove the claim that for any weight layers $V_i,V_j$ the expected time spent on $(A,B) \in \mathcal P_{\nu(i,j)}$ is $O(1 + \Ex[|E^{A,B}_{i,j}|])$. If $(A,B)$ is of type I, then any pair of vertices $(u,v) \in V_i^A\times V_j^B$ has probability $\Theta(1)$ to form an edge: Since the volume of $A$ and $B$ is $w_i w_j/\W$, their diameter is $(w_i w_j/\W)^{1/d}$ and we obtain $\|\x u - \x v\| \le (w_i w_j/\W)^{1/d} = O((\w u \w v / \W)^{1/d})$, which yields $p_{uv} = \Theta\big(\min\big\{\big(\frac{\w u \w v}{\|\x u - \x v\|^d \W}\big)^\alpha,1\big\}\big) = \Theta(1)$.
As we spend time $O(1)$ for any $(u,v) \in V_i^A\times V_j^B$, we stay in the desired running time bound $O(\Ex[|E^{A,B}_{i,j}|])$.
If $(A,B)$ is of type II, we first sample edges $\bar E$ with respect to the larger edge probability $\bar p$, and then for each edge $e \in \bar E$ sample whether it belongs to $E$. This takes total time $O(1 + |\bar E|)$. Note that any edge $e \in \bar E$ has constant probability $p_{uv} / \bar p = \Theta(1)$ to survive: It follows from $\w u = \Theta(w_i), \w v = \Theta(w_j)$, and $\|\x u - \x v\| = \Theta(d(A,B))$ that $p_{uv} = \Theta(\bar p)$. Hence, we obtain $\Ex[|\bar E|] = O(\Ex[|E^{A,B}_{i,j}|])$, and the running time $O(1 + |\bar E|)$ is therefore in expectation bounded by $O(1 + \Ex[|E^{A,B}_{i,j}|])$. This finishes the proof of the claim.
§.§ Sampling GIRGs in the Case $\alpha = \infty$
For $\alpha = \infty$, edges only exist between vertices in distance $\|\x u - \x v\| < c (\w u \w v / \W)^{1/d}$. We change Algorithm <ref> by setting $\nu(i,j) = \max\{1,c\}^d \cdot w_i w_j / \W$. Then for any $u \in V_i, v \in V_j$ and $(A,B) \in \mathcal P_{\nu(i,j)}$ of type II we have $d(A,B) \ge \Vol(A)^{1/d} \ge \nu^{1/d} \ge c (\w u \w v / \W)^{1/d}$, so there are no edges between $V_i^A$ and $V_j^B$ for type II. This allows to simplify the algorithm by completely ignoring type II pairs; the rest of the algorithm stays unchanged.
Additionally, we have to slightly change the running time analysis, since it no longer holds that all pairs of vertices $(u,v) \in V_i^A\times V_j^B$ satisfy $p_{uv} = \Theta(1)$. However, a variant of this property still holds: If we only uncovered that $\x u \in A$ and $\x v \in B$, but not yet where exactly in $A,B$ they lie, then the marginal probability of $(u,v)$ forming an edge is $\Theta(1)$, since for any $\eps > 0$ a constant fraction of all pairs of points in $A \times B$ are within distance $\eps (w_{i-1} w_{j-1} / \W)^{1/d}$, guaranteeing edge probability $\Theta(1)$ for sufficiently small $\eps$. This again allows to check all pairs of vertices in $V_i^A\times V_j^B$ whether they form an edge, which yields expected linear running time.
§ CLUSTERING
In this section we give a proof of Theorem <ref>. We start by stating the formal definition of the clustering coefficient.
In a graph $G=(V,E)$ the clustering coefficient of a vertex $v\in V$ is defined as
\[
\cc(v) := \cc_G(v) := \begin{cases} \#\big\{\{u,u'\} \subseteq \Gamma(v) \,\mid\, u \sim u'\big\} \big/ \binom{\deg v}{2}, & \text{if} \deg(v) \geq 2, \\ 0 ,& \text{otherwise,}\end{cases}
\]
and the (mean) clustering coefficient of $G$ is
\[
\cc(G) := \frac{\sum_{v\in V}\cc(v)}{|V|}.
\]
For the proof of Theorem <ref>, we need Le Cam's theorem which allows us to bound the total variation
distance of a binomial distribution to a Poisson distribution with the same
Suppose that $X_1,\ldots,X_n$ are independent Bernoulli random variables such that
$\Pr[X_i=1]=p_i$ for $i\in [n]$, $\lambda_n = \sum_{i\in [n]}p_i$ and
$S_n=\sum_{i\in [n]} X_i$. Then
\[
\sum_{k=0}^\infty\left|\Pr[S_n=k] - \frac{\lambda_n^k e^{-\lambda_n}}{k!}\right| < 2\sum_{i=1}^n p_i^2.
\]
In particular, if $\lambda_n = \Theta(1)$ and $\max_{i \in [n]}p_i = o(1)$, then $\Pr[S_n = k] = \Theta(1)$ for $k= O(1)$.
Furthermore, for showing concentration of the clustering coefficient we need a concentration inequality which bounds large deviations taking into account some bad event $\mathcal{B}$.
We use the following theorem.
Let $X_1,\ldots,X_m$ be independent random variables over $\Omega_1, \ldots, \Omega_m$. Let $X = (X_1,\ldots,X_m)$, $\Omega = \prod_{k=1}^m \Omega_k$ and let $f\colon \Omega \to \mathbb{R}$ be measurable with $0 \le f(\omega) \le M$ for all $\omega \in \Omega$. Let $\mathcal{B} \subseteq \Omega$ such that
for some $c > 0$ and for all $\omega \in \overline{\mathcal{B}}$, $\omega' \in \overline{\mathcal{B}}$ that differ in at most two components we have
\[
|f(\omega)-f(\omega')| \le c.
\]
Then for all $t \ge 2 M \Pr[\mathcal B]$
\[\Pr\big[\vert f(X)-\Ex[f(X)] \vert \ge t \big] \le 2e^{-\frac{t^2}{32mc^2}}+(2\tfrac{mM}{c} + 1)\Pr[\mathcal{B}].\]
Let $V' := V_{\le n^{1/8}}$ and $G'=G[V']$.
We first show that for the subgraph $G'$ we have $\Ex[\cc(G')] = \Omega(1)$.
Let $w_0 = \Theta(1)$ be a weight such that there are linearly many vertices with weight at most $w_0$.
Since $\cc(G') = \frac{1}{|V'|}\sum_{v\in V'} \cc_{G'}(v)=\Theta(\frac{1}{n}\sum_{v\in V'} \cc_{G'}(v))$, it suffices to show that a vertex $v$ of weight at most $w_0$ fulfills $\Ex[\cc_{G'}(v)] = \Omega(1)$.
For this we consider the set $\barV:=V_{\leq w_0}$ of vertices of weight at most $w_0$. Fix such a vertex $v$ at position $\x{v} \in \Space$, and let $U(v)$ be the ball around $\x{v}$ with radius $cn^{-1/d}$ for a sufficiently small constant $c>0$. Clearly the volume of this ball is $\Theta(n^{-1})$. Thus, the expected number of vertices in $\barV$ with position in $U(v)$ is $\Theta(1)$. Consider the event $\mathcal{E} = \mathcal{E}(v)$ that the following three properties hold.
* $v$ has at least two neighbors in $\barV$ with positions in $U(v)$.
* $v$ does not have neighbors in $\barV$ with positions in $\Space\setminus U(v)$.
* $v$ does not have neighbors in $[n]\setminus \barV$.
We claim that $\Pr[\mathcal{E}] = \Theta(1)$. For (i), note that the expected number of vertices in $\barV$ with position in $U(v)$ is $\Theta(1)$. Since the position of every vertex is independent, by Le Cam's theorem (Theorem <ref>) the probability that there are at least two vertices in $\barV$ with position in $U(v)$ is $\Theta(1)$. Moreover, by (<ref>) and (<ref>), for each such vertex the probability to connect to $v$ is $\Theta(1)$ (in the case $\alpha=\infty$ this follows because we have chosen $c$ small enough), so (i) holds with probability $\Theta(1)$. For (ii), for any vertex $u \in \barV\setminus\{v\}$, we can bound
\[
\Pr[v \sim u, \x{v} \in \Space\setminus U(v)] \leq \Pr[v \sim u] = \Theta(1/n).
\]
Hence, by Le Cam's theorem, (ii) holds with probability $\Theta(1)$, and this probability can only increase if we condition on (i). Finally, for every fixed position $x$, (iii) holds independently of (i) or (ii) with probability $\Theta(1)$, again by Le Cam's theorem. This proves the claim that $\Pr[\mathcal{E}] = \Theta(1)$.
Conditioned on $\mathcal{E}$, let $v_1$ and $v_2$ be two random neighbors of $v$. Then $\x{v_1}, \x{v_2} \in U(v)$, and $\w{v_1},\w{v_2} \leq w_0$. By the triangle inequality we obtain $\|\x{v_1}-\x{v_2}\| \le 2c n^{-1/d}$. For $c$ sufficiently small, we deduce from (<ref>) and (<ref>) that $v_1 \sim v_2$ holds with probability $\Theta(1)$. Thus we have shown that $\Ex[\cc_{G'}(v)\mid \mathcal{E}(v)] = \Omega(1)$ for all $v\in \barV$. Since $\Pr[\mathcal{E}(v)] = \Theta(1)$, this proves $\Ex[\cc_{G'}(v)] = \Omega(1)$ for all $v\in \barV$, which implies $\Ex[\cc(G')]=\Omega(1)$.
Next we show that $\cc(G')$ is concentrated around its expected value. We aim to do this via an Azuma-type inequality with error event, as given in Theorem <ref>. Note that in our graph model, we apply two different randomized processes to create the geometric graph. First, for every vertex $v$ we choose $\x{v}$ independently at random. Afterwards, every edge is present with some probability $p_{uv}$. Recall that we can apply the concentration bound only if all random variables are independent, which is not the case so far.
The $n$ random variables $\x{1},\ldots, \x{n}$ define the vertex set and the edge probabilities $p_{uv}$. We introduce a second set of $n-1$ independent random variables. For every $u \in \{2,\ldots,n\}$ we let $Y_u := (Y_u^1, \ldots, Y_u^{u-1})$, where every $Y_u^v$ is independently and uniformly at random from $[0,1]$. Then for $v<u$, we include the edge $\{u,v\}$ in the graph if and only if
\[p_{u v} > Y_u^v.\]
We observe that indeed this implies $\Pr[u \sim v \mid \x{u},\x{v}] = p_{u v}(\x u, \x v)$ as desired. Furthermore, the $2n-1$ random variables $\x{1},\ldots,\x{n},Y_2, \ldots Y_n$ are independent and define a probability space $\Omega$. Then $G$, $G'$ and $\cc(G')$ are all random variables on $\Omega$. Consider the following bad event:
\begin{equation} \label{eq:badevent}
\mathcal{B} := \{\omega \in \Omega: \text{ the maximum degree in }G'(\omega) \text{ is at least }n^{1/4}\}.
\end{equation}
We observe that $\Pr[\mathcal{B}]=n^{-\omega(1)}$, since whp every vertex $v \in V'$ has degree at most $O(\w{v} + \log^2 n) = o(n^{1/4})$ by Lemma <ref>. Let $\omega,\omega' \in \overline{\mathcal{B}}$ such that they differ in at most two coordinates. We observe that changing one coordinate $\x{i}$ or $Y_i$ can influence only the local clustering coefficients of $i$ itself and of the vertices which are neighbors of $i$ either before or after the coordinate change. Unless $\mathcal{B}$ holds, every vertex in $G'$ has degree at most $n^{1/4}$ and therefore every coordinate of the probability space has effect at most $2n^{1/4} / n$ onto $\cc(G')$. Thus, we obtain $|\cc(G'(\omega))-\cc(G'(\omega'))| \le 4n^{-3/4}$. We apply Theorem <ref> with $t=n^{-1/8}$ and $c := 4n^{-3/4}$ and deduce
\[\Pr\left[|\cc(G')-\Ex[\cc(G')]| \ge t \right] \le 2e^{-\Omega(t^2/n^{1-3/2})} + n^{O(1)}\Pr[\mathcal{B}] = n^{-\omega(1)},\]
where we used $\frac{t^2}{n^{1-3/2}}=n^{1/4}$. Hence, we have $\cc(G')=(1+o(1))\Ex[\cc(G')]=\Omega(1)$ whp.
In order to compare $\cc(G)$ with $\cc(G')$, we observe that every additional edge $e=\{u,v\}$ which we add to $G'$ can decrease only $\cc(u)$ and $\cc(v)$, both by at most one. Thus,
\[\cc(G) \ge \frac{|V'|}{n}\cc(G') - \frac{2}{n}\sum_{v \in V \setminus V'} \deg(v).\]
By Lemma <ref>,
$\frac{2}{n}\sum_{v \in V \setminus V'} \deg(v)=\Theta(n^{-1}\W_{> n^{1/8}})=o(1)$
whp. Together with $|V'|=\Theta(n)$, this concludes the argument and proves that $\cc(G)=\Omega(1)$ whp.
§ STABILITY OF THE GIANT, ENTROPY, AND COMPRESSION ALGORITHM
In this section we prove Theorems <ref> and <ref>. More precisely, we show that whp the graph (and its giant) has separators of sublinear size, and we make use of these small separators to devise a compression algorithm that can store the graph using a linear number of bits in expectation. Note that the compression maintains only the graph up to isomorphism, not the underlying geometry. The main idea is to enumerate the vertices in an ordering that reflects the geometry, and then storing for each vertex $i$ the differences $i-j$ for all neighbors $j$ of $i$. We start with a technical lemma that gives the number of edges intersecting an axis-parallel, regular grid. (For $\gamma > 0$ with $1/\gamma \in \N$, the axis-parallel, regular grid with side length $\gamma$ is the union of all $d-1$-dimensional hyperplanes that are orthogonal to an axis and that are in distance $k\gamma$ from the origin for a $k\in \Z$.) Both the existence of small separators and the efficiency of the compression algorithm follow easily from that formula.
Let $\eta > 0$. Let $1 \leq \mu \leq n^{1/d}$ be an integer, and consider an axis-parallel, regular grid with side length $1/\mu$ on $\Space$. Then the expected number of edges intersected by the grid is at most $O(n\cdot (n/\mu^d)^{2-\beta+\eta} + (n^{2-\alpha}\mu^{d(\alpha -1)}+n^{1-1/d}\mu)(1+\log(n/\mu^d)))$.
We defer the proof of Lemma <ref> to the end of this chapter. For the regime $2<\beta<3$, we immediately obtain from the statement that there is a sublinear set of vertices that disconnects the giant component.
By Lemma <ref> for $\mu = 2$, there are $m = O(n^{\max\{3-\beta,2-\alpha,1-1/d\}+\eta})$ edges intersecting a grid of side length $1/2$ in expectation, and two hyperplanes of this grid suffice to split $\Space$ into two halves. Whp there are $\Omega(n)$ vertices in each grid cell, and whp the weights of the vertices in each half satisfy a power law. If $2<\beta<3$ then whp each halfspace gives rise to a giant component of linear size, this follows from more general considerations in <cit.>. Hence, whp the two hyperplanes split the giant of $G$ into two parts of linear size, although almost surely they only intersect $n^{1-\Omega(1)}$ edges. Finally, since the bound $m = O(n^{\max\{3-\beta,2-\alpha,1-1/d\}+\eta})$ holds for all $\eta > 0$ we may conclude that it also holds with $\eta$ replaced by $o(1)$.
Compression algorithm: With Lemma <ref> at hand, we are ready to give a compression algorithm that stores the graph with $O(n)$ bits, i.e., with $O(1)$ bits per edge, proving Theorem <ref>.
We remark that our result does not directly follow from the general compression scheme on graphs with small separators in <cit.>, since our graphs only have small separators in expectation, in particular, small subgraphs of size $O(\sqrt{\log n})$ can form expanders and thus not have small separators. However, our algorithm loosely follows their algorithm as well as the practical compression scheme of <cit.>, see also <cit.>.
We first enumerate the vertices as follows.
Recall the definition of cells from Section <ref>, and consider all cells of volume $2^{-\ell_0 d}$, where $\ell_0 := \lfloor \log n /d\rfloor$. Note that the boundaries of these cells induce a grid as in Lemma <ref>.
Since each such cell has volume $\Theta(1/n)$, the expected number of vertices in each cell is constant. We fix a geometric ordering of these cells as in Lemma <ref>, and we enumerate the vertices in the order of the cells, breaking ties (between vertices in the same cell) arbitrarily. For the rest of the section we will assume that the vertices are enumerated in this way, i.e., we identify $V = [n]$, where $i \in [n]$ refers to the vertex with index $i$.
Having enumerated the vertices, for each vertex $i \in [n]$ we store a block of $1+\deg(i)$ sub-blocks. The first sub-block consists of a single dummy bit (to avoid empty sequences arising from isolated vertices). In the other $\deg(i)$ sub-blocks we store the differences $i-j$ using $\log_2 |i-j|+O(1)$ bits, where $j$ runs through all neighbors of $i$. We assume that the information for all vertices is stored in a big successive block $B$ in the memory. Moreover, we create two more blocks $B_V$ and $B_E$ of the same length. Both $B_V$ and $B_E$ have a one-bit whenever the corresponding bit in $B$ is the first bit of the block of a vertex, and $B_E$ has also a one-bit whenever the corresponding bit in $B$ is the first bit of an edge (i.e., the first bit encoding a difference $i-j$). All other bits in $B_V$ and $B_E$ are zero.
It is clear that with the data above the graph is determined. To handle queries efficiently, we replace $B_V$ and $B_E$ each with a rank/select data structure. This data structure allows to handle in constant time queries of the form “Rank($b$)”, which returns the number of one-bits up to position $b$, and “Select($i$)”, which returns the position of the $i$-th one-bit <cit.>. Given $i,s \in \N$, we can find the index of the $s$-th neighbor of $i$ in constant time by Algorithm <ref>, and the degree of $i$ by Algorithm <ref>. In particular, it is also possible for Algorithm <ref> to first check whether $s \leq \deg(i)$.
Finding the $s$-th neighbor of vertex $i$
$b := \Select(i,B_V)$
starting position of vertex $i$
$k := \Rank(b,B_E)$ number of edges and vertices before $b$
$b_1:= \Select(k+s,B_E)$ starting position of $s$-th edge of vertex $i$
$b_2 := \Select(k+s+1,B_E)$ bit after ending position of $s$-th edge of vertex $i$
return $B[b_1:b_2-1]$ block that stores $s$-th edge of vertex $i$
Finding the degree of vertex $i$
$b := \Select(i,B_V)$ starting position of vertex $i$
$b' := \Select(i+1,B_V)$ starting position of vertex $i+1$
$\Delta := \Rank(b',B_E)-\Rank(b,B_E)$ block in $B_E$ contains $\deg(i)+1$ one-bits
return $\Delta - 1$
We need to show that the data structure needs $O(n)$ bits in expectation. There are $n$ dummy bits, so we must show that we require $O(n)$ bits to store all differences $i-j$, where $ij$ runs through all edges of the graph. We need $2\log_2|i-j|+O(1)$ bits for each edge, as we store the edge $ij$ both in the block of $i$ and in the block of $j$. The $O(1)$ terms sum up to $O(|E|)$, which is $O(n)$ in expectation. Thus, it remains to prove the following.
Let the vertices in $V$ be enumerated by the geometric ordering. Then,
\begin{equation}\label{eq:compression}
\Ex\left[\sum_{ij \in E} \log(|i-j|)\right] = O(n).
\end{equation}
We abbreviate the expectation in (<ref>) by $R$.
Note that the geometric ordering puts all the vertices that are in the same cell of a $2^{-\ell}$-grid in a consecutive block, for all $1 \leq \ell \leq \ell_0$. Therefore, if $e = ij$ does not intersect the $2^{-\ell}$-grid then $|i-j| \leq \#\{\text{vertices in the cell of $e$}\}$.
For $1\leq \ell \leq \ell_0$, let $\mathcal{E}_\ell$ be the set of edges intersecting the $2^{-\ell}$-grid. For convenience, let $\mathcal{E}_{0} := \emptyset$, and let $\mathcal{E}_{\ell_0+1} := E$ be the set of all edges. Then, using concavity of $\log$ in the third step,
\begin{align*}
R & \leq \Ex\left[\sum_{\ell=0}^{\ell_0}\sum_{e=ij\in \mathcal{E}_{\ell+1}\setminus \mathcal{E}_{\ell}}\log(\#\{\text{vertices in the cell of $e$}\}) \right]\\
& = \sum_{\ell=0}^{\ell_0}\sum_{u<v}\Pr\left[uv\in \mathcal{E}_{\ell+1}\setminus \mathcal{E}_{\ell}\right]\Ex\left[\log(\#\{\text{vertices in the cell of $u$}\}) \mid uv \in \mathcal{E}_{\ell+1}\setminus \mathcal{E}_{\ell}\right] \\
& \leq \sum_{\ell=0}^{\ell_0}\sum_{u<v}\Pr\left[uv\in \mathcal{E}_{\ell+1}\right]\log\bigg(\underbrace{\Ex\left[\#\{\text{vertices in the cell of $u$}\} \mid uv \in \mathcal{E}_{\ell+1}\setminus \mathcal{E}_{\ell}\right]}_{=: T_\ell}\bigg)
\end{align*}
The term $T_\ell$ is at most $T_\ell \leq 2+(n-2)2^{-\ell d} \leq 3n2^{-\ell d}$ for $\ell \leq \ell_0$ (where we count 2 for $u$ and $v$ and use independence of the other vertex positions). Thus it remains to show that $\Ex\big[ \sum_{\ell=0}^{\ell_0}|\mathcal{E}_{\ell+1}|\log(3n2^{-\ell d}) \big] = O(n)$. Let $\eta>0$ be sufficiently small. From Lemma <ref> we know that $\Ex[|\mathcal{E}_{\ell}|] \leq E_\ell$, where $E_\ell= n \cdot (2^{d\ell}/n)^{\beta-2-\eta} + (n^{2- \alpha}2^{d\ell(\alpha -1)}+ n^{1-1/d}2^{\ell})(1+\log(n2^{-d\ell}))$. Since $E_\ell$ increases exponentially in $\ell$, we obtain
\[
\Ex\left[\sum_{\ell=0}^{\ell_0}|\mathcal{E}_{\ell+1}|\log(3n2^{-d\ell})\right] \leq O\left(\sum_{\ell=1}^{\ell_0+1}E_\ell \log(3n2^{-d\ell+d})\right) = O(E_{\ell_0+1}\log(3n2^{-d\ell_0})) = O(n),
\]
where the last equality follows since $1/n \leq 2^{-d\ell_0} \leq O(1/n)$ by our choice of $\ell_0$. This proves the lemma, and hence shows that we need $O(n)$ bits in expectation to store the graph.
This concludes the proof of Theorem <ref>, and it only remains to verify Lemma <ref>. In the proof of this lemma, we will use the following technical statement, which is a consequence of Fubini's theorem and allows us to replace certain sums by integrals.
Let $f:\R\to\R$ be a continuously differentiable function. Then for any weights $0 \leq w_0 \leq w_1$,
\[
\sum_{v \in V, w_0 \leq \w{v} \leq w_1} f(\w{v}) \;=\; f(w_0)\cdot |V_{\geq w_0}| \;-\; f(w_1) \cdot |V_{> w_1}| \;+\; \int_{w_0}^{w_1} f'(w) \cdot |V_{\geq w}| dw.
\]
In particular, if $f(0)=0$, then
\[\sum_{v \in V} f(\w{v}) = \int_{0}^{w_1} |V_{\geq w}| f'(w) dw=\int_{0}^{\infty} |V_{\geq w}| f'(w) dw.\]
Let $\nu$ be the sum of all Dirac measures given by the vertex weights between $w_0$ and $w_1$, i.e., for every set $A \subseteq \R$ we put $\nu(A):=|\{v \in V: \w{v} \in A, w_0 \le \w{v} \le w_1\}|$. Then
\begin{align*}
\sum_{v \in V, w_0 \leq \w{v} \leq w_1} f(\w{v}) &\;=\; \int_0^{\wmax} f(w) d\nu(w) \;=\; \int_0^{\wmax} \int_0^w f'(x) dx d\nu(w) \;+\; \int_0^{\wmax} f(0) d\nu(w)\\ &\;=\; \int_0^{\wmax} \int_0^{\infty} f'(x) \cdot \mathds{1}_{\{x \le w\}} dx d\nu(w) \;+\; f(0) \cdot \vert V_{\geq w_0} \setminus V_{>w_1}\vert.
\end{align*}
Notice that $[0,\wmax]$ is a compact set and $f'(x)$ is continuous by assumption. Hence the function $|f'(x)\cdot \mathds{1}_{\{x \le w\}}|$ is globally bounded on $[0,\wmax]$ and always zero for $x > \wmax$. Thus, $f'(x)\cdot \mathds{1}_{\{x \le w\}}$ is integrable and we can apply Fubini's theorem (see, e.g., <cit.>), which yields
\begin{align*}
\sum_{v \in V, w_0 \leq \w{v} \leq w_1} f(\w{v}) & \;=\; \int_0^{\infty} f'(x) \int_0^{\wmax}\mathds{1}_{\{w \ge x\}}d\nu(w) dx \;+\; f(0) \cdot \vert V_{\geq w_0} \setminus V_{>w_1}\vert \\
& \;=\; \int_0^{\infty} f'(x) \cdot \vert V_{\geq \max\{x,w_0\}} \setminus V_{>w_1}\vert dx \;+\; f(0) \cdot \vert V_{\geq w_0} \setminus V_{>w_1}\vert.
\end{align*}
The first summand (i.e. the integral) can be rewritten as
\[\int_0^{w_0} f'(x) \cdot \vert V_{\geq w_0} \setminus V_{>w_1}\vert dx \;\;+\; \int_{w_0}^{w_1} f'(x) \cdot \vert V_{\geq x} \setminus V_{>w_1}\vert dx ,\]
and then we obtain the first statement by using $\vert V_{\geq w_0} \setminus V_{>w_1}\vert=\vert V_{\geq w_0} \vert-\vert V_{>w_1}\vert$, calculating the integrals, and combining the resulting terms. The second statement follows directly by choosing $w_0=0$ and $w_1 > \wmax$.
We first observe that we can assume $\mu \ge 2$ as this implies the statement for smaller $\mu$ immediately. Thus let $2\le \mu \le n^{1/d}$, and consider an axis-parallel, regular grid with side length $1/\mu$ on $\Space$.
For $u,v \in V$, let $\rho_{u v}$ be the probability that the edge $uv$ exists and cuts the grid. Let $r_{\max} := 1/2$ be the diameter of $\Space$. We write
\begin{align} \label{eq:rhouv}
\rho_{u v} = \int_{0}^{r_{\max}} \Pr[\|\x u - \x v\| = r] \cdot p_{uv}(r) \cdot \Pr[\x u, \x v \text{ in different cells of $\mu$-grid}]\, dr.
\end{align}
Observe that $u$ and $v$ have distance $r$ with probability density $\Pr[\|\x u - \x v\| = r] = O(r^{d-1})$. Furthermore, setting $\gamma_{uv} := \min\{(\w{u}\w{v}/\W)^{1/d},r_{\max}\}$ we have
$$p_{uv}(r) = \begin{cases} \Theta(1), & \text{if } r \ge \gamma_{uv}, \\ \Theta( (\gamma_{uv} / r)^{\alpha d}), & \text{otherwise.} \end{cases}$$
Additionally, in the case $\alpha = \infty$, by increasing $\gamma_{uv}$ by at most a constant factor we may assume $p_{uv}(r) = 0$ for all $r \geq \gamma_{uv}$.
For the last term in (<ref>), for a fixed axis of $\Space$ consider the hyperplanes $\{h_i\}_{1 \leq i \leq \mu}$ of the grid perpendicular to that axis. If the edge $e= uv$ has length $\|\x u - \x v\| = r$, then after a random shift along the axis, the edge $e$ intersects one of the $h_i$ with probability at most $\min\{\mu r,1\}$. By symmetry of the underlying space, a random shift does not change the probability to intersect one of the $h_i$, so any edge of length $r$ has probability at most $\min\{\mu r,1\}$ to intersect one of the $h_i$. By the union bound over all (constantly many) axes, the probability for $u,v$ to lie in different cells of the grid is $O(\min\{\mu r,1\})$.
Now we distinguish several cases. For $\gamma_{uv}> 1/\mu$ and $\alpha < \infty$, we may estimate
\begin{align}\label{eq:hitgrid1}
\rho_{uv} %& \leq O\left(\int_{0}^{r_{\max}} r^{d-1} \cdot \min\{r,1\} \cdot p_{uv}(r) dr\right) \nonumber
& \leq O\bigg(\int_{0}^{1/\mu} r^{d-1}\cdot \mu r dr + \int_{1/\mu}^{\gamma_{uv}} r^{d-1} dr+ \underbrace{\int_{\gamma_{uv}}^{r_{\max}} r^{d-1-d\alpha}\gamma_{uv}^{d\alpha} dr}_{= O(\gamma_{uv}^d), \text{ since } d-d\alpha <0} \bigg) \leq O(\mu^{-d} + \gamma_{uv}^d) \leq O(\gamma_{uv}^d).
\end{align}
For $\gamma_{uv} > 1/\mu$ and $\alpha = \infty$, equation (<ref>) remains true, except that the third integral is replaced by $0$ by our choice of $\gamma_{uv}$. So in this case we still get $\rho_{uv} \leq O(\gamma_{uv}^d)$.
The case $\gamma_{uv} \leq 1/\mu$ is a bit more complicated. Again we consider first $\alpha <\infty$. Then we may bound
\begin{align}\label{eq:hitgrid2}
\rho_{uv} & \leq O\bigg(\underbrace{\int_{0}^{\gamma_{uv}} r^{d-1}\cdot \mu r dr}_{=:I_1} + \underbrace{\int_{\gamma_{uv}}^{1/\mu} r^{d-1}\cdot \mu r\cdot r^{-d\alpha}\gamma_{uv}^{d\alpha} dr}_{=:I_2}+ \underbrace{\int_{1/\mu}^{r_{\max}} r^{d-1-d\alpha}\gamma_{uv}^{d\alpha} dr}_{=:I_3} \bigg).
\end{align}
Similarly as before, $I_1 \leq O(\gamma_{uv}^{d+1}\mu)$ and $I_3 \leq O(\mu^{d\alpha-d}\gamma_{uv}^{d\alpha})$. Note that both terms are bounded from above by $O((\gamma_{uv} \mu)^{d\tilde \alpha}\mu^{-d})$, where $\tilde \alpha := \min\{\alpha, 1+1/d\}$, since $\gamma_{uv} \mu \leq 1$. For $I_2$, the inverse derivative of $r^{d-d\alpha}$ is either $\Theta(r^{1+d-d\alpha})$, or $\log r$, or $-\Theta(r^{1+d-d\alpha})$, depending on whether $1+d-d\alpha$ is positive, zero, or negative, respectively. Therefore, we obtain
\[
I_2 \leq
\left\{
\begin{aligned}
&O(\gamma_{uv}^{d\alpha}\mu^{d\alpha-d}) &&= O((\gamma_{uv}\mu)^{d\alpha}\mu^{-d}), && \text{if }d-d\alpha>-1,\\
&O(\gamma_{uv}^{d\alpha}\mu (\log(1/\mu)-\log(\gamma_{uv}))) && = O((\gamma_{uv}\mu)^{d+1}\mu^{-d}|\log(\gamma_{uv}\mu)|), && \text{if }d-d\alpha=-1,\\
&O(\gamma_{uv}^{d+1}\mu) && = O((\gamma_{uv}\mu)^{d+1}\mu^{-d}), && \text{if }d-d\alpha<-1.
\end{aligned}\right.
\]
In particular, we can upper-bound all terms (including $I_1$ and $I_3$) in a unified way by $O((\gamma_{uv} \mu)^{d\tilde \alpha}\mu^{-d})(1+|\log(\gamma_{uv} \mu)|)$. Moreover, since $\gamma_{uv} \geq (\wmin^2/\W)^{1/d} = \Omega(n^{-1/d})$, the second factor is bounded by $O(1+\log (n^{1/d}/\mu)) = O(1+\log (n/\mu^d))$. Also, in the case $\alpha = \infty$ the same calculation applies, except that $I_2$ and $I_3$ are replaced by $0$. Note that naturally $\tilde \alpha = 1+1/d$ for $\alpha = \infty$. So altogether we have shown that
\begin{equation*}
\rho_{uv} \leq
\begin{cases}
O(\gamma_{uv}^d), & \text{if }\gamma_{uv} \geq 1/\mu, \\
O((\gamma_{uv} \mu)^{d\tilde \alpha}\mu^{-d}(1+\log(n^{d}/ \mu))), & \text{if } \gamma_{uv} \leq 1/\mu.
\end{cases}
\end{equation*}
Therefore, the expected number of edges intersecting the grid is in $O(S_1 + S_2)$, where
\[
S_1 := \sum_{u,v\in V,\, \gamma_{uv} > 1/\mu} \gamma_{uv}^{d} \quad \text{ and } \quad S_2 := \sum_{u,v\in V,\, \gamma_{uv} \leq 1/\mu} (\gamma_{uv}\mu)^{d\tilde \alpha}\mu^{-d}(1+\log(n^{d}/\mu)).
\]
Let $0<\eta' <\eta <\beta-2$ be (sufficiently small) constants. Then we may use the power-law assumption (PL2), Lemma <ref>, and Lemma <ref> to bound $S_1$:
\begin{align*}
S_1 & \leq \sum_{u,v\in V,\, \w{u}\w{v} > \W/\mu^{d}} \frac{\w{u}\w{v}}{\W} = \sum_{u\in V} \frac{\w{u}}{\W}\cdot \W_{\geq \W/(\mu^d\w{u})} \stackrel{\ref{lem:totalweight}}{\leq} O\left(\sum_{u\in V} \frac{\w{u}}{\W} \cdot n\left(\frac{\W}{\mu^d\w{u}}\right)^{2-\beta+\eta}\right) \\
& \leq O\left(\left(\frac{\mu^d}{n}\right)^{\beta-2-\eta}\sum_{u\in V} \w{u}^{\beta-1-\eta}\right) \stackrel{\ref{lem:weightsums}}{\le} O\left(\left(\frac{\mu^d}{n}\right)^{\beta-2-\eta}\int_{\wmin}^{\infty} nw^{1-\beta+\eta'}w^{\beta-2-\eta}dw\right) \\
& = O\left(n\cdot \left(\mu^d/n\right)^{\beta-2-\eta}\right).
\end{align*}
To tackle $S_2$, we again use Lemma <ref>, let $\lambda_u := \W/(\w{u}\mu^d)$ and obtain
\begin{align}\label{eq:hitgridexpectation}
S_2':=\sum_{\substack{u,v\in V \\ \gamma_{uv} \leq 1/\mu}} (\gamma_{uv})^{d\tilde \alpha}& = \sum_{u\in V} \sum_{\substack{v \in V_{\leq \lambda_u}}} \left(\frac{\w{u}\w{v}}{\W}\right)^{\tilde \alpha}
\stackrel{\ref{lem:weightsums}}{\leq} O\left( \sum_{u\in V} \left(\frac{\w{u}}{n}\right)^{\tilde \alpha}\int_{\wmin}^{\lambda_u} nw^{1-\beta+\eta} w^{\tilde \alpha-1} dw\right)
%& \leq O\left(n^{1-\tilde \alpha} \sum_{u\in V} \w{u}^{\tilde \alpha}\int_{\wmin}^{\W/\w{u}} w^{\tilde \alpha-\beta+\eta}dw\right) .
\end{align}
Now we distinguish two cases, because the integral behaves differently for exponents larger or smaller than $-1$. If $\tilde \alpha \geq \beta -1$, then for $0< \eta' < \eta$ equation (<ref>) evaluates to
\begin{align*}
S_2' & \leq O\left( \sum_{u\in V} \left(\frac{\w{u}}{n}\right)^{\tilde \alpha}n\lambda_u^{1+\tilde \alpha-\beta+\eta} \right)
= O\left(\frac{n^{2-\beta+\eta}}{\mu^{d(1+\tilde \alpha-\beta+\eta)}} \sum_{u\in V} \w{u}^{\beta -1-\eta}\right) \\
& \stackrel{\ref{lem:weightsums}}{\leq} O\left(\frac{n^{2-\beta+\eta}}{\mu^{d(1+\tilde \alpha-\beta+\eta)}} \int_{\wmin}^{\infty} nw^{1-\beta+\eta'}w^{\beta -2-\eta}dw\right) = O\left(n\frac{n^{2-\beta+\eta}}{\mu^{d(1+\tilde \alpha-\beta+\eta)}}\right).
\end{align*}
Therefore, $S_2 = \mu^{d\tilde \alpha-d}(1+\log(n^{d}/\mu)) S_2' \leq O(n\cdot (n/\mu^d)^{2-\beta+\eta})$, which is one of the terms in the lemma. On the other hand, if $\tilde \alpha < \beta -1$ then for $0<\eta < \beta-\tilde \alpha -1$ we obtain from (<ref>)
\begin{align*}
S_2' & \leq O\left(n^{1-\tilde \alpha} \sum_{u\in V} \w{u}^{\tilde \alpha} \right) \stackrel{\ref{lem:weightsums}}{\leq} O\left(n^{1-\tilde \alpha}\int_{\wmin}^{\infty} nw^{1-\beta+\eta} w^{\tilde \alpha-1} dw\right) \leq O\left(n^{2-\tilde \alpha}\right),
\end{align*}
and again $S_2 = \mu^{d\tilde \alpha-d}(1+\log(n^{d}/\mu)) S_2'$ corresponds to terms in the lemma after plugging in $\tilde \alpha$. This concludes the proof.
§ COMPARISON WITH HYPERBOLIC RANDOM GRAPHS
In this section we show that hyperbolic random graphs are a special case of GIRGs.
We start by defining hyperbolic random graphs. There exist several different representations of hyperbolic geometry, all with advantages and disadvantages. For introducing this random graph model, it is most convenient to use the native representation. It can be described by a disk $H$ of radius $R$ around the origin $0$, where the position of every point $x$ is given by its polar coordinates $(r_x,\theta_x)$. The model is isotropic around the origin. The hyperbolic distance between two points $x$ and $y$ is given by the non-negative solution $d=d(x,y)$ of the equation
\begin{equation} \label{eq:hypdist}
\cosh(d) = \cosh(r_x)\cosh(r_y)-\sinh(r_x)\sinh(r_y)\cos(\phi_x-\phi_y).
\end{equation}
In the following definition, we follow the notation introduced by Gugelmann et al. <cit.>.
Let $\alpha_H>0, C_H\in \R,T_H>0,n\in \N$, and set $R=2\log n+C_H$. Then the random hyperbolic graph $G_{\alpha_H,C_H,T_H}(n)$ is a graph with vertex set $V=[n]$ and the following properties:
* Every vertex $v \in [n]$ independently draws random coordinates $(r_v,\phi_v)$, where the angle $\pi_v$ is chosen uniformly at random in $[0,2\pi)$ and the radius $r_v \in [0,R]$ is random with density $f(r) := \frac{\alpha_H\sinh(\alpha_H r)}{\cosh(\alpha_H R)-1}$.
* Every potential edge $e=\{u,v\}$, $u,v \in [n]$, is independently present with probability
\[p_H(d(u,v)) = \left(1+e^{\frac{1}{2T_H}(d(u,v)-R)}\right)^{-1}.\]
In the limit $T_H \rightarrow 0$, we obtain the threshold hyperbolic random graph $G_{\alpha_H,C_H}(n)$, where every edge $e=\{u,v\}$ is present if and only if $d(u,v) \le R$.
We will show that hyperbolic random graphs are almost surely contained in our general framework. To this end, we embed the disk of the native hyperbolic model into our model with dimension 1, hence we reduce the geometry of the hyperbolic disk to the geometry of a circle, but gain additional freedom as we can choose the weights of vertices. Notice that a single point on the hyperbolic disk has measure zero, so we can assume that no vertex has radius $r_v=0$. For the parameters, we put
\[d:=1,\quad\beta := 2\alpha_H + 1,\quad\alpha := 1/T_H.\]
Furthermore, we define the mapping
\[\w{v} := e^{\frac{R-r_v}{2}} \quad\text{and}\quad \x{v} := \frac{\phi_v}{2\pi}.\]
Since this is a bijection between $H \setminus \{0\}$ and $[1,e^{R/2}) \times \mathds{T}^1$, there exists as well an inverse function $g(\w{u},\x{u})=(r_u,\phi_u)$. Finally for any two vertices $u \neq v$ on the torus, we set
\[p_{uv} := p_H(d(g(\w{u},\x{u}),g(\w{v},\x{v}))).\]
This finishes our embedding. The following lemma, which we prove near the end of this section, demonstrates that under this mapping almost surely the weights will follow a power law.
Let $\alpha_H > \frac12$. Then for all $\eta=\eta(n)=\omega(\frac{\log\log n}{\log n})$, with probability $1-n^{-\Omega(\eta)}$ the induced weight sequence $\w{}$ follows a power law with parameter $\beta=2\alpha_H+1$.
Now we come to the main statement of this section.
In the following we assume that if we sample an instance of the hyperbolic random graph model, we first sample the radii, then the angles and at last the edges.
Let $\alpha_H > \frac12, n \in \N$ and fix a set of radii $(r_1, \ldots, r_n) \in [0,R]^n$ inducing a power-law weight sequence $\w{}$ with parameter $\beta =2\alpha_H+1$.
Then the random positions $\x u$ and the edge probabilities $p_{uv}(\x u, \x v)$ produced by our mapping satisfy the properties of the GIRG model, i.e., for fixed radii inducing power-law weights, hyperbolic random graphs are a special case of GIRGs.
Note that the precondition of Theorem <ref> on the weight sequence $\w{}$ holds for any $\eta=\eta(n)=\omega(\log\log n/\log n)$ with probability $1-n^{-\Omega(\eta)}$ by Lemma <ref>. Therefore an instance of random hyperbolic graphs is almost surely included in our GIRG model with parameters as set above. In particular, any property that holds with probability $1-q$ for GIRGs also holds for hyperbolic random graphs with probability at least, say, $1-q-n^{-o(1)}$.
Before proving Lemma <ref> and Theorem <ref>, we consider the following basic property of hyperbolic random graphs.
Let $\alpha_H > \frac12$. Then with probability $1-n^{-\Omega(1)}$ every vertex has radius at least $r_0 := (1-\frac{1}{2\alpha_H})\log n$. Furthermore, for all $r=\omega(1), r \le R$ and $v \in V$, we have
\[\Pr[r_v \le r] = e^{-\alpha_H(R-r)}(1+o(1)).\]
Let $v \in V$. By the given density $f$ it follows immediately that
\begin{align*}
\Pr[r_v \le r]&=\int_0^r f(x)dx =\alpha_H \int_0^r \frac{\sinh(\alpha_H x)}{\cosh(\alpha_H R)-1} dx = \frac{\cosh(\alpha_H r)-1}{\cosh(\alpha_H R)-1}\\ &= e^{-\alpha_H(R-r)}(1+o(1)),
\end{align*}
where we used $\cosh(x) = \frac{e^x+e^{-x}}{2} = \frac{e^x}{2}(1+o(1))$ whenever $x=\omega(1)$. Now let $X_{r_0}$ be the random variable counting the vertices of radius at most $r_0$. We observe that the above expression for $\Pr[r_v \le r]$ implies
\[\Ex[X_{r_0}] = n e^{-\alpha_H(R-r_0)}(1+o(1))=e^{-\alpha_H C_H}n^{1/2-\alpha_H}(1+o(1))=n^{-\Omega(1)}.\]
By Markov's inequality, with probability $1-n^{-\Omega(1)}$ we have $X_{r_0}=0$.
For every vertex of the random hyperbolic graph, the radius is chosen independently and uniformly according to $f(r)$. Hence under our mapping, we sample weights independently. We will prove that we fulfil the prerequisites of Lemma <ref>. Let $0<\eps<1$. By Lemma <ref>, the probability that a vertex $v$ has radius at most $r \ge \eps \log n$ is $\Theta(e^{-\alpha_H(R-r)})$. Let $1 \le z \le o(n^{1-\eps/2})$. Then $R-2\log z \ge \eps \log n$, and
\begin{align*}
F(z) := \Pr[\w{v} \le z] &= 1-\Pr[r_v \le R-2\log z]=1-\Theta(e^{-2\alpha_H\log z})\\&=1-\Theta(z^{-2\alpha_H})=1-\Theta(z^{1-\beta}).
\end{align*}
Furthermore, for $z < 1$ we get
\[F(z):= \Pr[\w{v} \le z] = \Pr[r_v \ge R-2\log{z}] = 0.\]
Clearly, $F(.)$ is non-decreasing and therefore satisfies the preconditions of Lemma <ref> by taking $\wmin=1$. Then it follows from this lemma that the weight sequence $\w{}$ follows a power law with parameter $\beta$ with sufficiently high probability.
Let us start by considering the sampling process of a random hyperbolic graph. First we sample the radii of the vertices, for which the precondition of the theorem assumes that they induce a power-law weight sequence. Next we sample the angles. This corresponds to coordinates chosen independently and uniformly at random on $\mathds{T}^1$. It remains to prove that $p_{uv}$ as defined above satisfies conditions (<ref>) and (<ref>).
Let $u \neq v$ be two vertices of the random hyperbolic graph with coordinates $(r_u,\phi_u)$ and $(r_v,\phi_v)$ and consider their mappings $(\w{u},\x{u})$ and $(\w{v},\x{v})$. Since the hyperbolic model is isotropic around the origin, we can assume without loss of generality that $r_u \ge r_v$, $\phi_v=0$ and $\phi_u \le \pi$.
Let us first consider the threshold model, corresponding to $\alpha = \infty$. We claim that there exist constants $M > m > 0$ such that whenever $\|\x{u}-\x{v}\|\ge M \frac{\w{u}\w{v}}{\W}$, then $p_{uv}=0$, and whenever $\|\x{u}-\x{v}\|\le m \frac{\w{u}\w{v}}{\W}$, then $p_{uv}=1$. This will imply (<ref>), as we set $d=1$.
Recall that in the threshold model, two vertices $u$ and $v$ are connected if and only if $d(u,v)\le R$. When $r_u+r_v \le R$, this is the case for all angles $\phi_u$ and $\phi_v$. Otherwise, for $\phi=0$ and $\phi_u \le \pi$, the distance between $u$ and $v$ is increasing in $\phi_u$ and there exists a critical value $\phi_0$ such that $d((r_u,\phi_u),(r_v,0))\le R$ if and only if $\phi_u \le \phi_0$. The following lemma estimates $\phi_0$.
Let $0 \le r_u \le R$, $r_u+r_v \ge R$ and assume $\phi_v=0$. Then
\[\phi_0 \;=\; 2e^{\frac{R-r_u-r_v}{2}}\left(1+\Theta(e^{R-r_u-r_v})\right).\]
Suppose $\|\x{u}-\x{v}\|\ge M \frac{\w{u}\w{v}}{\W}$. Notice that by our transformation we have $\|\x{u}-\x{v}\|=\frac{\phi_u}{2\pi}$ and
\begin{equation} \label{eq:weightcorresp}
\frac{\w{u}\w{v}}{\W} = \Theta\left(\frac{\w{u}\w{v}}{n}\right)=\Theta\left(\frac{e^{R-(r_u+r_v)/2}}{n}\right) = \Theta(e^{(R-r_u-r_v)/2}),
\end{equation}
where we used $\W=\Theta(n)$ by Lemma <ref>. Hence, we have $\frac{\phi_u}{2\pi M}=\Omega(e^{(R-r_u-r_v)/2})$. Since $\phi_u \le 1$, this implies $r_u+r_v > R$, if we choose the constant $M$ sufficiently large. Moreover, for sufficiently large $n$ we obtain $\phi_u > 2e^{\frac{R-r_u-r_v}{2}}\left(1+\Theta(e^{R-r_u-r_v})\right)$. Thus, by Lemma <ref> the two vertices $u$ and $v$ are not connected and indeed $p_{uv}=0$.
On the other hand, assume $\|\x{u}-\x{v}\|\le m \frac{\w{u}\w{v}}{\W}$. Then either $r_u+r_v < R$ and thus $\{u,v\} \in E$ follows directly, or $r_u+r_v \ge R$ and $\phi_u < 2e^{\frac{R-r_u-r_v}{2}}\left(1+\Theta(e^{R-r_u-r_v})\right)$, if $m$ is sufficiently small. In the second case, Lemma <ref> implies $p_{uv}=1$.
We now turn to the case $\alpha < \infty$. By our assumptions on $\phi_u$ and $\phi_v$ and by the identity $\cosh(x \pm y)=\cosh(x)\cosh(y) \pm \sinh(x)\sinh(y)$, we can rewrite (<ref>) as
\begin{equation} \label{eq:newhypdist}
\cosh(d)=\cosh(r_u-r_v)+(1-\cos(\phi_u))\sinh(r_u)\sinh(r_v).
\end{equation}
Next we observe that $\cosh(x)=\Theta(e^{|x|})$ for all $x$ and $\sinh(x)=\Theta(e^x)$ for all $x=\omega(1)$. Observe that (PL2) and $\w v = e^{(R-r_v)/2}$ imply $r_v = \Theta(\log n)$ for all vertices $v$. Furthermore, we perform a Taylor approximation of $1-\cos(\phi_u)$ around $0$ and get
$1-\cos(\phi_u)=\frac{\phi_u^2}{2}-\frac{\phi_u^4}{24}+\ldots = \Theta(\phi_u^2)$,
as $\phi_u$ is at most a constant. Combining these observations with (<ref>) and the assumption $r_u \ge r_v$, we deduce
\begin{equation} \label{eq:finalterm}
e^{d-R}=\Theta(\cosh(d)e^{-R})=\Theta\left(e^{r_u-r_v-R}+\phi_u^2 e^{r_u+r_v-R}\right).
\end{equation}
In the condition (<ref>) on $p_{uv}$ the minimum is obtained by the second term whenever $\|\x{u}-\x{v}\| \le \frac{\w{u}\w{v}}{\W}$. Mapping $u$ and $v$ to the hyperbolic disk, this implies $\phi_u =O(e^{(R-r_u-r_v)/2})$. We claim that whenever $\phi_u =O(e^{(R-r_u-r_v)/2})$, the two vertices $u$ and $v$ are connected with constant probability and therefore $p_{uv}=\Theta(1)$. Indeed, in this case by (<ref>) we have $e^{d-R}=O(1)$, and using Definition <ref> we deduce
\[p_{uv}=p_H(d(u,v))=\left(1+(e^{d-R})^{(1/(2T_H))}\right)^{-1}=\Theta(1).\]
On the other hand, suppose $\|\x{u}-\x{v}\| \ge \frac{\w{u}\w{v}}{\W}$, which implies $\phi_u =\Omega(e^{(R-r_u-r_v)/2})$. In this case by (<ref>) we have
$e^{d-R}=\Theta\left(\phi_u^2 e^{r_u+r_v-R}\right)=\Omega(1)$.
However, if $e^{d-R}=\Omega(1)$, we can use Definition <ref> and (<ref>) to obtain
\begin{align*}
\end{align*}
This finishes the case $\alpha < \infty$ and thus the proof.
§ CONCLUSION
To cope with the technical shortcomings of hyperbolic random graphs, we introduced a new model of scale-free random graphs with underlying geometry – geometric inhomogeneous random graphs – and theoretically analyzed their fundamental structural and algorithmic properties. Scale-freeness and basic connectivity properties of our model follow from more general considerations <cit.>. We established that (1) hyperbolic random graphs are a special case of GIRGs, (2) GIRGs have a constant clustering coefficient, and (3) GIRGs have small separators and are very well compressible. As our main result, (4) we presented an expected-linear-time sampling algorithm. This improves the best-known sampling algorithm for hyperbolic random graphs by a factor $O(\sqrt{n})$.
In this paper we laid the foundations for further experimental and theoretical studies on GIRGs. In particular, we hope that the model can be used for the analysis of processes such as epidemic spreading. We leave this to future work.
We thank Hafsteinn Einarsson, Tobias Friedrich, and Anton Krohmer for helpful discussions.
|
1511.00338
|
${}^{1}$ Instytut Fizyki Teoretycznej, Uniwersytet
Wrocławski, 50-204 Wrocław, Poland
${}^{2}$ Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, 141980 Dubna, Russia
${}^{3}$ National Research Nuclear University (MEPhI), 115409 Moscow, Russia
${}^{4}$ Matrosov Institute for System Dynamics and Control Theory, Irkutsk 664033, Russia
In the present work the Mott effect for pions and kaons is described within a Beth-Uhlenbeck approach on the basis of the PNJL model. The contribution of these degrees of freedom to the thermodynamics is encoded in the temperature dependence of their phase shifts. A comparison with results from $N_f=2+1$ lattice QCD thermodynamics is performed.
§ INTRODUCTION
One of the central problems in the investigation of the transition from hadronic to quark matter is a microphysical description of the dissociation of hadrons into their quark constituents. This Mott transition occurs under extreme conditions of high temperatures and densities as they are provided, e.g., in ultrarelativistic heavy-ion collisions or in the interiors of compact stars. Since an ab-initio description of QCD thermodynamics within simulations of lattice QCD (LQCD) is yet limited to finite temperatures and low chemical potentials only, the development of effective model descriptions is of importance. Here, we develop a relativistic Beth-Uhlenbeck approach to the description of mesonic bound and scattering states in a quark plasma <cit.> further by including the strange sector.
To this end we employ the PNJL model which is particularly suitable for addressing the appearance of pions and kaons as both, quasi Goldstone bosons of the broken chiral symmetry and pseudoscalar meson bound states.
Within this framework the confinement of colored quark states is effectively taken into account by coupling the chiral quark dynamics to the Polyakov loop and its effective potential. The model is widely used to describe quark-gluon thermodynamics in the meanfield approximation <cit.>, but has also been developed to address mesonic correlations
The relativistic Beth-Uhlenbeck approach is the appropriate tool to develop a unified description of quark-gluon and hadron thermodynamics including the transition between both asymptotic regimes of QCD.
In the next section the basic formulae for the thermodynamic potential, the phase shifts and the pressure in the Beth-Uhlenbeck approach are presented, followed by a discussion of numerical results compared to LQCD data in Sect. 3 and conclusions in Sect. 4.
§ RELATIVISTIC BETH-UHLENBECK APPROACH FOR THE PNJL MODEL
The Lagrangian for the 3 flavor PNJL model is given by
q̅[∂- m_0+γ_0(μ-A_4)]q+G_S∑^8_a=0[(q̅λ^a q)^2+(q̅γ_5λ^a q)^2]
-𝒰(Φ, Φ; T) .Here, $q$ denote the quark fields with 3 colors and flavors, $\lambda^a$ are the Gell-Mann matrices in flavor space $(a=0,1,2,3\ldots8)$, $G_S$ is dimensionful coupling constant. The Polyakov-loop potential $\mathscr{U}(\Phi, \Bar{\Phi}; T)$ is chosen in the polynomial form <cit.>, with the parameters taken from that reference.
The gluon background field in the Polyakov gauge is a diagonal matrix in color space
$A_4 = {\rm diag}(\phi_3+\phi_8,-\phi_3+\phi_8,-2\phi_8)$.
The Polyakov loop field $\Phi$ is defined via the color trace over the gauge-invariant average of the Polyakov line $L(\vec{x})$ <cit.>.
The thermodynamic potential in Gaussian approximation has the form <cit.>
Ω_Gauß= 𝒰(Φ, Φ; T)
+ Ω_Q
+ Ω_M ,
where in the quark $(\Omega_{\rm Q})$ and meson $(\Omega_{\rm M})$ contributions the zero-point energy terms are removed ("no sea" approximation).
The quark contribution is given by
-2N_c N_f/3∫dp/2π^2p^4/E_p
[f_Φ^+(E_p) + f_Φ^-(E_p)] ,
where $f_\Phi^{\pm}(E_p)$ are the generalized Fermi distribution functions
(Φ+2Φ̅Y̅)Y̅+Y̅^3/1+3(Φ+Φ̅Y̅)Y̅+Y̅^3 .
with the abbreviation $Y={\rm e}^{-(E_p-\mu)/T}$ and $\bar{Y}={\rm e}^{-(E_p+\mu)/T}$
(cf. Ref. <cit.>).
The meson contribution reads <cit.>
Ω_M =
-d_M ∫d^3q/(2π)^3∫dω/2π
[g^+(ω+μ_M) + g^-(ω-
μ_M)] δ_M(ω,q) .
Here $d_{M}$ are the meson degeneracy factors: $d_{\pi}=3$ for pions and $d_{K}=4$ for kaons; $g(E)=(\e^{\beta E}-1)^{-1}$ is the Bose function, $\mu_M = \mu_i-\mu_j$ is the chemical potential of a meson M composed of quark $i$ with chemical potential $\mu_i$ and
antiquark $j$ with chemical potential $-\mu_j$.
The phase shift at rest $\delta_{\rm M}(\omega, {\bf q=0})=\delta_{\rm M}(\omega)$ is determined by
\begin{equation}
\label{phaseshift}
\delta_{\rm M}(\omega)=
-\IM\ln\left[\beta^2S_{\rm M}^{-1}(\omega-\mu_M+\ii\eta)\right]~.
\end{equation}
The propagator and the polarization function in the present work should be understood as to be evaluated at the shifted energy $z=\omega-\mu_M+\ii\eta$
= G_S^-1 - Π_M(z) ,
Π_M(z) =
[ (ω-μ_M)^2-(M_i-M_j)^2]I^ij_2(ω-μ_M) ] ,
where the 1-loop integrals are given by
I^i_1= -N_c∫dp/4π^2p^2/E_i[f_Φ^+(E_i) + f_Φ^-(E_i)] ,
I^ij_2= -N_c∫d^3p/(2π)^3
- 1/2E_i
+1/2E_j1/(E_j-ω)^2-E^2_if_Φ^+(E_j)- 1/2E_j1/(E_j+ω)^2-E^2_if_Φ^-(E_j)] .
§ RESULTS
The parameters used for the numerical studies in the present work are the bare quark masses $m_{u,d} = 5.5$ MeV and $m_{s} = 138.6$ MeV, the three-momentum cutoff $\Lambda = 602$ MeV and the scalar coupling constant $G_{\rm S}\Lambda^2 = 2.317$. With these parameters one finds in vacuum a constituent quark mass of 367 MeV, a pion mass of 135 MeV and pion decay constant $f_\pi=92.4$ MeV.
We report here results for the case of vanishing chemical potentials.
The solution for the pion phase shift is shown in Figure <ref> as function of the energy variable
The jump of the phase shift from $0$ to $\pi$ indicates the position of a bound state in the spectrum below the threshold of the continuum states which is located where the phase shift starts decreasing
towards zero.
With increasing temperature the threshold moves to lower $\omega$-values and the phase shift jumps from $\pi$ to $0$ when the Mott temperature is reached where the bound state gets dissociated.
This behavior is in accordance with the Levinson theorem, for details see <cit.>.
Phase shift of pions (left panel) and kaons (right panel) as function of the energy for different temperatures from $T = 150$ MeV to $400$ MeV.
Left panel: pressure of pions (red dotted line) and kaons (blue solid line). Right panel: total pressure of $N_f=2+1$ flavor PNJL model with pseudoscalar meson correlations as a function of temperature (black solid line) and LQCD data (colored bands <cit.>).
In the kaon phase shift arises one more threshold at low energy because the dynamical masses of the quarks composing the kaon are different, for more information see Ref.<cit.>.
In the left panel of Figure <ref> the pressure of pions and kaons is presented. It shows a typical behavior: first increasing with temperature towards the Stefan-Boltzmann limit which isn't reached since due to the chiral phase transition the continuum threshold lowers and this induces a reduction of the meson gas pressure already before the Mott temperature is reached.
Above the Mott temperature, the growing meson width leads to a stronger reduction of the pressure with a rather sharp onset of this effect.
In the right panel of Figure <ref> we compare the total pressure of the $N_f=2+1$ flavor PNJL model with recent LQCD data from <cit.>.
While our results agree with LQCD below $T\simeq 150$ MeV, a discrepancy opens up beyond this temperature which is mainly due the fact that the chiral transition temperature in the PNJL model is too high ($\sim 250$ MeV) when compared to the pseudocritical temperature of $T_c=154 \pm 9$ MeV from LQCD
Above a temperature of $T \simeq 300$ MeV the model pressure reaches towards the Stefan-Boltzmann limit while the LQCD pressure stays below it.
§ CONCLUSIONS
The main result this work is the addition of the strange sector to the relativistic Beth-Uhlenbeck approach which allows to compare with $N_f=2+1$ LQCD data.
At the present level of description, there is a discrepancy between our results and the LQCD ones for temperatures $T\gtrsim 150$ MeV which can be reduced as follows.
First, by including higher lying hadronic states into the calculation of the total pressure.
Second, by lowering the Mott temperature.
Third, by accounting for virial corrections (perturbative) to the quark-gluon pressure corresponding to residual interactions of quarks and gluons in the plasma.
These improvements are subject of our current work.
§ ACKNOWLEDGEMENT
We gratefully acknowledge discussions about lattice QCD data with O. Kaczmarek.
The work of A.D. was supported by the Polish National Science Centre (NCN) under grant number UMO - 2013/09/B/ST2/01560 and by the Institute for Theoretical Physics of the University of Wroclaw under internal grant number 1439/M/IFT/15. D.B. and A.R. received support by NCN under grant number UMO-2011/02/A/ST2/00306.
§ REFERENCES
Hüfner J, Klevansky S P, Zhuang P and Voss H 1994 Annals Phys. 234 225
Blaschke D, Buballa M, Dubinin A, Röpke G and Zablocki D 2014 Annals Phys. 348 228
Fukushima K 2004 Phys. Lett. B 591, 277
Ratti C, Thaler M A and Weise W 2006 Phys. Rev. D 73, 014019
Hansen H, Alberico W M, Beraudo A, Molinari A, Nardi M and Ratti C 2007 Phys. Rev. D 75, 065004
Radzhabov A, Blaschke D, Buballa M and Volkov M K 2008 Phys. Atom. Nucl. 71, 1981
Radzhabov A, Blaschke D, Buballa M and Volkov M K 2011 Phys. Rev. D 83, 116004
Wergieluk A, Blaschke D, Kalinovsky Y L and Friesen A 2013 Phys. Part. Nucl. Lett. 10, 660
Yamazaki K and Matsui T 2013 Nucl. Phys. A 913, 19
Yamazaki K and Matsui T 2014 Nucl. Phys. A 922, 237
Blaschke D, Dubinin A and Buballa M 2015 Phys. Rev. D 91, 125040
Blaschke D, Dubinin A and Turko L 2015 Phys. Part. Nucl. 46, 732
Borsanyi A, Fodor Z, Hoelbling C, Katz S D, Krieg S and Szabo K K 2014 Phys. Lett. B 730, 99
Bazavov A, et al. [HotQCD Collaboration] 2014 Phys. Rev. D 90, 094503
Kaczmarek O, et al. 2011 Phys. Rev. D 83, 014504
R. Zimmermann and H. Stolz,
physica status solidi (b) 131, 151 (1985).
M. Schmidt, G. Röpke and H. Schulz,
Annals Phys. 202, 57 (1990).
|
1511.00147
|
We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations.
§ INTRODUCTION
Drift-diffusion equations with nonlocal dissipation naturally occur in hydrodynamics and in models of electroconvection. The study of these equations in bounded domains is hindered by a lack of explicit information on the kernels of the nonlocal operators appearing in them. In this paper we develop tools adapted for the Dirichlet boundary case: the Córdoba-Córdoba inequality (<cit.>) and a nonlinear lower bound in the spirit of (<cit.>), and commutator estimates. Lower bounds for the fractional Laplacian are instrumental in proofs of regularity of solutions to nonlinear nonlocal drift-diffusion equations. The presence of boundaries requires natural modifications of the bounds. The nonlinear bounds are proved using a representation based on the heat kernel and fine information regarding it (<cit.>, <cit.>, <cit.>). Nonlocal diffusion operators in bounded domains do not commute in general with differentiation. The commutator estimates are proved using the method of harmonic extension and results of (<cit.>). We apply these tools to linear drift-diffusion equations with nonlocal dissipation, where we obtain strong global bounds, and to global existence of weak solutions of the surface quasi-geostrophic equation (SQG) in bounded domains.
We consider a bounded open domain $\Omega\subset \Rr^d$ with smooth (at least $C^{2,\alpha}$) boundary. We denote by $\D$ the Laplacian operator with homogeneous Dirichlet boundary conditions. Its $L^2(\Omega)$ - normalized eigenfunctions are denoted $w_j$, and its eigenvalues counted with their multiplicities are denoted $\lambda_j$:
-w_j = λ_j w_j.
It is well known that $0<\lambda_1\le...\le \lambda_j\to \infty$ and that $-\D$ is a positive selfadjoint operator in $L^2(\Omega)$ with domain ${\mathcal{D}}\left(-\D\right) = H^2(\Omega)\cap H_0^1(\Omega)$.
The ground state $w_1$ is positive and
c_0d(x) ≤w_1(x)≤C_0d(x)
holds for all $x\in\Omega$, where
d(x) = dist(x,Ω)
and $c_0, \, C_0$ are positive constants depending on $\Omega$. Functional calculus can be defined using the eigenfunction expansion. In particular
(-)^αf = ∑_j=1^∞λ_j^α f_j w_j
\[
f_j =\int_{\Omega}f(y)w_j(y)dy
\]
for $f\in{\mathcal{D}}\left(\left (-\D\right)^{\alpha}\right) = \{f\left |\right. \; (\lambda_j^{\alpha}f_j)\in \ell^2(\mathbb N)\}$.
We will denote by
ł^s = (-)^α, s=2αlambdas
the fractional powers of the Dirichlet Laplacian, with $0\le \alpha\le 1$
and with $\|f\|_{s,D}$ the norm in ${\mathcal{D}}\left (\l^s\right)$:
f_s,D^2 = ∑_j=1^∞λ_j^sf_j^2.
It is well-known and easy to show that
\[
{\mathcal{D}}\left( \l \right) = H_0^1(\Omega).
\]
Indeed, for $f\in{\mathcal{D}}\left (-\D\right)$ we have
\[
\|\na f\|^2_{L^2(\Omega)} = \int_{\Omega}f\left(-\D\right)fdx = \|\l f\|_{L^2(\Omega)}^2 = \|f\|^2_{1,D}.
\]
We recall that the Poincaré inequality implies that the Dirichlet integral on the left-hand side above is equivalent to the norm in $H_0^1(\Omega)$
and therefore the identity map from the dense subset ${\mathcal{D}}\left(-\D\right)$ of $H_0^1(\Omega)$ to ${\mathcal D}\left(\l\right)$ is an isometry, and thus $H_0^1(\Omega)\subset {\mathcal{D}}\left(\l\right)$. But ${\mathcal{D}}\left(-\D\right)$ is dense in ${\mathcal D}\left(\l\right)$ as well, because finite linear combinations of eigenfunctions are dense in ${\mathcal D}\left(\l\right)$. Thus the opposite inclusion is also true, by the same isometry argument.
Note that in view of the identity
λ^α = c_α∫_0^∞(1-e^-tλ)t^-1-αdt,
\[
1 = c_{\alpha} \int_0^{\infty}(1-e^{-s})s^{-1-\alpha}ds,
\]
valid for $0\le \alpha <1$, we have the representation
((-)^αf)(x) = c_α∫_0^∞[f(x)-e^tf(x)]t^-1-αdt
for $f\in{\mathcal{D}}\left(\left (-\D\right)^{\alpha}\right)$.
We use precise upper and lower bounds for the kernel $H_D(t,x,y)$ of the heat operator,
(e^tf)(x) = ∫_ΩH_D(t,x,y)f(y)dy .
These are as follows (<cit.>,<cit.>,<cit.>).
There exists a time $T>0$ depending on the domain $\Omega$ and constants
$c$, $C$, $k$, $K$, depending on $T$ and $\Omega $ such that
cmin(w_1(x)|x-y|, 1)min(w_1(y)|x-y|, 1)t^-d2e^-|x-y|^2kt≤
min(w_1(x)|x-y|, 1)min(w_1(y)|x-y|, 1)t^-d2e^-|x-y|^2Kt
holds for all $0\le t\le T$. Moreover
|_x H_D(t,x,y)|H_D(t,x,y)≤C{
1d(x), √(t)≥d(x),
1√(t)(1 + |x-y|√(t)), √(t)≤d(x)
holds for all $0\le t\le T$.
Note that, in view of
H_D(t,x,y) = ∑_j=1^∞e^-tλ_jw_j(x)w_j(y) ,
elliptic regularity estimates and Sobolev embedding which imply uniform absolute convergence of the series (if $\pa\Omega$ is smooth enough), we have that
_1^βH_D(t,y,x) = _2^βH_D(t,x,y)
= ∑_j=1^∞e^-tλ_j_y^βw_j(y)w_j(x)
for positive $t$, where we denoted by $\pa_1^{\beta}$ and $\pa_2^{\beta}$ derivatives with respect to the first spatial variables and the second spatial variables, respectively.
Therefore, the gradient bounds (<ref>) result in
|_y H_D(t,x,y)|H_D(t,x,y)≤C{
1d(y), √(t)≥d(y),
1√(t)(1 + |x-y|√(t)), √(t)≤d(y).
§ THE CÓRDOBA - CÓRDOBA INEQUALITY
Let $\Phi$ be a $C^2$ convex function satisfying $\Phi(0)= 0$. Let $f\in C_0^{\infty}(\Omega)$ and let $0\le s\le 2$. Then
Φ'(f)ł^s f - ł^s(Φ(f))≥0
holds pointwise almost everywhere in $\Omega$.
In view of the fact that both $f\in H_0^1(\Omega)\cap H^2(\Omega)$ and
$\Phi(f)\in H_0^1(\Omega)\cap H^2(\Omega)$, the terms in the inequality (<ref>) are well defined. We
[(-)^αf]_ϵ(x)= c_α∫_ϵ^∞[f(x)-e^tf(x)]t^-1-αdt
and approximate the representation (<ref>):
((-)^αf)(x) = lim_ϵ→0[(-)^αf]_ϵ(x).
The limit is strong in $L^2(\Omega)$. We start the calculation with this approximation and then we rearrange terms:
\[
\ba
\Phi'(f(x))\left[\l^{2\alpha} f\right]_{\epsilon}(x) - \left[\l^{2\alpha} (\Phi(f))\right]_{\epsilon}(x) \\
= c_{\alpha}\int_\epsilon^{\infty} t^{-1-\alpha}dt\int_{\Omega}\left\{\Phi'(f(x))\left[\fr{1}{|\Omega|}f(x) -H_D(t,x,y)f(y)\right]
-\fr{1}{|\Omega|}\Phi(f(x)) + H_D(t,x,y)\Phi(f(y))\right\}dy\\
=c_{\alpha}\int_\epsilon^{\infty} t^{-1-\alpha}dt\int_{\Omega}H_D(t,x,y)\left[\Phi(f(y))-\Phi(f(x)) - \Phi'(f(x))(f(y)-f(x))\right]dy\\
+ c_{\alpha}\int_\epsilon^{\infty}t^{-1-\alpha}dt\int_{\Omega}\left[f(x)\Phi'(f(x))-\Phi(f(x))\right](\fr{1}{|\Omega|} - H_D(t,x,y))dy\\
= c_{\alpha}\int_\epsilon^{\infty} t^{-1-\alpha}dt\int_{\Omega}H_D(t,x,y)\left[\Phi(f(y))-\Phi(f(x)) - \Phi'(f(x))(f(y)-f(x))\right]dy\\
\ea
\]
Because of the convexity of $\Phi$ we have
\[
\Phi(b)-\Phi(a) -\Phi'(a)(b-a)\ge 0, \quad \forall\;\;a, b\in \Rr,
\]
and because $\Phi(0) =0$ we have
\[
a\Phi'(a)\ge \Phi(a), \quad \forall \;\; a\in\Rr.
\]
Consequently $f(x)\Phi'(f(x))-\Phi(f(x))\ge 0$ holds everywhere. The function
\[
\theta = e^{t\D}1
\]
solves the heat equation $\pa_t\theta -\D\theta =0$ in $\Omega$, with homogeneous Dirichlet boundary conditions, and with initial data equal everywhere to $1$. Although $1$ is not in the domain of $-\D$, $e^{t\D}$ has a unique extension to $L^2(\Omega)$ where $1$ does belong, and on the other hand, by the maximum principle $0\le\theta(x,t)\le 1$ holds for $t\ge 0$, $x\in\Omega$. It is only because $1\notin{\mathcal{D}}(-\D)$ that we had to use the $\epsilon$ approximation. Now we discard the nonnegative term
\[
\left[f(x)\Phi'(f(x))-\Phi(f(x))\right]c_{\alpha}\int_{\epsilon}^{\infty}(1-\theta(x,t))t^{-1-\alpha}dt
\]
in the calculation above, and deduce that
Φ'(f(x))[ł^2α f]_ϵ(x) - [ł^2α (Φ(f))]_ϵ(x) ≥0
as an element of $L^2(\Omega)$. (This simply means that its integral against any nonnegative $L^2(\Omega)$ function is nonnegative.) Passing to the limit $\epsilon\to 0$ we obtain the inequality (<ref>). If $\Phi $ and the boundary of the domain are smooth enough then we can prove that the terms in the inequality are continuous, and therefore the inequality holds everywhere.
§ THE NONLINEAR BOUND
We prove a bound in the spirit of (<cit.>). The nonlinear lower bound was
used as an essential ingredient in proofs of global regularity for drift-diffusion equations with nonlocal dissipation.
Let $f\in L^{\infty}(\Omega)\cap {\mathcal D}(\l^{2\alpha})$, $0\le \alpha<1$.
Assume that $f= \pa q$ with $q \in L^{\infty}(\Omega)$ and $\pa$ a first order derivative. Then there exist constants $c$, $C$ depending on $\Omega$ and $\alpha$ such that
fł^2α f -12ł^2α f^2 ≥cq_L^∞^-2α |f_d|^2+2α
holds pointwise in $\Omega$, with
|f_d(x)| = {
|f(x)|, |f(x)| ≥Cq_L^∞(Ω)max(1diam(Ω),1d(x)),
0, |f(x)| ≤Cq_L^∞(Ω)max(1diam(Ω),1d(x)).
Proof. We start the calculation using the inequality
fł^2αf - 12ł^2α f^2 ≥12c_α∫_0^∞ψ(tτ)t^-1-αdt∫_ΩH_D(t,x,y)(f(x)-f(y))^2dy
where $\tau>0$ is arbitrary and $0\le \psi(s)\le 1$ is a smooth function, vanishing identically for $0\le s\le 1$ and equal identically to $1$ for $s\ge 2$. This follows repeating the calculation of the proof of the Córdoba-Córdoba inequality with $\Phi(f)=\fr{1}{2}f^2$:
\[
\ba
f(x)\left[\l^{2\alpha}f\right]_{\epsilon}(x) - \fr{1}{2}\left[\l^{2\alpha} f^2\right]_{\epsilon}(x) \\
= c_{\alpha}\int_\epsilon^{\infty}t^{-1-\alpha}\int_{\Omega}\left\{\left[\fr{1}{|\Omega|} f(x)^2 - f(x)H_D(t,x,y)f(y)\right]- \fr{1}{2|\Omega|}f^{2}(x) + \fr{1}{2}H_D(t,x,y)f^2(y)\right\}dy\\
=c_{\alpha}\int_\epsilon^{\infty}t^{-1-\alpha}dt\int_{\Omega}\left\{\fr{1}{2}\left[H_D(t,x,y)(f(x)-f(y))^2\right] + \fr{1}{2}f^2(x)\left[\fr{1}{|\Omega|} -H_D(t,x,y)\right]\right\}dy \\
= c_{\alpha}\int_\epsilon^{\infty}t^{-1-\alpha}dt\int_{\Omega}\left\{\fr{1}{2}\left[H_D(t,x,y)(f(x)-f(y))^2\right]dy + \fr{1}{2}f^2(x)\left[1-e^{t\D}1\right](x)\right\}\\
\ge c_{\alpha}\int_\epsilon^{\infty}t^{-1-\alpha}dt\int_{\Omega}\fr{1}{2}H_D(t,x,y)\left(f(x)-f(y)\right)^2dy
\ea
\]
where in the last inequality we used the maximum principle again. Then, we choose $\tau>0$ and let $\epsilon<\tau$. It follows that
\[
f(x)\left[\l^{2\alpha}f\right]_{\epsilon}(x) - \fr{1}{2}\left[\l^{2\alpha} f^2\right]_{\epsilon}(x) \ge
\fr{1}{2}c_{\alpha}\int_0^{\infty}\psi\left(\fr{t}{\tau}\right)t^{-1-\alpha}dt\int_{\Omega}H_D(t,x,y)\left(f(x)-f(y)\right)^2dy.
\]
We obtain (<ref>) by letting $\epsilon\to 0$. We restrict to $t\le T$,
[fł^2αf - 12ł^2α f^2](x) ≥12c_α∫_0^Tψ(tτ)t^-1-αdt∫_ΩH_D(t,x,y)(f(x)-f(y))^2dy
and open brackets in (<ref>):
[fł^2αf - 12ł^2α f^2](x)
- f(x)c_α∫_0^Tψ(tτ)t^-1-αdt∫_ΩH_D(t,x,y)f(y)dy
≥|f(x)|[ 12|f(x)| I(x) - J(x)]
I(x) = c_α∫_0^Tψ(tτ)t^-1-αdt∫_ΩH_D(t,x,y)dy,
J(x) = c_α|∫_0^Tψ(tτ)t^-1-αdt∫_ΩH_D(t,x,y)f(y)dy|
= c_α|∫_0^Tψ(tτ)t^-1-αdt∫_Ω_yH_D(t,x,y)q(y)dy|.
We proceed with a lower bound on $I$ and an upper bound on $J$. For the lower bound on $I$ we note that
\[
\theta (x,t) = \int_{\Omega}H_D(t,x,y)dy\ge \int_{|x-y|\le \fr{d(x)}{2}} H_D(t,x,y)dy
\]
because $H_D$ is positive. Using the lower bound in (<ref>) we have that
$|x-y|\le \fr{d(x)}{2}$ implies
\[
\fr{w_1(x)}{|x-y|}\ge 2c_0,\quad \fr{w_1(y)}{|x-y|}\ge c_0,
\]
and then, using the lower bound in (<ref>) we obtain
\[
H_D(t,x,y) \ge 2cc_0^2t^{-\fr{d}{2}}e^{-\fr{|x-y|^2}{kt}}.
\]
Integrating it follows that
\[
\theta(x,t) \ge 2 cc_0^2\omega_{d-1}k^{\fr{d}{2}}\int_0^{\fr{d(x)}{2\sqrt{kt}}}\rho^{d-1}e^{-\rho^2}d\rho
\]
If $\fr{d(x)}{2\sqrt{kt}}\ge 1$ then the integral is bounded below by
$\int_0^1\rho^{d-1}e^{-\rho^2}d\rho$. If $\fr{d(x)}{2\sqrt{kt}}\le 1$ then
$\rho\le 1$ implies that the exponential is bounded below by $e^{-1}$ and so
θ(x,t)≥c_1min{1, (d(x)√(t))^d}
for all $0\le t\le T$ where $c_1$ is a positive constant, depending on $\Omega$. Because
\[
I(x) = \int_0^T\psi\left(\fr{t}{\tau}\right)t^{-1-\alpha}\theta(x,t)dt
\]
we have
\[
\ba
I(x)\ge c_1\int_0^{\min(T, d^2(x))}\psi\left(\fr{t}{\tau}\right)t^{-1-\alpha}dt\\
= c_1\tau^{-\alpha}\int_1^{\tau^{-1}(\min(T, d^2(x)))}\psi(s)s^{-1-\alpha}ds
\ea
\]
Therefore we have that
I(x)≥c_2 τ^-α
with $c_2 = c_1\int_1^2\psi(s)s^{-1-\alpha}ds$, a positive constant depending only on $\Omega$ and $\alpha$, provided $\tau$ is small enough,
τ≤12min(T, d^2(x)).
In order to bound $J$ from above we use the upper bound (<ref>) which yields
∫_Ω|_y H_D(t,x,y)|dy ≤C_1 t^-12
with $C_1$ depending only on $\Omega$. Indeed,
\[
\ba
\int_{d(y)\ge \sqrt{t}}|\na_y H_D(t,x,y)|dy \\\le C_2t^{-\fr{1}{2}}
\int_{\Rr^d}\left (1 + \fr{|x-y|}{\sqrt{t}}\right)t^{-\fr{d}{2}}e^{-\fr{|x-y|^2}{kt}}dy\\
= C_3t^{-\fr{1}{2}}
\ea
\]
and, in view of the upper bound in (<ref>), $\fr{1}{d(y)}w_1(y)\le C_0$ and the upper bound in (<ref>),
\[
\ba
\int_{d(y)\le \sqrt{t}}|\na_y H_D(t,x,y)|dy \\\le C_4\int_{\Rr^d}\fr{1}{|x-y|}t^{-\fr{d}{2}}e^{-\fr{|x-y|^2}{Kt}}dy = C_5t^{-\fr{1}{2}}
\ea
\]
\[
J\le \|q\|_{L^{\infty}(\Omega)}\int_0^T\psi\left(\fr{t}{\tau}\right)t^{-1-\alpha}dt\int_{\Omega}|\na_y H_D(t,x,y)|dy
\]
and therefore, in view of (<ref>)
\[
J\le C_1 \|q\|_{L^{\infty}(\Omega)}\int_0^T\psi\left(\fr{t}{\tau}\right )t^{-\fr{3}{2}-\alpha}dt
\]
and therefore
J ≤C_6q_L^∞(Ω)τ^-12-α
\[
C_6 = C_1\int_1^{\infty}\psi(s)s^{-\fr{3}{2}-\alpha}ds
\]
a constant depending only on $\Omega$ and $\alpha$. Now, because of the lower bound (<ref>),
if we can choose $\tau$ so that
\[
J(x) \le \fr{1}{4} |f(x)|I(x)
\]
then it follows that
[fł^2αf - 12ł^2α f^2](x) ≥14f^2(x)I(x).
Because of the bounds (<ref>), (<ref>) the choice
τ(x) = c_3q_L^∞^2|f(x)|^2
with $c_3= 16 C_6^2c_2^{-2}$ achieves the desired bound. The requirement (<ref>) limits the possibility of making this choice to the situation
c_3q_L^∞^2|f(x)|^2 ≤12min(T, d^2(x))
which leads to the statement of the theorem. Indeed, if (<ref>) is allowed then the lower bound in (<ref>) becomes
[fł^2αf - 12ł^2α f^2](x) ≥cq_L^∞^-2α |f_d|^2+2α
with $c=\fr{1}{4}c_2c_3^{-\alpha}$.
§ COMMUTATOR ESTIMATES
We start by considering the commutator $[\na,\l]$ in $\Omega = \Rr^d_+$. The heat kernel with Dirichlet boundary conditions is
\[
H(x,y,t) = ct^{-\fr{d}{2}}\left( e^{-\fr{|x-y|^2}{4t}} - e^{-\fr{|x-{\widetilde {y}}|^2}{4t}}\right)
\]
where $\widetilde{y} = (y_1,\dots, y_{d-1}, -y_d)$. We claim that
∫_Ω (_x+ _y)H(x,y,t) dy ≤Ct^-12e^-x_d^24t.
Indeed, the only nonzero component occurs when the differentiation is with respect to the normal direction, and then
\[
(\pa_{x_d} + \pa_{y_d})H(x,y,t) = ct^{-\fr{d}{2}}e^{-\fr{|x'-y'|^2}{4t}}\left(\fr{x_d+y_d}{t}\right) e^{-\fr{(x_d+y_d)^2}{4t}}
\]
where we denoted $x' = (x_1,\dots, x_{d-1})$ and $y'=(y_1, \dots, y_{d-1})$.
\[
\ba
\int_{\Omega} (\na_x+ \na_y)H(x,y,t) dy \le Ct^{-\fr{1}{2}}\int_0^{\infty}\left(\fr{x_d+y_d}{t}\right )e^{-\fr{(x_d+y_d)^2}{4t}}dy_d\\
= Ct^{-\fr{1}{2}}\int_{\fr{x_d}{\sqrt {t}}}^{\infty}\xi e^{-\fr{\xi^2}{4}}d\xi
\\
= Ct^{-\fr{1}{2}}e^{-\fr{x_d^2}{4t}}.
\ea
\]
\[
K(x,y) = \int_0^{\infty}t^{-\fr{3}{2}}(\na_x + \na_y) H(x,y,t)dt
\]
\[
\int_{\Omega}K(x,y) dy \le C\int_0^{\infty}t^{-2}e^{-\fr{x_d^2}{4t}}dt
= \fr{C}{x_{d}^{2}}.
\]
The commutator $[\na, \l]$ is computed as follows
\[
\ba
[\na, \l]f(x) = \int_0^{\infty}t^{-\fr{3}{2}}\int_{\Omega}\left[\na_xH_D(x,y,t)f(y) - H_D(x,y,t)\na_y f(y)\right]dydt\\
=\int_0^{\infty}t^{-\fr{3}{2}}\int_{\Omega}(\na_x+\na_y)H_{D}(x,y,t)f(y)dydt \\
\ea
\]
We have proved thus that the kernel $K(x,y)$ of the commutator obeys
∫_ΩK(x,y) dy ≤Cd(x)^-2
and therefore we obtain, for instance, for any $p,q\in [1,\infty]$ with $p^{-1}+q^{-1}=1$
\[
\left |\int_{\Omega} g[\na,\l]fdx\right | \le C\left(\int_{\Omega}d(x)^{-2}|f(x)|^pdx\right)^{\fr{1}{p}}\left(\int_{\Omega}d(x)^{-2}|g(x)|^qdx\right)^{\fr{1}{q}}.
\]
In general domains, the absence of explicit expressions for the heat kernel with Dirichlet boundary conditions requires a less direct approach to commutator estimates.
We take thus an open bounded domain $\Omega\subset\Rr^d$ with smooth boundary and describe the square root of the Dirichlet Laplacian using the harmonic extension. We denote
\[
Q= \Omega\times\Rr_+ =\{(x,z)\left|\right. \; x\in\Omega, z>0\}
\]
and consider the traces of functions in $H_{0,L}^1(Q)$,
\[
H_{0,L}^1(Q) = \{v\in H^1(Q)\; \left |\right. \; v(x,z)=0,\; x\in\pa\Omega, \; z>0 \}
\]
V_0(Ω) = {f |. ∃v∈H_0,L^1(Q), f(x)= v(x,0), x∈Ω}
where we slightly abused notation by referring to the images of $v$ under restriction operators as $v(x,z)$ for $x\in\pa\Omega$, and as $v(x,0)$ for $x\in \Omega$.
We recall from (<cit.>) that, on one hand,
V_0(Ω) = {f∈H^12(Ω) |. ∫_Ωf^2(x)d(x)dx <∞}
with norm
\[
\|f\|^2_{V_0}= \|f\|^2_{H^{\fr{1}{2}}(\Omega)} + \int_{\Omega}\fr{f^2(x)}{d(x)}dx,
\]
and on the other hand $V_0(\Omega) = {\mathcal D}(\l^{\fr{1}{2}})$, i.e.
V_0(Ω) = {f∈L^2(Ω) |. f = ∑_jf_jw_j, ∑_jλ_j^12f_j^2<∞}
with equivalent norm
\[
\|f\|^2_{\fr{1}{2}, D} = \sum_{j=1}^{\infty}\lambda_j^{\fr{1}{2}}f_j^2 = \|\l^{\fr{1}{2}}f\|_{L^2(\Omega)}^2.
\]
The harmonic extension of $f$ will be denoted $v_f$. It is given by
v_f(x,z) = ∑_j=1^∞f_je^-z√(λ_j)w_j(x)
and the operator $\l$ is then identified with
łf = -(_z v_f)_| . z=0
Note that if $f\in V_0(\Omega)$ then $v_f\in H^1(Q)$. Note also, that $v_f$ decays exponentially in the sense that
v_f_e^zlH^1(Q) = e^zlv_f_L^2(Q) + e^zlv_f_L^2(Q) ≤C f_V_0
holds with $\ell = \fr{\lambda_1}{4}$. We use a lemma in $Q$:
lemmaz Let $F\in H^{-1}(Q)$ (the dual of $H_0^1(Q)$).
Then the problem
-Δu = F, Q,
u = 0, Q
has a unique weak solution $u\in H_0^1(Q)$. If $F\in L^2(Q)$ and if there exists $l>0$ so that
\[
\|e^{zl}F\|_{L^2(Q)}^2=\int e^{2zl}|F(x,z)|^2dxdz <\infty
\]
then $u\in H_0^1(Q)\cap H^2(Q)$ and it satisfies
\[
\|u\|_{H^2(Q)}\le C\|e^{zl}F\|_{L^2(Q)}
\]
with $C$ a constant depending only on $\Omega$ and $l$.
Proof. We consider the domain $U= \Omega\times \Rr$ and take the odd extension of $F$ to $U$, $F(x,-z)=-F(x,z)$. The existence of a weak solution in $H_0^1(U)$ follows by variational methods, by minimizing
\[
I(v) = \int_U\left(\fr{1}{2}|\na v|^2 + vF\right)dxdz
\]
among all odd functions $v\in H_0^1(U)$. The domain $U$ has finite width, so the Poincaré inequality
\[
\|\na v\|^2_{L^2(U)}\ge c\|v\|^2_{L^2(U)}
\]
is valid for functions in $H_0^1(U)$. This allows to show existence and uniqueness of weak solutions.
If $F\in L^2(U)$ we obtain locally uniform elliptic estimates
\[
\|u\|_{H^2(U_j)}\le C\|F\|_{L^2(V_{j})}
\]
where $U_j= \{(x,z)\left|\right.\, x\in\Omega, z\in (j-1,j+1)\}$, $V_j= \{(x,z)\left|\right.\, x\in\Omega, z\in (j-2,j+2)\}$, and $j =\pm\fr{1}{2}, \pm 1,\pm\fr{3}{2},\dots$, i.e. $j\in \fr{1}{2}{\mathbb Z}$. The constant $C$ does not depend on $j$. Because of the decay assumption on $F$, the
estimates can be summed.
Let $a\in B(\Omega)$ where $B(\Omega) = W^{2,d}(\Omega)\cap W^{1,\infty}(\Omega)$, if $d\ge 3$, and $B(\Omega) = W^{2,p}(\Omega)$ with $p>2$, if $d=2$. There exists a constant $C$,
depending only on $\Omega$, such that
[a,ł]f_12, D≤Ca_B(Ω)f_12, D
holds for any $f\in V_0(\Omega)$, with
\[
\|a\|_{B(\Omega)} = \|a\|_{W^{2,d}(\Omega)} + \|a\|_{W^{1,\infty}(\Omega)}
\]
if $d\ge 3$ and
\[
\|a\|_{B(\Omega)} = \|a\|_{W^{2,p}(\Omega)}
\]
with $p>2$, if $d=2$.
In order to compute $v_{af}$, let us note that $av_f\in H_{0,L}^1(Q)$, and
\[
\Delta(av_f) = v_f\Delta_x a + 2\na_x a \cdot \na v_f
\]
and, because $v_f\in e^{zl}H^1(Q)$ and $a\in B(\Omega)$ we have that
\[
\|\Delta (av_f)\|_{L^2(e^{zl}dzdx)}\le C \|a\|_{B(\Omega)}\|v_f\|_{e^{zl}H^1(Q)}.
\]
\[
\left\{
\ba
\Delta u = \Delta (av_f) \quad {\mbox{in}}\; Q,\\
u = 0 \quad {\mbox{on}} \;\pa Q,
\ea
\right.
\]
we obtain $u\in H_0^1(Q)\cap H^2(Q)$. This follows from Lemma <ref> above. Note that $\pa_z u\in H_{0,L}^1(Q)$. Then
\[
v_{af} = av_f -u
\]
\[
a\l f-\l(af) = -a(\pa_z v_f)_{\left|\right. z=0} +\pa_z(av_f-u)_{\left|\right. z=0}= -\pa_z u_{\left |\right.z=0}.
\]
The estimate follows from elliptic estimates and restriction estimates
\[
\|\pa_zu_{\left|\right. z=0}\|_{V_0}\le C\|\pa_z u\|_{H^1(Q)}\le C\|a\|_{B(\Omega)}\|v_f\|_{e^{zl}H^1(Q)}\le C\|a\|_{B(\Omega)}\|f\|_{V_0}
\]
thmcomthm Let a vector field $a$ have components in $B(\Omega)$ defined above, $a\in \left(B(\Omega)\right)^d$. Assume that the normal component of the trace of $a$ on the boundary vanishes,
\[
a_{\left |\right. \pa \Omega}\cdot n = 0
\]
(i.e the vector field is tangent to the boundary).
There exists a constant $C$ such that
[a·,ł]f_12, D ≤Ca_B(Ω)f_32,D
holds for any $f$ such that $f\in {\mathcal{D}}\left(\l^{\fr{3}{2}}\right)$.
Proof. In order to compute $v_{a\cdot\na f}$ we note that
\[
\Delta (a\cdot\na v_f) = \Delta a\cdot \na v_f + \na a\cdot\na\na v_f,
\]
and because $v_f\in e^{zl}H^{2}(Q)$ and $a\in B(\Omega)$ we have that
\[
\|\Delta (a\cdot\na v_f)\|_{L^2(e^{zl}dzdx)}\le C\|a\|_{B(\Omega)}\|v_f\|_{e^{zl}H^2(Q)}.
\]
Then solving
\[
\left \{
\ba
\Delta u = \Delta (a\cdot\na v_f) \quad {\mbox{in}}\; Q,\\
u = 0 \quad {\mbox{on}} \; \pa Q,
\ea
\right.
\]
we obtain $u\in H^{2}(Q)$ (by Lemma <ref>) and therefore $\pa_z u\in H^1_{0,L}(Q)$. Consequently $-\pa_z u_{\left|\right. z=0}\in V_0(\Omega)$. Because $v_f$ vanishes on the boundary and $a\cdot\na$ is tangent to the boundary, it follows that $a\cdot\na v_f\in H_{0,L}^{1}(Q)$ (vanishes on the lateral boundary of $Q$ and is in $H^{1}(Q)$) and therefore
\[
v_{a\cdot\na f} = a\cdot\na v_f - u.
\]
\[
[a\cdot\na, \l]f = -\pa_z u_{\left |\right. z=0}.
\]
The estimate (<ref>) follows from the elliptic estimates and restriction estimates on $u$, as above:
\[
\|\pa_zu_{\left |\right. z=0}\|_{V_0}\le C \|\pa_z u\|_{H^1(Q)}
\le C\|a\|_{B(\Omega)}\|v_f\|_{e^{zl}H^2(Q)}\le C \|a\|_{B(\Omega)}\|f\|_{\fr{3}{2},D}
\]
§ LINEAR TRANSPORT AND NONLOCAL DIFFUSION
We study the equation
_tθ+ u·θ+ łθ= 0
with initial data
θ(x,0) = θ_0
in the bounded open domain $\Omega\subset\Rr^d$ with smooth boundary. We assume that $u=u(x,t)$ is divergence-free
·u = 0,
that $u$ is smooth
u∈L^2(0,T; B(Ω)^d),
and that $u$ is parallel to the boundary
u_|. Ω·n = 0.
We consider zero boundary conditions for $\theta$. Strictly speaking, because this is a first order equation, it is better to think of these as a constraint on the evolution equation. We satrt with initial data $\theta_0$ which vanish on the boundary, and maintain this property in time. The transport evolution
\[
\pa_t\theta + u\cdot\na\theta = 0
\]
and, separately, the nonlocal diffusion
\[
\pa_t\theta + \l \theta = 0
\]
keep the constraint of $\theta_{\left |\right. \pa \Omega}= 0$. Because the
operators $u\cdot\na $ and $\l$ have the same differential order, neither dominates the other, and the linear evolution needs to be treated carefully. We start by considering Galerkin approximations. Let
P_mf = ∑_j=1^m f_jw_j, f=∑_j=1^∞f_jw_j,
and let
θ_m (x,t) = ∑_j=1^mθ_j^(m)(t)w_j(x)
_tθ_m + P_m(u·θ_m) + łθ_m =0
with initial data
θ_m(x,0) = (P_mθ_0)(x).
These are ODEs for the coefficients $\theta_j^{(m)}(t)$, written conveniently. We prove bounds that are independent of $m$ and pass to the limit. Note that by construction
\[
\theta_m\in {\mathcal{D}}\left(\l^r\right), \quad \forall r\ge 0.
\]
We start with
12ddtθ_m^2_L^2(Ω) + θ_m^2_V_0 = 0
which implies
sup_0≤t≤T12θ_m(·, t)^2_L^2(Ω) + ∫_0^Tθ_m^2_V_0dt ≤12θ_0^2_L^2(Ω).
This follows because of the divergence-free condition and the fact that $u_{\left |\right. \pa\Omega}$ is parallel to the boundary. Next, we apply $\l$ to (<ref>). For convenience, we denote
[ł, u·]f = Γf
because $u$ is fixed throughout this section. Because $P_m$ and $\l$ commute, we have thus
_tłθ_m + P_m(u·łθ_m + θ_m) +ł^2θ_m= 0.
Now, we multiply (<ref>) by $\l^3\theta_m$ and integrate. Note that
\[
\int_{\Omega} P_m\left(u\cdot\na \l\theta_m + \G \theta_m\right) \l^3\theta_m dx=
\int_{\Omega} \left(u\cdot\na \l\theta_m + \G \theta_m\right) \l^3\theta_m dx
\]
because $P_m\theta_m = \theta_m$ and $P_m$ is selfadjoint. We bound the term
\[
\left |\int_{\Omega} \G\theta_m\l^3\theta_mdx\right |\le \|\G\theta_m\|_{V_0}\|\l^{2.5}\theta_m\|_{L^2(\Omega)}
\]
and use Theorem <ref> (<ref>) to deduce
\[
\left |\int_{\Omega} \G\theta_m\l^3\theta_mdx\right |\le C\|u\|_{B(\Omega)}\|\l\theta_m\|_{V_0}\|\l^{2.5}\theta_m\|_{L^2(\Omega)}.
\]
We compute
\[
\ba
\int_{\Omega}(u\cdot\na\l\theta_m)\l^3\theta_m dx = \int_{\Omega}\l^2(u\cdot\na\l\theta_m)\l\theta_m \\
=\int_{\Omega}\left [(-\Delta u)\cdot\na \l\theta_m - 2\na u\cdot\na\na\l\theta_m\right]\l\theta_mdx + \int_{\Omega}(u\cdot\na\l^3\theta_m)\l\theta_m dx\\
= \int_{\Omega}\left [(-\Delta u)\cdot\na \l\theta_m - 2\na u\cdot\na\na\l\theta_m\right]\l\theta_mdx - \int_{\Omega}\l^3\theta_m(u\cdot\na \l\theta_m)dx\\
=\int_{\Omega}\left [((-\Delta u)\cdot\na \l\theta_m)\l\theta_m + 2\na u\na\l\theta_m\na\l\theta_m\right]dx-\int_{\Omega}(u\cdot\na\l\theta_m)\l^3\theta_m dx.
\ea
\]
In the first integration by parts we used the fact that $\l^3\theta_m$ is a finite linear combination of eigenfunctions which vanish at the boundary. Then we use the fact that $\l^2 = -\Delta$ is local. In the last equality we integrated by parts using the fact that $\l \theta_m$ is a finite linear combination of eigenfunctions which vanish at the boundary and the fact that $u$ is divergence-free.
It follows that
\[
\int_{\Omega}(u\cdot\na\l\theta_m)\l^3\theta_m dx = \fr{1}{2}\int_{\Omega}\left [((-\Delta u)\cdot\na \l\theta_m)\l\theta_m + 2\na u\na\l\theta_m\na\l\theta_m\right]dx
\]
and consequently
\[
\left| \int_{\Omega}(u\cdot\na\l\theta_m)\l^3\theta_m dx\right| \le C\|u\|_{B(\Omega)}\|\l^2\theta_m\|_{L^2(\Omega)}^2
\]
We obtain thus
sup_0≤t≤Tł^2θ_m(·, t)^2_L^2(Ω) + ∫_0^Tł^2θ_m^2_V_0dt ≤Cł^2θ_0^2_L^2(Ω)e^C∫_0^Tu_B(Ω)^2dt.
Passing to the limit $m\to\infty$ is done using the Aubin-Lions Lemma (<cit.>). We obtain
thmlinthm Let $u\in L^2(0,T; B(\Omega)^d)$ be a vector field parallel to the boundary. Then the equation (<ref>) with initial data $\theta_0\in H_0^1(\Omega)\cap H^2(\Omega)$ has unique solutions belonging to
\[
\theta\in L^{\infty}(0,T; H^2(\Omega)\cap H_0^1(\Omega))\cap L^2(0,T; H^{2.5}(\Omega)).
\]
If the initial data $\theta_0\in L^p(\Omega)$, $1\le p\le \infty$, then
sup_0≤t ≤Tθ(·, t)_L^p(Ω) ≤θ_0_L^p(Ω)
The estimate (<ref>) holds because, by use of Proposition <ref> for the diffusive part and integration by parts for the transport part, we have for solutions of (<ref>)
\[
\fr{d}{dt}\|\theta\|^p_{L^p(\Omega)}\le 0,
\]
$1\le p<\infty$. The $L^{\infty}$ bound follows by taking the limit $p\to\infty$ in (<ref>).
§ SQG
We consider now the equation
_t θ+ u·θ+ łθ= 0
u = R_D^⊥θu
R_D = ł^-1
in a bounded open domain in $\Omega\subset \Rr^2$ with smooth boundary. Local existence of smooth solutions is possible to prove using methods similar to those developed above for linear drift-diffusion equations.
We will consider weak solutions (solutions which satisfy the equations in the sense of distributions).
thmweakgl Let $\theta_0\in L^2(\Omega)$ and let $T>0$. There exists a weak solution of (<ref>)
\[
\theta\in L^{\infty}(0,T; L^2(\Omega))\cap L^2(0,T; V_0(\Omega))
\]
satisfying $lim_{t\to 0}\theta(t) = \theta_0$ weakly in $L^2(\Omega)$.
Proof. We consider Galerkin approximations, $\theta_m$
\[
\theta_m (x,t) = \sum_{j=1}^m \theta_j(t)w_j(x)
\]
obeying the ODEs (written conveniently as PDEs):
\[
\pa_t\theta_m + P_m\left[R_D^{\perp}(\theta_m)\cdot\na \theta_m\right] + \l \theta_m = 0
\]
with initial datum
\[
\theta_m(0) = P_m(\theta_0).
\]
We observe that, multiplying by $\theta_m$ and integrating we have
\[
\fr{1}{2}\fr{d}{dt}\|\theta_m\|^2 + \|\theta_m\|^2_{\fr{1}{2}, D} = 0
\]
which implies that the sequence $\theta_m$ is bounded in
\[
\theta_m\in L^{\infty}(0,T; L^2(\Omega))\cap L^2(0,T; V_0(\Omega))
\]
It is known (<cit.>) that $V_0(\Omega)\subset L^4(\Omega)$ with continuous inclusion. It is also known (<cit.>) that
\[
R_D: L^4(\Omega)\to L^4(\Omega)
\]
are bounded linear operators. It is then easy to see that
$\pa_t \theta_m$ are bounded in $L^2(0,T; H^{-1}(\Omega))$. Applying the Aubin-Lions lemma, we obtain a subsequence, renamed $\theta_m$ converging strongly in $L^2(0,T; L^2(\Omega))$ and weakly in $L^2(0,T; V_0(\Omega))$ and in $L^2(0,T; L^4(\Omega))$. The limit solves the equation (<ref>) weakly. Indeed, this follows after integration by parts because the product $(R_D^{\perp}\theta_m)\theta_m$ is weakly convergent in $L^2(0,T; L^2(\Omega))$ by weak-times-strong weak continuity. The weak continuity in time at $t=0$ follows by integrating
\[
(\theta_m(t), \phi) - (\theta_m(0), \phi) = \int_0^t\fr{d}{ds}\theta_m(s)ds
\]
and use of the equation and uniform bounds. We omit further details.
Acknowledgment. The work of PC was partially supported by NSF grant DMS-1209394
cabre X. Cabre, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), no. 5, 2052-2093.
cv1 P. Constantin, V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, GAFA 22 (2012) 1289-1321.
cc A. Córdoba, D. Córdoba, A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249 (2004), 511–528.
davies1 E.B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Am. J. Math 109 (1987) 319-333.
jerisonkenig D. Jerison, C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Analysis 130 (1995), 161-212.
lions J.L. Lions, Quelque methodes de résolution des problèmes aux limites non linéaires, Paris, Dunod (1969).
qszhang1 Q. S. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians, J. Diff. Eqn 182 (2002), 416-430
qszhang2 Q. S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, IMRN (2006), article ID92314, 1-39.
|
1511.00221
|
xxxxx-xxx201XLM-CMA for Large Scale Black-box OptimizationI. Loshchilov
The limited memory BFGS method (L-BFGS) of Liu and Nocedal (1989) is often considered to be the method of choice for continuous optimization when first- and/or second- order information is available. However, the use of L-BFGS can be complicated in a black-box scenario where gradient information is not available and therefore should be numerically estimated. The accuracy of this estimation, obtained by finite difference methods, is often problem-dependent that may lead to premature convergence of the algorithm.
In this paper, we demonstrate an alternative to L-BFGS, the limited memory Covariance Matrix Adaptation Evolution Strategy (LM-CMA) proposed by Loshchilov (2014). The LM-CMA is a stochastic derivative-free algorithm for numerical optimization of non-linear, non-convex optimization problems. Inspired by the L-BFGS, the LM-CMA samples candidate solutions according to a covariance matrix reproduced from $m$ direction vectors selected during the optimization process. The decomposition of the covariance matrix into Cholesky factors allows to reduce the memory complexity to $O(mn)$, where $n$ is the number of decision variables. The time complexity of sampling one candidate solution is also $O(mn)$, but scales as only about 25 scalar-vector multiplications in practice. The algorithm has an important property of invariance w.r.t. strictly increasing transformations of the objective function, such transformations do not compromise its ability to approach the optimum. The LM-CMA outperforms the original CMA-ES and its large scale versions on non-separable ill-conditioned problems with a factor increasing with problem dimension. Invariance properties of the algorithm do not prevent it from demonstrating a comparable performance to L-BFGS on non-trivial large scale smooth and nonsmooth optimization problems.
LM-CMA, L-BFGS, CMA-ES, large scale optimization, black-box optimization.
§ INTRODUCTION
In a black-box scenario, knowledge about an objective function $f: \vc{X} \rightarrow \R$, to be optimized on some space $\vc{X}$, is restricted to the handling of a device that delivers the value of $f(\vc{x})$ for any input $\vc{x} \in \vc{X}$. The goal of black-box optimization is to find solutions with small (in the case of minimization) value $f(\vc{x})$, using the least number of calls to the function $f$ <cit.>. In continuous domain, $f$ is defined as a mapping $\R^n \rightarrow \R$, where $n$ is the number of variables.
The increasing typical number of variables involved in everyday optimization problems makes it harder to supply the search with useful problem-specific knowledge, e.g., gradient information, valid assumptions about problem properties.
The use of large scale black-box optimization approaches would seem attractive providing that a comparable
performance can be achieved.
The use of well recognized gradient-based approaches such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm <cit.> is complicated in the black-box scenario since gradient information is not available and therefore should be estimated by costly finite difference methods (e.g., $n+1$ function evaluations per gradient estimation for forward difference and $2n+1$ for central difference). The latter procedures are problem-sensitive and may require a priori knowledge about the problem at hand, e.g., scaling of $f$, decision variables and expected condition number <cit.>.
By the 1980s, another difficulty has become evident: the use of quasi-Newton methods such as BFGS is limited to small and medium scale optimization problems for which the approximate inverse Hessian matrix can be stored in memory. As a solution, it was proposed not to store the matrix but to reconstruct it using information from the last $m$ iterations <cit.>. The final algorithm called the limited memory BFGS algorithm (L-BFGS or LM-BFGS) proposed by <cit.> is still considered to be the state-of-the-art of large scale gradient-based optimization <cit.>.
However, when a large scale black-box function is considered, the L-BFGS is forced to deal both with a scarce information coming from only $m$ recent gradients and potentially numerically imprecise estimations of these gradients which scale up the run-time in the number of function evaluations by a factor of $n$. It is reasonable to wonder whether the L-BFGS and other derivative-based algorithms are still competitive in these settings or better performance and robustness can be achieved by derivative-free algorithms.
The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) seems to be a reasonable alternative, it is a derivative-free algorithm designed to learn dependencies between decision variables by adapting a covariance matrix which defines the sampling distribution of candidate solutions <cit.>. This algorithm constantly demonstrates good performance at various platforms for comparing continuous optimizers such as the Black-Box Optimization Benchmarking (BBOB) workshop <cit.> and the Special Session at Congress on Evolutionary Computation <cit.>. The CMA-ES was also extended to noisy <cit.>, expensive <cit.> and multi-objective optimization <cit.>.
The principle advantage of CMA-ES, the learning of dependencies between $n$ decision variables, also forms its main practical limitations such as $O(n^2)$ memory storage required to run the algorithm and $O(n^2)$ computational time complexity per function evaluation <cit.>. These limitations may preclude the use of CMA-ES for computationally cheap but large scale optimization problems if the internal computational cost of CMA-ES is greater than the cost of one function evaluation. On non-trivial large scale problems with $n>10,000$ not only the internal computational cost of CMA-ES becomes enormous but it is becoming simply impossible to efficiently store the covariance matrix in memory.
An open problem is how to extend efficient black-box approaches such as CMA-ES to $n\gg1000$ while keeping a reasonable trade-off between the performance in terms of the number of function evaluations and the internal time and space complexity. The low complexity methods such as separable CMA-ES (sep-CMA-ES by <cit.>), linear time Natural Evolution Strategy (R1-NES by <cit.>) and VD-CMA by <cit.> are useful when the large scale optimization problem at hand is separable or decision variables are weakly correlated, otherwise the performance of these algorithms w.r.t. the original CMA-ES may deteriorate significantly.
In this paper, we present a greatly improved version of the recently proposed extension of CMA-ES to large scale optimization called the limited memory CMA-ES (LM-CMA) by <cit.>. Instead of storing the covariance matrix, the LM-CMA stores $m$ direction vectors in memory and uses them to generate solutions.
The algorithm has $O(mn)$ space complexity, where $m$ can be a function of $n$.
The time complexity is linear in practice with the smallest constant factor among the presented evolutionary algorithms.
The paper is organized as follows. First, we briefly describe L-BFGS in Section <ref> and CMA-ES with its large scale alternatives in Section <ref>. Then, we present the improved version of LM-CMA in Section <ref>, investigate its performance w.r.t. large scale alternatives in Section <ref> and conclude the paper in Section <ref>.
§ THE L-BFGS
An early version of the L-BFGS method, at that time called the SQN method, was proposed by <cit.>.
During the first $m$ iterations the L-BFGS is identical to the BFGS method, but stores BFGS corrections separately until the maximum number of them $m$ is used up. Then, the oldest corrections are replaced by the newest ones. The approximate of the inverse Hessian of $f$ at iteration $k$, $\vc{H}_k$ is obtained by applying $m$ BFGS updates to a sparse symmetric and positive definite matrix $\vc{H}_0$ provided by the user <cit.>.
Let us denote iterates by $\vc{x}_k$, $\vc{s}_k=\vc{x}_{k+1}-\vc{x}_k$ and $\vc{y}_k=\vc{g}_{k+1}-\vc{g}_k$, where $\vc{g}$ denotes gradient.
The method uses the inverse BFGS formula in the form
\begin{equation}
\vc{H}_{k+1} = \vc{V}_k^T \vc{H}_k \vc{V}_k + \rho_k \vc{s}_k \vc{s}^T_k,
\end{equation}
where $\rho_k=1/\vc{y}^T_k \vc{s}_k$, and
$\vc{V}_k = \vc{I} - \rho_k \vc{y}_k \vc{s}_k^T$ <cit.>.
The L-BFGS method works as follows <cit.>:
Step 1. Choose $\vc{x}_0$, $m$, $0<\beta'<1/2$,
$\beta'<\beta<1$, and a symmetric and positive definite starting matrix $\vc{H}_0$. Set $k=0$.
Step 2. Compute
\begin{equation}
\vc{d}_{k} = -\vc{H}_k \vc{g}_k,
\end{equation}
\begin{equation}
\vc{x}_{k+1} = \vc{x}_k + \alpha_k \vc{d}_k,
\end{equation}
where $\alpha_k$ satisfies the Wolfe conditions <cit.>:
\begin{equation}
f(\vc{x}_k + \alpha_k \vc{d}_k) \leq f(\vc{x}_k) + \beta' \alpha_k \vc{g}_k^T \vc{d}_k,
\end{equation}
\begin{equation}
g(\vc{x}_k + \alpha_k \vc{d}_k)^T\vc{d}_k \geq \beta \vc{g}_k^T \vc{d}_k.
\end{equation}
The novelty introduced by <cit.> w.r.t. the version given in <cit.> is that the line search is not forced to perform at least one cubic
interpolation, but the unit steplength $\alpha_k=1$ is always tried first, and if it satisfies the Wolfe conditions, it is accepted.
Step 3. Let $\hat{m}=\min(k,m-1)$. Update $\vc{H}_0$ $\hat{m}+1$ times using the pairs $\left\{\vc{y}_j,\vc{s}_j \right\}^k_{j=k-\hat{m}}$ as follows:
\[ \begin{array}{lll}
\lefteqn{ \vc{H}_{k+1} = (\vc{V}^T_k \cdot \cdot \cdot \vc{V}^T_{k-\hat{m}}) \vc{H}_0 (\vc{V}_{k-\hat{m}} \cdot \cdot \cdot \vc{V}_{k})}\\
&& + \rho_{k-\hat{m}} (\vc{V}^T_k \cdot \cdot \cdot \vc{V}^T_{k-\hat{m}+1}) \vc{s}_{k-\hat{m}} \vc{s}^T_{k-\hat{m}} (\vc{V}_{k-\hat{m}+1} \cdot \cdot \cdot \vc{V}_k)\\
&& + \rho_{k-\hat{m}+1} (\vc{V}^T_k \cdot \cdot \cdot \vc{V}^T_{k-\hat{m}+2}) \vc{s}_{k-\hat{m}+1} \vc{s}^T_{k-\hat{m}+1} (\vc{V}_{k-\hat{m}+2} \cdot \cdot \cdot \vc{V}_k)\\
&& \vdots \\
&& + \rho_k \vc{s}_k \vc{s}_k^T
\end{array} \]
Step 4. Set $k=k+1$ and go to Step 2.
The algorithm space and time complexity scales as $O(mn)$ per iteration (not per function evaluation), where $m$ in order of 5-40 suggested in the original paper is still the most common setting. An extension to bound constrained optimization called L-BFGS-B has the efficiency of the original algorithm, however at the cost of a significantly more complex implementation <cit.>. Extensions to optimization with arbitrary constraints are currently not available. Satisfactory and computationally tractable handling of noise is at least problematic, often impossible.
Nevertheless, as already mentioned above, when gradient information is available, L-BFGS is competitive to other techniques <cit.> and often can be viewed as a method of choice <cit.> for large scale continuous optimization. However, in the black-box scenario when gradient-information is not available (direct search settings), the advantages of L-BFGS are becoming less obvious and derivative-free algorithms can potentially perform comparable. In this paper, we investigate this scenario in detail.
§ EVOLUTION STRATEGIES FOR LARGE SCALE OPTIMIZATION
Historically, first Evolution Strategies <cit.> were designed to perform the search without learning dependencies between variables which is a more recent development that gradually led to the CMA-ES algorithm <cit.>.
In this section, we discuss the CMA-ES algorithm and its state-of-the-art derivatives for large scale optimization. For a recent comprehensible overview of Evolution Strategies, the interested reader is referred to <cit.>.
More specifically, the analysis of theoretical foundations of Evolution Strategies is provided by <cit.>.
§.§ The CMA-ES
The ($\mu/\mu_{w},\lambda$)-CMA-ES
given $n \in \mathbb{N}_+$, $\lambda = 4 + \lfloor 3 \ln \, n \rfloor $, $\mu = \lfloor \lambda/2 \rfloor $,
$\vc{w}_i = \frac{ \ln(\mu + \frac{1}{2}) - \ln\,i}{ \sum^{\mu}_{j=1}(\ln(\mu + \frac{1}{2})-\ln\,j)} \; \mstr{for} \; i=1 \ldots \mu$,
$\mu_w = \frac{1}{\sum^{\mu}_{i=1} w^2_i}$,
$c_{\sigma} = \frac{\mu_w + 2}{n+\mu_w+3}$, $d_{\sigma} = 1 + c_{\sigma} +2 \, \mstr{max}(0,\sqrt{ \frac{\mu_w - 1}{n+1}}-1)$,
$c_c = \frac{4}{n+4}$, $c_1 = \frac{2 \, \mstr{\min}(1,\lambda/6)}{(n+1.3)^2 +\mu_w}$, $c_{\mu} = \frac{2 \, (\mu_w -2 + 1/{\mu_w})}{(n+2)^2+\mu_w}$
initialize $\vc{m}^{t=0} \in \R^{\dd}, \sigma^{t=0} > 0, \vc{p}^{t=0}_{\sigma} = \ma{0}, \vc{p}^{t=0}_{c} = \ma{0}, \C^{t=0} = \Id, t \leftarrow 0 $
$k \leftarrow 1,\ldots,\lambda$
$\vc{x}_k \leftarrow \vc{m}^t + \sigma^{t} {{\mathcal N} \hspace{-0.13em}\left({\ma{0},\C^{t}\,}\right)} $
$\vc{f}_k \leftarrow f(\vc{x}_k)$
$ \vc{m}^{t+1} \leftarrow \sum_{i=1}^{\mu} \vc{w}_i \vc{x}_{i:\lambda} \;$ // the symbol $i:\lambda$ denotes $i$-th best individual on $f$
$ \vc{p}^{t+1}_{\sigma} \leftarrow (1 - c_{\sigma}) \vc{p}^{t}_{\sigma} + \sqrt{c_{\sigma}(2-c_{\sigma})} \sqrt{\mu_w} {\C^t}^{-\frac{1}{2}} \frac{\vc{m}^{t+1}-\vc{m}^{t}}{\sigma^{t}} $
$ h_{\sigma} \leftarrow \ONE_{ \left\| p^{t+1}_{\sigma} \right\| < \sqrt{1 - (1-c_{\sigma})^{2(t+1)}}(1.4 + \frac{2}{n+1}) \, \mathbb{E} \left\| \NormOI \right\| } $
$ \vc{p}^{t+1}_{c} \leftarrow (1 - c_{c}) \vc{p}^{t}_{c} + h_{\sigma} \sqrt{c_{c}(2-c_{c})} \sqrt{\mu_w} \frac{\vc{m}^{t+1}-\vc{m}^{t}}{\sigma^{t}} $
$ \C_{\mu} \leftarrow \sum^{\mu}_{i=1} w_i \frac{\x_{i:\lambda} - \m^t}{\sigma^t} \times \frac{(\x_{i:\lambda} - \m^t)^T}{\sigma^t}$
$ \C^{t+1} \leftarrow (1 - c_1 - c_{\mu}) \C^t +
c_1 \underbrace{\vc{p}^{t+1}_c {\vc{p}^{t+1}_c}^T}_{\mstr{rank-one\,update}} +
c_{\mu} \hspace{-1.9em} \underbrace{\C_{\mu}}_{\mstr{rank-\mu \,update}}$
$ \sigma^{t+1} \leftarrow \sigma^{t} \mstr{exp}
( \frac{c_{\sigma}}{d_{\sigma}} ( \frac{\left\| \vc{p}^{t+1}_{\sigma} \right\|}{ \mathbb{E} \left\| \NormOI \right\| } - 1 )) $
$ t \leftarrow t + 1$
stopping criterion is met
The Covariance Matrix Adaptation Evolution Strategy <cit.> is probably the most popular and in overall the most efficient Evolution Strategy.
The ($\mu/\mu_{w},\lambda$)-CMA-ES is outlined in Algorithm <ref>.
At iteration $t$ of CMA-ES, a mean $\vc{m}^t$ of the mutation distribution (can be interpreted as an estimation of the optimum) is used to generate its $k$-th out of $\lambda$ candidate solution $\vc{x}_k \in \mathbb{R}^\dd$
(line <ref>) by adding a random Gaussian mutation defined by a (positive definite) covariance matrix $\C^t \in \R^{n \times n}$ and a mutation step-size $\sigma^t$ as follows:
\begin{equation}
\vc{x}^t_k \leftarrow {\mathcal N} \hspace{-0.13em}\left({\vc{m}^t,{\sigma^t}^2 {\C}^t}\right) \leftarrow \vc{m}^t + \sigma^t {\mathcal N} \hspace{-0.13em}\left({\ma{0},{\C}^t}\right)
\end{equation}
These $\lambda$ solutions then should be evaluated with an objective function $f$ (line <ref>).
The old mean of the mutation distribution is stored in $\vc{m}^{t}$ and a new mean $\vc{m}^{t+1}$ is computed as a weighted sum
of the best $\mu$ parent individuals selected among $\lambda$ generated offspring individuals (line <ref>).
The weights $\vc{w}$ are used to control the impact of the selected individuals, the weights are usually higher for better ranked individuals (line <ref>).
The procedure of the adaptation of the step-size $\sigma^t$ in CMA-ES is inherited from the Cumulative Step-Size Adaptation Evolution Strategy (CSA-ES) <cit.> and is controlled by evolution path $\vc{p}_{\sigma}^{t+1}$.
Successful mutation steps $\frac{\vc{m}^{t+1}-\vc{m}^{t}}{\sigma^{t}}$ (line <ref>) are tracked in the space of sampling, i.e., in the isotropic coordinate system defined by principal components of the covariance matrix $\C^t$. To update the evolution path $\vc{p}_{\sigma}^{t+1}$ a decay/relaxation factor $c_{\sigma}$ is used to decrease the importance of the previously performed steps with time.
The step-size update rule increases the step-size if the length of the evolution path $\vc{p}_{\sigma}^{t+1}$ is longer than
the expected length of the evolution path under random selection $\mathbb{E} \left\| \NormOI \right\|$,
and decreases otherwise (line <ref>). The expectation of $\left\| \NormOI \right\|$ is approximated by $\sqrt{n} (1 - \frac{1}{4 n} + \frac{1}{21 n^2} )$.
A damping parameter $d_{\sigma}$ controls the change of the step-size.
The covariance matrix update consists of two parts (line <ref>): rank-one update <cit.> and rank-$\mu$ update <cit.>. The rank-one update computes evolution path $\vc{p}_c^{t+1}$ of successful moves of the mean $\frac{\vc{m}^{t+1}-\vc{m}^{t}}{\sigma^{t}}$
of the mutation distribution in the given coordinate system (line <ref>),
in a similar way as for the evolution path $\vc{p}_{\sigma}^{t+1}$ of the step-size.
To stall the update of $\vc{p}_c^{t+1}$ when $\sigma$ increases rapidly, a $h_{\sigma}$ trigger is used (line <ref>).
The rank-$\mu$ update computes a covariance matrix $\C_{\mu}$ as a weighted sum of covariances of successful steps of $\mu$ best individuals (line <ref>).
The update of $\C$ itself is a replace of the previously accumulated information by a new one with corresponding weights of importance (line <ref>):
$c_1$ for covariance matrix $\vc{p}^{t+1}_c {\vc{p}^{t+1}_c}^T$ of rank-one update and
$c_{\mu}$ for $\C_{\mu}$ of rank-$\mu$ update <cit.> such that $c_1 + c_{\mu} \leq 1$.
It was also proposed to take into account unsuccessful mutations in the "active" rank-$\mu$ update <cit.>.
In CMA-ES, the factorization of the covariance $\C$ into $\A \A^T=\C$ is needed to sample the multivariate normal distribution (line <ref>). The eigendecomposition with $O(n^3)$ complexity is used for the factorization. Already in the original CMA-ES it was proposed to perform the eigendecomposition every $n/10$ generations (not shown in Algorithm <ref>) to reduce the complexity per function evaluation to $O(n^2)$.
§.§ Large Scale Variants
The original $O(n^2)$ time and space complexity of CMA-ES precludes its applications to large scale optimization with $n\gg1000$. To enable the algorithm for large scale optimization, a linear time and space version called sep-CMA-ES was proposed by <cit.>. The algorithm does not learn dependencies but the scaling of variables by restraining the covariance matrix update to the diagonal elements:
\begin{eqnarray}
c^{t+1}_{jj} = (1 - c_{cov}) c^t_{jj} + \frac{1}{\mu_{cov}}\left(\vc{p}_c^{t+1}\right)^2_j +
c_{ccov}\left(1-\frac{1}{\mu_{ccov}}\right) \sum_{i=1}^{\mu} w_i c^t_{jj} \left({z_{i:\lambda}}^{t+1}\right)^2_j, j=1,\ldots,n
\end{eqnarray}
where, for $j=1,\ldots,n$ the $c_{jj}$ are the diagonal elements of $\C^t$ and the $\left({z_{i:\lambda}}^{t+1}\right)_j = \left({x_{i:\lambda}}^{t+1}\right)_j/(\sigma^t \sqrt{c^t_{jj}})$.
This update reduces the computational complexity to $O(n)$ and allows to exploit problem separability. The algorithm demonstrated good performance on separable problems and even outperformed CMA-ES on non-separable Rosenbrock function for $n>100$.
A Natural Evolution Strategy (NES) variant, the Rank-One NES (R1-NES) by <cit.>,
adapts the search distribution according to the natural gradient with a particular low rank parametrization of the covariance matrix,
\begin{equation}
\C = \sigma^2 (\I + \vc{u}\vc{u}^T),
\end{equation}
where $u$ and $\sigma$ are the parameters to be adjusted. The adaptation of the predominant eigendirection $\vc{u}$ allows the algorithm to solve highly non-separable problems while maintaining only $O(n)$ time and $O(\mu n)$ space complexity.
The use of the natural gradient in the derivation of the NES algorithm motivated
a research which led to the formulation of the Information Geometric Optimization (IGO) framework by <cit.>.
The IGO framework was used to derive a similar to R1-NES algorithm called VD-CMA <cit.>
with the sampling distribution parametrized by a Gaussian model with the covariance matrix restricted as follows:
\begin{equation}
\label{VDCMAeq}
\C = D (\I + \vc{u}\vc{u}^T)D,
\end{equation}
where $D$ is a diagonal matrix of dimension $n$ and $\vc{u}$ is a vector in $\R^n$. This model is able to represent a scaling for each
variable by $D$ and a principal component, which is generally not parallel to an axis, by $Dv$ <cit.>.
It has $O(n)$ time and $O(\mu n)$ space complexity but i) typically demonstrates a better performance than sep-CMA-ES and R1-NES and
ii) can solve a larger class of functions <cit.>.
A version of CMA-ES with a limited memory storage also called limited memory CMA-ES (L-CMA-ES) was proposed by <cit.>. The L-CMA-ES uses the $m$ eigenvectors and eigenvalues spanning the $m$-dimensional
dominant subspace of the $n \times n$-dimensional covariance matrix .
The authors adapted a singular value decomposition updating
algorithm developed by <cit.> that allowed to avoid
the explicit computation and storage of the covariance matrix.
For $m < n$ the performance in terms of the number of function evaluations gradually decreases while enabling the search in $\R^n$ for $n>10,000$. However, the computational complexity of $O(m^2n)$ practically (for $m$ in order of $\sqrt{n}$ as suggested by <cit.>) leads to the same limitations of $O(n^2)$ time complexity as in the original CMA-ES.
The ($\mu/\mu_{w},\lambda$)-Cholesky-CMA-ES proposed by <cit.> is of special interest in this paper because the LM-CMA is based on this algorithm. The Cholesky-CMA represents a version of CMA-ES with rank-one update where instead of performing the factorization of the covariance matrix $\C^t$ into $\A^t {\A^t}^T=\C^t$, the Cholesky factor $\A^t$ and its inverse ${\A^t}^{-1}$ are iteratively updated. From Theorem 1 <cit.> it follows that if $\C^t$ is updated as
\begin{equation}
\C^{t+1} = \alpha \C^t + \beta \vc{v}^t {\vc{v}^t}^T,
\end{equation}
where $\vc{v} \in \R^n$ is given in the decomposition form $\vc{v}^t = \A^t \vc{z}^t$, and $\alpha,\beta \in \R^+$, then for $\vc{z} \neq \vc{0}$ a Cholesky factor of the matrix $\C^{t+1}$ can be computed by
\begin{equation}
\label{Aupdate}
\A^{t+1} = \sqrt{\alpha} \A^t + \frac{\sqrt{\alpha}}{{\left\| \vc{z}^t \right\|}^2} \left( \sqrt{1 + \frac{\beta}{\alpha} {{\left\| \vc{z}^t \right\|}^2}} -1 \right) [\A^t \vc{z}^t] {\vc{z}^t}^T,
\end{equation}
for $\vc{z}_t=\vc{0}$ we have $\A^{t+1}=\sqrt{\alpha}\A^t$. From the Theorem 2 <cit.> it follows that if ${\A^{-1}}^{t}$ is the inverse of $\A^t$, then the inverse of $\A^{t+1}$ can be computed by
\begin{equation}
\label{Ainvupdate}
{\A^{-1}}^{t+1} = \frac{1}{\sqrt{\alpha}} {\A^{-1}}^{t} - \frac{1}{\sqrt{\alpha}{\left\| \vc{z}^t \right\|}^2} \left( 1- \frac{1}{\sqrt{1 + \frac{\beta}{\alpha} {{\left\| \vc{z}^t \right\|}^2}}} \right) \vc{z}^t [{\vc{z}^t}^T {\A^{-1}}^{t}],
\end{equation}
for $\vc{z}^t \neq \vc{0}$ and by ${\A^{-1}}^{t+1}=\frac{1}{\sqrt{\alpha}} {\A^{-1}}^{t}$ for $\vc{z}^t = \vc{0}$.
The ($\mu/\mu_{w},\lambda$)-Cholesky-CMA-ES
given $n \in \mathbb{N}_+$, $\lambda = 4 + \lfloor 3 \ln \, n \rfloor $, $\mu = \lfloor \lambda/2 \rfloor $,
$w_i = \frac{ \ln(\mu + 1) - \ln(i)}{\mu \ln(\mu + 1) - \sum_{j=1}^{\mu} \ln(j)}; i=1 \ldots \mu$,
$\mu_w = \frac{1}{\sum^{\mu}_{i=1} w^2_i}$,
$c_{\sigma} = \frac{\sqrt{\mu_{w}}}{
\sqrt{n} + \sqrt{\mu_{w}}}$,
$d_{\sigma} = 1 + c_{\sigma} +2 \, \mstr{max}(0,\sqrt{ \frac{\mu_w - 1}{n+1}}-1)$,
$c_c = \frac{4}{n+4}$, $c_1 = \frac{2}{ {(n + \sqrt{2})}^2}$
initialize $\vc{m}^{t=0} \in \R^{\dd}, \sigma^{t=0} > 0, \vc{p}^{t=-1}_{\sigma} = \ma{0}, \vc{p}^{t=-1}_{c} = \ma{0}, \A^{t=0} = \Id, \A^{t=0}_{inv} = \Id, t \leftarrow 0 $
$k \leftarrow 1,\ldots,\lambda$
$\vc{z}_k \leftarrow {{\mathcal N} \hspace{-0.13em}\left({\ma{0},\I}\right)} $
$\vc{x}_k \leftarrow \vc{m}^t + \sigma^{t} \A \vc{z}_k $
$\vc{f}_k \leftarrow f(\vc{x}_k)$
$ \vc{m}^{t+1} \leftarrow \sum_{i=1}^{\mu} \vc{w}_i \vc{x}_{i:\lambda} \;$
$ \vc{z}_w \leftarrow \sum_{i=1}^{\mu} \vc{w}_i \vc{z}_{i:\lambda} \;$
$ \vc{p}^{t}_{\sigma} \leftarrow (1 - c_{\sigma}) \vc{p}^{t-1}_{\sigma} + \sqrt{c_{\sigma}(2-c_{\sigma})} \sqrt{\mu_w} \vc{z}_w$
$ \vc{p}^{t}_{c} \leftarrow (1 - c_{c}) \vc{p}^{t-1}_{c} + \sqrt{c_{c}(2-c_{c})} \sqrt{\mu_w} \A^t \vc{z}_w$
$ \vc{v}^t \leftarrow \A_{inv}^t \vc{p}^t_c $
$ \A^{t+1} \leftarrow \sqrt{1 - c_1} \A^t + \frac{\sqrt{1 - c_1}}{{\left\| \vc{v}^t \right\|}^2} \left( \sqrt{1 + \frac{c_1}{1 - c_1} {{\left\| \vc{v}^t \right\|}^2}} -1 \right) \vc{p}^t_c {\vc{v}^t}^T $
$ {\A_{inv}^{t+1}} \leftarrow \frac{1}{\sqrt{1 - c_1}} {\A_{inv}^t} -
\frac{1}{\sqrt{1 - c_1}{\left\| \vc{v}^t \right\|}^2}
\left( 1- \frac{1}{\sqrt{1 + \frac{c_1}{1 - c_1} {{\left\| \vc{v}^t \right\|}^2}}} \right)
{\vc{v}}^t [{{\vc{v}}^t}^T {\A_{inv}^t}],
$ \sigma^{t+1} \leftarrow \sigma^{t} \mstr{exp}
( \frac{c_{\sigma}}{d_{\sigma}} ( \frac{\left\| \vc{p}^{t}_{\sigma} \right\|}{ \mathbb{E} \left\| \NormOI \right\| } - 1 )) $
$ t \leftarrow t + 1$
stopping criterion is met
The ($\mu/\mu_{w},\lambda$)-Cholesky-CMA-ES is outlined in Algorithm <ref>.
As well as in the original CMA-ES, Cholesky-CMA-ES proceeds by sampling $\lambda$ candidate solutions (lines <ref> - <ref>) and taking into account the most successful $\mu$ out of $\lambda$ solutions in the evolution path adaptation (lines <ref> and
<ref>). However, the eigendecomposition procedure is not required anymore because the Cholesky factor and its inverse are updated incrementally (line <ref> and <ref>). This simplifies a lot the implementation of the algorithm and keeps its time complexity as $O(n^2)$. A postponed update of the Cholesky factors every $O(n)$ iterations would not reduce the asymptotic complexity further (as it does in the original CMA-ES) because the quadratic complexity will remain due to matrix-vector multiplications needed to sample new individuals.
The non-elitist Cholesky-CMA is a good alternative to the original CMA-ES and demonstrates a comparable performance <cit.>. While it has the same computational and memory complexity, the lack of rank-$\mu$ update may deteriorate its performance on problems where it is essential.
§ THE LM-CMA
The LM-CMA is inspired by the L-BFGS algorithm but instead of storing $m$ gradients for performing inverse Hessian requiring operations
it stores $m$ direction vectors to reproduce the Cholesky factor $\A$
and generate candidate solutions with a limited time and space cost $O(mn)$ (see Section <ref>).
These $m$ vectors are estimates of descent directions provided by evolution path vectors and
should be stored with a particular temporal distance to enrich $\A$ (see Section <ref>).
An important novelty introduced w.r.t. the original LM-CMA proposed by <cit.> is
a procedure for sampling from a family of search representations defined by Cholesky factors reconstructed from $m^* \leq m$ vectors (see Section <ref>)
and according to the Rademacher distribution (see Section <ref>).
These novelties allow to simultaneously reduce the internal time complexity of the algorithm and improve its performance in terms of the number of function evaluations.
Before describing the algorithm itself, we gradually introduce all the necessary components.
§.§ Reconstruction of Cholesky Factors
By setting $a=\sqrt{1-c_1}$, $b^t=\frac{\sqrt{1-c_1}}{{\left\| \vc{v}^t \right\|}^2} \left( \sqrt{1 + \frac{c_1}{1-c_1} {{\left\| \vc{v}^t \right\|}^2}} -1 \right)$ and considering the evolution path $\vc{p}^t_c$ (a change of optimum estimate $\vc{m}$ smoothed over iterations, see line <ref> of Algorithm <ref>) together with $\vc{v}^t=\A^{{-1}^t}{\vc{p}}^t_c$, one can rewrite Equation (<ref>) as
\begin{equation}
\label{AupdateRew}
\A^{t+1} = a \A^t + b^t \vc{p}_c^t {\vc{v}^t}^T,
\end{equation}
The product of a random vector $\vc{z}$ and the Cholesky factor $\A^t$ thus can be directly computed. At iteration $t=0$, $\A^0=\I$ and $\A^0 \vc{z}=\vc{z}$, the new updated Cholesky factor $\A^{1}=a \I + b^{0} \vc{p}_c^{0} {\vc{v}^{0}}^T$. At iteration $t=1$, $\A^{1} \vc{z} = (a \I + b^{0} \vc{p}_c^{0} {\vc{v}^{0}}^T) \vc{z} = a \vc{z} + b^0 \vc{p}_c^0 ({\vc{v}^0}^T \vc{z})$ and $\A^2=a (a \I + b^{0} \vc{p}_c^{0} {\vc{v}^{0}}^T) + b^1 \vc{p}_c^1 {\vc{v}^1}^T$. Thus, a very simple iterative procedure which scales as $O(mn)$ can be used to sample candidate solutions in $\RR^n$ according to the Cholesky factor $\A^t$ reconstructed from $m$ pairs of vectors $\vc{p}_c^t$ and $\vc{v}^t$.
Az(): Cholesky factor - vector update
given $\vc{z} \in \R^n, m \in \mathbb{Z}_+, \vc{j} \in \mathbb{Z}_+^m, \vc{i} \in \mathbb{Z}^{\left|\vc{i}\right|}_+, \vc{P} \in \R^{m \times n}, \vc{V} \in \R^{m \times n}, \vc{b} \in \R^m, a \in [0,1]$
initialize $\vc{x} \leftarrow \vc{z}$
$t \leftarrow 1,\ldots,\left|\vc{i}\right|$
$k \leftarrow \vc{b}^{\vc{j}_{\vc{i}_t}} \vc{V}^{(\vc{j}_{\vc{i}_t},:)} \cdot \vc{z} $
$\vc{x} \leftarrow a \vc{x} + k \vc{P}^{(\vc{j}_{\vc{i}_t},:)}$
return $\vc{x}$
Ainvz(): inverse Cholesky factor - vector update
given $\vc{z} \in \R^n, m \in \mathbb{Z}_+, \vc{j} \in \mathbb{Z}_+^m, \vc{i} \in \mathbb{Z}^{\left|\vc{i}\right|}_+, \vc{d} \in \R^m, c \in [0,1]$
initialize $\vc{x} \leftarrow \vc{z}$
$t \leftarrow 1,\ldots,\left|\vc{i}\right|$
$k \leftarrow \vc{d}^{\vc{j}_{\vc{i}_t}} \vc{V}^{(\vc{j}_{\vc{i}_t},:)} \cdot \vc{x} $
$\vc{x} \leftarrow c \vc{x} - k \vc{V}^{(\vc{j}_{\vc{i}_t},:)}$
return $\vc{x}$
The pseudo-code of the procedure to reconstruct $\vc{x}=\A^t \vc{z}$ from $m$ direction vectors
is given in Algorithm <ref>. At each iteration of reconstruction of $\vc{x}=\A^t \vc{z}$ (lines <ref> - <ref>), $\vc{x}$ is updated as a sum of $a$-weighted version of itself and ${b}^t$-weighted evolution path $\vc{p}_c^t$ (accessed from a matrix $\vc{P} \in R^{m \times n}$ as $\vc{P}^{(\vc{i}_t,:)}$ ) scaled by the dot product of $\vc{v}^t$ and $\vc{x}$. As can be seen, Algorithms <ref> and <ref> use $\vc{j}_{\vc{i}_t}$ indexation instead of $t$. This is a convenient way to have references to matrices $\vc{P}$ and $\vc{V}$ which store $\vc{p}_c$ and $\vc{v}$ vectors, respectively. In the next subsections, we will show how to efficiently manipulate these vectors.
A very similar approach can be used to reconstruct $\vc{x}={\A^t}^{-1} \vc{z}$, for the sake of reproducibility the pseudo-code is given in Algorithm <ref> for $c=1/\sqrt{1-c_1}$ and $d^t= \frac{1}{\sqrt{1-c_1}{\left\| \vc{v}^t \right\|}^2} \times \left( 1- \frac{1}{\sqrt{1 + \frac{c_1}{1-c_1} {{\left\| \vc{v}^t \right\|}^2}}} \right)$. The computational complexity of both procedures scales as $O(mn)$.
It is important to note that when a vector $\vc{p}^{\ell}$ from a set of $m$ vectors stored in $\vc{P}$ is replaced by a new vector $\vc{p}^{t+1}$ (see line <ref> in Algorithm <ref>),
all inverse vectors $\vc{v}^t$ for $t=\ell,\ldots,m$ should be iteratively recomputed <cit.>.
This procedure corresponds to line <ref> in Algorithm <ref>.
§.§ Direction Vectors Selection and Storage
The choice to store only $m\ll n$ direction vectors $\vc{p}_c$ to obtain a comparable amount of useful information as stored in the covariance matrix of the original CMA-ES requires a careful procedure of selection and storage. A simple yet powerful procedure proposed by <cit.> is to preserve a certain temporal distance in terms of number of iterations between the stored direction vectors. The procedure tends to store a more unique information in contrast to the case if the latest $m$ evolution path vectors would be stored. The latter case is different from the storage of $m$ gradients as in L-BFGS since the evolution path is gradually updated at each iteration
with a relatively small learning rate $c_c$ and from $\mu \ll n$ selected vectors.
UpdateSet(): direction vectors update
given $m \in \R^+, \vc{j} \in \mathbb{Z}_+^m, \vc{l} \in \mathbb{Z}_+^m, t \in \mathbb{Z}_+, \vc{N}\in \mathbb{Z}^m_+, \vc{P} \in \R^{m \times n}, \vc{p}_c \in \R^n$, $T \in \mathbb{Z}_+$
$t \leftarrow \left\lfloor t / T\right\rfloor$
$ t \leq m$
$ \vc{j}_t \leftarrow t$
$ i_{min} \leftarrow 1+argmin_i\left(\vc{l}_{\vc{j}_{i+1}}-\vc{l}_{\vc{j}_i} - \vc{N}_{i}\right),|1\leq i \leq (m-1)$
$ \vc{l}_{\vc{j}_{i_{min}}}-\vc{l}_{\vc{j}_{i_{min}-1}} - \vc{N}_{i} \geq 0$
$ i_{min} \leftarrow 1$
$ \vc{j}_{tmp} \leftarrow \vc{j}_{i_{min}}$
$i \leftarrow i_{min},\ldots,m-1$
$\vc{j}_i \leftarrow \vc{j}_{i+1}$
$\vc{j}_m \leftarrow \vc{j}_{tmp}$
$j_{cur} \leftarrow \vc{j}_{\texttt{min}(t + 1, m)}$
$\vc{l}_{j_{cur}} \leftarrow t T$
$\vc{P}^{(\vc{j}_{cur},:)} \leftarrow \vc{p}_c$
return: $\vc{j}_{cur}$, $\vc{j}$, $\vc{l}$
The selection procedure is outlined in Algorithm <ref> which outputs an array of pointers $\vc{j}$ such that $\vc{j}_1$ points out to a row in matrices $\vc{P}$ and $\vc{V}$ with the oldest saved vectors $\vc{p}_c$ and $\vc{v}$ which will be taken into account during the reconstruction procedure. The higher the index $i$ of $\vc{j}_i$ the more recent the corresponding direction vector is. The index $j_{cur}$ points out to the vector which will be replaced by the newest one in the same iteration when the procedure is called. The rule to choose a vector to be replaced is the following. Find a pair of consecutively saved vectors
($\vc{P}^{(\vc{j}_{i_{min}-1},:)},\vc{P}^{(\vc{j}_{i_{min}},:)})$ with the distance between them (in terms of indexes of iterations, stored in $\vc{l}$) closest to a target distance $N_{i}$ (line <ref>). If this distance is smaller than $N_{i}$ then the index $j_{i_{min}}$ will be swapped with last index
of $\vc{j}$ (lines <ref>-<ref>) and the corresponding
vector $\vc{P}^{(\vc{j}_{i_{min}},:)}$ will be replaced by the new vector
$\vc{p}_c$ (line <ref>), otherwise the oldest vector among $m$ saved vectors will be removed (as a result of line <ref>).
Thus, the procedure gradually replaces vectors in a way to keep them at the distance $N_i$ and with the overall time horizon for all vectors of at most $\sum_i^{m-1} N_i$ iterations.
The procedure can be called periodically every $T \in \mathbb{Z}_+$ iterations of the algorithm.
The values of $N_i$ are to be defined, e.g., as a function of problem dimension $n$ and direction vector index $i$.
Here, however, we set $N_i$ to $n$ for all $i$, i.e., the target distance equals to the problem dimension.
§.§ Sampling from a Family of Search Representations
At iteration $t$, a new $k$-th solution can be generated as
\begin{equation}
\label{newsol}
\vc{x}_k \leftarrow \vc{m}^t + \sigma^{t} Az( \vc{z}_k, \vc{i} ),
\end{equation}
where $\vc{z}_k \in \R^n$ is a vector drawn from some distribution and
transformed by a Cholesky factor by calling $Az(\vc{z}_k,\vc{i})$.
The $Az()$ procedure (see Algorithm <ref>) has an input $\vc{i}$
which defines indexes of direction vectors used to reconstruct
the Cholesky factor.
It is important to note that $\vc{P}^{(1,:)}$ refers to the first vector physically stored in matrix $\vc{P}$, $\vc{P}^{(\vc{j}_1,:)}$ refers to the oldest vector, $\vc{P}^{(\vc{j}_{\vc{i}_t},:)}$ refers to the $\vc{i}_t$-th oldest vector according to an array $\vc{i}$ with indexes of vectors of interest.
Thus, by setting $\vc{i} = 1,\ldots,m$ all $m$ vectors will be used in the reconstruction. Importantly, omission of some vector in $\vc{i}$ can be viewed as
setting of the corresponding learning rate in Equation (<ref>) to 0.
By varying $\vc{i}$, one can control the reconstruction of the Cholesky factor used for sampling and in this way explore a family of possible transformations of the coordinate system. The maximum number of vectors defined by $m$ can be associated with the number of degrees of freedom of this exploration.
While in the original LM-CMA <cit.> the value of $m$ is set to $4 + \lfloor 3 \ln \, n \rfloor $ to allow the algorithm scale up to millions of variables, we found that greater values of $m$, e.g., $\sqrt{n}$ often lead to better performance (see Section <ref> for a detailed analysis).
Thus, when memory allows, a gain in performance can be achieved.
However, due to an internal cost $O(mn)$ of $Az()$, the time cost then would scale as $O(n^{3/2})$ which is undesirable for $n \gg 1000$.
This is where the use of $m^{*}$ out of $m$ vectors can drastically reduce the time complexity. We propose to sample $m^{*}$ from a truncated half-normal distribution $\left| {{\mathcal N} \hspace{-0.13em}\left({{0},m^2_{\sigma}}\right)} \right|$ (see line <ref> of Algorithm <ref>) and set $\vc{i}$ to the latest $m^{*}$ vectors (line <ref>). For a constant $m_{\sigma}=4$, the time complexity of $Az()$ scales as $O(n)$. New value of $m^{*}$ is generated for each new individual. Additionally, to exploit the oldest information, we force $m^{*}$ to be generated with $10m_{\sigma}$ for one out of $\lambda$ individuals.
While for $m^{*}=0$ the new solution $\vc{x}_k$ appears to be sampled
from an isotropic normal distribution, the computation of $\vc{v}$ inverses is performed using all $m$ vectors.
SelectSubset(): direction vectors selection
given $m \in \mathbb{Z}_+, m_{\sigma} = 4,k \in \mathbb{Z}_+$
$k = 1$
$ m_{\sigma} \leftarrow 10 m_{\sigma}$
$ m^{*} \leftarrow \min(\left\lfloor m_{\sigma} \left| {{\mathcal N} \hspace{-0.13em}\left({{0},{1}}\right)} \right| \right\rfloor, m)$
$\vc{i} \leftarrow (m+1-m^{*}),\ldots,m$
return $\vc{i}$
§.§ Sampling from the Rademacher Distribution
Evolution Strategies are mainly associated with the multivariate normal distribution used to sample candidate solutions.
However, alternative distributions such as the Cauchy distribution can be used <cit.>.
Moreover, the Adaptive Encoding procedure proposed by <cit.> can be coupled with any sampling distribution as in <cit.>, where it was shown that completely deterministic adaptive coordinate descent on principal components obtained with the Adaptive Encoding procedure allows to obtain the performance comparable to the one of CMA-ES.
In this paper, inspired by <cit.>, we replace the original multivariate normal distribution used in LM-CMA by the Rademacher distribution, where a random variable has 50% chance to be either -1 or +1 (also can be viewed as a Bernoulli distribution). Thus, a pre-image vector of candidate solution $\vc{z}$ contains $n$ values which are either -1 or +1.
Our intention to use this distribution is three-fold: i) to reduce the computation complexity by a constant but rather significant factor,
ii) to demonstrate that the Rademacher distribution can potentially be an alternative to the Gaussian distribution at least in large scale
iii) to demonstrate that our new step-size adaptation rule (see next section), which does not make assumptions about the sampling distribution, can work well when used with non-Gaussian distributions.
As a support for this substitution, we recall that for a $n$-dimensional unit-variance spherical Gaussian,
for any positive real number $\beta \leq \sqrt{n}$, all but at most $3\exp^{-c \beta^2}$ of the mass lies within the annulus
$\sqrt{n-1}-\beta \leq r \leq \sqrt{n-1}+\beta$, where, $c$ is a fixed positive constant <cit.>.
Thus, when $n$ is large, the mass is concentrated in a thin annulus of width $O(1)$ at radius $\sqrt{n}$.
Interestingly, the sampling from the Rademacher distribution reproduces this effect of large-dimensional Gaussian sampling
since the distance from the center of a $n$-dimensional hypercube to its corners is $\sqrt{n}$.
§.§ Population Success Rule
The step-size used to define the scale of deviation of a sampled candidate solution from the mean of the mutation distribution can be adapted by the Population Success Rule (PSR) proposed for LM-CMA by <cit.>. This procedure does not assume that candidate solutions should come from the multivariate normal distribution as it is often assumed in Evolution Strategies including CMA-ES. Therefore, PSR procedure is well suited for the Rademacher distribution.
The PSR is inspired by the median success rule <cit.>. To estimate the success of the current population we combine fitness function values from the previous and current population into a mixed set
\begin{equation}
\label{Fmix}
\vc{f}_{mix} \leftarrow \vc{f}^{\,t-1} \cup \vc{f}^{\,t}
\end{equation}
Then, all individuals in the mixed set are ranked to define two sets $\vc{r}_{t-1}$ and $\vc{r}_{t}$ (the lower the rank the better the individual) containing ranks of individuals of the previous and current populations ranked in the mixed set.
A normalized success measurement is computed as
\begin{equation}
\label{ZPSR}
z_{PSR} \leftarrow \frac{\sum_{i=1}^{\lambda} \vc{r}^{t-1}(i) - \vc{r}^{t}(i)}{\lambda^2} - z^{*},
\end{equation}
where $z^*$ is a target success ratio and $\lambda^2$ accounts for the normalization of the sum term
and for different possible population size $\lambda$.
Then, for $s \leftarrow (1 - c_{\sigma}) s + c_{\sigma} z_{PSR}$, the step-size is adapted as
\begin{equation}
\label{MruleSigma}
\sigma^{t+1} \leftarrow \sigma^t \exp\left({s}\right / d_{\sigma}),
\end{equation}
where $d_{\sigma}$ is a damping factor which we set here to 1.
The ($\mu/\mu_{w},\lambda$)-LM-CMA
given $n \in \mathbb{N}_+$, $\lambda = 4 + \lfloor 3 \ln \, n \rfloor $, $\mu = \lfloor \lambda/2 \rfloor $,
$w_i = \frac{ \ln(\mu + 1) - \ln(i)}{\mu \ln(\mu + 1) - \sum_{j=1}^{\mu} \ln(j)}; i=1 \ldots \mu$,
$\mu_w = \frac{1}{\sum^{\mu}_{i=1} w^2_i}$,
$c_{\sigma} = 0.3$, $z^{*} = 0.25$,
$m = 4 + \lfloor 3 \ln \, n \rfloor $, $N_{steps} = n$,
$c_c = \frac{0.5}{\sqrt{n}}$, $c_1 = \frac{1}{10\ln(n+1)}$, $d_{\sigma}=1$,
$T=\left\lfloor log(n)\right\rfloor$
initialize $\vc{m}^{t=0} \in \R^{\dd}, \sigma^{t=0} > 0, \vc{p}^{t=0}_{c} = \ma{0}, s \leftarrow 0 , t \leftarrow 0 $
$k \leftarrow 1,\ldots,\lambda$
$k \pmod 2 = 1$
$ \vc{z}_k \leftarrow Rademacher() $
$ \vc{i} \leftarrow SelectSubset(k) $
$\vc{x}_k \leftarrow \vc{m}^t + \sigma^{t} Az( \vc{z}_k, \vc{i} ) $
$\vc{x}_k \leftarrow \vc{m}^t - (\vc{x}_{k-1} - \vc{m}^t) $
$\vc{f}^t_k \leftarrow f(\vc{x}_k)$
$ \vc{m}^{t+1} \leftarrow \sum_{i=1}^{\mu} \vc{w}_i \vc{x}_{i:\lambda} \;$
$ \vc{p}^{t+1}_{c} \leftarrow (1 - c_{c}) \vc{p}^{t}_{c} + \sqrt{c_{c}(2-c_{c})} \sqrt{\mu_w} (\vc{m}^{t+1} - \vc{m}^t) / {\sigma^t}$
$t \pmod{T} = 0$
$ UpdateSet(\vc{p}^{t+1}_{c}) $
$ UpdateInverses() $
$ \vc{r}^{t}, \vc{r}^{t-1} \leftarrow$ Ranks of $\vc{f}^{\,t}$ and $\vc{f}^{\,t-1}$ in $\vc{f}^{\,t} \cup \vc{f}^{\,t-1}$
$ z_{PSR} \leftarrow \frac{\sum_{i=1}^{\lambda} \vc{r}^{t-1}(i) - \vc{r}^{t}(i)}{\lambda^2} - z^{*} $
$ s \leftarrow (1 - c_{\sigma})s + c_{\sigma}z_{PSR} $
$ \sigma^{t+1} \leftarrow \sigma^{t} \mstr{exp}
( s / d_{\sigma}) $
$ t \leftarrow t + 1$
stopping criterion is met
§.§ The Algorithm
The improved LM-CMA is given in Algorithm <ref>.
At each iteration $t$, $\lambda$ candidate solutions are generated by mutation defined as a product of a vector $\vc{z}_k$ sampled from the Rademacher distribution and a Cholesky factor $\A^t$ reconstructed from $m^{*}$ out of $m$ vectors (line <ref>-<ref>) as described in Sections <ref>-<ref>. We introduce the mirrored sampling <cit.> to generate the actual $\vc{x}_k$ only every second time and thus decrease the computation cost per function evaluation by a factor of two
by evaluating $\vc{m}^t + \sigma^{t} Az( \vc{z}_k)$ and then its mirrored version
$\vc{m}^t - (\vc{x}_{k-1} - \vc{m}^t)$. The latter approach sometimes also improves the convergence rate.
The best $\mu$ out of $\lambda$ solutions are selected to compute the new mean $\vc{m}^{t+1}$ of the mutation distribution in line <ref>.
The new evolution path $\vc{p}^{t+1}_{c}$ is updated (line <ref>) from the change of the mean vector $\sqrt{\mu_w} (\vc{m}^{t+1} - \vc{m}^t) / {\sigma^t}$ and represents an estimate of descent direction.
One vector among $m$ vectors is selected and replaced by the new $\vc{p}^{t+1}_{c}$
in UpdateSet() procedure described in Section <ref>. All inverses $\vc{v}$ of evolution path vectors which are at least as recent as the direction vector to be replaced should be recomputed in the UpdateInverses() procedure as described in Section <ref>. The step-size is updated according to the PSR rule described in Section <ref>.
Test functions, initialization intervals and initial standard deviation (when applied). $\vc{R}$ is an orthogonal $n \times n$ matrix with each column vector $\vc{q}_i$ being a uniformly distributed unit vector implementing an angle-preserving transformation <cit.>.
Name Function Target $f(\vc{x})$ Init $\sigma^{0}$
Sphere (x)$=\sum_{i=1}^n \vc{x}^{2}_{i}$ $10^{-10}$ $[-5,5]^{n}$ 3
Ellipsoid (x)$=\sum_{i=1}^n 10^{6{\frac{i-1}{n-1}}} \vc{x}^{2}_{i}$ $10^{-10}$ $[-5,5]^{n}$ 3
Rosenbrock (x)$=\sum_{i=1}^{n-1}\left(100.(\vc{x}^{2}_{i}-x_{i+1})^{2}+(x_{i}-1)^{2}\right)$ $10^{-10}$ $[-5,5]^{n}$ 3
Discus (x)$=10^6\vc{x}^2_1 + \sum_{i=2}^n \vc{x}^{2}_{i}$ $10^{-10}$ $[-5,5]^{n}$ 3
Cigar (x)$=\vc{x}^2_1 + 10^6\sum_{i=2}^n \vc{x}^{2}_{i}$ $10^{-10}$ $[-5,5]^{n}$ 3
Different Powers (x)$=\sum_{i=1}^n \left|\vc{x}_i\right|^{2+4(i-1)/(n-1)}$ $10^{-10}$ $[-5,5]^{n}$ 3
Rotated Ellipsoid (x)=(Rx) $10^{-10}$ $[-5,5]^{n}$ 3
Rotated Rosenbrock (x)=(Rx) $10^{-10}$ $[-5,5]^{n}$ 3
Rotated Discus (x)=(Rx) $10^{-10}$ $[-5,5]^{n}$ 3
Rotated Cigar (x)=(Rx) $10^{-10}$ $[-5,5]^{n}$ 3
Rotated Different Powers (x)=(Rx) $10^{-10}$ $[-5,5]^{n}$ 3
§ EXPERIMENTAL VALIDATION
The performance of the LM-CMA is investigated comparatively to the L-BFGS <cit.>, the active CMA-ES by <cit.> and the VD-CMA by <cit.>. The sep-CMA-ES is removed from the comparison due to its similar but inferior performance w.r.t. the VD-CMA observed both in our study and by <cit.>.
We use the L-BFGS implemented in MinFunc library by <cit.> in its default parameter settings [<http://www.cs.ubc.ca/ schmidtm/Software/minFunc.html>],
the active CMA-ES (aCMA) without restarts in its default parametrization of CMA-ES MATLAB code version 3.61 [<http://www.lri.fr/ hansen/cmaes.m>].
For faster performance in terms of CPU time, the VD-CMA was (exactly) reimplemented in C language from the MATLAB code provided by the authors.
For the sake of reproducibility, the source code of all algorithms is available online [<http://sites.google.com/site/ecjlmcma/>]. The default parameters of LM-CMA are given in Algorithm <ref>.
We use a set of benchmark problems (see Table <ref>) commonly used in Evolutionary Computation, more specifically in the BBOB framework <cit.>. Indeed, many problems are missing including the ones where tested methods and LM-CMA fail to timely demonstrate reasonable performance in large scale settings. We focus on algorithm performance w.r.t. both the number of function evaluations used to reach target values of $f$, CPU time spent per function evaluation and the number of memory slots required to run algorithms. Any subset of these metrics can dominate search cost in large scale settings, while in low scale settings memory is typically of a lesser importance.
In this section, we first investigate the scalability of the proposed algorithm w.r.t. the existing alternatives.
While both the computational time and space complexities scale moderately with problem dimension, the algorithm is capable to preserve certain
invariance properties of the original CMA-ES. Moreover, we obtain unexpectedly good results on some well-known benchmark problems,
e.g., linear scaling of the budget of function evaluations to solve Separable and Rotated Ellipsoid problems.
We demonstrate that the performance of LM-CMA is comparable to the one of L-BFGS with exact estimation of gradient information.
Importantly, we show that LM-CMA is competitive to L-BFGS in very large scale (for derivative-free optimization) settings with 100,000 variables.
Finally, we investigate the sensitivity of LM-CMA to some key internal parameters such as the number of stored direction vectors $m$.
§.§ Space Complexity
Number of memory slots of floating point variables required to run different optimization algorithms versus problem dimension $n$.
The number of memory slots required to run optimization algorithms versus problem dimension $n$ referred to as space complexity can limit applicability of certain algorithms to large scale optimization problems. Here, we list the number of slots up to constant and asymptotically constant terms.
The presented algorithms store $\lambda$ generated solutions (LM-CMA, VD-CMA and aCMA with the default $\lambda=4 + \lfloor 3 \ln \, n \rfloor$ and $\mu=\lambda/2$, L-BFGS with $\lambda=1$) and some intermediate information (LM-CMA and L-BFGS with $m$ pairs of vectors, aCMA with at least two matrices of size $n\times n$) to perform the search.
Our implementation of VD-CMA requires $(max(\mu+15,\lambda) + 7)n$ slots compared to $(2.5\lambda +21) n$ of the original MATLAB code. The LM-CMA requires $(2m + \lambda + 6)n + 5m$ slots, the L-BFGS requires $(2m + 3) n$ slots and aCMA requires $(2n + \lambda + 3)n$ slots.
Figure <ref> shows that due to its quadratic space complexity aCMA
requires about $2\times 10^8$ slots (respectively, $8\times 10^8$ slots) for 10,000-dimensional (respectively, 20,000-dimensional) problems which with 8 bytes per double-precision floating point number would correspond to about 1.6 GB (respectively, 6.4 GB) of computer memory. This simply precludes the use of CMA-ES and its variants with explicit storage of the full covariance matrix or Cholesky factors to large scale optimization problems with $n>10,000$.
LM-CMA stores $m$ pairs of vectors as well as the L-BFGS. For $m=4 + \lfloor 3 \ln \, n \rfloor$ (as the default population size in CMA-ES), L-BFGS is 2 times and LM-CMA is 3 times more expensive in memory than VD-CMA, but they all basically can be run for millions of variables.
In this paper, we argue that additional memory can be used while it is allowed and is at no cost.
Thus, while the default $m=4 + \lfloor 3 \ln \, n \rfloor$, we suggest to use $m=\lfloor2\sqrt{n}\rfloor$ if memory allows (see Section <ref>). In general, the user can provide a threshold on memory and if, e.g., by using $m=\lfloor2\sqrt{n}\rfloor$ this memory threshold would be violated, the algorithm automatically reduces $m$ to a feasible $m_f$.
Timing results of LM-CMA and VD-CMA averaged over the whole run on the separable Ellipsoid compared to timing results of simple operations averaged over 100 seconds of experiments. The results for L-BFGS are not shown but for an optimized implementation would be comparable to one scalar-vector multiplication.
§.§ Time Complexity
The average amount of CPU time internally required by an algorithm per evaluation of some objective function $f \in \R^n$ (not per algorithm iteration) referred to as time complexity also can limit applicability of certain algorithms to large scale optimization problems. They simply can be too expensive to run, e.g., much more expensive than to perform function evaluations.
Figure <ref> shows how fast CPU time per evaluation scales for different operations measured on one 1.8 GHz processor of an Intel Core i7-4500U. Scalar-vector multiplication of a vector with $n$ variables scales linearly with ca. $4\cdot{10}^{-10}n$ seconds, evaluation of the separable Ellipsoid function is 2-3 times more expensive if a temporary data is used.
Sampling of $n$ normally distributed variables scales as ca. 60 vectors-scalar multiplications that defines the cost of sampling of unique candidate solutions of many Evolution Strategies such as separable CMA-ES and VD-CMA. However, the sampling of variables according to the Rademacher distribution is about 10 times cheaper. The use of mirrored sampling also decreases the computational burden without worsening the convergence. Finally, the internal computation cost of LM-CMA scales linearly as about 25 scalar-vector multiplications per function evaluation. It is much faster than the lower bound for the original CMA-ES defined by one matrix-vector multiplication required to sample one individual. We observe that the cost of one matrix-vector multiplications costs about $2n$ scalar-vector multiplications, the overhead is probably due to access to matrix members.
The proposed version of LM-CMA is about 10 times faster internally than the original version by <cit.> due to the use of mirrored sampling, the Rademacher sampling distribution and sampling with $m^*$ instead of $m$ direction vectors both for $m=4 + \lfloor 3 \ln \, n \rfloor$ and $m=\lfloor 2\sqrt{n} \rfloor$. For 8192-dimensional problems it is about 1000 times faster internally than CMA-ES algorithms with the full covariance matrix update (the cost of Cholesky-CMA-ES is given in <cit.>).
§.§ Invariance under Rank-preserving Transformations of the Objective Function
The LM-CMA belongs to a family of so-called comparison-based algorithms.
The performance of these algorithms is unaffected by rank-preserving (strictly monotonically increasing) transformations of the objective function, e.g.,
whether the function $f$, $f^3$ or $f \times \left| f \right|^{-2/3}$ is minimized <cit.>.
Moreover, this invariance property provides robustness to noise as far as this noise
does not impact a comparison of solutions of interest <cit.>.
In contrast, gradient-based algorithms are sensitive to rank-preserving transformations of $f$. While the availability of gradient information may
mitigate the problem that objective functions with the same contours can be solved with a different number of functions evaluations, the lack of gradient information
forces the user to estimate it with approaches whose numerical stability is subject to scaling of $f$. Here, we simulate L-BFGS in an idealistic black-box scenario when gradient information is estimated perfectly (we provide exact gradients) but at the cost of $n+1$ function evaluations per gradient that corresponds to the cost of the forward difference method. Additionally, we investigate the performance of L-BFGS with the central difference method ($2n+1$ evaluations per gradient) which is twice more expensive but numerically more stable. We denote this method as CL-BFGS.
§.§ Invariance under Search Space Transformations
The trajectories show the median of 11 runs of LM-CMA, L-BFGS with exact gradients provided at the cost of $n+1$ evaluations per gradient, CL-BFGS with central differencing, active CMA-ES and VD-CMA on 512- Separable/Original (Left Column) and Rotated (Right Column) functions.
The trajectories show the median of 11 runs of LM-CMA, L-BFGS with exact gradients, CL-BFGS with central differencing, active CMA-ES and VD-CMA on 512- Separable (Left Column) and Rotated (Right Column) functions.
Invariance properties under different search space transformations include translation invariance, scale invariance, rotational invariance and general linear invariance under any full rank matrix $\vc{R}$ when the algorithm performance on $f(\vc{x})$ and $f(\vc{R}\vc{x})$ is the same given that the initial conditions of the algorithm are chosen appropriately <cit.>.
Thus, the lack of the latter invariance is associated with a better algorithm performance for some $\vc{R}$ and worse for the others.
In practice, it often appears to be relatively simple to design an algorithm specifically for a set of problems with a particular $\vc{R}$, e.g., identity matrix, and then demonstrate its good performance. If this set contains separable problems, the problems where the optimum can be found with a coordinate-wise search, then even on highly multi-modal functions great results can be easily achieved <cit.>. Many derivative-free search algorithms in one or another way exploit problem separability and fail to demonstrate a comparable performance on, e.g., rotated versions of the same problems. This would not be an issue if most of real-world problems are separable, this is, however, unlikely to be the case and some partial-separability or full non-separability are more likely to be present.
Median (out of 11 runs) number of function evaluations required to find $f(\vc{x})=10^{-10}$ for LM-CMA, L-BFGS with exact gradients, CL-BFGS with central differencing, active CMA-ES and VD-CMA. Dotted lines depict extrapolated results.
The original CMA-ES is invariant w.r.t. any invertible linear transformation of the search space, $\vc{R}$, if
the initial covariance matrix $\C^{t=0} = \vc{R}^{-1} ( \vc{R}^{-1} )^{T}$, and the initial search point(s)
are transformed accordingly <cit.>.
However, $\vc{R}$ matrix is often unknown
(otherwise, one could directly transform the objective function) and cannot be stored in memory in large scale settings
with $n \gg 10,000$. Thus, the covariance matrix adapted by LM-CMA has at most rank $m$ and so the intrinsic coordinate
system cannot capture some full rank matrix $\vc{R}$ entirely. Therefore,
the performance of the algorithm on $f(\vc{R}\vc{x})$ compared to $f(\vc{x})$ depends on $\vc{R}$.
However, in our experiments, differences in performance on axis-aligned and rotated ill-conditioned functions were marginal.
Here, we test LM-CMA, aCMA, L-BFGS, CL-BFGS both on separable problems and their rotated versions (see Table <ref>).
It is simply intractable to run algorithms on large scale rotated problems with $n>1000$ due to the quadratic cost of involved matrix-vectors multiplications (see Figure <ref>).
Fortunately, there is no need to do it for algorithms that are invariant to rotations of the search space since their performance is the same as on the separable problems whose evaluation is cheap (linear in time).
Figures <ref>-<ref> show that the performance of aCMA on 512-dimensional (the dimensionality still feasible to perform full runs of aCMA) separable (left column) and rotated (right column) problems is very similar and the difference (if any) is likely due to a non-invariant initialization. The invariance in performance is not guaranteed but rather observed for LM-CMA, L-BFGS and CL-BFGS.
However, the performance of the VD-CMA degrades significantly on , and functions due to the restricted form of the adapted covariance matrix of Equation (<ref>). Both , and their rotated versions can be solved efficiently since they have a Hessian matrix whose inverse can be well approximated by Equation (<ref>) <cit.>.
An important observation from Figures <ref>-<ref> is that even the exact gradient information is not sufficient for L-BFGS to avoid numerical problems which lead to an imprecise estimation of the inverse Hessian matrix and premature convergence on and . The L-BFGS with the central difference method (CL-BFGS) experiences an early triggering of stopping criteria on and . While numerical problems due to imprecise derivative estimations are quite natural for L-BFGS especially on ill-conditioned problems, we assume that with a better implementation of the algorithm (e.g., with high-precision arithmetic) one could obtain a more stable convergence.
Therefore, we extrapolate the convergence curves of L-BFGS and CL-BFGS towards the target $f=10^{-10}$ after removing the part of the curve which clearly belongs to the stagnation, e.g., $f<10^{-7}$ on .
§.§ Scaling with Problem Dimension
The performance versus the increasing number of problem variables is given in Figure <ref>.
We exclude the results of VD-CMA on some problems because, as can be seen from Figures <ref>-<ref>, the algorithm does not find the optimum with a reasonable number of function evaluations or/and convergences prematurely. For algorithms demonstrating the same performance on separable and non-separable problems (see Figures <ref>-<ref>), we plot some results obtained on separable problems as obtained on rotated problems in Figure <ref> to avoid possible misunderstanding from designers of separability-oriented algorithms.
The results suggest that L-BFGS shows the best performance, this is not surprising given the form of the selected objective functions (see Table <ref>). We should also keep in mind that the exact gradients were provided and this still led to premature convergence on some functions (see Figures <ref>-<ref>). In the black-box scenario, one would probably use L-BFGS with the forward or central (CL-BFGS) difference methods. The latter is often found to lead to a loss by a factor of 2 (as expected due to $2n+1$ versus $n+1$ cost per gradient), except for the , where the loss is increasing with problem dimension.
Quite surprisingly, the LM-CMA outperforms VD-CMA and aCMA on . This performance is close to the one obtained for (1+1) Evolution Strategy with optimal step-size. Bad performance on anyway would not directly mean that an algorithm is useless, but could illustrate its performance in vicinity of local optima when variable-metric algorithms (e.g., CMA-like algorithms) may perform an isotropic search w.r.t. an adapted internal coordinate system. The obtained results are mainly due to the Population Success Rule which deserves an independent study similar to the one by <cit.>.
Nevertheless, we would like to mention a few key points of the PSR.
By design, depending on the target success ratio $z^*$, one can get either biased (for $z^*\neq 0$) or unbiased (for $z^*= 0$) random walk on random functions. It would be a bias to say that either biased or unbiased change of $\sigma$ "is better" on random functions, since the latter depends on the context. Due to the fact that the (weighted) mean of each new population is computed from the best $\mu$ out of $\lambda$ individuals, the $\lambda$ individuals of the new generation are typically as good as the (weighted) best $\mu$ individuals of the previous one, and, thus, if $z^*=0$ one may expect $z_{PSR}>0$ from Equation (<ref>). Typically, it is reasonable to choose $z^*\in(0,0.5)$ lower-bounded by 0 due to random functions and upper-bounded by 0.5 due to linear functions. In this study, we choose 0.3 which lies roughly in the middle of the interval. It is important to mention a striking similarity with the 1/5th success rule <cit.>. We consider the PSR to be its population-based version.
The trajectories show the median of 11 runs of LM-CMA, L-BFGS (with exact gradients provided at the cost of $n+1$ evaluations per gradient) on 100,000-dimensional Rosenbrock and Ellipsoid functions.
The performance of LM-CMA on is probably the most surprising result of this work.
In general, the scaling of CMA-ES is expected to be from super-linear to quadratic with $n$ on since the number of parameters of the full covariance matrix to learn is $(n^2+n)/2$ <cit.>. While aCMA demonstrates this scaling, LM-CMA scales linearly albeit with a significant constant factor. The performance of both algorithms coincides at $n\approx 1000$, then, LM-CMA outperforms aCMA (given that our extrapolation is reasonable) with a factor increasing with $n$. It should be noted that aCMA is slower in terms of CPU time per function evaluation by a factor of $n/10$ (see Figure <ref>). Another interesting observation is that the L-BFGS is only slightly faster than LM-CMA, while CL-BFGS is actually outperformed by the latter. An insight to these observations can be found in Figure <ref> where both LM-CMA and L-BFGS outperform aCMA by a factor of 10 in the initial part of the search, while aCMA compensates this loss by having the covariance matrix well adapted that allows to accelerate convergence close to the optimum. This might be explained as follows: a smaller number of internal parameters defining the intrinsic coordinate system can be learned faster and with greater learning rates, this allows a faster convergence but may slow down the search in vicinity of the optimum if the condition number cannot be captured by the reduced intrinsic coordinate system.
The LM-CMA is better or is as good as VD-CMA on and where it is expected to be outperformed by the latter due a presumably few principal components needed to be learnt to solve these problems. The scaling on suggests that the problem is more difficult (e.g., more difficult than ) than one could expect, mainly due to an adaptation of the intrinsic coordinate system required while following the banana shape valley of this function.
The results on 100,000-dimensional problems (see Figure <ref>) show that LM-CMA outperforms L-BFGS on the first $10n - 20n$ function evaluations which corresponds to the first 10-20 iterations of L-BFGS. This observation suggests that LM-CMA can be viewed as an alternative to L-BFGS when $n$ is large and the available number of function evaluations is limited. While it can provide a competitive performance in the beginning, it is also able to learn dependencies between variables to approach the optimum.
The trajectories show the median of 11 runs of LM-CMA in default settings,
CL-BFGS in default settings and tuned LM-CMA (all three algorithms are with restarts)
on the second nonsmooth variant of Nesterov-Chebyshev-Rosenbrock function in dimensions 128 and 2048.
Sensitivity of LM-CMA to different settings of $m$.
§.§ Performance on a nonsmooth variant of Nesterov's function
While designed for smooth optimization,
BFGS is known to work well for nonsmooth optimization too.
A recent study by <cit.> demonstrated
the difficulties encountered by BFGS on some nonsmooth
functions. We selected one of the test functions from <cit.> called
the second nonsmooth variant of Nesterov-Chebyshev-Rosenbrock
function defined as follows:
\begin{equation}
\label{Fnest}
\hat{N}(x) = \frac{1}{4} \left| x_1 - 1 \right| +
\sum_{i=1}^{n-1} \left| x_{i+1} - 2\left| x_i \right| + 1\right|
\end{equation}
This function is nonsmooth (though locally Lipschitz) as well as
nonconvex, it has $2^{n-1}$ Clarke stationary points <cit.>.
<cit.> showed that for $n=5$ BFGS starting from 1000
randomly generated points finds all 16 Clarke stationary points (for the definition of Clarke stationary points see <cit.>) and the probability to find the global minimizer is only by about a factor of 2 greater than to find any of the Clarke
stationary points. This probability dropped by a factor of 2 for $n=6$ while and since the number of Clarke stationary points doubled <cit.>. Clearly, the problem becomes extremely difficult for BFGS when $n$ is large.
We launched LM-CMA and CL-BFGS (L-BFGS performed worse) on $\hat{N}(x)$ for $n=128$ and $n=2048$.
Figure <ref> shows that CL-BFGS performs better than LM-CMA,
however, both algorithms in default settings and with restarts do not perform well.
We tuned both LM-CMA and CL-BFGS but report the results only for LM-CMA since we failed to improve
the performance of CL-BFGS by more than one order of magnitude of the objective function value.
The tuned parameters for LM-CMA are: i) doubled population size $\lambda$,
ii) increased learning rate by 15 to $c_1 = 15/(10\ln(n+1))$, iii) an extremely small learning rate for step-size adaptation
$c_{\sigma} = 0.3/n^2$ instead of $c_{\sigma} = 0.3$.
The last modification is probably the most important, practically, it defines the schedule how step-size decreases.
A similar effect can be achieved by reducing $z^{*}$ or increasing $d_{\sigma}$.
Faster learning of dependencies between variables and slower step-size decrease drastically improve the convergence
and the problem can be solved both for $n=128$ and $n=2048$ (Figure <ref>).
Interestingly, the number of function evaluations scales almost linearly with problem dimension.
We expected that tuning of CL-BFGS will lead to similar improvements. Surprisingly,
our attempts to modify its parameters, often in order to slow down the convergence (e.g., type and number of line-search steps, Wolfe conditions parameters) failed. We still expect that certain modifications should improve CL-BFGS and thus we leave this question open.
The settings tuned for $\hat{N}(x)$ function differ significantly from the default ones.
It is of great interest to find an online procedure to adapt them.
The next section is aimed at gaining some intuition on parameters importance in LM-CMA.
§.§ Sensitivity to Parameters
Eigenspectrums of $\C^t=\A^t {\A^t}^T$ for $t$ denoting iteration of LM-CMA with $m$ direction vectors ($m=5$, $m=4+\left\lfloor 3\ln 1024\right\rfloor=24$, $m=\left\lfloor 2\sqrt{n}\right\rfloor=64$) on 1024-dimensional problems. Darker (blue) lines correspond to later iterations. The number of function evaluations to reach $f(\vc{x})=10^{-10}$ is given in the title of each sub-figure.
The black-box scenario implies that the optimization problem at hand is not known,
it is therefore hard if even possible to suggest a "right" parametrization of our algorithm that works best on all problems.
Offline tuning in large scale optimization is also computationally expensive.
It is rather optimistic to believe that one always can afford enough computational resources to run algorithms till the optimum on very large real-world optimization problems. Nevertheless, we tend to focus on this scenario in order to gain an understanding about scalability on benchmark problems.
Our experience with parameter selection by exclusion of non-viable settings suggests that there exists a
dependency between the population size $\lambda$, number of stored vectors $m$,
the target temporal distance between them $N_{steps}$, the learning rate $c_c$ for the evolution path and learning rate $c_1$ for the Cholesky factor update. The main reason for this is that all of them impact how well the intrinsic coordinate system defined by the Cholesky factor reflects the current optimization landscape.
A posteriori, if $m\ll n$, it seems reasonable to store vectors with a temporal distance in order of $N_{steps}=n$ on problems where a global coordinate system is expected to be constant, e.g., on a class of problems described by the general ellipsoid model <cit.>. The learning rate for the evolution path is related to both $m$ and $n$, here, we set it to $c_c = \frac{0.5}{\sqrt{n}}$ which is roughly inversely proportional to the (if affordable) suggested $m=\lfloor2\sqrt{n}\rfloor$. We found that the chosen $c_c$ is still valid for the default $m = 4 + \lfloor 3 \ln \, n \rfloor $. We do not have a good interpretation for the learning rate $c_1 = \frac{1}{10\ln(n+1)}$. In general, we are not in favor of strongly arguing for some parameters settings against the others since as already mentioned above they are problem-dependent. A more appropriate approach would be to perform online adaptation of hyper-parameters as implemented for the original CMA-ES by <cit.>.
We present an analysis for $m$ which directly affects the amount of memory required to run the algorithm, and,
thus, is of special interest since the user might be restricted in memory on very large scale optimization problems with $n>10^6$. Figure <ref> shows that the greater the $m$ the better the performance.
The results obtained for the default $m = 4 + \lfloor 3 \ln \, n \rfloor $, i.e., the results demonstrated in the previous sections can be improved with $m=\lfloor2\sqrt{n}\rfloor$. The improvements are especially pronounced on functions, where the factor is increasing with $n$ and the overall cost to solve the function reaches the one extrapolated for aCMA at $n=8192$ (see Figure <ref>). It is surprising to observe that $m=5$ and even $m=2$ are sufficient to solve , and . The latter is not the case for , where small values of $m$ lead to an almost quadratic growth of run-time. The overall conclusion would be that on certain problems the choice of $m$ is not critical, while greater values of $m$ are preferable in general.
We investigated the eigenspectrum of the covariance matrix $\C^t$ constructed as $\A^t {\A^t}^T$ from the Cholesky factor $\A^t$. The results for single runs on different 1024-dimensional functions and for different $m$ are shown in Figure <ref>.
The evolution of the eigenspectrum during the run is shown by gradually darkening (blue) lines with increasing $t$. Clearly, the number of eigenvalues is determined by $m$. The profiles, e.g., the one of , also reflect the structure of the problems (see Table <ref>). The greater the $m$, the greater condition number can be captured by the intrinsic coordinate system as can be see for , and , that in turn leads to a better performance. However, this is not always the case as can be seen for that again demonstrates that optimal hyper-parameter settings are problem-dependent.
§ CONCLUSIONS
We adapt an idea from derivative-based optimization to extend best performing evolutionary algorithms such CMA-ES to large scale optimization. This allows to reduce the cost of optimization in terms of time by a factor of $n/10$ and memory by a factor between $\sqrt{n}$ and $n$. Importantly, it also often reduces the number of function evaluations required to find the optimum.
The idea to store a limited number of vectors and use them to adapt an intrinsic coordinate system is not the only but one of probably very few ways to efficiently search in large scale continuous domains.
We propose two quite similar alternatives: i) the storage of points and a later estimation of descent directions from differences of these points, and ii) the use of a reduced matrix $m \times n$ as in <cit.> but with a modified sampling procedure to obtain linear time complexity as proposed for the Adaptive Coordinate Descent by <cit.>.
The use of the Population Success Rule is rather optional and alternative step-size adaptation procedures can be applied.
However, we find its similarity with the 1/5-th rule quite interesting. The procedure does not make any assumption about the sampling distribution, this allowed to use the Rademacher distribution. When $n$ is large, the sampling from a $n$-dimensional Rademacher distribution resembles
the sampling from a $n$-dimensional Gaussian distribution since the probability mass of the latter is concentrated in a thin annulus of width $O(1)$ at radius $\sqrt{n}$.
The presented comparison shows that LM-CMA outperforms other evolutionary algorithms and is comparable to L-BFGS on non-trivial large scale optimization problems when the black-box (derivative-free) scenario is considered. Clearly, the black-box scenario is a pessimistic scenario but a substantial part of works that use finite difference methods for optimization deal with this scenario, and, thus, can consider LM-CMA as an alternative.
Importantly, LM-CMA is invariant to rank-preserving transformations of the objective function and therefore is potentially more robust than L-BFGS. The results shown in Figure 7 suggest that the use of a smaller number of direction vectors $m$ can be still efficient,
i.e., more efficient algorithms, e.g., with adaptive $m$ (or an adaptive $m \times n$ transformation matrix) can be designed.
It seems both promising and feasible to extend the algorithm to constrained, noisy and/or multi-objective optimization, the domains, which are both hardly accessible for L-BFGS and keenly demanded by practitioners.
As an important contribution to the success in this direction, it would be helpful to implement online adaptation of internal hyper-parameters as already implemented in the original CMA-ES <cit.>.
This would ensure an additional level of invariance and robustness on large scale black-box optimization problems.
I am grateful to Michèle Sebag and Marc Schoenauer for many valuable discussions and insights.
I also would like to thank Oswin Krause, Youhei Akimoto and the anonymous reviewers
whose interest and valuable comments helped to improve this work.
|
1511.00166
|
Algorithms and underlying mathematics are presented
for numerical computation with periodic
functions via approximations to machine precision by
trigonometric polynomials, including the solution of
linear and nonlinear periodic ordinary differential equations.
Differences from the nonperiodic Chebyshev case are highlighted.
Chebfun, Fourier series, trigonometric interpolation, barycentric
42A10, 42A15, 65T40
WRIGHT, JAVED, MONTANELLI AND TREFETHEN
EXTENSION OF CHEBFUN TO PERIODIC PROBLEMS
§ INTRODUCTION
It is well known that trigonometric representations of periodic
functions and Chebyshev polynomial representations of nonperiodic functions
are closely related. Table <ref> lists some of the
parallels between these two situations. Chebfun, a software
system for computing with functions and solving ordinary
differential equations <cit.>,
relied entirely on Chebyshev representations in its
first decade. This paper describes
its extension to periodic problems initiated by
the first author and released with Chebfun Version 5.1
in December 2014.
Some parallels between trigonometric and Chebyshev
settings. The row of contributors' names is just a sample
of some key figures.
Trigonometric Chebyshev
height 14 pt width 0 pt $t\in \pipi$ $x\in \ones$
periodic nonperiodic
$\exp(ikt)$ $T_k(x)$
trigonometric polynomials algebraic polynomials
equispaced points Chebyshev points
trapezoidal rule Clenshaw–Curtis quadrature
companion matrix colleague matrix
Horner's rule Clenshaw recurrence
Fast Fourier Transform Fast Cosine Transform
Gauss, Fourier, Zygmund$,\dots$ Bernstein, Lanczos, Clenshaw$,\dots$
Though Chebfun is a software product, the main focus of this
paper is mathematics and algorithms rather than software
per se. What makes this subject interesting is that
the trigonometric/Chebyshev parallel, though close, is not
an identity. The experience of building a software system
based first on one kind of representation and then extending
it to the other has given the Chebfun team a uniquely intimate
view of the details of these relationships. We begin this
paper by listing ten differences between Chebyshev
and trigonometric formulations that we have found important.
This will set the stage for presentations of the problems of
trigonometric series, polynomials, and projections (Section 2),
trigonometric interpolants, aliasing, and barycentric formulas
(Section 3), approximation theory and quadrature (Section 4), and various
aspects of our algorithms (Sections 5–7).
1. One basis or two. For working with polynomials on
$\ones$, the only basis functions one needs are the Chebyshev
polynomials $T_k(x)$. For trigonometric polynomials on $\pipi$,
on the other hand, there are two equally good equivalent
choices: complex exponentials $\exp(ikt)$, or sines and
cosines $\sin(kt)$ and $\cos(kt)$.
The former is mathematically
simpler; the latter is mathematically more elementary and
provides a framework for dealing with even and odd symmetries.
A fully useful software system for periodic functions needs
to offer both kinds of representation.
2. Complex coefficients.
In the $\exp(ikt)$ representation, the expansion coefficients
of a real periodic function are complex. Mathematically,
they satisfy certain symmetries, and a software system needs to
enforce these symmetries to avoid
imaginary rounding errors. Polynomial approximations
of real nonperiodic functions, by contrast, do not lead to
complex coefficients.
3. Even and odd numbers of parameters.
A polynomial of degree $n$ is determined by $n+1$ parameters,
a number that may be odd or even. A trigonometric
polynomial of degree $n$, by contrast, is determined by $2n+1$
parameters, always an odd number, as a consequence of the
$\exp(\pm inx)$ symmetry. For most purposes it is unnatural
to speak of trigonometric polynomials with an even number of
degrees of freedom. Even numbers make sense, on the other hand,
in the special case of trigonometric polynomials defined by
interpolation at equispaced points, if one imposes the symmetry
condition that the interpolant of the $(-1)^j$ sawtooth should
be real, i.e., a cosine rather than a complex exponential.
Here distinct formulas are needed for the even and odd cases.
4. The effect of differentiation.
Differentiation lowers the degree of an algebraic polynomial,
but it does not lower the degree of a trigonometric polynomial; indeed
it enhances the weight of its highest-degree components.
5. Uniform resolution across the interval.
Trigonometric representations have uniform properties across
the interval of approximation,
but polynomials are nonuniform, with much greater
resolution power near the ends of $\ones$ than near the
middle <cit.>.
6. Periodicity and translation-invariance.
The periodicity of trigonometric representations means
that a periodic chebfun constructed on $\pipi$, say, can be perfectly
well evaluated at $10\pi$ or $100\pi$; nonperiodic chebfuns have
no such global validity. Thus, whereas
interpolation and extrapolation are utterly different for
polynomials, they are not so different in the trigonometric case.
A subtler consequence of translation
invariance is explained in the footnote on p. footn.
7. Operations that break periodicity.
A function that is smooth and periodic may lose these properties
when restricted to a subinterval or subjected to
operations like rounding or absolute value.
This elementary fact has the consequence that
a number of operations on periodic chebfuns require
their conversion to nonperiodic form.
8. Good and bad bases.
The functions $\exp(ikt)$ or $\sin(kt)$ and $\cos(kt)$
are well-behaved by any measure, and nobody would normally
think of using any other basis functions for representing
trigonometric functions. For polynomials, however,
many people would reach for the basis of monomials $x^k$
before the Chebyshev polynomials $T_k(x)$. Unfortunately,
the monomials are exponentially ill-conditioned on $\ones$:
a degree-$n$ polynomial of size $1$ on $\ones$ will typically
have coefficients of order $2^n$ when expanded in the basis
$1,x,\dots,x^n$. Use of this basis will cause trouble in
almost any numerical calculation unless $n$ is very small.
9. Good and bad interpolation points.
For interpolation of periodic functions, nobody would normally
think of using any interpolation points other than equispaced.
For interpolation of nonperiodic functions by polynomials,
however, equispaced points lead to exponentially ill-conditioned
interpolation problems <cit.>. The mathematically
appropriate choice is not obvious until one learns it: Chebyshev
points, quadratically clustered near $\pm 1$.
10. Familiarity.
All the world knows and trusts Fourier analysis.
By contrast, experience with Chebyshev polynomials is often
the domain of experts, and it is not as widely appreciated that
numerical computations based on polynomials can be trusted.
Historically, points 8 and 9 of this list have led to
this mistrust.
The book Approximation Theory and Approximation Practice <cit.>
summarizes the mathematics and algorithms of Chebyshev technology
for nonperiodic functions. The present paper was written
with the goal in mind of compiling analogous information
in the trigonometric case. In particular,
Section 2 corresponds to Chapter 3 of <cit.>, Section 3 to
Chapters 2, 4, and 5, and Section 4 to Chapters 6, 7, 8, 10, and 19.
§ TRIGONOMETRIC SERIES, POLYNOMIALS, AND PROJECTIONS
Throughout this paper, we assume $f$ is
a Lipschitz continuous periodic function on
Here and in all our statements about periodic
functions, the interval $\pipi$ should be understood
periodically: $t=0$ and $t=2\pi$ are identified,
and any smoothness assumptions apply across this point in the
same way as for $t\in (0,2\pi)$ <cit.>.
It is known that $f$ has a unique trigonometric series,
absolutely and uniformly convergent, of the form
\begin{equation}
f(t) = \sum_{k=-\infty}^\infty \ck e^{ikt},
\label{series1}
\end{equation}
with Fourier coefficients
\begin{equation}
\ck = {1\over 2\pi} \int_0^{2\pi} f(t) e^{-ikt} dt.
\label{coeffs1}
\end{equation}
(All coefficients in our discussions are
in general complex, though in cases of certain symmetries
they will be purely real or imaginary.)
Equivalently, we have
\begin{equation}
f(t) = \sum_{k=0}^\infty \ak \cos(kt) + \sum_{k=1}^\infty \bk \sin(kt),
\label{series2}
\end{equation}
with $a_0^{} = c_0^{}$ and
\begin{equation}
\ak = {1\over \pi} \int_0^{2\pi} f(t) \cos(kt) dt, \quad
\bk = {1\over \pi} \int_0^{2\pi} f(t) \sin(kt) dt \qquad \rlap{$(k\ge 1).$}
\label{coeffs2}
\end{equation}
The formulas (<ref>) can be derived by matching
the $e^{ikt}$ and $e^{-ikt}$ terms of (<ref>) with
those of (<ref>), which yields the identities
\begin{equation}
\ck = {a_k^{}\over 2} + {\bk\over 2i},\quad
\cmk = {a_k^{}\over 2} - {\bk\over 2i} \qquad \rlap{$(k\ge 1),$}
\label{abccoeffs1}
\end{equation}
or equivalently,
\begin{equation}
\ak = \ck + \cmk, \quad \bk = i(\ck - \cmk) \qquad\rlap{$(k\ge 1).$}
\label{abccoeffs2}
\end{equation}
Note that if $f$ is real, then (<ref>) implies that
$\ak$ and $\bk$ are real. The coefficients
$\ck$ are generally complex, and (<ref>) implies that they
satisfy $\cmk = \overline{c}_k^{}$.
The degree $n$ trigonometric projection of $f$ is the function
\begin{equation}
\fn(t) = \sum_{k=-n}^n \ck e^{ikt},
\label{trigpoly1}
\end{equation}
or equivalently
\begin{equation}
\fn(t) = \sum_{k=0}^n \ak \cos(kt) + \sum_{k=1}^n \bk \sin(kt).
\label{trigpoly2}
\end{equation}
More generally, we say that a function of the form
(<ref>)–(<ref>) is a
trigonometric polynomial of degree $n$, and
we let $\Pn$ denote the $(2n+1)$-dimensional vector
space of all such polynomials.
The trigonometric projection $\fn$
is the least-squares approximant to $f$ in $\Pn$, i.e., the
unique best approximation to $f$ in the $L^2$ norm over $\pipi$.
§ TRIGONOMETRIC INTERPOLANTS, ALIASING, AND BARYCENTRIC FORMULAS
Mathematically, the simplest degree $n$ trigonometric approximation of
a periodic function $f$ is its trigonometric projection
(<ref>)–(<ref>). This approximation
depends on the values of $f(t)$ for all $t\in\pipi$ via
(<ref>) or (<ref>). Computationally, a simpler
approximation of $f$ is its degree $n$ trigonometric
interpolant, which only depends on the values at
certain interpolation points.
In our basic configuration, we wish to interpolate $f$ in equispaced
points by a function $\pn \in \Pn$.
Since the dimension of $\Pn$ is $2n+1$, there should be
$2n+1$ interpolation points. We take these
trigonometric points to be
\begin{equation}
\tk = {2\pi k\over N}, \qquad 0\le k \le N-1
\label{trigpts}
\end{equation}
with $N=2n+1$.
The trigonometric interpolation problem goes back at least to
the young Gauss's calculations of the orbit of the asteroid
Ceres in 1801 <cit.>.
It is known that there exists a unique interpolant
$\pn\in\Pn$ to any set of data values $\fk = f(\tk)$.
Let us write $\pn$ in the form
\begin{equation}
\pn(t) = \sum_{k=-n}^n \ckt e^{ikt},
\label{interp1}
\end{equation}
or equivalently
\begin{equation}
\pn(t) = \sum_{k=0}^n \akt \cos(kt) + \sum_{k=1}^n \bkt \sin(kt),
\label{interp2}
\end{equation}
for some coefficients $\tilde c_{-n}^{},\dots,\tilde c_n^{}$
or equivalently $\tilde a_0^{},\dots,\tilde a_n^{}$
and $\tilde b_1^{},\dots,\tilde b_n^{}$.
The coefficients $\ckt$ and $\ck$ are related by
\begin{equation}
\ckt = \sum_{j=-\infty}^\infty c_{k + jN}^{}
\qquad\rlap{$(|k| \le n)$}
\label{aliasing1}
\end{equation}
(the Poisson summation formula),
and similarly $\akt$/$\bkt$ and $\ak$/$\bk$ are related by
$\tilde a_0^{} = \sum_{j=0}^\infty a_{jN}^{}$ and
\begin{equation}
\akt = \ak + \sum_{j=1}^\infty (a_{k+jN}^{} + a_{-k+jN}^{}),
\quad \bkt = \bk + \sum_{j=1}^\infty (b_{k+jN}^{} - b_{-k+jN}^{})
\label{aliasing2}
\end{equation}
for $1\le k \le n$.
We can derive these formulas by considering the
phenomenon of aliasing. For all $j$, the functions
$\exp(i[k+jN]t)$ take the same
values at the trigonometric points (<ref>). This implies that
$f$ and the trigonometric polynomial (<ref>) with
coefficients defined by (<ref>) take the same
values at these points. In other words, (<ref>) is the
degree $n$ trigonometric interpolant to $f$.
A similar argument justifies (<ref>)–(<ref>).
Another interpretation of the coefficients $\ckt, \akt, \bkt$ is
that they are equal to the approximations to $\ck, \ak, \bk$
one gets if the integrals (<ref>) and (<ref>) are
approximated by the periodic trapezoidal quadrature rule with $N$
points <cit.>:
\begin{equation}
\ckt = {1\over N} \sum_{j=0}^{N-1} \fj e^{-ik\tj} ,
\label{coeffs3}
\end{equation}
\begin{equation}
\akt = {2\over N} \sum_{j=0}^{N-1} \fj \cos(k\tj), \quad
\bkt = {2\over N} \sum_{j=0}^{N-1} \fj \sin(k\tj) \qquad\rlap{$(k\ge 1).$}
\label{coeffs4}
\end{equation}
To prove this, we note that the trapezoidal rule computes
the same Fourier coefficients for $f$ as for $\pn$, since
they take the same values at the grid points;
but these must be equal to the true Fourier coefficients of $\pn$,
since the $N=(2n+1)$-point trapezoidal rule is exactly correct
for $e^{-2int}, \dots, e^{2int}$, hence for any
trigonometric polynomial of degree $2n$, hence in particular for any
trigonometric polynomial of degree $n$ times an exponential $\exp(-ikt)$
with $|k|\le n$.
From (<ref>)–(<ref>) it is evident that
the discrete Fourier coefficients
$\ckt$, $\akt$, $\bkt$ can be computed by
the Fast Fourier Transform (FFT), which, in fact, Gauss invented
for this purpose.
Suppose one wishes to evaluate the interpolant $\pn(t)$ at certain points
$t$. One good algorithm is to compute
the discrete Fourier coefficients and then apply them.
Alternatively, another good approach is to
perform interpolation directly by means of the barycentric
formula for trigonometric interpolation, introduced by
Salzer <cit.> and later simplified by Henrici <cit.>:
\begin{equation}
\pn(t) = \sum_{k=0}^{N-1} (-1)^k f_k \csc({t-\tk\over 2})
\left/\, \sum_{k=0}^{N-1} (-1)^k \csc({t-\tk\over 2}) \right.
\rlap{\quad ($N$ odd).}
\label{bary1}
\end{equation}
(If $t$ happens to be exactly equal to a grid point $\tk$,
one takes $\pn(t) = \fk$.)
The work involved in this formula
is just $O(N)$ operations per evaluation, and stability has
been established (after a small modification) in <cit.>.
In practice, we find the Fourier coefficients and barycentric
formula methods equally effective.
In the above discussion, we have assumed that the number
of interpolation points, $N$, is odd. However,
trigonometric interpolation, unlike trigonometric projection,
makes sense for an even number of degrees of freedom too
(see e.g. <cit.>);
it would be surprising if
FFT codes refused to accept input vectors of even lengths!
Suppose $n\ge 1$ is given and we wish to
interpolate $f$ in $N=2n$ trigonometric points (<ref>)
rather than $N=2n+1$. This is one data value less than usual for
a trigonometric polynomial of this degree,
and we can lower the number of degrees of
freedom in (<ref>) by imposing the condition
\begin{equation}
\tilde c_{-n}^{} = \tilde c_n^{}
\label{cond1}
\end{equation}
or equivalently in (<ref>) by imposing the condition
\begin{equation}
\tilde b_n^{} = 0.
\label{cond2}
\end{equation}
This amounts to prescribing that the trigonometric interpolant
through sawtoothed data of the form $\fk = (-1)^k$
should be $\cos(nt)$ rather than some other function such as
$\exp(int)$—the only choice that ensures that
real data will lead to a real interpolant.
An equivalent prescription is that an arbitrary number $N$
of data values, even or odd, will be interpolated by a linear
combination of the first $N$ terms of the sequence
\begin{equation}
1,\, \cos(t),\, \sin(t),\, \cos(2t),\, \sin(2t),\, \cos(3t),\, \dots.
\label{specialset}
\end{equation}
In this case of trigonometric interpolation with $N$ even, the formulas
(<ref>)–(<ref>) still hold, except that
(<ref>) and (<ref>) must be multiplied by $1/2$
for $k = \pm n$. FFT codes, however, do not
store the information that way. Instead, following (<ref>), they
compute $\tilde a_{-n}^{}$ by (<ref>) with $2/N$ instead
of $1/N$ out front—thus effectively storing
$\tilde c_{-n}^{} +\tilde c_n^{}$ in the place of $\tilde c_{-n}^{}$—and
then apply (<ref>) with the $k=n$ term omitted.
This gives the right result for values of $t$ on
the grid, but not at points in-between.
Note that the conditions (<ref>)–(<ref>) are
very much tied to
the use of the sample points (<ref>). If the grid
were translated uniformly, then different relationships
between $c_n^{}$ and $c_{-n}^{}$ or $a_n^{}/b_n^{}$ and
$a_{-n}^{}/b_{-n}^{}$ would be appropriate in
(<ref>)–(<ref>) and
different basis functions in (<ref>), and if the grid
were not uniform, then it would be hard to justify any particular
choices at all for even $N$.
For these reasons, even numbers of degrees of freedom make sense
in equispaced interpolation but not in other trigonometric
approximation contexts, in general.
Henrici <cit.> provides a modification
of the barycentric formula (<ref>) for the equispaced case $N=2n$.
§ APPROXIMATION THEORY AND QUADRATURE
The basic question of approximation theory is, will
approximants to a function $f$ converge as the degree
is increased, and how fast?
The formulas of the last two
sections enable us to derive theorems addressing this question
for trigonometric projection and interpolation.
(For finer points of trigonometric approximation theory,
see <cit.>.)
The smoother $f$ is, the faster its Fourier coefficients
decrease, and the faster the convergence of the approximants.
(If $f$ were merely continuous rather than
Lipschitz continuous, then the trigonometric version of the Weierstrass
approximation theorem <cit.>
would ensure that it could be approximated
arbitrarily closely by trigonometric polynomials,
but not necessarily by projection or interpolation.)
Our first theorem asserts that Fourier coefficients
decay algebraically if $f$ has a finite number of derivatives,
and geometrically
if $f$ is analytic. Here and in Theorem <ref> below,
we make use of the notion of the total variation, $V$, of
a periodic function $\varphi$ defined on $\pipi$, which is defined
in the usual way as the supremum of all sums $\sum_{i=1}^n
|\varphi(x_i)-\varphi(x_{i-1})|$, where $\{x_i\}$
are ordered points in $\pipi$ with $x_0 = x_n$; $V$ is equal to the
the $1$-norm of $f'$, interpreted if necessary
as a Riemann–Stieltjes integral <cit.>.
Thus $|\sin(t)|$ on $\pipi$, for example, corresponds to $\nu =1$,
and $|\sin(t)|^3$ to $\nu =3$.
All our theorems continue to assume that $f$ is $2\pi$-periodic.
If $f$ is $\nu\ge 0$ times differentiable and
$f^{(\nu)}$ is of bounded variation $V$ on $\pipi$, then
\begin{equation}
|\ck| \le {V\over 2\pi |k|^{\nu+1}}.
\label{est1}
\end{equation}
If $f$ is analytic with $|f(t)|\le M$ in the open strip of
half-width $\alpha$ around the real axis in the complex $t$-plane, then
\begin{equation}
|\ck| \le M e^{-\alpha|k|} .
\label{est2}
\end{equation}
The bound (<ref>) can be derived by integrating (<ref>) by
parts $\nu+1$ times.
Equation (<ref>) can be derived by shifting the interval
of integration $\pipi$ of (<ref>) downward in
the complex plane for $k>0$, or upward for $k<0$, by a
distance arbitrarily close to $\alpha$; see <cit.>.
To apply Theorem <ref> to trigonometric approximations, we
note that the error in the degree $n$ trigonometric
projection (<ref>) is
\begin{equation}
f(t) - \fn(t) = \sum_{|k|>n} \ck e^{ikt} ,
\label{error1}
\end{equation}
a series that converges absolutely and uniformly by the Lipschitz
continuity assumption on $f$. Similarly, (<ref>) implies
that the error in trigonometric interpolation is
\begin{equation}
f(t) - \pn(t) = \sum_{|k|>n} \ck (e^{ikt} - e^{ik'\kern -1pt t}),
\label{error2}
\end{equation}
where $k' = \hbox{mod}(k+n,2n+1)-n$ is the index that $k$ gets
aliased to on the $(2n+1)$-point grid, i.e., the integer of absolute value
$\le n$ congruent to $k$ modulo $2n+1$.
These formulas give us bounds on the error in trigonometric
projection and interpolation.
If $f$ is $\nu\ge 1$ times differentiable and
$f^{(\nu)}$ is of bounded variation $V$ on $\pipi$, then
its degree $n$ trigonometric projection and interpolant satisfy
\begin{equation}
\|f - \fn\|_\infty^{} \le {V\over \pi\kern .7pt \nu\kern .7pt n^\nu}, \qquad
\|f - \pn\|_\infty^{} \le {2V\over \pi\kern .7pt \nu \kern .7pt n^\nu}.
\label{est3}
\end{equation}
If $f$ is analytic with $|f(t)|\le M$ in the open strip of
half-width $\alpha$ around the real axis in the complex $t$-plane, they
\begin{equation}
\|f-\fn\|_\infty^{} \le {2M e^{-\alpha n}\over e^\alpha-1}, \qquad
\|f-\pn\|_\infty^{} \le {4M e^{-\alpha n}\over e^\alpha-1} .
\label{est4}
\end{equation}
The estimates (<ref>) follow by bounding the
tails (<ref>) and (<ref>) with (<ref>), and
(<ref>) likewise by bounding them with (<ref>).
A slight variant of this argument gives an estimate for quadrature.
If $I$ denotes the integral of a function $f$ over $\pipi$
and $\IN$ its approximation by the $N$-point
periodic trapezoidal rule, then from (<ref>) and (<ref>),
we have $I = 2\pi c_0^{}$ and $\IN = 2\pi \tilde c_0^{}$.
By (<ref>) this implies
\begin{equation}
\IN - I = 2\pi \sum_{j\ne 0} c_{jN}^{},
\label{trapest}
\end{equation}
which gives the following result.
If $f$ is $\nu\ge 1$ times differentiable and
$f^{(\nu)}$ is of bounded variation $V$ on $\pipi$, then
the $N$-point periodic trapezoidal rule
approximation to its integral over $\pipi$ satisfies
\begin{equation}
|\IN - I| \le {4 V\over N^{\nu+1}}.
\label{trap1}
\end{equation}
If $f$ is analytic with $|f(t)|\le M$ in the open strip of
half-width $\alpha$ around the real axis in the complex $t$-plane,
it satisfies
\begin{equation}
|\IN-I| \le {4\pi M \over e^{\alpha N}-1}.
\label{trap2}
\end{equation}
These results follow by bounding (<ref>) with
(<ref>) and (<ref>) as in the proof of Theorem <ref>.
From (<ref>), the bound one gets
is $2V\zeta(\nu+1)/N^{\nu+1}$, where $\zeta$ is the
Riemann zeta function, which we have simplified by
the inequality $\zeta(\nu+1)\le \zeta(2) < 2$ for $\nu\ge 1$.
The estimate (<ref>) originates with Davis <cit.>;
see also <cit.>.
Finally, in a section labeled “Approximation theory” we must mention another
famous candidate for periodic function
approximation: best approximation in the $\infty$-norm.
Here the trigonometric version of the Chebyshev alternation theorem holds,
assuming $f$ is real.
This result is illustrated below in Figure <ref>.
Let $f$ be real and continuous on
the periodic interval $\pipi$. For each $n\ge 0$,
$f$ has a unique best approximant $\pns\in \Pn$ with
respect to the norm $\|\cdot\|_\infty^{}$, and $\pns$ is characterized
by the property that the error curve $(f-\pns)(t)$ equioscillates on
$[\kern .5pt 0,2\pi)$ between at least $2n+2$ equal extrema $\pm\|f-\pns\|_\infty^{}$ of
alternating signs.
See <cit.>.
§ TRIGFUN COMPUTATIONS
Building on the mathematics of the past three sections,
Chebfun was extended in 2014 to incorporate trigonometric
representations of periodic functions alongside its traditional
Chebyshev representations.
(Here and in
the remainder of the paper, we assume the reader is familiar
with Chebfun.)
Our convention is that a trigfun is a
representation via coefficients $\ck$ as in (<ref>)
of a sufficiently smooth
periodic function $f$ on an interval by a trigonometric
polynomial of adaptively determined degree, the aim always being
accuracy of 15 or 16 digits relative
to the $\infty$-norm of the function on the interval.
This follows the same pattern as traditional Chebyshev-based
chebfuns, which are representations of nonperiodic functions
by polynomials, and a trigfun is not a distinct object from a
chebfun but a particular type of chebfun. The default interval,
as with ordinary chebfuns, is $\ones$, and other intervals are
handled by the obvious linear
one aspect of the transplantation is not obvious, an indirect
consequence of the translation-invariance
of trigonometric functions.
The nonperiodic function $f(x) = x$ defined on $[-1,1]$, for
example, has Chebyshev coefficients $a_0^{}=0$ and $a_1^{} = 1$,
corresponding to the expansion $f(x) = 0T_0^{}(x) + 1T_1^{}(x)$.
Any user will expect the transplanted function $g(x) = x-1$
defined on $[0,2]$ to have the same coefficients $a_0^{}=0$
and $a_1^{} = 1$, corresponding to the transplanted expansion
$g(x) = 0T_0^{}(x-1) + 1T_1^{}(x-1)$, and this is what Chebfun
delivers. By contrast, consider the periodic function $f(t)
= \cos t$ defined on $[-\pi,\pi]$ and its transplant $g(t) =
\cos(t-\pi) = -\cos t$ on $\pipi$. A user will expect the
expansion coefficients of $g$ to be not the same as those
of $f$, but their negatives! This is because we expect to
use the same basis functions $\exp(ikx)$ or $\cos(kx)$ and
$\sin(kx)$ on any interval of length $2\pi$, however translated.
The trigonometric part of Chebfun is designed accordingly.]
For example, here we construct and plot a trigfun for $\cos(t) +
\sin(3t)/2$ on $\pipi$:
The trigfun representing
$f(t) = \cos(t)+ \sin(3t)/2$ on $\pipi$. One can evaluate
$f$ with f(t), compute its definite integral with sum(f)
or its maximum with max(f),
find its roots with roots(f), and so on.
The plot appears in Figure <ref>, and
the following text output is produced, with the flag trig signalling
the periodic representation.
We see that Chebfun has determined that this function
$f$ is of length $N=7$. This means
that there are $7$ degrees of freedom, i.e., $f$ is a trigonometric polynomial
of degree $n=3$, whose coefficients we can extract with
c = trigcoeffs(f), or in cosine/sine form
with [a,b] = trigcoeffs(f).
Note that the Chebfun constructor does not analyze its input
symbolically, but just evaluates the function at trigonometric points
(<ref>), and from this information the degree
and the values of the coefficients are determined.
The constructor also detects when a function is real.
A trigfun constructed in the ordinary manner
is always of odd length $N$, corresponding to a trigonometric
polynomial of degree $n = (N-1)/2$, though it is
possible to make even-length trigfuns by explicitly specifying $N$.
To construct a trigfun, Chebfun samples the function on grids of
size $16, 32, 64,\dots$ and tests the resulting discrete Fourier
coefficients for convergence down to relative machine
precision. (Powers of 2 are used
since these are particularly efficient for the FFT, even though
the result will ultimately be trimmed to an odd number
of points. As with non-trigonometric Chebfun,
the engineering details are complicated and under ongoing
development.) When convergence is achieved, the
series is chopped at an appropriate point and the degree reduced
Once a trigfun has been created, computations can be carried out
in the usual Chebfun fashion via overloads of familiar
MATLAB commands. For example,
This number is computed by integrating the trigonometric
representation of $f^2$, i.e., by returning the number $2\pi c_0^{}$
corresponding to the trapezoidal rule applied to $f^2$ as
described around Theorem <ref>.
The default 2-norm is the square root of this result,
Derivatives of functions are computed
by the overloaded command diff.
(In the unusual case where a trigfun has been constructed of
even length, differentiation will increase its length by $1$.)
The zeros of $f$ are found with roots:
and Chebfun determines maxima and minima by first computing
the derivative, then checking all of its roots:
Concerning the algorithm used for periodic rootfinding,
one approach would be to solve a companion
matrix eigenvalue problem, and $O(n^2)$ algorithms for
this task have recently been developed <cit.>.
When development of these methods settles down, they may be incorporated
in Chebfun. For the moment, trigfun rootfinding
is done by first converting the problem to nonperiodic Chebfun form using
the standard Chebfun constructor,
whereupon we take advantage of Chebfun's $O(n^2)$ recursive interval
subdivision strategy <cit.>.
This shifting to subintervals for rootfinding is an
example of an operation that breaks periodicity as mentioned in
item 7 of the introduction.
The main purpose of the periodic part of Chebfun is to
enable machine precision computation with periodic
functions that are not exactly trigonometric polynomials.
For example,
$\exp(\sin t)$ on $\pipi$ is represented by a trigfun
of length $27$, i.e., a
trigonometric polynomial of degree 13:
The coefficients can be plotted on a log scale with the
command plotcoeffs(f),
and the in
Figure <ref> reveals the faster-than-geometric decay
of an entire function.
Absolute values of the Fourier coefficients of
the trigfun for $\exp(\sin t)$ on $\pipi$.
This is an entire function (analytic throughout
the complex $t$-plane), and in accordance with Theorem $\ref{thm1}$, the
coefficients decrease faster than geometrically.
Figure <ref> shows trigfuns and coefficient plots
for $f(t)=\tanh(5\cos(5t))$ and
$g(t)=\exp(-1/\max\{0, 1-t^2/4\})$ on
$[-\pi, \pi]$. The latter is $C^\infty$ but not analytic.
Figure <ref> shows a further pair of examples that
we call an “AM signal” and an “FM signal”. These are
among the preloaded functions available with
cheb.gallerytrig, Chebfun's trigonometric analogue
of the MATLAB gallery command.
Trigfuns of
$\tanh(5\sin t)$ and $\exp(-100(t+.3)^2)$
(upper row) and corresponding absolute values
of Fourier coefficients (lower row).
Trigfuns of the “AM signal”
$\cos(50t)(1+\cos(5t)/5)$ and the “FM signal”
$\cos(50t+4\sin(5t))$ (upper row) and corresponding absolute values
of Fourier coefficients (lower row).
Computation with trigfuns, as with nonperiodic chebfuns,
is carried out by
a continuous analogue of floating point arithmetic <cit.>.
To illustrate the “rounding” process involved,
the degrees of the trigfuns above are
555 and 509, respectively. Mathematically,
their product is of degree 1064. Numerically, however,
Chebfun achieves 16-digit accuracy with degree 556.
Here is a more complicated example of Chebfun rounding adapted
from <cit.>, where it is computed with nonperiodic
This program takes 15 steps of an iteration that in principle
quadruples the degree at each step, giving a function
$s$ at the end of degree
$4^{15} = \hbox{1,073,741,824}$. In actuality,
however, because of the rounding to 16 digits,
the degree comes out one million times smaller as 1148. This function
is plotted in Figure <ref>. Following <cit.>,
we can compute the roots of $s-8$
in half a second on a desktop machine:
The integral with sum(s) is
$15.265483825826763$, correct except in the last two digits.
After fifteen steps of an iteration,
this periodic function has degree $1148$ in its Chebfun representation
rather than the mathematically exact figure 1,073,741,824.
If one tries to construct a trigfun by sampling
a function that is not smoothly periodic,
Chebfun will by default go up to length $2^{16}$ and then
issue a warning:
On the other hand, computations that are known
to break periodicity or smoothness will result in the representation
being automatically cast from a trigfun to a chebfun.
For example, here we define $g$ to be the
absolute value of the function
$f(t) = \cos(t) + \sin(3t)/2$ of Figure <ref>.
The system detects that $f$ has zeros,
implying that $g$ will probably not be smooth, and
accordingly constructs it not as a trigfun but as an ordinary chebfun with
several pieces:
Similarly, if you add or multiply a trigfun and a chebfun,
the result is a chebfun.
When the absolute value of
the trigfun $f$ of Figure $\ref{fig1a}$ is computed,
the result is a nonperiodic chebfun with three smooth pieces.
§ APPLICATIONS
Analysis of periodic functions and signals is one of the oldest
topics of mathematics and engineering. Here we give
six examples of how a system for automating such computations may be useful.
Complex contour integrals.
Smooth periodic integrals arise ubiquitously in complex analysis.
For example, suppose we wish to determine the number of zeros of
$f(z) = \cos(z) - z$ in the
complex unit disk. The answer is given by
\begin{equation}
m = {1\over 2\pi i} \int {f'(z)\over f(z) } dz
= {1\over 2\pi i} \int {1\over f(z) } {df\over dt} dt
\label{contourint}
\end{equation}
if $z = \exp(it)$ with $t\in \pipi$.
With periodic Chebfun, we can compute $m$ by
Changing the integrand from $f'(z)/f(z)$ to $z f'(z)/f(z)$
gives the location of the zero, correct to all digits displayed.
(The real commands are included to remove imaginary
rounding errors.)
For wide-ranging extensions of calculations like these, including
applications to matrix eigenvalue problems, see <cit.>.
Linear algebra.
Chebfun does not work from explicit formulas: to construct a
function, it is only necessary to be able to evaluate it.
This is an extremely useful feature for linear algebra calculations.
For example, the matrix
\begin{equation}
\def\r#1{\phantom{xx}\llap{$#1$}}
\def\rr#1{\phantom{xx,}\llap{$#1$}}
A = {1\over 3}
\pmatrix{
\r{2} & \rr{-2i} & \r{1} & \r{1} \cr
\r{2i} & \rr{-2} & \r{0} & \r{2} \cr
\r{-2} & \rr{0} & \r{1} & \r{2} \cr
\r{0} & \rr{i} & \r{0} & \r{2}
\end{equation}
has all its eigenvalues in the unit disk. A question with
the flavor of control and stability theory is, what is the
maximum resolvent norm $\|(zI-A)^{-1}\|$ for $z$ on the unit
circle? We can calculate the answer with the code below, which
constructs a periodic chebfun of degree $n=569$. The maximum is
$27.68851$, attained with $z = \exp(0.454596\kern.5pt i)$.
Resolvent norm $\|(zI-A)^{-1}\|$
for a $4\times 4$ matrix $A$ with $z= e^{it}$ on
the unit circle.
Circular convolution and smoothing.
The circular or periodic convolution of two functions
$f$ and $g$ with period $T$ is defined by
\begin{equation}
(f*g)(t) := \int_{t_0}^{t_0 + T} g(s)f(t-s)ds,
\end{equation}
where $t_0$ is aribtrary. Circular convolutions can be
computed for trigfuns with the circconv function, whose algorithm
consists of coefficientwise multiplication in Fourier space.
For example, here is a trigonometric interpolant through
$201$ samples of a smooth function plus noise, shown
in the upper-left panel of Figure <ref>.
The high wave numbers can be smoothed by convolving
$f$ with a mollifier. Here we use
a Gaussian of standard deviation $\sigma=0.1$ (numerically
periodic for $\sigma\le 0.35$). The result is
shown in the upper-right panel of the figure.
Circular convolution of a noisy function with
a smooth mollifier.
Fourier coefficients of non-smooth functions.
A function $f$ that is not smoothly periodic will at best
have a very slowly converging trigonometric series, but still,
one may be interested in its Fourier coefficients. These can
be computed by applying trigcoeffs to a chebfun representation
of $f$ and specifying how many coefficients are required; the
integrals (<ref>) are then evaluated numerically by
Chebfun's standard method of Clenshaw–Curtis quadrature.
For example,
Figure <ref> shows a portrayal of the Gibbs phenomenon
from Runge's 1904 book together with its Chebfun equivalent computed
in a few seconds with the commands
On the left, a figure from
Runge's $1904$ book
Theorie und Praxis der Reihen <cit.>.
On the right, the
equivalent computed with periodic Chebfun. Among other things, this
figure illustrates that a trigfun can be accurately evaluated outside
its interval of definition.
Interpolation in unequally spaced points.
Very little attention has been given to trigonometric
interpolation in unequally spaced points, but the
barycentric formula (<ref>) for odd $N$ and Henrici's
generalization for even $N$ have been generalized
to this case by Salzer and Berrut <cit.>. Chebfun
makes these formulas available through the
command chebfun.interp1, just as has long been
true for interpolation by algebraic polynomials. For
example, the code
interpolates the function $|t|$ on $[-\pi,\pi]$
in the 9 points indicated by a trigonometric
polynomial of degree $n=4$.
The interpolant is shown
in Figure <ref> together with the analogous
curve for equispaced points.
Trigonometric interpolation of $|t|$ in unequally spaced
points with
the generalized barycentric formula implemented in chebfun/interp1.
Best approximation, CF approximation, and rational functions.
Chebfun has long had a dual role: it is a tool
for computing with functions, and also a tool for exploring
principles of approximation theory, including
advanced ones. The trigonometric side of
Chebfun extends this second aspect to periodic problems. For example,
Chebfun's new
trigremez command can compute best approximants
with equioscillating error curves as described in
Theorem <ref> <cit.>.
Here is an example that generates the error curve displayed
in Figure <ref>, with error $12.1095909$.
Chebfun is also acquiring other capabilities for trigonometric
polynomial and rational approximation, including
Carathéodory–Fejér (CF) near-best approximation via singular
values of Hankel matrices, and these will be described elsewhere.
Error curve in degree $n=10$ best
trigonometric approximation to
$f(t) = 1/(1.01-\sin(t-2))$ over $\pipi$. The curve equioscillates between
$2n+2 = 22$ alternating extrema.
§ PERIODIC ODES, OPERATOR EXPONENTIALS, AND EIGENVALUE PROBLEMS
A major capability of Chebfun is
the solution of linear and nonlinear ordinary differential
equations (ODEs), as well as integral equations,
by applying the backslash command to a “chebop” object.
We have extended these capabilities to periodic problems, both
scalars and systems.
See <cit.>
for the theory of existence and uniqueness
of solutions to periodic ODEs, which goes back to Floquet in the 1880s,
a key point being the avoidance of nongeneric
configurations corresponding to eigenmodes.
Chebfun's algorithm for linear ODEs amounts to
an automatic spectral collocation method wrapped up so that
the user need not be aware of the discretization. With
standard Chebfun, these are Chebyshev spectral methods, and now
with the periodic extension, they are Fourier
spectral methods <cit.>.
The problem is solved on grids of size 32, 64, and so on until
the system judges that the Fourier coefficients have converged
down to the level of noise, and the series is then truncated
at an appropriate point.
For example, consider the problem
\begin{equation}
0.001(u'' + u') - \cos(t) u = 1, \qquad 0\le t \le 6\pi
\label{odelin}
\end{equation}
with periodic boundary conditions.
The following Chebfun code produces the solution
plotted in Figure <ref> in half a second
on a laptop. Note that the trigonometric discretizations are
invoked by the flag .
Solution of the linear periodic ODE
$(\ref{odelin})$ as a trigfun of degree $168$, computed by
an automatic Fourier spectral method.
This trigfun is of degree 168,
and the residual reported by norm(L*u-1) is
$1\times 10^{-12}$.
As always, $u$ is a chebfun; its maximum, for example, is
$\hbox{\tt max(u)} =66.928$.
For periodic nonlinear ODEs, Chebfun applies trigonometric
analogues of the algorithms developed by Driscoll and Birkisson
in the Chebshev case <cit.>. The whole solution is carried out
by a Newton or damped Newton iteration formulated
in a continuous mode (“solve then discretize” rather than
“discretize then solve”), with Jacobian matrices replaced
by Fréchet derivative operators implemented by means of automatic
differentiation and automatic spectral discretization.
For example, suppose we seek a solution of the nonlinear problem
\begin{equation}
0.004u'' + uu' - u = \cos(2\pi t), \qquad t\in [-1,1]
\label{nonlinprob}
\end{equation}
with periodic boundary conditions. After seven Newton
steps, the Chebfun commands below
produce the result shown in Figure <ref>, of
degree $n = 362$, and
the residual norm norm(N(u)-rhs,'inf') is reported as
$8\times 10^{-9}$.
Solution of the nonlinear periodic ODE
$(\ref{nonlinprob})$ computed by iterating the Fourier
spectral method within a continuous form of
Newton iteration. Executing
max(diff(u)) shows that the maximum of $u'$ is
Chebfun's overload of the MATLAB eigs command solves
linear ODE eigenvalue problems by, once again, automated spectral
collocation discretizations <cit.>. This too has
been extended to periodic problems, with Fourier discretizations
replacing Chebyshev. For example, a famous periodic eigenvalue
problem is the Mathieu equation
\begin{equation}
-u'' + 2 q \cos(2t) u = \lambda u, \qquad t\in \pipi,
\label{mathieueq}
\end{equation}
where $q$ is a parameter. The commands below give
the plot shown in Figure <ref>.
First five eigenfunctions of the Mathieu
equation $(\ref{mathieueq})$ with $q=2$, computed with eigs.
So far as we know, Chebfun is the only system offering
such convenient solution of ODEs and related
problems, now in the periodic as well as nonperiodic case.
We have also implemented a periodic analogue of Chebfun's
expm command for computing
exponentials of linear operators, which
we omit discussing here for reasons of space. All the capabilities
mentioned in this section can be explored with Chebgui, the graphical
user interface written by Birkisson, which now invokes
trigonometric spectral discretizations when periodic boundary
conditions are specified.
§ DISCUSSION
Chebfun is an open-source project written in MATLAB and
hosted on GitHub; details and the user's guide can be found at
www.chebfun.org <cit.>.
About thirty people have contributed to
its development over the years, and at present there are about
ten developers based mainly at the University of Oxford.
During 2013–2014 the code was redesigned and rewritten as
version 5 (first released June 2014) in
the form of about 100,000 lines of code realizing
about 40 classes. The aim of this redesign was to enhance Chebfun's
modularity, clarity, and extensibility, and the introduction
of periodic capabilities, which had not been planned in
advance, was the first big test of this extensibility.
We were pleased to
find that the modifications proceeded smoothly.
The central new feature is a new
class @trigtech in parallel to the existing
@chebtech1 and @chebtech2, which work with
polynomial interpolants in first- and second-kind Chebyshev points,
About half the classes of Chebfun are concerned with representing
functions, and the remainder are mostly
concerned with ODE discretization and automatic
differentiation for solution of nonlinear problems, whether
scalar or systems, possibly with nontrivial block structure.
The incorporation of periodic problems into this second, more
advanced part of Chebfun was achieved by introducing
a new class @trigcolloc matching
@chebcolloc1 and @chebcolloc2.
About a dozen software projects in various computer languages have been
modeled on Chebfun, and a partial
list can be found at www.chebfun.org. One of
these, Fourfun, is a MATLAB system for periodic functions
developed independently of the
present work by Kristyn McLeod, a student of former Chebfun
developer Rodrigo Platte <cit.>. Another that also has
periodic and differential equations
capabilities is ApproxFun, written in Julia by Sheehan
Olver and former Chebfun developer
Alex Townsend <cit.>.[Platte created
Chebfun's edge detection algorithm for fast splitting
of intervals. Townsend extended Chebfun
to two dimensions.] We think the enterprise of
numerical computing with functions is here to stay,
but cannot predict what systems or languages may be dominant,
say, twenty years from now. For the moment, only Chebfun offers
the breadth of capabilities entailed in the vision of
MATLAB-like functionality for continuous functions and operators in
analogy to the long-familiar methods for discrete
vectors and matrices.
In this article we
have not discussed Chebfun computations with two-dimensional
periodic functions,
which are under development. For example, we are
investigating capabilities for solution of time-dependent PDEs on a
periodic spatial domain and for PDEs in two space dimensions, one or
both of which are periodic. A particularly interesting prospect is
to apply such representations to computation with functions
on disks and spheres.
For computing with vectors and matrices, although MATLAB codes are
rarely the fastest in execution, their convenience makes them
nevertheless the best tool for many applications.
We believe that Chebfun, including now its extension to periodic problems,
plays the same role for numerical computing with functions.
§ ACKNOWLEDGEMENTS
This work was carried out in collaboration with the rest of
the Chebfun team, whose names are listed at
www.chebfun.org. Particularly active in this phase of the project have
been Anthony Austin, Ásgeir Birkisson,
Toby Driscoll, Nick Hale, Hrothgar (an Oxford graduate
student who has just a single name),
Alex Townsend, and Kuan Xu.
We are grateful to all of these people for their suggestions
in preparing this paper.
The first author would like to thank the Oxford University Mathematical
Institute, and in particular the Numerical Analysis Group, for hosting
and supporting his sabbatical visit in
2014, during which this research was initiated.
J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins,
Fast and backward stable computation of roots of polynomials,
SIAM J. Matr. Anal. Applics., to appear.
A. P. Austin, P. Kravanja and L. N. Trefethen,
Numerical algorithms based on analytic function values
at roots of unity, SIAM J. Numer. Anal. 52 (2014),
A. P. Austin and K. Xu,
On the numerical stability of the second-kind barycentric
formula for trigonometric inteprolation in equispaced points,
submitted, 2015.
Z. Battles and L. N. Trefethen,
An extension of MATLAB to continuous functions and operators,
SIAM J. Sci. Comp. 25 (2004), 1743–1770.
J.-P. Berrut, Baryzentrische Formeln zur trigonometrischen
Interpolation (I), J. Appl. Math. Phys. 35
(1984), 91–105.
J.-P. Berrut and L. N. Trefethen,
Barycentric Lagrange interpolation,
SIAM Rev. 46 (2004), 501–517.
A. Birkisson and T. A. Driscoll,
Automatic Fréchet differentiation for
the numerical solution of boundary-value problems,
ACM Trans. Math. Softw. 38 (2012), 1–26.
A. Birkisson and T. A. Driscoll,
Automatic linearity dection, preprint, eprints.maths.ox.ac.uk, 2013.
J. P. Boyd, Chebyshev and Fourier Spectral Methods,
2nd ed., Dover, 2001.
P. J. Davis, On the numerical integration of periodic
analytic functions, in R. E. Langer, ed., On Numerical Integration,
Math. Res. Ctr., U. of Wisconsin, 1959, pp. 45–59.
T. A. Driscoll, Automatic spectral collocation for
integral, integro-differential, and integrally reformulated differential
equations, J. Comput. Phys. 229 (2010), 5980–5998.
T. A. Driscoll, N. Hale, and L. N. Trefethen,
Chebfun Guide, Pafnuty Publications, Oxford, 2014.
Most recent version freely available at www.chebfun.org.
M. S. P. Eastham, The Spectral Theory of Periodic
Differential Equations, Scottish Academic Press, Edinburgh, 1973.
G. Faber, Über stetige Funktionen,
Math. Ann. 69 (1910), 372–443.
C. F. Gauss, Theoria interpolationis methodo nova tractata,
Werke, v. 3, Königl. Ges. Gött., 1866, pp. 265–327.
P. Henrici, Barycentric formulas for interpolating
trigonometric polynomials and their conjugates,
Numer. Math., 33 (1979), 225–234.
M. Javed and L. N. Trefethen,
The Remez algorithm for trigonometric approximation
of periodic functions, submitted, 2015.
Y. Katznelson, An Introduction to Harmonic Analysis,
Dover, 1968.
R. Kress, Ein ableitungsfreies Restglied
für die trigonometrische Interpolation
periodischer analytischer Funktionen,
Numer. Math. 16 (1971), 389–396.
K. McLeod, Fourfun: A new system for automatic computations
Fourier expansions, manuscript, 2014.
G. Meinardus, Approximation of Functions:
Theory and Numerical Methods, Springer, 1967.
S. Olver and A. Townsend, A practical framework
for infinite-dimensional linear algebra,
arXiv:1409.5529, 2014.
R. B. Platte, L. N. Trefethen, and A. B. J. Kuijlaars,
Impossibility of fast stable approximation of
analytic functions from equispaced samples,
SIAM Rev. 53 (2011), 308–318.
C. Runge, Über empirische Funktionen und die
Interpolation zwischen äquisitanten Ordinaten, Z. Math. Phys.
46 (1901), 224–243.
C. D. T. Runge, Theorie und Praxis der Reihen,
Sammlung Schebert, 1904, reprinted by VKM Verlag, Saarbrücken,
H. E. Salzer, Coefficients for facilitating trigonometric
interpolation, J. Math. Phys. 27 (1948), 274–278.
L. N. Trefethen, Numerical computation with functions instead
of numbers, Commun. ACM, to appear.
L. N. Trefethen, Approximation Theory
and Approximation Practice, SIAM, 2013.
L. N. Trefethen and J. A. C. Weideman, The exponentially
convergent trapezoidal rule, SIAM Rev. 56 (2014),
A. Zygmund, Trigonometric Series, Cambridge U. Press, 1959.
|
1511.00228
|
Department of Mathematics, Institute for Advanced Studies
in Basic Science (IASBS), P.O.Box 45195-1159, Zanjan, Iran
Department of Mathematics, Institute for Advanced Studies
in Basic Science (IASBS), P.O.Box 45195-1159, Zanjan, Iran
-0.4 true cm MSC(2010): Primary: 5E40; Secondary: 5C69, 5C75.
Keywords: $r$-partite graph, well-covered, unmixed,
perfect matching, clique.
Unmixed bipartite graphs have been characterized by Ravadra and
Villarreal independently. Our aim in this paper is to
characterize unmixed $r$-partite graphs under a certain condition,
witch is a generalization of villarreal's theorem on bipartite
graphs. Also we give some examples and counterexamples in relevance this subject.
0.2 true cm
§ INTRODUCTION
0.4 true cm
In the sequel, we use <cit.> as reference for terminology and
notation on graph theory.
Let $G$ be a simple finite graph with vertex set $V(G)$ and edge set $E(G)$.
A subset $C$ of $V(G)$ is said to be a vertex cover of G if every
edge of $G$, is adjacent with some vertices in $C$. A vertex cover
$C$ is called minimal, if there is no proper subset of $C$ which is a vertex cover. A graph is called unmixed, if all minimal
vertex covers of $G$ have the same number of elements. A subset $H$
of $V(G)$ is said to be independent, if $G$ has not
any edge $\{x, y\}$ such that $\{x, y\}\subseteq H$. A maximal
independent set of $G$, is an independent set $I$ of $G$, such
that for every $H\supsetneqq I$, $H$ is not an independent set of
$G$. Notice that $C$ is a minimal vertex cover if and only if
$V(G)\setminus C$ is a maximal independent set. A graph $G$ is called
well-covered if all the maximal independent sets of $G$ have the
same cardinality. Therefore a graph is unmixed if and only if it
is well-covered. The minimum cardinality of all minimal vertex
covers of $G$ is called the covering number of $G$, and the
maximum cardinality of all maximal independent sets of G is
called the independence number of $G$. For determining the
independence number see <cit.>. For relation between unmixedness of a graph and other graph properties see <cit.>.
Well-covered graphs were introduced by Plummer.
See <cit.> for a survey on well-covered graphs and properties
of them. For an integer $r\geq 2$, a graph $G$ is said to
be $r$-partite, if $V(G)$ can be partitioned into $r$
disjoint parts such that for every $\{x, y\}\in E(G)$, $x$ and
$y$ do not lie in the same part. If $r=2, 3$, $G$ is
said to be bipartite and tripartite, respectively. Let G be an $r$-partite graph. For a vertex $v\in V(G)$, let $N(v)$ be the set of all vertices $u\in
V(G)$ where $\{u, v\}$ be an edge of $G$. Let $G$ be a bipartite
graph, and let $e=\{u, v\}$ be an edge of $G$. Then $G_{e}$ is the
subgraph induced on $N(u)\cup N(v)$. If $G$ is connected, the distance between $x$ and $y$ where $x,
y\in V(G)$, denoted by $d(x, y)$, is the length of the shortest
path between $x$ and $y$. A set $M\subseteq E(G)$ is said to be a
matching of $G$, if for any two $\{x, y\}, \{x', y'\}\in M$,
$\{x, y\}\cap \{x', y'\}= \emptyset$. A matching $M$ of $G$ is called
perfect if for every $v\in V(G)$, there exists an edge $\{x,
y\}\in M$ such that $v\in \{x, y\}$. A clique in G is a set Q of
vertices such that for every $x, y\in Q$, if $x\neq y$, $x, y$
lie in an edge. An $r$-clique is a clique of size r.
Unmixed bipartite graphs have already been characterized by
Ravindra and villarreal in a combinatorial way independently
<cit.>. Also these graphs have been characterize in an
algebraic method <cit.>.
In 1977, Ravindra gave the following criteria for unmixedness of
bipartite graphs.
<cit.> Let $G$ be a connected bipartite graph. Then $G$ is
unmixed if and only if $G$ contains a perfect matching $F$ such
that for every edge $e=\{x, y\}\in F$, the induced subgraph
$G_{e}$ is a complete bipartite graph.
Villarreal in 2007, gave the following characterization of
unmixed bipartite graphs.
<cit.> Let $G$ be a bipartite graph without
isolated vertices. Then $G$ is unmixed if and only if there is a
bipartition $V_{1}=\{x_{1}, \ldots , x_{g}\}, V_{2}=\{y_{1},
\ldots , y_{g}\}$ of $G$ such that: (a) $\{x_{i}, y_{i}\}\in
E(G)$, for all i, and (b) if $\{x_{i}, y_{j}\}$ and $\{x_{j},
y_{k}\}$ are in $E(G)$, and $i, j, k$ are distinct, then
$\{x_{i}, y_{k}\}\in E(G)$.
H. Haghighi in <cit.> gives the following characterization
of unmixed tripartite graphs under certain conditions.
<cit.> Let $G$ be a tripartite graph which satisfies the condition $(\ast )$. Then the graph $G$ is unmixed
if and only if the following conditions hold:
(1) If $\{u_{i}, x_{q}\}, \{v_{j}, y_{q}\}, \{w_{k}, z_{q}\}\in E(G)$, where no two vertices of $\{x_{q}, y_{q}, z_{q}\}$ lie in one of the tree parts of $V(G)$ and $i, j, k, q$ are distinct, then the set $\{u_{i}, v_{j}, w_{k}\}$ contains an edge of $G$.
(2) If $\{r, x_{q}\}, \{s, y_{q}\}, \{t, z_{q}\}$ are edges of $G$, where $r$ and $S$ belong to one of the three parts of $V(G)$ and $t$ belongs to another part, then the set $\{r, s, t\}$ contains an edge of $G$(here $r$ and $s$ may be equal).
In the above theorem, he has considered the condition $(\ast )$ as:
being a tripartite graph with partitions
\[U=\{u_{1}, \ldots u_{n}\}, V=\{v_{1}, \ldots v_{n}\}, W=\{w_{1}, \ldots w_{n}\},\]
in which $\{u_{i}, v_{i}\}, \{u_{i}, w_{i}\}, \{v_{i}, w_{i}\}\in E(G)$, for all $i=1, \ldots , n$.
Also to simplify the notations, he has used $\{x_{i}, y_{i}, z_{i}\}$ and $\{r_{i}, s_{i}, t_{i}\}$ as two permutations of $\{u_{i}, v_{i}, w_{i} \}$.
We give a characterization of unmixed $r$-partite graphs under certain condition which we name it $(\ast )$(see Theorem 2.3).
In both theorems 2.1 and 2.2 in an unmixed connected bipartite
graph, there is a perfect matching, with cardinality equal
to the cardinality of a minimal vertex cover, i.e. $\frac{|V(G)|}{2}$. An unmixed graph with $n$ vertices such that its
independence number is $\frac{n}{2}$, is said to be very well-covered. The unmixed
connected bipartite graphs are contained in the class of very well-covered
graphs. A characterization of very well-covered graphs is given in
§ A GENERALIZATION
0.4 true cm
By the following proposition, bipartition in connected bipartite graphs is unique.
Let $G$ be a connected bipartite graph with bipartition $\{A,
B\}$, and let $\{X, Y\}$ be any bipartition of $G$. Then $\{A,
B\}=\{X, Y\}$.
Let $x\in A$ be an arbitrary vertex of $G$. Then
$x\in X$ or $x\in Y$. without loss of generality let
$x$ be in $X$. Let $a\in A$.
then $d(x,a)$ is even. Then $a$ and $x$ are in the same part (of partition
$\{X,Y\}$). Then $A\subseteq X$, and by the same argument we have $X\subseteq A$. Therefore $A=X$, and
then $\{A, B\}=\{X, Y\}$.
The above fact for bipartite graphs, is not true in case of tripartite graphs, as shown in the following example.
[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
(0.,2.) rectangle (4.,5.);
(1.8,4.44)– (3.68,3.58);
(1.8,4.44)– (0.54,4.4);
(1.8,4.44)– (0.54,3.66);
(0.54,3.66)– (1.8,4.44);
(1.8,4.44)– (0.54,4.4);
(3.68,3.58)– (1.82,2.68);
(1.82,2.68)– (0.58,2.66);
(0.36,5.02) node[anchor=north west] $a_{1}$;
(0.36,4.32) node[anchor=north west] $a_{2}$;
(0.4,3.26) node[anchor=north west] $a_{3}$;
(1.66,5.06) node[anchor=north west] $a_{4}$;
(1.64,3.28) node[anchor=north west] $a_{5}$;
(3.52,4.22) node[anchor=north west] $a_{6}$;
[fill=qqqqff] (1.8,4.44) circle (1.5pt);
[fill=qqqqff] (3.68,3.58) circle (1.5pt);
[fill=qqqqff] (0.54,4.4) circle (1.5pt);
[fill=qqqqff] (0.54,3.66) circle (1.5pt);
[fill=qqqqff] (1.82,2.68) circle (1.5pt);
[fill=qqqqff] (0.58,2.66) circle (1.5pt);
In the above graph there are two different tripartitions:
\[\{\{a_{1}, a_{2}, a_{3}\}, \{a_{4}, a_{5}\}, \{a_{6}\}\}\]
\[\{\{a_{1}, a_{2}\}, \{a_{4}, a_{5}\}, \{ a_{3}, a_{6}\}\}.\]
A natural question refers to find criteria which
characterize a special class of unmixed $r$-partite $(r\geq 2)$
In the above two characterizations of bipartite graphs, having a
perfect matching is essential in both proofs. This motivates us to impose the following condition.
We say a graph $G$ satisfies the condition $(\ast )$ for an integer $r\geq 2$, if $G$ can be partitioned to $r$ parts
\ldots , x_{ni}\}$,$(1\leq i\leq r)$, such that for all $1\leq j\leq n$, $\{x_{j1}, \ldots , x_{jr}\}$ is a clique.
Let $G$ be a graph which satisfies $(\ast )$ for $r\geq 2$. If $G$ is unmixed,
then every minimal vertex cover of $G$, contains $(r-1)n$
vertices. Moreover the independence number of $G$ is
Let $C$ be a minimal vertex cover of $G$. Since for every $1\leq
j\leq n$, the vertices $x_{j1}, \ldots , x_{jr}$ are in a clique, $C$ must contain at least $r-1$ vertices in $\{x_{j1},
\ldots , x_{jr}\}$.
Therefore $C$ contains at least $(r-1)n$ vertices. By hypothesis
$\bigcup_{i=1}^{r-1} V_{i}$ is minimal vertex cover with $(r-1)n$
vertices, and $G$ is unmixed. Then every minimal vertex cover of
$G$ contains exactly $(r-1)n$ elements. The last claim can be
concluded from this fact that the complement of a minimal vertex
cover, is an independent set.
Now we are ready for the main theorem.
Let $G$ be an $r$-partite graph which satisfies the
condition $(\ast )$ for $r$. Then $G$ is unmixed if and only if the
following condition hold:
For every $1\leq q\leq n$, if there is a set $\{x_{k_{1}s_{1}}, \ldots , x_{k_{r}s_{r}}\}$ such that
\[x_{k_{1}s_{1}}\thicksim x_{q1}, \ldots , x_{k_{r}s_{r}}\thicksim x_{qr},\]
then the set $\{x_{k_{1}s_{1}}, \ldots , x_{k_{r}s_{r}}\}$ is not independent.
Let $G$ be an arbitrary $r$-partite graph which satisfies the
condition $(\ast )$ for $r$.
Let $G$ be unmixed. We prove that mentioned condition holds. Assume the contrary.
\[x_{k_{1}s_{1}}\thicksim x_{q1}, \ldots , x_{k_{r}s_{r}}\thicksim x_{qr},\]
but the set $\{x_{k_{1}s_{1}}, \ldots , x_{k_{r}s_{r}}\}$ is independent.
Then there is a maximal independent set $M$, such that
$M$ contains this set. Since $M$ is maximal, $C=V(G)\backslash M$
is a minimal vertex cover of $G$. Since the set $\{x_{k_{1}s_{1}}, \ldots , x_{k_{r}s_{r}}\}$
is contained in $M$, then its elements are not in $C$, and since
$C$ is a cover of $G$, then all vertices $x_{qi}$, $(1\leq i\leq r)$ are in $C$. But
by Lemma 3.2, every minimal vertex cover, contains $n-1$ vertices
of clique $q$ th, a contradiction.
Conversely let the condition hold. We have to prove that $G$ is
unmixed. We show that all minimal vertex covers of $G$, intersect the set
$\{x_{q1}, \ldots, x_{qr}\}$ in exactly $r-1$ elements (for every
$1\leq q\leq n$). Let $C$ be a minimal vertex cover and $q$ be
arbitrary. Since $C$ is a vertex cover and $\{x_{q1}, \ldots,
x_{qr}\}$ is a clique, then $C$ intersects this set at least in
$r-1$ elements. Let the contrary. Let the cardinality of $C\cap
\{x_{q1}, \ldots, x_{qr}\}$ be $r$. Attending to minimality of
$C$, for every $1\leq i\leq r, N(x_{qi})$ contains at least one
element, distinct from the elements of $\{x_{q1}, \ldots,
x_{qr}\}\backslash\{x_{qi}\}$, which is not in $C$, because we
can not remove $x_{qi}$ of cover. Let this element be
$x_{k_{i}s_{i}}$ where $s_{i}\neq i$ and $k_{i}\neq q$. Then
$x_{k_{i}s_{i}}\notin C$ and $\{x_{k_{i}s_{i}}, x_{qi},\}$ is
in $E(G)$. There is at least two elements $i$ and $j$ such that $1\leq i< j\leq r$ and $s_{i}\neq s_{j}$, because $x_{qi}$ can not choose its adjacent vertex from the part $i$. Therefore the set $\{x_{k_{1}s_{1}}, \ldots , x_{k_{r}s_{r}}\}$ contain at least two elements. Then by hypothesis, at least two elements, say $a, b$ of $\{x_{k_{1}s_{1}}, \ldots , x_{k_{r}s_{r}}\}$ are adjacent by an edge. Now $C$ is a cover but $a, b$ are not in $C$, a contradiction.
Villareal's theorem (Theorem 1.2) for bipartite graphs, and Haghighi's theorem (Theorem 1.3) for tripartite graphs, are special cases of Theorem 2.3 (where $r=2$, and $r=3$).
§ EXAMPLES AND COUNTEREXAMPLES
0.4 true cm
In this section, we give examples of two classes of unmixed graphs, and an example which shows that it is not necessary that an unmixed $r$-partite graph satisfies condition $(\ast )$.
By Theorem 2.3, the following 4-partite graphs are unmixed.
[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
(-3.,1.) rectangle (7.,5.8);
(5.86,4.26)– (3.26,4.26);
(3.26,4.26)– (3.26,1.86);
(5.86,4.26)– (5.9,1.84);
(5.9,1.84)– (3.26,1.86);
(3.92,3.68)– (5.22,3.68);
(5.22,3.68)– (5.22,2.44);
(5.22,2.44)– (3.88,2.46);
(3.92,3.68)– (3.88,2.46);
(5.22,2.44)– (5.9,1.84);
(5.22,3.68)– (5.86,4.26);
(3.92,3.68)– (3.26,4.26);
(3.88,2.46)– (3.26,1.86);
(3.92,3.68)– (5.22,2.44);
(5.22,3.68)– (3.88,2.46);
[shift=(4.2284210526315675,3.4192105263158026)] plot[domain=0.4514648196483709:4.156601207784886,variable=](1.*1.835477267784604*cos(r)+0.*1.835477267784604*sin(r),0.*1.835477267784604*cos(r)+1.*1.835477267784604*sin(r));
[shift=(4.91,3.41)] plot[domain=-1.0081993171568522:2.6899926544431825,variable=](1.*1.8560711193270627*cos(r)+0.*1.8560711193270627*sin(r),0.*1.8560711193270627*cos(r)+1.*1.8560711193270627*sin(r));
(0.6,1.9)– (0.6,4.18);
(0.6,4.18)– (-1.96,4.16);
(-1.96,4.16)– (-1.94,1.88);
(0.6,1.9)– (-1.94,1.88);
(-1.34,3.54)– (0.02,3.56);
(0.02,3.56)– (0.02,2.42);
(-1.34,3.54)– (-1.32,2.44);
(-1.32,2.44)– (0.02,2.42);
(0.02,2.42)– (0.6,1.9);
(0.02,3.56)– (0.6,4.18);
(-1.34,3.54)– (-1.96,4.16);
(-1.32,2.44)– (-1.94,1.88);
(0.6,1.9)– (0.02,3.56);
(0.6,4.18)– (-1.34,3.54);
(-1.96,4.16)– (-1.32,2.44);
(-1.94,1.88)– (0.02,2.42);
(-1.34,3.54)– (0.02,2.42);
(0.02,3.56)– (-1.32,2.44);
[shift=(-0.3952032520325195,3.35260162601626)] plot[domain=-0.9701283912621932:2.6906444134786236,variable=](1.*1.7608182747692012*cos(r)+0.*1.7608182747692012*sin(r),0.*1.7608182747692012*cos(r)+1.*1.7608182747692012*sin(r));
[shift=(-0.9754200542005421,3.367289972899729)] plot[domain=0.4510333247708006:4.137034697971628,variable=](1.*1.7726945408971515*cos(r)+0.*1.7726945408971515*sin(r),0.*1.7726945408971515*cos(r)+1.*1.7726945408971515*sin(r));
(-2.5,4.5) node[anchor=north west] $t_{2}$;
(-2.52,2.03) node[anchor=north west] $x_{2}$;
(0.56,2.04) node[anchor=north west] $y_{2}$;
(0.52,4.5) node[anchor=north west] $z_{2}$;
(-1.87,2.78) node[anchor=north west] $y_{1}$;
(-1.6,4.03) node[anchor=north west] $x_{1}$;
(-0.,3.8) node[anchor=north west] $t_{1}$;
(-0.3,2.42) node[anchor=north west] $z_{1}$;
(2.69,2.00) node[anchor=north west] $x_{2}$;
(5.84,1.98) node[anchor=north west] $y_{2}$;
(2.72,4.68) node[anchor=north west] $t_{2}$;
(5.8,4.62) node[anchor=north west] $z_{2}$;
(3.38,3.82) node[anchor=north west] $x_{1}$;
(3.38,2.87) node[anchor=north west] $y_{1}$;
(5.17,2.8) node[anchor=north west] $z_{1}$;
(5.18,3.82) node[anchor=north west] $t_{1}$;
[fill=qqqqff] (5.86,4.26) circle (1.5pt);
[fill=qqqqff] (3.26,4.26) circle (1.5pt);
[fill=qqqqff] (3.26,1.86) circle (1.5pt);
[fill=qqqqff] (5.9,1.84) circle (1.5pt);
[fill=qqqqff] (3.92,3.68) circle (1.5pt);
[fill=qqqqff] (5.22,3.68) circle (1.5pt);
[fill=qqqqff] (5.22,2.44) circle (1.5pt);
[fill=qqqqff] (3.88,2.46) circle (1.5pt);
[fill=qqqqff] (5.88,4.22) circle (1.5pt);
[fill=qqqqff] (3.24,4.22) circle (1.5pt);
[fill=qqqqff] (0.6,1.9) circle (1.5pt);
[fill=qqqqff] (0.6,4.18) circle (1.5pt);
[fill=qqqqff] (-1.96,4.16) circle (1.5pt);
[fill=qqqqff] (-1.94,1.88) circle (1.5pt);
[fill=qqqqff] (-1.34,3.54) circle (1.5pt);
[fill=qqqqff] (0.02,3.56) circle (1.5pt);
[fill=qqqqff] (0.02,2.42) circle (1.5pt);
[fill=qqqqff] (-1.32,2.44) circle (1.5pt);
[fill=qqqqff] (-1.98,4.12) circle (1.5pt);
[fill=qqqqff] (0.62,4.14) circle (1.5pt);
In each of the above graphs, there are two complete graphs of order 4 and some edges between them.
For $r>4$, also $r=3$, using two complete graphs of order $r$, we can construct $r$-partite unmixed graphs which are natural generalization of the above graphs.
For every $n$, $n\geq 3$, the complete graph $K_{n}$, is an $n$-partite graph which satisfies the condition $(\ast )$. By Theorem 2.3, $K_{n}$ is unmixed.
Theorem 2.3 dose not characterize all unmixed $r$-partite graphs.
More precisely, the condition $(\ast )$ is not valid for all unmixed graphs. In the following, we give an example of an
unmixed $r$-partite graph which dose not satisfy the condition
$(\ast )$.
The following graph is a 4-partite graph with partition
$\{y_{1}\}$, $\{y_{2}, y_{4}\}$, $\{y_{3}\}$, and $\{y_{5},
y_{6}\}$. This graph dose not satisfy the condition $(\ast )$
because 6 is not a multiple of 4.
[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
(1.3,0.7) rectangle (5.6,5.);
(4.46,3.88)– (2.66,3.88);
(4.46,3.88)– (5.06,2.34);
(5.06,2.34)– (3.72,1.36);
(2.66,3.88)– (2.2,2.2);
(3.72,1.36)– (2.2,2.2);
(2.2,2.2)– (4.46,3.88);
(2.2,2.2)– (5.06,2.34);
(3.6,2.72)– (4.46,3.88);
(3.6,2.72)– (5.06,2.34);
(3.6,2.72)– (2.66,3.88);
(1.66,2.48) node[anchor=north west] $y_{1}$;
(2.30,4.4) node[anchor=north west] $y_{2}$;
(4.34,4.4) node[anchor=north west] $y_{3}$;
(5.10,2.62) node[anchor=north west] $y_{4}$;
(3.58,1.35) node[anchor=north west] $y_{5}$;
(3.20,2.74) node[anchor=north west] $y_{6}$;
[fill=qqqqff] (2.66,3.88) circle (1.5pt);
[fill=qqqqff] (4.46,3.88) circle (1.5pt);
[fill=qqqqff] (5.06,2.34) circle (1.5pt);
[fill=qqqqff] (3.72,1.36) circle (1.5pt);
[fill=qqqqff] (2.2,2.2) circle (1.5pt);
[fill=qqqqff] (3.6,2.72) circle (1.5pt);
We show that this graph is unmixed. Let $C$ be an arbitrary
minimal vertex cover of $G$. We show that $C$ is of size 4.
Since $C$ is a cover, it selects at least one element of
$\{y_{4},y_{6}\}$. Now we consider the following cases:
case 1: $y_{6}\in C$ and $y_{4}\notin C$. In this case,
since $C$ is a vertex cover, $y_{1}, y_{3}, y_{5}\in C$. Now
$\{y_{1}, y_{3}, y_{5}, y_{6}\}$ is a vertex cover of $G$, and
since $C$ is minimal, $C=\{y_{1}, y_{3}, y_{5}, y_{6}\}$.
case 2: $y_{4}\in C$ and $y_{6}\notin C$. In this case, $y_{2},y_{3}\in C$, and at least one
vertex of $y_{1},y_{5}$ and by minimality, only one is in $C$.
Now since $\{y_{2}, y_{3}, y_{4}, y_{i}\}$ where $i\in\{1, 5\}$
is one of two vertices $y_{1}$ and $y_{5}$, is a cover of $G$, by
minimality of $C$, $C=\{y_{2}, y_{3}, y_{4}, y_{i}\}$.
case 3: $y_{4}, y_{6}\in C$. In this case, at least one
of two vertices $y_{1}, y_{5}$ and by minimality of $C$, only one
is in $C$. Now if $y_{5}\in C$, $y_{3}$ should be in $C$ (because
the edge $\{y_{1}, y_{3}\}$ should be covered). Also $y_{2}\in C$
(because the edge $\{y_{1}, y_{2}\}$ should be covered). Now
$\{y_{2}, y_{3}, y_{5}, y_{4}, y_{6}\}$ is a cover, and since $C$
is minimal, $C= \{y_{2}, y_{3}, y_{5}, y_{4}, y_{6}\}$, that is a
contradiction because $y_{6}$ can be removed. If $y_{1}\in C$, at
least one of $y_{2}$ and $y_{3}$, and by minimality only one, is
in $C$. Now since $\{y_{1}, y_{4}, y_{6}, y_{j}\}$, where
$j\in\{2, 3\}$ is one of two vertices $y_{2}$ and $y_{3}$, is a
vertex cover, by minimality of $C$, $C=\{y_{1}, y_{4}, y_{6},
0.4 true cm
EE M. Estrada and R. H. Villarreal, Cohen-Macaulay bipatite graphs, Arc. Math., 68 (1997), 124-128.
ET O. Fanaron, Very well covered graphs, Discrete. Math., 42 (1982), no. 2-3, 177-187.
Ha H. Haghighi, A generalization of Villarreal's result
for unmixed tripartite graphs, Bull. Iranian Math. Soc.,
40 (2014), no. 6, 1505-1514.
EV F. Harary, Graph Theory, Addison-Wesley, Reading, MA,
HH J. Herzog and T. Hibi, Distributive lattices, bipartite graphs, and Alexander duality, J. Algebraic Combin., 22 (2005), no. 3, 289-302.
K R. M. Karp, Complexity of computer computation, Plenum Press,
New York, (1972), 85-103.
P M. D. Plummer, Well-covered graphs: A survay, Questions Math., 16 (1993), no. 3, 253-287.
R G. Ravindra, Well-covered graphs, J. Combin. Inform. , System Sci.
2 (1977), no. 1, 20-21.
V R. H. Villarreal, Cohen-Macaulay graphs, Manuscripta
Math., 66 (1990), 277-293.
V1 R. H. Villarreal, Monomial Algebras, Marcel Dekker, Inc. New York, 2001.
V2 R. H. Villarreal, Unmixed bipartite graphs, Rev. Colombiana
Mat., 41 (2007), no. 2, 393-395.
Z R. Zaree-Nahandi, Pure simplicial
complexes and well-covered graphs, Rocky Mountain Journal
of Mathematics, 45 (2015), no. 2, 695-702.
|
1511.00559
|
Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
We report the continuous and partially nondestructive measurement of optical photons. For a weak light pulse traveling through a slow-light optical medium (signal), the associated atomic-excitation component is detected by another light beam (probe) with the aid of an optical cavity. We observe strong correlations of $g^{(2)}_{sp}=4.4(5)$ between the transmitted signal and probe photons. The observed (intrinsic) conditional nondestructive quantum efficiency ranges between 13% and 1% (65% and 5%) for a signal transmission range of 2% to 35%, at a typical time resolution of 2.5 $\si{\micro}$s. The maximal observed (intrinsic) device nondestructive quantum efficiency, defined as the product of the conditional nondestructive quantum efficiency and the signal transmission, is 0.5% (2.4%). The normalized cross-correlation function violates the Cauchy-Schwarz inequality, confirming the non-classical character of the correlations.
Photons are unique carriers of quantum information that can be strongly interfaced with atoms for quantum state generation and processing <cit.>. Quantum state detection, a particular type of processing, is at the heart of quantum mechanics and has profound implications for quantum information technologies. Photons are standardly detected by converting a photon's energy into a measurable signal, thereby destroying the photon. Nondestructive photon detection, which is of interest for many quantum optical technologies <cit.>, is possible through strong non-linear interactions <cit.> that ideally form a quantum non-demolition (QND) measurement <cit.>. To date, QND measurement of single microwave photons bound to cooled cavities has been demonstrated with high fidelity using Rydberg atoms <cit.>, and in a circuit cavity quantum electrodynamics system using a superconducting qubit <cit.>.
(a), (b), The $\pi$-polarized signal light travels with slow group velocity through the atoms by means of EIT on the $|g\rangle\leftrightarrow|c\rangle\leftrightarrow|d\rangle$ transitions. The associated atomic-excitation component is nondestructively detected via cavity light in the geometric overlap between the atomic ensemble and the cavity mode. Input cavity light is linearly polarized such that, in the absence of signal photons, the probe port of the polarization beam splitter (PBS) ideally remains dark. Whenever a signal photon traverses the atomic medium in the cavity, the transmission of the $\sigma^+$ polarized light through the cavity is blocked. The atomic levels are $|g\rangle$= $|6S_{1/2}; F=3, m_F=3\rangle$, $|c\rangle$= $|6P_{3/2}; 3, 3\rangle$, $|d\rangle$=$|6S_{1/2}; 4, 4\rangle$, $|e\rangle$= $|6P_{3/2}; 5, 5\rangle$ and $|f\rangle$= $|6P_{3/2}; 5, 3\rangle$, where $F,m_F$ are the hyperfine quantum numbers.
For quantum communication and many other photonics quantum information applications <cit.>, it is desirable to detect traveling optical photons instead of photons bound to cavities. Previously, a single-photon transistor was realized using an atomic ensemble inside a high finesse cavity where one stored photon blocked the transmission of more than one cavity photon and could still be retrieved <cit.>. Such strong cross-modulation <cit.> can be used for all-optical destructive detection of the stored optical photon, but the parameters in that experiment did not allow nondestructive detection with any appreciable efficiency. High-efficiency pulsed nondestructive optical detection has recently been achieved using a single atom in a cavity <cit.>. In that implementation, the atomic state is prepared in 250 $\si{\micro}$s, altered by the interaction with an optical pulse reflected from the cavity, and read out in 25 $\si{\micro}$s.
In this Letter, we realize partially nondestructive, continuous detection of traveling optical photons with micro-second time resolution. The signal photons to be detected propagate through an atomic ensemble as slow-light polaritons <cit.> under conditions of electromagnetically induced transparency (EIT) <cit.>. The signal polariton's atomic-excitation component is nondestructively detected via the polarization change on another light field (probe), enhanced by an optical cavity. We observe positive correlations between the signal and probe photons of $g_{sp}^{(2)}=4.4(5)$, and use the measured correlation function to calculate the conditional nondestructive quantum efficiency Q. We achieve efficiencies Q between 13% and 1% at a signal transmission $T_s$ between 2% and 35%, with a maximum device nondestructive quantum efficiency $Q\times T_s$ of 0.47% at a maximum signal input rate of 300 kHz.
(a) Signal-probe correlation function, $g_{sp}^{(2)}$, is plotted as a function of separation time, $\tau$, between signal ($t_s$) and probe ($t_p$) photons. The decay time constant for negative (positive) times $\tau_{<}=1.2(2)$ $\si{\micro}$s ($\tau_{>}=1.3(2)$ $\si{\micro}$s) is consistent with the cavity decay time (EIT lifetime) <cit.>. This measurement is done with mean input cavity photon number $\langle n^{in}_c \rangle=R^{s=0}_c\tau_c/q_c=3.7$, cavity path detection efficiency $q_c=0.2$, $\tau_c=(\kappa/2)^{-1}=2~\si{\micro}$s and Rabi frequency $\Omega/2\pi=1.9$ MHz. The inset shows the cross-correlation function for signal and $\sigma^+$-polarized cavity photons, measured for $\langle n^{in}_c \rangle=0.1$ and $\Omega/2\pi= 2.6$ MHz. The observed signal-probe anti-correlation is $g^{(2)}_{s\sigma^+}(0)=0.41(7)$. In this and all following figures, statistical errorbars are plotted when they are larger than the points and indicate one standard deviation. (b) Normalized detected probe rate $R_p/R_c^{s=0}$, plotted against input signal photon number $\langle n^{in}_s \rangle=R^{in}_s\tau_{_{\text{EIT}}}/q_s$. The slope of the solid fitted line is $0.20(1)$. The dashed line represents the maximum possible probe rate with a slope of $\varepsilon_{id}=0.25$. For this measurement, $\langle n^{in}_c \rangle=1.2$, $q_s=0.3$, and $\Omega/2\pi=1.3$ MHz, giving $\tau_{_{\text{EIT}}}=1.4$ $\si{\micro}$s.
The nondestructive measurement scheme and atomic level structure are shown in Fig. <ref>. A laser-cooled atomic ensemble of $^{133}$Cs atoms is held in a cigar-shaped dipole trap that partly overlaps with the fundamental mode of the optical cavity.
A signal light resonant with the $|g\rangle\rightarrow|c\rangle$ transition propagates orthogonal to the cavity axis through the ensemble. A control laser induces an EIT transmission window that slows down the signal light to a typical group velocity of $300~m/s$ and reversibly maps it onto a collective atomic excitation in state $|d\rangle$ <cit.>. This atomic population couples strongly to the $\sigma^+$ polarized light which is simultaneously resonant with the optical cavity and the $|d\rangle\rightarrow|e\rangle$ transition, blocking its transmission through the cavity <cit.>. To generate a useful positive detection signal in transmission, we add $\sigma^-$ reference light and probe the cavity continuously with horizontally polarized light. The reference light interacts only weakly with the atoms: the atomic coupling strength on the $|d\rangle\rightarrow|f\rangle$ transition is 45 times smaller than the strength of the $\sigma^+$ transition and is also detuned from resonance by $\Delta/2\pi=6$ MHz by the $5.2$ G magnetic field along the cavity axis ($z$). Light transmitted through the cavity is then analyzed in a horizontal/vertical basis. Vertically polarized light (probe port) corresponds to detection, as the probe port is dark in the absence of signal photons.
Quantum correlations between detected outgoing signal and probe photons are the signature of nondestructive detection. The cross-correlation function $g^{(2)}_{sp}=\langle n_sn_p\rangle/\langle n_s\rangle\langle n_p\rangle$ can be understood as the likelihood of measuring the signal twice: first measuring it nondestructively with our cavity QED system, which results in a detected probe photon ($n_p=1$), and then checking the first measurement by measuring the signal photon again destructively ($n_s=1$). The cross-correlation function in Fig. <ref>a with zero-time value $g^{(2)}_{sp}(0)=4.0(3)$ demonstrates that simultaneous nondestructive and destructive measurements of the signal photon occur four times more often than randomly. This value also agrees well with the directly observed blocking of $\sigma^+$-polarized cavity photons by a signal photon (inset to Fig. <ref>a), and with the theoretical expectations for our system's cooperativity $\eta=4.3$ and relevant optical depth $\mathcal{D}\simeq3$ (see S.M.). The increased $\mathcal{D}$ accounts for the improvement over previously published results with the same apparatus <cit.>.
To confirm the linearity of the system, we plot the probe rate normalized to the empty cavity output rate, $R_p/R_c^{s=0}$, against the average input signal photon number per EIT lifetime $\langle n_s^{in}\rangle=R_s^{in}\tau_{\text{EIT}}/q_s$ in Fig. <ref>b. Here, $R_s^{in}/q_s$ is measured input rate corrected for the finite detection efficiency $q_s=0.3$. Under ideal circumstances, an incident cavity photon emerges in the probe port with probability $\varepsilon_{id}=1/4$ in the presence of a signal photon, indicated in the figure as a dashed line. Achieving this limit requires a strong single-atom-cavity coupling (cooperativity $\eta\gg1$) <cit.>, large ensemble optical depth inside the cavity region $\mathcal{D}\gg1$, and sufficiently slowly traveling signal photons $\tau_{\text{EIT}}/\tau_c>1$, where $\tau_c$ is the cavity lifetime. Even with finite cooperativity $\eta$ and optical depth, we measure $\varepsilon=0.20(1)$. This number is the detection probability per input cavity photon and includes both nondestructive and destructive detection of the signal photon. The nonzero offset in Fig. <ref>b at $\langle n_s^{in}\rangle=1$ corresponds to the background noise in the average measurement. The observed linear increase in probe rate for $\langle n_s^{in}\rangle<1$ also confirms the sensitivity of our experiment at the single photon level. However, unlike output correlations, this average signal neither distinguishes between destructive and nondestructive detection events nor does it reveal the time resolution of the detector. Destructive detection events correspond to decohered polaritons, i.e. atomic population in state $|d\rangle$, and hence have the same effect on the cavity light as traveling signal photons.
To study only those events when we preserve the signal photon, we define the conditional nondestructive quantum efficiency, $\text{Q}$, to be the conditional probability for a correlated photon to be detected in the probe port when a signal photon is present: $\text{Q}=\langle n_sn_p\rangle/\langle n_s\rangle-\langle n_p\rangle$ for $\langle n_s\rangle\ll1$. (Note that the second term $\langle n_s\rangle=\langle n_p\rangle\langle n_s\rangle/\langle n_s\rangle$ is necessary to remove uncorrelated (random) coincidences between signal and probe photons.) The time scale for this conditioning is defined by the typical correlation time: this conditional nondestructive quantum efficiency $\text{Q}$ is precisely the area under $g^{(2)}(\tau)-1$ (the shaded area in Fig. <ref>a) multiplied by the average rate of detected photons at the probe port, $R_p$: $\text{Q}= R_p\int{(g^{(2)}(\tau)-1)d\tau}$. $\text{Q}$ evaluates to 10% for the cross-correlation function plotted in Fig. <ref>a. The time resolution is the sum of the positive- and negative- correlation times, $\tau_>+\tau_<=2.5(4)~\si{\micro}$s <cit.>. Since $\text{Q}$ scales with the detected rate at the probe port, finite probe photon detection efficiency $q_p$ directly reduces $\text{Q}$. The total detection efficiency for probe photons, $q_p=0.2$, is the product of detector efficiency (0.45), fiber coupling and filter losses (0.7) and cavity outcoupling losses (0.66). Correcting for these linear losses gives the intrinsic conditional nondestructive quantum efficiency $\text{Q}/q_p=50\%$. Single photon detectors with better than 0.99 efficiency exist at our wavelength, so only improving the optics and detectors outside of our vacuum chamber would already allow us to achieve a $\text{Q}$ of 30%.
(a) The observed conditional nondestructive quantum efficiency $\text{Q}$ is plotted against mean cavity photon number, $\langle n^{in}_c\rangle$, with mean $R^{in}_s=2.8\times10^{5}$ s$^{-1}$. The slope of the fitted curves (solid lines) is $\frac{d\text{Q}}{d\langle n^{in}_c\rangle}=\{10(2), 5(1), 1.9(5)\}$% for $\Omega/2\pi= \{1.8, 2.9, 3.5\}$ MHz (top to bottom) and represents the observed detection efficiency per input cavity photon. (b) Signal transmission $T_s$ for the same data presented in (a). Exponential fits give $1/e$ transmission at cavity photon numbers of $\{1.2(1),1.9(1), 2.1(1)\}$ for $\Omega/2\pi= \{1.8, 2.9, 3.5\}$ MHz (bottom to top), respectively. The inset displays the nondestructive quantum efficiency Q as a function of signal transmission for the same Rabi frequencies as in (a) and (b).
We define the device nondestructive quantum efficiency as the probability for an input photon to be nondestructively detected. It is equal to $\text{Q}\times T_s$, the product of the conditional nondestructive quantum efficiency and the signal transmission. Fig. <ref> explores the tradeoff between these two factors. $\text{Q}$ scales linearly with the input cavity photon number (Fig. <ref>a), as with increasing cavity input rate it becomes more likely for a randomly arriving cavity photon to “hit" a signal photon and perform the detection. At the same time, the signal transmission, $T_s$, degrades exponentially with input cavity rate due to cavity-induced decoherence of the signal polariton, as seen in Fig. <ref>b. Slower signal polaritons (smaller control Rabi frequency $\Omega$) are more likely to be “hit" by a cavity photon, and thus have a larger nondestructive quantum efficiency but also have a lower transmission due to greater decoherence for a given cavity photon number. The choice of Rabi frequency changes the detector speed but does not improve the tradeoff between efficiency and transmission; the inset of Fig. <ref>b shows that observed quantum efficiency as a function of signal transmission collapses to a single curve for all measured Rabi frequencies.
Fig. <ref> plots the device nondestructive quantum efficiency and the error probability, $P_{err}$ as a function of input photon number. The maximal observed (intrinsic) device nondestructive quantum efficiency is 0.47% (2.4%). $P_{err}$ is the probability of having a false detection event when no signal photon is present. Considering these together, the detector achieves a nondestructive signal to noise ratio of 2.4. We further characterize the performance for single input photons by calculating by the four probabilities, $P_{sp}$, to detect $s=\{0,1\}$ signal and $p=\{0,1\}$ probe photons given one input signal photon. These probabilities can be obtained from measured quantities in the limit $\langle n^{in}_s\rangle\ll1$ using the relations $P_{11}/(P_{11}+P_{10})=\text{Q}$, $P_{11}+P_{10}=T_s$, $P_{01}+P_{11}=\langle n_p\rangle/\langle n^{in}_s\rangle$ and $\sum{P_{sp}}=1$. These probabilities describe different aspects of the nondestructive detection. In particular, the device nondestructive quantum efficiency is $P_{11}$ and the state preparation probability, $P_{11}/(P_{11}+P_{01})\simeq4\%$, represents the probability of having an outgoing signal photon if a photon is present at the probe port.
Device nondestructive quantum efficiency ($P_{11}=\text{Q}\times T_s$) and error probability $P_{\text{err}}$ for joint detection of signal and probe outputs for $\Omega/2\pi=2.9$ MHz. $P_{11}$ is calculated assuming one input signal photon. $P_{\text{err}}$ represents the false detection of probe photons in absence of signal photons. Inset displays all four characterizing probabilities $P_{sp}$ with $s,p=\{0,1\}$ (see text) and $P_{\text{err}}$ under the same conditions.
In our system, the transmission of detected signal photons is limited to about 70% by the standing wave nature of our cavity probe, which imprints a grating onto the detected polariton and reduces its readout efficiency in the original mode. In addition, since the atomic medium extends outside the cavity mode, the detection localizes the signal polariton in a finite region of the ensemble, and the corresponding spectral broadening outside the EIT transmission window reduces the signal transmission by 30% (see S.M.). Finally, cavity photon scattering into free-space, which destroys the signal polariton, occurs with a finite probability $2\eta/(1+\eta)^2=0.3$. The combination of these effects explains the observed transmission reduction for the signal.
To further enhance the effective optical density of the medium, we also carried out an experiment where the signal propagates twice through the medium (see S.M.). In this case, we observed slightly stronger correlations of $g^{(2)}_{sp}(\tau=0)=4.4(5)$ due to the larger effective optical depth. To remove classical correlations from the observed cross-correlation, we normalize the cross-correlation function to the auto-correlations measured at the signal and probe ports of $g_{ss}^{(2)}=1.6(3)$ and $g_{pp}^{(2)}=5.6(1)$. The resulting normalized quantum correlation $G_{sp}=\left(g_{sp}^{(2)}\right)^2\left/\left(g_{ss}^{(2)}g_{pp}^{(2)}\right)\right.=2.7(8)$ at $\tau=0$ violates the Cauchy-Schwarz inequality <cit.>, $G<1$, and confirms that our interactions are non-classical.
Key to the nondestructive photon measurement scheme demonstrated here is the strong interaction between one atom and a cavity photon <cit.> (large single-atom cooperativity $\eta\gg1$), in combination with the strong collective interaction of atoms with signal photons (large optical depth $\mathcal{D}\gg1$ inside the cavity). Both quantities can be further improved in our experiment. For realistic values $\mathcal{D}=10$ and $\eta=20$, we expect a device nondestructive quantum efficiency exceeding 55% with a conditional nondestructive quantum efficiency of about 80% and a signal transmission of about 70%. An interaction of this kind enables many quantum applications such as the projection of a coherent state of a light pulse into a photon number state <cit.>, the implementation of nearly deterministic photonic quantum gates through nondestructive measurement and conditional phase shift <cit.>, engineering exotic quantum states of light <cit.> or non-deterministic noiseless amplification for entanglement distillation <cit.>.
The authors would like to thank Arno Rauschenbeutel for insightful discussions. This work was supported by NSF and the Air Force Office of Scientific Research. K.M.B. acknowledges support from NSF IGERT under grant 0801525.
§ SUPPLEMENTAL MATERIAL
Each second-long experimental cycle has a 12 ms detection period, which consists of 20 $\si{\micro}$s measurement times, a time window arbitrarily chosen to be much longer than the EIT lifetime to allow the continuous measurement of signal photons, interleaved with 20 $\si{\micro}$s preparation times that ensure the atoms are optically pumped to the $|g\rangle$ state. For cross correlation measurements such as Fig.<ref>(a) an average of approximately 8000 experimental cycles were used.
The temperature of the cloud in the dipole trap is about 120 $\si{\micro}$K corresponding to a measured atomic decoherence rate of $\gamma_0/2\pi\simeq100$ kHz, dominated by the Doppler broadening. The signal path detection efficiency is $q_s\simeq0.3$ including the fiber coupling efficiency and photodetector quantum efficiency. The optical cavity has a waist size of $35~\si{\micro}$m, length of 13.7 mm, and out-coupling efficiency of 66%.
The single-photon Rabi frequency for $\sigma^+$ polarized light is $2g=2\pi\times 2.5$ MHz. Thus, the single-atom cooperativity for an atom on the cavity axis (along $z$) at an antinode of the cavity standing wave is given by $\eta=4g^2/\kappa\Gamma=8.6>1$, i.e. the system operates in the strong coupling regime of cavity quantum electrodynamics. The cavity resonance frequency matches the atomic frequency $|d\rangle\to|e\rangle$
Detection and transmission probabilities.
The probability to observe a probe photon when a cavity photon is present and a signal photon is propagating through the EIT window at $\tau=0$ is given by <cit.>
\begin{eqnarray}
\varepsilon_0& = &\frac{1}{4}\frac{\eta^2}{(1+\eta)^2} \left[1-\exp\left(-\mathcal{D}/2\zeta\right)\right]^2,
\label{eq: epsilon}
\end{eqnarray}
\begin{eqnarray}
\zeta & = & \left(1+\frac{\gamma \Gamma}{\Omega^2}\right)\left(1+\frac{\Omega^2/\kappa\Gamma+\gamma/\kappa}{1+\eta}\right).
\end{eqnarray}
Here, $\eta=4.3$ is the spatially-averaged cavity cooperativity, $\mathcal{D}$ is the effective optical density that overlaps with the cavity mode, $\Omega$ is the control Rabi frequency, $\kappa= 2\pi\times140$ kHz is the decay rate of the cavity, $\gamma\simeq \gamma_c+\gamma_0$, $\gamma_0\approx2\pi\times100$ kHz is atomic decoherence rate in the absence of cavity photons, $\gamma_c$ is cavity-induced decoherence, and $\Gamma$ is the Cs excited-state decay rate. The decoherence rate, $\gamma_c$, caused by cavity light scattering manifests itself as: (1) loss of atomic coherence given by $\langle n^{in}_c \rangle \kappa\eta/(1+\eta)^2$ where $\langle n^{in}_c \rangle$ is the mean $\sigma^+$-polarized input cavity photon number, (2) reduction of signal transmission as a result of inhomogenous coupling of cavity light to atoms (see below). For the anti-correlation data shown in the inset of Fig. 2a, when we take into account the cavity blocking due to an atom in state $|d\rangle$, we obtain $4\varepsilon_0=0.1$ and a blocking probability for $\sigma^+$ light of $P=1-(1-\sqrt{4\varepsilon_0})^2=0.55$. This is in good agreement with the measured probability of $1-g^{(2)}_{s\sigma^+}(0)=0.59(7)$. A detailed theoretical treatment of the cavity interaction with atomic ensemble is given in Ref. [25].
In the nondestructive detection where horizontally-polarized cavity light is used, the detection probability is defined as the field amplitude of the transmitted $\sigma^+$ light, which interacts with atoms in state $|d\rangle$ as described in Ref. [23], combined with the field amplitude of $\sigma^-$ light on the output polarization beamsplitter. The field amplitude addition results in the factor 1/4 in Eq. <ref>. In principle, this reduction can be avoided by impinging only $\sigma^+$ light onto an impedance-matched cavity and measuring the reflected photons. In our present lossy cavity, the reflection in the absence of signal photons causes a large background for the probe light.
Cavity-induced decoherence reduces the transmission probability of the signal photon and the EIT coherence time <cit.>. The signal transmission in the presence of cavity photons is given by:
\begin{eqnarray}
T_s = T_0\exp\left(-\frac{\mathcal{D}}{1+\Omega^2/\Gamma\gamma}\right)
\end{eqnarray}
where $T_0=\exp\left(-\frac{\mathcal{D}'}{1+\Omega^2/\Gamma\gamma_0}\right)$ is the EIT transmission corresponding to atoms outside the cavity waist and $\mathcal{D}'$ is the corresponding optical density.
An additional limit to the signal transmission is caused by the standing-wave nature of the cavity light in combination with the uniform distribution of atoms between nodes and antinodes of the cavity. Once the signal is detected, the spatial mode of the polation is projected onto the cavity mode resembling a grating imprinted onto the polariton structure. This effect leads to reduction in transmission of the signal. The overlap between the polariton before and after detection of a probe photon can be calculated as
\begin{eqnarray}
\mathcal{F}_p=\frac{1}{\pi}\int_0^{\pi}\frac{\eta\cos^2(\theta)}{1+\eta\cos^2(\theta)}d\theta=1- \frac{1}{\sqrt{1+\eta}}
\end{eqnarray}
where $\theta=kz$, $k$ is the wave-number of cavity light and $z$ is the position along the cavity axis. At large cooperativity, $\eta\gg1$, the expected maximum transmission approaches 100%. For our system parameters this evaluates to about 70%.
Also, the atomic cloud extended beyond the cavity region introduces additional signal transmission loss. This is because the signal photon wave-packet is localized inside the cavity region upon detection of a probe photon and therefore its spectral bandwidth exceeds the EIT bandwidth. Hence, after detection via the cavity, the signal photon propagating through the EIT window experiences dispersion and loss. Our numerical simulations predicts a loss of 30% in signal transmission given the experimental parameters. In principle, this loss can be eliminated by removing atoms outside the cavity region.
Quantum correlation between probe and signal photons.
The mean photon rate entering the cavity can be calculated from the total detected photon rate exiting the cavity, $R^{s=0}_c$, in absence of signal photons as
\begin{eqnarray}
\langle R^{in}_c \rangle= \frac{R^{s=0}_c}{q_d(\frac{\mathcal{T}}{\mathcal{T}+\mathcal{L}})}
\end{eqnarray}
where $q_d=0.3$ accounts for detection losses including fiber coupling, filter losses and photodetector quantum efficiency and $\frac{\mathcal{T}}{\mathcal{T}+\mathcal{L}}=0.66$ is the cavity out-coupling efficiency with $\mathcal{L}$ and $\mathcal{T}$ being mirror loss and transmissivity, respectively. In the following, we combine $q_d$ and the cavity out-coupling efficiency into a single parameter $q_p$. The mean cavity photon number in absence of signal photons is then $\langle n^{in}_c \rangle =R^{in}_c \tau_c$ where $\tau_c=(\kappa/2)^{-1}$. The mean signal photon number in the relevant time window, i.e. the EIT life time $\tau_{_{EIT}}=(\Omega^2/(\Gamma\mathcal{D})+\gamma_0)^{-1}$, is given by $\langle n^{in}_s \rangle= R^{in}_s\tau_{_{EIT}}/q_s$ where $R^{in}_s/q_s$ is the signal photon rate entering the medium and $q_s=0.3$ accounts for detection losses. In absence of population in state $|d\rangle$, the linearly polarized cavity light is rotated by atoms in state $|g\rangle$ due to the differences in the coupling strengths for $\sigma^+$ and $\sigma^-$ polarized light interacting with state $|g\rangle$ and excited states. Ideally, this rotation is constant and we compensate for it with a waveplate at the output of the cavity. However, the shot-to-shot atom number fluctuation during loading provides a varying background, $\alpha q_p\langle n^{in}_c \rangle$, that dominates the probe port at low signal photon rates. We typically measure a maximum fractional background of $\alpha\approx3\times 10^{-3}$ of the total detected cavity photons. The detection events consists of a background given by $\langle b \rangle = \alpha q_p \langle n^{in}_c \rangle +\langle r_p \rangle$, where $\langle r_p \rangle$ denotes the dark counts of the probe detector $D_p$. We define the detected mean signal photon number $\langle n_s \rangle$, true detection events $\langle t \rangle$ and total detected mean probe photon number $ \langle n_p \rangle$ as
\begin{eqnarray}
\langle n_s \rangle & = & q_s T_s\langle n^{in}_s \rangle + \langle r_s\rangle\\
\langle t \rangle & = & (\varepsilon_0 +\epsilon_b) q_p \langle n^{in}_c \rangle \langle n^{in}_s \rangle \\
\langle n_p \rangle & = & \langle t \rangle + \langle b \rangle= (\varepsilon_0 \langle n^{in}_s \rangle+\epsilon_b \langle n^{in}_s \rangle+\alpha)q_p \langle n^{in}_c \rangle \nonumber\\
&&+\langle r_p\rangle
\end{eqnarray}
where $\langle r_s\rangle$ denotes the dark-counts of the signal detector $D_s$ and $\epsilon_b=\varepsilon_d f_s$ is the probability of detecting a probe photon for a decohered atoms in state $|d\rangle$, $\varepsilon_d$, multiplied by the fraction of signal photons, $f_s$, incoherently mapped to state $|d\rangle$ via absorption. The coincidence counts are
\begin{eqnarray}
\langle n_sn_p \rangle & = & \varepsilon_0 q_p\langle n^{in}_c \rangle\times T_s q_s \langle n^{in}_s \rangle +\\ \nonumber
& & (\alpha+\epsilon_b \langle n^{in}_s \rangle) q_p\langle n^{in}_c \rangle\times T_s q_s \langle n^{in}_s \rangle +\nonumber \\
&& T_s q_s\langle n^{in}_s \rangle\langle r_p \rangle +((\varepsilon_0+\epsilon_b) \langle n^{in}_s \rangle+\nonumber \\
&& \alpha)\times q_p \langle n^{in}_c \rangle \langle r_s \rangle + \langle r_p \rangle \langle r_s \rangle.
\end{eqnarray}
Here, we assume that the conditional signal transmission is approximately equal to the mean signal transmission, $T_s$. Note that all terms, except the first, are caused by background sources. The cross-correlation function, neglecting the detectors' dark counts, can be approximated as
\begin{eqnarray}
g^{(2)} (\tau=0)& = & \frac{ \langle n_sn_p \rangle }{ \langle n_s \rangle \langle n_p \rangle } \simeq \frac{1+\beta}{\beta+\langle n^{in}_s \rangle}
\label{eq: g2}
\end{eqnarray}
where $\beta=\frac{\alpha+\epsilon_b \langle n^{in}_s \rangle}{\varepsilon_0}$. When background processes are negligible ($\alpha, \epsilon_b, \langle r_s \rangle, \langle r_p \rangle \ll1 $), the maximum cross-correlation function at $\tau=0$ is simply approximated by $ g^{(2)} \simeq 1/\langle n^{in}_s\rangle$ for $\langle n^{in}_s\rangle<1$. Note that in the regime where $\langle r_{s} \rangle, \langle r_{p} \rangle\ll 1$, the correlation function $g^{(2)}$ is independent of the cavity photon number as both the detection probability and background scale linearly with it. However, the measured $g^{(2)}(\tau=0)$ drops at low cavity photon numbers where probe-part dark counts, $\langle r_p \rangle$, are not negligible compared to the detected cavity mean photon number.
Observed cross-correlation for double-pass signal beam, measured with $\langle n^{in}_c \rangle=4.4$ and $\Omega/2\pi= 2.9$ MHz. The fitted values are $g^{(2)}=4.4(5)$, $\tau_{<}=1.3(3)$ $\si{\micro}$s, and $\tau_{>}=0.5(2)$ $\si{\micro}$s.
To further increase the photon-photon interaction, we carried out an experiment to increase the effective optical density by transmitting the signal through the atomic ensemble twice. The retro-reflected signal is collected by a 90/10 fiber-beam splitter used at the signal input. We simultaneously measure auto-correlations of $g_{ss}^{(2)}=1.6(3)$, $g_{pp}^{(2)}=5.6(1)$ and the cross-correlation as plotted in Fig. <ref>.
Quantum efficiency.
The conditional nondestructive quantum efficiency of detecting a signal photon with mean input photon number $\langle n^{in}_s \rangle\ll1$ can be written as
\begin{eqnarray}
\text{Q} & = & \varepsilon q_p\langle n^{in}_c \rangle \simeq \frac{\langle n_sn_p \rangle- \langle n_p \rangle\langle n_s \rangle}{ \langle n_s \rangle}
\end{eqnarray}
where $\varepsilon$ is the total probability of having a probe photon given a signal photon traveling through the medium. It can be obtained from the asymptotic quantum efficiency and integrating the area under the $g^{(2)}$ function as
\begin{eqnarray}
\varepsilon&=&\frac{\text{Q}}{q_p\langle n^{in}_c \rangle}= \frac{1}{q_p\langle n^{in}_c \rangle(1-\langle n^{in}_s \rangle)} \int{(g^{(2)}(\tau)-1) R_p d\tau} \nonumber \\
&=& \varepsilon_0\frac{\tau_c+\tau_{_{EIT}}}{\tau_c}.
\label{eq: epsilon2}
\end{eqnarray}
The probability $\varepsilon$ is calculated from the slope of the fitted lines in Fig. 4c and is plotted for different control Rabi frequencies in Fig. <ref>. These extracted probabilities agree with theoretical predictions.
The total probability $\varepsilon$ calculated from the slope of the linear fits to the data in Fig. 3. The fitted curve represents the theory using Eq.<ref> with fitted optical density $\mathcal{D}=4(2)$.
Detection probabilities and QND requirements.
The QND requirements can be quantified using the measurement error, $\Delta X$, the transfer coefficient of input signal to meter (probe), $\mathcal{T}_M$, and transfer coefficient of input signal to output signal, $\mathcal{T}_S$ <cit.>. Using the formalism provided by Ralph et al. [Phys. Rev. A 73, 012113 (2006)], one can link the measurement probabilities in the discrete variable (DV) regime and $\mathcal{T}_S$ and $\mathcal{T}_M$ in continuous variable (CV) regime through different fidelity measures. The transfer coefficients in terms of measurement fidelity, $F_M$, and QND fidelity, $F_{QND}$, can be written as
\begin{eqnarray*}
\mathcal{T}_M &= & (\frac{2}{F_M^2} -1)^{-1}\\
\mathcal{T}_S &=&(\frac{2}{F_{QND}^2} - 1)^{-1}
\end{eqnarray*}
\begin{eqnarray*}
F_M & = & P_{11}+P_{01} = \frac{\langle n_p\rangle}{\langle n^{in}_s\rangle}\\
F_{QND} & = & P_{11}+P_{10} = T_s.
\end{eqnarray*}
To estimate the measurement error in the CV regime, the conditional variance of the signal is measured and is compared to the shot-noise limit. In the DV regime, however, as the particle aspect of photons are detected and not the wave aspect, the conditional correlation function, $g^{(2)}_{ss|m}$ (signal auto-correlation function conditioned on detecting a meter photon), can be used instead to quantify the measurement error. In particular, a QND measurement satisfies $g^{(2)}_{ss|m}<1$ (quantum state preparation)and $\mathcal{T}_S+\mathcal{T}_M>1$.
|
1511.00455
|
$^{1}$Department of Physics and Institute of Theoretical Physics and
Astrophysics, Xiamen University, Xiamen 361005, Fujian, China
$^{2}$Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, Anhui, China
$^{3}$Collaborative Innovation Center of Chemistry for Energy Materials,
Xiamen University, Xiamen 361005, Fujian, China
The hard-disk model plays a role of touchstone for testing and developing
the transport theory. By large scale molecular dynamics simulations of this
model, three important autocorrelation functions, and as a result the
corresponding transport coefficients, i.e., the diffusion constant, the
thermal conductivity and the shear viscosity, are found to deviate
significantly from the predictions of the conventional transport theory
beyond the dilute limit. To improve the theory, we consider both the kinetic
process and the hydrodynamic process in the whole time range, rather than
each process in a seperated time scale as the conventional transport theory
does. With this consideration, a unified and coherent expression free of any
fitting parameters is derived succesfully in the case of the velocity
autocorrelation function, and its superiority to the conventional
`piecewise' formula is shown. This expression applies to the whole time
range and up to moderate densities, and thus bridges the kinetics and
hydrodynamics approaches in a self-consistent manner.
05.60.Cd, 51.10.+y,51.20.+d,47.85.Dh
For a system with translation invariance, the transport theory predicts that
the autocorrelation function (ACF) of a physical quantity, denoted by $C(t)$
, generally decays as <cit.>
\begin{equation}
\frac{C(t)}{C(0)}=\left\{
\begin{array}{ll}
e^{-\xi t}, & \text{kinetics stage;} \\
b_{h}t^{-d/2}, & \text{hydrodynamics stage.}%
\end{array}%
\right.
\end{equation}
Here $t$ is the correlation time, $\xi $ is a characterizing constant, $d$
is the dimension of the system, $b_{h}$ is the amplitude of the power-law
decay function. Given $C(t)$, the transport coefficient of the corresponding
physical quantity can be obtained with the Green-Kubo formula <cit.>. The most important ACFs are those of the
velocity, the energy current and the viscosity current, which will be
referred to as the VACF, the EACF and the VisACF in the following.
Nevertheless, the transport theory has not been fully established. On one
hand, the theoretical predictions have not been well verified yet. Though a
lot of numerical studies have been done since the 1970's <cit.>, the results are
not accurate enough to conclude until 2008 <cit.>, Isobe computed
the VACF of the two-dimensional hard-disk fluid and found that the tail is
of the power-law $\sim t^{-1}$ at low densities but logarithmic $\sim (t%
\sqrt{\ln t})^{-1}$ at moderate densities. It implies that the power-law
decay prediction may not be always correct. The logarithmic decay agrees
with the self-consistent mode coupling prediction <cit.>.
However, the transition form $\sim t^{-1}$ to $\sim (t\sqrt{\ln t})^{-1}$
has not been characterized. For the EACF and VisACF, up to now simulation
results are rare and those allow for drawing a conclusion still lack. On the
other hand, dividing the time dependence of $C(t)$ into the kinetics and
hydrodynamics stages is an expedient measure. To quantitatively calculate
the transport coefficient, a coherent and unified expression is
indispensable. Particularly, for a two dimensional system the power-law tail
of $C(t)$ makes the transport coefficient diverge in the thermodynamical
limit, but for a real system, the measured coefficient should be finite. To
predict the coefficient theoretically, one needs to evaluate the influence
of the long-time tail to reveal when it can be ignored comparing with the
kinetic contribution and when it becomes dominant<cit.>. This requires to know the crossover time from the
kinetics stage to the hydrodynamics stage. Numerically, parameter fitting
<cit.> may allow one to construct a unified $C(t)$
within the time period investigated, but it is risky to extend it
out or to use it in other parameter regimes.
In this work we revisit the hard-disk fluid. First, by large scale
simulations, we calculate the three ACFs and show their deviations from the
theoretical predictions. We then derive a unified expression for the VACF.
The model consists of $N$ disks of unitary mass $m=1$ moving in an $%
L_{x}\times L_{y}$ rectangular area with the periodic boundary conditions.
The system is evolved with the event-driven algorithm <cit.> at
the dimensionless temperature $T=1$ (the Boltzmann constant is set to be $%
k_{B}=1$). The disk number density is fixing at $n=N/(L_{x}L_{y})=0.01$
throughout and the disk diameter, $\sigma $, is adopted to control the
packing density $\phi =n\pi \sigma ^{2}/4$ (referred to as the density for
short in the following). Three cases, $\sigma =2$, $4$, and $6$
corresponding to $\phi \approx 0.03$, $0.13$, and $0.28$, respectively, are
studied intensively. As a reference, the crystallization density is $\phi
=0.71$, hence our study covers the moderate density regime. Applying the
Enskog formula with the first Sonine polynomial approximation <cit.>, the diffusion coefficient $D$, the thermal
conductivity $\lambda $, the sheer viscosity $\eta $, and the sound speed $%
u_{s}$ are, respectively, $D=13.4$, $5.70$, $2.76$, $\lambda =0.59$, $0.35$
, $0.36$, $\eta =0.14$, $0.077$, $0.063$, and $u_{s}=1.5$, $1.8$, $2.7$, for
the three densities.
The VACF is defined as $C_{u}(t)=$ $\langle u_{x}(t)u_{x}(0)\rangle $, where
$u_{x}(t)$ is the $x$-component of the velocity of a tagged disk. Figure
1(a)-(c) show the simulated results obtained with $10^{10}$ ensemble samples
for $L_{x}=L_{y}=2000$ ($N=40000$). The time range free from the finite-size
effects is $0\leq t<t_{f}=L_{x}/(2u_{s})$ <cit.>. For
the three densities, $t_{f}=667$, $556$, and $370$, respectively. In this
time range, the initial exponential decaying stage and the long-time tail
can be observed in all the three cases. The hydrodynamics prediction $%
C_{u}(t)/C_{u}(0)=[8\pi (D+\nu )n]^{-1}t^{-1}$ <cit.> is also plotted for comparison,
where $\nu =\eta m/n$ is the viscosity diffusivity. It can be seen that the
predicted $\sim t^{-1}$ tail is close to the simulation result, but as the
density increases, the deviation grows. The diffusion coefficients
calculated following the Green-Kubo formula, $D(t)=\int_{0}^{t}C_{u}(t^{%
\prime })dt^{\prime }$, are shown in Fig. 1(d).
(Color online) (a)-(c) The VACFs obtained by, respectively,
simulations (black solid lines), the unified formula Eq. (5) (green dashed
lines), and the hydrodynamics theory (red dotted lines). (d) The diffusion
coefficients calculated with the VACFs obtained by simulations (black solid
lines) and by the unified formula Eq. (5). For the latter, the green dashed
lines are for the results with the kinetic transport coefficients given by
the Enskog equation and the magenta dash-dotted lines are for the results
with the corrected coefficients.
Calculating the EACF $C_{J}(t)=\langle J_{x}(t)J_{x}(0)\rangle $ is $N$
times harder than calculating the VACF. Here $J_{x}(t)=\sum_{i}j_{x}^{i}(t)$
, where $j_{x}^{i}(t)=|\mathbf{v}^{i}|^{2}v_{x}^{i}$ is the $x$-component of
the energy current of the $i$th disk. For a given simulation run we can
obtain $N$ ensemble samples for calculating $C_{u}(t)$ as every disk can be
taken as the tagged disk but only one for calculating $C_{J}(t)$ because the
total current, $J_{x}$, involves the contributions of all the disks <cit.>. This is the reason why the hydrodynamics prediction of the
EACF has not been conclusively tested. To decrease the simulation difficulty
we consider a smaller size, i.e., $L_{x}\times L_{y}=3000\times 400$ $%
(N=12000)$. Correspondingly, $t_{f}=1000$, $833$, and $535$ for the three
densities. Figure 2(a)-(c) show the results of $C_{J}(t)$ calculated with $%
10^{9}$ ensemble samples. For $\sigma =2$, a perfect $\sim t^{-1}$ tail is
observed, but the value of $C_{J}(t)$ at the tail is one time larger than
the hydrodynamics prediction <cit.> that $C_{J}(t)/C_{J}(0)=[4\pi (\eta /m+\lambda
/2k_{B})]^{-1}t^{-1}$. For $\sigma =4$ and $6$, the EACF shows a multistage
decaying behavior – after the initial exponential decaying stage there
appears another fast decaying stage, before a power-law tail slower than $%
\sim t^{-1}$ follows. Figure 2(d) shows the corresponding thermal
conductivity calculated following the Green-Kubo formula $\lambda (t)=\frac{1%
}{k_{B}T^{2}L_{x}L_{y}}\int_{0}^{t}C_{J}(t^{\prime })dt^{\prime }$.
(Color online) (a)-(c) The EACFs obtained by simulations (black
solid lines) and by the hydrodynamics theory (red dotted lines). (d) The
thermal conductivity calculated based on the EACFs obtained by simulations.
Calculating the VisACF $C_{vis}(t)=\langle J_{vis}(t)J_{vis}(0)\rangle $
suffers from the same difficulty. Here $J_{vis}(t)=%
\sum_{i}u_{x}^{i}(t)u_{y}^{i}(t)$. Taking $L_{x}=L_{y}=1000$ $(N=10^{4})$
and $\sigma =4$, we show in Fig. 3(a) the VisACF calculated with $10^{10}$
ensemble samples. Though for $t<t_{f}=278$ the VisACF decays fast for
several orders, it is still uncertain if a power-law tail follows. Indeed,
the VisACF may drop to be negative from $t\approx 70$ to $100$. The demanded
huge amount of samples make the computation so difficult that we can only
provide the results for one case ($\sigma =4$) as an example. The
hydrodynamics prediction $C_{vis}(t)/C_{vis}(0)=(32\pi )^{-1}[m/\eta +(\eta
/m+\lambda /2k_{B})^{-1}]t^{-1}$ <cit.> is also plotted for comparison. Figure 3(b)
shows the shear viscosity by following the Green-Kubo formula $\eta
(t)=m\int_{0}^{t}C_{vis}(t^{\prime })dt^{\prime }$.
(Color online) (a) The simulation result of the VisACF and (b) the
corresponding sheer viscosity for $\protect\sigma =4$.
Next, we derive the unified expression for the VACF. Our key consideration
is that the VACF is governed by two physical processes simultaneously. One
is collisions of the tagged disk with other surrounding disks, referred to
as the kinetic process, through which $C_{u}(0)$ will be transferred to
other disks from the tagged disk. Let $C_{u}^{k}(t)$ denote the portion of $%
C_{u}(0)$ that has not been transferred at time $t$. Following the kinetics
theory, it decays exponentially: $C_{u}^{k}(t)=C_{u}(0)\exp ({-\frac{k_{B}T}{%
mD^{k}}t})$ <cit.>,
where $D^{k}$ is the kinetic diffusion constant. Meanwhile, it is possible
for the transferred portion to feedback to the tagged disk by ring collisions
<cit.>, which is referred to as the hydrodynamic process. The amount
returns to the tagged disk at time $t$, denoted by $C_{u}^{h}(t)$,
contributes to the hydrodynamics diffusion constant $D^{h}$. The VACF is
thus a sum of these two portions,
\begin{equation}
\end{equation}
and the diffusion constant is divided into the kinetic and hydrodynamic
parts as $D=D^{k}+D^{h}$ accordingly. To obtain $C_{u}^{h}(t)$, it is
necessary to investigate how $C_{u}(0)$ is delivered to the surroundings.
This reduces to investigating the relaxation process of the momentum $%
\mathbf{p}_{c}=m\mathbf{u}_{c}$ initially carried by the tagged disk, which
can be approached with the spatiotemporal correlation function <cit.>
\begin{equation}
c(\mathbf{r},t)=\frac{\langle \mathbf{p}_{c}\cdot \mathbf{p}(\mathbf{r}%
,t)\rangle }{\langle |\mathbf{p}_{c}|^{2}\rangle }+\frac{n}{N-1}.
\end{equation}
Here $\mathbf{p}(\mathbf{r},t)$ is the momentum density of the system. It is
found that $c(\mathbf{r},t)$ is axisymmetric with respect to $\mathbf{r}=0$;
it has one center peak surrounded by a `crater' (see Fig. 4(a)-(b) for the
intersection of $c(\mathbf{r},t)$ with $y=0$) and the center peak can be
well fitted by the Gaussian function $c^{center}(\mathbf{r},t)=\frac{a_{\nu }%
}{4\pi \widetilde{\nu }t}\exp ({-\frac{r^{2}}{4\widetilde{\nu }t}})$ with $%
a_{\nu }=1/2$ and $\widetilde{\nu }=14.3$, $8.3$, and $7.9$ for $\sigma =2$,
$4$, and $6$, respectively. The function $c(\mathbf{r},t)$ gives the portion
of $C_{u}(0)$, that transfers to a unit area centering $\mathbf{r}$ at time $%
t$. There are $n$ disks on average in this area, and each of them carries a
portion, i.e., $c(\mathbf{r},t)/n$, of $C_{u}(0)$. Suppose that the tagged
disk appears in this area with the probability $\rho (\mathbf{r},t)$, then
on average the portion of $C_{u}(0)$ it carries is $\rho (\mathbf{r},t)c(%
\mathbf{r},t)/n$. The total amount carried by it is therefore $%
C_{u}^{h}(t)=(1/n)C_{u}(0)\int c(\mathbf{r},t)\rho (\mathbf{r},t)d\mathbf{r}$
(Color online) (a)-(b) The intersection of $c(\mathbf{r},t)$ and
(c)-(d) the intersection of $\protect\rho (\mathbf{r},t)$ at $t=100$ and $%
300 $ (black solid lines). The red dotted line in each panel is the best
Gaussian fitting to the center peak of $c(\mathbf{r},t)$ or $\protect\rho (%
\mathbf{r},t)$. $\protect\sigma =4$.
The probability function $\rho (\mathbf{r},t)$ can be measured directly by
tracing the tagged disk. It is found to overlap perfectly with a Gaussian
function (see Fig. 4(c)-(d) for its intersection) with a time dependent
diffusion coefficient, i.e., $\rho (\mathbf{r},t)=\frac{1}{4\pi D(t)t}\exp ({%
-\frac{r^{2}}{4D(t)t}})$. As $\rho (\mathbf{r},t)$ decays exponentially as $%
r^{2}$, we can replace $c(\mathbf{r},t)$ with $c^{center}(\mathbf{r},t)$ in
the integrand for calculating $C_{u}^{h}(t)$. With this simplification we
\begin{equation}
C_{u}^{h}(t)/C_{u}(0)=a_{\nu }[4\pi (D(t)+\widetilde{\nu })n]^{-1}t^{-1}.
\end{equation}
Noting that Fig. 4 represents the case that the hydrodynamics effects become
completely dominant. For $t>100$, $C_{u}^{k}(t)$ has decayed to a negligibly
small value($<10^{-8}$ with $D^{k}=5.70$), implying that $C_{u}(0)$ has
transferred to the surrounding disks almost completely. Equation (4) thus
characterizes the situation at large times. At short times, the portion of $%
C_{u}(0)$ the hydrodynamics process accounts for is $C_{u}(0)[1-\exp (-\frac{%
k_{B}T}{mD^{k}}t)]$; Assuming that $c(\mathbf{r},t)$ for this portion has
the same structure as shown in Fig. 4(a)-4(b), it is straightforward to have
\begin{equation}
C_{u}^{h}(t)/C_{u}(0)=a_{\nu }[1-\exp (-\frac{k_{B}T}{mD^{k}}t)][4\pi (D(t)+%
\widetilde{\nu })n]^{-1}t^{-1}.
\end{equation}
This extended expression applies in both the kinetics and the hydrodynamics
The parameter $a_{v}$ and $\widetilde{\nu }$ can be connected to the
properties of the hydrodynamic modes analytically <cit.>. Let $n(\mathbf{%
r},t)$ and $\mathbf{u}(\mathbf{r},t)$ be the disk number density and the
velocity density, we have $\mathbf{p}(\mathbf{r},t)=mn(\mathbf{r},t)\mathbf{u%
}(\mathbf{r},t)=m\mathbf{j}(\mathbf{r},t)$, considering the hydrodynamics
assumption <cit.> that local deviations of hydrodynamic variables
from their average values are small. Here $\mathbf{j}(\mathbf{r},t)$ is the
local disk current. Applying the hydrodynamics analysis to solve the
linearized conservation laws for the disk number, the energy, and the
momentum with initial conditions of $\delta $-function impulses of $\delta (%
\mathbf{r})\Delta n$, $\delta (\mathbf{r})\Delta T$ and $\delta (\mathbf{r})%
\mathbf{p}_{c}$, we obtain $c(\mathbf{k},t)=(k_{y}^{2}/k^{2})\exp (-\nu
k^{2}t)$ in the wave-vector space, where $\Delta n$ and $\Delta T$ represent
the deviation of the disk density and the temperature induced by the tagged
particle, and $\mathbf{k}$ is the wave vector in the Fourier space. With a
rough estimation of $k_{y}^{2}/k^{2}\sim 1/2$, it appears $c(\mathbf{k}%
,t)=(1/2)\exp (-\nu k^{2}t)$ and gives $c(\mathbf{r},t)=(1/8\pi \nu t)\exp
(-r^{2}/4\nu t)$ in the real space. The expression of $c(\mathbf{r},t)$
implies $a_{v}=1/2$ and $\widetilde{\nu }=\nu $. With this connection,
Eq. (4) is exactly the same as that of the hydrodynamics theory.
In principle, $\nu $ is time-dependent according to the hydrodynamics
theory, but based on our numerical observation of the relaxation of $%
c^{center}(\mathbf{r},t)$ and the fact that $\eta $ converges in time [see
Fig. 3(d)], it can be assumed to be a time-independent constant up to
moderate densities. Previous numerical studies using the Helfand-Einstein
formula have also shown that the shear viscosity does not depend on the
system size either <cit.>, which supporting the
constant $\nu $ assumption as well.
Inserting $C_{u}^{h}(t)$ into the Green-Kubo formula, we have
\begin{equation}
D^{h}(t)=\int_{0}^{t}C_{u}^{h}(t^{\prime })dt^{\prime }.
\end{equation}
It is interesting to note that the self-consistent solutions <cit.>, i.e., $C_{u}^{h}(t)/C_{u}(0)=\sqrt{1/16\pi n}(t\sqrt{\ln
(t)})^{-1}$ and $D^{h}(t)=\sqrt{k_{B}T\ln (t)/4\pi mn}$, are asymptotic
solutions of Eq. (5) and (6) in the long-time limit where $D^{h}(t)\gg
D^{k}+\nu $, i.e., $t>\exp [(4\pi mn/k_{B}T)(D^{k}+\nu )^{2}]$. Using the
Enskog results of $D^{k}$ and $\nu $, it can be estimated that this time
scale is about $10^{23}$, $10^{10}$, and $10^{4}$, respectively, for $\sigma
=2$, $4$, and $6$. During the transition process which may contribute a
dominant part to the diffusion constant, the self-consistent asymptotic
solutions are not exact.
In order to solve the coupled equations (5) and (6) accurately, we turn to
the iterative algorithm: We set $D^{h}(t)=0$ as the first trial solution and
substitute it into Eq. (5) to get $C_{u}^{h}(t)$, then put it into Eq. (6)
to get the next trial solution of $D^{h}(t)$, and so on. In general if $%
D^{h}(t)$ increases, $C_{u}^{h}(t)$ will decrease and make $D^{h}(t)$
decrease, and vice versus, hence the convergence of the iteration is
guaranteed. Indeed, usually the solutions converge after only several
iterations. The predicted VACFs [Fig.1(a)-(c)] and the corresponding
diffusion coefficients [Fig.1(d)] agree with the simulation results quite
well, except that the diffusion coefficient $D(t)$ show a shift from the
simulation result as the density increases.
This deviation should be induced by the inaccuracy of the kinetics transport
coefficients that we have employed. With the simulation data of $C_{u}(t)$,
we can estimate the kinetics diffusion constant $D^{k}$. The kinetics
process plays a role mainly before the time, denoted as $\tau $, at which $%
C_{u}(t)$ turns from the exponential decay to the followed tail. For
example, for $\sigma =2$, $\tau \approx 110$, at which $C_{u}^{k}$ has
decayed to $C_{u}^{k}(\tau )/C_{u}(0)\sim 10^{-4}$. Truncating the
Green-Kubo integration at $\tau $, we have $D(\tau )=13.97$, $6.40$, and $%
2.85$ for $\sigma =2$, $4$, and $6$, respectively. These values can be
considered as the upper bound of $D^{k}$. Subtracting $D^{h}(\tau )$ from
it, we get the estimated $D^{k}$; i.e., $D^{k}=13.61$, $5.50$, and $2.35$,
correspondingly. These values are close to the Enskog approximations, within
maximally $13\%$ errors. Meanwhile, the relation $\widetilde{\nu }=\nu $ is
also not accurate; it is a result of ignoring the anisotropic feature of the
momentum diffusion <cit.>. More properly, $\widetilde{\nu }$ measured by
the direct simulation should be employed to characterize the momentum
diffusion instead of $\nu $. With these corrections of $D^{k}$ and $\nu $,
the shift of $D(t)$ can be well suppressed [see Fig. 1(d)].
Similarly, we can estimate the upper bounds for the heat conductivity. From
Fig. 2(d) we have the heat conductivity $\lambda \leq 0.580$, $0.258$, and $%
0.139$ for $\sigma =2$, $4$, and $6$, respectively, which deviates at least $%
2\%$, $36\%$, and $160\%$ from the Enskog approximations. Therefore, the
Enskog equation is relatively precise for the kinetic diffusion constant,
but lacks accuracy for the heat conductivity and the sheer viscosity at
higher densities.
With the unified expression of $C_{u}(t)$, we can estimate the hydrodynamics
contribution to the diffusion constant to systems of macroscopic sizes. For
example, the average distance between two neighboring molecules in the air
is about $10^{-9}$ meter, implying that if our model has a macroscopic size,
say one centimeter, we have $L_{x}$, $L_{y}\sim 10^{8}$. For such a size,
the time a disk diffuses freely without being influenced by the boundaries
is $t\sim L_{x}/(2u_{s})\sim 10^{7}$. Taking this time as the truncation
time of integration in Eqs. (5) and (6), our iteration algorithm gives $%
D^{h}(t)/D^{k}\approx 0.15$, $0.5$, and $10$ for $\sigma =2$, $4$,and $6$,
respectively, suggesting that in a dilute system it is the kinetics
contribution that dominates, but as the density increases, the hydrodynamics
contribution increases dramatically and the kinetics contribution turns to
be negligible.
In summary, beyond the dilute limit, the accuracy of the hydrodynamics
theory is not sufficient in describing the ACFs at least in the transient
stage that is essential for calculating the transport coefficients. For the
VACF, the numerically observed tail is between $\sim t^{-1}$ and $\sim (t%
\sqrt{\ln (t)})^{-1}$. For the EACF, we have evidenced the power-law tail
but the exponent agrees with the hydrodynamics prediction only in a very
dilute system. As the density increase, a multistage decaying phenomenon is
observed, and the long-time tail is slower than $\sim t^{-1}$. The VisACF
decays much faster than the VACF and the EACF. The long-time tail has not
been observed in our example. In addition, we have estimated the upper
bounds of transport coefficients using the simulated ACFs, and reveal that
the Enskog equation generally used for approximating the kinetics transport
coefficients need be improved particularly at higher densities.
For the VACF, our intuitive representation of the ring-collision mechanism
and the iterative algorithm lead us to a unified and coherent expression
valid in the whole time range. The key point is to distinguish the kinetics
and the hydrodynamics processes and investigate them respectively over the
whole time range. Particularly, we emphasize that the hydrodynamic
contribution at short times should not be ignored. This is different from
the traditional treatments that divide the relax process into separated
stages. Extension of our method to the EACF and VisACF is open.
Very useful discussions with J. Wang, Y. Zhang and Dahai He are gratefully
acknowledged. This work is supported by the National Natural Science
Foundation of China (Grant No. 11335006), and the NSCC-I computer system of
theorySP J. P. Hansen and I. R. McDonald, Theory of Simple
Liquids, 3rd ed. (Academic, London, 2006).
Lebowitz F. Bonetto, J. L. Lebowitz, and L. Rey-Bellet,
Fourie's Law: A Chanllenge to Theorists, Mathematical Physics 2000
(Imperial College Press, London, 2000).
Leener P. Reusibois and M. de Leener, Classical Kinetic
Theory of Fluids Wiley-Interscience, New York, 1977.
Alder B. S. Alder and T. E. Wainwright, Phys. Rev. Lett. 18
, 988 (1967)
Alder2 B. S. Alder and T. E. Wainwright, Phys. Rev. A 1
,18 (1970).
Alder3 B. S. Alder, D. M. Gass, and T. E. Wainwright, J. Chem.
Phys. 53, 3813 (1970).
wain-tlnt T. E. Wainwright, B.J. Alder, and D. M. Gass, Phys. Rev.
A 4, 233 (1971).
Dorfman1 S. R. Dorfman and E. G. D. Cohen, Phys. Rev. Lett.
25, 1257(1970)
Dorfman2 J. R. Dorfman and E. G. D. Cohen, Phys. Rev. A 6
, 776 (1972)
Dorfman3 S. R. Dorfman and E. G. D. Cohen, Phys. Rev. A.12, 292
Ernst M. H. Ernst, E. H. Hauge, and J. M. J. van Leeuwen, Phys.
Rev. A 4, 2055 (1971)
tlnt D. Forster, D. R. Nelson, and M. J. Stephen, Phys. Rev. A
16 732 (1977).
cv-longtail J. J. Erpenbeck and W. W. Wood, Phys. Rev. A 26
, 1648 (1982); Phys. Rev. A 32, 412 (1985).
review-l-t-coupling Y. Pomeau and P. Reuibois, Physics Reports,
19, 63 (1975).
high-density J. W. Dufty, Mol. Phys. 100, 2331, (2002);
J. W. Dufty and M. H. Ernst, ibid, 102, 2123 (2004).
Kubo R. Kubo, M. Toda, and N. Hashitsume, Statistical
Physics II: Nonequilibrium Statistical Mechanics (Springer, New York, 1991);
Andrieux D. Andrieux and P. Gaspard, J. Stat. Mech. P02006 (2007).
formula-2d R. Garcia-Rojo, S. Luding, and J. J. Brey, Phys. Rev. E
74, 061305 (2006).
2d-tailnotimport S. Viscardy and P. Gaspard, Phys. Rev. E
68, 041204 (2003).
Erpenbeck J. J. Erpenbeck and W. W. Wood, Phys. Rev. A 26
, 1648(1982)
pre2008 M. Isobe, Phys. Rev. E 77, 021201 (2008).
pre2015 S. Bellissima, M. Neumann, E. Guarini, U. Bafile, and F.
Barocchi, Phys. Rev. E 92, 042166 (2015).
cal D. C. Rapaport, J. Comput. Phys. 34, 184 (1980)
Gass D. M. Gass, J. Chem. Phys. 54, 1898 (1971).
pl-eskog J. J. Erpenbeck and W. W. Wood, Phys. Rev. A 43,
4254 (1991).
chenfinite S. Chen, Y. Zhang, J. Wang, and H. Zhao, Phys. Rev. E
89, 022111 (2014)
chen2016 S. Chen, Y. Zhang, J. Wang, and H. Zhao, J. Stat. Mech.,
033205 (2016)
zhao2006 H. Zhao, Phys. Rev. Lett. 96, 140602 (2006)
chendiffusion S. Chen, Y. Zhang, J. Wang, and H. Zhao, Phys. Rev.
E 87, 032153 (2013)
SM See the Supplied Materials with this submission.
|
1511.00459
|
Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
Earth-Life Science Institute, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
No Hadean rocks have ever been found on Earth's surface except for zircons—evidence of continental crust, suggesting that Hadean continental crust existed but later disappeared. One hypothesis for the disappearance of the continental crust is excavation/melting by the Late Heavy Bombardment (LHB), a concentration of impacts in the last phase of the Hadean eon. In this paper, we calculate the effects of LHB on Hadean continental crust in order to investigate this hypothesis. Approximating the size-frequency distribution of the impacts by a power-law scaling with an exponent $\alpha$ as a parameter, we have derived semi-analytical expressions for the effects of LHB impacts. We calculated the total excavation/melting volume and area affected by the LHB from two constraints of LHB on the moon, the size of the largest basin during LHB, and the density of craters larger than 20 km. We also investigated the effects of the value of $\alpha$. Our results show that LHB does not excavate/melt all of Hadean continental crust directly, but over 70% of the Earth's surface area can be covered by subsequent melts in a broad range of $\alpha$. If there have been no overturns of the continental crust until today, LHB could be responsible for the absence of Hadean rocks because most of Hadean continental crust is not be exposed on the Earth's surface in this case.
§ INTRODUCTION
Hadean rocks have not been found on Earth until today, and the age of the oldest rock is about 4.0 Ga. No continental crust may have existed on the Hadean Earth. However, some zircons that formed during the Hadean eon were found in Jack Hills sedimentary rocks <cit.>. Since zircons are formed by igneous activity at the same time as granite, this discovery suggests that the Hadean continental crust existed but later disappeared. The disappearance of the continental crust could be accounted for by geological activity such as reworking or plate tectonics, and erosion of the crusts <cit.>. Another possibility is excavation/melting of the continental crust by the Late Heavy Bombardment (LHB), a concentration of impacts considered to have existed in the last phase of the Hadean eon. In this paper, we calculate the effects of LHB on Hadean continental crust in order to investigate this hypothesis.
§.§ Late Heavy Bombardment
A classic scenario of LHB has been based on the fact that radiometric dates of lunar basins' impact melts were concentrated at 3.9 Ga <cit.>. In this model, about 15 lunar basins are considered to be formed between 3.9 and 3.8 Ga <cit.>. <cit.> argued from Ar-geochronology of lunar meteorites that the concentration at 3.9 Ga is not required and <cit.> claimed that the decrease of lunar crater density is just the tail of the planetesimal accretion. The most recent model proposed a sawtooth-like timeline of impact flux <cit.>. If LHB was caused by a disturbance in the ancient main belt asteroids, the impact flux decreased exponentially after the onset of LHB <cit.>. This exponential curve fits the lunar crater density curve very well if the disturbance occurred at 4.1 Ga <cit.>. It is adjusted in order to explain the accreted mass of highly siderophile elements (HSE) of the moon after the moon formation giant impact. <cit.> argued that the cause of LHB was the migration of Jupiter and other giant planets predicted in the Nice model. The migration of the giant planets moved some resonances and scattered the main belt asteroids and outer comets.
§.§ Previous works
The estimate for coverage of the Earth's surface by impact craters depends on the impactors' size-frequency distribution (SFD). The estimates with unconstrained SFD predicted diverse results <cit.>. Furthermore, stochastic one or two huge impacts could give considerable effects <cit.>.
Recently, using a more constrained SFD corresponding to the current main belt asteroids with the total LHB mass of $2\times10^{23}$ g, <cit.> showed that LHB may not have melted all of Hadean continental crust. They computed 3D temperature distributions of the crust using an analytical shock-heating model with effects of impact melt generation, uplift, and ejecta heating. The result is that 1.5–2.5% of the upper 20 km of the crust was melted during LHB, and only 0.3–1.5% was melted through LHB period. They also indicated that 5–10% of the Earth's surface area was covered by over 1 km depth of impact melt sheets, and the entire surface was covered by impact ejecta close to 1 km deep.
On the other hand, <cit.> suggested that Hadean impacts could explain the absence of early terrestrial rocks based on the sawtooth-like timeline, with the current main belt asteroids' SFD. The key point of <cit.> is the “stochastic” nature of LHB. They showed that the melting volume deeply depended on whether very large impactors hit Earth or not. Another key point is the effect of “impact-induced decompression” and subsequent adiabatic melting of rising material in the mantle, which increases the total melting volume. They claimed that these flood melts from under the crust emerge when the impactors' diameters were greater than 100 km. The melts flowed on the Earth's surface like spherical caps. The melt sheets' diameters were about 20–30 times as large as that of the impactors, for an assumed melt thickness of 3 km. The result is that 70–100% of the Earth's surface area was covered by the melt sheets since 4.15 Ga, and 400–600% was covered during the period 4.5–4.15 Ga. This value of the total melting area on the Earth's surface is about ten times as large as that of <cit.>.
§.§ Aims of this work
Because these previous works were carried out based on respective model of the impactor's SFD and the effects of impact, the results were inconsistent with each other. This work is going to investigate the effects of the slope of the impactor's SFD, which is the main new contribution of this work.
The SFD of LHB impactors is still controversial. Some studies claim that the lunar craters' SFD is consistent with the current main belt asteroids' SFD <cit.>. Figure <ref> shows that the current main belt asteroids' SFD can be approximated as $\alpha=1.71$, where (minus) $\alpha$ is the power index of the mass-frequency distribution <cit.>. On the other hand, analytical theory of the evolution of SFD in a collision cascade predicted that $\alpha=1.83$ <cit.>.
Corresponding to these uncertainties in the LHB impactors' SFD, we here consider a wide variety of SFDs. In order to reveal intrinsic physics more clearly, we approximate the SFD as a power-law function with a broad range of the power index $\alpha$, rather than adopting a single more detailed SFD.
The number of impactors heavier than $m$, $N_{\rm sfd}(>m)$, is defined as
\begin{equation}
\frac{\mathrm{d} N_{\rm sfd}}{\mathrm{d}m}=Am^{-\alpha},
\label{dN(m)}
\end{equation}
where $A$ is a proportional constant. Then,
\begin{eqnarray}
N_{\rm sfd}\!\!\!\!&=&\!\!\!\!\int_{m}^{\infty} Am'^{-\alpha}dm' \nonumber \\
\label{N(m)}
\end{eqnarray}
when $\alpha>1$ <cit.>.
Current main belt asteroids' SFD and its power-law approximation Red dots are derived from observations <cit.>. The bin width increases by a factor of 0.5. The black line is a power-law approximation of these plots and $\alpha =1.71$. We transformed the SFD into the mass-frequency distribution as the asteroids' densities are $\rho_{\rm i}=2.6$ g/cm$^{3}$; the value of density is consistent with the total mass of the main belt asteroids <cit.>.
We estimate the effects of LHB in following way. In Section 2, we derive the semi-analytical expressions for the effects of LHB impacts. We estimate the total excavation/melting volume and area by integrating the effects of individual impacts, assuming a power-law SFD. In Section 3, we evaluate total effects with a fixed total mass of impactors and the two lunar constraints—the maximum size of basin formed during LHB and the small craters' density. We also investigate the dependence of the power index $\alpha$ of SFD. In Section 4, we discuss whether LHB can explain the absence of Hadean rocks. Then, we compare our results with those of the previous works for a nominal value of $\alpha =1.61$. We also discuss the validity of our assumptions and models. In Section 5, we summarize this paper. Appendix <ref> is devoted to the calculation of the value of $\alpha$ which fulfills both the lunar constraints.
§ BASIC METHODS
§.§ Effects of a single impact
We consider a single impact causes direct excavation/melting and subsequent excavation/melting. Formation of transient crater and melting by shock heating cause “direct excavation/melting,” while formation of final crater and uplifted or excavated molten rock spreading on and beyond the final crater cause “ subsequent excavation/melting,” respectively. In this work, these effects are denoted $V_{\rm xcav}$, $V_{\rm melt}$, $S_{\rm xcav}$ and $S_{\rm melt}$ for direct effects, and $S_{\rm xcav,f}$ and $S_{\rm melt,f}$ for subsequent effects (Fig. <ref>). The volume of evaporated rocks would be sufficiently small so that we did not include it in this study.
We assume the direct excavation volume, $V_{\rm xcav}$, is the volume of the transient crater <cit.>,
\begin{eqnarray}
V_{\rm xcav}\!\!\!\!&=&\!\!\!\!0.146\left(\frac{gL}{v^2}\right)^{-0.66}\biggl(\frac{\rho_{\rm t}}{m}\biggr)^{-1}\mathrm{sin}^{1.3}\theta \nonumber \\
&=&\!\!\!\!0.127\rho_{\rm i}^{0.22}\rho_{\rm t}^{-1}v^{1.3}g^{-0.66}\mathrm{sin}^{1.3}\theta m^{0.78},
\label{Vxcav}
\end{eqnarray}
where $\rho_{\rm i}$ and $\rho_{\rm t}$ are the densities of the impactor and the crust, and $v$, $\theta$, $L$, $m$, and $g$ are the impact speed and angle, impactor's diameter and mass, and gravity, respectively. Eq. (<ref>) is estimated from impact experiments <cit.> and is consistent with recent computer simulations. This volume includes the crust not excavated but displaced, and the depth of the transient crater would be deeper than the thickness of Hadean continental crust. So, the excavated crust volume would be overestimated in this work.
According to <cit.>, the diameter of the transient crater $D_{\rm t}$ is
\begin{eqnarray}
D_{\rm t}\!\!\!\!&=&\!\!\!\!1.44\left(\frac{gL}{v^2}\right)^{-0.22}\biggl(\frac{\rho_{\rm t}}{m}\biggr)^{-1/3} \nonumber \\
&=&\!\!\!\!1.37\rho_{\rm i}^{0.22/3}\rho_{\rm t}^{-1/3}v^{0.44}g^{-0.22}m^{0.26}.
\label{Dtm}
\end{eqnarray}
So the direct excavation (circular) area on the Earth's surface (i.e. horizontal cross section of the direct excavation region) is
\begin{eqnarray}
S_{\rm xcav}\!\!\!\!&=&\!\!\!\!1.63\left(\frac{gL}{v^2}\right)^{-0.44}\biggl(\frac{\rho_{\rm t}}{m}\biggr)^{-2/3} \nonumber \\
&=&\!\!\!\!1.48\rho_{\rm i}^{0.44/3}\rho_{\rm t}^{-2/3}v^{0.88}g^{-0.44}m^{0.52}.
\label{Sxcav}
\end{eqnarray}
The direct melting volume is
\begin{equation}
V_{\rm melt}=0.42\left(\frac{v^{2}}{\epsilon_{\rm m}}\right)^{0.84}\left(\frac{m}{\rho_{\rm t}}\right)\mathrm{sin}^{1.3}\theta,
\label{Vmelt}
\end{equation}
where $\epsilon_{m}$ is the specific internal energy of the target <cit.>. The direct melting volume is proportional to the impactors' mass. When the depth of the melting region is deeper than the thickness of the crust, $h$, we use the following equation in place of Eq. (<ref>).
\begin{equation}
V_{\rm melt,h}=\pi\left\{\left(\dfrac{3V_{\rm melt}}{2\pi}\right)^{2/3}h-\dfrac{h^{3}}{3}\right\}.
\label{Vmelth}
\end{equation}
The melting region's shape is considered to be a hemisphere. From Eq. (<ref>), the direct melting area on the Earth's surface (i.e. horizontal cross section of the direct melting region) is
\begin{equation}
S_{\rm melt}=\pi\left(\dfrac{3V_{\rm melt}}{2\pi}\right)^{2/3}=1.08\left( \frac{v^{2}}{\epsilon_{\rm m}}\right)^{0.56}\left(\frac{m}{\rho_{\rm t}}\right)^{2/3}\mathrm{sin}^{2.6/3}\theta.
\label{Smelt}
\end{equation}
Equations (<ref>), (<ref>) and (<ref>) implicitly assume that the target crusts have no geothermal gradient and a homogeneous initial temperature of 0$^\circ$C.
Then, we consider the subsequent effects. As craters collapse from transient craters to final craters due to gravity, their diameters become larger. There is a relationship between these diameters <cit.>,
\begin{equation}
D_{\rm t}=(D_{\rm c}^{0.15}D_{\rm f}^{0.85}) /1.2,
\label{Dt}
\end{equation}
where $D_{\rm t}$, $D_{\rm f}$, and $D_{\rm c}$ are the transient crater's diameter, final crater's diameter, and critical diameter between simple and complex craters, respectively. The final crater's diameter and the excavation area including gravitational collapse are
\begin{equation}
D_{\rm f}=1.79\rho_{\rm i}^{0.22/2.55}\rho_{\rm t}^{-0.1/2.55}v^{0.44/0.85}g^{-0.22/0.85}D_{\rm c}^{-0.15/0.85}m^{0.26/0.85},
\label{Dfm}
\end{equation}
\begin{equation}
S_{\rm xcav,f}=2.52\rho_{\rm i}^{0.44/2.55}\rho_{\rm t}^{-4/2.55}v^{0.88/0.85}g^{-0.44/0.85}D_{\rm c }^{-0.3/0.85}m^{0.52/0.85}.
\label{Sxcavf}
\end{equation}
Excavated or uplifting melts spread around the final crater <cit.>. We assume the melting area including the area covered by melts, $S_{\rm melt,f}$, is equal to $S_{\rm xcav,f}$. When the impactor is larger than 100 km, the melting area including the area covered by melts can be expressed as the following equation,
\begin{eqnarray}
S_{\rm melt,f}\!\!\!\!&=&\!\!\!\!\frac{\pi}{4}f^{2}L^{2} \nonumber \\
&=&\!\!\!\!1.21f^{2}\left(\frac{m}{\rho_{\rm i}}\right)^{2/3},
\label{Smeltf}
\end{eqnarray}
where $f$ is the proportion of the diameter of the melt region to that of the impactor. The effects of decompression and adiabatic melting are included, and $f$ reaches 20–30 <cit.>.
Vertical cross sections of single impact effects Excavation regions are framed by dashed curves, and melting regions are framed by solid curves. We defined four direct effects, “direct excavation volume, $V_{\rm xcav}$,” “direct melting volume, $V_{\rm melt}$,” “direct excavation area, $S_{\rm xcav}$,” and “direct melting area, $S_{\rm melt}$.” $S_{\rm xcav}$ and $S_{\rm melt}$ are the horizontal cross sections of the direct excavation/melting regions. When the depth of the melting region is deeper than the thickness of the crust, we cut the volume beneath the crust. We also defined two subsequent effects, “excavation area including gravitational collapse, $S_{\rm xcav,f}$" and “melting area including the area covered by melts, $S_{\rm melt,f}$.” $S_{\rm xcav,f}$ is the area of the final crater. The rim of the transient crater collapses with gravity and the final crater is formed. $S_{\rm melt,f}$ includes both the direct melting area and the area covered by melts. The direct melting area is surrounded by the area covered by melts. When the diameter of the impactor is smaller than 100 km, $S_{\rm melt,f}$ is equal to $S_{\rm xcav,f}$. When the diameter is larger than 100 km, $S_{\rm melt,f}$ is calculated by Eq. (<ref>).
§.§ Integrating the effects of a single impact
The definition of $N_{\rm sfd}(>m_{\rm max})$ is
\begin{eqnarray}
N_{\rm sfd}(>m_{\rm max})\!\!\!\!&=&\!\!\!\!\frac{A}{\alpha-1}m_{\rm max}^{1-\alpha} \nonumber \\
\label{N(mmax)}
\end{eqnarray}
\begin{equation}
A\approx (\alpha-1)m_{\rm max}^{\alpha-1},
\label{Ammax}
\end{equation}
\begin{equation}
m_{\rm max}\approx \left(\frac{A}{\alpha-1}\right)^{1/(\alpha-1)},
\label{mmaxA}
\end{equation}
where $m_{\rm max}$ is the maximum mass of the impactors <cit.>. This $m_{\rm max}$ just represents the maximum mass value with the highest possibility, so it would have larger or smaller values actually.
The ratio of the collision probability with Earth and the moon is 23:1 <cit.>. The proportional constants $A$ have the following relationship,
\begin{equation}
A_{\rm e}=23A_{\rm m},
\label{AeAm}
\end{equation}
where $A_{\rm e}$ and $A_{\rm m}$ are the $A$ values for Earth and the moon, respectively.
The effects of all LHB impacts are obtained by calculating the integral from $m_{\rm min}$ to $m_{\rm max}$ of the effects of a single impact. We describe all of the effects of a single impact (Eqs. (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>)) as one general equation form, $I_{\rm vs}$, in order to easily understand their characteristics,
\begin{equation}
I_{\rm vs}=B_{j}m^{b_{j}},
\label{Ivs}
\end{equation}
where $b_{j}$ is the power-law index of each $m$, and $B_{j}$ is a constant fixed by impact velocity and density (see Table <ref>). The general form of the effects of all LHB impacts is
\begin{eqnarray}
I_{\rm vs,T}\!\!\!\!&=&\!\!\!\!\int_{m_{\rm min}}^{m_{\rm max}}\frac{\mathrm{d} N_{\rm sfd}}{\mathrm{d}m}I_{\rm vs}dm \nonumber \\
&=&\!\!\!\!B_{j}\frac{A_{\rm e}}{1+b_{j}-\alpha}(m_{\rm max}^{1+b_{j}-\alpha}-m_{\rm min}^{1+b_{j}-\alpha}).
\label{IvsT}
\end{eqnarray}
When $m_{\rm min}$ is small enough,
\begin{equation}
I_{\rm vs,T}\approx\begin{cases}
B_{j}\dfrac{A_{\rm e}}{1+b_{j}-\alpha}m_{\rm max}^{1+b_{j}-\alpha} & (0<\alpha<1+b_{j}) \\
B_{j}\dfrac{A_{\rm e}}{\alpha-b_{j}-1}m_{\rm min}^{1+b_{j}-\alpha} & (1+b_{j}<\alpha). \\
\end{cases}
\label{IvsTapp}
\end{equation}
The total effects of the impacts are dependent on the impactor's maximum mass $m_{\rm max}$ when $0<\alpha<1+b_{j}$, and on the impactor's minimum mass $m_{\rm min}$ when $1+b_{j}<\alpha$. We only use Eq. (<ref>) to understand the dependence of the effects on the impactor's mass, and always use Eq. (<ref>) to estimate the scales of the effects.
The total melting volume is dependent on the impactor's maximum mass, $m_{\rm max}$ (when $\alpha<2$) and is proportional to the total mass, $M_{\rm T}$ (see Eq. (<ref>)). The total direct melting area is proportional to the total cross section of the impactors, and the total cross section is proportional to $m^{5/3-\alpha}$. It has often been stressed that the effects of large impacts are greater than those of small impacts <cit.>. However, we found that large impacts and their effects are important only when $\alpha$ is small, and small impacts are more important than large impacts when $1.52<\alpha$ and $5/3<\alpha$ because the number of such small impacts is very large. In this case, thin melt sheets formed by small impacts would have covered the entire surface of the Earth, and it can explain the absence of Hadean rocks (see Section <ref>).
Especially when $\alpha$ is large, the total melting volume and area are strongly dependent on the minimum mass, $m_{\rm min}$. In this work, $m_{\rm min}$ is determined from the minimum mass $\mu_{\rm e}$ which can survive a fall through the Earth's atmosphere (see Section <ref>). Assuming the current atmosphere, $\mu_{\rm e}=10^{11.5}$ g <cit.>.
Effects $1+b_{j} $ $B_{j} $
$V_{\rm xcav}$ 1.78 $0.127\rho_{\rm i}^{0.22}\rho_{\rm t}^{-1}v^{1.3}g^{-0.66}\mathrm{sin}^{1.3}\theta$
$V_{\rm melt}$ 2 $0.42\rho_{\rm t}^{-1}\left(\dfrac{v^{2}}{\epsilon_{m}}\right)^{0.84}\mathrm{sin}^{1.3}\theta$
$S_{\rm xcav}$ 1.52 $1.48\rho_{\rm i}^{0.44/3}\rho_{\rm t}^{2/3}v^{0.88}g^{-0.44}$
$S_{\rm melt}$ 5/3 $1.08\rho_{\rm t}^{-2/3}\left(\dfrac{v^{2}}{\epsilon_{m}}\right)^{0.56}\mathrm{sin}^{2.6/3}\theta$
$S_{\rm xcav,f}$, $S_{\rm melt,f}$ 1.61 $2.52\rho_{\rm i}^{0.17}\rho_{\rm t}^{-0.78}v^{1.04}g^{-0.52}D_{\rm c}^{-0.35}$
$S_{\rm melt,f}$ ($L>100$ km) 5/3 $1.21f^{2}\rho_{\rm i}^{-2/3}$
These estimates do not include overlapping of craters. A better estimate including overlappings for the total excavation/melting area is expressed in the following equation <cit.>,
\begin{equation}
S_{\rm r}=1-\mathrm{exp}\left(-\frac{S_{\rm T}}{S_{\rm e}}\right),
\label{Sr}
\end{equation}
where $S_{\rm T}$ is $S_{\rm xcav,T}$, $S_{\rm melt,T}$, $S_{\rm xcav,f,T}$ or $S_{\rm melt,f,T}$, and $S_{\rm e}$ is the total surface area of Earth. We use this correction in this paper. On the other hand, estimating the excavation/melting volumes including overlapping is difficult, so we do not use any corrections for them and they are probably overestimated.
§ EFFECTS OF LHB IMPACTS
§.§ Effects of LHB impacts with fixed total impactors' mass
We estimate the effects of LHB and their dependence on $\alpha$, where $M_{\rm T}$, the total mass of impactors, is fixed. We assume the $\alpha$ to be $1<\alpha<2$. From Eq. (<ref>), $M_{\rm T}$ is
\begin{eqnarray}
M_{\rm T}\!\!\!\!&=&\!\!\!\!\int_{m_{\rm min}}^{m_{\rm max}} \frac{\mathrm{d} N_{\rm sfd}}{\mathrm{d}m}mdm \nonumber \\
&=&\!\!\!\!\frac{A}{2-\alpha}(m_{\rm max}^{2-\alpha}-m_{\rm min}^{2-\alpha}).
\label{MT1}
\end{eqnarray}
\begin{equation}
M_{\rm T}\approx\frac{Am_{\rm max}^{2-\alpha}}{2-\alpha},
\label{MT}
\end{equation}
where $m_{\rm min}$ is small enough and $\alpha<2$ <cit.>. This equation shows that the total mass depends on the maximum mass. The following equations derived from Eqs. (<ref>) and (<ref>),
\begin{equation}
m_{\rm max} \approx \frac{2-\alpha}{\alpha-1}M_{\rm T},
\label{mmaxMT}
\end{equation}
\begin{equation}A=(2-\alpha)^{\alpha-1}(\alpha-1)^{2-\alpha}M_{\rm T}^{\alpha-1}.
\label{AeMT}
\end{equation}
The dependence of the effects on $M_{\rm T}$ is derived from Eqs. (<ref>), (<ref>) and (<ref>),
\begin{equation}
I_{\rm vs,T}=B_{j}\frac{(2-\alpha)^{\alpha-1}(\alpha-1)^{2-\alpha}}{1+b_{j}-\alpha} \left\{\left(\dfrac{2-\alpha}{\alpha-1}M_{\rm T}\right)^{1+b_{j}-\alpha}-\mu_{\rm e}^{1+b_{j}-\alpha}\right\} M_{\rm T}^{\alpha-1}.
\label{IvsMT}
\end{equation}
The most typical value of $M_{\rm T}$ is about 1–5$\times10^{23}$ g <cit.>. <cit.> also used the value $M_{\rm T}=2\times10^{23}$ g. In this paper, we use the value $M_{\rm T}=$ 1, 2 and $5\times10^{23}$ g to calculate the effects of LHB.
Figure <ref> (a) shows the estimated total excavation and melting volumes relative to the total Hadean continental crust's volume. We considered the thickness of the crust (30 km) for the estimates of the melting volume (see Section <ref>). We also consider the cut-off of the heavier side of the SFD to avoid the mass becoming unrealistically large. We chose the cut-off value as $m_{\rm ceres}=9.4\times10^{23}$ g, the mass of Ceres. Ceres is the largest object among the main belt asteroids. For example, when $M_{\rm T}=5\times10^{23}$ g, we have to cut off the SFD where $\alpha<1.35$. In this $\alpha$ region, $m_{\rm e,max}= m_{\rm ceres}$. The estimated total excavation/melting volumes are normalized with the total current continental crust's volume, $7.18\times10^{9}$ km$^{3}$ <cit.>. According to geochemical constrains of the mantle, we assumed that the continental crust was formed at 4.5 Ga for the first time and there was about 12% of the total current continental crust's volume in the last phase of the Hadean <cit.>. We assumed $\rho_{\rm i}=2.6$ g/cm$^{3}$, $\rho_{\rm t}=2.7$ g/cm$^{3}$, $v=21$ km/s and $\epsilon_{\rm m}=5.2$ MJ/kg. In the case of $M_{\rm T}=2\times10^{23}$ g in Fig. <ref> (a), the total melting volume is smaller than the total volume of Hadean continental crust in all $\alpha$ ranges. LHB impacts do not melt all of Hadean continental crust in this case. On the other hand, the total excavation volume exceeds that of the continental crust when $\alpha$ is larger than about 1.8, though we note that excavation volume of the continental crust would be overestimated (see Section <ref>). LHB can not excavate/melt all of the continental crust by direct effects when $\alpha$ =1.61.
Figure <ref> (b) shows the direct excavation/melting area using Table <ref>, and Eqs. (<ref>), (<ref>), and (<ref>). This figure shows that LHB does not excavate/melt all of the Earth's surface area by direct effects when $\alpha$ =1.61. If there were many small impacts (i.e., when $\alpha$ was large), the excavation area would be able to expand the whole surface of the Earth.
Figure <ref> shows the total excavation/melting area including gravitational collapse and the area covered by melts. In particular, the total melting area including the area covered by melts (red and blue solid curves) is dramatically increased compared to the total direct melting area and approaches 60–70% of the Earth's surface area where $\alpha=1.61$ and $f=$20–30 <cit.>. As $\alpha$ becomes larger, the number of small impactors dramatically increases. For small impactors, the area covered by melts is comparable to the excavated area on the Earth's surface including gravitational collapse (Table <ref>). Therefore, the total melting area including the area covered by melts (the red and blue solid curves) approaches the total excavation area including gravitational collapse (the red dashed curve) when $\alpha$ is large enough.
Total direct effects of LHB with fixed total impactors' mass Panel (a) and (b) show the total direct excavation/melting volumes and areas, respectively. Green, blue, and red curves show the effects when $M_{\rm T}=1$, 2, and $5\times10^{23}$ g, respectively. Solid curves show the total melting volumes and areas, dashed curves are those of excavation. The black line in panel (a) shows the total Hadean continental crust's volume. We cut off the SFD larger than the size of Ceres when $\alpha<1.35$ ($M_{\rm T}=5\times10^{23}$ g), left side of the aqua vertical line. In panel (a), the solid curves (i.e., melting volumes) include the correction of the thickness of the crust, $h=30$ km, but the dashed curves (i.e., excavation volumes) do not.
Total excavation/melting areas with fixed total impactors' mass Dashed and solid green curve show the direct excavation and melting areas, respectively. Red and blue curves show the total melting area including the area covered by melts when $f=30$ and 20, respectively. Dashed red curve shows the total excavation area including gravitational collapse. The dashed and solid green curves are consistent with dashed and solid blue curves in Fig. <ref> (b), respectively. Including the area covered by melts, the total melting area increases dramatically. The total mass that hit Earth, $M_{\rm T}$, is fixed at $2\times10^{23}$ g. We cut off the SFD larger than the size of Ceres when $\alpha<1.18$, left side of the aqua vertical line.
Figure <ref> shows the total melting area for each impactor's size. Figure <ref> (a) shows the total direct melting area. It is mainly dependent on the maximum mass when $\alpha>5/3$ and on the minimum mass when $\alpha<5/3$ (also see $S_{\rm melt}$ in Table <ref>). In other words, as $\alpha$ becomes larger, the number of small impactors dramatically increases, and as $\alpha$ becomes smaller, the maximum mass to hit Earth increases. This trend is the same as that of the total impactors' cross sections. Also, even if we estimate the total direct melting area from the size of the maximum LHB basin on the moon or the lunar crater density, their dependence on $m$ is not changed, so that this trend of the total melting area is not changed (see Section <ref>).
When the area covered by melts is included, this trend is changed. Figure <ref> (b) suggests that the trend of the contributions can be divided into three ranges. Small impactors mainly melt the Earth's surface when $\alpha$ is larger than about 1.7. Impactors of about 100 km in size mainly melt the surface when $\alpha$ is about 1.5–1.7. Impactors of over 500 km in size mainly melt the surface when $\alpha$ is less than about 1.3. Considering the “stochastic” view, an impactor of such size is the largest one in most cases. When $\alpha=1.61$, the effects of impactors of about 100 km in size are dominant.
Total melting areas for each impactor's size with fixed total impactor's mass Panel (a) shows the total direct melting area; panel (b) shows the total melting area where $f=30$. Black curves in panels (a) and (b) show the total melting areas of all size impactors, consistent with the green and red solid curves in Fig. <ref>, respectively. In panel (b), red, gray and yellow curves show the total melting areas by impactors larger than 100, 300 and 500 km, respectively. Dashed pink curve shows the total direct melting area by impactors smaller than 1 km. $M_{\rm T}$ is fixed at $2\times10^{23}$ g.
§.§ Estimate from lunar constraints
While constraints of LHB on Earth have been erased, they remain on the moon. We estimate the value of $A_{\rm m}$ from lunar constraints when $1<\alpha<2$. First, we estimate it based on the largest basin formed by LHB impacts. The following equation is derived from Eqs. (<ref>), where $m_{\rm m, max}$ is the maximum mass to hit the moon,
\begin{equation}
A_{\rm m}=(\alpha-1)m_{\rm m, max}^{\alpha-1} \label{aemmmax}.
\end{equation}
Because the impactor formed the Imbrium basin is at the edge of the SFD where the statistical fluctuation is the largest, this estimate has uncertainty.
We assumed the largest impactor ($m_{\rm m, max}$) formed the Imbrium basin at 3.85 Ga. Although the South Pole-Aitken is the largest lunar basin, it was considered to be formed before the onset of LHB. The impactor's mass $m$ is estimated from $D_{\rm t}$ (Eq. (<ref>)),
\begin{equation}
m=0.149\rho_{\rm i}^{-0.11/0.35}\rho_{\rm t}^{1/0.78}v^{-0.22/0.13}g^{0.11/0.13}D_{\rm c}^{0.15/0.26}D_{\rm f}^{0.85/0.26}.
\label{m}
\end{equation}
Using this equation, the mass of the impactor formed the Imbrium is $m_{\rm m, max}=8.02\times10^{20}$ g, where $\rho_{\rm i}=2.6$ g/cm$^{3}$, $\rho_{\rm t}=$2.9 g/cm$^{3}$, $v=$18 km/s, $D_{\rm c}=$18 km on the moon, and the Imbrium basin's diameter, $D_{\rm f}$, is 1160 km <cit.>.
Then, we estimate the value of $A_{\rm m}$ from the crater density on the moon. According to <cit.>, the following differential equation represents the number of impacts to hit the moon:
\begin{equation}
\frac{\mathrm{d} N_{20}}{\mathrm{d}t}=2.7\times10^{-16}\mathrm{exp}(6.93t)+5.9\times10^{-7},
\label{NI94Sawtooth}
\end{equation}
where $N_{20}$ is the lunar crater density whose diameters are larger than 20 km, and $t$ is the age. The units are [$/\rm km^{2}$] and [Ga (Gyr ago)]. In this expression, LHB started at 4.1 Ga (a sawtooth-like timeline).
The number of impacts to hit area $S$ between the age $t$ and $t+\mathrm{d}t$ forming craters larger than $D$ is
\begin{equation}
n_{t,t+\mathrm{d}t}(>D)=\frac{\mathrm{d} N_{D}}{\mathrm{d}t}S.
\label{ntD}
\end{equation}
Then, the proportional constant $A$ is derived from Eq. (<ref>),
\begin{equation}
\label{At}
\end{equation}
where $m_{D}$ is the impactor's mass that forms a crater whose diameter is $D$. The total number of impacts which hit the moon to form a crater larger than 20 km during LHB is
\begin{eqnarray}
N_{\rm m,20}\!\!\!\!&=&\!\!\!\!S_{\rm m}N_{20} \nonumber \\
&=&\!\!\!\!S_{\rm m}\int_{t_{\rm f}}^{t_{0}}\dfrac{\mathrm{d} N_{20}}{\mathrm{d}t}dt \nonumber \\
&=&\!\!\!\!3.79\times10^{7}\times8.76\times10^{-5} \nonumber \\
\label{Nm}
\end{eqnarray}
where $S_{\rm m}$ is the moon's surface area, $t_{0}=4.5$ and $t_{\rm f}=0$. This estimate of $N_{20}=8.76\times10^{-5}$ km$^{-2}$ is exactly consistent with the real lunar crater density in the Nectaris basin <cit.>. The key point is that we use the timeline only for calculating the lunar crater density $N_{20}$ and so this estimate does not depend on a specific LHB model. Therefore, $A_{\rm m}$ can be directly estimated only by observing the lunar crater density. Then,
\begin{equation}
A_{\rm m}=(\alpha-1)m_{\rm m,20}^{\alpha-1}N_{\rm m,20},
\label{AmDensity}
\end{equation}
where $m_{\rm m, 20}$ is the impactor's mass which can make a 20-km-diameter crater on the moon. Using Eq. (<ref>), $m_{\rm m,20}=1.38\times10^{15}$ g. We summarize how to estimate $A_{\rm e}$ and $m_{\rm e, max}$ from lunar constraints in Table <ref>.
How to calculate each value
Constraints $A_{\rm e}$ $m_{\rm e, max}$
Total mass $(2-\alpha)^{\alpha-1}(\alpha-1)^{2-\alpha}M_{\rm T}^{\alpha-1}$ $\dfrac{2-\alpha}{\alpha-1}M_{\rm T}$
Imbrium size $23(\alpha-1)m_{\rm m,max}^{\alpha-1}$ $23^{1/(\alpha-1)}m_{\rm m,max}$
Crater density $23(\alpha-1)m_{\rm m,20}^{\alpha-1}N_{\rm m,20}$ $\left(23m_{\rm m,20}^{\alpha-1}N_{\rm m,20}\right)^{1/(\alpha-1)}$
Figures <ref> and <ref> show the total excavation/melting volume and area estimated from the size of the Imbrium. Fig. <ref> (b) includes the subsequent effects, while Fig. <ref> (a) does not. On the other hand, Figs. <ref> and <ref> show those estimated from the crater density on the moon. Fig. <ref> (b) includes the subsequent effects, while Fig. <ref> (a) does not.
Figures <ref> (a) and <ref> (a) show that LHB does not excavate/melt all of Hadean continental crust in almost all $\alpha$ ranges. This is the same result as that of the excavation/melting volume estimated from the total mass to hit the Earth (Fig. <ref> (a)). Figures <ref> (b) and <ref> (b) show that, although LHB does not excavate/melt all of the Earth's surface directly, most of the surface is covered by the melts from the subsequent effect in almost all $\alpha$ ranges. Figures <ref> (a) and <ref> (a) show the total direct melting areas for each impactor's size. They show that the total melting areas are mainly dependent on the maximum mass when $\alpha>5/3$ and on the minimum mass when $\alpha<5/3$ (also see $S_{\rm melt}$ in Table <ref>). Figures <ref> (b) and <ref> (b) suggest that the trend in contributions to covering the surface with melts are divided into three $\alpha$ ranges like the contribution derived from the total mass (Fig. <ref> (b)). The three $\alpha$ ranges are about 1.0–1.5, 1.5–1.7, and 1.7–2.0 for both the estimates derived from the size of the Imbrium basin and the lunar crater density. When $\alpha=1.61$, the effects of impacts of about 100 km in size are dominant.
Total effects of LHB estimated from the size of the Imbrium basin Panel (a) and (b) show the total excavation/melting volumes and areas, respectively. In panel (a), red and blue curves show the total excavation and melting volumes, respectively. The black line shows the total Hadean continental crust's volume. The melting volume includes the correction of the thickness of the crust, $h=30$ km, but the excavation volume does not. In panel (b), dashed and solid green curves show the total direct excavation and melting areas, respectively. Dashed red curve shows the total excavation areas including gravitational collapse. Red and blue solid curves show the total melting areas including the area covered by melts where $f=30$ and 20, respectively. The Imbrium mass is $m_{\rm m, max}=8.02\times10^{20}$ g. We cut off the SFD larger than the size of Ceres when $\alpha<1.44$.
Total melting areas for each impactor's size estimated from the size of the Imbrium basin Panel (a) represents the direct total melting area; panel (b) shows the total melting area including the area covered by melts where $f=30$. Black curves in panels (a) and (b) show the total melting areas of all size impactors, consistent with the green and red solid curves in Fig. <ref> (b), respectively. In panel (b), red, gray, and yellow curves show the total melting areas by impactors larger than 100, 300, and 500 km, respectively. Dashed pink curve shows the total direct melting area by impactors smaller than 1 km. Imbrium mass is $m_{\rm m, max}=8.02\times10^{20}$ g.
Total effects of LHB estimated from the lunar crater density Panel (a) and (b) show the total excavation/melting volumes and areas, respectively. In panel (a), red and blue curves show the total excavation and melting volumes, respectively. The black line shows the total Hadean continental crust's volume. The melting volume includes the correction of the thickness of the crust, $h=30$ km, but the excavation volume does not. In panel (b), dashed and solid green curves show the total direct excavation and melting areas, respectively. Dashed red curve shows the total excavation area including gravitational collapse. Red and blue solid curves show the total melting areas including the area covered by melts where $f=30$ and 20, respectively. The density of lunar craters larger than 20 km is $N_{20}=8.76\times10^{-5}$ km$^{-2}$. We cut off the SFD larger than the size of Ceres when $\alpha<1.55$.
Total melting ares for each impactor's size estimated from the lunar crater density Panel (a) represents the total direct melting area; panel (b) shows the total melting area including the area covered by melts where $f=30$. Black curves in panels (a) and (b) show the total melting areas of all size impactors, consistent with the green and red solid curves in Fig. <ref> (b), respectively. In panel (b), red, gray, and yellow curves show the total melting areas by impactors larger than 100, 300, and 500 km, respectively. Dashed pink curve shows the direct melting rate by impactors smaller than 1 km. The density of lunar craters larger than 20 km is $N_{20}=8.76\times10^{-5}$ km$^{-2}$.
We summarize the above results in Fig. <ref>. The red and blue arrows represent the $\alpha$ ranges of LHB estimated from the size of the Imbrium basin and the lunar crater density that can excavate/melt 70% of Hadean continental crust's volume or the Earth's surface area. Considering only the direct excavation/melting of the continental crust, it is difficult for LHB impacts for most $\alpha$ ranges to excavate/melt all of the continental crust. According to the estimate from the size of the Imbrium basin, there is enough excavation/melting only for limited conditions, when $\alpha>1.7$ (excavation) and $\alpha>1.9$ (melting), the $\alpha$ ranges where the total effects are enough for both the volume and area. According to the estimate from the lunar crater density, there is enough excavation and melting only for $\alpha<1.3$. When $\alpha=1.61$, all estimated direct effects of LHB impacts are not enough. On the other hand, when we consider the subsequent effects, the $\alpha$ ranges in which LHB can excavate/melt the continental crust expand. In particular, the subsequent melting can cover over 70% of the Earth's surface in all $\alpha$ ranges (estimated from the size of the Imbrium) and when about $\alpha<1.6$ and $\alpha>1.9$ (estimated from the lunar crater density). When $\alpha=1.61$, the subsequent melting covers over 70% of the surface in both the estimates. In conclusion, our results show that LHB would not excavate/melt all of the Hadean continental crust directly, but most of the Earth's surface area could be covered by melts.
The $\alpha$ ranges where LHB excavates (a) and melts (b) 70% of total Hadean continental crust Red arrows show the $\alpha$ ranges of LHB estimated from the size of Imbrium basin that can excavate/melt 70% of the total Hadean continental crust's volume (i.e., 8.4% of the total current continental crust's volume) or the Earth's surface area. Blue arrows show the $\alpha$ ranges estimated from the lunar crater density. Black solid lines show $\alpha=1.61$. In the estimate of $S_{\rm melt,f,T}$, we assumed $f=30$. When we consider the subsequent effects, the $\alpha$ ranges in which LHB can excavate/melt most of the continental crust expand.
§ DISCUSSION
§.§ LHB and the absence of Hadean rocks
Can LHB explain the absence of Hadean rocks? When a crust is excavated by impacts, it is broken into small pieces and scattered. This helps Hadean continental crust to subduct with oceanic plates and disappear from the Earth's surface. When the crusts melt, their radiometric ages of zircons are reset <cit.>. It erased the record of Hadean rocks. If the melts covered the entire Earth's surface area, and the stratigraphic succession has been preserved until today, the absence of Hadean rocks on the surface can be explained. Our result that LHB is not able to excavate/melt all of Hadean continental crust directly but can cover most of the Earth's surface with 3-km thick melts suggests that if there has not been a large-scale folding or overturn of the crust until today, LHB could explain why we have never found any Hadean rocks except the Jack Hills zircons because most of Hadean continental crust is not be exposed on the Earth's surface in this case.
§.§ Comparison of results to previous works
We compare our results derived from the lunar constraints with previous works with $\alpha=1.61$. In Table <ref>, we summarized SFDs and the excavation/melting volumes and areas of our calculations, <cit.> and <cit.>. Without the subsequent effects, both our calculation and <cit.> show that a few crusts are excavated/melted by LHB. Not considering crater overlapping, the total melting area including the area covered by melts, $S_{\rm melt,f, T}$, is over 90%. This estimate is consistent with that of <cit.>. However, if we take into account crater overlapping, the coverage is reduced to 60-74%.
Comparison between our work and previous works
Our work <cit.> <cit.>
SFD $\alpha$=1.61 MBAa MBAa
$v$ [km/s] 21 20 25
$\rho_{\rm i} \ \rm [g/cm^{3}$] 2.6 2.7 3.314
$\rho_{\rm t} \ \rm [g/cm^{3}$] 2.7 2.7 "granite"
$\mu_{\rm e}$ [g] $10^{11.5}$ $1.41\times10^{15}$b $5.86\times10^{18}$c
$T_{\rm surf} \ \rm [C^{\circ}$] 0 20 20
d$T$/d$z$ [C$^{\circ}$/km] 0 12, 70 11.25
$h$ [km] 30 - 30
$M_{\rm T}$ [g] $2.2\times10^{23}$ $2\times10^{23}$ -
$V_{\rm xcav,T}$ (%)d 3.6 $\sim$7e -
$V_{\rm melt,T}$ 2.7 $\sim$2.1-3.6f -
$S_{\rm xcav,T}$ (%)g 20 $\sim$25 -
$S_{\rm melt,T}$ 2.2 $\sim$5-10 -
$S_{\rm xcav,f,T}$ 51 - -
$S_{\rm melt,f,T}$ 60-74 - -
$S_{\rm melt,f,T}$h 92-130 - 70-100
aMain belt asteroids' SFD
dPercent of the total current continental crust's volume
e“Impact ejecta covered the entire surface of the LHB-era Earth to a depth close to 1 km.”
f“$\sim$1.5–2.5 vol.% of the upper 20km of Earth's crust was melted in the LHB.”
gPercent of the Earth's surface area
hNot considering crater ovarlapping
Note that while we analytically estimated the values of LHB effects at the case of highest probability, <cit.> adopted Monte Carlo simulations so that their results have a “stochastic” nature. To roughly take into account the stochastic nature in our calculations, we have the maximum mass fluctuate by a value from $1/e$ to $e$ because the fluctuation of maximum mass impact has a large contribution to results. Figure <ref> shows the total melting area including the area covered by melts for $M_{\rm T}=2\times10^{23}$ g. When $\alpha$ is small, the fluctuation affects the melting rate significantly because the contribution from the maximum mass is larger for smaller $\alpha$.
Total melting areas in a stochastic scenario with fixed total impactors' mass Blue and green dashed curves show the total melting areas including the area covered by melts when the maximum masses are $1/e$ and $e$ times as heavy as the previous maximum mass, $m_{\rm e,max}$, respectively. Red solid curve shows our analytical estimate of the case with highest possibility. We did not consider the cut off of the SFD by the mass of Ceres.
§.§ Validity of assumptions and models
Here we discuss the validity of the assumptions we adopt. First, we did not consider the geothermal heat. We obtained $V_{\rm melt}$, that is consistent with the analytical shock-heating model used in <cit.>, where the target has no geothermal gradient and the initial temperature is homogenized to be $0^{\circ}$C. Because geothermal heat increases the melting volume, $V_{\rm melt}$ that we have obtained could be an underestimate. If the Hadean Earth had a higher geothermal gradient than that of the present Earth, the melting volume could increase by a factor of two or three at most <cit.>. On the other hand, the subsequent melting includes the gradient's effect. However, <cit.> claimed that if the Hadean mantle potential temperature was higher or the lithosphere was thinner than today, the melting volume may increase by 50–75%.
Second, our estimates of the minimum impactor mass may be overestimated because the Hadean atmosphere may have had higher pressure than the current atmosphere. The minimum diameter $L_{\rm min}$ can be shown analytically as
\begin{equation}
L_{\rm min}=0.15\frac{P_{\rm surf}}{\rho_{\rm i}g_{\rm surf}\rm sin\theta},
\label{Lmin}
\end{equation}
where $P_{\rm surf}$ and $g_{\rm surf}$ are, respectively, the atmospheric pressure at the surface and the acceleration of gravity <cit.>. In this case, our estimates of the particular excavation/melting areas decrease when $\alpha$ is large (Fig. <ref>).
Dependence of the total direct excavation/melting areas on $\mu_{\rm e}$ estimated from the lunar crater density Red, blue and green curves show the total direct excavation/melting areas when $\mu_{\rm e}=10^{10}$, $10^{11.5}$ and $10^{13}$ g, respectively. Dashed and solid curves show the total direct excavation and melting areas, and blue ones are consistent with the dashed and solid green curves in Fig. <ref> (b), respectively. These estimates are very dependent on the minimum size of the impactor, especially when $\alpha$ is large.
Third, our estimate assumes impactors' density, $\rho_{\rm i}$, of 2.6 g/cm$^{3}$ and velocity, $v$, of 21 km/s during LHB, and these values imply that the impactors were asteroids. However, some previous works claimed that the source of LHB impacts were comets or icy planetesimals <cit.>. In these cases, the impactor's density should be changed to $\rho_{\rm i}\sim1$ g/cm$^{3}$ and the velocity to $v\sim30$ km/s. However, this change would not affect the results significantly. According to Table <ref>, $V_{\rm xcav}\propto \rho_{\rm i}^{0.22}v^{1.3}m^{0.78}$, $S_{\rm xcav}\propto \rho_{\rm i}^{0.44/3}v^{0.88}m^{0.52}$, $S_{\rm xcav,f}\propto \rho_{\rm i}^{0.17}v^{1.0}m^{0.61}$, $V_{\rm melt}\propto v^{1.68}m$ and $S_{\rm melt}\propto v^{1.12}m^{2/3}$, so these effects should increase by only $(1.0/2.6)^{0.22+0.78}\times (30/21)^{1.3}=0.61$, $(1.0/2.6)^{0.44/3+0.52}\times (30/21)^{0.88}=0.72$, $(1.0/2.6)^{0.17+0.61}\times (30/21)^{1.0}=0.68$, $(30/21)^{1.68}\times(1.0/2.6)=0.70$ and $(30/21)^{1.12}\times(1.0/2.6)^{2/3}=0.79$ times, respectively. $S_{\rm melt,f}$ does not depend on $\rho_{\rm i}$ and $v$. If the effect is regulated by the impactor's kinetic energy, $f$ should be multiplied $(1.0/2.6\times (30/21)^2)^{(1/3)}=0.92$ times, and the minimum size impactor that can form the flood melt should be multiplied $(1.0/2.6\times (30/21)^2)^{(-1/3)}=1.1$ times.
Finally, we discuss the effects of pre-LHB impacts. In the sections above, we only estimated the effects of LHB. However, several impacts occurred during the middle of the Hadean eon and much more impacts occurred at the beginning of it. In the first few hundred million years of the Hadean eon, impacts melted (or excavated) Hadean continental crust right after they formed, and the continental crust must not have had time to grow sufficiently.
§ SUMMARY
We have investigated by analytical arguments the possibility for LHB impacts to excavate/melt Hadean continental crust. In order to reveal intrinsic physics, we adopt simple power-law impactors' SFD with various exponents $\alpha$, rather than a single detailed SFD. We divided the effects of impactors into two phases, and derived general formulas of excavation/melting volume and area as functions of $\alpha$ and the impactor's mass multiplied by a factor determined by impact velocity, planetary gravity, bulk density of impactors and the target planet. We estimated the total LHB effects from the total mass of LHB impacts and two types of constraints on the moon, the size of the largest basin during LHB and the small crater density.
With the fixed total LHB mass, the total direct melting area on the Earth's surface is generally regulated by small (large) impacts, for large (small) $\alpha$. The estimates from the lunar constraints suggest that LHB can excavate/melt almost all of Hadean continental crust in narrow $\alpha$ ranges. Estimating from the size of the Imbrium basin, LHB can remove the continental crust only at $\alpha > 1.7$ (excavation) and $\alpha > 1.9$ (melting), while estimating from the lunar crater density, only at $\alpha < 1.3$ (excavation and melt). In contrast, the subsequent melts which spread on and beyond the final craters can cover over 70% of the Earth's surface in all $\alpha$ ranges (estimated from the size of the Imbrium basin) or when about $\alpha < 1.6$ (estimated from the lunar crater density). However, the most likely value of $\alpha$ is 1.6–1.7. We conclude that LHB impacts would not excavate/melt all of Hadean continental crust directly, but most of the Earth's surface could be covered by melts of subsequent impact effects. It suggests the absence of Hadean rocks could be explained by LHB if the stratigraphic succession has been preserved until today because most of Hadean continental crust is not be exposed on the Earth's surface in this case.
We thank S. Marchi and O. Abramov for very useful comments as reviewers. Their comments helped us improve the paper a lot. We thank S. Maruyama, K. Kurosawa, and H. Sawada for valuable discussions. T. S. was supported by a Grant-in-Aid for Young Scientists (B), JSPS KAKENHI Grant Number 24740120, and Grant-in-Aid for Scientific Research on Innovative Areas Number 2605, MEXT. This research was supported by a grant for JSPS (23103005) Grant-in-aid for Scientific Research on Innovative Areas.
§ VALUE OF $\ALPHA$ THAT FULFILLS BOTH LUNAR CONSTRAINTS
We calculated the $\alpha$ value that fulfills both lunar constraints of the size of the Imbrium basin and the lunar crater density. From Table <ref>,
\begin{equation}
23(\alpha-1)m_{\rm m, max}^{\alpha-1} =23(\alpha-1)m_{\rm m,20}^{\alpha-1}N_{\rm m,20},
\label{AemmmaxDensity}
\end{equation}
\begin{eqnarray}
\alpha\!\!\!\!&=&\!\!\!\!1+\dfrac{\mathrm{ln}N_{\rm m,20}} {\mathrm{ln}(m_{\rm m, max}/m_{\rm m,20})} \nonumber \\
\label{fulfillalpha}
\end{eqnarray}
where $N_{\rm m,20}=3.32\times10^{3}$, $m_{\rm m, max}=8.02\times10^{20}$ g, and $m_{\rm m,20}=1.38\times10^{15}$ g (see Section <ref>).
[Abramov et al.(2012)]abr12 Abramov, O., Wong, S.M., & Kring, D.A., 2012. Differential melt scaling for oblique impacts on terrestrial planets. Icarus, 218, 906-916, doi:10.1016/j.icarus.2011.12.022.
[Abramov et al.(2013)]abr13 Abramov, O., Kring, D.A., & Mojzsis, S.J., 2013. The impact environment of the Hadean Earth. Chemie der Erde 73, 227-248, doi:10.1016/j.chemer.2013.08.004.
[Bottke et al.(2005)]bot05 Bottke, W.F., Durda, D.D., Nesvorný, D., Jedicke, R., Morbidelli, A., Vokrouhlický, D., & Levison, H., 2005. The fossilized size distribution of the main asteroid belt. Icarus 175, 111-140, doi:10.1016/j.icarus.2004.10.026.
[Bottke et al.(2012)]bot12 Bottke, W.F., Vokrouhlický, D., Minton, D.A., Nesvorný, D., Morbidelli, A., Brasser, R., Simonson, B., & Levison, H.F., 2012. An Archaean heavy bombardment produced by a destabilized extension of the asteroid belt. Nature 485, 78-81, doi:10.1038/nature10967.
[Bland and Artemieva(2003)]bla03 Bland, P. A. & Artemieva, N. A., 2003. Efficient disruption of small asteroids by Earth’s atmosphere. Nature 424, 288-291, doi:10.1038/nature01757.
[Cogley(1984)]cog84 Cogley, J.G., 1984. Continental margins and the extent and number of the continents. Reviews of Geophysics and Space Physics 22, 101-122, doi:10.1029/RG022i002p00101.
[Cohen et al.(2005)]coh05 Cohen, B.A., Swindle, T. D., & Kring, D.A., 2005. Geochemistry and 40 Ar-39 Ar geochronology of impact-melt clasts in feldspathic lunar meteorites: implications for lunar bombardment history. Meteorit. Planet. Sci. 40, 755, doi:10.1111/j.1945-5100.2005.tb00978.x.
[Dohnanyi(1969)]doh69 Dohnanyi, J. S., 1969. Collisional model of asteroids and their debris. J. Geophys. Res. 74, 2531-2554, doi:10.1029/JB074i010p02531.
[Frey(1977)]fre77 Frey, H., 1977. Origin of the Earth's ocean basins. Icarus, 32, 235-250, doi:10.1016/0019-1035(77)90064-1.
[Frey(1980)]fre80 Frey, H., 1980. Crustal evolution of the early Earth: The role of major impacts. Precambrian Res., 10, 195–216, doi:10.1016/0301-9268(80)90012-1.
[Gomes et al.(2005)]gom05 Gomes, R., Levison, H.F., Tsiganis, K., & Morbidelli, A., 2005. Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets. Nature 435, 466-469, doi: 10.1038/nature03676.
[Grieve(1980)]gri80 Grieve, R.A.F., 1980. Impact bombardment and its role in proto-continental growth on the early Earth. Precambrian Res. 10, 217-247.
[Hartmann et al.(2003)]har03 Hartmann, W.K., 2003. Megaregolith evolution and cratering cataclysm models—Lunar cataclysm as a misconception (28 years later). Meteorit. Planet. Sci. 38, 579-593, doi:10.1111/j.1945-5100.2003.tb00028.x.
[Ito and Malhotra(2006)]ito06 Ito, T., & Malhotra, R., 2006. Dynamical transport of asteroid fragments from the $\nu\_{6}$ resonance. Adv. Space Res. 38, 817-825, doi:10.1016/j.asr.2006.06.007.
[Jørgensen et al.(2009)]jor09 Jørgensen, U.G., Appel, P.W.U., Hatsukawa, Y., Frei, R., Oshima, M., Toh, Y., & Kimura, A., 2009. The Earth-Moon system during the late heavy bombardment period - Geochemical support for impacts dominated by comets. Icarus 204, 368-380, doi:10.1016/j.icarus.2009.07.015.
[Kawai et al.(2009)]kaw09 Kawai, K., Tsuchiya, T., Tsuchiya, J., & Maruyama, S., 2009, Lost primordial continents. Gondwana Research 16, 581-586, doi:10.1016/j.gr.2009.05.012.
[Krasinsky(2002)]kra02 Krasinsky, G.A., Pitjeva, E.V., Vasilyev, M.V., & Yagudina, E.I., 2002, Hidden mass in the asteroid belt. Icarus 158, 98-105, doi:10.1006/icar.2002.6837.
[Levison et al.(2001)]lev01 Levison, H.F., Dones, L., Chapman, C.R., Stern, S.A., Duncan, M.J., & Zahnle, K., 2001. Could the lunar “Late Heavy Bombardment” have been triggered by the formation of Uranus and Neptune? Icarus 151, 286-306, doi:10.1006/icar.2001.6608.
[Marchi et al.(2012)]mar12 Marchi, S., Bottke, W.F., Kring, D.A., Morbidelli, A., & 2012. The onset of the lunar cataclysm as recorded in its ancient crater populations. Earth Planet. Sci. Lett. 325, 27-38, doi:10.1016/j.epsl.2012.01.021.
[Marchi et al.(2014)]mar14 Marchi, S., Bottke, W.F., Elkins-Tanton, L.T., Bierhaus, M., Wuennemann, K., Morbidelli, A., & Kring, D.A., 2014. Widespread mixing and burial of Earth's Hadean crust by asteroid impacts. Nature 511, 578-582, doi:10.1038/nature13539.
[McCulloch and Bennett(1994)]mcc94 McCulloch, M.T., & Bennett, V.C., 1994. Progressive growth of the Earth's continental crust and depleted mantle: geochemical traces. Geochimica Cosmochimica Acta 58, 4717-4738, doi:10.1038/nature13539.
[Melosh(1989)]mel89 Melosh, H.J., 1989. Impact Cratering: A Geologic Process. Oxford University Press, Oxford, UK.
[Morbidelli et al.(2012)]mor12 Morbidelli, A., Marchi, S., Bottke, W.F., & Kring, D.A., 2012. A sawtooth-like timeline for the first billion years of lunar bombardment. Earth Planet. Sci. Lett. 355-356, 144-151, doi:10.1016/j.epsl.2012.07.037.
[Osinski et al.(2011)]osi11 Osinski, G.R., Tornabene, L.L., & Grieve, R.A.F., 2011. Impact ejecta emplacement on terrestrial planets. Earth Planet. Sci. Lett. 310, 167-181, doi:10.1016/j.epsl.2011.08.012.
[Ryder et al.(2000)]ryd00 Ryder, G., Koeberl, C., & Mojzsis, S.J., 2000. Heavy Bombardment on the Earth at 3.85 Ga: The Search for Petrographic and Geochemical Evidence. In: Canup, R., Righter, K. (Eds.), Origin of the Earth and Moon. University of Arizona Press, Tucson, 475-492.
[Ryder(2002)]ryd02 Ryder, G., 2002. Mass flux in the ancient Earth-Moon system and benign implications for the origin of life on Earth. J. Geophys. Res. 107, doi:10.1029/2001JE001583.
[Schmidt and Housen(1987)]sch87 Schmidt, R.M., & Housen, K.R., 1987. Some recent advances in the scaling of impact and explosion cratering. Int. J. Impact Eng. 5, 543-560.
[Spudis(1993)]spu93 Spudis, P. D., 1993. The Geology of Multi-ring Impact Basins: The Moon and Other Planets, Cambridge Univ. Press, Cambridge, New York, 37-41.
[Strom et al.(2005)]str05 Strom, R.G., Malhotra, R., Ito, T., Yoshida, F., & Kring, D.A., 2005. The origin of planetary impactors in the inner solar system. Science 309, 1847-1850, doi:10.1126/science.1113544.
[Tanaka et al.(1996)]tan96 Tanaka, H., Inaba, S., & Nakazawa, K., 1996. Steady-state size distribution for the self-similar collision cascade. Icarus 123, 450-455, doi:10.1006/icar.1996.0170.
[Tera et al.(1974)]ter74 Tera, F., Papanastassiou, D.A., & Wasserburg, G.J., 1974. Isotopic evidence for a terminal lunar cataclysm. Earth Planet. Sci. Lett. 22, 1-21, doi:10.1016/0012-821X(74)90059-4.
[Wilde et al.(2001)]wil01 Wilde, S. A., Valley, J. W., Peck, W. H., & Graham, C. M., 2001. Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago. Nature 409, 175-178, doi:10.1038/35051550.
[Zahnle and Sleep(1997)]zah97 Zahnle, K.J., & Sleep, N.H., 1997. Impacts and the early evolution of life. In: Thomas, P.J., Chyba, C.F., McKay, C.P. (Eds.), Comets and the Origin and Evolution of Life. Springer-Verlag, New York, 175-208.
|
1511.00048
|
Given a multi-armed bandit problem it may be desirable to achieve a smaller-than-usual worst-case regret for some special actions.
I show that the price for such unbalanced worst-case regret guarantees is rather high. Specifically, if an algorithm enjoys a worst-case
regret of $B$ with respect to some action, then there must exist another action for which the worst-case regret is at least $\Omega(nK/B)$, where $n$ is
the horizon and $K$ the number of actions. I also give upper bounds in both the stochastic and adversarial settings showing that this result cannot
be improved. For the stochastic case the pareto regret frontier is characterised exactly up to constant factors.
§ INTRODUCTION
The multi-armed bandit is the simplest class of problems that exhibit the exploration/exploitation dilemma.
In each time step the learner chooses one of $K$ actions and receives a noisy reward signal for the chosen action.
A learner's performance is measured in terms of the regret, which is the (expected) difference between the rewards it actually received
and those it would have received (in expectation) by choosing the optimal action.
Prior work on the regret criterion for finite-armed bandits has treated all actions uniformly and has aimed for bounds on the regret that
do not depend on which action turned out to be optimal. I take a different approach and ask what can be achieved if some actions are given
special treatment. Focussing on worst-case bounds, I ask whether or not it is possible to achieve improved worst-case regret for some
actions, and what is the cost in terms of the regret for the remaining actions. Such results may be useful in a variety of cases. For example,
a company that is exploring some new strategies might expect an especially small regret if its existing strategy turns out to be (nearly) optimal.
This problem has previously been considered in the experts setting where the learner is allowed to observe the reward for all actions in every round, not only for the
action actually chosen. The earliest work seems to be by <cit.> where it is shown that the learner can assign a prior weight to each action and pays
a worst-case regret of $O(\sqrt{-n \log \rho_i})$ for expert $i$ where $\rho_i$ is the prior belief in expert $i$ and $n$ is the horizon. The uniform
regret is obtained by choosing $\rho_i = 1/K$, which leads to the well-known $O(\sqrt{n \log K})$ bound achieved by the exponential weighting algorithm <cit.>.
The consequence of this is that an algorithm can enjoy a constant regret with respect to a single action while suffering minimally on the remainder.
The problem was studied in more detail by <cit.> where (remarkably) the author was able to exactly describe the pareto regret frontier when $K=2$.
Other related work (also in the experts setting) is where the objective is to obtain an improved regret against a mixture of available experts/actions
<cit.>. In a similar vain, <cit.> showed that algorithms for prediction with expert advice can be combined with minimal cost to obtain the
best of both worlds. In the bandit setting I am only aware of the work by <cit.> who study the effect of the prior on the regret of Thompson sampling in
a special case. In contrast the lower bound given here applies to all algorithms in a relatively standard setting.
The main contribution of this work is a characterisation of the pareto regret frontier (the set of achievable worst-case regret bounds)
for stochastic bandits.
Let $\mu_i \in \R$ be the unknown mean of the $i$th arm and assume that $\sup_{i,j} \mu_i - \mu_j \leq 1$. In each time step the learner chooses an
action $I_t \in \set{1,\ldots, K}$ and receives reward $g_{I_t,t} = \mu_i + \eta_t$ where $\eta_t$ is the noise term that I assume to be
sampled independently from a $1$-subgaussian distribution that may depend on $I_t$. This model subsumes both Gaussian and Bernoulli (or bounded)
rewards. Let $\pi$ be a bandit strategy, which is a function from histories of observations to an action $I_t$. Then the $n$-step
expected pseudo regret with respect to the $i$th arm is
R^π_μ,i = n μ_i - ∑_t=1^n μ_I_t ,
where the expectation is taken with respect to the randomness in the noise and the actions of the policy.
Throughout this work $n$ will be fixed, so is omitted from the notation.
The worst-case expected pseudo-regret with respect to arm $i$ is
R^π_i = sup_μR^π_μ, i .
This means that $R^\pi \in \R^K$ is a vector of worst-case pseudo regrets with respect to each of the arms.
Let $\calB \subset \R^K$ be a set defined by
= B ∈[0,n]^K : B_i ≥minn, ∑_j ≠i n/B_j for all i .
The boundary of $\calB$ is denoted by $\frontier$.
The following theorem shows that $\frontier$ describes the pareto regret frontier up to constant factors.
[draw,text width=12.5cm,inner sep=5pt] at (0,0)
There exist universal constants $c_1 = 8$ and $c_2 = 252$ such that:
Lower bound: for $\eta_t \sim \mathcal N(0,1)$ and all strategies $\pi$ we have $c_1(R^\pi + K) \in \calB$
Upper bound: for all $B \in \calB$ there exists a strategy $\pi$ such that $R^\pi_i \leq c_2 B_i$ for all $i$
Observe that the lower bound relies on the assumption that the noise term be Gaussian while the upper bound holds for subgaussian
noise. The lower bound may be generalised to other noise models such as Bernoulli, but does not hold for all subgaussian noise models. For example,
it does not hold if there is no noise ($\eta_t = 0$ almost surely).
The lower bound also applies to the adversarial framework where the rewards may be chosen arbitrarily.
Although I was not able to derive a matching upper bound in this case, a simple modification of the Exp-$\gamma$ algorithm <cit.> leads to an algorithm
R^π_1 ≤B_1 and R^π_k ≲nK/B_1 log(nK/B_1^2) for all k ≥2 ,
where the regret is the adversarial version of the expected regret.
The details may be found in the Appendix.
Details are in the supplementary material.
The new results seem elegant, but disappointing. In the experts setting we have seen that the learner can distribute a prior amongst the actions and
obtain a bound on the regret depending in a natural way on the prior weight of the optimal action. In contrast, in the bandit setting the learner
pays an enormously higher price to obtain a small regret with respect to even a single arm. In fact, the learner must essentially choose a single arm to favour,
after which the regret for the remaining arms has very limited flexibility. Unlike in the experts setting, if even a single arm enjoys constant worst-case
regret, then the worst-case regret with respect to all other arms is necessarily linear.
§ PRELIMINARIES
I use the same notation as <cit.>.
Define $T_i(t)$ to be the number of times action $i$ has been chosen after time step $t$ and $\hat \mu_{i,s}$ to be the empirical estimate
of $\mu_i$ from the first $s$ times action $i$ was sampled. This means that $\hat \mu_{i,T_i(t-1)}$ is the empirical estimate of $\mu_i$ at the start of the $t$th round.
I use the convention that $\hat \mu_{i,0} = 0$.
Since the noise model is $1$-subgaussian we have
∀ϵ> 0 ∃s ≤t : μ̂_i,s - μ_i ≥ϵ/ s ≤exp(-ϵ^2/2t ) .
This result is presumably well known, but a proof is included in <ref> for convenience.
This result is presumably well known, but a proof is included in the supplementary material for convenience.
The optimal arm is $i^* = \argmax_i \mu_i$ with ties broken in some arbitrary way. The optimal reward is $\mu^* = \max_i \mu_i$.
The gap between the mean rewards of the $j$th arm and the optimal arm is
$\Delta_j = \mu^* - \mu_j$ and $\Delta_{ji} = \mu_i - \mu_j$.
The vector of worst-case regrets is $R^\pi \in \R^K$ and has been defined already in <ref>.
I write $R^\pi \leq B \in \R^K$ if $R^\pi_i \leq B_i$ for all $i \in \set{1,\ldots,K}$.
For vector $R^\pi$ and $x \in \R$ we have $(R^\pi + x)_i = R^\pi_i + x$.
§ UNDERSTANDING THE FRONTIER
Before proving the main theorem I briefly describe the features of the regret frontier.
First notice that if $B_i = \sqrt{n(K-1)}$ for all $i$, then
= √(n(K-1))
= ∑_j≠i √(n/(K-1))
= ∑_j≠i n/B_j .
Thus $B \in \calB$ as expected. This particular $B$ is witnessed up to constant factors by MOSS <cit.> and OC-UCB <cit.>, but
not UCB <cit.>, which suffers $R^\text{ucb}_i \in \Omega(\sqrt{n K \log n})$.
Of course the uniform choice of $B$ is not the only option. Suppose the first arm is special, so $B_1$ should be chosen especially small.
Assume without loss of generality that $B_1 \leq B_2 \leq \ldots \leq B_K \leq n$.
Then by the main theorem we have
B_1 ≥∑_i=2^K n/B_i ≥∑_i=2^k n/B_i ≥(k-1)n/B_k .
B_k ≥(k-1)n/B_1 .
This also proves the claim in the abstract, since it implies that $B_K \geq (K-1)n / B_1$.
If $B_1$ is fixed, then choosing $B_k = (k-1)n / B_1$ does not lie on the frontier because
∑_k=2^K n/B_k = ∑_k=2^K B_1/k-1 ∈Ω(B_1 logK)
However, if $H = \sum_{k=2}^K 1/(k-1) \in \Theta(\log K)$, then choosing $B_k = (k-1)nH / B_1$
does lie on the frontier and is a factor of $\log K$ away from the lower bound given in <ref>.
Therefore up the a $\log K$ factor, points on the regret frontier are characterised entirely by a permutation determining
the order of worst-case regrets and the smallest worst-case regret.
Perhaps the most natural choice of $B$ (assuming again that $B_1 \leq\ldots \leq B_K$) is
B_1 = n^p and B_k = (k-1)n^1-pH for k > 1 .
For $p = 1/2$ this leads to a bound that is at most $\sqrt{K} \log K$ worse than that obtained by MOSS and OC-UCB while being a factor of $\sqrt{K}$ better for
a select few.
§.§.§ Assumptions
The assumption that $\Delta_i \in [0,1]$ is used to avoid annoying boundary problems caused by the fact that time is discrete. This means that if $\Delta_i$ is extremely large, then
even a single sample from this arm can cause a big regret bound. This assumption is already quite common, for example a worst-case regret of $\Omega(\sqrt{Kn})$ clearly does not
hold if the gaps are permitted to be unbounded. Unfortunately there is no perfect resolution to this annoyance. Most elegant would be to allow time to be continuous with actions
taken up to stopping times. Otherwise you have to deal with the discretisation/boundary problem with special cases, or make assumptions as I have done here.
§ LOWER BOUNDS
Assume $\eta_t \sim \mathcal N(0, 1)$ is sampled from a standard Gaussian.
Let $\pi$ be an arbitrary strategy, then $\displaystyle 8(R^\pi + K) \in \calB$.
Assume without loss of generality that $R^\pi_1 = \min_i R^\pi_i$ (if this is not the case, then simply re-order the actions).
If $R^\pi_1 > n/8$, then the result is trivial. From now on assume $R^\pi_1 \leq n/8$.
Let $c = 4$ and define
ϵ_k = min1/2, c R^π_k/n ≤1/2 .
Define $K$ vectors $\mu_1,\ldots, \mu_K \in \R^K$ by
(μ_k)_j = 1/2 +
0 if j = 1
ϵ_k if j = k ≠1
-ϵ_j otherwise .
Therefore the optimal action for the bandit with means $\mu_k$ is $k$.
Let $A = \set{k : R^\pi_k \leq n / 8}$ and $A' = \set{k : k \notin A}$ and assume $k \in A$. Then
(b)≥ϵ_k ^π_μ_k [∑_j ≠k T_j(n)]
(c)= ϵ_k (n - ^π_μ_k T_k(n))
(d)= c R^π_k (n - ^π_μ_k T_k(n))/n ,
where (a) follows since $R^\pi_k$ is the worst-case regret with respect to arm $k$,
(b) since the gap between the means of the $k$th arm and any other arm is at least $\epsilon_k$ (Note that this
is also true for $k = 1$ since $\epsilon_1 = \min_k \epsilon_k$.
(c) follows from the fact that $\sum_i T_i(n) = n$ and
(d) from the definition of $\epsilon_k$.
n(1 - 1/c)
≤^π_μ_k T_k(n) .
Therefore for $k \neq 1$ with $k \in A$ we have
n(1 - 1/c)
≤^π_μ_k T_k(n)
(a)≤^π_μ_1 T_k(n) + nϵ_k √(^π_μ_1 T_k(n))
(b)≤n - ^π_μ_1 T_1(n) + nϵ_k √(^π_μ_1 T_k(n))
(c)≤n/c + n ϵ_k √(^π_μ_1 T_k(n)) ,
where (a) follows from standard entropy inequalities and a similar argument as used by <cit.>
(details given in <ref>),
(details in supplementary material),
(b) since $k \neq 1$ and $\E^\pi_{\mu_1} T_1(n) + \E^\pi_{\mu_1} T_k(n) \leq n$, and
(c) by <ref>.
^π_μ_1 T_k(n)
≥1 - 2/c/ϵ_k^2 ,
which implies that
= ∑_k=2^K ϵ_k ^π_μ_1 T_k(n)
≥∑_k ∈A - 1 1 - 2/c/ϵ_k
= 1/8 ∑_k ∈A - 1 n/R^π_k .
Therefore for all $i \in A$ we have
8R^π_i ≥∑_k ∈A - 1 n/R^π_k ·R^π_i/R^π_1
≥∑_k ∈A - i n/R^π_k .
8R^π_i + 8K ≥∑_k≠i n/R^π_k + 8K - ∑_k ∈A' - i n/R^π_k
≥∑_k ≠i n/R^π_k ,
which implies that $8(R^\pi + K) \in \calB$ as required.
§ UPPER BOUNDS
I now show that the lower bound derived in the previous section is tight up to constant factors.
The algorithm is a generalisation MOSS <cit.> with two modifications.
First, the width of the confidence bounds are biased in a non-uniform way, and second, the upper confidence bounds
are shifted.
The new algorithm is functionally identical to MOSS in the special case that $B_i$ is uniform.
Define $\log_{+}\!(x) = \max\set{0, \log(x)}$.
Unbalanced MOSS
Input: $n$ and $B_1,\ldots,B_K$
$n_i = n^2 / B_i^2$ for all $i$
$t \in 1,\ldots,n$
$\displaystyle I_t = \argmax_i \hat \mu_{i,T_i(t-1)} + \sqrt{\frac{4}{T_i(t-1)} \logp\left(\frac{n_i}{T_i(t-1)}\right)} - \sqrt{\frac{1}{n_i}}$
Let $B \in \calB$, then the strategy $\pi$ given in <ref> satisfies
$R^\pi \leq 252B$.
For all $\mu$ the following hold:
* $R^\pi_{\mu,i^*} \leq 252 B_{i^*}$.
* $R^\pi_{\mu,i^*} \leq \min_i (n\Delta_i + 252 B_i)$
The second part of the corollary is useful when $B_{i^*}$ is large, but there exists an arm
for which $n\Delta_i$ and $B_i$ are both small.
The proof of <ref> requires a few lemmas.
The first is a somewhat standard concentration inequality
that follows from a combination of the peeling argument and Doob's maximal inequality.
The proof may be found in the supplementary material.
Let $\displaystyle Z_i = \max_{1 \leq s \leq n} \mu_{i} - \hat \mu_{i,s} - \sqrt{\frac{4}{s} \logp\left(\frac{n_{i}}{s}\right)}$.
Then $\P{Z_i \geq \Delta} \leq \frac{20}{n_{i} \Delta^2}$ for all $\Delta > 0$.
Using the peeling device.
Z_i ≥Δ
(a)= ∃s ≤n : μ_i - μ̂_i,s ≥Δ+ √(4/s (n_i/s))
(b)≤∑_k=0^∞∃s < 2^k+1 : s(μ_i - μ̂_i,s) ≥2^k Δ+ √(2^k+2 (n_i/2^k+1))
(c)≤∑_k=0^∞exp(-2^k-2 Δ^2 ) min1, 2^k+1/n_i
(d)≤(8/log(2) + 8) ·1/n_i Δ^2
≤20/n_iΔ^2 ,
where (a) is just the definition of $Z_i$,
(b) follows from the union bound and re-arranging the equation inside the probability,
(c) follows from <ref> and the definition of $\log_{+}$ and
(d) is obtained by upper bounding the sum with an integral.
In the analysis of traditional bandit algorithms the gap $\Delta_{ji}$ measures how quickly the algorithm can detect the
difference between arms $i$ and $j$. By design, however, <ref> is negatively biasing
its estimate of the empirical mean of arm $i$ by $\sqrt{1/n_i}$. This has the effect of shifting the gaps, which I denote
by $\bar \Delta_{ji}$ and define to be
Δ̅_ji = Δ_ji + √() - √(1/n_i)
= μ_i - μ_j + √() - √(1/n_i) .
Define stopping time $\tau_{ji}$ by
τ_ji = mins : μ̂_j,s + √(4/s (n_j/s)) ≤μ_j + Δ̅_ji / 2 .
If $Z_i < \bar \Delta_{ji} / 2$, then $T_j(n) \leq \tau_{ji}$.
Let $t$ be the first time step such that $T_j(t-1) = \tau_{ji}$. Then
μ̂_j,T_j(t-1) + √(4/T_j(t-1) (n_j/T_j(t-1))) - √()
≤μ_j + Δ̅_ji / 2 - √()
=μ_j + Δ̅_ji - Δ̅_ji /2 - √()
=μ_i - √(1/n_i) - Δ̅_ji/2
<μ̂_i,T_i(t-1) + √(4/T_i(t-1) (n_i/T_i(t-1))) - √(1/n_i) ,
which implies that arm $j$ will not be chosen at time step $t$ and so also not for any subsequent time steps by the same argument and induction.
Therefore $T_j(n) \leq \tau_{ji}$.
If $\bar \Delta_{ji} > 0$,
then $\displaystyle \E \tau_{ji} \leq \frac{40}{\bar \Delta_{ji}^2} + \frac{64}{\bar \Delta_{ji}^2} \plog\left(\frac{n_j \bar\Delta_{ji}^2}{64}\right)$.
Let $s_0$ be defined by
s_0 = 64/Δ̅_ji^2 (n_jΔ̅_ji^2/64)
√(4/s_0 (n_j/s_0)) ≤Δ̅_ji/4 .
= ∑_s=1^n τ_ji ≥s
≤1 + ∑_s=1^n-1 μ̂_i,s - μ_i,s ≥Δ̅_ji/2 - √(4/s (n_j/s))
≤1 + s_0 + ∑_s=s_0+1^n-1 μ̂_i,s - μ_i,s ≥Δ̅_ji/4
≤1 + s_0 + ∑_s=s_0+1^∞exp(-s Δ̅_ji^2/32)
≤1 + s_0 + 32/Δ̅_ji^2
≤40/Δ̅_ji^2 + 64/Δ̅_ji^2 (n_j Δ̅_ji^2/64) ,
where the last inequality follows since $\bar \Delta_{ji} \leq 2$.
Let $\Delta = 2/\sqrt{n_i}$ and $A = \set{j : \Delta_{ji} > \Delta}$. Then for $j \in A$ we have
$\Delta_{ji} \leq 2 \bar\Delta_{ji}$ and $\bar \Delta_{ji} \geq \sqrt{1/n_i} + \sqrt{\fixnj}$.
Letting $\Delta' = \sqrt{1/n_i}$ we have
= [∑_j=1^K Δ_ji T_j(n)]
≤nΔ+ [∑_j ∈A Δ_ji T_j(n)]
(a)≤2B_i + [∑_j ∈A Δ_ji τ_ji + n max_j ∈A Δ_ji : Z_i ≥Δ̅_ji/2 ]
(b)≤2B_i + ∑_j ∈A (80/Δ̅_ji + 128/Δ̅_ji(n_j Δ̅_ji^2/64)) + 4n [Z_i Z_i ≥Δ']
(c)≤2B_i + ∑_j ∈A 90 √(n_j) + 4n [Z_i Z_i ≥Δ'] ,
where (a) follows by using <ref> to bound $T_j(n) \leq \tau_{ji}$ when $Z_i < \bar \Delta_{ji}$.
On the other hand, the total number of pulls for arms $j$ for which $Z_i \geq \bar \Delta_{ji} / 2$ is at most $n$.
(b) follows by bounding $\tau_{ji}$ in expectation using <ref>.
(c) follows from basic calculus and because for $j \in A$ we have $\bar \Delta_{ji} \geq \sqrt{1/n_i}$.
All that remains is to bound the expectation.
4n [Z_i Z_i ≥Δ']
≤4n Δ' Z_i ≥Δ' + 4n ∫^∞_Δ' Z_i ≥z dz
≤160n/Δ' n_i
= 160n/ √(n_i) = 160B_i ,
where I have used <ref> and simple identities.
Putting it together we obtain
R^π_μ,i ≤2 B_i + ∑_j ∈A 90 √(n_j) + 160B_1
≤252 B_i ,
where I applied the assumption $B \in \calB$ and so $\sum_{j \neq 1} \sqrt{n_j} = \sum_{j \neq 1} n/B_j \leq B_i$.
The above proof may be simplified in the special case that $B$ is uniform where we recover the minimax regret of MOSS, but with perhaps a simpler proof
than was given originally by <cit.>.
§.§ On Logarithmic Regret
In a recent technical report I demonstrated empirically that MOSS suffers sub-optimal problem-dependent regret in terms of the minimum
gap <cit.>. Specifically, it can happen that
R^moss_μ,i^* ∈Ω(K/Δ_min logn) ,
where $\Delta_{\min} = \min_{i : \Delta_i > 0} \Delta_i$.
On the other hand, the order-optimal asymptotic regret can be significantly smaller. Specifically, UCB by <cit.> satisfies
R^ucb_μ,i^* ∈O(∑_i : Δ_i > 0 1/Δ_i logn) ,
which for unequal gaps can be much smaller than <ref> and is asymptotically order-optimal <cit.>.
The problem is that MOSS explores only enough to obtain minimax regret, but sometimes obtains
minimax regret even when a more conservative algorithm would do better. It is worth remarking that this effect is harder to observe than one might think.
The example given in the afforementioned technical report is carefully tuned to exploit this failing, but still
requires $n = 10^9$ and $K = 10^3$ before significant problems arise.
In all other experiments MOSS was performing admirably in comparison to UCB.
All these problems can be avoided by modifying UCB rather than MOSS. The cost is a factor of $O(\sqrt{\log n})$.
The algorithm is similar to <ref>, but chooses the action that maximises the following index.
I_t = _i μ̂_i,T_i(t-1) + √((2 + ϵ) logt/T_i(t-1)) - √(logn/n_i) ,
where $\epsilon > 0$ is a fixed arbitrary constant.
If $\pi$ is the strategy of unbalanced UCB with $n_i = n^2 / B_i^2$ and $B \in \calB$, then the regret of the unbalanced UCB satisfies:
* (problem-independent regret). $R^\pi_{\mu,i^*} \in O\left(B_{i^*} \sqrt{\log n}\right)$.
* (problem-dependent regret). Let $A = \set{i : \Delta_i \geq 2\sqrt{1/n_{i^*} \log n}}$. Then
R^π_μ,i^* ∈O(B_i^* √(logn) A ≠∅ + ∑_i ∈A 1/Δ_i logn) .
The proof may be found in <ref>.
The proof is deferred to the supplementary material.
The indicator function in the problem-dependent bound vanishes for sufficiently large $n$ provided $n_{i^*} \in \omega(\log(n))$, which is equivalent to
$B_{i^*} \in o(n / \sqrt{\log n})$.
Thus for reasonable choices of $B_1,\ldots,B_{K}$ the algorithm is going to enjoy the same asymptotic performance as UCB.
<ref> may be proven for any index-based algorithm for which it can be shown that
T_i(n) ∈O(1/Δ_i^2 logn) ,
which includes (for example) KL-UCB <cit.> and Thompson sampling (see analysis by <cit.> and original paper by <cit.>), but not OC-UCB <cit.> or MOSS <cit.>.
§.§ A Note on Constants
The constants in the statement of <ref> can be improved by carefully tuning all thresh-holds, but the proof would
grow significantly and I would not expect a corresponding boost in practical performance. In fact, the reverse is true, since the “weak” bounds used
in the proof would propagate to the algorithm.
Also note that the $4$ appearing in the square root
of the unbalanced MOSS algorithm is due to the fact that I am not assuming rewards are
bounded in $[0,1]$ for which the variance is at most $1/4$. It is possible to replace the $4$ with $2 + \epsilon$ for any $\epsilon > 0$ by
changing the base in the peeling argument in the proof of <ref> as was done by <cit.> and others.
§.§ Experimental Results
[comment chars=%]data/fig1.txt
[comment chars=%]data/fig2.txt
cycle list=blue, red,dotted, green!50!black,dashdotdotted, black
every axis plot/.append style=line width=1.5pt
I compare MOSS and unbalanced MOSS in two simple simulated examples, both with horizon $n = 5000$.
Each data point is an empirical average of $\sim\!\!10^4$ i.i.d. samples, so error bars are too small to see. Code/data is available in the supplementary material.
The first experiment has $K = 2$ arms and $B_1 = n^{\frac{1}{3}}$ and $B_2 = n^{\frac{2}{3}}$.
I plotted the results for $\mu = (0, -\Delta)$ for varying $\Delta$. As predicted, the new algorithm performs significantly better than MOSS
for positive $\Delta$, and significantly worse otherwise (<ref>).
The second experiment has $K = 10$ arms. This time $B_1 = \sqrt{n}$ and $B_k = (k-1)H \sqrt{n}$ with $H = \sum_{k=1}^9 1/k$.
Results are shown for $\mu_k = \Delta \ind{k = i^*}$ for $\Delta \in [0,1/2]$ and $i^* \in \set{1,\ldots,10}$.
Again, the results agree with the theory. The unbalanced algorithm is superior to MOSS for $i^* \in \set{1,2}$ and inferior otherwise (<ref>).
+[] table[x index=0,y index=1] ;
+[] table[x index=0,y index=2] ;
U. MOSS;
+[blue,solid,mark=none,restrict x to domain=0:0.5] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=0.5:1] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=1:1.5] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=1.5:2] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=2:2.5] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=2.5:3] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=3:3.5] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=3.5:4] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=4:4.5] table[x index=0,y index=1] ;
+[blue,solid,mark=none,restrict x to domain=4.5:5] table[x index=0,y index=1] ;
+[red,mark=none,dotted,restrict x to domain=0:0.5] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=0.5:1] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=1:1.5] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=1.5:2] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=2:2.5] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=2.5:3] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=3:3.5] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=3.5:4] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=4:4.5] table[x index=0,y index=2] ;
+[red,mark=none,dotted,restrict x to domain=4.5:5] table[x index=0,y index=2] ;
[thin,samples=50,smooth] coordinates (0.5,0)(0.5,2500);
[thin,samples=50,smooth] coordinates (1.5,0)(1.5,2500);
[thin,samples=50,smooth] coordinates (2.5,0)(2.5,2500);
[thin,samples=50,smooth] coordinates (3.5,0)(3.5,2500);
[thin,samples=50,smooth] coordinates (4.5,0)(4.5,2500);
[thin,samples=50,smooth] coordinates (1,0)(1,2500);
[thin,samples=50,smooth] coordinates (2,0)(2,2500);
[thin,samples=50,smooth] coordinates (3,0)(3,2500);
[thin,samples=50,smooth] coordinates (4,0)(4,2500);
[thin,samples=50,smooth] coordinates (5,0)(5,2500);
$\theta = \Delta + (i^*-1) / 2$
Sadly the experiments serve only to highlight the plight of the biased learner, which suffers significantly worse results than
its unbaised counterpart for most actions.
§.§ Adversarial Bandits
By modifying the Exp3-$\gamma$ in a trivial way it is possible to obtain a strategy $\pi$ depending on $B_1 > 0$
such that
R^π_g = max_i [∑_t=1^n g_i,t - g_I_t,t]
B_1 if _i ∑_t=1^n g_i,t = 1
B_1/2 + 2Kn/B_1 log(4Kn(K-1)/B_1^2) otherwise .
where $g_{i,t}$ is the gain/reward from choosing action $i$ in time step $t$.
Details may be found in <ref>.
Details may be found in the supplementary material.
§ DISCUSSION
I have shown that the cost of favouritism for multi-armed bandit algorithms is rather serious. If an algorithm exhibits a small worst-case
regret for a specific action, then the worst-case regret of the remaining actions is necessarily significantly larger than the well-known
uniform worst-case bound of $\Omega(\sqrt{Kn})$. This unfortunate result is in stark contrast to the experts setting for which there exist
algorithms that suffer constant regret with respect to a single expert at almost no cost for the remainder.
Surprisingly, the best achievable (non-uniform) worst-case bounds are determined up to a permutation almost entirely by the value of the smallest worst-case regret.
There are some interesting open questions. Most notably, in the adversarial setting I am not sure if the upper or lower bound is tight (or neither).
It would also be nice to know if the constant factors can be determined exactly asymptotically, but so far this has not been done even
in the uniform case. For the stochastic setting it is natural to ask if the OC-UCB algorithm can also be modified. Intuitively one would expect this to be
possible, but it would require re-working the very long proof.
§.§.§ Acknowledgements
I am indebted to the very careful reviewers who made many suggestions for improving this paper. Thank you!
§ TABLE OF NOTATION
$n$ time horizon
$K$ number of available actions
$t$ time step
$k,i$ actions
$\calB$ set of achievable worst-case regrets defined in <ref>
$\frontier$ boundary of $\calB$
$\mu$ vector of expected rewards $\mu \in [0,1]^K$
$\mu^*$ expected return of optimal action
$\Delta_j$ $\mu^* - \mu_j$
$\Delta_{ji}$ $\mu_i - \mu_j$
$\pi$ bandit strategy
$I_t$ action chosen at time step $t$
$R_{\mu,k}^\pi$ regret of strategy $\pi$ with respect to the $k$th arm
$R_k^\pi$ worst-case regret of strategy $\pi$ with respect to the $k$th arm
$\hat \mu_{k,s}$ empirical estimate of the return of the $k$ action after $s$ samples
$T_k(t)$ number of times action $k$ has been taken at the end of time step $t$
$i^*$ optimal action
$\log_{+}\!(x)$ maximum of $0$ and $\log(x)$
$\mathcal N(\mu, \sigma^2)$ Gaussian with mean $\mu$ and variance $\sigma^2$
§ PROOF OF <REF>
Recall that the proof of UCB depends on showing that
T_i(n) ∈O(1/Δ_i^2 logn) .
Now unbalanced UCB operates exactly like UCB, but with shifted rewards.
Therefore for unbalanced UCB we have
T_i(n) ∈O(1/Δ̅_i^2 logn) ,
Δ̅_i ≥Δ_i + √(logn/n_i) - √(logn/n_i^*) .
Define :
A = i : Δ_i ≥2√(logn/n_i^*)
If $i \in A$, then $\Delta_i \leq 2 \bar \Delta_i$ and $\bar \Delta_i \geq \sqrt{\frac{\log n}{n_i}}$.
Δ_i T_i(n)
∈O(Δ_i/Δ̅_i^2 logn)
⊆O(1/Δ̅_i logn)
⊆O(√(n_i logn))
⊆O(n/B_i √(logn)) .
For $i \notin A$ we have $\Delta_i < 2\sqrt{\frac{\log n}{n_{i^*}}}$ thus
[∑_i ∉A Δ_iT_i(n) ]
∈O(n √(logn/n_i^*))
⊆O(B_i^* √(logn)) .
= ∑_i=1^K Δ_i T_i(n)
∈O((B_i^* + ∑_i ∈A n/B_i) √(logn))
= O(B_i^* √(logn))
as required.
For the problem-dependent bound we work similarly.
= ∑_i=1^K Δ_i T_i(n)
∈O(∑_i ∈A 1/Δ̅_i logn + A ≠∅ B_i^*√(logn))
∈O(∑_i ∈A 1/Δ_i logn + A ≠∅ B_i^*√(logn)) .
§ KL TECHNIQUES
Let $\mu_1, \mu_k \in \R^K$ be two bandit environments as defined in the proof of <ref>.
Here I prove the claim that
^π_μ_k T_k(n) - ^π_μ_1 T_k(n) ≤nϵ_k √(^π_μ_1 T_k(n)) .
The result follows along the same lines as the proof of the lower bounds given by <cit.>.
Let $\set{\calF_t}_{t=1}^n$ be a filtration where $\calF_t$ contains information about rewards and actions chosen
up to time step $t$. So $g_{I_t,t}$ and $\ind{I_t = i}$ are measurable with respect to $\calF_t$.
Let $P_1$ and $P_k$ be the measures on $\calF$ induced by bandit problems $\mu_1$ and $\mu_k$ respectively.
Note that $T_k(n)$ is a $\calF_n$-measurable random variable bounded in $[0,n]$. Therefore
^π_μ_k T_k(n) - ^π_μ_1 T_k(n)
(a)≤n sup_A |P_1(A) - P_2(A) |
(b)≤n √(1/2 (P_1, P_k)) ,
where the supremum in (a) is taken over all measurable sets (this is the total variation distance) and (b) follows from
Pinsker's inequality.
It remains to compute the KL divergence.
Let $P_{1,t}$ and $P_{k,t}$ be
the conditional measures on the $t$th reward. By the chain rule for the KL divergence we have
(P_1, P_k) = ∑_t=1^n _P_1 (P_1,t, P_k,t)
(a)= 2ϵ_k^2 ∑_t=1^n _P_1 I_t = k
= 2ϵ_k^2 ^π_μ_1 T_k(n) ,
where (a) follows by noting that if $I_t \neq k$, then the distribution of the rewards at time step $t$ is the same for both
bandit problems $\mu_1$ and $\mu_k$. For $I_t = k$ we have the difference in means is $(\mu_k)_k - (\mu_1)_k = \epsilon_k$ and
since the distributions are Gaussian the KL divergence is $2\epsilon_k^2$.
For Bernoulli random noise the KL divergence is also $\Theta(\epsilon_k^2)$ provided $(\mu_k)_k \approx (\mu_1)_k \approx 1/2$ and so
a similar proof works for this case. See the work by <cit.> for an example.
§ ADVERSARIAL BANDITS
In the adversarial setting I obtain something similar. First I introduce some new notation.
Let $g_{i,t} \in [0,1]$ be the gain/reward from choosing action $i$ at time step $t$. This is chosen in an arbitrary way by the adversary with $g_{i,t}$
possibly even dependent on the actions of the learner up to time step $t$. The regret difference between the gains obtained by the learner and those
of the best action in hindsight.
R_g^π= max_i ∈1,…,K [∑_t=1^n g_i,t - g_I_t,t] .
I make the most obvious modification to the Exp3-$\gamma$ algorithm, which is to bias the prior towards the special action and tune the learning rate accordingly.
The algorithm accepts as input the prior $\rho \in [0,1]^K$, which must satisfy $\sum_i \rho_i = 1$, and the learning rate $\eta$.
Input: $K$, $\rho \in [0,1]^K$, $\eta$
$w_{i,0} = \rho_i$ for each $i$
$t \in 1,\ldots,n$
Let $p_{i,t} = \frac{w_{i,t-1}}{\sum_{i=1}^K w_{i,t-1}}$
Choose action $I_t = i$ with probability $p_{i,t}$ and observe gain $g_{I_t,t}$
$\tilde \ell_{t,i} = \frac{(1 - g_{t,i}) \ind{I_t = i}}{p_{i,t}}$
$w_{i,t} = w_{i,t-1} \exp\left(-\eta \tilde \ell_{t,i}\right)$
The following result follows trivially from the standard proof.
Does Exp3-$\gamma$ even appear in this reference? First reference for Exp-$\gamma$?
Let $\pi$ be the strategy determined by <ref>, then
R_g^π≤ηKn + 1/η log1/ρ_i^* .
If $\rho$ is given by
ρ_i =
exp(-B_1^2/4Kn) if i = 1
(1 - ρ_1) / (K-1) otherwise
and $\eta = B_1 / (2Kn)$, then
B_1 if i^* = 1
B_1/2 + 2Kn/B_1 log(4Kn(K-1)/B_1^2) otherwise .
The proof follows immediately from <ref> by noting that for $i^* \neq 1$ we have
= log(K-1/1 - exp(-B_1^2/4Kn))
as required.
§ CONCENTRATION
The following straight-forward concentration inequality is presumably well known and the proof of an almost identical result is available by <cit.>, but
an exact reference seems hard to find.
Let $X_1,X_2,\ldots, X_n$ be independent and $1$-subgaussian, then
∃t ≤n : 1/t ∑_s ≤t X_s ≥ϵ/t ≤exp(-ϵ^2/2n) .
Since $X_i$ is $1$-subgaussian, by definition it satisfies
(∀λ∈) [exp(λX_i)] ≤exp(λ^2/2) .
Now $X_1,X_2,\ldots$ are independent and zero mean, so by convexity of the exponential function $\exp(\lambda \sum_{s=1}^t X_s)$ is a sub-martingale.
Therefore if $\epsilon > 0$, then by Doob's maximal inequality
∃t ≤n : ∑_s=1^t X_s ≥ϵ
= inf_λ≥0 ∃t ≤n : exp(λ∑_s=1^t X_s) ≥exp(λϵ)
≤inf_λ≥0 exp(λ^2 n/2 -λϵ)
= exp(-ϵ^2/2n)
as required.
|
1511.00044
|
Department of Chemistry, University of California, Irvine, CA 92697
The Milestoning algorithm created by Ron Elber et al. is a method for determining the time scale of processes too complex to be studied using brute force simulation methods. The fundamental objects implemented in the Milestoning algorithm are the first passage time distributions $K_{AB}(\tau)$ between adjacent protein configuration milestones $A$ and $B$. The method proposed herein aims to further enhance Milestoning by employing an artificial applied force, akin to wind, which pushes the trajectories from their initial states to their final states, and subsequently re-weights the trajectories to yield the true first passage time distributions $K_{AB}(\tau)$ in a fraction of the computation time required for unassisted calculations. The re-weighting method, rooted in Ito's stochastic calculus, was adapted from previous work by Andricioaei et al. The theoretical basis for this technique and numerical examples are presented.
Valid PACS appear here
§ INTRODUCTION
The task of calculating kinetic properties from molecular dynamics simulations is a complex problem of considerable interest <cit.> <cit.>. In contrast to computational methods designed for equilibrium calculations, in which the basic observables are thermodynamic averages over conformational points (structures) generated over an invariant measure without the need to obey exact dynamical equations, studies of kinetics require physically correct time-ordered trajectories to obtain time-correlation functions as the basic objects <cit.>. Since each time-correlation function describes the relaxation under investigation as an average over all relevant trajectories, adequate sampling for accurate calculation of long-time kinetics can quickly become computationally intractable via direct simulation <cit.>. This is because a direct, brute force method of this type would require sufficiently long simulation times such that the system would be able to transition between states of interest enough times that a statistically significant distribution of first passage times could be generated. Several computational methods have been developed to address the challenge of calculating chemical kinetics, starting with the venerable transition state theory (TST) <cit.> <cit.>, and continuing with more recent developments, such as, transition path sampling (TPS) <cit.>, transition path theory (TPT) <cit.>, and transition interface sampling (TiS) <cit.>. Although transition state theory has been successfully used in the determination of the kinetics for many systems with well-defined reactant and product states, for which the “dynamical bottleneck" can be identified <cit.>, there are many interesting problems in biophysics, and elsewhere, for which these assumptions do not hold. In contrast, transition path sampling approaches require no intuition for reaction mechanisms or advance knowledge of transition state, although the requirement of a "dynamical bottleneck" does persist <cit.> <cit.>. In the same category of methods is the milestoning algorithm created by Ron Elber et al., which is a method for calculating kinetic properties, where the fundamental objects are the first passage time distributions $K_{AB}(\tau)$ between adjacent protein configuration milestone states (configurations $A$ and $B$ in this case), where the milestone states do not necessarily need to be meta-stable states as in transition state theory. The key feature of the milestoning method is that long trajectory pathways for large scale configuration changes can be broken up into shorter trajectories for which a linear network of transition probabilities between milestones can be devised. The aforementioned linear networks of transition probabilities can then be solved for such quantities as first passage time between any pair of milestones, including those at the extreme ends of the space, and the flux through a given milestone, $s$, as a function of time, written as $P_s(t)$ (equation <ref>). Some of the key gains from this treatment are that breaking up these long trajectory pathways into a network of shorter trajectories leads to increased sampling of the would-be under-sampled areas, and that gains in computational efficiency are possible due to the capacity to run these short trajectories in parallel <cit.>. In practice, previous milestoning calculations have been limited to calculating the constant flux values representative of the system at equilibrium, which can be thought of as the long time flux values $\lim_{t \to \infty}P_s(t)$. A method for calculating the time-dependent flux through a given milestone $P_s(t)$ can be found in the companion paper to this article, also in this publication <cit.>. The aim of the technique we present in this paper is to increase the computational speed of the milestoning method via the addition of an artificial constant force ($\mathscr{F}_{wind}$) along the vector pointing from the initial state to the final state for each pair of milestones in the simulation, causing the system to arrive at the destination configuration in far fewer time steps than if it were left to Brownian dynamics alone. The key idea which makes this possible is the use of a re-weighting function we have introduced previously <cit.> <cit.> <cit.> <cit.>, which generates a re-weighting coefficient for each trajectory, thus allowing the true distribution of first passage times to be recovered from the artificially accelerated trajectories. Preliminary calculations conducted on model systems, described in the Numerical Demonstration section, have demonstrated a computation time speedup by a factor of nearly $40$ using this method.
§ THEORY
The quantity of most fundamental importance in milestoning is the flux through a given milestone, for which the equation is <cit.>:
\begin{equation}
\label{m}
%q_{\alpha}(t) = p_{\alpha}\delta(t - 0^+) + \displaystyle\sum\limits_{\beta}\int_0^t q_{\beta}(t')K_{\beta\alpha}(t - t')dt'
P_s(t) = \int_0^t Q_s(t')\left[ 1-\int_0^{t-t'}K_s(\tau)d\tau \right]dt' , \nonumber
\end{equation}
\begin{equation}
Q_s(t) = 2 \delta(t)P_s(0) + \int_0^t Q_{s\pm1}(t'')K^{\mp}_{s\pm1}(t-t'')dt''
\end{equation}
where $P_{s}(t)$ is the probability of being at milestone $s$ at time $t$, (or, more specifically, arriving at time $t'$ and not leaving before time $t$ <cit.>), and $Q_{s}(t)$ is the probability of a transition to milestone $s$ at time t. $K_s(\tau)$ indicates the probability of transitioning out of milestone $s$ given an incubation time of $\tau$, thus $\int_0^{t-t'}K_s(\tau)d\tau$ is the probability of an exit from milestone $s$ anytime between $0$ and $t-t'$, which makes $1-\int_0^{t-t'}K_s(\tau)d\tau$ the probability of there not being an exit from milestone $s$ over that same time period. Since the probability of two independent simultaneous events happening concurrently is the product of the two events, the equation for $P_s(t)$ is simply integrating the concurrent probabilities of arriving at milestone $s$ and not leaving over the time frame from time $0$ to $t$. Dissecting the meaning of $Q_{s}(t)$ (14), the first term, $2 \delta(t)P_s(0)$, simply represents the probability that the system is already occupying milestone $s$ at time $t = 0$, where the factor of 2 is present since the $\delta$ function is centered at zero, meaning only half of its area would be counted without this factor. $Q_{s\pm1}(t'')$ is the probability that the system transitioned into one of the two milestones adjacent to $s$ at an earlier time $t''$. $K^{\mp}_{s\pm1}(t-t'')$ is the probability of a transition from milestones $s\pm 1$ into milestone $s$. Thus the second term of the second line of equation 14 is another concurrent probability: the probability of the system entering an adjacent milestone at an earlier time, and then transitioning into milestone $s$ between time $t$ and $0$. It is important to note that all functions $P_s(t)$ and $Q_s(t)$ are calculated using the respective values of $K_s(\tau)$ between adjacent milestones, thus the set of $K_s(\tau)$ between all milestones of interest contains all the information needed to calculate kinetics using the milestoning method.
The important connection to make in regard to combining the milestoning method with re-weighting of artificially accelerated trajectories is that a $K$ function between two milestones $x = A$ and $x = B$, $K_{AB}(\tau)$, is nothing more than a probability distribution as a function of lifetime describing the conditional probability that a system found in state $A$ at time $t = 0$ will be found, for the first time, in state $B$ at time $t = \tau$:
\begin{equation}
K_{AB}(\tau) = P(x_B, \tau | x_A, 0)
\end{equation}
Given this relationship, we can now begin to make the connection between milestoning and re-weighting of artificially accelerated trajectories. Assuming Langevin dynamics with the addition of a $wind$ force:
\begin{equation}
m\ddot{x} = -\gamma m \dot{x} -\nabla V(x) + \xi (t) + \mathscr{F}_{wind}
\end{equation}
where $\gamma$ is the friction coefficient, $V(x)$ is the potential, $\xi (t)$ is the random force, and $\mathscr{F}_{wind}$ is a constant force applied in the direction of the goal milestone for each run; conditional probabilities reflecting first passage transitions from milestone $A$ to $B$ can be expressed as:
\begin{equation}
P(x_B, \tau | x_A, 0) = \int D \xi W[\xi] \delta (x(\tau) - x_B)
\end{equation}
In this equation, $W[\xi(t)]$ is the probability distribution representing the joint probabilities of all possible series of random kicks, so multiplying by the delta function $\delta (x(\tau) - x_B)$ and integrating selects for only the portion of the distribution which represents a series of random kicks which results in a transition from state $A$ to state $B$ given an incubation time $\tau$. It then follows suit that the integral in this equation is simply the expectation value for the probability of a transition from $A$ to $B$ for each incubation time point $\tau$, which, again, is equivalent to $K_{AB}(\tau)$. Because of fluctuation dissipation, $\langle \xi(t) \xi(t') \rangle = 2 k_B T m \gamma$, the random force in Langevin dynamics is a Gaussian distribution with variance $w \equiv 2 k_B T m \gamma$. Thus it can be show that:
\begin{equation}
W[\xi(t)] = \exp (-\frac{1}{2w}\int_0^t \xi(t')^2 dt')
\end{equation}
With $W[\xi(t)]$, our weighting function for joint probabilities of random kick sequences in terms of our random force $\xi(t)$ in hand, we can now write the noise history $\xi(t)$ in terms of the trajectory $x(t)$ it generates:
\begin{equation}
\xi(t)^2 = (m\ddot{x} + \gamma m \dot{x} + \nabla V(x) - \mathscr{F}_{wind})^2
\end{equation}
We are ultimately interested in measuring conditional probability distributions in configuration space, $x(t)$, not random force space, $\xi(t)$, but since the Jacobian is built into the measure, $x$, we can define $S[x(t)]$ thusly:
\begin{equation}
S[x(t)] \equiv \xi(t)^2 = (m\ddot{x} + \gamma m \dot{x} + \nabla V(x) - \mathscr{F}_{wind})^2
\end{equation}
Then, using the Ito formalism for stochastic calculus, we can express our conditional probability using the Wiener formalism of path integrals <cit.> as:
\begin{equation}
P(x_B, \tau | x_A, 0) = \int_{(x_A, 0)}^{(x_B, \tau)} D x \exp \left( -\frac{S[x(t)]}{2w} \right)
\end{equation}
In this form, it is clear that the exponential function represents the weighting function for the trajectory $x(t)$:
\begin{equation}
W[x(t)] \equiv \exp \left( -\frac{S[x(t)]}{2w} \right)
\end{equation}
With the weight of each trajectory, $W[x(t)]$, now formally defined, we can define the re-weighting factor for obtaining the true weight of an artificially accelerated trajectory as:
\begin{equation}
\frac{W[x(t)]}{W_f[x(t)]} = \exp \left( -\frac{S[x(t)] - S_f[x(t)]}{2w}\right)
\end{equation}
where the $f$ subscript indicates a function generated under the influence of the artificial $\mathscr{F}_{wind}$ force. In practice, once a trajectory $x(t)$ is generated (in the presence of the wind force), the actions are calculated in discrete numerical form using:
\begin{equation}
S_f[x(t)] \approx \sum_i \left( m\frac{\Delta \nu_i}{\Delta t} + m\gamma \frac{\Delta x_i}{\Delta t} + \nabla V_i - \mathscr{F}_{wind} \right)^2 \Delta t \nonumber
\end{equation}
\begin{equation}
S[x(t)] \approx \sum_i \left( m\frac{\Delta \nu_i}{\Delta t} + m\gamma \frac{\Delta x_i}{\Delta t} + \nabla V_i \right)^2 \Delta t
\label{x}
\end{equation}
The re-weighting factor is calculated from Eq. (10) and stored in an array. When post-processing to compute the $K_{AB}(\tau)$ distribution for a particular pair of milestones $A$ and $B$ by histogramming trajectories by lifetime $\tau$, instead of adding 1 to a particular bin each time the lifetime of a particular trajectory falls within the bounds of that bin, the weight $W[x(t)]$ corresponding to that trajectory is instead added. It is clear from equation 10 that as $S_f[x(t)]$ for the artificially accelerated trajectory approaches $S[x(t)]$, the weight of the trajectory approaches unity, thus the method reduces to an unweighted histogram for $\mathscr{F}_{wind} = 0$ as it should.
§ NUMERICAL DEMONSTRATION
§.§ Model System in One Dimension
A simple two well potential (see inset of figure 7) of equation $y = (x-1)^2(x+1)^2$ was chosen to be the model system upon which the wind-assisted milestoning methodology could be developed. In running wind-assisted milestoning, the potential to which the particle is being subjected is first divided into any number of milestones, in this case, 7 milestones, thus 6 separate spaces. Next, numerous Langevin trajectories are run both from left to right and right to left between each pair of adjacent milestones. The number of time steps required to go from the starting milestone to the destination milestone for each trial of each pair and the weight of each trajectory is then stored in an array as mentioned in the theory section. As shown in figures 1 and 2, this method has brought about a more than tenfold speedup in computation time with very little sacrifice in terms of accuracy.
Shown here is calculation time as a function of the $\mathscr{F}_{wind}$ force for all $K(\tau)$ distributions in both directions for six subspaces ranging from -2 to 2 on the bistable harmonic potential. The calculation took 507 seconds for $\mathscr{F}_{wind} = 0 pN$ and just 44 seconds for $\mathscr{F}_{wind} = 12 pN$.
Shown in this figure is the transition probability distribution $K_{23}(\tau)$, i.e. the transition probability from milestone 2 to milestone 3 as a function of lifetime, calculated using $\mathscr{F}_{wind}$ forces ranging from 0 to 12 pN. The plots indicate that the rapid decrease in computation time due to the added $\mathscr{F}_{wind}$ force has almost no effect on accuracy.
§.§ Model System in One Dimension with Distortion
Thus far, we have approached the WARM method from the standpoint of speeding up the calculation by pushing $\mathscr{F}_{wind}$ until the $K(\tau)$ functions begin to distort. Here we will explore the possibility that even slightly distorted $K(\tau)$ functions can yield useful information, allowing for even greater computational speedup. The flux value for a given milestone $s$, $P_s(t)$, should approach the probability predicted by the Boltzmann distribution generated from configurational partition function as time approaches infinity. Given a discrete space in $x$, subject to our 1D potential $y = (x-1)^2(x+1)^2$, the Boltzmann distribution function can be obtained in the usual way, shown in equation <ref>, below:
\begin{equation}
\lim_{t \to \infty} P_s(t) = \frac{e^{-\beta U(x_s)}}{ \sum_{n=1}^{N_S} e^{-\beta U(x_n)}}
\label{B}
\end{equation}
where $N_s$ is the total number of milestone configurations, and $x_n$ signifies the spatial position of each milestone. This discrete space approximation for the equilibrium flux values is utilized below as a test for accuracy in figure <ref> (dashed lines). The numerical demonstration in this section consists of dividing the space for the bistable 1D potential between $x = -2$ and $x = 2$ into 11 subspaces bounded by 12 milestones. First hitting trajectories were run between each pair of adjacent milestones, and then each pair of $K(\tau)$ functions describing a transition away from each milestone were normalized (e.g. for milestone 3, all trajectories must terminate at either milestone 2 or milestone 4, therefore $\int_0^\infty K_{32}(\tau) d\tau + \int_0^\infty K_{34}(\tau) d\tau = 1$). The normalized $K(\tau)$ functions are then integrated over all time, and these values are placed in a matrix, K, of equilibrium transition probabilities. The equilibrium flux values for the vector representing the set of milestones, P, is then found by numerically solving for the eigenvector: P$\cdot$K = P <cit.>. By using this method to determine equilibrium flux values, it is demonstrated in figures <ref> and <ref>, that even when $\mathscr{F}_{wind}$ is set to a value strong enough to distort the $K(\tau)$ functions, accurate equilibrium flux values can still be calculated.
The plot at the top of this figure shows plots for one of the transition probability distributions $K(\tau)$ for the bistable 1D potential with different values of $\mathscr{F}_{wind}$ implemented. Note that although the distributions distort considerably for higher values of $\tau$ when the system is pushed with high magnitude $\mathscr{F}_{wind}$, the equilibrium flux values in the plot below remain fairly constant. The color scheme legend applies to both plots.
Here we show effects of applying higher magnitude $\mathscr{F}_{wind}$ which are strong enough to significantly distort the $K(\tau)$ functions. This figure facilitates a direct comparison of gain in computational speed with the accuracy of the equilibrium flux values (measured as $X^2$). Note that while there is no appreciable change in accuracy, calculation time drops from 1109 s to 26 s, a speedup by a factor of nearly 40.
§.§ Model Systems in Two Dimensions
Two additional test systems for the WARM technique were implemented for further validating the method in two dimensions. Both systems have double well shapes, however for one well, the barrier to transition from one well to the other is primarily energetic, while the other is primarily entropic (see figure 3 below). The potential with the energetic barrier is a generalization of the 1D potential from the previous section to two dimensions, and the potential with the entropic barrier is the same potential implemented by Elber and Faradjian in their original paper on milestoning <cit.>. As can be seen in the data below, the WARM method successfully re-weighted first passage time distributions ($K(\tau)$) generated using artificially accelerated trajectories to yield the true first passage time distributions which would have resulted from trajectories in the absence of the $wind$ force. In both cases, the method achieved more than 60% faster computation times with very little sacrifice in terms of accuracy.
Show here are the potentials used in the 2D WARM calculations. In the first case, the primary barrier to crossing from one well to the other is the height of the barrier relative to the strength of the “kicks" from the random force in the Langevin equation. In the second potential <cit.>, the barrier to crossing between wells is entropic, in that a trajectory which results in a transition between wells must find its way through the gap at the center, i.e. the likelihood of a transition is not limited by any sort of uphill battle, but instead by decreased degeneracy in the number of possible trajectories which result in a transition.
Shown in this figure is the transition probability distribution $K_{12}(\tau)$, i.e. the transition probability from milestone 1 (the line $x = -1$) to milestone 2 (the line $x = 0$) on the the 2D potential with the energetic barrier as a function of lifetime, calculated using $\mathscr{F}_{wind}$ forces ranging from 0 to 12 pN. The plots indicate that the rapid decrease in computation time due to the added $wind$ force has almost no effect on accuracy.
Shown here is calculation time as a function of the $\mathscr{F}_{wind}$ force for all $K(\tau)$ distributions in both directions for two subspaces ranging from -1 to 1 on the x axis of the 2D potential with the energetic barrier. All trajectories were run using $\beta = .123$. The highest value of $\mathscr{F}_{wind}$ yielded a faster computation time by a factor of 4.17 than the unassisted calculation with very little distortion to the $K(\tau)$ function.
Shown in this figure is the transition probability distribution $K_{12}(\tau)$, i.e. the transition probability from milestone 1 (the line $x = -.5$) to milestone 2 (the line $x = 0$) on the the 2D potential with the entropic barrier as a function of lifetime, calculated using $\mathscr{F}_{wind}$ forces ranging from 0 to 1 pN. The plots indicate that the rapid decrease in computation time due to the added $\mathscr{F}_{wind}$ force has almost no effect on accuracy.
Shown here is calculation time as a function of the $\mathscr{F}_{wind}$ force for all $K(\tau)$ distributions in both directions for two subspaces ranging from -.5 to .5 on the x axis of the 2D potential with the entropic barrier. All trajectories were run using $\beta = 3.0$ so as to ensure that transitions over the barrier instead of through the small gap were highly unlikely. The highest value of $\mathscr{F}_{wind}$ yielded a faster computation time by a factor of 4.78 than the unassisted calculation with almost no distortion to the $K(\tau)$ function..
§.§ Model System in Eleven Dimensions
In order to demonstrate that the WARM method possesses no inherent limitations due to scaling, the method was applied to an $11$ dimensional hyperspace. For this model, the $11D$ potential was defined as:
\begin{equation}
V(x_1, x_2,..., x_{11}) = (x_1 - 1)^2(x_1+1)^2 - \frac{1}{2}\sum_{n = 2}^{11} x_n^2 x_1^2 + \sum_{n = 2}^{11} x_n^4
\end{equation}
where the first term is the same bistable potential in $x_1$ used in the first one dimensional example, the second term couples motion in the $10$ dimensions orthogonal to barrier height in $x_1$, and the third term simply confines the system to a reasonably sized configurational space using a quartic potential. In order to develop some intuition for this potential, see figure <ref>, then just imagine that there are nine other dimensions which have the same effect as $y$ on barrier height in $x_1$.
Show here is a 2D representation of the 11D coupled potential. The $y$ in the second term (red) has been left as a parameter in this plot. The surfaces shown are for values for the parametric $y$ of $0, \pm 1,$ and $\pm 1.5$, where the deepest well corresponds to $y = 1.5$ and the shallowest corresponds to parametric $y = 0$. Just as the well becomes deeper, the further from the system wanders from the origin in the $y$ direction in this 2D model, the 11D system also encounters deeper wells in the $x_n$ dimensions the further it wanders from the origin in each $x_n$ dimension.
Accordingly, the milestones must be defined as hyperplanes, given the general definition of a hyperplane:
\begin{equation}
a_1 x_1 + a_2 x_2 + a_3 x_3 ... a_n x_n = b
\end{equation}
To keep things simple, we set $a_2$ through $a_{11}$ equal to zero, and $a_1 = 1$, allowing us to define two hyperplanes as $x_1 = -1$ and $x_1 = 1$. In this scenario, the features of interest are the transitions between the wells at $x_1 = -1$ and $x_1 = 1$, thus the 11D $\mathscr{F}_{wind}$ is applied with zero components in all dimensions except for $x_1$ where it is used to push the system over the barrier between wells. The WARM method was successfully applied to this 11 dimensional potential, and a speedup by a factor of 4.5 was observed (figures <ref> and <ref>).
This plot shows CPU time as a function of the magnitude of the $\mathscr{F}_{wind}$ in 11D. The maximum speedup measured was a factor of 4.5.
Shown in this figure are the $K(\tau)$ functions generated for each data point in the CPU time vs. $\mathscr{F}_{wind}$ plot for the 11D system.
§.§ Wind Force as a Vector Field
In all of the preceding examples, $\mathscr{F}_{wind}$ was applied to the system as a constant force applied in a straight line, perpendicular to the parallel milestone hyperplanes, but $\mathscr{F}_{wind}$ can be defined any way we choose. This section demonstrates a method whereby the directionality of $\mathscr{F}_{wind}$ is defined by a vector field which allows $\mathscr{F}_{wind}$ to blow in a curved path between two nearly orthogonal milestones (see figures <ref> and <ref>), i.e. our wind has become a tornado! In order to define this vector field, the point of intersection between the two planes was determined, then a function was created which finds the straight line connecting the current position to this point of intersection, and then defines $\mathscr{F}_{wind}$ at that point to be a vector both orthogonal to that straight line and pointing in a clockwise direction. When first passage times were calculated going from the milestone shown in red toward the milestone shown in green (figure <ref>), the vector field is simply multiplied by $-1$ to cause our tornado to spin counterclockwise. Using a wind force defined in this manner, we obtain an efficient directionality for $\mathscr{F}_{wind}$ which biases the system toward both leaving its initial milestone in the right direction and approaching its destination milestone, regardless of the positioning of the milestones in configuration space. Another advantage of this scheme is that our curved vector field of $\mathscr{F}_{wind}$ can be defined without any knowledge of the system itself, we only need to know the positions of the milestones, which are always known in milestoning calculations. Figures <ref> through <ref> illustrate the application and results of this method using two different 2D potentials, the Muller-Brown potential <cit.>, and a simpler Muller-inspired potential with two Gaussian wells we'll call our Gaussian potential. The Gaussian potential is defined as:
\begin{gather}
V(x, y) = -exp[-(2(x-.8)^2+y^2)] \\ \nonumber
-1.3exp[-((x+1)^2+(y-1.5)^2)] \\ \nonumber
+ .2x^2+.2y^2
\label{gp}
\end{gather}
and the Muller potential is defined as:
\begin{gather}
V(x,y) = h \sum_{k = 1}^4 exp[a_k(x-x_k^0)^2 \\ \nonumber
+b_k(x-x_k^0)(y-y_k^0)+c_k(y-y_k^0)^2] \\ \nonumber
\label{mp}
\end{gather}
\begin{gather*}
A = (-200, -100, -170, 15), a =(-1, -1, -6.5, .7) \\ \nonumber
b = (0, 0, 11, .6), c =(-10, -10, -6.5, .7) \\ \nonumber
x^0 = (1, 0, -.5, -1), y^0 =(0, .5, 1.5, 1) \\ \nonumber
h = .005
\end{gather*}
A speedup factor of 4 was observed in both the Muller potential and the Gaussian potential, although the $K(\tau)$ functions in the Gaussian potential example displayed less distortion than those produced in the calculations performed using the Muller potential.
Shown here is a representation of the vector field approach to applying $\mathscr{F}_{wind}$ to push milestoning trajectories between two nearly orthogonal planes, subject to our Gaussian potential. The green milestone is defined as the plane for which $\frac{y}{44} - x = -.7$ and the red milestone is defined as the plane for which $y = 1.5$. The vector wind is configured to show the $\mathscr{F}_{wind}$ scheme for accelerating trajectories going from red to green.
This plot shows the same milestone placement and $\mathscr{F}_{wind}$ scheme as the Gaussian potential example applied to the Muller potential and with a directionality for accelerating trajectories from the green milestone to the red one.
This plot shows CPU time as a function of $\mathscr{F}_{wind}$ magnitude for the Gaussian potential.
This plot shows the $K(\tau)$ functions corresponding to different magnitudes of $\mathscr{F}_{wind}$ as applied to the Gaussian potential.
This plot shows CPU time as a function of $\mathscr{F}_{wind}$ magnitude for the Muller potential.
This plot shows the $K(\tau)$ functions corresponding to different magnitudes of $\mathscr{F}_{wind}$ as applied to the Muller potential.
§ CONCLUDING DISCUSSION
We have presented and tested a method for accelerating milestoning calculations, whereby the true probability density functions for first passage transition time between milestones, $K_{AB}(\tau)$, are recovered from artificially accelerated trajectories via the re-weighting method described in the Theory section. These $K_{AB}(\tau)$ functions are central to milestoning calculations, thus the WARM method presented herein shows potential for broad application.
Our method has been shown to be effective on one and two dimensional potentials with both energetic and entropic barriers, as well as an $11$ dimensional hyperspace, implying that the method should have no scaling limitations, thus the next step will be to test the method on chemical systems. The simplest application would be to apply a single force vector to a single atom which pushes the system toward a configurational change of interest. In this case, the re-weighting factors $S[x(t)]$ and $S_f[x(t)]$ could be calculated by summing the force components in the $x$, $y$, and $z$ directions both with and without the components of the applied force, respectively.
The main limitation of the WARM method, regardless of the number dimensions are present, is obtaining good re-weighting in the longer $\tau$ range. This is simply a matter of under-sampling. If $\mathscr{F}_{wind}$ is pushing the system to the next milestone so quickly that longer values of $\tau$, relevant to the true $K_{AB}(\tau)$ distribution, are not being sampled, then there just isn't enough density present (or even none at all) to re-weight. This is why the accuracy in the low tau regime is often still quite good when too high of an $\mathscr{F}_{wind}$ has caused the latter portion of the distribution to turn to noise. Thus far, the limitations on the WARM method appear to be solely dependent on whether or not we've pushed the force so hard that trajectories in the longer $\tau$ region of the true $K_{AB}(\tau)$ are even being sampled. For this reason, systems whose true $K_{AB}(\tau)$ distribution functions possess fat tails place the greatest limitations on the degree of computational speedup achievable by WARM. This issue can be addressed in a couple different ways. One approach is to simply define more milestones in the space, the other is to combine the WARM method with some sort of artificial heating method, both modifications which will yield $K_{AB}(\tau)$ functions which decay more rapidly after their peak.
It should be noted that our application of the WARM method to both the high dimensional model, and our vector field-based $\mathscr{F}_{wind}$ implementation of demonstrate that this technique can be applied to systems too complex to intuit the placement of the artificial forces. Given an initial and a final milestone configuration, one could determine the vectors pointing from each atom's initial position to it's final position. Artificial forces, $\mathscr{F}_{wind}$, could then be placed upon all atoms in the system pointing in the direction of these vectors and with a magnitude proportional to the length of the vectors. A zero cutoff could also be added so as not to waste computational resources on applying and accounting for $\mathscr{F}_{wind}$ forces atoms which are beginning at a position fairly close to their destination. We believe that, upon implementation into a molecular dynamics package such as MOIL <cit.>, the WARM method has the potential to be a useful tool for the determination of the kinetic properties of macromolecules.
§ ACKNOWLEDGMENTS
IA acknowledges funds from an NSF CAREER award (CHE-0548047).
|
1511.00457
|
This paper is devoted to advancing the theoretical understanding of
the iterated immediate snapshot (IIS) complexity of the Weak Symmetry
Breaking task (WSB). Our rather unexpected main theorem states that
there exist infinitely many values of $n$, such that WSB for
$n$ processes is solvable by a certain explicitly constructed
$3$-round IIS protocol. In particular, the minimal number of rounds,
which an IIS protocol needs in order to solve the WSB task, does not go
to infinity, when the number of processes goes to infinity.
Our methods can also be used to generate such values of $n$.
We phrase our proofs in combinatorial language, while avoiding using
topology. To this end, we study a certain class of graphs, which we
call flip graphs. These graphs encode adjacency structure in certain
subcomplexes of iterated standard chromatic subdivisions of a simplex.
While keeping the geometric background in mind for an additional
intuition, we develop the structure theory of matchings in flip graphs
in a purely combinatorial way. Our bound for the IIS complexity is
then a corollary of this general theory.
As an afterthought of our result, we suggest to change the overall
paradigm. Specifically, we think, that the bounds on the IIS
complexity of solving WSB for $n$ processes should be formulated in
terms of the size of the solutions of the associated Diophantine
equation, rather than in terms of the value $n$ itself.
§ INTRODUCTION
§.§ Solvability of Weak Symmetry Breaking
The Theoretical
Distributed Computing revolves around studying solvability and
complexity of the so-called distributed tasks. Roughly speaking
these are specifications of sets of required outputs for all legal
inputs. One of the classical and central tasks is the so-called Hard $M$-Renaming. For this task, $n$ processes with unique names
in a large name space of size $N$ must cooperate in a wait-free manner
to choose unique names from a typically much smaller name space of
size $M$.
In order to talk about solvability and complexity of various tasks,
one needs to specify the computational model. A standard one, called
iterated immediate snapshot, has been in the center of attention
of many papers, including this one. In this model the processes use
two atomic operations being performed on shared memory. These
operations are: write into the register assigned to that
process, and snapshot read, which reads entire memory in one
atomic step. Furthermore, it is assumed that the executions are
well-structured in the sense that they must satisfy the two following
conditions. First, it is only allowed that at each time a group of
processes gets active, these processes perform a write operation
together, and then they perform a snapshot read operation together; no
other interleaving in time of the write and read operations is
permitted. Such executions are called immediate snapshot
executions. Second, each execution can be broken up in rounds, where
in every round each non-faulty process gets activated precisely once,
alternatively, this can be phrased as each process using fresh memory
every time its gets activated.
Historically, the main focus, when studying the Hard $M$-Renaming in
the iterated immediate snapshot model, has been on finding the lower
bounds for $M$. About 20 years ago it was shown that $M\geq
2n-1$. Inconveniently, the proof only worked for the case when $n$ is
a prime power. Since no other classical problem in Distributed
Computing depends on the number-theoretic properties of $n$, one was
tempted to believe that the appearance of the prime power condition
was an artefact of the method of the proof, rather than that of the
underlying question. Most surprisingly, the exact opposite was shown
to be true. A long and complex construction was proposed to
demonstrate that algorithms exist for $M = 2n$, when $n$ is not
a prime power. Unfortunately, it was difficult to calculate the
communication round complexity using that construction.
A further task, the so-called Weak Symmetry Breaking task (WSB)
for $n$ processes, is an inputless task with possible outputs $0$ and
$1$. A distributed protocol is said to solve the WSB if in any
execution without failed processes, there exists at least one process
which has value $0$ as well as at least one process which has
value $1$. WSB for $n$ processes is equivalent to the Hard
$(2n-2)$-Renaming task, providing one of the mains reasons to study
its solvability and its complexity.
In the classical setting, the processes trying to solve WSB know their
id's, and are allowed to compare them. It is however not allowed that
any other information about id's is used. The protocols with this
property are called comparison-based.[Alternative
terminology rank-symmetric is also used in the literature.] In
practice this means that behavior of each process only depends on the
relative position of its id among the id's of the processes it
witnesses and not on its actual numerical value. In this way, trivial,
uninteresting solutions can be avoided. As a special case, we note
that each process must output the same value in case he does not
witness other processes at all.
One of the reasons the Iterated Immediate Snapshot model is used
extensively in Distributed Computing, which is also its major
advantage, is that the protocol complexes have a comparatively simple
simplicial structure, and are amenable to mathematical
analysis. Specifically, the existence of a distributed protocol
solving WSB in $r$ rounds is equivalent to the existence of a certain
$0/1$-labeling of the vertices of the $r$th iterated standard
chromatic subdivision of an $(n-1)$-simplex.
§.§ Previous work
The Iterated Immediate Snapshot model is due to Gafni&Borowsky,
see <cit.>; in <cit.> this model goes under the name layered immediate snapshot. Several groups of researchers have
studied the solvability of the WSB by means of comparison-based IIS
protocols. Due primarily to the work of Herlihy&Shavit, <cit.>, as
well as Castañeda&Rajsbaum, <cit.>, it is known that
the WSB is solvable if and only if the number of processes is not
a prime power, see also <cit.> for a counting-based argument for
the impossibility part. This makes $n=6$ the smallest number of
processes for which this task is solvable.
The combinatorial structures arising in related questions on
subdivisions of simplex paths have been studied in <cit.>.
The specific case $n=6$ has been studied in <cit.>, who has proved
the existence of the distributed protocol which solves the WSB task in
17 rounds. This bound was recently improved to 3 rounds in <cit.>,
where also an explicit protocol was given.
We recommend <cit.> as a general reference for Theoretical
Distributed Computing, and <cit.> as a general reference for
combinatorial topology. Furthermore, our book <cit.> contains
all the standard terminology, which we are using here.
A broader framework of symmetry breaking tasks can be found
in <cit.>. The topological description for the IIS model can be
found in <cit.>; in addition, topological descriptions of
several other computational models have also been studied,
see <cit.>.
§.§ Our results
Our main result states that, surprisingly, there is an infinite set of
numbers of processes for which WSB can be solved in $3$ rounds in the
comparison-based IIS model. Specifically, we prove that this is the
case when the number of processes is divisible by $6$. There is
$O(n^2)$ overhead cost to translate an IIS protocol to an IS protocol,
so the resulting IS complexity is $O(n^2)$, which is of course less
Let $\msb(n)$ denote the minimal number of rounds which is needed for
an IIS protocol to solve WSB for $n$ processes, then our main theorem
can succinctly be stated as follows.
For all $t\geq 1$, we have $\msb(6t)\leq 3$.
Our proof is based on combinatorial analysis of certain matchings in
the so-called flip graphs, and strictly speaking does not need any
§ INFORMAL SKETCH OF THE PROOF
§.§ The situation prior to this work.
It has long been understood, that there is a 1-to-1 correspondence
between IIS protocols solving WSB on one hand and binary assignments
$\lambda$ to the vertices of the iterated chromatic subdivision of
a simplex, on the other hand, where these assignments must satisfy
certain technical boundary conditions, and have no monochromatic
top-dimensional simplices, see, e.g., <cit.>. Furthermore,
Herlihy$\,$&$\,$Shavit, see <cit.>, found an obstruction to the existence
of such an assignment in case the number of vertices of that simplex
(i.e., the number of processes) is a prime power. This obstruction is
a number which only depends on the values of $\lambda$ on the boundary
of the subdivided simplex, and which must be $0$ if there are no
monochromatic maximal simplices.
Thus the construction of IIS protocols solving WSB has been reduced to
finding such $\lambda$, where the number of IIS rounds is equal to the
number of iterations of the standard chromatic subdivision. The
construction of $\lambda$, when the number is not a prime power, was
then done by Castañeda$\,$&$\,$Rajsbaum, see <cit.>, using the
following method. First, boundary values are assigned, making sure
that this obstruction value is $0$. After that the rest of the values
are assigned, taking some care that only few monochromatic maximal
simplices appear in the process. This is followed by a sophisticated
and costly reduction procedure, during which the monochromatic
simplices are connected by paths, and eventually eliminated. This
elimination procedure is notoriously hard to control, leading to
exponential bounds.
§.§ The main ideas of our approach
The idea which we introduce in this paper is radically different.
Just as Castañeda$\,$&$\,$Rajsbaum we produce a boundary labeling
making sure the invariant is $0$. However after that our approaches
diverge in a crucial way. We assign value $0$ to all internal
vertices. This is quite counter-intuitive, as we are trying to get rid
of the monochromatic simplices in the long run, while such an
assignment on the contrary will produce an enormous amount of
them. However, the following key observation comes to our rescue: if
we can match the monochromatic simplices with each other so that any
pair of matched simplices shares a boundary simplex of one dimension
lower, then we can eliminate them all in one go using one more round.
Next, we make a bridge to combinatorics. We have a graph, whose
vertices are all the monochromatic maximal simplices, connected by an
edge if they share a boundary simplex of one dimension lower; we shall
call such graphs flip graphs. What we are looking for is a perfect matching on graphs of this type. This is a simple reduction,
but it is very fruitful, since the matching theory on graphs is a very
well-developed subject and we find ourselves having many tools at our
disposal. A classical method to enlarge existing matchings is that of
augmenting paths. The idea is elementary but effective: connect
the unmatched (also called critical) vertices by a path $p$, such that
all other vertices on the path are matched by the edges along $p$, and
then make all the non-matching edges of $p$ matching and vice versa.
This trick will keep all the internal vertices of $p$ matched,
while also making end vertices matched. In particular, if we
have a matching, and we succeeded to connect critical vertices in
pairs by non-intersection augmenting paths, then applying this trick
to all these paths simultaneously, we will end up with a perfect
§.§ The blueprint of the proof
This set of ideas leads to the following blueprint for constructing
the 3-round IIS protocol to solve WSB for $n$ processes:
Step 1. Find a good boundary assignment for the second
standard chromatic subdivision of the simplex with $n$ vertices,
making sure the Herlihy-Shavit obstruction vanishes. Assign value
$0$ to all internal vertices.
Step 2. Decompose the resulting flip graph of monochromatic
simplices into pieces corresponding to the maximal simplices of the first
chromatic subdivision. Describe an initial matching on each of these
pieces, and combine them to a total matching.
Step 3. Construct a system of non-intersecting augmenting
paths with respect to that total matching. Changing our initial
matching along these paths produces the desired perfect matching.
Step 4. Eliminate all monochromatic maximal simplices in
one go, producing a binary assignment for the third standard
chromatic subdivision of the simplex with $n$ vertices, which now
has no maximal monochromatic simplices.
This is a general scheme, and if the technical details work out, it
can be used for various values of $n$ and also for various numbers of
rounds. In this paper we restrict ourselves to the values
$n=6,12,18,\dots$, mainly because this is the case in which we can
provide complete rigorous details. The techniques of this paper can
further be extended to discover other values of $n$ for which WSB can
be solved in $3$ rounds, see <cit.>. In that paper, progress has
been made using techniques of Sperner theory, see <cit.>,
specifically a variation of local LYM inequality, to cover values
$n=15,20,21,\dots$, see Theorem <ref>.
To start with, for $n=6t$, there is a quite special arithmetic
identity (<ref>), which has a stronger set-theoretic version,
see the proof of Theorem <ref>, stating the existence of
a certain bijection $\Phi$. We produce a quite special labeling of the
vertices on the boundary of $\mych^2(\da^{n-1})$, and we put the label
$0$ on all the internal vertices of $\mych^2(\da^{n-1})$. Since any
top-dimensional simplex has at least one internal vertex, we will have
no $1$-monochromatic simplices at that point. This corresponds to the
step 1 above.
We then proceed with steps 2 and 3, which are at the technical core of
our proof. We start with a rough approximation to the matching which
we want to get at the end. In this approximation, called the standard matching, most of the simplices will get matched. There
will be a small number of critical simplices left, concentrated around
barycenters of certain boundary simplices. We then find a system of
augmenting paths which connect all the critical simplices in
pairs. Our idea of how to find these paths is to use the system of
non-intersection paths in $\gn$ which we construct along the bijection
$\Phi$, like a ”system of tunnels” between areas of
$\mych^2(\da^{n-1})$ which contain the monochromatic simplices. Within
each such tunnel we use our analysis of the combinatorial structure of
the flip graphs, namely certain properties, which we call conductivity of these graphs, to connect the critical simplices by
augmenting paths, see Figure <ref>. This yields the desired
result, allowing us to produce a perfect matching on the set of
monochromatic simplices. Step 4 is an easy and standard step which has
been used before, we do not make any original contribution there.
§.§ Note on the language we use to formulate our argument
As mentioned above, it is by now a classical knowledge that executions of
a distributed protocol in IIS model can be encoded using the simplicial
structure of the standard chromatic subdivision and its iterations.
Consequently, various tools of topology have been used in the past to gain
information on the distributed computing tasks. In contrast, our argument does
not need any implicit or non-constructive topological results, such as,
for instance, existance of fixed points. All that is required is the incidence structure
if the underlying subdivisions and various labeling techniques for the vertices.
To underline this fact, we shall mostly omit any mentioning of simplicial
structures, and formulate everything using only the language of graph theory.
It is certainly very helpful for the intuition to keep the simplicial
picture in mind, and we invite the reader to do so when going through
the text. Also, many of our illustrations refer to the simplicial picture.
However, we feel it is of value to have our proof phrased exclusively in
terms of combinatorics of graphs.
§ BASIC CONCEPTS
§.§ Graph theory concepts
We start by recalling some graph terminology, which we need
throughout the paper. For a graph $G$ we let $V(G)$ denote the set of
its vertices and we let $E(G)$ denote the set of its edges. Two
different edges are called adjacent if they share a vertex.
An edge coloring of a graph $G$ with colors from a set $C$ is an
assignment $c:E(G)\to C$, such that adjacent edges get different
Assume $G$ is a graph and $A$ is a subset of $V(G)$. We say that the
graph $H$ is the subgraph of $G$ induced by $A$, if the set of
vertices of $H$ is $A$, and two vertices of $H$ are connected by an
edge in $H$ if and only if they are connected by an edge in $G$.
A matching on a graph $G$ is a set of edges, such that no two of
these edges are adjacent. The vertices of these edges are said to be
matched, while the rest of the vertices are called critical.
To underline that not all vertices are matched we often
say partial matching.
The matching is called perfect if all vertices are matched, and
it is called near-perfect if it has exactly one critical vertex.
§.§ Set theory concepts
For all natural numbers $n$, we let $[n]$ denote the set
$\{1,\dots,n\}$. Furthermore, throughout the paper we shall skip
curly brackets when the set consists of a single element. When we say
a tuple we mean any ordered sequence. An order $R$ on
a set $S$ is any tuple $R=(x_1,\dots,x_d)$, satisfying
Assume $A$ is an arbitrary nonempty finite set of natural numbers, and
let $k$ denote the cardinality of $A$. Then there exists a unique
order-preserving bijection $\varphi:A\ra[k]$. We call $\varphi$ the
normalizer of $A$.
Note, that if $A$ and $B$ are equicardinal nonempty finite sets of
natural numbers, $\varphi$ is a normalizer of $A$, and $\gamma:A\ra B$
is the unique order-preserving bijection between $A$ and $B$, then
$\varphi\circ\gamma^{-1}$ is the normalizer of $B$. On the other hand,
if $\psi:B\ra[k]$ is a normalizer of $B$, then $\psi\circ\gamma$ is
a normalizer of $A$.
When talking about normalizers in the rest of the paper, we shall
typically include the information on the set cardinality in
normalizers definition. In other words, when we say $\varphi:A\ra[k]$
is a normalizer of $A$, we mean let $k$ denote the
cardinality of $A$ and let the map $\varphi:A\ra[k]$ denote
the unique order-preserving bijection.
§.§ $S$-tuples
For any finite set $S$, an $S$-tuple is a tuple
$(A_1,\dots,A_t)$ of disjoint non-empty subsets of $S$. We call $t$
the length of this $S$-tuple. We shall use the short-hand
notation $A_1\s\dots \s A_t$. An $S$-tuple $A_1\s\dots\s A_t$ is
called full if $A_1\cup\dots\cup A_t=S$. For a full $S$-tuple
$\sigma=A_1\s\dots\s A_t$, we let $V(\sigma)$ denote the set
$A_1\cup\dots\cup A_{t-1}=S\sm A_t$.
Note, that in <cit.> full $S$-tuples were called ordered set partitions of the set $S$.
Clearly, if $T\supseteq S$, then any $S$-tuple can be interpreted as
a $T$-tuple as well. Furthermore, if $A=A_1\s\dots\s A_k$ is an
$S$-tuple and $B$ is another $S$-tuple, we say that $A$ is a truncation of $B$, if $B$ has the form $A_1\s\dots\s A_k\s
A_{k+1}\s\dots\s A_t$, for some $t\geq k$.
Full $[n]$-tuples are mathematical objects which encode possible
executions of the standard one round protocol for $n$ processes
in the immediate snapshot model.
Given a set $S$, and two $S$-tuples of the same length
$\sigma=A_1\s\dots\s A_t$ and $\tau=B_1\s\dots\s B_t$, we call the
ordered pair $(\sigma,\tau)$ a coherent pair of $S$-tuples, if we
have the set inclusion $B_i\subseteq A_i$, for all $i=1,\dots,t$.
We find it convenient to view a coherent pair of $S$-tuples as a $2\times
t$ table of sets
\[(\sigma,\tau)=
\begin{array}{|c|c|c|c|c|}
\hline
A_1 & \dots & A_t \\ \hline
B_1 & \dots & B_t \\
\hline
\end{array}.
\]
An arbitrary $S$-tuple $A_1\s\dots\s A_t$ can alternatively be viewed
as a coherent pair of $S$-tuples $(A_1\s\dots\s A_t,A_1\s\dots\s A_t)$,
and we shall use the two interchangeably. We also recall the following
terminology from <cit.>: for an $S$-tuple $\sigma=A_1\s\dots\s
A_t$ we set its carrier set to be
$\carr(\sigma):=A_1\cup\dots\cup A_t$; for a coherent pair of $S$-tuples
$(\sigma,\tau)$, we set $\carr(\sigma,\tau):=\carr(\sigma)$, and we
set $\col(\sigma,\tau):=\carr(\tau)$; the latter is called the color set of $(\sigma,\tau)$.
The terminology of coherent pairs of $S$-sets comes from the need to have combinatorial
language to describe partial views on the executions of distributed protocols.
It is related to the previous work of the author, see <cit.>.
The sets $B_1,\dots,B_t$ contain the id's of the active processes whose view
on the execution is available to us. Accordingly, the color set contains
id's of all active processes. The sets $A_1,\dots,A_t$ contain the id's of all
processes which are seen by the active ones. These include the active processes,
as any process sees itself, but it may include more than that.
Accordingly, the carrier set contains id's of a possibly larger set of processes,
which are seen by the active ones.
Assume that we are given a coherent pair of $S$-tuples, say
$(\sigma,\tau)=(A_1\s\dots\s A_t,B_1\s\dots\s B_t)$, together with
a non-empty proper subset $T\subset\col(\sigma,\tau)$. We define a new
coherent pair of $S$-tuples $(\tilde\sigma,\tilde\tau)$, which we call
a restriction of $(\sigma,\tau)$ to $T$. To do this, we first
decompose $T=T_1\cup\dots\cup T_d$ as a disjoint union of non-empty
subsets such that $T_1\subseteq B_{i_1}$, $\dots$, $T_d\subseteq
B_{i_d}$, for some $1\leq i_1<\dots<i_d\leq t$. Then, we set $\tilde
A_1:=A_1\cup\dots\cup A_{i_1}$, $\tilde A_2=A_{i_1+1}\cup\dots\cup
A_{i_2}$, $\dots$, $\tilde A_d=A_{i_{d-1}+1}\cup\dots\cup
A_{i_d}$. Finally, we set $\tilde\sigma:=\tilde A_1\s\dots\s\tilde
A_d$, and $\tilde\tau:=T_1\s\dots\s T_d$. We denote this new coherent
pair of $S$-tuples by $(\sigma,\tau)\dar T$.
A restriction of $(\sigma,\tau)$ to $T$ is always uniquely defined by
the above.
Note, that since an arbitrary $S$-tuple can be viewed as a coherent
pair of $S$-tuples, we are able to talk about restrictions of
$S$-tuples. However, the set of $S$-tuples, unlike the set of coherent
pairs of $S$-tuples, is not closed under taking restrictions, and the
result of restricting an $S$-tuple will be a coherent pair of
As an example, let $S=[6]$,
$\sigma=\{1,2\}\s 3\s 4\s 5\s 6$, and $T=\{1,2,3,5\}$, then
\[\sigma\dar T=
\begin{array}{|c|c|c|c|c|}
\hline
1,2 & 3 & 4,5 \\ \hline
1,2 & 3 & 5 \\
\hline
\end{array}.
\]
Furthermore, taking $\widetilde T=\{1,5\}$, we get
\[\sigma\dar\widetilde T=(\sigma\dar T)\dar\widetilde T=
\begin{array}{|c|c|c|c|}
\hline
1,2 & 3,4,5 \\ \hline
1 & 5 \\
\hline
\end{array}.
\]
Dually, we introduce an operation of deletion by setting
$\dl((\sigma,\tau),T):=(\sigma,\tau)\dar(S\sm T)$, for an arbitrary
coherent pair of $S$-tuples $(\sigma,\tau)$, and an arbitrary
non-empty proper subset $T\subset S$. In this case we say that the
coherent pair of $S$-tuples $\dl((\sigma,\tau),T)$ is obtained from
$(\sigma,\tau)$ by deleting $T$.
The restriction operation described in Definition <ref> may sound
complicated. However, it faithfully describes what happens to the combinatorial
labels of partial executions, when the set of active processe is reduces,
cf. Distributed Computing Context <ref>. Some of the sets of processes
may be merged, which happens when those processes which distinguished between
the two groups are no longer active. Other processes may stop to be seen
altogether; their id's are then deleted from the description. The need to
phraze in combinatorial language what happens to the partial execution description,
led the author to formulate the Definition <ref>.
§ FLIP GRAPHS
§.§ The graphs $\gn$
When $\sigma=A_1\s\dots\s A_t$ is a full $[n]$-tuple, we will use
the following short-hand notation:
\[F(\sigma):=
\begin{cases}
[n]\sm A_t, & \text{ if } |A_t|=1; \\
[n], & \text{ otherwise. }
\end{cases}\]
The set $F(\sigma)$ is called the flippable set of $\sigma$.
Assume, we are given a full $[n]$-tuple $\sigma=A_1\s\dots\s A_t$, and
$x\in F(\sigma)$. Let $k$ be the index, such that $x\in A_k$. We let
$\flip(\sigma,x)$ denote the full $[n]$-tuple obtained by using the
following rule.
Case 1. If $|A_k|\geq 2$, then set
\[\flip(\sigma,x):=A_1\s\dots\s A_{k-1}\s x \s A_k\sm x\s A_{k+1}\s\dots\s A_t.\]
Case 2. If $|A_k|=1$ (that is $A_k= x $), then we
must have $k<t$. We set
\[\flip(\sigma,x):=A_1\s\dots\s A_{k-1}\s x \cup A_{k+1}\s A_{k+2}\s\dots\s A_t.\]
Due to background geometric intuition, we think of the process of
moving from a full $[n]$-tuple $\sigma$ to the full $[n]$-tuple
$\tau=\flip(\sigma,x)$ as a flip. If the first case of
Definition <ref> is applicable, we say that $\tau$ is obtained
from $\sigma$ by splitting off the element $x$, else we say that
$\tau$ is obtained from $\sigma$ by merging in the element $x$.
The operation $\flip(-,x)$ behaves as a flip operation is expected to
behave. Namely, for any $[n]$-tuple $\sigma$ and $x\in F(\sigma)$, we
have $x\in F(\flip(\sigma,x))$, and importantly
\begin{equation}\label{eq:flfl}
\flip(\flip(\sigma,x),x)=\sigma.
\end{equation}
Assume $\sigma$ is a full $[n]$-tuple and $x\in F(\sigma)$, then we
\begin{equation}\label{eq:vf}
V(\cf(\sigma,x))\cap([n]\sm x)=V(\sigma)\cap([n]\sm x),
\end{equation}
where $V(-)$ is as in Definition <ref>. In other words, barring
$x$, the set of elements, which are not contained in the last set
of $\sigma$, does not change during the flip.
Assume $\sigma=A_1\s\dots\s A_t$, and $\cf(\sigma,x)=B_1\s\dots\s
B_m$. We have one of the 3 cases: $B_m=A_t$, $B_m=A_t\sm\{x\}$, or
$B_m=A_t\cup\{x\}$. Either way, we have
\[B_m\cap([n]\sm x)= A_t\cap([n]\sm x),\]
so (<ref>) follows. 5pt
Assume $\sigma=A_1\s\dots\s A_t$, and assume $x\in A_k$. Let us
consider two cases.
Case 1. Assume that $|A_k|\geq 2$. This corresponds to
Case 1 of Definition <ref>. If $k\neq t$, then we simply have
$V(\cf(\sigma,x))=V(\sigma)$, and (<ref>) holds trivially. If
$k=t$, then $V(\cf(\sigma,x))=V(\sigma)\cup x$, and so (<ref>)
is still valid.
Case 2. Assume now that $A_k=x$, $k<t$. This corresponds to
Case 2 of Definition <ref>. If $k<t-1$, then we again have
$V(\cf(\sigma,x))=V(\sigma)$, while if $k=t-1$, we have
$V(\cf(\sigma,x))\cup x=V(\sigma)$. Either way (<ref>) is true.
We now define the basic flip graphs.
Let $n$ be any natural number. We define a graph $\gn$ as follows.
The vertices of $\gn$ are indexed by all full $[n]$-tuples.
Two vertices $\sigma$ and $\tau$ are connected by an edge if and only
if there exists $x\in F(\sigma)$, such that $\tau=\flip(\sigma,x)$.
We can color the edges of the graph $\gn$ by elements of $[n]$. To do
this we simply assign color $x$ to the edge connecting full
$[n]$-tuple $\sigma$ with the full $[n]$-tuple $\flip(\sigma,x)$. It
follows from (<ref>) that this assignment yields
a well-defined edge coloring of the graph $\gn$; meaning that we
will assign the same color to an edge independently from which of its
endpoints we take as $\sigma$, and furthermore, any two edges which
share a vertex will get different colors under this assignment.
Indeed, if two edges share a vertex, then they must correspond to
flips with respect to different vertices, meaning that they also must
have different colors. Furthermore, consider an edge $(\sigma,\tau)$.
Take $x\in F(\sigma)$, such that $\tau=\cf(\sigma,x)$. By the
equality (<ref>), we have $\cf(\tau,x)=\sigma$, so the color
of the edge does not depend on the choice of the endpoint used to
define that color.
Combinatorially, the edges of $\gn$ are indexed by all coherent pairs
of $[n]$-tuples $(\sigma,\tau)=(A_1\s\dots\s A_t,\ab B_1\s\dots\s
B_t)$, satisfying the conditions: $\carr(\sigma,\tau)=[n]$, and
$|\col(\sigma,\tau)|=n-1$. In this case we have $B_1\cup\dots\cup
B_t=[n]\sm x$, where $x$ is the color of that edge. Pick index $k$
such that $x\in A_k$. The vertices adjacent to that edge are
$A_1\s\dots\s A_{k-1}\s x\s A_k\sm x\s A_{k+1}\s\dots\s A_t$ and
$A_1\s\dots\s A_t$. On the other hand, given a vertex
$\sigma=A_1\s\dots\s A_t$ of $\gn$, and $x\in F(\sigma)$, the edge
with color $x$ which is adjacent to $\sigma$ is indexed by
As an example, the $[3]$-tuples $\sigma_1=1\s 23$ and $\sigma_2=1\s
3\s 2$ index vertices of $\Gamma_3$. These vertices are connected by
an edge labeled with $3$ and indexed by the coherent pair of
$[3]$-tuples $(1|23,1|2)=\dl(\sigma_1,3)=\dl(\sigma_2,3)$. See
Figure <ref>.
The standard chromatic subdivision and the flip graph.
Two executions are connected by an edge if and only if there exists
exactly one process $x$ which has different views under these
executions; all other processes have the same view. The color of that
edge is $x$. The two cases of Definition <ref> correspond to
the situation where process $x$ either stops seing processes which
executed “at the same step”, or starts seeing processes, which
executed in the subsequent step. The views of other processes on $x$
or on each other are unaffected by that change.
§.§ The graphs $\gn^2$
As our next step, we consider ordered pairs of full $[n]$-tuples.
Given two full $[n]$-tuples $\sigma$ and $\tau$, we let
$\sigma\ds\tau$ denote the ordered pair $(\sigma,\tau)$. We shall also
combine this with our previous notations, so if $\sigma=A_1\s\dots\s
A_t$, and $\tau=B_1\s\dots\s B_q$, then $\sigma\ds\tau=A_1\s\dots\s
A_t\ds B_1\s\dots\s B_q$.
We define $F(\sigma\ds\tau):=F(\sigma)\cup F(\tau)$.
Furthermore, for any $x\in F(\sigma\ds\tau)$ we define
\begin{equation}\label{eq:flip2}
\flip(\sigma\ds\tau,x):=
\begin{cases}
\sigma\ds\flip(\tau,x),& \text{ if } x\in F(\tau); \\
\flip(\sigma,x)\ds\tau,& \text{ if } x\notin F(\tau),\,\, x\in F(\sigma).
\end{cases}
\end{equation}
Note, that since $F(\sigma\ds\tau):=F(\sigma)\cup F(\tau)$, one of the
cases in equation (<ref>) must occur. We also remark that
$F(A_1\s\dots\s A_t\ds B_1\s\dots\s B_q)=[n]$, unless $A_t=B_q=x$, for
some $x\in[n]$, in which case we would have $F(A_1\s\dots\s A_t\ds
B_1\s\dots\s B_q)=[n]\sm x$.
Let $n$ be any natural number. We define a graph $\gn^2$ as follows.
The vertices of $\gn^2$ are indexed by all ordered pairs of full
$[n]$-tuples. Two vertices $\sigma_1\ds\tau_1$ and
$\sigma_2\ds\tau_2$ are connected by an edge if and only if there
exists $x\in F(\sigma_1\ds\tau_1)$, such that
Figure <ref> shows the example $\Gamma_3^2$.
The flip graph $\Gamma_3^2$ shown in solid color.
Lurking in the background is the second standard chromatic subdivision
of a triangle.
To describe the edges of $\gn^2$, we extend our $\ds$-notation and
write $(\sigma,\sigma')\ds(\tau,\tau')$, to denote an ordered pair of
coherent pairs of $[n]$-tuples, subject to an important additional
\[\col(\sigma,\sigma')=\carr(\tau,\tau').\]
Now, the edges in $\gn^2$ are of two different types, corresponding to
the two cases of (<ref>):
* either they are indexed by $\sigma\ds(\tau,\tau')$, where
$\sigma$ is a full $[n]$-tuple and $(\tau,\tau')$ indexes an edge
in $\gn$;
* or they are indexed by $(\sigma,\sigma')\ds\tau$, where
$(\sigma,\sigma')$ indexes an edge in $\gn$, and $\tau$ is a full
As an example consider the following vertices of the graph
$\Gamma_3^2$: $v_1=1\s 23\ds 1\s 2\s 3$, $v_2=1\s 23\ds 12\s 3$, and
$v_3=1\s 3\s 2\ds 1\s 2\s 3$. The vertices $v_1$ and $v_2$ are
connected by an edge labeled by $1$ and indexed by $1\s 23\ds(12\s
3,2\s 3)$. The vertices $v_1$ and $v_3$ are connected by an edge
labeled by $3$ and indexed by $(1\s 23,1\s 2)\ds 1\s 2$.
We let $\gn^2(\sigma)$ denote the subgraph of $\gn^2$ induced by the
vertices of the form $\sigma\ds\tau$. Clearly, mapping $\sigma\ds\tau$
to $\tau$ gives an isomorphism between $\gn^2(\sigma)$ and $\gn$.
The vertices of $\gn^2$ correspond to $2$-round executions, and the
edges connect two executions where exactly one process changes it
view. The intuition behind the two cases of Definition <ref>
is as follows. If the process $x$ is seen by someone in the second
round, then what he saw in the first round is know to someone else, so
$x$ cannot change his first round view without affecting the others.
All the process $x$ can do, is to change its view of the second round
in exactly the same way as in the $1$-round model. This is the first
line of (<ref>). The second line of (<ref>)
corresponds to the case when $x$ is the last one to act in the second
round, so nobody sees it. The execution we get, if $x$ changes its
view now, must come from $x$ changing its view of the first round
§.§ Higher flip graphs, support and subdivision maps
Since the graphs $\gn$ and $\gn^2$ are by far the main characters
of this paper, we have chosen to present them separately and in fine
detail. However, the constructions from subsections <ref>
and <ref> can easily be generalized to define the graphs
$\gn^d$, for arbitrary $d\geq 1$. Though we will only need these
briefly for $d=3$, we include the general definitions for
completeness. The concepts in this subsection were previously
introduced in <cit.>.
Let us fix $d\geq 1$ and consider all $d$-tuples of full $[n]$-tuples,
$v=\sigma_1\ds\dots\ds\sigma_d$. Definition <ref> can be
generalized to such $d$-tuples as follows. Given $v$ as above we set
\[F(v):=F(\sigma_1)\cup\dots\cup F(\sigma_d).\]
Effectively this means that $F(v)=[n]$, unless
$F(\sigma_1)=\dots=F(\sigma_d)=[n]\sm p$, for some $p$, in which
case we have $F(v)=[n]\sm p$.
Assume $v$ is a $d$-tuple of $[n]$-tuples, and $x\in F(v)$. Let $k$ be
the maximal index such that $x\in F(\sigma_k)$, by the definition of
$F(v)$, such $k$ must exist. We define $\flip(v,x)$ to be the
following $d$-tuple of full $[n]$-tuples:
\begin{equation}\label{eq:dflip}
\flip(v,x):=(\sigma_1\ds\dots\ds\sigma_{k-1}\ds \flip(\sigma_k,x)\ds
\sigma_{k+1}\ds\dots\ds\sigma_n).
\end{equation}
Again, it is easy to see that for any $x\in F(v)$, we have the
identities $F(\flip(v,x))=F(v)$ and
\begin{equation}\label{eq:ffx}
\flip(\flip(v,x),x)=v.
\end{equation}
For an arbitrary $d\geq 1$ we define graph $\gn^d$ as follows. The
vertices of $\gn^d$ are indexed by all $d$-tuples of full $[n]$-tuples.
Two vertices $v$ and $w$ are connected by an edge if and only if there
exists $x\in F(v)$, such that $w=\flip(v,x)$.
In this context, the graph $\gn^1$ is the same as the graph
$\gn$. The edges of $\gn^d$ are indexed by all $d$-tuples of coherent
pairs of $[n]$-tuples, which for some $1\leq k\leq d$ have the special
\ds\dots\ds\sigma_d$,
* $\sigma_1,\dots,\sigma_{k-1}$ are full $[n]$-tuples,
* $(\sigma_k,\sigma_k')$ indexes an edge in $\gn$,
* $\sigma_{k+1},\dots,\sigma_d$ are full
We call the graphs $\gn^d$ higher flip graphs. They are related
to each other by means of the so-called support
maps. Specifically, given $c<d$, the support map $\supp_d^c$ goes
from the set of vertices of $\gn^d$ to the set of vertices $\gn^c$, it
takes the $d$-tuple $\sigma_1\ds\dots\ds\sigma_d$ to the $c$-tuple
$\sigma_1\ds\dots\ds\sigma_c$. If two vertices of $v$ and $w$ of
$\gn^d$ are connected by an edge, then either
$\supp_d^c(v)=\supp_d^c(w)$ or vertices $\supp_d^c(v)$ and
$\supp_d^c(w)$ are connected by an edge in $\gn^c$.
Furthermore, assume we are given an arbitrary vertex of $\gn^c$,
$v=\sigma_1\ds\dots\ds\sigma_c$, and an arbitrary number $d>c$. We
let $\gn^d(v)$ denote the subgraph of $\gn^d$ induced by all vertices
$w$, for which $\supp_d^c w=v$. Clearly, we have a graph isomorphism
$\varphi:\gn^d(v)\simeq\gn^{d-c}$, given by
\tau_1\ds\dots\ds\tau_{d-c}$.
In the higher flip graph $\gn^d$ the vertices correspond to the
$d$-round executions. Again, two executions are connected by an edge
if and only if there exists exactly one process $x$ which has
different views under these executions; all other processes have the
same view. The color of that edge is $x$. Our flip operation describes
precisely what happens to the label encoding the execution. Here, $k$
is the latest round in which someone has seen process $x$.
Furthermore, a vertex $v$ of $\gn^c$ corresponds to the initial $c$
rounds of an execution, the so-called prefix, and the graph $\gn^d(v)$
encodes all executions which start with that prefix $v$.
§ STANDARD MATCHINGS ON GRAPHS $\GN(\OMEGA,V)$
At this point we would like to issue a word of warning to the
reader. While our initial mathematical concepts are closely related
to the Distributed Computing, this connection will weaken from this
point on. Once one has a mathematical model, which is equivalent to
the questions in Distributed Computing which we would like to study,
we need to start exploring purely mathematical structures, in order to
be able to arrive at the resolution of our initial questions.
In particular, the concepts such as forbidden sets, prefixes,
standard matchings, in this section, or sets of patterns and
other notions, in the subsequent sections, do not have a distributed
computing interpretation, which is easy to grasp. Or perhaps the right
way to phrase this is that such an interpretation is yet to be
invented. We will however still provide distributed computing context
where possible, such as for example for the case of nodes in
Section <ref>.
§.§ Forbidden sets and graphs $\gn(V)$
Let $n$ be an arbitrary natural number, and let $V$ be any subset of
$[n]$. We let $\gn(V)$ denote the subgraph of $\gn$ induced by the
vertices indexed by those full $[n]$-tuples $A_1\s\dots\s A_t$, for
which $A_1\not\subseteq V$.
We think of $V$ as a ”forbidden set”, in which case the condition in
Definition <ref> simply says that the first block of the full
$[n]$-tuple must contain an element which is not
forbidden. Clearly, when nothing is forbidden, we have no
restrictions, hence $\gn(\emptyset)=\gn$, and when everything is
forbidden, we have no full $[n]$-tuples satisfying that condition,
hence $\gn([n])=\emptyset$. Further examples are provided in
Figure <ref>.
The graphs $\Gamma_3(\{2,3\})$ (left) and $\Gamma_3(\{3\})$
§.§ Prefixes and graphs $\gn(\Omega,V)$
Let $V$ be any subset of $[n]$, and let $\sigma=A_1\s\dots\s A_t$ be
any full $[n]$-tuple. Let $t-1\geq k\geq 0$ denote the minimal number
for which $A_{k+1}\not\subseteq V$; if no such number exists, we set
$k:=t$. Then the $V$-tuple $A_1\s\dots\s A_k$ is called the $V$-prefix of $\sigma$, and is denoted by $\pre_V(\sigma)$.
Note, that we allow the $V$-prefix of $\sigma$ to be empty, which is
the case if $k=0$, or equivalently $A_1\not\subseteq V$, meaning that
$\sigma$ is a vertex of $\gn(V)$.
Let $n$ be an arbitrary natural number, let $V$ be a subset of $[n]$,
and let $\Omega$ be a family of $V$-tuples. The graph $\gn(\Omega,V)$
is the subgraph of $\gn$ induced by all vertices $\sigma$, such that
Note how this relates to our previously used notations:
$\gn(V)=\gn(\emptyset,V)$. In line with thinking about the set $V$ as
a forbidden set, we think about $\Omega$ as a set of allowed
prefixes. See Figure <ref> for an example.
The graph $\Gamma_3(\Omega,V)$, for $V=\{2,3\}$,
$\Omega=\{2,2\s 3\}$.
The next proposition states a simple, but important property of
Let $\sigma$ be some full $[n]$-tuple, and let $V$ be any subset of
$[n]$. Assume that we are given $x\in F(\sigma)$, such that $x\notin
V$. Then, flipping with respect to $x$ does not change the $V$-prefix,
in other words, we have
\begin{equation}\label{eq:st1}
\pre_V(\sigma)=\pre_V(\cf(\sigma,x)).
\end{equation}
Assume $\sigma=A_1\s\dots\s A_k\s A_{k+1}\s\dots\s A_t$, such that
$A_1\s\dots\s A_k=\pre_V(\sigma)$, so for $1\leq i\leq k$, we have
$A_i\subseteq V$, and furthermore $A_{k+1}\not\subseteq V$. Since
$x\not\in V$, we have $x\in A_{k+1}\cup\dots\cup A_t$, so we can pick
$k+1\leq l\leq t$, such that $x\in A_l$. We now consider different
First, if $l\geq k+2$, then
\[\cf(\sigma,x)=A_1\s\dots\s A_k\s A_{k+1}\s B_{k+2}\s\dots\s B_{\tilde t},\]
for some sets $B_{k+2},\dots,B_{\tilde t}$, where $\tilde t=t+1$ or
$\tilde t=t-1$. Clearly, we then have
$\pre_V(\cf(\sigma,x))=A_1\s\dots\s A_k$.
Now assume $l=k+1$ and $|A_{k+1}|=1$, i.e., $A_{k+1}=x$. Since $x\in
F(\sigma)$, we have $k+2\leq t$. The Case 2 of
Definition <ref> applies, and we have
\[\cf(\sigma,x)=A_1\s\dots\s A_k\s x\cup A_{k+2}\s A_{k+3}\s\dots\s
Hence again $\pre_V(\cf(\sigma,x))=A_1\s\dots\s A_k$, since
$x\notin V$.
Finally, assume $l=k+1$, and $|A_{k+1}|\geq 2$. The Case 1 of
Definition <ref> applies, and we have
\[\cf(\sigma,x)=A_1\s\dots\s A_k\s x\s A_{k+1}\sm x\s
A_{k+2}\s\dots\s A_t.\]
Since $x\notin V$, we get
$\pre_V(\cf(\sigma,x))=A_1\s\dots\s A_k$ here as well.
§.§ Standard matchings
We shall now describe a set of partial matchings on the graphs
$\gn(\Omega,V)$, which we shall call the standard matchings. To
start with, note that formally a matching is a function $\mu$ defined
on some of the vertices of $G$, which has vertices of $G$ as values,
and which satisfies the following conditions:
* if $\mu(\sigma)$ is defined, then the vertices $\sigma$ and
$\mu(\sigma)$ are connected by an edge, called the matching
* when $\mu(\sigma)$ is defined, then $\mu(\mu(\sigma))$ is also
defined and is equal to $\sigma$.
When we consider matchings in the specific case of the flip graphs, we
can record the labels of the matching edges. Assuming $\mu(\sigma)$ is
defined, we let $\id_\mu(\sigma)$ denote the label of the matching
edge $(\sigma,\mu(\sigma))$; it is uniquely determined by the identity
$\mu(\sigma)=\cf(\sigma,\id_\mu(\sigma))$. By (<ref>), we
have $\id_\mu(\mu(\sigma))=\id_\mu(\sigma)$.
Let $V$ be any subset of $[n]$, and let $R=(x_1,\dots,x_d)$ be an
order on its complement $[n]\sm V$; in particular, $d=n-|V|$. Assume
$\sigma=A_1\s\dots\s A_t$ is a full $[n]$-tuple. We set $h_R(\sigma)$
to be the index $1\leq h\leq d$, such that $\sigma=A_1\s\dots\s A_k\s
x_{h+1}\s\dots\s x_d$, and $A_k\neq x_h$. Clearly, if such an index
exists, it is unique. If it does not exist, we have
$\sigma=A_1\s\dots\s A_k\s x_1\s\dots\s x_d$, in which case we set
$h_R(\sigma):=0$. We call $h_R(\sigma)$ the height of $\sigma$
with respect to $R$.
By Definition <ref>, we have $0\leq h_R(\sigma)\leq d$,
where $d=n-|V|$. The maximum $d$ is achieved if and only if $A_t\neq
x_d$. The full $[n]$-tuples of height $0$ with respect to some fixed
order $R$ are called critical with respect to $R$.
The critical full $[n]$-tuples all begin by some full $V$-tuple,
followed by the full $([n]\sm V)$-tuple $x_1\s\dots\s x_d$.
We now have the necessary terminology to define the standard
Assume $V$ is a subset of $[n]$, and $R$ is an order on its complement
$[n]\sm V$. We define a partial matching on the vertices of $\gn$,
denoted by $\mu_R$. For an arbitrary full $[n]$-tuple $\sigma$, set
$h:=h_R(\sigma)$. If $h\neq 0$, we set
\begin{equation}\label{eq:mu}
\mu_R(\sigma):=\cf(\sigma,x_h),
\end{equation}
else $\mu_R(\sigma)$ is undefined. We call $\mu_R$ the standard
matching associated to $R$.
Note, that in the above Definition <ref>, we might as well ask
$V$ to be a proper subset of $[n]$. This is because the case $V=[n]$
is rather degenerate: any $R$ is an empty order, and so the standard
matching $\mu_R$ is empty as well, with all vertices being critical
with respect to $R$. We refer the reader to <cit.>, for the illustration of the standard matching for
Whenever $V$ is a subset of $[n]$, and $R$ is any order on $[n]\sm V$,
the partial matching $\mu_R$ on the set of vertices of $\Gamma_n$ is
well-defined. Furthermore, when $\mu_R(\sigma)$ is defined, we have
\begin{equation} \label{eq:st2-1}
\pre_V(\sigma)=\pre_V(\mu_R(\sigma))
\end{equation}
\begin{equation} \label{eq:st2-2}
\end{equation}
To say that $\mu_R$ is well-defined is equivalent to the following
(1) if $\mu_R(\sigma)$ is defined, then $\sigma$ and
$\mu_R(\sigma)$ are connected by an edge;
(2) $\mu_R(\mu_R(\sigma))$ is also defined;
(3) $\mu_R(\mu_R(\sigma))=\sigma$.
The statement (1) is obvious, since $\mu_R(\sigma)$ is a certain flip
of $\sigma$. To verify the rest, assume $R=(x_1,\dots,x_d)$, and
$\sigma=A_1\s\dots\s A_k\s x_{h+1}\s\dots\s x_d$, where
$h=h_R(\sigma)$. Since $\mu_R(\sigma)$ is defined, we have
$h_R(\sigma)>0$, and $A_k\neq x_h$. By Definition <ref>, we have
$\mu_R(\sigma)=\cf(\sigma,x_h)$. By construction, we have $x_h\notin
V$, so by Proposition <ref> we get
and (<ref>) is proved.
Pick $1\leq l\leq k$, such that $x_h\in A_l$. Assume first $|A_l|=1$,
i.e., $A_l=x_h$. In this case we must have $l\leq k-1$. If $l\leq
k-2$, then
\[\mu_R(\sigma)=A_1\s\dots\s A_{l-1}\s x_h\cup A_{l+1}\s A_{l+2}\s\dots\s
A_k\s x_{h+1}\s\dots\s x_d,\]
and, since $x_h\neq A_k$, we get
$h_R(\sigma)=h_R(\mu(\sigma))$. If $l=k-1$ instead, we have
\[\mu_R(\sigma)=A_1\s\dots\s A_{k-2}\s x_h\cup A_k\s
x_{h+1}\s\dots\s x_d,\]
and, since $x_h\neq x_h\cup A_k$, we again
get $h_R(\sigma)=h_R(\mu(\sigma))$.
Assume now that $|A_l|\geq 2$. In this case $\mu_R(\sigma)$ is
obtained from $\sigma$ by splitting off the element $x_h$. If $l\leq
k-2$, then
\[\mu_R(\sigma)=A_1\s\dots\s A_{l-1}\s x_h\s A_l\sm x_h\s A_{l+1}\s\dots\s
A_k\s x_{h+1}\s\dots\s x_d,\]
and, since $x_h\neq A_k$, we get
$h_R(\sigma)=h_R(\mu(\sigma))$. Finally, if $l=k-1$, we have
\[\mu_R(\sigma)=A_1\s\dots\s A_{k-1}\s x_h\s A_k\sm x_h\s
x_{h+1}\s\dots\s x_d,\]
and, since $x_h\neq A_k\sm x_h$, we again
get $h_R(\sigma)=h_R(\mu(\sigma))$.
The equality (<ref>) has now been proved for all $\sigma$.
In particular, if $h_R(\sigma)\neq 0$, then $h_R(\mu_R(\sigma))\neq 0$,
so $\mu_R(\mu_R(\sigma))$ is defined.
Finally, the equalities (<ref>), (<ref>), and
definition of $\mu_R$, combine to
\[\mu_R(\mu_R(\sigma))=\cf(\cf(\sigma,x_h),x_h)=\sigma,\]
finishing the proof of the proposition.
Note that Proposition <ref>, including the
equality (<ref>), implies that $\mu_R$ restricts to a partial
matching on $\gn(\Omega,V)$, for any $\Omega$. The next theorem states
the key properties of standard matchings in flip graphs.
(1) Let $R=(x_1,\dots,x_n)$ be an arbitrary order on the set
$[n]$. The standard matching $\mu_R$ on $\gn$ is near-perfect, it
has a unique critical vertex, indexed by $x_1\s\dots\s x_n$.
(2) Let $V$ be an arbitrary non-empty subset of $[n]$, and let
$R$ be any order on $[n]\sm V$. The associated standard matching
$\mu_R$ on $\gn(V)$ is perfect.
(3) Assume $V$ is a non-empty subset of $[n]$,
$R=(x_1,\dots,x_d)$ an order on $[n]\sm V$, and $\Omega$ a family of
$V$-tuples. The associated standard matching $\mu_R$ is a partial
matching on $\gn(\Omega,V)$, with critical vertices of the form
$A_1\s\dots\s A_k\s x_1\s\dots\s x_d$, where $A_1\s\dots\s A_k$ is
a full $V$-tuple in $\Omega$. In particular, if $\Omega$ has no full
$V$-tuples, then $\mu_R$ is a perfect matching on $\gn(\Omega,V)$.
(1) We have $V=\emptyset$, hence $\pre_V(\sigma)=\emptyset$, for
all $\sigma$. By Remark <ref> all critical vertices are
indexed by concatenations of full $V$-tuples with $x_1\s\dots\s x_d$,
where $d=n-|V|$. Here that description reduces to the existence of
a single critical vertex $x_1\s\dots\s x_n$.
(2) Let $\sigma$ be a vertex of $\Gamma_n$ which is critical with
respect to $\mu_R$. We have $\sigma=A_1\s\dots\s A_k\s x_1\s\dots\s
x_d$, where $A_1\s\dots\s A_k$ is a full $V$-tuple. Since $V\neq
\emptyset$, we have $k\geq 1$ and $A_1\subseteq V$. By
Definition <ref> this vertex does not belong to $\gn(V)$, so
$\gn(V)$ has no critical vertices. Clearly, this is the same as to say
that $\mu_R$ restricts to a perfect matching on the vertices
of $\gn(V)$. Identical argument shows the more general statement (3)
as well.
§ CONDUCTIVITY IN THE FLIP GRAPHS
§.§ Previous work
While the standard matchings defined in subsection <ref> are
very useful, they do not always yield perfect matchings in the
situations we will be interested in. It is therefore practical to have
a procedure to modify a partial matching so as to decrease the number
of critical vertices. To start with, let us recall the following
additional terminology from graph theory.
Assume we are given a matching on a graph $G$. An edge path is called
alternating if its edges are alternating between matching and
non-matching ones. It is called properly alternating if, in
addition, it starts and ends either with a matching edge, or with
a critical vertex.
A properly alternating path is called augmenting if it starts
and ends with critical vertices, it is called non-augmenting if
it starts and ends with matching edges, and, finally, it is called
semi-augmenting if it is neither augmenting nor non-augmenting.
Next definition describes a classical technique for modifying
Assume we are given a matching $\mu$ on a graph $G$, and a properly
alternating non-self-intersecting edge path $\gamma$. We define
$D(\mu,\gamma)$ as a new matching on $G$ consisting of all edges from
$\mu$ which do not belong to $\gamma$ together with all edges from
$\gamma$ which do not belong to $\mu$.
When trying to modify a matching one is looking for existence of such
properly alternating non-self-intersecting edge paths $\gamma$. It
turns out that when the underlying graph is bipartite, the condition
for the path to be non-self-intersecting can be dropped.
Assume $G$ is a bipartite graph, $\mu$ is a matching on $G$, $v$ and
$w$ are different vertices of $G$, and $\gamma$ is a properly
alternating edge path from $v$ to $w$. Then there exists a properly
alternating non-self-intersecting edge path from $v$ to $w$.
If $\gamma$ does have self-intersections, then it contains
cycles. Any such cycle is of even length, since the graph is
bipartite. Deleting a cycle of even length from a properly alternating
edge path yields another properly alternating edge path. If we keep
removing the cycles, we will eventually make our properly alternating
edge path non-self-intersecting.5pt
Assume $\gamma$ is a semi-augmenting path with endpoints $v$ and $w$,
where $v$ is a critical vertex, and $w$ is not. It is easy to see that
the set of critical vertices with respect to $D(\mu,\gamma)$ is
obtained by taking the critical vertices with respect to $\mu$, and
then replacing $v$ with $w$. For this reason, we shall intuitively
view the process of replacing $\mu$ with $D(\mu,\gamma)$ as transporting $v$ to $w$ along the path $\gamma$. We think of the
corresponding property of the graph as its conductivity. The
following result has been proved in <cit.>.
(<cit.>) $\,$
Assume $n$ is an arbitrary natural number.
* The graph $\gn$ is a bipartite graph with a unique bipartite
decomposition $(A,B)$ such that $|A|=|B|+1$. For any vertex $v\in A$
there exists a perfect matching on $\gn\sm v$.
* Assume $[n]\supset V\neq\emptyset$, then the graph $\gn(V)$ is
a bipartite graph with a bipartite decomposition $(A,B)$ such that
$|A|=|B|$. If furthermore $|V|\leq n-2$, then for any vertices
$v\in A$, $w\in B$, there exists a perfect matching on
The proof of the first part of Proposition <ref> in
<cit.> was based on the fact that given a near-perfect matching
(recall Definition <ref>) of $\gn$ with a critical vertex
$v\in A$, for any other vertex $w\in A$ there would exist
a semi-augmenting path from $v$ to $w$. For the proof of the second
part, we constructed in <cit.> non-augmenting paths for any pair
of vertices $v\in A$ and $w\in B$.
Rather than directly generalizing the techniques used in <cit.>
to prove Proposition <ref>, we take a slightly different
approach. Namely, instead of seeking to connect any pair of arbitrary
vertices, we single out a special group of vertices, which we call
connectors and only try to conduct between them.
Any vertex of $\gn$, which is indexed by a full $[n]$-tuple
$a_1\s a_2\s\dots\s a_n$ is called an $a_n$-connector of the
first type, whereas any vertex indexed by a full $[n]$-tuple
$\{a_1,a_2\}\s a_3\s\dots\s a_n$ is called an $a_n$-connector of
the second type.
Given a connector $\tau=A_1\s\dots\s A_t$ of any of the two types,
and a full $[n]$-tuple $\sigma$, we say that $\tau$ is proper
with respect to $\sigma$, if $A_1\notin V(\sigma)$, in other words,
$\tau$ is a well-defined vertex of $\gn(V(\sigma))$.
In the rest of this section we assume that $n\geq 5$, and that
$n-1\geq |V|\geq 1$.
§.§ Conductivity in $\gn(V,\Omega)$ when $\Omega$ has no
full $V$-tuples
Assume we are given a family of $V$-tuples $\Omega$ which
has no full $V$-tuples. In this case, according to
Theorem <ref>(3) the standard matching associated to
any order is perfect.
Let $\sigma\in\gn(V,\Omega)$, $\sigma=a_1\s\dots\s a_n$, be
an arbitrary $a_n$-connector of the first type, such that $a_1\notin
(1) For any given element $f\neq a_1$, there exists an order
$R$ on $[n]\sm V$, and an edge path $p$ in $\gn(V,\Omega)$, which is
non-augmenting with respect to $\mu_R$, starting from $\sigma$, and
terminating at some $f$-connector of the second type $\tau=\{a_1,
y_2\}\s\dots\s y_{n-1}\s f$.
(2) If $|V|\leq n-2$, there exists an order $R$ on $[n]\sm V$,
and an edge path $p$ in $\gn(V,\Omega)$, which is non-augmenting
with respect to $\mu_R$, starting from $\sigma$, and terminating at
some $a_1$-connector of the second type $\tau=\{y_1,y_2\}\s\dots\s
y_{n-1}\s a_1$, with $\{y_1,y_2\}\not\subseteq V$.
After renaming, we can assume without loss of generality that
$V=\{1,\dots,d\}$, where $d\leq n-1$, and that $a_1=n$, i.e.,
$\sigma=n\s a_2\s\dots\s a_n$. We now set $R:=(d+1,\dots,n)$.
We start by proving (1), i.e., we are given $f\neq a_1$. Assume first
that $f=a_l$, for some $l\geq 3$. Using the alternating path
$\swap^I_k$ shown on left-hand side of the Figure <ref> we
can swap the $k$-th and the $(k+1)$-st parts of our full $[n]$-tuple,
for any $k\geq 3$. We concatenate the paths $\swap^I_l$,
$\swap^I_{l+1}$, $\dots$, $\swap^I_{n-1}$ to obtain a new alternating
path. This path ends at a vertex of the form $\tilde\tau=n\s a_2\s
b_3\s\dots\s b_{n-1}\s f$, for some $b_3,\dots,b_{n-1}$, which is
an $f$-connector of the first type. Set $\tau=\{n,a_2\}\s b_3\s\dots\s
b_{n-1}\s f$, and add the matching edge $(\tilde\tau,\tau)$ to our
path. We now have a non-augmenting path between $\sigma$ and $\tau$,
with the latter being an $f$-connector of the second type of the
required form. Thus the statement (1) is proved in this case.
Assume now that $f=a_2$, i.e., $\sigma=n\s f\s a_3\s\dots\s a_n$. In
this case we first follow the somewhat more complicated alternating
path $\swap^I_2$ shown on the left-hand side of the
Figure <ref>, and then proceed as in the case $l\geq 3$,
by concatenating the alternating paths $\swap_3^I$, $\dots$,
$\swap_{n-1}^I$. Again, we will end up with a non-augmenting path
between $\sigma$ and the $f$-connector of the second type
$\tau=\{n,a_3\}\s a_4\s\dots\s a_n\s a_2$. Note, that we use here the
fact that $n\geq 5$, implying $n-1>3$. This finishes the proof
of (1).
To prove (2) assume now that $|V|\leq n-2$, in particular, we have
$n-1\notin V$. Let $l\geq 2$ be such that $a_l=n-1$. If $l\geq 3$, we
start by concatenating the alternating paths $\swap^I_{l-1}$,
$\swap^I_{l-2}$, $\dots$, $\swap^I_2$, to arrive at the vertex of the
form $n\s n-1\s b_3\s\dots\s b_n$, for some $b_3,\dots,b_n$; if $l=2$
then we are at that vertex to start with. Note, that the alternating
paths $\up^I_k$, for $1\leq k\leq n-1$, allow in certain situations to
move the element $n$ from being the $k$-th set of our full $[n]$-tuple
to being its $(k+1)$-st set. We now concatenate the alternating paths
$\up^I_1$, $\up^I_2$, $\dots$, $\up^I_{n-1}$ to arrive at the vertex
$\tilde\tau= n-1\s b_3\s\dots\s b_n\s n$. To finish, set
$\tau=\{n-1,b_3\}\s\dots\s b_n\s n$, and add the matching edge
$(\tilde\tau,\tau)$ to our path. We now have a non-augmenting path
between $\sigma$ and $\tau$, where $\tau$ is a $a_1$-connector of the
second type satisfying the desired conditions. This finishes the proof
of Lemma <ref>.
Assume we are given a connector of second type
$\sigma=\{a_1,a_2\}\s a_3\s\dots\s a_n$, such that $\{a_1,a_2\}\not\subseteq
V$, say $a_1\notin V$.
(1) For any $f\neq a_1$, there exists an order $R$ on $[n]\sm
V$, and an edge path $p$ in $\gn(V,\Omega)$, which is non-augmenting
with respect to $\mu_R$, starting from $\sigma$, and terminating at
some $f$-connector of the first type $\tau=a_1\s y_2\s\dots\s
y_{n-1}\s f$.
(2) If $|V|\leq n-2$, there exists an order $R$ on $[n]\sm V$,
and an edge path $p$ in $\gn(V,\Omega)$, which is non-augmenting
with respect to $\mu_R$, starting from $\sigma$, and terminating at
some $a_1$-connector of the first type $\tau=y_1\s y_2\s\dots\s
y_{n-1}\s a_1$, with $y_1\notin V$.
Again, we can assume without loss of generality, that
$V=\{1,\dots,d\}$, where $d\leq n-1$, and that $a_1=n$, i.e.,
$\sigma=\{n,a_2\}\s a_3\s\dots\s a_n$. We now set $R:=(d+1,\dots,n)$.
First, we prove the statement (1). Assume $f=a_k$, $k\geq 3$. We can
concatenate the paths $\swap^{II}_k$, $\dots$, $\swap^{II}_{n-1}$,
which are shown in the right hand side of Figures <ref>,
and <ref>. This will get us to the vertex
$\tilde\tau=\{n,y_2\}\s\dots\s y_{n-1}\s f$, for some
$y_2,\dots,y_{n-1}$. We set $\tau:=n\s y_2\s\dots\s y_{n-1}\s f$ and
note that $(\tilde\tau,\tau)$ is a matching edge. Adding that edge to
the path which we have up to now yields a non-augmenting path
connecting $\sigma$ with $\tau$.
Let us now show (2). We have assumed that $|V|\leq n-2$, i.e.,
$n-1\notin V$. Here we have $\sigma=\{n,a_2\}\s a_3\s\dots\s a_n$, and
we pick index $k$ such that $a_k=n-1$. Assume first that $k\geq 3$. If
$k=3$, then we have $\sigma=\{n,a_2\}\s n-1\s a_4\s\dots\s a_n$. If
$k\geq 4$, then we can concatenate paths $\swap^{II}_{k-1}$,
$\swap^{II}_{k-2}$, $\dots$, $\swap^{II}_3$. This will yield
an alternating path starting at $\sigma$ and terminating at
$\{n,a_2\}\s n-1\s a_4\s\dots\s a_n$. After that we concatenate with
the path $\specup^{II}$ shown on the left hand side of the
Figure <ref>. The obtained path terminates at the vertex
$\{n-1,a_2\}\s n\s a_4\s\dots$. Further, we concatenate with the
alternating paths $\up^{II}_3$, $\up^{II}_4$, $\dots$,
$\up^{II}_{n-1}$, see the Figure <ref>, to arrive at the
vertex $\{n-1,a_2\}\s a_4\s\dots\s a_n\s n$. We finish by
concatenating with the matching edge between $\{n-1,a_2\}\s
a_4\s\dots\s a_n\s n$ and $\tau=n-1\s a_2\s a_4\s\dots\s a_n\s n$, to
obtain a non-augmenting path from $\sigma$ to the appropriate
$n$-connector of the first type $\tau$.
It remains to consider the case $k=2$, that is $\sigma=\{n-1,n\}\s
a_3\s a_4\s\dots$. In this situation we start with the alternating
path $\up^{II}_2$, see the right hand side of the
Figure <ref>, and arrive at the vertex $\{n-1,a_3\}\s n\s
a_4\s\dots$. We can then proceed just as in the case before with the
alternating paths $\up^{II}_3$, $\up^{II}_4$, $\dots$,
$\up^{II}_{n-1}$, followed up with the matching edge between
$\{n-1,a_3\}\s a_4\s\dots\s a_n\s n$ and $\tau=n-1\s a_3\s
a_4\s\dots\s a_n\s n$, to again obtain a non-augmenting path from
$\sigma$ to the appropriate $n$-connector of the first
type $\tau$. This is the last case to be considered and we have now
shown the statement (2).
Clearly, the Lemmata <ref> and <ref> allow us to extend
augmenting paths across $\gn^2$ as shown on the Figure <ref>.
Concatenating augmenting paths.
Note that if $|V|\leq n-2$, then there are $x$-connectors of both
types for all $x$. If $|V|=n-1$, then there are $x$-connectors of
both types if and only if $x\neq[n]\sm V$. If $|V|=n-1$ and
$[n]=V\cup\{x\}$, then there are no $x$-connectors, but we also do not
need any.
§.§ Conductivity in $\gn(V,\Omega)$ in some special cases
Let us first consider the case when $\Omega$ has a unique
full $V$-tuple. Again, according to Theorem <ref>(3)
the standard matching associated to any order has a unique critical
Assume that the family $\Omega$ contains a unique full $V$-tuple
$v_1\s\dots\s v_d$ together with all of its truncations. Assume
furthermore, that we are given some connector of the first type
$\tau=y_1\s\dots\s y_n$, such that $y_1\notin V$. Then there exists an
order $R$ on $[n]\sm V$, and an edge path in $\gn(V,\Omega)$, which is
semi-augmenting with respect to $\mu_R$, and which connects the
critical vertex $\sigma$ to $\tau$.
After suitable renaming we can assume, without loss of generality,
that $V=\{1,\dots,d\}$, and that the unique full $V$-tuple is
$1\s\dots\s d$. Furthermore, we can make sure that $y_1=n$ after that
renaming. We now choose the order $R:=(d+1,\dots,n)$, hence the
unique, critical with respect to $\mu_R$, vertex is $\sigma=1\s
2\s\dots\s n$. We need to find a semi-augmenting path from $\tau=n\s
y_2\s\dots\s y_n$ to $\sigma$.
To start with, we can concatenate paths $\swap^I_k$, for $2\leq k\leq
n-1$ in an appropriate order, so as to obtain an alternating path
starting at $\tau$ and terminating at $n\s 1\s 2\s\dots\s n-1$. After
this, we concatenate the paths $\up_1^I,\dots,\up_{n-1}^I$. Note, that
these paths lie within the graph $\gn(V,\Omega)$, since we assumed
that $\Omega$ contains all truncations of $1\s\dots\s d$. The total
path terminates at $\sigma$, which is exactly what we are looking
We start our path from $\sigma$ with the segment shown on
Figure <ref>. Next, we want to move $y_2$ in the second
position. If $y_2=1$ then we are done and we can skip this step, so
assume $2\leq y_2\leq n-1$. The path shown on Figure <ref>
allows us to swap entries in positions $k$ and $k+1$, provided $k>2$,
much like the path shown on Figure <ref>. Concatenating
such paths for $k=y-1,\dots,1$ we end up at the vertex $n\s 1\s y_2\s
2\s\dots\s y_2-1\s y_2+1\s\dots\s n-1$. We now follow the edge path on
Figure <ref> and arrive at the vertex $n\s y_2\s 1\s\dots\s
y_2-1\s y_2+1\s\dots\s n-1$.
We have now placed $y_1$ and $y_2$ into the first and the second
positions, and it remains to place the elements $y_3$, $\dots$, $y_n$
into their slots. This can be done by concatenating the edge paths
shown on Figure <ref> as it allows arbitrary swaps of the
elements, as long as they are not in the positions 3 or higher.
Let us now consider the second special case. This time we assume that
$\Omega$ has three full $V$-prefixes: $(v_1\s v_2\s v_3\s\dots\s
v_d)$, $(\{v_1,v_2\}\s v_3\s\dots\s v_d)$, and $(v_1\s \{v_2,v_3\}
\s\dots\s v_d)$. In this case the standard matching $\mu_R$ associated
to any order has three critical vertices. We shall extend $\mu_R$ by
matching two of the critical vertices to each other. After this we
find an augmenting path from the third critical vertex to
a $y$-connector, as in Lemma <ref>.
Assume we are given set $V$ and a family of $V$-tuples $\Omega$, which
contains three full $V$-tuples as above, together with all of their
truncations. Assume, furthermore, we are given $f\in [n]$, such that
$[n]\sm V\neq f$. Then there exists an order $R$ on $[n]\sm V$, such
that the standard matching $\mu_R$ can be extended by matching two of
the critical vertices to each other, and, furthermore, there exists
an edge path in $\gn(V,\Omega)$, which is semi-augmenting with respect
to that extended matching, and which connects the remaining critical
vertex $\sigma$ to some $f$-connector of the second type
$\{y_1,y_2\}\s y_3\s\dots\s y_{n-1}\s f$, such that
$\{y_1,y_2\}\not\subseteq V$.
Again, after suitable renaming, we can assume, without loss of
generality, that $V=\{1,\dots,d\}$, and that the full $V$-tuples are
$(1\s 2\s 3\s\dots\s d)$, $(\{1,2\}\s 3\s\dots\s d)$, and
$(1\s\{2,3\}\s\dots\s d)$. Since $[n]\sm V\neq f$, we can pick
an element of $[n]\sm V$ different from $f$. Without loss of
generality we can make sure, that after renaming that element is
called $n$. We set $R:=(d+1,\dots,n)$, so the three critical vertices
are now $\sigma=\{1,2\}\s 3\s 4\s\dots\s n$, $\alpha_1=1\s 2\s 3\s
4\s\dots\s n$, and $\alpha_2=1\s\{2,3\}\s 4\s\dots\s n$. We extend the
standard matching $\mu_R$ by matching $\alpha_1$ with $\alpha_2$.
By our construction, $f\neq n$, and we set $\tau:=\{n,1\}\s 2\s\dots\s
f-1\s f+1\s\dots\s n-1\s f$. This is an $f$-connector of the second
type satisfying necessary conditions, since $n\notin V$. We now
describe how to find a semi-augmenting path from $\tau$
to $\sigma$. To start with, we concatenate the paths
$\swap_{n-1}^{II}$, $\dots$, $\swap_{f+1}^{II}$, to arrive at the
vertex $\{n,1\}\s 2\s 3\s\dots\s n-1$. After this, we concatenate with
the path on Figure <ref> to get to the desired
semi-augmenting path to $\sigma$.
We now start our path by the segment shown on Figure <ref>.
This augmented path is legal since the $V$-tuples of all vertices on
that path are truncations of the $V$-tuple $1\s 2\s 3\s\dots\s d$, or
of the $V$-tuple $1, 2\s 3\s\dots\s d$, hence are allowed.
To decide how to continue, let us first assume that $y\geq 3$. The
segment shown in Figure <ref>, which is highly similar to the
one in Figure <ref>, allows us to swap entries in positions
$k$ and $k+1$, provided $k>2$. Repeating these as above, we will
eventually arrive at a $y$-connector.
We are now left with considering the cases $y=1$ and $y=2$. We need
to continue starting at the vertex $n\s 1,2\s 3\s\dots\s n-1$. To do
this, we concatenate with the segment shown on Figure <ref>;
we take the left hand side if $y=1$, and the right hand side if $y=2$.
After this we use the swaps shown on Figure <ref>, which are
very similar to the swaps we used before. Again, if $y=1$ that we use
the segments shown on the left hand side of Figure <ref>,
else we use the right hand side of that figure. Either way, we will
eventually arrive at a $y$-connector.
§ NODES
§.§ Definition of $n$-nodes of the $d$-th level
It is now time to define the nodes, which, after the flip
graphs, constitute the second main combinatorial concept of this
paper. On the geometric side the nodes correspond to vertices of
iterated chromatic subdivisions, while on distributed computing side
they correspond to local views of the processes.
Let $n$ and $d$ be arbitrary natural numbers. A $d$-tuple
$v=\nu_1\ds\dots\ds\nu_d$ of coherent pairs of $[n]$-tuples
is called an $n$-node of the $d$-th level if it
satisfies the following properties:
(1) $\col(\nu_i)=\carr(\nu_{i+1})$, for all $1\leq i\leq d-1$;
(2) $|\col(\nu_d)|=1$, in other words, there exists $S\subseteq
[n]$ and $x\in S$, such that $\nu_d=(S,x)$.
We set $\carr(v):=\carr(\nu_1)$, and call it the carrier of $v$;
we set $\col(v):=\col(\nu_d)$, and call it the color of $v$.
Finally, let $\cn_n^d$ denote the set of all $n$-nodes of the $d$-th level.
The $3$-nodes of the first level juxtaposed on $\Gamma_3$.
The special cases $d=1$ and $d=2$ are the ones most used in this paper,
therefore it makes sense to unwind the Definition <ref> to see
explicitly what it says for these values of $d$.
* An $n$-node of the first level is simply a pair $(S,x)$,
where $S\subseteq[n]$ and $x\in S$.
* An $n$-node of the second level is a pair $\sigma\ds\tau$,
where $\sigma$ is a coherent pair of $[n]$-tuples, and $\tau=(S,x)$ is
an $n$-node of the first level, such that $\col(\sigma)=S$.
Assume we have a bijective set map $\varphi:S\ra T$. Then we have an
induced map taking $S$-tuples to $T$-tuples, it is simply given by
$\varphi(A_1\s\dots\s A_t)=\varphi(A_1) \s\dots\s\varphi(A_t)$, that
is we apply $\varphi$ to each set separately. In the same way, the
function $\varphi$ extends to coherent pairs of $S$-tuples, as well as
to tuples of coherent pairs of $S$-tuples.
In particular, assume $v$ is an $n$-node of $d$-th level, set
$S:=\supp(v)$, and assume we are given a bijective map $\varphi:S\ra
T$. Then $\varphi(v)$ is well-defined, it is an $n$-node of $d$-th
level, and $\supp(\varphi(v))=T$.
Let $v$ be an $n$-node of the $d$-th level, and let $\varphi$ be the
normalizer of $\carr(v)$. We call $\varphi(v)$ the normal form
of $v$.
Let $n$ and $d$ be arbitrary natural numbers. Given an $n$-node of the
$(d+1)$-st level $v=v_1\ds\dots\ds v_{d+1}$, we define a new $n$-node
$w=w_1\ds\dots\ds w_d$ of the $d$-th level as follows:
$w_d:=v_d\dar\col(v)$, $w_{d-1}:=v_{d-1}\dar\carr(w_d)$, $\dots$,
We call the obtained node $w$ the parent of $v$ and denote it by
Let us note a few special cases. If $d=1$, we have $v=v_1\ds (S,x)$,
and we set $\parent(v):=(T,x)=v_1\dar x$. If $d=2$, we have $v=v_1\ds
v_2\ds (S,x)$, and we set $\parent(v):=w_1\ds (T,x)$, where
$(T,x):=v_2\dar x$, and $w_1:=v_1\dar T$.
The nodes are “local views” when $n$ processes run a standard
protocol for $d$ rounds.
§.§ Adjacency of nodes and vertices of the flip graphs
The $n$-nodes of the $d$-th level and vertices of $\gn^d$ are
related by means of adjacency. We start by giving the general
Let $n$ and $d$ be arbitrary natural numbers. Assume we are given an
$n$-node $v=\nu_1\ds\dots\ds\nu_d$ of the $d$-th level and a vertex
$\sigma=\sigma_1\ds\dots\ds\sigma_d$ of $\gn^d$. We say that $v$ and
$\sigma$ are adjacent if $\nu_d=\sigma_d\dar\col(v)$ and
\begin{equation} \label{eq:adj}
\nu_i=\sigma_i\dar\carr(\nu_{i+1}),
\end{equation}
for all $i=1,\dots,d-1$.
It is again instructive to describe explicitly the cases $d=1$ and
$d=2$. When $d=1$, we have $v=(S,x)$, and $\sigma=A_1\s\dots\s
A_t$. Let $k$ be the index $1\leq k\leq t$, such that $x\in A_k$. Then
the vertex $\sigma$ and the node $(S,x)$ are adjacent if and
only if $S=A_1\cup\dots\cup A_k$.
On the other hand, when $d=2$, we have an $n$-node of the second level
$v=(\alpha,\beta)\ds\ab (S,x)$, and a vertex of $\gn^2$,
$\sigma=\sigma_1\ds\sigma_2$. Let $\sigma_2=B_1\s\dots\s B_q$, and
let $k$ be the index $1\leq k\leq q$, such that $x\in B_k$. We say
that the vertex $\sigma$ and the node $v$ are adjacent if the
following conditions are satisfied:
* $S=B_1\cup\dots B_k$;
* $(\alpha,\beta)=\sigma_1\dar S$.
It is easy to see that every vertex of $\gn^d$ is adjacent to exactly
$n$ nodes of $d$-th level. This is because, once a vertex of $\gn^d$
is fixed, the color of the node $v$ defines the node $v$ uniquely by
means of equations (<ref>).
The adjacency encodes correspondence between local views and global
executions. Namely, an $n$-node of the $d$-th level $v$ is adjacent to
a vertex $\sigma$ of $\gn^d$ if and only if the local view of
a process encoded by $v$ is a view contained in the execution of the
$d$-round protocol encoded by $\sigma$.
§.§ Node labelings
The main result of this paper is a construction of a function on
the set of the nodes satisfying certain constraints.
Assume we are given arbitrary natural numbers $n$ and $d$. A labeling
of the $n$-nodes of the $d$-th level, or simply a node labeling,
is a function $\lambda:\cn_n^d\ra\Lambda$, where $\Lambda$ is an
arbitrary set. A binary node labeling is a function
Unless explicitly stated otherwise, all our node labelings will be
binary, so we will frequently omit that word.
An $n$-node $v$ is called internal if $\carr(v)=[n]$. Any node
which is not internal is called a boundary node.
Note that in particular $\supp(v)\supseteq\carr(\nu_i)$, for all
$i=1,\dots,d$, so the carrier of $v$ is sort of a universe, containing
all the sets needed to define $v$.
It is easy to rephrase Definition <ref> in the special cases
$d=1$ and $d=2$. An $n$-node of the first level $(S,x)$ is internal if
and only if $S=[n]$, indeed its carrier is simply given by $S$. On the
other hand, an $n$-node of the second level $(A_1\s\dots\s A_t,
B_1\s\dots\s B_t) \ds (S,x)$ is internal if and only if
$A_1\cup\dots\cup A_t=[n]$.
A binary node labeling $\lambda:\cn_n^d\ra\{0,1\}$ is called blank
if $\lambda(v)=0$ whenever $v$ is an internal node.
We want to look at the blank binary node labelings which satisfy a
certain condition on the boundary.
The binary node labeling $\lambda:\cn_n^d\ra\{0,1\}$ is called compliant if the following property is satisfied. Assume we are
given two $n$-nodes of the $d$-th level, say $v$ and $w$, such that
$|\supp(v)|=|\supp(w)|$. Let $\varphi:\supp(v)\ra\supp(w)$ be the
unique order-preserving bijection, and assume furthermore that
$w=\varphi(v)$. Then we have $\lambda(v)=\lambda(w)$.
Note, that when $|\supp(v)|=n$, i.e., when $v$ is an internal node,
the condition in Definition <ref> is empty, since $\varphi$
must be the identity map. Thus being compliant is really a condition
on the boundary nodes in $\cn_n^d$.
Assume we are given a binary node labeling
$\lambda:\cn_n^d\ra\{0,1\}$. A vertex $\sigma\in V(\gn^d)$ is called
$0$-monochromatic if $\lambda(w)=0$ for any node $w\in\cn_n^d$
which is adjacent to $\sigma$. Analogously, a vertex $\sigma\in
V(\gn^d)$ is called $1$-monochromatic if $\lambda(w)=1$ for any
node $w\in\cn_n^d$ which is adjacent to $\sigma$.
The next definition describes the most important class of node labelings
in this paper.
A binary node labeling is called symmetry breaking if it is
compliant and does not have monochromatic vertices.
Any vertex of $\gn^d$ is adjacent to some internal node. In
particular, $\gn^d$ has no $1$-monochromatic vertices under a blank
binary labeling.
Assume $\sigma=\sigma_1\ds\dots\ds\sigma_d$ is a vertex of
$\gn^d$. Let us say $\sigma_d=A_1\s\dots\s A_t$. Take any $x\in A_t$,
and let $v$ be the unique $n$-node of $d$-th level whose id is $x$ and
which is adjacent to $\sigma$. It is easy to see, using
Definition <ref>, that $v=\sigma_1\ds\dots\ds\sigma_{d-1}\ds
([n],x)$. Clearly, this node is internal. Finally, this implies that
we have no $1$-monochromatic vertices, since a blank binary node
labeling evaluates to $0$ on any internal node.
Given a binary node labeling $\lambda:\cn_n^d\ra\{0,1\}$, let $\cm_\lambda$
denote the subgraph of $\gn^d$ induced by the $0$-monochromatic vertices.
Furthermore, for any natural number $q<d$, and whenever $\sigma$ is
a vertex of $\gn^q$, we let $\cm_\lambda(\sigma)$ denote the
intersection of $\gn^d(\sigma)$ with $\cm_\lambda$.
The graph $\cm_\lambda$ for the case when the only
nodes labeled with $1$ are $(12,1)$ and $(23,2)$.
When dealing with blank labelings we shall automatically have no
$1$-monochromatic vertices. Our next step will be to eliminate
$0$-monochromatic vertices as well, by looking for matchings on the
graph $M_\lambda$.
§ SETS OF PATTERNS
§.§ Definition and some specific sets of patterns
For an arbitrary natural number $n$, a set of patterns in $[n]$
is a union $\cb_1\cup\dots\cup\cb_{n-1}$, where for each $1\leq k\leq
n-1$, $\cb_k$ is some set of the $k$-nodes of the first level.
As an example we consider the set of patterns which has been
instrumental in our previous work, <cit.>, when we analyzed the
case $n=6$. Rephrasing the construction from <cit.> in the
language of this paper, yields the following set of patterns in $[6]$:
\[\begin{array} {lcl}
\cb_1&=&\{(\{1\},1)\}, \\
\cb_2&=&\{(\{1\},1),(\{1,2\},2)\}, \\
\cb_3&=&\{(\{1\},1),(\{1,2\},1),(\{1,2\},2),(\{1,2,3\},2),(\{1,2,3\},3)\}.
\end{array}\]
The case-by-case analysis which we did in <cit.> can be derived
from the general structure results which we prove in this paper.
For future reference we define a certain special set of patterns.
Assume $n$ is an arbitrary natural number and
$\fatx=(x_1,\dots,x_{n-1})$ is a vector, where $x_1,x_2\in\{0,1\}$,
$x_i\in\{-1,0,1\}$, for all $3\leq i\leq n-1$. The set of patterns
$\cb_\fatx=(\cb_1,\dots,\cb_{n-1})$ is now defined by the following
\[\cb_k:=
\begin{cases}
\cp^+_k&\text{ if }x_k=1;\\
\cp^-_k&\text{ if }x_k=-1;\\
0&\text{ otherwise;}
\end{cases}\]
for all $k=1,\dots,n-1$, where we set
\[\cp_k^+:=\{(\{1\},1),(\{1,2\},2),\dots,(\{1,\dots,k\},k)\},
\text{ for all } 1\leq k\leq n-1,\]
\[\cp_k^-:=\cp_k^+\cup\{(\{1,2\},1),(\{1,2,3\},2)\},
\text{ for all } 3\leq k\leq n-1.\]
We are not aware of any nice interpretation or intuition behind the
sets of patterns $\cp^+_k$ and $\cp^-_k$. For us, these are technical
constructions, which are needed to emulate the appearance of signs
in the solutions of the associated Diophantine equations.
We say that the set of patterns $\cb_\fatx$ is associated to the
vector $\fatx$.
The set of patterns $\cp^-_3$ viewed geometrically.
§.§ Node labeling associated to sets of patterns
Whenever $\cb$ is some set of patterns in $[n]$, we define a certain
binary node labeling $\lambda_\cb:\cn_n^2\ra\{0,1\}$, which we say is
associated to $\cb$. Pick $v\in\cn_n^2$, $v=(A_1\s\dots\s
A_t,B_1\s\dots\s B_t)\ds(S,x)$. One of the following 3 cases must
Case 1. The $n$-node $v$ is internal. In that case, we set
Case 2. The $n$-node $v$ is a boundary node, such that
$t\geq 2$. In that case, we set $\lambda_\cb(v):=1$.
Case 3. The $n$-node $v$ is a boundary node, such that
$t=1$, in other words, we can write $v=(A,S)\ds(S,x)$, where $A\neq
[n]$. Let $\varphi$ be the normalizer of $A$. We set
\[\lambda_\cb(v):=\begin{cases}
0,& \text{ if } (\varphi(S),\varphi(x))\in\cb; \\
1,& \text{ otherwise.}
\end{cases}\]
Definition <ref> provides us with a method of generating a large
family of blank and compliant binary node labelings as the next
proposition shows.
The binary node labeling associated to an arbitrary set of patterns
is blank and compliant.
Assume $\cb$ is some given set of patterns in $[n]$. The binary
node labeling $\lambda_\cb$ is set to be $0$ on the internal vertices
by definition, so it is blank. To see that it is also compliant, pick
two $n$-nodes of the second level, say $v$ and $w$, such that
$|\carr(v)|=|\carr(w)|$. Let $\varphi:\carr(v)\ra\carr(w)$ be the
unique order-preserving bijection, and assume that $w=\varphi(v)$; in
other words, if $v=(A_1\s\dots\s A_t,B_1\s\dots\s B_t)\ds(S,x)$, then
\ds(\varphi(S),\varphi(x))$.
If $v$ is an internal vertex, then $v=w$, so
$\lambda_\cb(v)=\lambda_\cb(w)=0$. Assume now $v$ is a boundary
vertex. If $t\geq 2$, then Case 2 of Definition <ref> applies
both to $v$ and to $w$, so we get $\lambda_\cb(v)=\lambda_\cb(w)=1$.
Assume finally $t=1$. Let $\psi$ be the normalizer of $A_1$, then
$\psi\circ\varphi^{-1}$ is a normalizer of $\varphi(A_1)$. Note, that
$(\gamma(\varphi(S)),\gamma(\varphi(x)))=(\psi(S),\psi(x))$. In
particular $\gamma(\varphi(S),\varphi(x))\in\cb$ if and only if
$\psi(S,x)\in\cb$, implying that $\lambda_\cb(v)=\lambda_\cb(w)$ in
this final case as well.
Let $\cb$ be an arbitrary set of $n$-nodes of the first level. We let
$\cp(\cb)$ denote the set of $[n]$-tuples $C_1\s\dots\s C_t$, such that
$(C_1\cup\dots\cup C_i,x)\in\cb$, for any $1\leq i\leq t$, and any
$x\in C_i$. We say that $\cp(\cb)$ consists of all $[n]$-tuples which
can be composed from $\cb$.
Note, that if an $[n]$-tuple $C_1\s\dots\s C_t$ can be composed then
then any of its truncations can be composed as well.
The notion of being composed has an interesting simplicial
interpretation. Recall, that the $n$-nodes of first level correspond
to vertices of $\chi(\da^{n-1})$, which is the standard chromatic
subdivision of an $(n-1)$-simplex. Given $\cb$ as in
Definition <ref>, we let $K_\cb$ denote the simplicial
complex induced by $\cb$, that is consisting of all simplices from
$\chi(\da^{n-1})$ whose vertices are in $\cb$. Call a simplex of
$\chi(\da^{n-1})$ essential if it is contained in a simplex of
$\da^{n-1}$ of the same dimension. Then $\cp(\cb)$ consists of
(indexes of) all essential simplices of $K_\cb$.
As an example, we have
\[\cp(\cp_3^-)=\{1,1\s 2,12,1\s 2\s 3,1\s 23,12\s 3\}.\]
Simplicially, these correspond to the $6$ essential simplices on
Figure <ref>: $1$ vertex, $2$ edges, and $3$ triangles.
Assume we are given $\cb=\cb_1\cup\dots\cup\cb_{n-1}$ - a set of
patterns in $[n]$, and $\sigma=A_1\s\dots\s A_t$ - a full
$[n]$-tuple. We shall now give an algorithm producing a set of
$V$-tuples, where $V=[n]\sm A_t$. This set of $V$-tuples will be
denoted $\Omega(\cb,\sigma)$.
Pick some $1\leq l\leq t-1$. Let $\varphi:A_1\cup\dots\cup A_l\ra[k]$
be the normalizer of $A_1\cup\dots\cup A_l$. We define a set of
$V$-tuples $\Omega_k$ by saying that a $V$-tuple $C_1\s\dots\s C_q$
belongs to $\Omega_k$ if and only if
$\varphi(C_1)\s\dots\s\varphi(C_q)$ belongs to $\cp(\cb_k)$ and
$C_1\cup\dots\cup C_q\subseteq A_l$. We now set
Let us note for future reference two important properties of
(1) The set $\Omega(\cb,\sigma)$ is closed under taking
(2) If $t\geq 3$, then $\Omega(\cb,\sigma)$ does not contain
any full $V$-tuples.
Assume $C_1\s\dots\s C_q\in \Omega(\cb,\sigma)$. Picking $l$ as in
Definition <ref> we see that $C_1\cup\dots\cup C_{q-1}\subseteq
A_l$, and $\varphi(C_1)\s\dots\s\varphi(C_{l-1})\in\cp(\cb_k)$; which
shows (1). Property (2) follows from the condition that whenever
$C_1\s\dots\s C_q$ belongs to $\Omega_k$, we have $C_1\cup\dots\cup
C_q\subseteq A_l$, for some $1\leq l\leq t-1$.
§.§ Flip graphs associated to sets of patterns
The next theorem allows us to understand the combinatorial structure
of the subgraphs $\cm_{\lambda(\cb)}(\sigma)$.
Assume $\cb=\cb_1\cup\dots\cup\cb_{n-1}$ is an arbitrary set of
patterns in $[n]$ and $\sigma=S_1\s\dots\s S_t$ is a full
$[n]$-tuple. We have an isomorphism
\begin{equation}\label{eq:pat}
\cm_{\lambda_\cb}(\sigma)\cong\gn(\Omega(\cb,\sigma),V(\sigma)),
\end{equation}
given by $(\sigma\ds\tau)\mapsto\tau$.
Let us take a vertex $\sigma\ds\tau$ of the graph
$\cm_{\lambda_\cb}(\sigma)$ and show that $\tau$ is a vertex of
$\gn(\Omega(\cb,\sigma),V(\sigma))$. Assume $\tau=T_1\s\dots\s T_q$,
and choose $1\leq k\leq q$ to be the minimal index such that
$T_{k+1}\cap S_t\neq\emptyset$. We need to show that $T_1\s\dots\s
Pick an arbitrary $1\leq i\leq k$, and $x\in T_i$. Set
$T:=T_1\cup\dots\cup T_i$. The node of the second level $w=(\sigma\dar
T)\ds(T,x)$ is obviously adjacent to the vertex $\sigma\ds\tau$, so
$\lambda_\cb(w)=0$. It cannot be internal since $T\cap S_t=\emptyset$,
so we assume it is a boundary node. Since $\lambda_\cb(w)=0$, and $w$
is a boundary node, we must have $w=(S,T)\ds(T,x)$, where $S\supseteq
T\ni x$. This means that there exists an index $1\leq d\leq t$, such
that $S_d\supseteq T$. Since this is true for any $i$, we get
$S_d\supseteq T_1\cup\dots\cup T_k$. Furthermore, we have
$S=S_1\cup\dots\cup S_d$, and we set $m:=|S|$. Let $\varphi:S\ra[m]$
be the normalizer of $S$. Since $\lambda_\cb(w)=0$, we must have
$\varphi(T,x)\in\cb_m$, for all $x$, which means precisely that
$\varphi(T_1)\s\dots\s\varphi(T_k)\in \cp(\cb_k)$. By
Definition <ref> we conclude that $T_1\s\dots\s T_k
\in\Omega(\cb,\sigma)$.
This argument can easily be reversed to show that for any vertex
$\tau$ in $\gn(\Omega(\cb,\sigma),V(\sigma))$, the vertex
$\sigma\ds\tau$ belongs to $\cm_{\lambda_\cb}(\sigma)$. Indeed, pick
$\tau=T_1\s\dots\s T_q\in\gn(\Omega(\cb,\sigma),V(\sigma))$, and let
$k$ be the minimal index such that $T_{k+1}\cap S_t\neq\emptyset$. We
need to show that all the nodes adjacent to $(\sigma\ds\tau)$ have
a label $0$. This is clearly the cases for internal nodes, so let us
consider a boundary node. This means we need to pick some $1\leq i\leq
k$, and some $x\in T_i$. The corresponding node is $w=(\sigma\dar
T)\ds(T,x)$, where $T=T_1\cup\dots\cup T_i$. Since $T_1\s\dots\s T_k
\in\Omega(\cb,\sigma)$, there exists $d$ such that $T\subseteq S_d$.
Setting $S:=S_1\cup\dots\cup S_d$, we get $w=(S,T)\ds(T,x)$. If
$\varphi$ is a normalizer of $S$, then
$(\varphi(T),\varphi(x))\in\cb$, and we conclude that the node $w$ has
label $0$.
Finally, we get a graph isomorphism, since graphs on both sides
of (<ref>) are induced by their respective sets of vertices.
§ SETS OF DISJOINT PATHS IN $\GN$
A well-ordered pair of sets is a pair of sets $(S,T)$, such that
$\emptyset\neq S\subset T\subset [n]$, together with some fixed order
on the set $T$, under which all the elements of $S$ come before all
the other elements of $T$. Two well-ordered pairs of sets $(S,T)$ and
$(S',T')$ are called nested if either $S\subset S'\subset
T'\subset T$ or $S'\subset S\subset T\subset T'$.
For an arbitrary $S\subseteq[n]$, we let $\fatb_S$ denote the vertex
of $\gn$ indexed by $S\s [n]\sm S$.
Given a well-ordered pair of sets $(S,T)$, the edge path in $\gn$
which is shown on Figure <ref> connects the vertices
$\fatb_S$ and $\fatb_T$. We denote this path by $p_{S,T}$ and call it
the standard path associated to the well-ordered pair $(S,T)$.
We shall say that two well-ordered pairs of sets $(S,T)$ and $(S',T')$
are disjoint if $\{S,T\}\cap\{S',T'\}=\emptyset$. Clearly, two
well-ordered pairs of sets are disjoint if and only if the
corresponding paths $p_{S,T}$ and $p_{S',T'}$ have no endpoints in
common. The following theorem shows that a much stronger statement is
Assume $(S,T)$ and $(S',T')$ are disjoint well-ordered pairs of sets,
which are not nested, then the corresponding standard paths $p_{S,T}$
and $p_{S',T'}$ are disjoint.
The informal idea of the proof is that we want to see that one of
the endpoints of the standard path is detectable from any vertex on
the path. Hence, roughly speaking, if two standard paths have a vertex
in common, then they would have to have an endpoint in common.
To start with, we define an operation $ds(-)$. Given a full
$[n]$-tuple $\sigma=A_1\s\dots\s A_t$, pick the indices $1\leq
i_1<\dots<i_k\leq t$, such that $|A_j|\geq 2$ if and only if
$j\in\{i_1,\dots,i_k\}$. We now set
\[\dd(\sigma):=A_1\cup\dots\cup A_{i_1}\s A_{i_1+1}\cup\dots\cup A_{i_2}
\s\dots\s A_{i_{k-1}+1}\cup\dots\cup A_{i_k}\s A_{i_k+1}\cup\dots\cup
In the degenerate case $|A_1|=\dots=|A_t|=1$, we set
Clearly, $\dd(\sigma)$ is again a full $[n]$-tuple, which either
does not have any singletons, or has exactly one singleton as the last
set. In general, $\sigma=\dd(\sigma)$ if and only if $\sigma$ either
does not have any singletons, or its last set is the only singleton.
Let us now consider a well-ordered pair of sets $(S,T)$. As the first
case we assume that $|S|\geq 2$ and $|T|\geq |S|+2$. We now apply
$\dd(-)$ to the vertices of the standard path $p_{S,T}$. The obtained
full $[n]$-tuples are: $(S\s [n]\sm S)$, $(T\s [n]\sm T)$, and $(S\s
T\sm S\cup [n]\sm T)$. In all cases, the first set in that full
$[n]$-tuple is indexing one of the endpoints of $p_{S,T}$, thus if
$p_{S,T}$ and $p_{S',T'}$ have a vertex in common, then they also have
one of the endpoints in common.
In the remaining cases we still get the same possible patterns for
$\dd(\sigma)$ with one additional pattern: $\dd(\sigma)=[n]$. We get
this pattern in two cases:
* when $s=1$ and $\sigma=x_1\s\dots\s x_k\s
x_{k+1},\dots,x_t,y_1,\dots,y_{n-t}$ with some $1\leq k\leq t$;
* when $t=s+1$ and $\sigma=x_1\s\dots\s x_s\s x_{s+1}\s
In other words, given $\sigma$ from $p_{S,T}$ we can always determine
either $S$ or $T$, except for one case. In this case, we have
$\sigma=a_1\s\dots\s a_k\s b_1,\dots,b_{n-k}$. There are two
possibilities for the well-ordered pair of sets $(S,T)$. Either $S=\{a_1\}$
and $\{a_1\,\dots,a_k\}\subseteq T$, or $S=\{a_1,\dots,a_{k-1}\}$ and
Assume now that the paths $p_{S,T}$ and $p_{S',T'}$ do intersect.
Since $(S,T)$ and $(S',T')$ are disjoint, the paths must intersect at
an internal point. By what is said above we can assume without loss of
generality that $S=\{a_1\}$, $\{a_1,\dots,a_k\}\subseteq T$,
$S'=\{a_1,\dots,a_{k-1}\}$, and $T'=\{a_1,\dots,a_k\}$. However, this
means that the pairs $(S,T)$ and $(S',T')$ are nested, contradicting
our assumptions.
§ FROM COMPARABLE MATCHINGS TO SYMMETRY BREAKING LABELINGS
§.§ Perfect matchings induce symmetry breaking labelings
(Theorem A). $\,$
Let $n$ be an arbitrary natural number, and let
$\lambda:\cn_n^2\ra\{0,1\}$ be a blank and compliant binary labeling
on the $n$-nodes of second level. Assume that there exists a perfect
matching on the graph $\cm_\lambda$, then there exists a symmetry
breaking labeling on the $n$-nodes of third level.
Let $\mu$ denote the perfect matching on the graph
$\cm_\lambda$. We now proceed to give a rule defining a binary node
labeling $\rho:\cn_n^3\ra\{0,1\}$ on the $n$-nodes of the third level.
Take $v\in\cn_n^3$, $v=v_1\ds v_2\ds (S,x)$. To start with we set
\[\rho^\defa(v):=\lambda(\parent(v)),\]
and call this a default value of $\rho$. The rule for defining
the value of $\rho$ distinguishes 3 cases.
Case 1. Assume $|S|\leq n-2$. In this case, we set
Case 2. Assume $|S|=n-1$. In this case, there exists
$y\in [n]$, such that $S=[n]\sm y$. Since $\carr(v_2)\supseteq S$, we
have $|\carr(v_2)|\geq n-1$. If $|\carr(v_2)|=n-1$, then we set
$\rho(v):=\rho^\defa(v)$. Else, we must have $\carr(v_2)=[n]$. Since
$\col(v_2)=S$, we see that $v_2$ is an edge in $\gn$. At the same time
$v_1$ is a full $[n]$-tuple, so $v_1\ds v_2$ is an edge in $\gn^2$. If
the vertices of $\gn^2$ connected by this edge are matched under
$\mu$, then we set $\rho(v):=\rho^\defa(v)$, else we set $\rho(v):=1$.
Case 3. Assume $|S|=n$. In other words, we have
$S=[n]$. In this case $\alpha=v_1\ds v_2$ is a vertex of $\gn^2$. If
this vertex is not monochromatic with respect to $\lambda$, then we
set $\rho(v):=\rho^\defa(v)$. If $\alpha$ is monochromatic, then we
know that it has been matched, since we assumed that the matching
$\mu$ is perfect. In particular, the label $\id_\mu(\alpha)$ is
well-defined. We now complete our definition of $\rho$ by setting
\[\rho(v):=\begin{cases}
0, &\text{ if }x=\id_\mu(\alpha);\\
1, &\text{ if }x\neq\id_\mu(\alpha).
\end{cases}\]
The value $\rho(v)$ has now been defined for all $v\in\cn_n^3$, and we
would like to summarize by saying that $\rho(v)$ may be different from
the default value $\rho^\defa(v)$ only in the following two cases:
* if $|S|=n-1$ and $v_1\ds v_2$ is a matching edge;
* if $S=[n]$, $v_1\ds v_2$ is a monochromatic vertex, and
$x\neq\id_\mu(v_1\ds v_2)$.
To see that the node labeling $\rho$ is symmetry breaking, we need to
verify that it is compliant, and that it does not have any
monochromatic vertices. We start with proving that $\rho$ is
compliant. Assume, we have two boundary nodes $v,w\in\cn_n^3$,
$v=v_1\ds v_2 \ds (S,x)$, $w=w_1\ds w_2\ds (T,y)$, such that
$|\supp(v)|=|\supp(w)|\leq n-1$. By definition of the carrier this
means that $|\carr(v)|=|\carr(w)|$. Let $\varphi:\supp(v)\ra\supp(w)$
be the unique order-preserving bijection. Assume furthermore that
$\varphi(v)=w$. Specifically, this means that $w_1=\varphi(v_1)$,
$w_2=\varphi(v_2)$, $T=\varphi(S)$, and $y=\varphi(x)$.
We now show that under these conditions, we have
$\rho(v)=\rho^\defa(v)$ and $\rho(w)=\rho^\defa(w)$. First, since
$T=\varphi(S)$, we have $|S|=|T|$. Second, we have
$S\subseteq\supp(v)$, so $|S|\leq |\supp(v)|\leq n-1$. If $|S|=|T|\leq
n-2$, then $\rho(v)=\rho^\defa(v)$ and $\rho(w)=\rho^\defa(w)$ by the
Case 1 of our rule. If, on the other hand, $|S|=|T|=n-1$, then
$|\supp(v)|=|\supp(w)|=n-1$, and this time $\rho(v)=\rho^\defa(v)$ and
$\rho(w)=\rho^\defa(w)$ by the Case 2 of our rule for defining $\rho$.
Next, let us show that $\varphi(\parent(v))=\parent(w)$. By the
calculation after Definition <ref>, we have
$\parent(v)=\gamma_1\ds (\wti S,x)$, where $(\wti S,x)=v_2\dar x$, and
$\gamma_1=v_1\dar\wti S$. Clearly, the operation $\dar$ commutes with
$\varphi$, so we have
\[\varphi(\wti S,x)=\varphi(v_2\dar x)=\varphi(v_2)\dar\varphi(x)=w_2\dar y\]
\[\varphi(\gamma_1)=\varphi(v_1\dar\wti S)=\varphi(v_1)\dar\varphi(\wti S)=
w_1\dar (w_2\dar y),\]
\[\varphi(\gamma_1\ds(\wti S,x))=\varphi(\gamma_1)\ds(\varphi(\wti S),\varphi(x)))
Let us now verify that $\rho$ has no monochromatic vertices. Let
$\sigma=\sigma_1\ds\sigma_2\ds\sigma_3$ be an arbitrary vertex of $\gn^3$.
We consider two cases.
Case 1. Assume the vertex $\sigma_1\ds\sigma_2$ is not
monochromatic. We show that $\rho(v)=\rho^\defa(v)$ whenever $v$
is a node of the third level adjacent to the vertex $\sigma_1 \ds
\sigma_2\ds\sigma_3$. Assume $\sigma_3=A_1\s\dots\s A_t$. If $x\in
A_k$, such that $|A_1\cup\dots\cup A_k|\leq n-2$, then $\rho(v)=
\rho^\defa(v)$ by definition. If, on the other hand, $x\in A_t$, then
$\sigma\dar x=\sigma_1\ds\sigma_2\ds([n],x)$. Since the vertex
$\sigma_1 \ds\sigma_2$ is not monochromatic, we again get
$\rho(v)=\rho^\defa(v)$. The last remaining case is when $x\in A_k$,
such that $|A_1\cup\dots\cup A_k|=n-1$. This is only possible if
$\sigma_3=A_1\s\dots\s A_{t-1}\s y$, and $x\in A_{t-1}$. If that
happens we have $\sigma\dar x=\tau_1\ds\tau_2\ds ([n]\sm y,x)$. If now
$\tau_1\ds\tau_2$ is not an edge, we must have $\rho(v)=
\rho^\defa(v)$. Finally, if $\tau_1\ds\tau_2$ is an edge, then
$\sigma_1\ds\sigma_2$ is one of its endpoints. However, the vertex
$\sigma_1\ds\sigma_2$ is not monochromatic by our assumption, and so
$\tau_1\ds\tau_2$ cannot be a matching edge; hence again we get
$\rho(v)= \rho^\defa(v)$.
We have now proved that $\rho(v)=\rho^\defa(v)$ whenever $v$ is a node
of the third level adjacent to the vertex $\sigma_1\ds\sigma_2 \ds
\sigma_3$. On the other hand, the set of parents of these nodes is
precisely the set of nodes of the second level adjacent to $\sigma_1
\ds \sigma_2$, because $\parent(\sigma\dar x)= (\sigma_1\ds \sigma_2)
\dar x$. Since we assumed that the vertex $\sigma_1\ds \sigma_2$ is
not monochromatic, we conclude that neither is the vertex $\sigma$.
Case 2. Assume the vertex $\sigma_1\ds\sigma_2$ is
monochromatic. Since $\mu$ is a perfect matching, there exists
an edge in $\gn^2$ matching $\sigma_1\ds\sigma_2$ to some other
monochromatic vertex. Set $c$ to be the label of that edge, and assume
$\sigma_3=A_1\s\dots\s A_t$. We now distinguish three further
Case 2a. Assume $|A_t|\geq 2$. If $x\in A_t$, then set
$v_r:=\sigma\dar x=\sigma_1\ds\sigma_2\ds ([n],x)$. This is the node
with color $x$ adjacent to $\sigma$. By our rule, if $x\neq c$, then
$\rho(v_r)=1$. Since $|A_t|\geq 2$, there exists $x\in A_t$ such that
$x\neq c$, so at least one of the nodes adjacent to $\sigma$ has
label $1$. On the other hand, let $v_c$ be the node of color $c$
adjacent to $\sigma$. If $c\in A_t$, then this node has label $0$.
Otherwise, we have $c\in A_k$, for some $1\leq k<t$. In this case, we
have $v_c:=\sigma\dar c=\ti\sigma_1\ds\ti\sigma_2\ds (A_1\cup\dots\cup
A_k,c)$. Since $|A_1\cup\dots\cup A_k|\leq n-2$, we get $\rho(v_c)=0$
again. Either way, we have nodes with different labels adjacent to
$\sigma$, so $\sigma$ is not monochromatic.
Case 2b. Assume $|A_t|=1$, $A_t=\{y\}$, with $y\neq c$. As
above we calculate $\rho(v_y)=1$. Take now $x\in A_{t-1}$. If
$\rho(x)=\rho^\defa(x)$, then $\rho(x)=0$, since the vertex
$\sigma_1\ds\sigma_2$ is $0$-monochromatic. Otherwise, we have
$\sigma\dar x=\ti\sigma_1\ds\ti\sigma_2\ds([n]\sm y,x)$, and
$\ti\sigma_1\ds\ti\sigma_2$ is a matching edge. This is impossible,
since that edge would be labeled $y$, and have $\sigma_1\ds\sigma_2$
as one of its endpoints, contradicting the assumption $y\neq c$.
In either case we have two nodes adjacent to $\sigma$ with different
values of $\rho$, so $\sigma$ is not monochromatic.
Case 2c. Assume $|A_t|=1$, $A_t=\{c\}$. By the calculation
in Case 2a, we get $\rho(v_c)=0$. Take any $x\in A_{t-1}$, so
$v_x=\sigma\dar x=\ti\sigma_1\ds\ti\sigma_2\ds([n]\sm c,x)$. Now
$\ti\sigma_1\ds\ti\sigma_2$ is an edge labeled $c$ which has
$\sigma_1\ds\sigma_2$ as an endpoint. By assumptions above this means
that $\ti\sigma_1\ds\ti\sigma_2$ is a matching edge, and so we get
$\rho(v_x)=1$. Again, we have different values of $\rho$ on the nodes
adjacent to $\sigma$, so $\sigma$ is not monochromatic.
This finishes the proof of the theorem.
Let $n$ be an arbitrary natural number, and let $\cb$ be an arbitrary
set of patterns in $[n]$. If there exists a perfect matching on the
graph $\cm_{\lambda_\cb}$, then there exists a symmetry breaking
labeling on the $n$-nodes of third level.
The binary node labeling $\lambda_\cb$ associated to the set of
patterns $\cb$ is always blank and compliant, see
Proposition <ref>, hence the statement follows from
Theorem <ref>.
§.§ Non-intersecting path systems induce perfect matchings
Let $n$ be an arbitrary natural number, $n\geq 2$. The linear
Diophantine equation in $n-1$ variables
\begin{equation}
\end{equation}
is called binomial Diophantine equation associated to $n$.
A solution $(x_1,\dots,x_{n-1})$ to the binomial Diophantine equation
associated to $n$ is called primitive if $x_1=1$,
$x_2\in\{0,1\}$, and $x_i\in\{-1,0,1\}$, for all $i=3,\dots,n-1$. For
example $(1,1,-1,0,0)$ is a primitive solution for $n=6$, and
$(1,0,-1,1,0,0,0,0,-1,-1,0)$ is a primitive solution for $n=12$.
We shall now consider families of proper subsets of the set $[n]$,
i.e., $\Sigma\subseteq 2^{[n]}\sm\{\emptyset,[n]\}$, which we call
proper families. Let $C^n_t$ be the family of all subsets of
$[n]$ of cardinality $t$. It is a proper family if $1\leq t\leq n-1$.
A proper family $\Sigma\subseteq 2^{[n]}$ is called cardinal if
the following is satisfied: whenever $S\in\Sigma$, we have
$C^n_{|S|}\subseteq\Sigma$. In other words, if $\Sigma$ contains one
set with $k$ elements, then it contains all sets with $k$ elements,
which are subsets of $[n]$.
A cardinal family $\Sigma$ can be described simply by specifying the
list of cardinalities $C(\Sigma)\subseteq\{1,\dots,n-1\}$ of the sets
in $\Sigma$, namely $\Sigma=\cup_{t\in C(\Sigma)}C^n_t$. Two proper
families $\Sigma$ and $\Lambda$ are disjoint if and only if they
are disjoint as sets. Clearly, cardinal families $\Sigma$ and
$\Lambda$ are disjoint if and only if the corresponding cardinality
sets $C(\Sigma)$ and $C(\Lambda)$ are disjoint.
Assume now that we are given $n\geq 2$, and that
$\fatx=(x_1,\dots,x_{n-1})$ is a primitive solution to the binomial
Diophantine equation associated to $n$. Set
$I_\fatx:=\{i\,|\,i\in[n],\,x_i=1\}$ and
$J_\fatx:=\{j\,|\,j\in[n],\,x_j=-1\}$, in particular $1\in
I_\fatx$. Furthermore, set $\Sigma_\fatx:=\{S\,|\,S\subset[n],\,|S|\in
I_\fatx\}$ and $\Lambda_\fatx:=\{T\,|\,T\subset[n],\,|T|\in
J_\fatx\}$. Clearly, $\Sigma_\fatx$ and $\Lambda_\fatx$ are proper
families of subsets. They are disjoint because $I_\fatx$ and $J_\fatx$
are disjoint. Finally, since $\fatx$ is a primitive solution to the
binomial Diophantine equation associated to $n$ we have
$|\Sigma_\fatx|=|\Lambda_\fatx|+1$. We say that the proper families of
subsets $\Sigma_\fatx$ and $\Lambda_\fatx$ are associated
to $\fatx$.
Assume we are given two proper set families $\Sigma$ and $\Lambda$,
such that $|\Sigma|=|\Lambda|$. A non-intersecting path system
between $\Sigma$ and $\Lambda$ consists of a bijection
$\varphi:\Sigma\ra\Lambda$ together with a set of disjoint edge paths
$\{q_{S,\varphi(S)}\}_{S\in\Sigma}$, such that each path
$q_{S,\varphi(S)}$ connects $\fatb_S$ with $\fatb_{\varphi(S)}$.
(Theorem B).
Assume $\fatx=(x_1,\dots,x_{n-1})$ is a primitive solution to the
binomial Diophantine equation associated to some $n\geq 2$, and
$\Sigma_\fatx$ and $\Lambda_\fatx$ are the associated proper families
of subsets of $[n]$. If there exists a non-intersecting path system
between $\Sigma_\fatx\sm\{n\}$ and $\Lambda_\fatx$, then there exists
a compliant symmetry breaking binary labeling on the $n$-nodes of the
third level.
Set as above $I_\fatx:=C(\Sigma_\fatx)$ and
$J_\fatx:=C(\Lambda_\fatx)$. We have a bijection
$\varphi:\Sigma_\fatx\sm\{n\}\ra\Lambda_\fatx$, and a family of
disjoint edge paths in $\gn$,
$\{q_{S,\varphi(S)}\}_{S\in\Sigma_\fatx\sm\{n\}}$, such that each path
$q_{S,\varphi(S)}$ connects $\fatb_S$ with $\fatb_{\varphi(S)}$. Let
$\cb_\fatx=(\cb_1,\dots,\cb_{n-1})$ be the set of patterns associated
to vector $\fatx$, and consider the associated node labeling
We have a bipartite graph $\cm_\lambda$, and we are looking for
a perfect matching on this graph. This graph consists of subgraphs
$\cm_\lambda(\sigma)$ where $\sigma=A_1\s\dots\s A_t$ ranges through
vertices of $\gn$, We can start by taking some matchings on these
subgraphs and then eliminating the remaining critical vertices.
Theorem <ref> describes $\cm_\lambda(\sigma)$ combinatorially,
for each $\sigma$. All these graphs are isomorphic to
$\gn(\Omega(\cb_\fatx,\sigma),V)$. By Remark <ref>, the set
$\Omega(\cb_\fatx,\sigma)$ does not contain any full $V$-tuples
whenever $t\geq 3$, so in these cases $\cm_\lambda(\sigma)$ will have
a perfect matching: we can take the standard matching with respect to
any order on $[n]\sm V$ and then apply
Theorem <ref>(3).
When $t=2$, we have $\sigma=S\s[n]\sm S=\fatb_S$, say
$S=\{x_1,\dots,x_k\}$, for $x_1<\dots<x_k$. In this case the standard
matchings on $\cm_\lambda(\sigma)$ are not perfect. They have
critical vertices which are in a bijective correspondence with full
$V$-prefixes. Going back to the definition of the set of patterns
associated to $\fatx$, we distinguish 3 cases.
Case 1. If $|S|\in I_\fatx$, then the set
$\Omega(\cb_\fatx,\sigma)$ has a unique full $V$-prefix, namely
$x_1\s\dots\s x_k$. Thus for any order on $[n]\sm V$ the standard
matching will have a unique critical vertex.
Case 2. If $|S|\in J_\fatx$, then the set
$\Omega(\cb_\fatx,\sigma)$ has 3 full $V$-prefixes, namely
$x_1\s\dots\s x_k$, $x_1,x_2\s x_3\s\dots\s x_k$, and $x_1\s
x_2,x_3\s \dots\s x_k$. Note, that in this case we must have $k\geq
3$. Thus for any order on $[n]\sm V$ the standard matching will have
the corresponding 3 critical vertices.
Case 3. If $|S|\notin I_\fatx\cup J_\fatx$, then
$\Omega(\cb_\fatx,\sigma)$ is empty, and it follows from
Theorem <ref>(2) that the standard matching is
Finally, when $t=1$, we have $\sigma=[n]$. In this case
Theorem <ref>(1) applies and we have a unique critical
vertex which depends on the chosen order.
We now use conductivity in flip graphs, as developed in
Section <ref>, to find edge paths in $\gn^2$ connecting all the
critical vertices. Let us fix $S\in\Sigma_\fatx\sm\{n\}$, and take the
corresponding path $q_{S,\varphi(S)}$. We shall be traversing that
path starting from $\fatb_{\varphi(S)}$ and going towards $\fatb_S$,
so we let $w_1:=\fatb_{\varphi(S)},w_2,\dots,w_{d-1},w_d:= \fatb_S$
denote the vertices on the path listed in that order. Note that $d$
must be odd. For $k=1,\dots,d-1$, let $y_k$ be the label of the edge
between $w_k$ and $w_{k+1}$. By Lemma <ref> one can choose the
order $R$, so that the standard matching can be extended to match two
of the critical vertices to each other, and there will exist
a semi-augmenting edge path connecting the remaining critical vertex
to some $y_1$-connector of the second type $\tau_1^f$, which is proper
with respect to $\fatb(\varphi(S))$.
Let $\tau_2^s$ be the unique vertex connected to $\tau_1^f$ by the
edge with label $y$. Clearly, $\tau_2^s$ is a $y_1$-connector of the
second type, which by the identity (<ref>) is proper with
respect to $w_2$. By Lemma <ref>, there exists an order $R$ and
a non-augmenting edge path in the graph $\cm_\lambda(w_2)$ with
respect to $\mu_R$, connecting $\tau_2^s$ to some $y_2$-connector of
the first type $\tau_2^f$, which is proper with respect to $w_2$. We
now let $\tau_3^s$ be the unique vertex connected to $\tau_2^f$ by the
edge with the label $y_2$, which is a $y_2$-connector of the first
type proper with respect to $w_3$. We then repeat that argument for
the graph $\cm_\lambda(w_3)$, using Lemma <ref> instead.
Eventually, we will arrive at a vertex $\tau_d^s$ in
$\cm_\lambda(\fatb_S)$. Since $d$ is odd, $\tau_d^s$ is
a $y_{d-1}$-connector of the first type, and it is proper with respect
to $\fatb_S$. We now employ Lemma <ref>, which tells us that
there exists an order $R$ and a semi-augmenting path with respect to
$\mu_R$ which connects $\tau_d^s$ with the unique critical vertex in
$\cm_\lambda(\fatb_S)$. Concatenating all these paths will yield an
augmenting path which connects the two critical vertices in
$\cm_\lambda(\fatb_{\varphi(S)})$ and $\cm_\lambda(\fatb_S)$. Applying
the transformation from Definition <ref> to that path will
yield a new matching, where these two critical vertices are now
matched. Doing this for all paths $q_{S,\varphi(S)}$, when $S$ ranges
over all subsets in $\Sigma_\fatx\sm\{n\}$ will yield a matching on
$\cm_\lambda$, with two of the critical vertices remaining: one in
$\cm_\lambda(n\s[n-1])$, and one in $\cm_\lambda([n])$.
Note, that by Theorem <ref>, we have
$\cm_\lambda([n])\cong\gn$ and $\cm_\lambda(n\s [n-1]) \cong
\gn(\Omega,n)$, where $\Omega=\{n\}$. To start with, consider the
standard matching in $\cm_\lambda([n])$ with respect to the order
$(1,\dots,n)$, by Theorem <ref>, we have a unique
critical vertex $v$ indexed by $[n]\ds 1\s 2\s\dots\s n$. Let $w$ be
the vertex of $\cm_\lambda(n\s [n-1])$ indexed by $n\s[n-1]\ds 1\s
2\s\dots\s n$. Clearly, the vertices $v$ and $w$ are connected in
$\cm_\lambda$ by an edge labeled $n$. By Lemma <ref>, there
exists an order on $[n-1]$, such that there exists a semi-augmenting
path connecting the unique critical vertex with $w$. In fact the order
$(1,\dots,n-1)$ will do, in which case the unique critical vertex will
be $n\s[n-1]\ds n\s 1\s\dots\s n-1$, and the semi-augmenting path can
be given explicitly: $n\s 1\s\dots\s n-1\ra 1,n\s 2\s\dots\s n-1\ra
1\s n\s 2\s\dots\s n-1\ra\dots\ra 1\s 2\s\dots\s n-1,n\ra
1\s 2\s\dots\s n$. Concatenating this path with the edge between $v$
and $w$ yields an augmenting path which eliminates the last two
critical vertices, resulting in a perfect matching on $\cm_\lambda$.
§.§ Comparable matchings induce non-intersection path systems
Producing a non-intersecting path system for $n=6$, and
$\fatx=(1,1,-1,0,0)$ has been done by hand in <cit.>.
Unfortunately, doing it directly appears prohibitive for larger values
of $n$. We now look for further structures which will help us
construct non-intersecting path systems.
A comparable matching between disjoint proper families $\Sigma$
and $\Lambda$ is a bijection $\varphi:\Sigma\ra\Lambda$, such that for
any $S\in\Sigma$, either $(S,\varphi(S))$ or $(\varphi(S),S)$ is
a well-ordered pair. We say that this well-ordered pair is associated to $S$.
The comparable matching $\varphi$ is called non-nested if for
any $S,T\in\Sigma$ the associated well-ordered pairs are not nested.
Given disjoint proper families $\Sigma$ and $\Lambda$, the set of
comparable matchings $\varphi:\Sigma\ra\Lambda$ can be partially
ordered as follows. Assume we have two subsets $S,T\subseteq[n]$, such
that one contains the other one. If $S\subset T$, then we set
$l(S,T):=|T\sm S|$, else set $l(S,T):=|S\sm T|$; this is a distance
between $S$ and $T$. Let $L_\varphi$ denote the multiset of distances
$\{l(S,\varphi(S))\,|\,S\in\Sigma\}$. We define an associated function
$\dist_\varphi$ on the set of natural numbers, by setting
$\dist_\varphi(d)$ to be the number of occurrences of $d$ in
$L_\varphi$. Since $L_\varphi$ is a finite multiset, the value
$\dist_\varphi(d)$ is different from $0$ for only finitely many values
of $d$.
Assume now we are given two comparable matchings
$\varphi,\psi:\Sigma\ra\Lambda$. If the functions $\dist_\varphi$ and
$\dist_\psi$ are identical, we say that $\varphi$ and $\psi$ are
incomparable. Otherwise, let $k$ be the maximal index such that
$\dist_\varphi(d)\neq\dist_\psi(d)$. We now say that
$\varphi\prec\psi$ if $\dist_\varphi(d)<\dist_\psi(d)$, and we say
that $\varphi\succ\psi$ if $\dist_\varphi(d)>\dist_\psi(d)$. Clearly,
this is a well-defined partial order on the set of all comparable
matchings between $\Sigma$ and $\Lambda$, which we call distance-lexicographic order.
Assume we have proper families $\Sigma$ and $\Lambda$, and
a comparable matching $\varphi:\Sigma\ra\Lambda$, then there exists
a non-nested comparable matching $\psi:\Sigma\ra\Lambda$.
Without loss of generality, we can assume that $\varphi$ is chosen
to be a comparable matching which is minimal with respect to the
distance-lexicographic order defined above. If $\varphi$ is
non-nested, then we are done, so assume this is not the case and take
any pair $S,T\in\Sigma$ such that the associated well-ordered pairs
are nested. Without loss of generality, swapping $\Sigma$ and
$\Lambda$ if necessary, we can assume that $S\subset\varphi(S)$,
$S\subset T$, and $S\subset\varphi(T)$. We then have two cases, either
we have $S\subset T\subset \varphi(T)\subset \varphi(S)$, or
$S\subset\varphi(T)\subset T \subset\varphi(S)$.
Define a new bijection $\psi:\Sigma\ra\Lambda$ as follows:
$\psi(A):=\varphi(A)$, for $A\neq S,T$, $\psi(S):=\varphi(T)$, and
$\psi(T):=\varphi(S)$. Clearly, $\psi$ is again a comparable matching,
which precedes $\varphi$ in the distance-lexicographic order. This
contradicts the choice of $\varphi$.
(Theorem C).
Assume $\fatx=(x_1,\dots,x_{n-1})$ is a primitive solution to the
binomial Diophantine equation associated to some $n\geq 2$, and
$\Sigma_\fatx$ and $\Lambda_\fatx$ are the associated proper families
of subsets of $[n]$. If there exists a comparable matching between
$\Sigma_\fatx\sm\{n\}$ and $\Lambda_\fatx$, then there exists
a compliant symmetry breaking binary labeling on the $n$-nodes of the
third level.
Consider a comparable matching
$\varphi:\Sigma_\fatx\sm\{n\}\ra\Lambda_\fatx$. By
Proposition <ref> we might as well assume that $\varphi$ is
non-nested. By Theorem <ref> the family
$\{p_{S,\varphi(S)}\}_{S\in\Sigma_\fatx\sm\{n\}}$ is
a non-intersecting path system, so the result follows from
Theorem <ref>.
Theorem <ref> means that in the standard computational model
the existence of a comparable matching between disjoint cardinal
proper families of subsets of $[n]$ implies the existence of
a wait-free protocol solving Weak Symmetry Breaking in $3$ rounds.
§ NEW UPPER BOUNDS FOR $\MSB(N)$
§.§ The formulation of the main theorem and some set theory notations
Our goal now is to use Theorem <ref> to improve upper
bounds for the symmetry breaking function $\msb(n)$. Our most definite
result is to show that there are infinitely many values of $n$ for
which $\msb(n)\leq 3$.
We now return to considering Theorem <ref>. The case $t=1$ has
been previously settled in <cit.>. To deal with the case $t=2$,
we need to start with an appropriate Diophantine equation. It just so happens
that we have the arithmetic identity:
\begin{equation}
\label{eq:507}
\binom{12}{1}+\binom{12}{4}=\binom{12}{0}+
\binom{12}{3}+\binom{12}{9}+\binom{12}{10}.
\end{equation}
Indeed, both sides of (<ref>) are equal to $507$. That
particular number has only technical significance - it shows the
number of augmenting paths which we will need to fix in the initial
standard matching, in order to arrive at a perfect matching. Once we
have the identity (<ref>), we can use computer search to show
the existence of a comparable matching between the corresponding
disjoint cardinal proper families of subsets of $[12]$. Clearly, this
approach will only work for small values of $n$, and to deal with the
general case, we need to move beyond the direct computer search.
Before proceeding with the proof of Theorem <ref> we need
a little bit of terminology. We shall think about subsets of the set
$[n]$ in terms of their support vector, i.e., we identify a subset
$S\subseteq [n]$ with an $n$-tuple $\chi_S=(a_1,\dots,a_n)$, where
$a_i=1$ if $i\in S$ and $a_i=0$ otherwise.
Let $\alpha$ be any tuple of length at most $n$, consisting of $0$'s
and $1$'s. We let $\suf{\alpha}_n$ denote the set of all subsets $S$
whose support vector ends with $\alpha$. If $n$ does not matter, we
shall drop it, and simply write $\suf\alpha$.
So, if $\alpha=(a_1,\dots,a_k)$, then the set $\suf\alpha$ consists of
all subsets $S$, for which we have $\chi_S=(b_1,\dots,b_{n-k},\ab
a_1,\dots,a_k)$, or, in other words, for $i=n-k+1,\dots n$, we have
$i\in S$ if and only if $a_{i+k-n}=1$. If $k=0$, i.e., $\alpha$ is an
empty tuple, we have $\suf\alpha_n=2^{[n]}$, consistently with the
standard notation. As another example $\suf{(0,1)}_n$ denotes the set
of all subsets $S$ such that $n\in S$, but $n-1\notin S$.
We shall use the short-hand notation skipping the commas and the
brackets, and write $\suf{01}$ instead of $\suf{(0,1)}$. Furthermore,
we shall use the square brackets to encode the repetitions: when
$\alpha$ is any tuple of $0$'s and $1$'s, $[\alpha]^k$ denotes the
tuple obtained by repeating $\alpha$ $k$ times. For example, $0[01]^3$
stands for $0010101$. We shall also use the notation $[\alpha]^*$ to
say that $\alpha$ is repeated a certain number of times, without
specifying the number of repetitions, which can also be $0$. For
\[\suf{0[01]^*}=\suf 0\cup\suf{001}\cup\suf{00101}\cup\dots,\]
and we have
We let $[\alpha]^\infty_{\leq n}$ denote the tuple obtained by first
repeating $\alpha$ infinitely many times, and then truncating it at
position $n$; it is as much of repeated $\alpha$ as is possible to fit
in the first $n$ slots. For example, for $\alpha=01$, we get
$[01]^\infty_{\leq 4}=0101$ and $[01]^\infty_{\leq 3}=101$.
§.§ Some useful set decompositions
In order to define a bijection, which is crucial for the proof of
Theorem <ref>, we need a number of specific set
decompositions. In the formulations below, we use the symbol
$\,\,\,\bar{}\,\,\,$ to denote negation, so $\bar 0=1$ and $\bar 1=0$.
Whenever $\alpha=(\alpha_1,\dots,\alpha_t)$ is a tuple consisting of
$0$'s and $1$'s, where $n\geq t$, we have the following decomposition
into disjoint subsets:
\begin{equation}\label{eq:alpha}
2^{[n]}=[\alpha]^\infty_{\leq n}\cup\bigcup_{i=1}^t
\suf{\bar\alpha_i\alpha_{i+1}\dots\alpha_t[\alpha]^*}.
\end{equation}
Take any $S\in 2^{[n]}$, and read its support vector
$\chi_S=(a_1,\dots,a_n)$ from right to left, starting with $a_n$. The
first position, where $\chi_S$ deviates from $[\alpha]^\infty_{\leq
n}$ will show in which of the disjoint sets of the right hand side
of (<ref>) the set $S$ lies.
For arbitrary $p\geq 1$, we have following identities:
\[2^{[2p]}=\suf{0[01]^*}\cup\suf{11[01]^*}\cup [01]^p=\suf{1[10]^*}\cup
\suf{00[10]^*}\cup [10]^p\]
\[2^{[2p+1]}=\suf{0[01]^*}\cup\suf{11[01]^*}\cup 1[01]^p=\suf{1[10]^*}\cup
\suf{00[10]^*}\cup 0[10]^p\]
Follows from Lemma <ref> by substituting $\alpha=01$ and
$\alpha=10$ into the equation (<ref>) and considering the two
cases when $n$ is even or odd.
For any $p\geq 2$ we have the following identities, where all unions
on the right hand side are disjoint
\begin{multline}\label{eq:001even}
\suf{0[01]^*}_{2p}= \suf{10[01]^*}\cup\suf{1[10]^* 00[01]^*}
\cup\suf{00[10]^* 00[01]^*}\cup \\
\cup\{[10]^k00[01]^{p-k-1}\,|\,0\leq k\leq p-1\},
\end{multline}
\begin{multline}\label{eq:001odd}
\suf{0[01]^*}_{2p+1}=\suf{10[01]^*}\cup\suf{1[10]^* 00[01]^*}
\cup\suf{00[10]^* 00[01]^*}\cup 0[01]^p\cup\\
\cup\{0[10]^k00[01]^{p-k-1}\,|\,0\leq k\leq p-1\}.
\end{multline}
Throughout the proof all the unions will be disjoint. We start
with the identity
\begin{equation}\label{eq:c1}
\suf{0[01]^*}_{2p}=\suf{00[01]^*}_{2p}\cup\suf{10[01]^*}_{2p}.
\end{equation}
We expand the first term
\begin{equation}\label{eq:c2}
\suf{00[01]^*}_{2p}=\suf{00}_{2p}\cup\suf{0001}_{2p}\cup\dots\cup
\suf{00[01]^k}_{2p}\cup\dots\cup\suf{00[01]^{p-1}}_{2p}.
\end{equation}
By Corollary <ref>, for all $0\leq k\leq p-2$ we get
\begin{equation}\label{eq:c3}
\suf{00[01]^k}_{2p}=\suf{1[10]^*00[01]^k}_{2p}\cup
\suf{00[10]^*00[01]^k}_{2p}\cup [10]^{p-k-1}00[01]^k,
\end{equation}
and, furthermore, we have $\suf{00[01]^{p-1}}_{2p}=00[01]^{p-1}$.
Taking the union of this equation with the equation (<ref>) for
all $k$, we get the identity
\begin{multline}\label{eq:c4}
\suf{00[01]^*}_{2p}=\suf{1[10]^*00[01]^*}_{2p}\cup
\suf{00[10]^*00[01]^*}_{2p}\cup\\\cup
\{\suf{[10]^{p-k-1}00[01]^k}_{2p}\,|\,0\leq k\leq p-1\}.
\end{multline}
Substituting this into (<ref>) proves (<ref>).
Proving the odd case (<ref>) needs a little modification.
The decomposition (<ref>) gets replaced with
\begin{equation}\label{eq:c1odd}
\suf{0[01]^*}_{2p+1}=\suf{00[01]^*}_{2p+1}\cup\suf{10[01]^*}_{2p+1}
\cup 0[01]^p,
\end{equation}
while the decomposition (<ref>) gets replaced with
\begin{equation}\label{eq:c3odd}
\suf{00[01]^k}_{2p+1}=\suf{1[10]^*00[01]^k}_{2p+1}\cup
\suf{00[10]^*00[01]^k}_{2p+1}\cup 0[10]^{p-k-1}00[01]^k,
\end{equation}
for $0\leq k\leq p-2$, and $\suff{00[01]^{p-1}}_{2p+1}=
\{000[01]^{p-1},100[01]^{p-1}\}$. Taking the union over all $k$, and
then substituting back into (<ref>) will now
yield (<ref>).
We have the following identities, where all unions on the right hand
side are disjoint
\begin{multline}\label{eq:110even}
\suf{1[10]^*}_{2p}=\suf{11[01]^*}\cup\suf{1[10]^* 10[01]^*}
\cup\suf{00[10]^* 01[01]^*}\cup\\ \cup\{[10]^k [01]^{p-k}\,|\,0\leq
k\leq p-1\},\text{ for all }p\geq 3,
\end{multline}
\begin{multline}\label{eq:110odd}
\suf{1[10]^*}_{2p+1}= \suf{11[01]^*}\cup\suf{1[10]^* 10[01]^*}
\cup\suf{00[10]^* 01[01]^*}\cup 1[01]^p\cup \\
\cup\{0[10]^k[01]^{p-k}\,|\,0\leq k\leq p-1\},\text{ for all }p\geq 2.
\end{multline}
Throughout the proof all the unions will be disjoint. We start
with proving (<ref>). By definition of $[]^*$-notation, we
have the identity
\begin{equation}\label{eq:p0}
\suf{1[10]^*}_{2p}=\suf{1}_{2p}\cup\suf{1[10]^*10}_{2p}.
\end{equation}
On the other hand, we have
$\suff{1}_{2p}=\suff{11}_{2p}\cup\suff{01}_{2p}$. Substituting this
in (<ref>) we arrive at
\begin{equation}\label{eq:p1}
\suf{1[10]^*}_{2p}=\suff{11}_{2p}\cup\suff{01}_{2p}\cup\suf{1[10]^*10}_{2p}.
\end{equation}
Next, we shall derive a formula for the term $\suff{01}_{2p}$. We
start with the identity
\begin{equation}\label{eq:p1b}
\end{equation}
which was proved in Corollary <ref>. By definition of $[]^*$,
we have
\begin{equation}\label{eq:p1c}
\suff{0[01]^*}_{2p-2}=\suff{0}_{2p-2}\cup\suff{001}_{2p-2}\cup\dots
\cup\suff{0[01]^k}_{2p-2}\cup\dots\cup\suff{0[01]^{p-2}}_{2p-2}.
\end{equation}
Using Corollary <ref> again, we derive the following identity
for all $0\leq k\leq p-3$
\begin{equation}\label{eq:p2}
\suff{0[01]^k}_{2p-2}=\suff{11[01]^*0[01]^k}_{2p-2}\cup
\suff{0[01]^*0[01]^k}_{2p-2}\cup 1[01]^{p-k-2}0[01]^k.
\end{equation}
For future reference, note that
\[\{1[01]^{p-k-2}0[01]^k\,|\,0\leq k\leq p-3\}=\{[10]^k [01]^{p-k-1}
\,|\,2\leq k\leq p-1\}.\]
For $k=p-2$ we simply have
$\suff{0[01]^{p-2}}_{2p-2}=\{00[01]^{p-2}, 10[01]^{p-2}\}$. Taking the
union of that last identity with the identities (<ref>), for all
$k=0,\dots,p-3$, we arrive at the formula
\begin{multline}\label{eq:p3}
\suff{0[01]^*}_{2p-2}= \suff{11[01]^*0[01]^*}_{2p-2}
\cup\suff{0[01]^*0[01]^*}_{2p-2}\cup \\\cup\{[10]^k[01]^{p-k-1}\,|\,1\leq k\leq p-1\},
\end{multline}
where the element $00[01]^{p-2}$ went into the second term and the
element $10[01]^{p-2}$ went into the third term on the right hand
side. We now substitute (<ref>) into (<ref>) to get the
\begin{multline*}
2^{[2p-2]}= \suff{11[01]^*}_{2p-2}\cup\suff{11[01]^*0[01]^*}_{2p-2}
\cup\suff{0[01]^*0[01]^*}_{2p-2}\cup \\\cup\{[10]^k[01]^{p-k-1}\,|\,0\leq k\leq p-1\}.
\end{multline*}
Combining this with the suffix $01$ we get
\begin{multline}\label{eq:p4}
\suff{01}_{2p}=\suff{11[01]^*01}_{2p}\cup\suff{11[01]^*0[01]^*01}_{2p}\cup
\suff{0[01]^*0[01]^*01}_{2p}\cup\\\cup\{[10]^k[01]^{p-k}\,|\,1\leq k\leq p-1\}.
\end{multline}
Substituting this into (<ref>) we arrive at the formula
\begin{multline}\label{eq:p5}
\suff{1[10]^*}_{2p}=\suff{11}_{2p}\cup\suff{1[10]^*10}_{2p}\cup
\suff{11[01]^*01}_{2p}\cup\suff{11[01]^*0[01]^*01}_{2p}\cup\\\cup
\suff{0[01]^*0[01]^*01}_{2p}\cup\{[10]^k[01]^{p-k}\,|\,0\leq k\leq p-1\}.
\end{multline}
We now note following identities: $\suff{11}_{2p}\cup\suff{11[01]^*01}_{2p}=
\suff{11[01]^*}_{2p}$,
\begin{multline*}
\suff{1[10]^*10}_{2p}\cup\suff{11[01]^*0[01]^*01}_{2p}=
\suff{1[10]^*10}_{2p}\cup\suff{1[10]^*10[01]^*01}_{2p}=\\
\end{multline*}
and $\suff{0[01]^*0[01]^*01}_{2p}=\suff{00[10]^*01[01]^*}_{2p}$.
Substituting these back into (<ref>) will
yield (<ref>).
No new ideas are needed to show (<ref>). All we have to do
is the retrace the argument used to show (<ref>).
Throughout the argument, all $\suff{}_{2p}$ and $\suff{}_{2p-2}$
should be replaced with $\suff{}_{2p+1}$ and $\suff{}_{2p-1}$.
Then (<ref>) and (<ref>) remain the same, subject to the
subscript change we just mentioned, while (<ref>) gets replaced
\begin{equation}\label{eq:p1bodd}
2^{[2p-1]}=\suff{11[01]^*}_{2p-1}\cup\suff{0[01]^*}_{2p-1}\cup 1[01]^{p-1}.
\end{equation}
The identity (<ref>) becomes
\[
\suff{0[01]^*}_{2p-1}=\suff{0}_{2p-1}\cup\suff{001}_{2p-1}\cup\dots\cup
\suff{0[01]^{p-2}}_{2p-1}\cup\suff{0[01]^{p-1}}_{2p-1},
\]
and we get
\[
\suff{0[01]^k}_{2p-1}=\suff{11[01]^*0[01]^k}_{2p-1}\cup
\suff{0[01]^*0[01]^k}_{2p-1}\cup [01]^{p-k-1}0[01]^k,
\]
for $0\leq k\leq p-2$, and $\suff{0[01]^{p-1}}_{2p-1}=\{0[01]^{p-1}\}$.
The identity (<ref>) becomes
\begin{multline}\label{eq:p3odd}
\suff{0[01]^*}_{2p-1}= \suff{11[01]^*0[01]^*}_{2p-1}
\cup \suff{0[01]^*0[01]^*}_{2p-1}\cup \\\cup
\{0[10]^k[01]^{p-k-1}\,|\,0\leq k\leq p-1\}.
\end{multline}
Substituting this into (<ref>) yields
\begin{multline*}
2^{[2p-1]}= \suff{11[01]^*}_{2p-1}\cup 1[01]^{p-1}\cup\suff{11[01]^*0[01]^*}_{2p-1}
\cup\suff{0[01]^*0[01]^*}_{2p-1}\cup \\\cup\{0[10]^k[01]^{p-k-1}\,|\,0\leq k\leq p-1\}.
\end{multline*}
and combining with the suffix $01$ as above, and then substituting it
into the analog of (<ref>) gives the analog of the
formula (<ref>), which now says the following
\begin{multline}\label{eq:p5odd}
\suff{1[10]^*}_{2p+1}=\suff{11}_{2p+1}\cup\suff{1[10]^*10}_{2p+1}\cup
\suff{11[01]^*01}_{2p+1}\cup1[01]^p\cup\\\cup\suff{11[01]^*0[01]^*01}_{2p+1}\cup
\suff{0[01]^*0[01]^*01}_{2p+1}\cup\{0[10]^k[01]^{p-k}\,|\,0\leq k\leq p-1\}.
\end{multline}
Repeating the same transformations as in the $2p$-case we
derive (<ref>).
We have now used an explicit constructive argument to prove the
Proposition <ref>. We feel, it is well-suited for explaining
the inner mechanics of the formulae (<ref>)
and (<ref>). However, it is also possible to give a much
shorter implicit argument. To save space, we restrict ourselves to
giving a sketch of how to show (<ref>) in two
steps. Step 1: count the number of elements on both side and derive
that they are both equal to $\frac{2}{3}(4^p-1)$. Note that we do not
know yet that the sets on the right hand side are disjoint, so
overlaps are counted multiple times. Step 2: show that every element
from the left hand side can be found on the right hand side. This can
be done by considering several different cases. Once this is done we
know both that the two sides are equal and that the sets on the right
hand side are disjoint. The identity (<ref>) can be shown
exactly the same, with the number of elements on both sides being
equal to $\frac{1}{3}(4^p-1)$.
§.§ The main bijection and the proof of our main theorem
We let $\Sigma\suf\alpha$ denote the set of all subsets from
a family $\Sigma$ which end on $\alpha$, in other words
$\Sigma\suf\alpha:=\Sigma\cap\suf\alpha$. Clearly, all the
decompositions above can be intersected with $\Sigma$. This will give
a number of different decompositions of $\Sigma$, such as
\[\Sigma=\Sigma\suf{0[01]^*}\cup\Sigma\suf{11[01]^*}\cup\{[01]^{n/2}\}\]
For any integer $n\geq 5$, and any $0\leq t\leq n-1$, there exists
a bijection
\[\Phi_t^n:C^n_t\suf{0[01]^*}\ra C^n_{t+1}\suf{1[10]^*},\]
such that for all $S\in\suf{0[01]^*}_n$, we have $S\subseteq\Phi_t^n(S)$.
Assume first that $n=2p$ is even, and compare the
formula (<ref>) with (<ref>). We see a strong
similarity, and define the bijection $\Phi_t^n$ by the rules
\[
\begin{array}{rcl}
\alpha 10[01]^k&\mapsto& \alpha 11[01]^k,\\
\alpha1[10]^m 00[01]^k &\mapsto &\alpha 1[10]^m 10[01]^k,\\
\alpha00[10]^m 00[01]^k &\mapsto &\alpha 00[10]^m 01[01]^k,\\
\,[10]^k 00[01]^{p-k-1}&\mapsto & [10]^k 01[01]^{p-k-1},
\end{array}
\]
for all $k,m\geq 0$, and all strings $\alpha$. When $n=2p+1$ is odd,
we compare the formula (<ref>) with (<ref>)
instead, and the last rule of the bijection gets changed to
\[
\begin{array}{rcl}
0[10]^k 00[01]^{p-k-1}& \mapsto &0[10]^k 01[01]^{p-k-1},\\
0[01]^p& \mapsto &1[01]^p,
\end{array}
\]
for all $k$.
Assume $2\leq t\leq n$, and consider bijections
\[
\begin{array}{rclcrcl}
\gamma:C_t^n\suf{11[01]^*} & \ra & C_{t-1}^{n-1}(1[10]^*) & \quad &
\rho:C_{t-2}^{n-1}\suff{0[01]^*} & \ra & C_{t-2}^n\suff{00[10]^*} \\
\alpha 1[10]^k 1 & \mapsto & \alpha 1[10]^k & \quad &
\alpha 0[01]^* & \mapsto & \alpha0[01]^*0
\end{array}
\]
where $\alpha$ is any string, and $k$ is any positive
integer.[Note how we use the facts that $11[01]^k=1[10]^k1$
and $00[10]^k=0[01]^k0$.] We then define a bijection
\[\Psi^n_t:C_t^n\suf{11[01]^*}\ra C_{t-2}^n\suf{00[10]^*}\]
as a composition
Let $M_r^n=\{S\subseteq[n]\,|\, |S|\equiv r\mod 3\}$.
For all $t\geq 1$, there exists a bijection
\[\Lambda:M_0^{6t}\sm[01]^{3t}\ra M_1^{6t},\]
such that either $S\subset\Lambda(S)$ or $\Lambda(S)\subset S$. Under
this bijection we have $\Lambda([1]^{6t})=[1]^{6t-2}00$.
By Corollary <ref> we have
\[M_0^{6t}\sm[01]^{3t}=\bigcup_{k=0}^{2t-1}C_{3k}^n\suff{0[01]^*}\cup
\bigcup_{k=1}^{2t}C_{3k}^n\suff{11[01]^*},\]
\[M_1^{6t}=\bigcup_{k=0}^{2t-1}C_{3k+1}^n\suff{1[10]^*}\cup
\bigcup_{k=1}^{2t}C_{3k-2}^n\suff{00[10]^*},\]
where all the unions
are disjoint. We now define $\Lambda$ by saying that the restriction
of $\Lambda$ to $C_{3k}^n\suff{0[01]^*}$ is equal to $\Phi_{3k}^n$,
and the restriction of $\Lambda$ to $C_{3k}^n\suff{11[01]^*}$ is equal
to $\Psi_{3k}^n$. By what is proved until now, this is clearly
a bijection.
We are now ready to prove our main theorem.
Proof of Theorem <ref>. We have a matching where the
only unmatched sets are $[01]^{3t}$ and $[1]^{6t-2}00$. We fix that by
using an augmenting path
\[[01]^{3t}\rightsquigarrow [01]^{3t-1}11\ra[01]^{3t-1}10\rightsquigarrow
[01]^{3t-2}1110\ra[01]^{3t-2}1100\rightsquigarrow [1]^{6t-2}00,\]
where the edges $[01]^{3t-1}11\ra[01]^{3t-1}10$
and $[01]^{3t-2}1110\ra[01]^{3t-2}1100$ are matching edges.
Theorem <ref> means that whenever the number of processes is
divisible by $6$, the Weak Symmetry Breaking task can be solved in 3
rounds. In particular, there are infinitely many values for the number
of processes, for which this task can be solved using a constant
number of rounds.
The smallest values of $n$ which are not covered by
Theorem <ref> are $n=10,14,15$. The binomial Diophantine
equations associated to $n=10$ and to $n=14$ do not have primitive
solutions. For $n=15$ we do find several primitive solutions, for
example $x_1=x_3=x_5=x_{10}=1$, $x_4=x_6=x_{13}=-1$, and for all other
$i$ we take $x_i=0$. This corresponds to the following arithmetic
\[\binom{15}{1}+\binom{15}{3}+\binom{15}{5}+\binom{15}{10}=
\binom{15}{0}+\binom{15}{4}+\binom{15}{6}+\binom{15}{13}=6476.\]
A computer search can then be used to find a comparable
matching between disjoint cardinal proper families of subsets of
$[15]$, implying $\msb(15)\leq 3$.
The binomial Diophantine equation associated to $n$ has solutions if
and only if $n$ is not a prime power. Furthermore, there are
infinitely many values of $n$, say $n=6t$, for arbitrary natural
number $m$, for which the binomial Diophantine equation associated to
$n$ has a primitive solution.
If $n=p^m$, then all the binomial coefficients
$\binom{p}{1},\dots,\binom{p}{p-1}$ are divisible by $p$, so obviously
the binomial Diophantine equation associated to $n$ has no
solutions. Otherwise, the greatest common divisor of these
coefficients is equal to $1$, and so a solution can be found by
Euclidean algorithm.
For $n=6t$ we have an identity
\begin{equation}\label{eq:6t}
\binom{6t}{0}+\binom{6t}{3}+\binom{6t}{6}+\dots+\binom{6t}{6t-3}=
\binom{6t}{1}+\binom{6t}{4}+\binom{6t}{7}+\dots+\binom{6t}{6t-2},
\end{equation}
so the following is a primitive solution: $x_1=x_4=\dots=x_{6t-2}=1$,
$x_3=x_6=\dots=x_{6t-3}=-1$, and all other coefficients are equal
to $0$.
§.§ Example $t=1$
When $t=1$ we are dealing with the subsets of the set $[6]$. We
have $|M_0^6\sm[01]^3|=|M_1^6|=21$ and we need to match the elements
of these two sets with each other. To start with we have
$C_0^6\suff{0[01]^*}=000000$, $C_1^6\suff{1[10]^*}=000001$,
furthermore $000000\in C_0^6\suff{00[10]^*00[01]^*}$, and hence
$\Phi^6_0(000000)=000001$. Similarly, $C_6^6\suff{11[01]^*}=111111$,
$C_4^6\suff{00[10]^*}=111100$, and $\Psi_6^6(111111)=111100$. It
remains to mutually match the $14$-element sets $C_3^6\suff{0[01]^*}$
and $C_4^6\suff{1[10]^*}$, and the $5$-element sets
$C_3^6\suff{11[01]^*}$ and $C_1^6\suff{00[10]^*}$.
We start with the two $14$-element sets. In this case the
formula (<ref>) simplifies to
\[C_3^6\suff{0[01]^*}=C_3^6\suff{10[01]^*}\cup C_3^6\suff{1[10]^*00[01]^*},\]
where the first set in the union has $9$ elements, and the second one
has $5$ elements. Similarly, the formula (<ref>)
simplifies to
\[C_4^6\suff{1[10]^*}=C_4^6\suff{11[01]^*}\cup C_4^6\suff{1[10]^*10[01]^*}.\]
Our matching rule is now the following
\[
\begin{array}{ccccccc}
C_3^6\suff{1{\bf 0}[01]^*}& \! &C_4^6\suff{1{\bf 1}[01]^*}&\,&
C_3^6\suff{1[10]^*{\bf 0}0[01]^*}&\! &C_4^6\suff{1[10]^*{\bf 1}0[01]^*}\\
1{\bf 0}0101 & \mapsto & 1{\bf 1}0101&\quad&1101{\bf 0}0&\mapsto&1101{\bf 1}0\\
011{\bf 0}01 & \mapsto & 011{\bf 1}01&\quad&0111{\bf 0}0&\mapsto&0111{\bf 1}0\\
101{\bf 0}01 & \mapsto & 101{\bf 1}01&\quad&1011{\bf 0}0&\mapsto&1011{\bf 1}0\\
00111{\bf 0} & \mapsto & 00111{\bf 1}&\quad&1110{\bf 0}0&\mapsto&1110{\bf 1}0\\
01011{\bf 0} & \mapsto & 01011{\bf 1}&\quad&11{\bf 0}001&\mapsto&11{\bf 1}001\\
10011{\bf 0} & \mapsto & 10011{\bf 1}&&&&\\
01101{\bf 0} & \mapsto & 01101{\bf 1}&&&&\\
10101{\bf 0} & \mapsto & 10101{\bf 1}&&&&\\
11001{\bf 0} & \mapsto & 11001{\bf 1}&&&&
\end{array}
\]
For the above-mentioned $5$-element sets we need to use the bijection
$\Psi_3^6$, whose definition is somewhat more complicated. The
composition from Definition <ref> yields in our case the following maps:
\[
\begin{array}{ccccccc}
100011&\longrightarrow&1000{\bf 1}&\longrightarrow&1000{\bf 0}&\longrightarrow&100000\\
010011&\longrightarrow&0100{\bf 1}&\longrightarrow&0100{\bf 0}&\longrightarrow&010000\\
001101&\longrightarrow&001{\bf 1}0&\longrightarrow&001{\bf 0}0&\longrightarrow&001000\\
000111&\longrightarrow&0001{\bf 1}&\longrightarrow&0001{\bf 0}&\longrightarrow&000100\\
001011&\longrightarrow&00{\bf 1}01&\longrightarrow&00{\bf 0}01&\longrightarrow&000010
\end{array}
\]
Finally, we need to alter our matching one time since $010101$ and
$111100$ are not matched, alternatively, we can think that $111100$ is
matched to $111111$, which needs the same modification. This is done
as is described in the proof of Theorem <ref>. We break down
the bonds $010110\mapsto 010111$ and $011100\mapsto 011110$, and take
the following 3 edges as the new matching edges: $010101\mapsto 010111$,
$010110\mapsto 011110$, and $011100\mapsto 111100$.
§ CURRENT BOUNDS FOR THE SYMMETRY BREAKING NUMBER
We strongly believe that the techniques developed in this paper can be
extended to deal with many other values of $n$. This has recently been
confirmed as follows.
<cit.>. $\,$
Assume that $n$ is a natural number and that for some numbers $0\leq
a_1<\dots<a_k\leq n$ and $0\leq b_1<\dots<b_m\leq n$, we have an
\begin{equation} \label{eq:bin}
\binom{n}{a_1}+\dots+\binom{n}{a_k}=\binom{n}{b_1}+\dots+\binom{n}{b_m}.
\end{equation}
We call such an identity a binomial identity.
Let $\Sigma$, resp. $\Lambda$, be set of all subsets of $[n]$, with
cardinalities $a_1,\dots,a_k$, resp. $b_1,\dots,b_m$. We say that the
binomial identity (<ref>) is orderable if there exists
a bijection $\Phi:\Sigma\ra\Lambda$, such that for all $S\in\Sigma$ we
either have $S\subseteq\Phi(S)$ or $S\supseteq\Phi(S)$.
In this paper we have constructed a complicated explicit bijection for
the case $n=6t$, $k=m=2t$, $\{a_1,\dots,a_{2t}\}=\{0,3,\dots,6t-3\}$,
and $\{b_1,\dots,b_{2t}\}=\{1,4,\dots,6t-2\}$, see
Section <ref>. It would be interesting to see whether
a simpler bijection can be found.
Recently, we proved the following combinatorial theorem.
All binomial identities are orderable.
It has been shown in <cit.> that, together with
Theorem <ref>, this implies the following bound on the
symmetry breaking number.
Assume the binomial Diophantine equation associated to $n$ has
a primitive solution, then we have $\msb(n)\leq 3$.
Our current knowledge about $\msb(n)$ is summarized in
the Table <ref>.
\[\begin{array}{l|l}
\text{Bound} & \text{Source}\\ \hline
\msb(n)=\infty \text{ if and only if } n \text{ is a prime power } &
\text{\cite{CR0,CR1,CR2}}\\ [0.2cm]
\msb(n)=O(n^{q+3}),
\text{ if } n \text{ is not a~prime power and } & \text{\cite{ACHP}}\\
q \text{ is the largest prime power in
the prime factorization of } n &\\ [0.2cm]
\msb(n)\geq 2& \text{\cite{wsb6}} \\ [0.2cm]
\msb(6t)\leq 3,\text{ for all }t\geq 1& \text{Theorem~\ref{thm:6n} above;} \\
&\text{the case $t=1$ in \cite{wsb6}}\\ [0.2cm]
\msb(n)\leq 3,\text{ if the binomial Diophantine equation associated } & \text{\cite{bid}}\\
\text{to $n$ has
a~primitive solution}&
\end{array}\]
The known bounds of $\msb(n)$.
In general, we feel that the work presented in this paper is
suggesting that we need to change our paradigm. When looking for lower
bounds for the complexity of the distributed protocols solving Weak
Symmetry Breaking, the focus needs to shift from the number of
processes $n$ itself to the sizes of the coefficients in the solution
of the binomial Diophantine equation associated to $n$.
Acknowledgments. I would like to thank Maurice Herlihy and
Sergio Rajsbaum for discussions and encouragement for writing up these
[An]An I. Anderson, Combinatorics of finite sets,
Corrected reprint of the 1989 edition. Dover Publications, Inc.,
Mineola, NY, 2002, xvi+250 pp.
[AC11]AC H. Attiya, A. Castañeda,
A Non-topological Proof for the Impossibility of k-Set Agreement,
Stabilization, Safety, and Security of Distributed Systems,
Lecture Notes in Computer Science 6976, (2011), 108–119.
[ACHP13]ACHP H. Attiya, A. Castañeda, M. Herlihy,
A. Paz, Upper bound on the complexity of solving hard
renaming, PODC 2013: 190–199.
[AP12]AP H. Attiya, A. Paz, Counting-based impossibility proofs for renaming and set agreement,
26th International Symposium, DISC 2012, Salvador, Brazil, Lecture
Notes in Computer Science 7611, (2012), 356–370.
[AW]AW H. Attiya, J. Welch, Distributed Computing:
Fundamentals, Simulations, and Advanced Topics, Wiley Series on
Parallel and Distributed Computing, 2nd Edition, Wiley-Interscience,
2004. 432 pp.
[BR15]BR F. Benavides, S. Rajsbaum, The read/write
protocol complex is collapsible, preprint, 19 pages, arXiv:1512.05427 [cs.DC]
[BG93]BG1 E. Borowsky, E. Gafni, Immediate Atomic
Snapshots and Fast Renaming (Extended Abstract), PODC 1993, 41–51.
[BG97]BG2 E. Borowsky, E. Gafni, A Simple
Algorithmically Reasoned Characterization of Wait-Free Computations
(Extended Abstract), PODC 1997, 189–198.
[CR08]CR0 A. Castañeda,
S. Rajsbaum, New combinatorial topology upper and lower bounds
for renaming, Proceeding of the 27th Annual ACM Symposium on
Principles of Distrib. Comput. ACM, New York, (2008), 295–304.
[CR10]CR1 A. Castañeda,
S. Rajsbaum, New combinatorial topology bounds for renaming:
the lower bound, Distrib. Comput. 25, No. 5, (2010),
[CR12a]CR2 A. Castañeda,
S. Rajsbaum, New combinatorial topology bounds for renaming:
the upper bound, J. ACM 59, No. 1, (2012), Article
No. 3.
[HKR]HKR M. Herlihy, D.N. Kozlov, S. Rajsbaum, Distributed Computing through Combinatorial Topology, Elsevier,
2014, 336 pp.
[HS]HS M. Herlihy, N. Shavit, The topological
structure of asynchronous computability, J. ACM 46 (1999),
no. 6, 858–923.
[IRR11]IRR D. Imbs, S. Rajsbaum, M. Raynal,
The universe of symmetry breaking tasks, Structural
Information and Communication Complexity, Lecture Notes in Computer
Science 6796, (2011), 66–77.
[Ko07]book D.N. Kozlov, Combinatorial
Algebraic Topology, Algorithms and Computation in Mathematics 21, Springer-Verlag Berlin Heidelberg, 2008, XX, 390 pp. 115 illus.
[Ko12]subd D.N. Kozlov, Chromatic subdivision of
a simplicial complex, Homology, Homotopy and Applications 14(2) (2012), 197–209.
[Ko14a]k2 D.N. Kozlov, Topology of immediate snapshot
complexes, Topology Appl. 178 (2014), 160–184.
[Ko14b]k1 D.N. Kozlov, Witness structures and
immediate snapshot complexes, preprint, 26 pages, arXiv:1404.4250 [cs.DC]
[Ko15a]view D.N. Kozlov, Topology of the view complex,
Homology Homotopy Appl. 17 (2015), no. 1, 307–319.
[Ko15b]paths D.N. Kozlov, Weak Symmetry Breaking and
abstract simplex paths, Math. Structures in Computer
Science 25, no. 6, (2015), 1432–1462.
[Ko15c]wsb6 D.N. Kozlov, Combinatorial topology of the
standard chromatic subdivision and weak symmetry breaking for 6
processes, in: Configuration Spaces, Springer INdAM series 14,
F. Callegaro et al. (eds.), Springer International Publishing
Switzerland, 2016, pp. 155–195.
[Ko16]bid D.N. Kozlov, All binomial identities are orderable,
J. Europ. Combinatorics, in print.
§ APPENDIX: PATH BUILDING KIT
In this appendix we list different edge paths in $\gn$ which are
used elsewhere in the paper. These paths are alternating with respect
to some give matching $\mu$, and we use $\stackrel\mu\longrightarrow$
to denote edges belonging to the matching, while $\rightsquigarrow$
denotes all other edges.
n x_2…x_kx_k+1…dμ
n , x_2…x_kx_k+1…dμ
n, x_2…x_k x_k+1…[squiggly]d
nx_2…x_k x_k+1…[squiggly]d
n, x_2…x_k,x_k+1…dμ
n, x_2…x_k,x_k+1…[squiggly]d
n, x_2…x_k+1x_k…
Paths $\swap^I_k$ and $\swap^{II}_k$, for $3\leq k\leq n-1$.
n x_2x_3…x_n-1x_n dμ
n,x_2x_3…x_n-1x_n dμ
n,x_2x_3…x_n-1x_n [squiggly]d
nx_2x_3…x_n-1x_n [squiggly]d
n, x_2x_3…x_n-1, x_n dμ
nx_2, x_3…x_n-1x_n dμ
nx_2x_3…x_n-1, x_n [squiggly]d
n, x_2, x_3…x_n-1x_n [squiggly]d
nx_2, x_3…x_n-1, x_n dμ
n, x_2, x_3…x_n-1, x_n dμ
n, x_2, x_3…x_n-1, x_n [squiggly]d
nx_2, x_3…x_n-1, x_n [squiggly]d
n, x_2, x_3…x_n-1x_n dμ
nx_3 x_2…x_n-1, x_n dμ
nx_2, x_3…x_n-1x_n [squiggly]d
n,x_3x_2…x_n-1, x_n [squiggly]d
Paths $\swap^I_2$ and $\swap^{II}_2$.
n x_2x_3…dμ
n, x_2x_3…[squiggly]d
On the left hand side we have the alternating path $\up^I_1$,
which is legal if either $x_2\notin V$ or if $x_2\in\Omega$. On the
right hand side we have alternating path $\up^I_k$, for $2\leq k\leq
n-1$, which is legal if either $x_1\notin V$, or if $x_1\s\dots\s
x_{k-1}\in\Omega$ and $x_1\s\dots\s x_{k-1}\s x_{k+1}\in\Omega$.
x_1,x_2x_3…x_k-1x_k+1,n …[squiggly]d
Path $\up^{II}_k$, for $3\leq k\leq n-1$: legal if either
$\{x_1,x_2\}\not\subseteq V$, or if $\{x_1,x_2\}\s x_3\s\dots\s
x_{k-1} \in\Omega$ and $\{x_1,x_2\}\s x_3\s\dots\s x_{k-1}\s
n,a_2 n-1a_4…dμ
na_2,n-1 a_4…dμ
nn-1a_3, a_4…dμ
n,a_2,n-1 a_4…[squiggly]d
n-1,na_3, a_4…[squiggly]d
n-1n,a_2 a_4…dμ
n-1na_3, a_4…dμ
n-1n, a_3, a_4…[squiggly]d
n-1n,a_2,a_4 …[squiggly]d
n-1, a_3na_4…
Paths $\specup^{II}$ and $\up^{II}_2$.
n-1,na_3a_4…a_n-1a_n dμ
nn-1a_3a_4…a_n-1a_n [squiggly]d
na_2n-1a_4…a_n-1a_n [squiggly]d
nn-1a_3a_4…a_n-1, a_n dμ
n-1,na_3a_4…a_n-1, a_n [squiggly]d
a_2, n-1,na_4…a_n-1a_n [squiggly]d
n-1na_3a_4…a_n-1, a_n dμ
n-1a_2,n a_4…a_n-1a_ndμ
n-1a_3,na_4…a_n-1, a_n [squiggly]d
na_2n-1a_4…a_n-1a_n [squiggly]d
n-1a_3na_4…a_n-1, a_n dμ
n-1a_3a_4,n…a_n-1, a_n [squiggly]d
na_2n-1a_4…a_n-1a_n [squiggly]d
n-1,na_3a_4…a_n-1a_n dμ
12…n-1n [squiggly]d
12…n-1,n dμ
12…nn-1 [squiggly]d
Move $n$ to the front.
n,12 34 …n-1 dμ
n1234…n-1 [squiggly]d
n12,34 …n-1 dμ
n,1 2,34 …n-1 [squiggly]d
1n2,34 …n-1 dμ
1n,2,3 4 …n-1 [squiggly]d
12n,34 …n-1 dμ
12n34…n-1 [squiggly]d
1,2 n34…n-1 dμ
1,2n,3 4 …n-1 [squiggly]d
1,23n 4 …n-1 dμ
1,234, n…n-1 [squiggly]d
1,234n…n-1 dμ
The final part for the path in Lemma <ref>.
n12…x_k,x_k+1 …dμ
1,n 2…x_k,x_k+1 …[squiggly]d
1,n 2…x_k+1x_k…dμ
n1 2…x_k+1x_k…
Swap until $y$ is at the end.
n1y_22…y_2-1y_2+1…n-2n-1 [squiggly]d
n1, y_22…y_2-1y_2+1…n-2n-1 dμ
1, y_2, n2…y_2-1y_2+1…n-2n-1 [squiggly]d
1, y_2, n2…y_2-1y_2+1…n-2, n-1 dμ
n1, y_22…y_2-1y_2+1…n-2,n-1 [squiggly]d
ny_212…y_2-1y_2+1…n-2,n-1 dμ
y_2,n12…y_2-1y_2+1…n-2,n-1 [squiggly]d
y_2,n12…y_2-1y_2+1…n-2n-1 dμ
Moving $y_2$ into the second position.
123…n-1n [squiggly]d
123…n-1,n dμ
123…nn-1 [squiggly]d
1, 2n3…n-1[squiggly]d
12, n3…n-1[squiggly]d
1, 2, n3…n-1dμ
n1, 23…n-1
Move $n$ to the front.
n1, 23…x_kx_k+1…[squiggly]d
n1, 23…x_k,x_k+1 …dμ
1,2,n 3…x_k,x_k+1 …[squiggly]d
1,2,n 3…x_k+1x_k…dμ
n1,2 3…x_k+1x_k…
Swap until $y$ is at the end.
n1,23 …n-1 [squiggly]d
n1,23 …n-1 [squiggly]d
n213…n-1 dμ
n123…n-1 dμ
2,n1 3 …n-1
1,n2 3 …n-1
2,n3…1,y …dμ
1,n3…2,y …dμ
n2 3…1,y …[squiggly]d
n1 3…2,y …[squiggly]d
n2 3…y1…dμ
n1 3…y2…dμ
Swap until $1$ or $2$ is at the end.
x_1,x_2,…,x_s-1,x_s,x_s+1,x_s+2,…,x_t-1,x_t y_1,…,y_n-t d
x_1x_2…x_s-1x_sx_s+1,x_s+2,…,x_t-1,x_t y_1,…,y_n-t d
x_1x_2…x_s-1,x_sx_s+1,x_s+2,…,x_t-1,x_t y_1,…,y_n-t d
x_1,x_2,…,x_s-1,x_sx_s+1,x_s+2,…,x_t-1,x_t y_1,…,y_n-t d
x_1,x_2,…,x_s-1,x_sx_s+1x_s+2,…,x_t-1,x_t y_1,…,y_n-t d
x_1,x_2,…,x_s-1,x_sx_s+1x_s+2…x_t-1x_t y_1,…,y_n-t d
x_1,x_2,…,x_s-1,x_sx_s+1x_s+2…x_t-1x_t, y_1,…,y_n-t d
x_1,x_2,…,x_s-1,x_sx_s+1, x_s+2,…,x_t-1,x_t, y_1,…,y_n-t
The standard path $p_{S,T}$ for $S=(x_1,\dots,x_s)$, $T=(x_1,\dots,x_t)$.
|
1511.00442
|
We formulate the conditional Kolmogorov complexity of $x$ given $y$ at precision $r$, where $x$ and $y$ are points in Euclidean spaces and $r$ is a natural number. We demonstrate the utility of this notion in two ways.
* We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set $E$ in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of $E$. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension $2$.
* We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions $\dim(x|y)$ and $\Dim(x|y)$ of $x$ given $y$, where $x$ and $y$ are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of $x$ conditioned on the information in $y$. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions $\dim(x)$ and $\Dim(x)$ and the mutual dimensions $\mdim(x:y)$ and $\Mdim(x:y)$.
§ INTRODUCTION
This paper concerns the fine-scale geometry of algorithmic information in Euclidean spaces. It shows how new ideas in algorithmic information theory can shed new light on old problems in geometric measure theory. This introduction explains these new ideas, a general principle for applying these ideas to classical problems, and an example of such an application. It also describes a newer concept in algorithmic information theory that arises naturally from this work.
Roughly fifteen years after the mid-twentieth century development of the Shannon information theory of probability spaces <cit.>, Kolmogorov recognized that Turing's mathematical theory of computation could be used to refine the Shannon theory to enable the amount of information in individual data objects to be quantified <cit.>. The resulting theory of Kolmogorov complexity, or algorithmic information theory, is now a large enterprise with many applications in computer science, mathematics, and other sciences <cit.>. Kolmogorov proved the first version of the fundamental relationship between the Shannon and algorithmic theories of information in <cit.>, and this relationship was made exquisitely precise by Levin's coding theorem <cit.>. (Solomonoff and Chaitin independently developed Kolmogorov complexity at around the same time as Kolmogorov with somewhat different motivations <cit.>.)
At the turn of the present century, the first author recognized that Hausdorff's 1919 theory of fractal dimension <cit.> is an older theory of information that can also be refined using Turing's mathematical theory of computation, thereby enabling the density of information in individual infinite data objects, such as infinite binary sequences or points in Euclidean spaces, to be quantified <cit.>. The resulting theory of effective fractal dimensions is now an active enterprise with a growing array of applications <cit.>. The paper <cit.> proved a relationship between effective fractal dimensions and Kolmogorov complexity that is as precise as — and uses — Levin's coding theorem.
Most of the work on effective fractal dimensions to date has concerned the (constructive) dimension $\dim(x)$ and the dual strong (constructive) dimension $\Dim(x)$ <cit.> of an infinite data object $x$, which for purposes of the present paper is a point in a Euclidean space $\R^n$ for some positive integer $n$.[These constructive dimensions are $\Sigma^0_1$ effectivizations of Hausdorff and packing dimensions <cit.>. Other effectivizations, e.g., computable dimensions, polynomial time dimensions, and finite-state dimensions, have been investigated, but only the constructive dimensions are discussed here.]
The inequalities
\[0\leq\dim(x)\leq \Dim(x)\leq n\]
hold generally, with, for example, $\Dim(x) = 0$ for points $x$ that are computable and $\dim(x) = n$ for points that are algorithmically random in the sense of Martin-Löf [Mart66].
How can the dimensions of individual points—dimensions that are defined using the theory of computing—have any bearing on classical problems of geometric measure theory? The problems that we have in mind here are problems in which one seeks to establish lower bounds on the classical Hausdorff dimensions $\dim_H(E)$ (or other fractal dimensions) of sets $E$ in Euclidean spaces. Such problems involve global properties of sets and make no mention of algorithms.
The key to bridging this gap is relativization. Specifically, we prove here a point-to-set principle saying that, in order to prove a lower bound $\dim_H(E)\geq \alpha$, it
suffices to show that, for every $A \subseteq \N$ and every $\ve>0$, there is a point $x \in E$ such that $\dim^A(x)\geq\alpha-\ve$, where $\dim^A(x)$ is the dimension of $x$ relative to the oracle $A$. We also prove the analogous point-to-set principle for the classical packing dimension $\dim_P(E)$ and the relativized strong dimension $\Dim^A(x)$.
We illustrate the power of the point-to-set principle by using it to give a new proof of a known theorem in geometric measure theory. A Kakeya set in a Euclidean space $\R^n$ is a set $K \subseteq \R^n$ that contains a unit line segment in every direction. Besicovitch <cit.> proved that Kakeya sets can have Lebesgue measure $0$ and asked whether Kakeya sets in the Euclidean plane can have dimension less than $2$ <cit.>. The famous Kakeya conjecture asserts a negative answer to this and to the analogous question in higher dimensions, i.e., states that every Kakeya set in a Euclidean space $\R^n$ has Hausdorff dimension $n$.[Statements of the Kakeya conjecture vary in the literature. For example, the set is sometimes required to be compact or Borel, and the dimension used may be Minkowski instead of Hausdorff. Since the Hausdorff dimension of a set is never greater than its Minkowski dimension, our formulation is at least as strong as those variations.] This conjecture holds trivially for $n=1$ and was proven by Davies <cit.> for $n=2$. A version of the conjecture in finite fields has been proven by Dvir <cit.>. For Euclidean spaces of dimension $n\geq 3$, it is an important open problem with deep connections to other problems in analysis <cit.>.
In this paper we use our point-to-set principle to give a new proof of Davies's theorem. This proof does not resemble the classical proof, which is not difficult but relies on Marstrand's projection theorem <cit.> and point-line duality. Instead of analyzing the set $K$ globally, our proof focuses on the information content of a single, judiciously chosen point in $K$. Given a Kakeya set $K\subseteq\R^2$ and an oracle $A\subseteq\N$, we first choose a particular line segment $L\subseteq K$ and a particular point $(x,mx+b)\in L$, where $y=mx+b$ is the equation of the line containing $L$.[One might naïvely expect that for independently random $m$ and $x$, the point $(x,mx+b)$ must be random. In fact, in every direction there is a line that contains no random point <cit.>.] We then show that $\dim^A(x,mx+b)\geq2$. By our point-to-set principle this implies that $\dim_H(K)\geq2$.
Our proof that $\dim^A(x,mx+b)\geq2$ requires us to formulate a concept of conditional Kolmogorov complexity in Euclidean spaces. Specifically, for points $x\in\R^m$ and $y\in\R^n$ and natural numbers $r$, we develop the conditional Kolmogorov complexity $K_r(x|y)$ of $x$ given $y$ at precision $r$. This is a “conditional version” of the Kolmogorov complexity $K_r(x)$ of $x$ at precision $r$ that has been used in several recent papers (e.g., <cit.>).
In addition to enabling our new proof of Davies's theorem, conditional Kolmogorov complexity in Euclidean spaces enables us to fill a gap in effective dimension theory. The fundamental quantities in Shannon information theory are the entropy (information content) $H(X)$ of a probability space $X$, the conditional entropy $H(X|Y)$ of a probability space $X$ given a probability space $Y$, and the mutual information (shared information) $I(X;Y)$ between two probability spaces $X$ and $Y$ <cit.>. The analogous quantities in Kolmogorov complexity theory are the Kolmogorov complexity $K(u)$ of a finite data object $u$, the conditional Kolmogorov complexity $K(u|v)$ of a finite data object $u$ given a finite data object $v$, and the algorithmic mutual information $I(u:v)$ between two finite data objects $u$ and $v$ <cit.>. The above-described dimensions $\dim(x)$ and $\Dim(x)$ of a point $x$ in Euclidean space (or an infinite sequence $x$ over a finite alphabet)
are analogous by limit theorems <cit.> to $K(u)$ and hence to $H(X)$. Case and the first author have recently developed and investigated the mutual dimension $\mdim(x:y)$ and the dual strong mutual dimension $\Mdim(x:y)$, which are densities of the algorithmic information shared by points $x$ and $y$ in Euclidean spaces <cit.> or sequences $x$ and $y$ over a finite alphabet <cit.>. These mutual dimensions are analogous to $I(u:v)$ and $I(X;Y)$.
What is conspicuously missing from the above account is a notion of conditional dimension. In this paper we remedy this by using conditional Kolmogorov complexity in Euclidean space to develop the conditional dimension $\dim(x|y)$ of $x$ given $y$ and its dual, the conditional strong dimension $\Dim(x|y)$ of $x$ given $y$, where $x$ and $y$ are points in Euclidean spaces. We prove that these conditional dimensions are well behaved and that they have the correct information theoretic relationships with the previously defined dimensions and mutual dimensions. The original plan of our proof of Davies's theorem used conditional dimensions, and we developed their basic theory to that end. Our final proof of Davies's theorem does not use them, but conditional dimensions (like the conditional entropy and conditional Kolmogorov complexity that motivate them) are very likely to be useful in future investigations.
The rest of this paper is organized as follows. Section <ref> briefly reviews the dimensions of points in Euclidean spaces. Section <ref> presents the point-to-set principles that enable us to use dimensions of individual points to prove lower bounds on classical fractal dimensions. Section <ref> develops conditional Kolmogorov complexity in Euclidean spaces. Section <ref> uses the preceding two sections to give our new proof of Davies's theorem. Section <ref> uses Section <ref> to develop conditional dimensions in Euclidean spaces. Most proofs are deferred to the optional technical appendix.
§ DIMENSIONS OF POINTS IN EUCLIDEAN SPACES
This section reviews the constructive notions of dimension and mutual dimension in Euclidean spaces. The presentation here is in terms of Kolmogorov complexity. Briefly, the conditional Kolmogorov complexity $K(w|v)$ of a string $w \in \{0,1\}^*$ given a string $v\in\{0,1\}^*$ is the minimum length $|\pi|$ of a binary string $\pi$ for which $U(\pi,v) = w$, where $U$ is a fixed universal self-delimiting Turing machine. The Kolmogorov complexity of $w$ is $K(w|\lambda)$, where $\lambda$ is the empty string. We write $U(\pi)$ for $U(\pi,\lambda)$. When $U(\pi) = w$, the string $\pi$ is called a program for $w$. The quantity $K(w)$ is also called the algorithmic information content of $w$. Routine coding extends this definition from $\{0,1\}^*$ to other discrete domains, so that the Kolmogorov complexities of natural numbers, rational numbers, tuples of these, etc., are well defined up to additive constants. Detailed discussions of self-delimiting Turing machines and Kolmogorov complexity appear in the books <cit.> and many papers.
The definition of $K(q)$ for rational points $q$ in Euclidean space is lifted in two steps to define the dimensions of arbitrary points in Euclidean space. First, for $x \in \R^n$ and $r \in \N$, the Kolmogorov complexity of $x$ at precision $r$ is
\begin{equation}\label{eq:Kr}
K_r(x)=\min\{K(q)\,:\,q\in\Q^n\cap B_{2^{-r}}(x)\}\,,
\end{equation}
where $B_{2^{-r}}(x)$ is the open ball with radius $2^{-r}$ and center $x$. Second, for $x \in \R^n$, the dimension and strong dimension of $x$ are
\begin{equation}\label{eq:dimDim}
\dim(x)=\liminf_{r\to\infty}\frac{K_r(x)}{r}\qquad\textrm{and}\qquad\Dim(x)=\limsup_{r\to\infty}\frac{K_r(x)}{r}\,,
\end{equation}
respectively.[We note that $K_r(x)=K(x\upharpoonright r)+o(r)$, where $x\upharpoonright r$ is the binary expansion of $x$, truncated $r$ bits to the right of the binary point. However, it has been known since Turing's famous correction <cit.> that binary notation is not a suitable representation for the arguments and values of computable functions on the reals. (See also <cit.>.) Hence, in order to make our definitions useful for further work in computable analysis, we formulate complexities and dimensions in terms of rational approximations, both here and later.]
Intuitively, $\dim(x)$ and $\Dim(x)$ are the lower and upper asymptotic densities of the algorithmic information in $x$. These quantities were first defined in Cantor spaces using betting strategies called gales and shown to be constructive versions of classical Hausdorff and packing dimension, respectively <cit.>. These definitions were explicitly extended to Euclidean spaces in <cit.>, where the identities (<ref>) were proven as a theorem. Here it is convenient to use these identities as definitions. For $x \in \R^n$, it is easy to see that
\[0\leq\dim(x)\leq\Dim(x)\leq n\,,\]
and it is known that, for any two reals $0\leq\alpha\leq\beta\leq n$, there exist uncountably many points $x \in \R^n$ satisfying $\dim(x)=\alpha$ and $\Dim(x)=\beta$ <cit.>. Applications of these dimensions in Euclidean spaces appear in <cit.>.
§ FROM POINTS TO SETS
The central message of this paper is a useful point-to-set principle by which the existence of a single high-dimensional point in a set $E\subseteq\R^n$ implies that the set $E$ has high dimension.
To formulate this principle we use relativization. All the algorithmic information concepts in Sections <ref> and <ref> above can be relativized to an arbitrary oracle $A\subseteq\N$ by giving the Turing machine in their definitions oracle access to $A$. Relativized Kolmogorov complexity $K_r^A(x)$ and relativized dimensions $\dim^A(x)$ and $\Dim^A(x)$ are thus well defined. Moreover, the results of Section <ref> hold relative to any oracle $A$.
We first establish the point-to-set principle for Hausdorff dimension. Let $E\subseteq\R^n$. For $\delta>0$, define $\mathcal{U}_\delta(E)$ to be the collection of all countable covers of $E$ by sets of positive diameter at most $\delta$. That is, for every cover $\{U_i\}_{i\in\N}\in\mathcal{U}_\delta(E)$, we have $E\subseteq\bigcup_{i\in\N}U_i$ and $|U_i|\in(0,\delta]$ for all $i\in\N$, where for $X\in\R^n$, $|X|=\sup_{p,q\in X}|p-q|$. For $s\geq0$, define
\[H_\delta^s(E)=\inf\bigg\{\sum_{i\in\N}\left|U_i\right|^s\,:\,\{U_i\}_{i\in\N}\in\mathcal{U}_\delta(E)\bigg\}\,.\]
Then the $s$-dimensional Hausdorff outer measure of $E$ is
\[H^s(E)=\lim_{\delta\to 0^+}H_\delta^s(E)\,,\]
and the Hausdorff dimension of $E$ is
\[\dim_H(E)=\inf\left\{s>0:H^s(E)=0\right\}\,.\]
More details may be found in standard texts, e.g., <cit.>.
For every set $E\subseteq\mathbb{R}^n$,
\[\dim_H(E)=\adjustlimits\min_{A\subseteq\N}\sup_{x\in E}\,\dim^A(x)\,.\]
Three things should be noted about this principle. First, while the left-hand side is the classical Hausdorff dimension, which is a global property of $E$ that does not involve the theory of computing, the right-hand side is a pointwise property of the set that makes essential use of relativized algorithmic information theory. Second, as the proof shows, the right-hand side is a minimum, not merely an infimum. Third, and most crucially, this principle implies that, in order to prove a lower bound $\dim_H(E)\geq\alpha$, it suffices to show that, for every $A\subseteq\N$ and every $\ve>0$, there is a point $x\in E$ such that $\dim^A(x)\geq\alpha-\ve$.[The $\ve$ here is useful in general but is not needed in some cases, including our proof of Theorem <ref> below.]
For the $(\geq)$ direction of this principle, we construct the minimizing oracle $A$. The oracle encodes, for a carefully chosen sequence of increasingly refined covers for $E$, the approximate locations and diameters of all cover elements. Using this oracle, a point $x\in\R^n$ can be approximated by specifying an appropriately small cover element that it belongs to, which requires an amount of information that depends on the number of similarly-sized cover elements. We use the definition of Hausdorff dimension to bound that number. The $(\leq)$ direction can be shown using results from <cit.>, but in the interest of self-containment we prove it directly.
Theorem <ref>
Let $E\subseteq\R^n$, and let $d=\dim_H(E)$. For every $s>d$ we have $H^s(E)=0$, so there is a sequence $\{\{U^{t,s}_i\}_{i\in\N}\}_{t\in\N}$ of countable covers of $E$ such that $\big|U_i^{t,s}\big|\leq2^{-t}$ for every $i,t\in\N$, and for every sufficiently large $t$ we have
\begin{equation}\label{eq:diam}
\sum_{i\in\N} \left|U^{t,s}_i\right|^s<1\,.
\end{equation}
Let $D=\N^3\times(\Q\cap(d,\infty))$.
Our oracle $A$ encodes functions
such that for every $(i,t,r,s)\in D$, we have
\[f_A(i,t,r,s)\in B_{2^{-r-1}}(u)\]
for some $u\in U_i^{t,s}$ and
\begin{equation}\label{eq:diamerr}
\Big|g_A(i,t,r,s)-\big|U_i^{t,s}\big|\Big|<2^{-r-4}\,.
\end{equation}
We will show, for every $x\in E$ and rational $s>d$, that $\dim^A(x)\leq s$.
Fix $x\in E$ and $s\in\Q\cap(d,\infty)$. If for any $i_0,t_0\in\N$ we have $x\in U_{i_0}^{t_0,s}$ and $\left|U_{i_0}^{t_0,s}\right|=0$, then $U_{i_0}^{t_0,s}=\{x\}$, so $f_A(i_0,t_0,r,s)\in B_{2^{-r}}(x)$ for every $r\in\N$. In this case, let $M$ be a prefix Turing machine with oracle access to $A$ such that, whenever $U(\iota)=i\in\N$, $U(\tau)=t\in\N$, $U(\rho)=r\in\N$, and $U(\sigma)=q\in\Q\cap(d,\infty)$,
\[M(\iota\tau\rho\sigma)=f_A(i,t,r,q)\,.\]
Now for any $r\in\N$, let $\iota$, $\tau$, $\rho$, and $\sigma$ be witnesses to $K(i_0)$, $K(t_0)$, $K(r)$, and $K(s)$, respectively.
Since $i_0$, $t_0$, and $s$ are all constant in $r$ and $|\rho|=o(r)$, we have $|\iota\tau\rho\sigma|=o(r)$. Thus $K^A_r(x)=o(r)$, and $\dim^A(x)=0$. Hence assume that every cover element containing $x$ has positive diameter.
Fix sufficiently large $t$, and let $U_{i_x}^{t,s}$ be some cover element containing $x$.
Let $M^\prime$ be a self-delimiting Turing machine with oracle access to $A$ such that whenever $U(\kappa)=k\in\N$, $U(\tau)=\ell\in\N$, $U(\rho)=r\in\N$, and $U(\sigma)=q\in\Q\cap(d,\infty)$,
\[M^\prime(\kappa\tau\rho\sigma)=f_A(p,\ell,r,q)\,,\]
where $p$ is the $k$th index $i$ such that
$g_A(i,t,r,q)\geq 2^{-r-3}$.
Now fix $r\geq t-1$ such that
\[\big|U_{i_x}^{t,s}\big|\in \left[2^{-r-2},2^{-r-1}\right)\,.\]
Notice that $g_A(i_x,t,r,s)\geq 2^{-r-3}$. Hence there is some $k$ such that, letting $\kappa$, $\tau$, $\rho$, and $\sigma$ be witnesses to $K(k)$, $K(t)$, $K(r)$, and $K(s)$, respectively,
\[M^\prime(\kappa\tau\rho\sigma)\in B_{2^{-r-1}}(u)\,,\]
for some $u\in U^{t,s}_{i_x}$.
Because $\big|U_{i_x}^{t,s}\big|<2^{-r-1}$ and $x\in U_{i_x}^{t,s}$, we have
\[M^\prime(\kappa\tau\rho\sigma)\in B_{2^{-r}}(x)\,.\]
\[K^A_{r}(x)\leq K(k)+K(t)+K(s)+K(r)+c\,,\]
where $c$ is a machine constant for $M^\prime$. Since $s$ is constant in $r$ and $t<r$, Observation <ref> tells us that this expression is $K(k)+o(r)\leq\log(k)+o(r)$. By (<ref>), there are fewer than $2^{(r+4)s}$ indices $i\in\N$ such that
\[\left|U_i^{t,s}\right|\geq2^{-r-4}\,,\]
hence by (<ref>) there are fewer than $2^{(r+4)s}$ indices $i\in\N$ such that $g_A(i,t,r,s)\geq 2^{-r-3}$,
so $\log(k)<{(r+4)s}$.
Therefore $K_{r}^A(x)\leq rs+o(r)$.
There are infinitely many such $r$, which can be seen by replacing $t$ above with $r+2$. We have shown
\[\dim^A(x)=\liminf_{r\to\infty}\frac{K_r^A(x)}{r}\leq s\,,\]
for every rational $s>d$, hence $\dim^A(x)\leq d$. It follows that
\[\adjustlimits\min_{A\subseteq\N}\sup_{x\in E}\,\dim^A(x)\leq d\,.\]
For the other direction, assume for contradiction that there is some oracle $A$ and $d^\prime<d$ such that
\[\sup_{x\in E}\,\dim^A(x)= d^\prime\,.\]
Then for every $x\in E$, $\dim^A(x)\leq d^\prime$. Let $s\in(d^\prime,d)$. For every $r\in\N$, define the sets
\[\mathcal{B}_r=\left\{B_{2^{-r}}(q)\,:\,q\in\Q\textrm{ and }K^A(q)\leq rs\right\}\]
\[\mathcal{W}_r=\bigcup_{k=r}^\infty\mathcal{B}_k\,.\]
There are at most $2^{ks+1}$ balls in each $\mathcal{B}_k$, so for every $r\in\N$ and $s^\prime\in(s,d)$,
\begin{align*}
\sum_{W\in \mathcal{W}_r}|W|^{s^\prime}&=\sum_{k=r}^\infty\sum_{W\in\mathcal{B}_k}|W|^{s^\prime}\\
&\leq\sum_{k=r}^\infty 2^{ks+1}(2^{1-k})^{s^\prime}\\
&=2^{1+s'}\cdot\sum_{k=r}^\infty 2^{(s-s^\prime)k}\,,
\end{align*}
which approaches $0$ as $r\to\infty$. As every $\mathcal{W}_r$ is a cover for $E$, we have $H^{s^\prime}(E)=0$, so $\dim_H(E)\leq s^\prime<d$, a contradiction.
The packing dimension $\dim_P(E)$ of a set $E\subseteq\R^n$, defined in the appendix and standard texts, e.g., <cit.>, is a dual of Hausdorff dimension satisfying $\dim_P(E)\geq\dim_H(E)$, with equality for very “regular” sets $E$. We also have the following.
For every set $E\subseteq\R^n$,
\[\dim_P(E)=\adjustlimits\min_{A\subseteq\N}\sup_{x\in E}\,\Dim^A(x)\,.\]
§ CONDITIONAL KOLMOGOROV COMPLEXITY IN EUCLIDEAN SPACES
We now develop the conditional Kolmogorov complexity in Euclidean spaces.
For $x \in \R^m$, $q\in\Q^n$, and $r\in\N$, the conditional Kolmogorov complexity of $x$ at precision $r$ given $q$ is
\begin{equation}\label{eq:Krxq}
\hat{K}_r(x|q)=\min\left\{K(p|q)\,:\,p\in\Q^m\cap B_{2^{-r}}(x)\right\}\,.
\end{equation}
For $x\in\R^m$, $y\in\R^n$, and $r,s\in\N$, the conditional Kolmogorov complexity of $x$ at precision $r$ given $y$ at precision $s$ is
\begin{equation}\label{eq:Krsxy}
K_{r,s}(x|y)=\max\big\{\hat{K}_r(x|q)\,:\,q\in\Q^n\cap B_{2^{-s}}(y)\big\}\,.
\end{equation}
Intuitively, the maximizing argument $q$ is the point near $y$ that is least helpful in the task of approximating $x$.
Note that $K_{r,s}(x|y)$ is finite, because $\hat{K}_r(x|q)\leq K_r(x)+O(1)$. For $x\in\R^m$, $y\in\R^n$, and $r\in\N$, the conditional Kolmogorov complexity of $x$ given $y$ at precision $r$ is
\begin{equation}\label{eq:Krxy}
\end{equation}
For all $x\in\R^m$ and $y\in\R^n$,
\[K_r(x,y)=K_r(x|y) + K_r(y)+o(r)\,.\]
We also consider the Kolmogorov complexity of $x\in\R^m$ at precision $r$ relative to $y\in\R^n$. Let $K^y_r(x)$ denote $K^{A_y}_r(x)$, where $A_y\subseteq\N$ encodes the binary expansions of $y$'s coordinates. The following lemma reflects the intuition that oracle access to $y$ is at least as useful as any bounded-precision estimate for $y$.
For each $m,n\in\N$ there is a constant $c\in\N$ such that, for all $x\in\R^m$, $y\in\R^n$, and $r,s\in\N$,
\[K_r^y(x)\leq K_{r,s}(x|y)+K(s)+c\,.\]
In particular, $K_r^y(x)\leq K_r(x|y)+K(r)+c$.
§ KAKEYA SETS IN THE PLANE
This section uses the results of the preceding two sections to give a new proof of the following classical theorem. Recall that a Kakeya set in $\R^n$ is a set containing a unit line segment in every direction.
Every Kakeya set in $\mathbb{R}^2$ has Hausdorff dimension 2.
Our new proof of Theorem <ref> uses a relativized version of the following lemma.
Let $m\in[0,1]$ and $b\in\mathbb{R}$. Then for almost every $x\in[0,1]$,
\begin{equation}\label{eq:mainlem}
\liminf_{r\to\infty}\frac{K_r(m,b,x)-K_r(b|m)}{r}\leq\dim(x,mx+b)\,.
\end{equation}
We build a program that takes as input a precision level $r$, an approximation $p$ of $x$, an approximation $q$ of $mx+b$, a program $\pi$ that will approximate $b$ given an approximation for $m$, and a natural number $h$.
In parallel, the program considers each multiple of $2^{-r}$ in [0,1] as a possible approximate value $u$ for the slope $m$, and it checks whether each such $u$ is consistent with the program's inputs. If $u$ is close to $m$, then $\pi(u)$ will be close to $b$, so $up+\pi(u)$ will be close to $mx+b$. Any $u$ that satisfies this condition is considered a “candidate” for approximating $m$.
Some of these candidates may be “false positives,” in that there can be values of $u$ that are far from $m$ but for which $up+\pi(u)$ is still close to $mx+b$. Thus the program is also given an input $h$ so that it can choose the correct candidate; it selects the $h$th candidate that arises in its execution. We will show that this $h$ is often not large enough to significantly affect the total input length.
Formally, let $M$ be a Turing machine that runs the following algorithm on input $\rho\pi\sigma\eta$ whenever $U(\rho)=r\in\N$, $U(\eta)=h\in\N$, and $U(\sigma)=(p,q)\in\mathbb{Q}^2$:
for $i=0,1,\ldots,2^r$, in parallel:
do atomically:
if $v_i\in\R$ and $|u_ip+v_i-q|<2^{2-r}$, then $candidate:=candidate+1$
if $candidate = h$, then return $(u_i,v_i,p)$ and halt
Fix $m\in[0,1]$ and $b\in\mathbb{R}$. For each $r\in\mathbb{N}$, let $m_r=2^{-r}\lfloor m\cdot2^{r}\rfloor$, and fix $\pi_r$ testifying to the value of $\hat{K}_r(b|m_r)$ and $\sigma_r$ testifying to the value of $K_r(x,mx+b)$.
Proofs of the following four claims appear in the appendix. Intuitively, Claim <ref> says that no point in $B_{2^{-r}}(m)$ gives much less information about $b$ than $m_r$ does. Claim <ref> states that there is always some value of $h$ that causes this machine to return the desired output. Claim <ref> says that for almost every $x$, this value does not grow too quickly with $r$, and Claim <ref> says that (<ref>) holds for every such $x$.
For every $r\in\N$, $K_r(b|m)=\hat{K}_r(b|m_r)+o(r)$.
For each $x\in[0,1]$ and $r\in\mathbb{N}$, there exists an $h\in\mathbb{N}$ such that
\[M(\rho\pi_r\sigma_r\eta)\in B_{2^{1-r}}(m,b,x)\,,\]
where $U(\rho)=r$ and $U(\eta)=h$.
For every $x\in[0,1]$ and $r\in\N$, define $h(x,r)$ to be the minimal $h$ satisfying the conditions of Claim <ref>.
For almost every $x\in[0,1]$, $\log(h(x,r))=o(r)$.
For every $x\in[0,1]$, if $\log(h(x,r))={o(r)}$, then
\[\liminf_{r\to\infty}\frac{K_r(m,b,x)-K_r(b|m)}{r}\leq\dim(x,mx+b)\,.\]
The lemma follows immediately from Claims <ref> and <ref>.
Theorem <ref>
Let $K$ be a Kakeya set in $\R^2$. By Theorem <ref>, there exists an oracle $A$ such that $\dim_H(K)=\sup_{p\in K}\dim^A(p)$.
Let $m\in[0,1]$ such that $\dim^A(m)=1$; such an $m$ exists by Theorem 4.5 of <cit.>. $K$ contains a unit line segment $L$ of slope $m$. Let $(x_0,y_0)$ be the left endpoint of such a segment. Let $q\in\Q\cap[x_0,x_0+1/8]$, and let $L^\prime$ be the unit segment of slope $m$ whose left endpoint is $(x_0-q,y_0)$. Let $b=y_1+qm$, the $y$-intercept of $L^\prime$.
By a relativized version of Lemma <ref>, there is some $x\in[0,1/2]$ such that $\dim^{A,m,b}(x)=1$ and
\[\liminf_{r\to\infty}\frac{K^A_r(m,b,x)-K^A_r(b|m)}{r}\leq\dim^A(x,mx+b)\,.\]
(This holds because almost every $x\in[0,1/2]$ is algorithmically random relative to $(A,m,b)$ and hence satisfies $\dim^{A,m,b}(x)=1$.)
Fix such an $x$, and notice that $(x,mx+b)\in L^\prime$. Now applying a relativized version of Theorem <ref>,
\begin{align*}
\dim^A(x,mx+b)&\geq\liminf_{r\to\infty}\frac{K^A_r(m,b,x)-K^A_r(b|m)}{r}\\
\end{align*}
By Lemma <ref>, $K^A_r(x|b,m)\geq K^{A,b,m}_r(x)+o(r)$, so we have
\begin{align*}
\dim^A(x,mx+b)&\geq\liminf_{r\to\infty}\frac{K^{A,b,m}_r(x)}{r}+\liminf_{r\to\infty}\frac{K^A_r(m)}{r}\\
\end{align*}
which is $2$ by our choices of $m$ and $x$.
By Observation <ref>,
\[\dim^A(x,mx+b)=\dim^A(x+q,mx+b)\,.\]
Hence, there exists a point $(x+q,mx+b)\in K$ such that $\dim^A(x+q,mx+b)\geq 2$. By Theorem <ref>, the point-to-set principle for Hausdorff dimension, this completes the proof.
It is natural to ask what prevents us from extending this proof to higher-dimensional Euclidean spaces. The point of failure in a direct extension would be Claim <ref> in the proof of Lemma <ref>. Speaking informally, the problem is that the total number of candidates may grow as $2^{(n-1)r}$, meaning that $\log(h(x,r))$ could be $\Omega((n-2)r)$ for every $x$.
§ CONDITIONAL DIMENSIONS IN EUCLIDEAN SPACES
The results of Section <ref>, which were used in the proof of Theorem <ref>, also enable us to give robust formulations of conditional dimensions.
For $x\in\R^m$ and $y\in\R^n$, the lower and upper conditional dimensions of $x$ given $y$ are
\begin{equation}\label{eq:conddimDim}
\dim(x|y)=\liminf_{r\to\infty}\frac{K_r(x|y)}{r}\qquad\textrm{and}\qquad\Dim(x|y)=\limsup_{r\to\infty}\frac{K_r(x|y)}{r}\,,
\end{equation}
The use of the same precision bound $r$ for both $x$ and $y$ in (<ref>) makes the definitions (<ref>) appear arbitrary and “brittle.” The following theorem shows that this is not the case.
Let $s:\N\to\N$. If $|s(r)-r|=o(r)$, then, for all $x\in\R^m$ and $y\in\R^n$,
\[\dim(x|y)=\liminf_{r\to\infty}\frac{K_{r,s(r)}(x|y)}{r}\,,\]
\[\Dim(x|y)=\limsup_{r\to\infty}\frac{K_{r,s(r)}(x|y)}{r}\,.\]
The rest of this section is devoted to showing that our conditional dimensions have the correct information theoretic relationships with the previously developed dimensions and mutual dimensions.
Mutual dimensions were developed very recently, and Kolmogorov complexity was the starting point. The mutual (algorithmic) information between two strings $u,v \in \{0,1\}^*$ is
\[I(u:v) = K(v) - K(v|u)\,.\]
Again, routine coding extends $K(u|v)$ and $I(u:v)$ to other discrete domains. Discussions of $K(u|v)$, $I(u:v)$, and the correspondence of $K(u)$, $K(u|v)$, and $I(u:v)$ with Shannon entropy, Shannon conditional entropy, and Shannon mutual information appear in <cit.>.
In parallel with (<ref>) and (<ref>), Case and J. H. Lutz <cit.> lifted the definition of $I(p:q)$ for rational points $p$ and $q$ in Euclidean spaces in two steps to define the mutual dimensions between two arbitrary points in (possibly distinct) Euclidean spaces. First, for $x \in \R^m$, $y \in \R^n$, and $r \in \N$, the mutual information between $x$ and $y$ at precision $r$ is
\begin{equation}\label{eq:Ir}
I_r(x:y)=\min\left\{I(p:q)\,:\,p\in B_{2^{-r}}(x)\cap\Q^m\textrm{ and }q\in B_{2^{-r}}(y)\cap\Q^n\right\}\,,
\end{equation}
where $B_{2^{-r}}(x)$ and $B_{2^{-r}}(y)$ are the open balls of radius $2^{-r}$ about $x$ and $y$ in their respective Euclidean spaces. Second, for $x \in \R^m$ and $y \in \R^n$, the lower and upper mutual dimensions between $x$ and $y$ are
\begin{equation}\label{eq:mdimMdim}
\mdim(x:y)=\liminf_{r\to\infty}\frac{I_r(x:y)}{r}\quad \textrm{and}\quad\Mdim(x:y)=\limsup_{r\to\infty}\frac{I_r(x:y)}{r}\,,
\end{equation}
respectively. Useful properties of these mutual dimensions, especially including data processing inequalities, appear in <cit.>.
For all $x\in\R^m$ and $y\in\R^n$,
\[I_r(x:y)=K_r(x)-K_r(x|y)+o(r)\,.\]
The following bounds on mutual dimension follow from Lemma <ref>.
For all $x\in\R^m$ and $y\in\R^n$, the following hold.
* $\mdim(x:y)\geq \dim(x)-\Dim(x|y)$.
* $\Mdim(x:y)\leq \Dim(x)-\dim(x|y)$.
Our final theorem is easily derived from Theorem <ref>.
For all $x\in\R^m$ and $y\in\R^n$,
\begin{align*}
\dim(x)+\dim(y|x)&\leq\dim(x,y)\\
\end{align*}
§ CONCLUSION
This paper shows a new way in which theoretical computer science can be used to answer questions that may appear unrelated to computation. We are hopeful that our new proof of Davies's theorem will open the way for using constructive fractal dimensions to make new progress in geometric measure theory, and that conditional dimensions will be a useful component of the information theoretic apparatus for studying dimension.
§.§ Acknowledgments
We thank Eric Allender for useful corrections and three anonymous reviewers of an earlier version of this work for helpful input regarding presentation.
§ APPENDIX
§.§ Packing dimension
Let $E\subseteq\R^n$. For $\delta>0$, define $\mathcal{V}_\delta(E)$ to be the collection of all countable packings of $E$ by disjoint open balls of diameter at most $\delta$. That is, for every packing $\{V_i\}_{i\in\N}\in\mathcal{V}_\delta(E)$ and every $i\in\N$, we have $V_i=B_{\ve_i}(x_i)\subseteq E$ for some $x_i\in E$ and $\ve_i\in[0,\delta/2]$.
For $s\geq0$, define
\[P_\delta^s(E)=\sup\bigg\{\sum_{i\in\N}\left|V_i\right|^s\,:\,\{V_i\}_{i\in\N}\in\mathcal{V}_\delta(E)\bigg\}\,,\]
and let
\[P_0^s(E)=\lim_{\delta\to0^+}P^s_\delta(E)\,.\]
Then the $s$-dimensional packing measure of $E$ is
\[P^s(E)=\inf\bigg\{\sum_{i\in\N} P_0^s(E_i)\,:\,E\subseteq\bigcup_{i\in\N}E_i\bigg\}\,,\]
and the packing dimension of $E$ is
\[\dim_P(E)=\inf\left\{s:P^s(E)=0\right\}\,.\]
For every set $E\subseteq\mathbb{R}^n$,
\[\dim_P(E)=\adjustlimits\min_{A\subseteq\N}\sup_{x\in E}\,\Dim^A(x)\,.\]
Let $E\subseteq\R^n$, and let $d=\dim_P(E)$. For every $s>d$ we have $P^s(E)=0$, so there is a cover $\big\{E^{s}_j\big\}_{j\in\N}$ for $E$ such that
\begin{equation}\label{eq:thm:packing:1}
\sum_{j\in\N}\lim_{\delta\to 0^+}P_\delta^s(E^{s}_j)<1\,.
\end{equation}
For every $r,j\in\N$, let
\[\big\{V_i^{r,s,j}\big\}_{i\in\N}\in\mathcal{V}_{2^{-r-2}}(E_j^s)\]
be a maximal packing of $E_j^s$ by open balls of radius exactly $2^{-r-2}$ (and higher-indexed balls of radius $0$).
Let $D=\N^3\times(\Q\cap(d,\infty))$. Our oracle $A$ encodes a function $f_A:D\to\Q^n$
such that for every $(i,j,r,s)\in D$ we have
\[f_A(i,j,r,s)\in V_i^{r,s,j}\,.\]
We will show, for every $x\in E$ and rational $s>d$, that $\Dim^A(x)\leq s$.
Let $M$ be a self-delimiting Turing machine with oracle access to $A$ such that, whenever $U(\iota)=i\in\N$, $U(\kappa)=j\in\N$, $U(\rho)=r\in\N$, and $U(\sigma)=q\in\Q\cap(d,\infty)$,
\[M(\iota\kappa\rho\sigma)=f_A(i,j,r,s)\,.\]
Fix $x\in E$ and $s\in\Q\cap(0,\infty)$, and let $k\in\N$ be such that $x\in E_{k}^{s}$.
Notice that by our choice of packing, for every $r\in\N$ there must be some $i_r\in\N$ such that
\[V_{i_r}^{r,s,k}\subseteq B_{2^{-r}}(x)\,.\]
Thus, for every $r\in\N$, letting $\iota$, $\kappa$, $\rho$, $\sigma$ testify to $K(i_r)$, $K(k)$, $K(r)$, and $K(s)$, respectively,
\begin{align*}
&\in V_{i_r}^{r,s,k}\\
&\subseteq B_{2^{-r}}(x)\,,
\end{align*}
hence $K_r^A(x)\leq K(i_r)+K(k)+K(r)+K(s)+c$, where $c$ is a machine constant for $M$.
Because $k$ and $s$ are constant in $r$, $K(r)=o(r)$, and $K(i_r)\leq\log i_r+o(r)$, we have
\[K_{r}^A(x)\leq \log i_r + o(r)\,.\]
By (<ref>), $\lim_{\delta\to 0^+}P^s_\delta(E_k^s)<1$, so there is some $R\in\N$ such that, for every $r>R$, $P^s_{2^{-r}}(E_k^s)<1$. Then for every $r>R$,
\[\sum_{i\in\N}\big|V_i^{r,s,k}\big|^s<1\,,\]
hence there are fewer than $2^{(r+2)s}$ balls of radius $2^{-r-2}$ in the packing, and $\log i_r<(r+2)s$. We conclude that $K_r^A(x)\leq rs+o(r)$ for every $r>R$, so
\[\dim^A(x)=\limsup_{r\to\infty}\frac{K_r^A(x)}{r}\leq s\,.\]
Since this holds for every rational $s>d$, we have shown $\Dim^A(x)\leq d$ and thus
\[\adjustlimits\min_{A\subseteq\N}\sup_{x\in E}\,\Dim^A(x)\leq d\,.\]
For the other direction, assume for contradiction that there is some oracle $A$ and $d^\prime<d$ such that
\[\sup_{x\in E}\,\Dim^A(x)= d^\prime\,.\]
Then for every $x\in E$, $\Dim^A(x)\leq d^\prime$. Let $s\in(d^\prime,d)$. For every $k\in\N$, define the set
\[C_k=\bigcup\left\{B_{2^{-k}}(q)\,:\,q\in\Q\textrm{ and }K^A(q)\leq ks\right\}\,,\]
and for every $i\in\N$, define
\[E_i=\bigcap_{k=i}^\infty C_k\,.\]
For $r\geq i$, consider any packing in
$\mathcal{V}_{2^{-r}}\left(E_i\right)$. Let $B_\ve(x)$ be an element of the packing, and let $k=\lceil-\log\ve\rceil$. Then $k\geq r+1>i$, so $B_\ve(x)\subseteq E_i\subseteq C_k$. In particular $x\in C_k$, meaning that there is some $q\in\Q$ such that $K^A(q)\leq ks$ and $x\in B_{2^{-k}}(q)$. As $2^{-k}\leq\ve$, we also have $q\in B_\ve(x)$. Thus, every packing element of radius at least $2^{-k}$ contains a (distinct) member of the set $\{q\in\Q:K^A(q)\leq ks\}$. It follows that for every $k\geq r+1$, the packing includes at most $2^{ks+1}$ elements with diameters in the range $[2^{1-k},2^{2-k})$.
Now let $s'\in(s,d)$. For every $i\in\N$ and $r\geq i$, we have
\begin{align*}
&\leq\sum_{k=r+1}^\infty 2^{ks+1}(2^{2-k})^{s'}\\
&=2^{1+2s'}\cdot\sum_{k=r+1}^\infty 2^{(s-s')k}\,.
\end{align*}
This approaches $0$ as $r\to\infty$, so $P_0^{s'}(E_i)=0$. Observe now that
\[E\subseteq\bigcup_{i\in\N}E_i\,.\]
\[P^{s'}(E)\leq\sum_{i\in\N}P^{s'}_0(E_i)=0\,,\]
meaning that $\dim_P(E)\leq s^\prime<d$, a contradiction. We conclude that for every oracle $A$,
\[\sup_{x\in E}\,\Dim^A(x)\geq d\,.\]
§.§ Chain rule for $K_r$
For all $x\in\R^m$, $y\in\R^n$, and $r\in\N$,
\[K_r(x,y)=K_r(x|y) + K_r(y)+o(r)\,.\]
Theorem 4.10 of <cit.> tells us that
\[I_r(x:y)=K_r(x)+K_r(y)-K_r(x,y)+o(r)\,.\]
Combining this with Lemma <ref>, we have
\[K_r(x)+K_r(y)-K_r(x,y)+o(r)=K_r(x)-K_r(x|y)+o(r)\,.\]
The theorem follows immediately.
§.§ Proof of Lemma <ref>
For each $m,n\in\N$ there is a constant $c\in\N$ such that, for all $x\in\R^m$, $y\in\R^n$, and $r,s\in\N$,
\[K_r^y(x)\leq K_{r,s}(x|y)+K(s)+c\,.\]
In particular, $K_r^y(x)\leq K_r(x|y)+K(r)+c$.
Let $m,n\in\N$, and let $U$ be the optimal Turing machine fixed for the definition of conditional Kolmogorov complexity. Let $M$ be an oracle Truing machine that, on input $\pi\in\{0,1\}^*$ with oracle $g:\N\to\Q^n$, does the following. If $\pi$ is of the form $\pi=\pi_1\pi_2$, where $U(\pi_1,\lambda)=t\in\N$, then $M$ simulates $U(\pi_2,g(t))$. Let $c$ be an optimality constant for the oracle Turing machine $M$.
To see that $c$ affirms the lemma, let $x\in\R^m$, $y\in\R^n$, and $r,s\in\N$. Let $q=y\upharpoonright (s+\log\sqrt{n})$, the truncation of the binary expansions of each of $y$'s coordinates to $s+\log\sqrt{n}$ bits to the right of the binary point. Let $\pi_s\in\{0,1\}^*$ testify to the value of $K(s)$, and let $\pi_x$ testify to the value of $\hat{K}_r(x|q)$. Then
\[q\in\Q^n\cap B_{2^{-s}}(y)\]
\[M^{y}(\pi_s\pi_x)=U(\pi_x,q)\in\Q^m\cap B_{2^{-r}}(x)\,,\]
\begin{align*}
K_r^y(x)&\leq K_{M,r}^y(x)+c\\
&\leq |\pi_s\pi_x|+c\\
&\leq K_{r,s}(x|y)+K(s)+c\,.
\end{align*}
§.§ Claims in proof of Lemma <ref>
For every $r\in\N$, $\hat{K}_r(b|m)=K_r(b|m_r)+o(r)$, where $m_r=2^{-r}\lfloor m\cdot2^r\rfloor$.
$K_r(b|m)\geq \hat{K}_r(b|m_r)$ by definition, since $m_r\in B_{2^{-r}}(m)$.
Let $\hat{b}\in B_{2^{-r}}(b)$ be such that $K(\hat{b}|m_r)=\hat{K}_r(b|m_r)$. Then
\[|(\hat{b},m_r)-(b,m)|\leq\sqrt{2}\cdot2^{-r}<2^{1-r}\,,\]
\[K(\hat{b},m_r)\geq K_{r-1}(b,m)=K_r(b,m)+o(r)\,,\]
by Corollary 3.9 of <cit.>.
Let $\mu$ testify to the value of $K_r(m)$, and let $\hat{m}=U(\mu)$. Then $|\hat{m}-m|<2^{-r}$, so $|\hat{m}-m_r|<2^{1-r}$. Thus once $\hat{m}$ and $r$ have been specified, there are at most four possible values for $m_r$. Therefore there is a self-delimiting Turing machine that takes as input $\mu$, an encoding of $r$ of length $o(r)$, and $O(1)$ additional bits and outputs $m_r$. We conclude that $K(m_r)\leq K_r(m)+o(r)$. Therefore we have
\begin{align*}
\hat{K}_r(b|m_r)&=K(\hat{b}|m_r)\\
&\geq K_r(b,m)+o(r)-(K_r(m)+o(r))+o(r)\\
\end{align*}
by Theorem <ref>.
For each $x\in[0,1]$ and $r\in\mathbb{N}$, there exists an $h\in\mathbb{N}$ such that $M$ halts on input $(\rho\pi_r\sigma_r\eta)$ with $M(\rho\pi_r\sigma_r\eta)\in B_{2^{1-r}}(m,b,x)$, where $U(\rho)=r$ and $U(\eta)=h$.
Fix $x\in[0,1]$ and $r\in\mathbb{N}$. It is clear that for some $j\in\{0,1,\ldots,2^r\}$, $|u_j-m|<2^{-r}$. By the definition of $K_r(b|m)$, $u_j\in\mathbb{Q}\cap B_{2^{-r}}(m)$ implies that $U(\pi_r,u_j)$ halts and outputs $v_j\in\mathbb{Q}\cap B_{2^{-r}}(b)$. $U(\sigma_r)\in B_{2^{-r}}(x,mx+b)$ by the definition of $\sigma_r$, so $|p-x|<2^{-r}$. It follows that
\[|(u_j,v_j,p)-(m,x,b)|<\sqrt{3(2^{-r})^2}<2^{1-r}\,.\]
It remains to show that $|u_ip+v_j-q|<2^{2-r}$. To do so, we repeatedly apply the triangle inequality and use the fact that $x,m\in[0,1]$:
\begin{align*}
\end{align*}
For almost every $x\in[0,1]$, $\log(h(x,r))=o(r)$.
By the countable additivity of Lebesgue measure, it suffices to show for every $k\in\N$ that the set
\[D_k=\left\{x\in[0,1]:\exists\textrm{ infinitely many }r\in\N\textrm{ such that }\log(h(x,r))>r/k\right\}\]
has Lebesgue measure $0$. For each $r\in\mathbb{N}$, let $D_{k,r}=\{x:h(x,r)>2^{r/k}\}$. We now estimate $\lambda(D_{k,r})$, the Lebesgue measure of $D_{k,r}$.
For fixed $x$ and $r$, the algorithm run by the Turing machine $M$ entails
\[h(x,r)\leq\left|\left\{i:|u_ip+v_i-q|<2^{2-r}\right\}\right|\,.\]
For fixed $i$,
\begin{align*}
\end{align*}
That is,
\[\left\{i:|u_ip+v_i-q|<2^{2-r}\right\}\subseteq\left\{i:|u_ix+v_i-(mx+b)|-2^{1-r}<2^{2-r}\right\}\,,\]
\[h(x,r)\leq\left|\left\{i:|u_ix+v_i-(mx+b)|<2^{3-r}\right\}\right|\,.\]
For fixed $r$ and $i=0,1,\ldots,2^r$, define
\[C^r_i=\{x\in[0,1]:|u_ix+v_i-(mx+b)|<2^{3-r}\}\,,\]
For each $i$, if $m=u_i$, then $C^r_i$ is either $[0,1]$ or empty; otherwise, $C^r_i$ is an interval of length
\[\lambda(C^r_i)\leq\min\left\{\frac{2^{3-r}}{|u_i-m|},1\right\}\,.\]
Notice that for each $k=0,\ldots,2^r$, there are at most 2 values of $i$ for which
so we have
\begin{align*}
\int_0^1 h(x,r)dx&\leq\sum\limits_{i=0}^{2^r}\lambda(C^r_i)\\
&\leq 2+\sum\limits_{k=1}^{2^r}2\frac{2^{3-r}}{2^{-r}k}\\
\end{align*}
Thus, as $h(x,r)>2^{r/k}$ for all $x\in D_{k,r}$,
\[\lambda(D_{k,r})<\frac{r2^6}{2^{r/k}}=r2^{6-r/k}\,.\]
This implies that
\[\sum_{r=1}^\infty\lambda(D_{k,r})<\infty\,,\]
so the Borel-Cantelli Lemma tells us that $\lambda(D_k)=0$.
For every $x\in[0,1]$, if $\log(h(x,r))=o(r)$, then
\[\liminf_{r\to\infty}\frac{K_r(m,b,x)-K_r(b|m)}{r}\leq\dim(x,mx+b)\,.\]
For fixed $r$, Claim <ref> gives
\[K_{r-1}(m,b,x)\leq K(u_i,v_i,p)\leq K_M(u_i,v_i,p)+c_M\,,\]
where $c_M$ is an optimality constant for $M$. Let $\rho$ and $\eta$ testify to the values of $K(r)$ and $K(h(x,r))$, respectively. Then $K_M(u_i,v_i,p)\leq|\rho\pi_r\sigma_r\eta|$. By our choices of $\rho, \pi_r, \sigma_r$, and $\eta$,
\begin{align*}
\end{align*}
by Claim <ref>.
By Corollary 3.9 of <cit.>,
\begin{align*}
\end{align*}
Applying Observation <ref>, for some constant $c$,
\begin{align*}
\limsup_{r\to\infty}\frac{K(h(x,r))}{r}&\leq\limsup_{r\to\infty}\frac{\log(1+h(x,r))+2\log\log(2+h(x,r))+c}{r}\\
\end{align*}
If $\log(h(x,r))={o(r)}$, then this is
\[\limsup_{r\to\infty}\frac{o(r)+2\log(o(r))}{r}=0\,.\]
§.§ Observations about Kolomogorov Complexity in
Euclidean Space
For every open ball $B\subseteq\R^m$ of radius $2^{-r}$,
\[B\cap2^{-\left(r+\left\lfloor\frac{1}{2}\log m\right\rfloor+1\right)}\Z^m\neq\emptyset\,.\]
For $a\in\Z^m$, let $|a|$ denote the distance from the origin to $a$.
There is a constant $c_0\in\N$ such that, for all $j\in\N$,
\[K(j)\leq\log(1+j)+2\log\log(2+j)+c_0\,.\]
There is a constant $c\in\N$ such that, for all $a\in\Z^m$,
\[K(a)\leq m\log(1+|a|)+\varepsilon(|a|)\,,\]
where $\varepsilon(t)=c+2\log\log(2+t)$.
Observation <ref> holds by a routine technique <cit.>. The proof of Observation <ref> is also routine:
Fix a computable, nonrepeating enumeration $a_0,a_1,a_2,\ldots$ of $\Z^m$ in which tuples $a_j$ appear in nondecreasing order of $|a_j|$. Let $M$ be a Turing machine such that, for all $\pi\in\{0,1\}^*$, if $U(\pi)\in\N$, then $M(\pi)=a_{U(\pi)}$. Let $c=c_0+c_M+m+\lceil2\log m\rceil+2$, where $c_0$ is as in Observation <ref> and $c_M$ is an optimality constant for $M$.
To see that $c$ affirms Observation <ref>, let $a\in\Z^m$. Let $j\in\N$ be the index for which $a_j=a$, and let $\pi\in\{0,1\}^*$ testify to the value of $K(j)$. Then $M(\pi)=a_{U(\pi)}=a_j=a$, so
\[K(a)\leq K_M(a)+c_M\leq |\pi|+c_M=K(j)+c_M\,.\]
It follows by Observation <ref> that
\begin{equation}\label{eq:A}
\end{equation}
We thus estimate $j$.
Let $B$ be the closed ball of radius $|a|$ centered at the origin in $\Z^m$, and let $Q$ be the solid, axis-parallel $m$-cube circumscribed about $B$. Let $B^\prime=B\cap\Z^M$ and $Q^\prime=Q\cap\Z^m$. Then
\[j\leq|B^\prime|-1\leq|Q^\prime|-1\leq(2|a|+1)^m-1\,,\]
so (<ref>) tells us that
\begin{align*}
K(a)&\leq m\log(2|a|+1)+2\log\log(1+(2|a|+1)^m)+c+c_M\\
&\leq m\log(2|a|+2)+2\log(m\log(2|a|+4))+c+c_M\,.
\end{align*}
\[m\log(2|a|+2)=m+m\log(1+|a|)\]
\begin{align*}
\log(m\log(2|a|+4))&=\log m+\log(1+\log(2+|a|))\\
&\leq\log m+1+\log\log(2+|a|)\,,
\end{align*}
it follows that $K(a)\leq m\log(1+|a|)+\varepsilon(|a|)$.
For every $r,n\in\N$, $x\in\R^n$, and $q\in\Q^n$,
\[K_r(x+q)=K_r(x)+O(1)\,.\]
Let $M$ be a self-delimiting Turing machine such that $M(\pi\kappa)=U(\pi)+U(\kappa)$ whenever $U(\pi),U(\kappa)\in\Q^n$. If $\pi$ is a witness to $K_r(x)$ and $\kappa$ is a witness to $q$, then $M(\pi\kappa)=p+q$ for some $p\in B_{2^{-r}}(x)$, so $M(\pi\kappa)\in B_{2^{-r}}(x+q)$. Thus
\[K_r(x+q)\leq K_r(x)+K(q)+c\,,\]
where $c$ is a machine constant for $M$. Since $K(q)$ is constant in $r$, we have $K_r(x+q)\leq K_r(x)+O(1)$. Applying the same argument with $-q$ replacing $q$ completes the proof.
§.§ Linear Sensitivity of $\hat{K}_r(x|q)$ to $r$
There is a constant $c_1\in\N$ such that, for all $x\in\R^m$, $q\in\Q^n$, and $r,\Delta r\in\N$,
\[\hat{K}_r(x|q)\leq \hat{K}_{r+\Delta r}(x|q)\leq \hat{K}_r(x|q)+m\Delta r+\varepsilon_1(r,\Delta r)\,,\]
where $\varepsilon_1(r,\Delta r)=2\log(1+\Delta r)+K(r,\Delta r)+c_1$.
Let $M$ be a Turing machine such that, for all $\pi_1,\pi_2,\pi_3\in\{0,1\}^*$ and $q\in\Q^n$, if $U(\pi_1,q)=p\in\Q^m$, $U(\pi_2)=(r,\Delta r)\in\N^2$, and $U(\pi_3)=a\in\Z^m$, then $M(\pi_1\pi_2\pi_3,q)=p+2^{-r^*}a$, where $r^*=r+\Delta r+\left\lfloor\tfrac{1}{2}\log m\right\rfloor+1$. Let $c_1=c+c_M+3m+m\left\lfloor\tfrac{1}{2}\log m\right\rfloor+\left\lceil 2\log(3+\left\lfloor\tfrac{1}{2}\log m\right\rfloor)\right\rceil$, where $c$ is the constant from Observation <ref> and $c_M$ is an optimality constant for $M$.
To see that $c_1$ affirms the lemma, let $x$, $q$, $r$, and $\Delta r$ be as given. The first inequality holds trivially. To see that the second inequality holds, let $\pi_1,\pi_2\in\{0,1\}^*$ testify to the values of $\hat{K}_r(x|q)$ and $K(r,\Delta r)$, respectively. Let $B=B_{2^{-r}}(x)$, $B^\prime=B_{2^{-(r+\Delta r)}}(x)$ and $p=U(\pi_1,q)$, noting that $p\in\Q^m\cap B$. Applying Observation <ref> to the ball $B^\prime-p$ tells us that
\[(B^\prime-p)\cap2^{-r^*}\Z^m\neq\emptyset\,,\]
i.e., that
\[B^\prime\cap(p+2^{-r^*}\Z^m)\neq\emptyset\,.\]
So fix a point $p^\prime\in B^\prime\cap(p+2^{-r^*}\Z^m)$, say, $p^\prime=p+2^{-r^*}a$, where $a\in\Z^m$, and let $\pi_3\in\{0,1\}^*$ testify to the value of $K(a)$. Then
\[M(\pi_1\pi_2\pi_3,q)=p^\prime\in\Q\cap B^\prime\,,\]
\begin{align*}
\hat{K}_{r+\Delta r}(x|q)&\leq K(p^\prime|q)\\
&\leq \hat{K}_M(p^\prime|q)+c_M\\
&\leq |\pi_1\pi_2\pi_3|+c_M\,.
\end{align*}
By our choice of $\pi_1$, $\pi_2$, and $\pi_3$, this implies that
\begin{equation}\label{eq:B}
\hat{K}_{r+\Delta r}(x|q)\leq \hat{K}_r(x|q)+K(r,\Delta r)+K(a)+c_M\,.
\end{equation}
We thus estimate $K(a)$.
\begin{align*}
|a|&= 2^{r^*}|p^\prime-p|\\
&\leq 2^{r^*}(|p^\prime-x|+|p-x|)\\
&< 2^{r^*}\left(2^{-(r+\Delta r)}+2^{-r}\right)\\
&= 2^{1+\left\lfloor\frac{1}{2}\log m\right\rfloor}\left(1+2^{\Delta r}\right)\,,
\end{align*}
Observation <ref> tells us that
\begin{align*}
K(a)&\leq m\log\left(1+2^{\left\lfloor\frac{1}{2}\log m\right\rfloor}\left(1+2^{\Delta r}\right)\right)+\varepsilon(|a|)\\
&\leq m\log\left(2^{\Delta r+3+\left\lfloor\frac{1}{2}\log m\right\rfloor}\right)+\varepsilon(|a|)\,,
\end{align*}
i.e., that
\begin{equation}\label{eq:C}
K(a)\leq m\Delta r+3m+m\left\lfloor\tfrac{1}{2}\log m\right\rfloor+\varepsilon(|a|)\,,
\end{equation}
\begin{align*}
\varepsilon(|a|)&\leq c+2\log\log\left(2+2^{1+\left\lfloor\frac{1}{2}\log m\right\rfloor}\left(1+2^{\Delta r}\right)\right)\\
&\leq c+2\log\log\left(2^{\Delta r+3+\left\lfloor\frac{1}{2}\log m\right\rfloor}\right)\\
&= c+2\log\left(\Delta r+\left\lfloor\tfrac{1}{2}\log m\right\rfloor+3\right)\\
&\leq c+2\log\left(\left(1+\Delta r\right)\left(3+\left\lfloor\tfrac{1}{2}\log m\right\rfloor\right)\right)\\
&= c+2\log(1+\Delta r)+2\log\left(3+\left\lfloor\tfrac{1}{2}\log m\right\rfloor\right)\,.
\end{align*}
It follows by (<ref>) and (<ref>) that
\[\hat{K}_{r+\Delta r}(x|q)\leq \hat{K}_r(x|q)+m\Delta r+\varepsilon_1(r,\Delta r)\,.\]
§.§ Linear Sensitivity of $K_{r,s}(x|y)$ to $s$
There is a constant $c_2\in\N$ such that, for all $x\in\R^m$, $y\in\R^n$, and $r,s,\Delta s\in\N$,
\[K_{r,s}(x|y)\geq K_{r,s+\Delta s}(x|y)\geq K_{r,s}(x|y)-n\Delta s-\varepsilon_2(s,\Delta s)\,,\]
where $\varepsilon_2(s,\Delta s)=2\log(1+\Delta s)+K(s,\Delta s)+c_2$.
Let $M$ be a Turing machine such that, for all $\pi_1,\pi_2,\pi_3\in\{0,1\}^*$ and $q\in\Q^n$, if $U(\pi_1)=(s,\Delta s)\in\N^2$ and $U(\pi_2)=a\in\Z^m$, then $M(\pi_1\pi_2\pi_3,q)=U(\pi_3,q+2^{-s^*}a)$, where $s^*=s+\Delta s+\left\lceil\tfrac{1}{2}\log n\right\rceil$. Let $c_2=c+c_M+3n+n\left\lfloor\tfrac{1}{2}\log n\right\rfloor+2\left\lceil2\log(3+\left\lfloor\tfrac{1}{2}\log n\right\rfloor)\right\rceil$, where $c$ is the constant from Observation <ref> and $c_M$ is an optimality constant for $M$.
To see that $c_2$ affirms the lemma, let $x$, $y$, $r$, $s$, and $\Delta s$ be as given The first inequality holds trivially. To see that the second inequality holds, let $B=B_{2^-s}(y)$, $B^\prime=B_{2^{-(s+\Delta s)}}(y)$, and $q\in\Q^n\cap B$. It suffices to prove that
\begin{equation}\label{eq:D}
\hat{K}_r(x|q)\leq K_{r,s+\Delta s}(x|y)+n\Delta s+\varepsilon_2(s,\Delta s)\,.
\end{equation}
Let $\pi_1\in\{0,1\}^*$ testify to the value of $K(s,\Delta s)$. Applying Observation <ref> to the ball $B^\prime-q$ tells us that
\[(B^\prime-q)\cap2^{-s^*}\Z^n\neq\emptyset\,,\]
i.e., that
\[B^\prime\cap(q+2^{-s^*}\Z^n)\neq\emptyset\,.\]
So fix a point $q^\prime\in B^\prime\cap(q+2^{-s^*}\Z^n)$, say, $q^\prime=q+2^{-s^*}a$, where $a\in\Z^n$. Note that
\begin{equation}\label{eq:E}
\hat{K}_r(x|q^\prime)\leq K_{r,s+\Delta s}(x|y).
\end{equation}
Let $\pi_2,\pi_3\in\{0,1\}^*$ testify to the values of $K(a)$ and $\hat{K}_r(x|q^\prime)$, respectively, noting that $U(\pi_3,q^\prime)=p$ for some $p\in\Q^m\cap B_{2^{-r}}(x)$. Then
\[M(\pi_1\pi_2\pi_3,q)=U(\pi_3,q^\prime)=p\in\Q^m\cap B_{2^{-r}}(x)\,,\]
\begin{align*}
\hat{K}_r(x|q)&\leq K(p|q)\\
&\leq K_M(p|q)+c_M\\
&\leq |\pi_1\pi_2\pi_3|+c_M\,.
\end{align*}
By our choice of $\pi_1$, $\pi_2$, and $\pi_3$, and by (<ref>), this implies that
\begin{equation}\label{eq:F}
\hat{K}_r(x|q)\leq K_{r,s+\Delta s}(x|y)+K(a)+K(s,\Delta s)+c_M\,.
\end{equation}
We thus estimate K(a).
\begin{align*}
|a|&= 2^{s^*}|q^\prime-q|\\
&\leq 2^{s^*}(|q^\prime-y|+|q-y|)\\
&< 2^{s^*}(s^{-(s+\Delta s)}+2^{-s})\\
&= 2^{1+\left\lfloor\frac{1}{2}\log n\right\rfloor}(1+2^{\Delta s})\,,
\end{align*}
Observation <ref> tells us that
\begin{align*}
K(a)&\leq n\log(1+2^{1+\left\lfloor\frac{1}{2}\log n\right\rfloor}(1+2^{\Delta s}))+\varepsilon(|a|)\\
&\leq n\log(2^{\Delta s+3+\left\lfloor\frac{1}{2}\log n\right\rfloor})+\varepsilon(|a|)\,,
\end{align*}
i.e., that
\begin{equation}\label{eq:G}
K(a)\leq n\Delta s+3n+n\left\lfloor\tfrac{1}{2}\log n\right\rfloor+\varepsilon(|a|),
\end{equation}
\begin{align*}
\varepsilon(|a|)&\leq c+2\log\log(2+2^{1+\left\lfloor\frac{1}{2}\log n\right\rfloor}(1+2^{\Delta s}))\\
&\leq c+2\log\log(2^{\Delta s+3+\left\lfloor\frac{1}{2}\log n\right\rfloor})\\
&= c+2\log(\Delta s+3+\left\lfloor\tfrac{1}{2}\log n\right\rfloor)\\
&\leq c+2\log\big((1+\Delta s)(3+\left\lfloor\tfrac{1}{2}\log n\right\rfloor)\big)\\
&= c+2\log(1+\Delta s)+2\log(3+\left\lfloor\tfrac{1}{2}\log n\right\rfloor)\,.
\end{align*}
It follows by (<ref>) and (<ref>) that (<ref>) holds.
§.§ Proof of Theorem <ref>
Let $s:\N\to\N$. If $|s(r)-r|=o(r)$, then, for all $x\in\R^m$ and $y\in\R^n$,
\[\dim(x|y)=\liminf_{r\to\infty}\frac{K_{r,s(r)}(x|y)}{r}\,,\]
\[\Dim(x|y)=\limsup_{r\to\infty}\frac{K_{r,s(r)}(x|y)}{r}\,.\]
Assume the hypothesis. Define $s^-,s^+:\N\to\N$ by
\[s^-(r)=\min\{r,s(r)\},\;s^+(r)=\max\{r,s(r)\}\,.\]
Lemma <ref> tells us that, for all $x\in\R^m$ and $y\in\R^n$,
\begin{align*}
K_{r,s^-(r)}(x|y)&\geq K_{r,r}(x|y)\\
&\geq K_{r,s^+(r)}(x|y)\\
&\geq K_{r,s^-(r)}(x|y)-O(s^+(r)-s^-(r))-o(r)\\
&= K_{r,s^-(r)}(x|y)-O(|s(r)-r|)-o(r)\\
&= K_{r,s^-(r)}(x|y)-o(r)\,.
\end{align*}
\[K_{r,s^-(r)}(x|y)\geq K_{r,s(r)}(x|y)\geq K_{r,s^+(r)}(x|y)\,,\]
it follows that
\[\big|K_{r,s(r)}(x|y)-K_{r,r}(x|y)\big|=o(r)\,.\]
The theorem follows immediately.
§.§ Proof of Lemma <ref>
For all $x\in\R^m$, $y\in\R^n$, and $r\in\N$,
\[I_r(x:y)=K_r(x)-K_r(x|y)+o(r)\,.\]
Let $B_x=B_{2^{-r}}(x)\cap\Q^m$ and $B_y= B_{2^{-r}}(y)\cap\Q^n$. Let $p_0$ and $q_0$ be $K$-minimizers for $B_x$ and $B_y$, respectively, such that
\begin{equation}\label{eq:MI1}
\end{equation}
These exist by Theorem 4.6 of <cit.>. Then
\begin{align*}
K_r(x)-K_r(x|y)&=\min_{p\in B_x}K(p)-\max_{q\in B_y}\min_{p\in B_x}K(p|q)\\
&\geq\min_{p\in B_x}K(p)-\min_{p\in B_x}\max_{q\in B_y}K(p|q)\\
&=\min_{p\in B_x}K(p)-\min_{p\in B_x}K(p|q_0)+o(r)\,,
\intertext{by Lemma 4.2 and Observation 3.7 of~\cite{CasLut15}.}
&=K(p_0)-\min_{p\in B_x}K(p|q_0)+o(r)\\
&\geq K(p_0)-K(p_0|q_0)+o(r)\\
\end{align*}
For the other direction, let $p_1\in B_x$ be such that
\[K(p_1|q_0)=\min_{p\in B_x}K(p|q_0)\,.\]
By Lemma 4.5 of <cit.>,
\begin{align*}
I(p_0:q_0)&\geq K(p_1)-K(p_1|p_0,K(p_0))-K(p_1|q_0,K(q_0))+o(r)\\
&\geq K(p_1)-K(p_1|p_0,K(p_0))-K(p_1|q_0)+o(r)\numberthis\label{eq:MI2}\,.
\end{align*}
\begin{align*}
&\leq K(p_1)+K(p_0|p_1)+o(r)\\
\end{align*}
by Corollary 4.4 of <cit.>. So
\[K(p_1)-K(p_1|p_0,K(p_0))\geq K(p_o)+o(r)\,,\]
thus by (<ref>),
\begin{align*}
I(p_0:q_0)&\geq K(p_0)-K(p_1|q_0)+o(r)\\
&=K(p_0)-\min_{p\in B_x}K(p|q_0)+o(r)\\
&=K_r(x)-\min_{p\in B_x}K(p|q_0)+o(r)\\
&\geq K_r(x)-\max_{q\in B_y}\min_{p\in B_x}K(p|q)+o(r)\\
\end{align*}
Then by (<ref>), $I_r(x:y)\geq K_r(x)-K_r(x|y)+o(r)$, so equality holds.
|
1511.00301
|
Predicted Higgs-related spin 1/2 particles as a new
dark matter candidate
Joshua Stenzel, Johannes Kroll, Minjie Lei, and Roland E. Allen
The theory at arXiv:1101.0586 [hep-th] predicts new fundamental spin $1/2$ particles which can be produced in pairs through their couplings to vector bosons or fermions. The lowest-energy of these should have a mass $m_{1/2}$ comparable to the mass $m_h$ of the recently discovered Higgs boson, with $m_{1/2} = m_h$ in the simplest model. These particles should therefore be detectable in collider experiments, perhaps in Run 2 or 3 of the LHC. They cannot decay through any obvious mechanisms in standard physics, making them a new dark matter candidate. In the simplest model, annihilations would produce a well-defined signature with photons, positrons, and excess electrons at about 125 GeV, and the mass would also be well-defined for direct dark matter detection.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
The particle discovered by the ATLAS and CMS collaborations at the LHC is almost certainly a Higgs boson <cit.>. After the electron was discovered in 1897, and the photon was introduced by Einstein in 1905, the richness of behavior associated with spin 1/2 fermions and spin 1 gauge bosons emerged
slowly during the following decades. More than a century later, the third kind of Standard Model particle, with spin 0, has finally been discovered, and one should not be completely surprised if some of its implications are
yet to be determined.
In an earlier paper <cit.>, based on a novel fundamental picture, the following
action for spin 1/2 fermions and scalar bosons was obtained as a low energy
\[
\hspace{-0.2cm}S_{f}+S_{sb}=\int d^{4}x\,\bigg[\psi _{f}^{\dagger }\left(
x\right) \,ie_{\alpha }^{\mu }\,\sigma ^{\alpha }{D}_{\mu }\,\psi _{f}\left(
x\right) -g^{\mu \nu }\left( {D}_{\mu }\phi _{b}\left( x\right) \right)
^{\dagger }{D}_{\nu }\phi _{b}\left( x\right) +F_{b}^{\dagger }\left(
x\right) F_{b}\left( x\right) \bigg]
\]
\[
g^{\mu \nu }=\eta ^{\alpha \beta }e_{\alpha }^{\mu }e_{\beta }^{\nu }\;.
\]
Here $e_{\alpha }^{\mu }$ is the vierbein and $\eta ^{\alpha \beta }$ is the
Minkowski metric tensor in the $(-1,1,1,1)$ convention. The familiar form
for a Lorentz-invariant and supersymmetric action was thus found to follow
automatically from a picture that is intially quite unfamilar. The spin 1/2
fermion fields in $\psi _{f}$, the scalar boson fields in $\phi _{b}$, and
the auxiliary fields in $F_{b}$ span the various physical representations of
the fundamental gauge group, which must be $SO(N)$ in the present theory.
(More precisely, the group is $Spin(N)$, but $SO(N)$ is conventional
terminology.) One unfamiliar feature remains: There is no factor of $e=\left(
-\det g_{\mu \nu }\right) ^{1/2}$ in the integral, and this is related to
the more general fact that the usual cosmological constant vanishes in the
theory of Ref. <cit.>. (The fields $\psi_{f}$, $\phi _{b}$, and $F_{b}$ are required to transform under general coordinate transformations as scalars with weight $1/2$ rather than $0$. Under a Lorentz transformation in the tangent space, they transform in the usual way, as ordinary spinors and scalars respectively.)
At higher energies, including those currently achieved at the LHC, the
theory implies that the above form for the action is no longer a valid
approximation, because internal degrees of freedom can be excited in a
4-component field
\[
\Phi _{b}=\left(
\begin{array}{c}
\Phi \\
\Phi _{c}^{\dag }
\end{array}
\right) \; .
\]
This can be written as the inner product of two $N_{g}$-component fields
$\phi _{b}$ and $\chi _{b}$, where each component of $\phi _{b}$ is a complex
scalar and each component of $\chi _{b}$ is a 4-component bispinor (and
where $N_{g}$ is the number of fields spanning all the physical gauge representations):
\begin{equation}
\Phi _{b}=\phi _{b}\chi _{b}=\phi _{b}^{r}\chi _{b}^{r}
\end{equation}
with the usual summation over the repeated index $r$. The amplitude of each
component $\Phi _{b}^{r}$ is given by $\phi _{b}^{r}$, and the
“spin configuration” by $\chi _{b}^{r}$.
If $\chi _{b}$ is constant, it is convenient to choose the normalization
\begin{equation}
\chi _{b}^{r\,\dag }\chi _{b}^{r}=1\quad \mathrm{[no\;sum\;on\;}r\mathrm{]\;.
\end{equation}
The more general form of the Lagrangian corresponding to scalar bosons includes the internal
degrees of freedom which are “hidden” at low energy. In a locally inertial frame of reference it is
\begin{equation}
\mathcal{L}_{\Phi }=\Phi _{b}^{\dag }\left( x\right) D^{\mu }D_{\mu }\Phi
_{b}\left( x\right) -\frac{1}{2}\left[ \Phi _{b}^{\dag }\left( x\right)
\,S^{\mu \nu }F_{\mu \nu }\,\Phi _{b}\left( x\right) +h.c.\right]
\end{equation}
where $F_{\mu \nu }$ is the field strength tensor and the $S^{\mu \nu
}=\sigma ^{\mu \nu }/2$ are the Lorentz generators which act on Dirac
spinors. When the second term above is written out explicitly, it involves
$\phi _{b}^{r\,\dag }\phi _{b}^{r^{\prime }}\chi _{b}^{r\,\dag }\sigma
^{k}\chi _{b}^{r^{\prime }}$ interacting with the “magnetic” field strengths in $F_{\mu \nu }$ (and is thus
analogous to the interaction of an electron spin with a magnetic field).
Some experimental implications are discussed in Appendix E of Ref. <cit.>. In
particular, the theory predicts new fundamental spin 1/2 particles which can
be produced in pairs through their couplings to vector bosons or fermions.
The lowest-energy of these should have a mass $m_{1/2}$ comparable to the
mass $m_{h}$ of the recently discovered Higgs boson, with $m_{1/2}=m_{h}$ in
the simplest model.
There are two unconventional features in the Lagrangian $\mathcal{L}_{\Phi }$: Each field
$\Phi _{b}^{r}$ has four components rather than one, and there is a second
term involving the gauge field strengths $F_{\mu \nu }$. One can read off
the general Feynman-diagram vertices for virtual and real processes from the
interactions in each term. These are relevant for all the $\Phi _{b}^{r}$
that correspond to scalar boson fields in standard physics, but let us now
focus on the one $\Phi _{h}$ that corresponds to a single neutral Higgs
The vacuum expectation value of $\Phi _{h}$ has the form
\[
\left\langle \Phi _{h}^{0}\right\rangle =\frac{v}{\sqrt{2}}\left(
\begin{array}{c}
1 \\
0 \\
0 \\
\end{array}
\right) \;.\;
\]
In Ref. <cit.> it is shown that the condensate then has zero angular momentum and
also no coupling to the gauge fields beyond that in the Standard Model
(since the second term in $\mathcal{L}_{\Phi }$ vanishes when the internal
degrees of freedom in $\Phi _{h}$ are not excited).
The simplest model for excitations of $\Phi _{h}$ has a mass term Lagrangian
\[
\mathcal{L}_{h}^{\mathrm{mass}}=m_{h}^{2}\,\left( \Delta \Phi _{h}\right)
^{\dag }\Delta \Phi _{h}\;.
\]
When the internal degrees of freedom are not excited, so that $\Delta \Phi
_{h}=h\chi _{0}$ with $\chi _{0}^{\dag }\chi _{0}=1$, the mass term is
$m_{h}^{2}\,h^{2}$ (for $h$ real). I.e., $m_{h}$ is the mass of the scalar Higgs boson.
For a spin 1/2 excitation with the form
\[
\Delta \Phi _{h}=\left(
\begin{array}{c}
h_{+} \\
\end{array}
\right) \quad \mathrm{or}\quad \Delta \Phi _{h}=\left(
\begin{array}{c}
0 \\
\end{array}
\right)
\]
we obtain
\[
\mathcal{L}_{+}^{\mathrm{mass}}=m_{h}^{2}\,h_{+}^{\dag }h_{+}\quad
\mathrm{or}\quad \mathcal{L}_{-}^{\mathrm{mass}}=m_{h}^{2}\,h_{-}^{\dag }h_{-}\;.
\]
In other words, in the simplest model the spin 1/2 particles $h_{+}$ and
$h_{-}$ have the same mass $m_{h}$ as the scalar Higgs boson $h$. More
generally, the masses $m_{1/2}$ of these particles should be comparable to
$m_{h}$. A suggestive
analogy is s-wave superconductivity, where there are single-particle excitations, two-particle excitations, and “Higgs
mode” excitations with minimum energies $\Delta $, $2\Delta $, and $2\Delta $ respectively.
According to the spin-statistics theorem, spin $1/2$ bosonic excitations are
impossible, but the requirements of this theorem are not satisfied in this
one specific context, since $\mathcal{L}_{\Phi }$ is not fully Lorentz
invariant: It is invariant under a rotation, but not a Lorentz boost with
respect to the original (cosmological) coordinate system. The present theory
is, however, fully Lorentz invariant (as well as initially supersymmetric)
if the internal degrees of freedom in $\Phi _{b}$ are not excited – and
these excitations can be observed only at the high energies that are now
becoming available. Furthermore, the extremely weak virtual effects of these
excitations are irrelevant to the many existing tests of Lorentz invariance,
which probe those phenomena in various areas of physics and astrophysics
where the present theory is fully Lorentz invariant.
The spin $1/2$ excitations of $\Phi _{b}$ can
be produced in pairs through the coupling to gauge boson fields in $\mathcal{L}_{\Phi }$ – for
example, by the coupling to virtual or real Z and W bosons. In addition, the
Higgs-related spin $1/2$ particles should have the same basic Yukawa
couplings to fermions as a Higgs boson, since $\Phi _{h}=\phi _{h}\chi _{h}$.
Once a lowest-mass particle of this kind has left the region where it was created, it is unable to decay through Standard-Model mechanisms without
violating lepton number or baryon number conservation, since the net decay
products must have angular momentum $1/2$. This implies that these (weakly-interacting)
particles are dark matter candidates.
In the simplest model described above, annihilations would produce a well-defined signature with photons, positrons, and excess electrons at about 125 GeV, and the mass would also be well-defined for direct dark matter detection.
ATLAS ATLAS Collaboration, “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC”, Phys. Lett. B716, 1 (2012), arXiv:1207.7214 [hep-ex].
CMS CMS Collaboration, “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC”, Phys. Lett. B716, 30 (2012), arXiv:1207.7235 [hep-ex].
pdg See e.g. K. A. Olive et al. (Particle Data Group), “The Review of Particle Physics”, Chin. Phys. C 38, 090001 (2014), updated at http://pdg.lbl.gov/.
allen-2015 R. E. Allen, arXiv:1101.0586 [hep-th].
|
1511.00121
|
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
A-1090 Vienna, Austria
[2010]35Q70, 35Q41, 42C40, 81S10, 81S30
We consider Hamiltonian deformations of Gabor systems, where the window
evolves according to the action of a Schrödinger propagator and the
phase-space nodes evolve according to the corresponding Hamiltonian flow. We
prove the stability of the frame property for small times and Hamiltonians
consisting of a quadratic polynomial plus a potential in the Sjöstrand
class with bounded second order derivatives. This answers a question raised
in [de Gosson, M. Symplectic and Hamiltonian Deformations of Gabor Frames.
Appl. Comput. Harmon. Anal. Vol. 38 No.2, (2015) p.196–221.]
§ INTRODUCTION
Let $\mathcal{H}(x,p)$ be a Hamiltonian on ${\mathbb{R}^{2d}}$ and
$H:=\mathcal{H}^w$ its Weyl quantization. The solution to the Schrödinger
\begin{align*}
&i \partial_t u(t,\cdot) = H u(t,\cdot), \qquad t\in {\mathbb{R}}, \\
\end{align*}
is given by the propagation formula $u(t,\cdot)=e^{-i t H}f$. The model case
is the one of a real quadratic (homogeneous) Hamiltonian: $\mathcal{H}(x,p)
= \left<M (x,p),(x,p)\right> $, with $M \in {\mathbb{R}}^{2d\times 2d}$
symmetric. In this case, the evolution operator $e^{-i t H}$ and the
(symmetric) time-frequency shift operators
\begin{align} \label{eq_tfs}
\rho(z) f := e^{-\pi i x\xi} e^{2 \pi i \xi} f(\cdot-x), \qquad z=(x,\xi)
\in {\mathbb{R}^{2d}},
\end{align}
satisfy the symplectic covariance relation
\begin{align} \label{eq_cov}
e^{-itH} \rho(z) = \rho(e^{2tJM} z) e^{-itH},
\end{align}
where $J=\big(
\begin{smallmatrix}
0 & I \\ -I & 0
\end{smallmatrix}\big)
$ is the standard symplectic form (see for example <cit.>). Thanks to (<ref>), the action of
the evolution operator on a state
$f$, can be understood by considering an expansion into coherent states:
\begin{align} \label{eq_exp}
f = \sum_{\lambda \in \Lambda} c_\lambda \rho(\lambda) g,
\end{align}
where $g$ is a smooth, fast decaying function, $\Lambda \subseteq {\mathbb{R}^d}$ is a set of phase-space nodes and
$c_\lambda \in {\mathbb{C}}$. Such an
expansion is a discrete version of the continuous coherent state
representation <cit.>, and the canonical choice for $g$ is a
Gaussian function. The evolution generated by the quadratic Hamiltonian $\mathcal{H}$ is then given by
\begin{align} \label{eq_ev}
e^{-i t H} f = \sum_{\lambda \in \Lambda} c_\lambda \rho(e^{2tJM} \lambda)
e^{-i t H} g,
\end{align}
and therefore the description of the evolution of an arbitrary state $f$ is
reduced to the one of $g$. If $\mathcal{H}(x,p) = x^2 + p^2$ is the
harmonic oscillator and $g$ is chosen to
be an adequate Gaussian function, then $e^{-i t H}g=g$, and (<ref>)
amounts to a rearrangement of the time-frequency content of $f$. The case of
higher order Hermite functions is also important since these correspond to
higher energy Landau levels (see <cit.>).
The collection of coherent states
\begin{align*}
\mathcal{G}(g, \Lambda) := \big\{ \, \rho(\lambda)g \, : \, \lambda \in
\Lambda \, \big\}
\end{align*}
is called a Gabor system, and it is a Gabor frame if every $f \in L^2({\mathbb{R}^d})$ admits an expansion as in
(<ref>) with $\lVert
c\rVert_2 \asymp \lVert f\rVert_2$. In this case, several properties of $f$
can be read from the coefficients $c$. The theory of Gabor frames - also
called Weyl–Heisenberg frames - plays an increasingly important role in
physics; see for instance <cit.> and the references
Recently one of us started the investigation of the relation between the
theory of Gabor frames and Hamiltonian and quantum mechanics <cit.>
and introduced the notion of a Hamiltonian deformation of a Gabor
system. For a (time-independent) Hamiltonian $\mathcal{H}(x,p)$ we let $\Phi_t(x,p)$ be the flow given by the Hamilton
\begin{equation*}
\left\{
\begin{array}{ll}
\dot{x} & = \mathcal{H}_p(x,p), \\
\dot{p} & = -\mathcal{H}_x(x,p),
\end{array}
\right.
\end{equation*}
and let $H:=\mathcal{H}^w$ be the Weyl quantization of $\mathcal{H}$. Given
a Gabor system $\mathcal{G}(g, \Lambda)$, we consider the time-evolved
\begin{align} \label{eq_ham_def}
\mathcal{G}_t(g, \Lambda) := \mathcal{G}(e^{-it H}g, \Phi_t\Lambda),
\end{align}
and investigate the stability of the frame property under the evolution $\mathcal{G}(g, \Lambda) \mapsto
\mathcal{G}_t(g, \Lambda)$.
When $\mathcal{H}
$ is a quadratic form $\mathcal{H}(x,p) = \left<M (x,p),(x,p)\right> $, its
flow is given by the linear map $\Phi_t(x,p) = e^{2t JM} (x,p)$ and (<ref>) expresses the fact that the evolution
operator $e^{-i t H}$
is the metaplectic operator associated with the linear map $e^{2t JM}$. As a consequence, $\mathcal{G}_t(g,
\Lambda)$ is the image of $\mathcal{G}(g, \Lambda)$ under the unitary map $e^{-i t H}$ and hence it enjoys the
same spanning properties (in particular, the frame property is preserved).
This observation is called the symplectic covariance of Gabor frames
For more general Hamiltonians $\mathcal{H}$, no strict covariance property
holds, and the analysis of the deformation $t\mapsto\mathcal{G}_t(g, \Lambda)
$ is difficult. In <cit.>, one of us analyzed a linearized version of
this problem and established some stability estimates (see also <cit.> for a higher-order approximation to
the deformation
Based on these results, <cit.> conjectured that the evolution
$\mathcal{G}(g, \Lambda) \mapsto \mathcal{G}_t(g, \Lambda)$ preserves
the frame property for more general Hamiltonians. In particular, one
expect perturbations of
quadratic Hamiltonians to exhibit a certain approximate symplectic
covariance, in the form of stability of the frame property of $\mathcal{G}_t(g, \Lambda)$ for a certain range of time.
In this article we solve the deformation problem for small times. More
precisely, we consider a perturbation of a quadratic Hamiltonian by an
element of the Sjöstrand class $M^{\infty, 1}({\mathbb{R}^{2d}})$ with
bounded second order derivatives. We also consider a Gabor frame with window
in the Feichtinger algebra $M^1({\mathbb{R}^d})$ of functions with
integrable Wigner distribution. (See Section <ref> and <ref>
for precise definitions). The following is our main result.
Let $a$ be a real-valued, quadratic, homogeneous
polynomial on ${\mathbb{R}^{2d}}$ and let $\sigma \in M^{\infty, 1}({\mathbb{R}^{2d}})\cap C^2({\mathbb{R}^{2d}})$ have
bounded second order derivatives.
Consider the Hamiltonian $\mathcal{H}(t,x,p) := a(x,p)+\sigma(x,p)$. Let $H
:= \mathcal{H}^w(x,D)$ be the Weyl quantization of $\mathcal{H}$ and let $(\Phi_t)_{t \in {\mathbb{R}}}$ be the flow of
Let $g \in M^1({\mathbb{R}^d})
$ and $\Lambda \subseteq {\mathbb{R}^{2d}}$, such that
$\mathcal{G}(g,\Lambda)$ is a Gabor frame. Then there
exists $t_0 >0$ such that for all $t \in [-t_0,t_0]$, $\mathcal{G}(e^{-itH}g, \Phi_t(\Lambda))$ is a Gabor frame.
To see what is at stake, we consider once more the symplectic
covariance property (<ref>). It links the classical Hamiltonian flow
$e^{2tJM}$ on phase space to the quantum mechanical evolution. If
$\mathcal{H}$ is not quadratic, then the flow $\Phi _t$ is no longer
linear, and, in general, there is no explicit and exact formula for the quantum
mechanical evolution. We therefore have to understand the classical
evolution of the set $\Lambda $ under $\Phi _t$ separately from the
quantum mechanical evolution of the state (window) $g$ under
The stability of the frame property of $\mathcal{G}(g, \Phi _t(\Lambda
))$ is part of the deformation theory of Gabor frames.
While there is a significant literature on the
stability of Gabor frames under linear distortions of the time-frequency
nodes $\Lambda$ (covering perturbation of lattice parameters <cit.> on the one hand, and general point sets <cit.>), only
recently a fully non-linear deformation theory of Gabor systems was
developed in <cit.>. It turns out that the concept of
Lipschitz deformation is precisely the right tool to treat non-linear
Hamiltonian flows, and we will use the main result of
<cit.> in a decisive manner.
The second ingredient in Theorem <ref> is the assumption
$g\in M^1(\mathbb{R}^d)$. This is an essential assumption for Gabor
frames to be useful in phase space analysis. In particular, most
stability results for Gabor frames under perturbations of the window require
that $g\in M^1(\mathbb{R}^d)$. Outside $M^1$ one encounters quickly
pathologies <cit.>. In regard to our problem it is therefore
important to understand whether $M^1(\mathbb{R}^d)$ is invariant under
the evolution of the Schrödinger equation. This is indeed the case
for certain classes of Hamiltonians <cit.>, and will be the second
important tool used to prove Theorem <ref>.
The rest of the article is organized as follows. In Section <ref>
we provide some definitions and background results. Section <ref>
collects the essential tools and derives some auxiliary estimates. Finally,
the proof of Theorem <ref> is presented in Section <ref>.
§ BACKGROUND
§.§ Time-frequency analysis
Given a function $g \in L^2({\mathbb{R}^d})$, with $\lVert
g\rVert_2=1$, the short-time Fourier transform of a function $f \in
L^2({\mathbb{R}^d})$ with respect to the window $g$ is defined as
\begin{align} \label{eq_def_stft}
V_g f(x,\xi) := \left<f,e^{\pi i x\xi} \rho(x,\xi) g\right> , \qquad (x,\xi)
\in {\mathbb{R}^{2d}},
\end{align}
where $\rho(x,\xi)$ is the (symmetric) time-frequency shift defined in (<ref>). The function $g$ is often called
window and the
normalization $\lVert g\rVert_2=1$ implies that
\begin{align} \label{eq_stft_l2}
\lVert V_g f\rVert_{{L^2({\mathbb{R}^{2d}})}}=\lVert f\rVert_{{L^2({\mathbb{R}^d})}}, \qquad f \in
\end{align}
The standard choice for $g$ is the Gaussian $\phi(x) := 2^{d/4}
e^{-\pi\left| x \right| ^2}$. Analogously, the Feichtinger algebra,
originally introduced in <cit.>,
is defined to be
\begin{align*}
M^1({\mathbb{R}^d}) := \big\{ \, f \in L^2({\mathbb{R}^d}) \, : \, \lVert
f\rVert_{M^1} := \lVert V_\phi f\rVert_{L^1({\mathbb{R}^{2d}})} < +\infty \, \big\},
\end{align*}
and is used as a standard reservoir for windows $g$. Equivalently, $f \in M^1(\mathbb{R}^d)$, if the Wigner
distribution $Wf(x,\xi ) = \int f(x+t/2) \overline{f(x-t/2)} e^{-2\pi i \xi \cdot t} \,
dt$ of $f$ is integrable on $\mathbb{R}^{2d}$. When $g \in M^1({\mathbb{R}^d})$, the map $f \mapsto V_g f$ can be
extended beyond $L^2({\mathbb{R}^d})$.
We define the modulation spaces as follows: fix a non-zero $g\in
\mathcal{S} ({\mathbb{R}^d} )$ and let $1 \leq p,q \leq \infty$. Then $M^{p,q}({\mathbb{R}^d})$ is the class of all
distributions $f \in \mathcal{S}^{\prime
}({\mathbb{R}^d})$ such that
\begin{align} \label{eq_def_mp}
\lVert f\rVert_{M^{p,q}({\mathbb{R}^d})} := \left( \int_{{\mathbb{R}^d}}
\left( \int_{{\mathbb{R}^d}} \left| V_g f(x,\xi) \right| ^p dx
\right)^{q/p} d\xi \right)^{1/q} < \infty,
\end{align}
with the usual modification when $p$ or $q$ is $\infty$. Different choices
of non-zero windows $g \in \mathcal{S}({\mathbb{R}^d})$ yield the same space
with equivalent norms, see <cit.> and <cit.>. In addition, for $g \in M^1({\mathbb{R}^d})$, the short-time Fourier transform is well-defined on
all $M^{p,q}({\mathbb{R}^d})$. Originally introduced by Feichtinger in <cit.>, modulation spaces combine smoothness and integrability conditions. In this article, we will be mainly concerned with Feichtinger's algebra $M^1(\Rdst)$, as a window class for Gabor systems, and $M^{\infty,1}(\Rtdst)$ - also known as Sjöstrand's class, as a symbol class for pseudodifferential operators.
§.§ Sampling the short-time Fourier transform
A set $\Lambda \subseteq {\mathbb{R}^d}$ is called relatively separated if
\begin{align} \label{eq_rel}
\mathop{\mathrm{rel}}(\Lambda) := \sup \{ \#(\Lambda \cap (\{x\}+[0,1]^d)) :
x \in {\ \mathbb{R}^d} \} < \infty.
\end{align}
The assumption that $g \in M^1({\mathbb{R}^d})$ implies certain sampling
estimates for the short-time Fourier transform. We quote the following
standard result (see for example <cit.>.)
Let $g \in M^1({\mathbb{R}^d})$ and let $\Lambda
\subseteq {\mathbb{R}^{2d}}$. Then
\begin{align*}
\left(\sum_{\lambda \in \Lambda} \left| V_g f(\lambda) \right|
^2\right)^{1/2} \leq C \mathop{\mathrm{rel}}(\Lambda) \lVert g\rVert_{M^1}
\lVert f\rVert_2, \qquad f \in {L^2({\mathbb{R}^d})},
\end{align*}
where the constant $C$ depends only on the dimension $d$.
§.§ Gabor frames
Given a window $g \in M^1({\mathbb{R}^d})$ and a relatively separated set $\Lambda \subseteq {\mathbb{R}^{2d}}$, the
collection of functions
\begin{align*}
\mathcal{G}(g,\Lambda) := \left \{ \rho(\lambda)g : \lambda \in \Lambda
\right \}
\end{align*}
is called the Gabor system generated by $g$ and $\Lambda$. It is a
Gabor frame, if there exist constants $A,B>0$ such that
\begin{align} \label{eq_frame}
A \lVert f\rVert_2^2 \leq \sum_{\lambda \in \Lambda} \left|
\left<f,\rho(\lambda)g\right> \right| ^2 \leq B \lVert f\rVert_2^2,\qquad
f \in L^2({\mathbb{R}^d}).
\end{align}
The constants $A,B$ are called frame bounds for $\mathcal{G}(g,\Lambda)$. We
remark that the definition of Gabor system given here is slightly
non-standard. In signal processing, it is more common to define the
time-frequency shifts by
\begin{align*}
\pi(z)f(t) := e^{2\pi i \xi t} f(t-x), \qquad z=(x,\xi) \in {\mathbb{R}^d}\times {\mathbb{R}^d}, t \in {\mathbb{R}^d}.
\end{align*}
Since $\pi(x,\xi) = e^{\pi i x\xi} \rho(x,\xi)$, the choice $\rho$ has no
impact on the frame inequality in (<ref>). Note that the sum
in (<ref>) is the same as $\|V_gf |_\Lambda \|_2^2$. The use
of $\rho$ instead of $\pi $ in
this article is motivated by the symplectic covariance property in (<ref>), which would require additional phase
factors if $\pi$ was
used instead of $\rho$.
The following basic fact can be found for example in <cit.>.
If $\mathcal{G}(g,\Lambda)$ is a frame, then $\Lambda$
is relatively separated.
§ THE ESSENTIAL TOOLS
§.§ Schrödinger operators on modulation spaces
The Weyl transform of a distribution $\sigma \in
\mathcal{S}^{\prime }({\mathbb{R}^d} \times {\mathbb{R}^d})$ is an operator $\sigma^w$ that is formally defined on
functions $f:{\mathbb{R}^d} \to {\mathbb{C}}$ as
\begin{align*}
\sigma^w (f)(x) := \int_{{\mathbb{R}^d} \times {\mathbb{R}^d}} \sigma\left(\frac{x+y}{2},\xi\right) e^{2\pi i(x-y)\xi}
f(y) dy d\xi, \qquad x \in {\mathbb{R}^d}.
\end{align*}
The fundamental results in the theory of pseudodifferential operators
provide conditions on $\sigma$ for the operator $\sigma^w$ to be
well-defined and bounded on various function spaces. In particular,
Sjöstrand proved that if $\sigma \in M^{\infty, 1}({\mathbb{R}^{2d}})$,
then $\sigma^w$ is bounded on $L^2({\mathbb{R}^d})$ <cit.>. See
also <cit.> for extensions of these results to weighted symbol
classes and modulations spaces.
The following result is one of our main tools. It shows that
perturbing a quadratic Hamiltonian with a potential in the Sjöstrand's class $M^{\infty, 1}({\mathbb{R}^{2d}})$ gives rise to propagators that
are strongly continuous on $M^1({\mathbb{R}^d})$.
Let $a$ be a real-valued, quadratic, homogeneous
polynomial on ${\mathbb{R}^{2d}}$ and let $\sigma \in M^{\infty, 1}({\mathbb{\ R}^{2d}})$. Let $H := a^w(x,D) +
\sigma^w(x,D)$. Then $e^{i t H}$ is a
strongly continuous one-parameter group of operators on $M^1({\mathbb{R}^d})$. In other words:
(a) for all $t \in {\mathbb{R}}$, $e^{i t H}: M^1({\mathbb{R}^d}) \to
(b) for each $g \in M^1({\mathbb{R}^d})$,
\begin{align}
e^{i t H} g \longrightarrow g \mbox{ in }M^1({\mathbb{R}^d}), \mbox{ as } t
\longrightarrow 0.
\end{align}
§.§ Deformation of Gabor frames
Our second essential tool is a description of the stability of the frame
property of a Gabor frame $\mathcal{G}(g,\Lambda)$ under small deformations
of $\Lambda$. Our general assumption is that $g \in M^1({\mathbb{R}^d})$.
(Without this assumption the frame property might be very unstable under
perturbation of $\Lambda$, even for lattices <cit.>).
The classical results in signal processing describe the stability of the
frame property under the so-called jitter perturbations: if $\mathcal{G}(g,\Lambda)$ $\sup _{\lambda \in \Lambda } \inf
_{\lambda ^{\prime }\in
\Lambda ^{\prime }} |\lambda - \lambda ^{\prime }|<\epsilon$ and $\sup
_{\lambda^{\prime }\in \Lambda^{\prime }} \inf _{\lambda \in \Lambda}
|\lambda - \lambda ^{\prime }| < \epsilon $, for sufficiently small $\varepsilon$, then $\mathcal{G} (g, \Lambda
^{\prime })$ is also a frame. A
much deeper property is the stability of the frame condition under linear
maps $\Lambda \mapsto A \Lambda$, where $A$ is a matrix that is sufficiently
close to the identity (but possibly not symplectic!). Such results have been
derived first for lattices <cit.> and then for general sets
<cit.>. In order to deal with Hamiltonian flows, we will resort to
a recent fully non-linear stability theory <cit.>.
Let $\Lambda \subseteq {\mathbb{R}^d}$ be a set. We consider a sequence $\left \{ \Lambda_n: n \geq 1 \right \} $ of
subsets of ${\mathbb{R}^d}$
produced in the following way. For each $n \geq 1$, let $\tau_n: \Lambda \to
{\mathbb{R}^d}$ be a map and let $\Lambda_n := \tau_n(\Lambda) = \left \{
\tau_n(\lambda): \lambda \in \Lambda \right \} $. We assume that $\tau_n(\lambda) \longrightarrow \lambda$, as $n
\longrightarrow \infty$, for
all $\lambda \in \Lambda$. The sequence of sets $\left \{ \Lambda_n: n \geq
1 \right \} $ together with the maps $\left \{ \tau_n: n \geq 1 \right \} $
is called a deformation of $\Lambda$. We think of each sequence of
points $\left \{ \tau_n(\lambda): n \geq 1 \right \} $ as a (discrete) path
moving towards the endpoint $\lambda$.
We will say that $\left \{ \Lambda_n: n \geq 1 \right \} $ is a deformation
of $\Lambda$, with the understanding that a sequence of underlying maps $\left \{ \tau_n: n \geq 1 \right \} $ is also
We now describe a special class of deformations.
A deformation $\left \{ \Lambda_n: n \geq 1 \right \} $ of $\Lambda $ is called Lipschitz,
denoted by $\Lambda_n \xrightarrow{Lip}
\Lambda$, if the following two conditions hold:
(L1) Given $R>0$,
\begin{align*}
\sup_{ \overset{\lambda, \lambda^{\prime }\in \Lambda}{\left|
\lambda-\lambda^{\prime }\right| \leq R}} \left| (\tau_n(\lambda) -
\tau_n(\lambda^{\prime })) - (\lambda - \lambda^{\prime }) \right|
\rightarrow 0, \quad \mbox {as } n \longrightarrow \infty.
\end{align*}
(L2) Given $R>0$, there exists $R^{\prime }>0$ and $n_0 \in {\
\mathbb{N}}$ such that if $\left| \tau_n(\lambda) - \tau_n(\lambda^{\prime
}) \right| \leq R$ for some $n \geq n_0$ and some $\lambda,
\lambda^{\prime }\in \Lambda$, then $\left| \lambda-\lambda^{\prime }\right|
\leq R^{\prime }$.
The following results shows that the frame property of a Gabor system is
stable under Lipschitz deformations.
Let $g \in M^1({\mathbb{R}^d})$ and $\Lambda \subseteq
{\mathbb{R}^{2d}}$. Assume that $\mathcal{G}(g,\Lambda)$ is a (Gabor)
frame and that $\Lambda_n \xrightarrow{Lip} \Lambda$. Then there exist $A,B>0$
and $n_0 \in {\mathbb{N}}$ such that $\mathcal{G}(g,\Lambda_n)$ is a frame with uniform bounds $A,B$ for
all $n \geq n_0$.
We will also need the following technical lemma concerning Lipschitz
convergence and relative separation.
Let $\Lambda_n \xrightarrow{Lip} \Lambda$ and assume that $\Lambda$ is relatively separated. Then
$\limsup_n \mathop{\mathrm{rel}}
(\Lambda_n) < \infty$.
The following corollary enables us to combine the stability of Gabor frames
under deformations of $\Lambda$ with small perturbations of the window $g$
on $M^1$-norm.
Assume that $g_n \longrightarrow g$ in $M^1({\mathbb{R}^d })$ and that $\Lambda_n \xrightarrow{Lip}
\Lambda$. Then $\mathcal{G}
(g_n,\Lambda_n)$ is a frame for all sufficiently large $n$. (Moreover, the
corresponding frame bounds can be taken to be uniform in $n$).
By Theorem <ref>, there exist $A,B>0$ and $n_0 \in {\mathbb{N}}
$ such that for all $n \geq n_0$
\begin{align} \label{eq_AB}
A \lVert f\rVert_2 \leq \lVert V_g f| \Lambda_n\rVert_2 \leq B \lVert
f\rVert_2, \qquad f \in {L^2({\mathbb{R}^d})}.
\end{align}
(Here $A,B$ are the square roots of the frame bounds.) By Lemma <ref>, $\Lambda$ and all $\Lambda_n$ with
$n\gg 0$ are relatively
separated. Using Proposition <ref> we deduce that for all $f
\in {L^2({\mathbb{R}^d})}$
\begin{align*}
&\Big| \| V_g f| \Lambda_n\|_2- \| V_{g_n} f| \Lambda_n\|_2
\Big| \leq \lVert V_{g-g_n} f| \Lambda_n\rVert_2 \\
&\qquad \leq C \lVert g-g_n\rVert_{M^1} \mathop{\mathrm{rel}}(\Lambda_n)
\lVert f\rVert_2.
\end{align*}
\begin{align*}
&A_n := A - C \lVert g-g_n\rVert_{M^1} \mathop{\mathrm{rel}}(\Lambda_n), \\
&B_n := B + C \lVert g-g_n\rVert_{M^1} \mathop{\mathrm{rel}}(\Lambda_n),
\end{align*}
we deduce from (<ref>) and the triangle inequality that
\begin{align} \label{eq_ABn}
A_n \lVert f\rVert_2 \leq \lVert V_{g_n} f| \Lambda_n\rVert_2 \leq B_n
\lVert f\rVert_2, \qquad f \in {L^2({\mathbb{R}^d})}.
\end{align}
By Lemma <ref> and the fact that $g_n \longrightarrow g$ in $M^1$
it follows that $A_n \longrightarrow A$ and $B_n \longrightarrow B$.
Combining this with (<ref>) we conclude that for all sufficiently
large $n$
\begin{align*}
A/2 \lVert f\rVert_2 \leq \lVert V_{g_n} f| \Lambda_n\rVert_2 \leq B/2
\lVert f\rVert_2, \qquad f \in {L^2({\mathbb{R}^d})}.
\end{align*}
Hence, for $n \gg 1$, $\mathcal{G}(g_n,\Lambda_n)$ is a frame with bounds $A^2/4, B^2/4$.
§.§ Flows and Lipschitz convergence
A function $F: {\mathbb{R}} \times {\mathbb{R}^d} \to {\mathbb{R}^d}$ is
Lipschitz in the second variable if there exists $L>0$ such that
\begin{align*}
\left| F(t,x)-F(t,y) \right| \leq L \left| x-y \right| , \mbox{ for all }
(t,x) \in {\mathbb{R}}\times{\mathbb{R}^d}.
\end{align*}
Under this assumption, we let $(\Phi_t)_{t\in {\mathbb{R}}} $ denote the
flow of $F$ (associated with time 0). This means that for each $x \in {\mathbb{R}^d}$, ${\mathbb{R}} \ni t \mapsto
\Phi_t(x) \in {\mathbb{R}^d}$ is
a $C^1$ function and that
(a) $\Phi_0(x)=x$,
(b) $\frac{d}{dt}\Phi_t(x)=F(t,\Phi_t(x))$.
The theory of ODEs implies that the flow exists and it is uniquely
determined by properties $(a)$ and $(b)$ above. Moreover, the flow satisfies
the following distortion estimate: given $T>0$, there exist constants $c_t,
C_T>0$ such that
\begin{align} \label{eq_dist}
c_T \left| x-y \right| \leq \left| \Phi_t(x)-\Phi _t(y) \right| \leq C_T
\left| x-y \right| , \qquad x,y \in {\mathbb{R}}, \quad t \in [-T,T].
\end{align}
The previous estimate is normally proved using the following useful lemma.
Let $I \subseteq {\mathbb{R}}$ be an interval and $a \in I
$. Let $g: I \to [0,+\infty)$ be a continuous function that satisfies
\begin{align*}
g(t) \leq A + B\left| \int_a^t g(s) ds \right| , \qquad t \in I,
\end{align*}
for some constants $A,B \in {\mathbb{R}}$. Then
\begin{align*}
g(t) \leq Ae^{B\left| t-a \right| }, \qquad t \in I.
\end{align*}
(The reason for the absolute value outside the integral is that $t-a$ can be
We now show that the flows of ODEs provide examples of Lipschitz
Let $F: {\mathbb{R}} \times {\mathbb{R}^d} \to {\mathbb{R }^d}$ be Lipschitz in the second variable
and let $(\Phi_t)_{t\in {\mathbb{R} }}$ be the corresponding flow. Let $\Lambda \subseteq {\mathbb{R}^d}$ be a
relatively separated set. Then $\Phi_t(\Lambda) \xrightarrow{Lip} \Lambda$,
as $t \longrightarrow 0$.
(More precisely, for each sequence $t_n \longrightarrow 0$, $\Phi_{t_n}(\Lambda) \xrightarrow{Lip} \Lambda$.)
Let $L>0$ be the Lipschitz constant of $F$ (in the second variable). We
first check condition $(L1)$ from Definition <ref>. From the
definition of the flow it follows that
\begin{align*}
\Phi_t(x) = x + \int_0^t F(s,\Phi_s(x)) ds,\qquad t \in {\mathbb{R}}.
\end{align*}
\begin{align}
\Phi_t(\lambda)-\Phi_t(\lambda^{\prime })-(\lambda - \lambda^{\prime }) &=
\int_0^t \left( F(s,\Phi_s(\lambda)) - F(s,\Phi_s(\lambda^{\prime }))
\right) ds.
\end{align}
As a consequence,
\begin{align*}
&\left| \Phi_t(\lambda)-\Phi_t(\lambda^{\prime })-(\lambda - \lambda^{\prime
}) \right| \leq \left| \int_0^t \left| F(s,\Phi_s(\lambda)) -
F(s,\Phi_s(\lambda^{\prime })) \right| ds \right| \\
&\qquad \leq \left| \int_0^t L \left| \Phi_s(\lambda)-\Phi_s(\lambda^{\prime
}) \right| ds \right| \\
&\qquad \leq L \left| \int_0^t \left| \Phi_t(\lambda)-\Phi_t(\lambda^{\prime
})-(\lambda - \lambda^{\prime }) \right| ds \right| + L\left| t \right|
\left| \lambda-\lambda^{\prime }\right| .
\end{align*}
Applying Gronwall's Lemma <ref> to $g(t):=\left\vert \Phi
_{t}(\lambda )-\Phi _{t}(\lambda ^{\prime })-(\lambda -\lambda ^{\prime
})\right\vert $ we deduce that
\begin{equation*}
\left\vert \Phi _{t}(\lambda )-\Phi _{t}(\lambda ^{\prime })-(\lambda
-\lambda ^{\prime })\right\vert \leq L\left\vert t\right\vert \left\vert
\lambda -\lambda ^{\prime }\right\vert e^{L\left\vert t\right\vert }.
\end{equation*}
Condition $(L1)$ follows from here.
To check condition $(L2)$, we consider only $t \in [-1,1]$ and (<ref>) to obtain a constant $C$ such that
\begin{align*}
C^{-1} \left| x-y \right| \leq \left| \Phi_t(x)-\Phi_t(y) \right| \leq C
\left| x-y \right| , \qquad t \in (-1,1).
\end{align*}
Hence, if for some instant $t_0$ we know that $\left|
\Phi_{t_0}(\lambda)-\Phi_{t_0}(\lambda^{\prime }) \right| \leq R$, then we
can deduce that $\left| \lambda-\lambda^{\prime }\right| \leq R^{\prime
}:=CR $.
This completes the proof.
§ HAMILTONIAN DEFORMATIONS: DÉNOUEMENT
We finally combine all tools from the previous section
and prove the main result.
Let us define $F: {\mathbb{R}} \times {\mathbb{R}^{2d}} \to {\mathbb{R}^{2d}}
$ by
\begin{align*}
F(t,x,p):=(\partial_p \mathcal{H}(x,p), -\partial_x \mathcal{H}(x,p)).
\end{align*}
Then $F$ is a $C^1$ function with bounded derivatives and,
consequently, $F$ is
Lipschitz in the second set of variables $(x,p)$. Let $t_n \longrightarrow 0$ and define $\Lambda_n :=
\Phi_{t_n}(\Lambda)$. Theorem <ref> implies that $\Lambda_n \xrightarrow{Lip} \Lambda$, while Theorem
<ref> implies
that $e^{-it_nH}g \longrightarrow g$ in $M^1$. Hence, Corollary <ref> yields the desired conclusion.
The proof shows that, under the conditions of Theorem
<ref>, the
Gabor systems $\mathcal{G}(e^{-ith}g, \Phi_t(\Lambda))$ admit uniform frame
bounds for $t \in [-t_0,t_0]$.
We do not know whether the conclusion of Theorem <ref> remains
valid for arbitrary times. Moreover, we do not know of any example of a
Hamiltonian deformation that does not preserve the frame
MR3203099 L. D. Abreu and H. G. Feichtinger. Function
spaces of polyanalytic functions. In Harmonic and complex
analysis and its applications, Trends Math., pages 1–38. Birkhäuser/Springer, Cham, 2014.
alanga00 S. Ali, J.-P. Antoine, and J.-P. Gazeau. Coherent States, Wavelets and their Generalizations. Graduate Texts in Contemporary Physics. Springer, New York,
asfeka13 G. Ascensi, H. G. Feichtinger, and N. Kaiblinger.
Dilation of the Weyl symbol and Balian-Low theorem.
Trans. Amer. Math. Soc., 366(7):3865–3880, 2014.
J. Bellissard.
Lipshitz continuity of gap boundaries for Hofstadter-like spectra.
Comm. Math. Phys., 160(3):599–613, 1994.
Á. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers.
Unimodular Fourier multipliers for modulation spaces.
J. Funct. Anal., 246(2):366–384, 2007.
bebuconi15 M. Berra, M. Bulai, E. Cordero, F. Nicola, Gabor Frames of Gaussian Beams
for the Schrödinger
equation. Appl. Comput. Harmon. Anal. To appear.
chdehe99 O. Christensen, B. Deng, and C. Heil. Density of Gabor frames. Appl. Comput. Harmon. Anal.,
7(3):292–304, 1999.
cogrniro13 E. Cordero, K. Gröchenig, F. Nicola, and L.
Rodino. Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class. J. Math. Phys. 55, 081506 (2014); doi: 10.1063/1.4892459.
MR3317554 E. Cordero, F. Nicola, and L. Rodino.
Propagation of the Gabor wave front set for Schrödinger
equations with non-smooth potentials. Rev. Math. Phys.,
27(1):1550001, 33, 2015.
rosu15 X. R. Dai, and Q. Sun, The abc problem for Gabor
systems, Preprint. arXiv:1304.7750.
dego11 M. de Gosson.
Symplectic methods in harmonic analysis and in mathematical
physics, volume 7 of Pseudo-Differential Operators. Theory and
Applications. Birkhäuser/Springer Basel AG, Basel, 2011.
dego13 M. de Gosson.
Hamiltonian deformations of Gabor frames: first steps.
Appl. Comput. Harmon. Anal., 38(2):196–221, 2015.
ams R. S. Doran, C. C. Moore, R. J. Zimmer. Group Representations,
Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey :
AMS Special Session Honoring the Memory of George W. Mackey, January 7-8,
2007, New Orleans, Louisiana. Contemporary Mathematics, American
Mathematical Soc., 2008.
H. G. Feichtinger.
On a new Segal algebra.
Monatsh. Math., 92:269–289, 1981.
H. G. Feichtinger.
Modulation spaces on locally compact Abelian groups.
Technical report, January 1983.
fe83 H. G. Feichtinger. Banach convolution algebras
of Wiener type. In Proc. Conf. on Functions, S
eries, Operators, Budapest 1980, volume 35 of Colloq. Math. Soc. Janos Bolyai, pages 509–524.
North-Holland, Amsterdam, E
ds. B. Sz.-Nagy and J. Szabados. edition, 1983.
H. G. Feichtinger.
Modulation Spaces: Looking Back and Ahead.
Sampl. Theory Signal Image Process., 5(2):109–140, 2006.
feka04 H. G. Feichtinger and N. Kaiblinger. Varying the time-frequency lattice of Gabor
frames. Trans.
Amer. Math. Soc., 356(5):2001–2023, 2004.
G. B. Folland.
Harmonic Analysis in Phase Space.
Princeton Univ. Press, Princeton, NJ, 1989.
gr01 K. Gröchenig. Foundations of
Time-Frequency Analysis. Appl. Numer. Harmon. Anal.
Birkhäuser Boston, Boston, MA, 2001.
gr06-3 K. Gröchenig. Composition and spectral
invariance of pseudodifferential operators on modulation spaces. J. Anal. Math., 98:65–82, 2006.
gr06 K. Gröchenig. Time-Frequency analysis
of Sjörstrand's class. Rev. Mat. Iberoam.,
22(2):703–724, 2006.
grorro15 K. Gröchenig, J. Ortega-Cerdà, and
J. L. Romero. Deformation of Gabor systems. Adv. Math. Vol. 277 No.4, (2015) p.388-425.
ja03-1 A. J. E. M. Janssen. On generating tight Gabor frames at critical density. J. Fourier Anal. Appl.,
9(2):175–214, 2003.
sj94 J. Sjöstrand. An algebra of
pseudodifferential operators. Math. Res. Lett.,
1(2):185–192, 1994.
sj95 J. Sjöstrand. Wiener type algebras of
pseudodifferential operators.
Séminaire sur les équations aux Dérivées
Partielles, 1994-1995, École Polytech, Palaiseau, Exp. No.
IV, 21, 1995.
|
1511.00132
|
Center for Theoretical Physics of the Universe
Institute for Basic Science (IBS), Daejeon 34051, Korea
We discuss a scheme to implement the relaxion solution to the hierarchy problem with multiple axions, and present a UV-completed model realizing the scheme.
All of the $N$ axions in our model are periodic
with a similar decay constant $f$ well below the Planck scale. In the limit $N\gg 1$, the relaxion $\phi$ corresponds to an exponentially long multi-helical flat direction
which is shaped by a series of mass mixing between nearby axions
in the compact field space of $N$ axions.
With the length of flat direction given by $\Delta \phi =2\pi f_{\rm eff} \sim e^{\xi N} f$ for $\xi={\cal O}(1)$,
both the scalar potential driving the evolution of $\phi$ during the inflationary epoch and the $\phi$-dependent Higgs boson mass vary with an exponentially large periodicity of ${\cal O}(f_{\rm eff})$, while the back reaction potential stabilizing the relaxion has a periodicity of ${\cal O}(
f)$. A natural UV completion of our scheme can be found in high scale or (mini) split supersymmetry (SUSY) scenario with the axion scales
generated by SUSY breaking as $f\sim \sqrt{m_{\rm SUSY}M_*}$,
where the soft SUSY breaking scalar mass $m_{\rm SUSY}$ can be well above the weak scale, and the fundamental scale $M_*$ can be identified as the Planck scale or the GUT scale.
§ INTRODUCTION
Recently a new approach to address the hierarchy problem has been proposed in <cit.>. The scheme introduces a scalar degree of freedom, the relaxion
$\phi$, making
the Higgs boson mass a dynamical field depending on $\phi$. During the inflationary epoch, the Higgs boson mass-square $\mu_h^2(\phi)$ is scanned by the rolling $\phi$ from a large positive initial value to zero.
Right after the relaxion crosses the point $\mu_h(\phi)=0$, so that $\mu_h^2(\phi)$ becomes negative, a nonzero Higgs vacuum expectation value (VEV) is developed and a Higgs-dependent
back reaction potential begins to operate to stabilize the relaxion[A mechanism to cosmologically relax the Higgs boson mass down to a small value through a nucleation of domain wall bubbles has been discussed in <cit.>.]. One can then arrange the model parameters in a technically natural way to result in the relaxion stabilized at a point where the corresponding
Higgs VEV is much smaller than the initial Higgs boson mass.
An intriguing feature of the relaxion mechanism is that the relaxion potential involves two very different scales. One is the period of the back reaction potential, and the other is the excursion range of the relaxion necessary to scan $\mu_h(\phi)$ from a large initial value to zero.
To see this, let us consider
the relaxion potential given by
V(ϕ, h) = V_0(ϕ) + μ_h^2(ϕ) |h|^2 + V_br(ϕ, h)
$V_0$ is the potential driving the rolling of $\phi$ during the inflationary epoch
and $V_{\rm br}$ is the periodic back reaction potential stabilizing $\phi$ right after it crosses $\mu_h(\phi)=0$.
In fact, the key feature of the mechanism can be read off from the following form of potential:
V_0 = ϵ_0 f^3 ϕ+ ...., μ_h^2 = M_h^2 + ϵ_h f ϕ+ ..., V_br = Λ^4_br(h) cos(ϕ/f),where $M_h$ denotes the initial Higgs boson mass, $\epsilon_0$ and $\epsilon_h$ are small dimensionless parameters
describing the explicit breaking of the relaxion shift symmetry in $V_0$ and $\mu^2_h$, respectively, and finally $f$ is the relaxion decay constant in the back reaction potential.
In non-supersymmetric theory, the Higgs mass parameter $M_h$ is naturally of the order of the cutoff scale of the model.
On the other hand, in supersymmetric theory, it corresponds to the scale of soft supersymmetry (SUSY) breaking mass which can be well below the cutoff scale of the model. In any case, we are interested in the case that $M_h$ is much larger than the weak scale:
M_h ≫ v≡⟨h⟩=174 GeV,which might be explained by the relaxion mechanism.
Let us now list the conditions for the relaxion mechanism to work. First of all,
in order for the rolling relaxion to cross $\mu_h(\phi)=0$ without a fine tuning of the initial condition,
it should experience a field excursion
Δϕ/f ≳ M^2_h/ϵ_h f^2.In order for the scalar potential to be technically natural under radiative corrections, the symmetry breaking parameters $\epsilon_0$ and $\epsilon_h$ should obey
ϵ_0 ≳ ϵ_h M_h^2/f^2.
On the other hand, from the stability condition $\partial_\phi V=0$, one finds ϵ_0 ∼ Λ^4_br/f^4,and therefore
Δϕ/f ≳ M_h^4/Λ_br^4.As for the back reaction potential, generically $\Lambda_{\rm br}(h=0)$ may not be vanishing, and then
one needs
Λ^4_br(h=v) ≫ Λ^4_br(h=0).Also, in order not to destabilize the weak scale size of the Higgs VEV, its magnitude should be bounded as
Λ_br(h=v) ≲ O(v).
An immediate consequence of the above conditions is that
the relaxion should experience a field excursion much bigger than $f$ in the limit $M_h\gg v$:
Δϕ/f ≳ M_h^4/v^4.The required excursion is huge in the case that the back reaction potential is generated by the QCD anomaly, in which
$\Lambda^4_{\rm br}\,\sim\, f_\pi^2 m_\pi^2$ and therefore
Δϕ/f ≳ O(M_h^4/f^2_πm_π^2) ∼ 10^12(M_h/v)^4.Even when the scale of the back reaction potential saturates the bound (<ref>), the required relaxion excursion
is still much larger than $f$ as long as $M_h$ is higher than the weak scale
by more than a few orders of magnitudes.
Note that the natural size of $M_h$
is the cutoff scale of the model for non-SUSY case, while it is the soft SUSY breaking scalar mass for SUSY case.
Therefore, in the relaxion scenario, the hierarchy $M_h/v \gg 1$ is replaced with a much bigger hierarchy $\Delta \phi/f \gtrsim M^4_h/v^4$. Although $\Delta \phi \gg f$ might be stable against radiative corrections, it is still crying for an explanation with a sensible UV completion.
To incorporate a huge relaxion excursion, one may simply assume that the relaxion is a non-compact field variable.
See <cit.> for recent discussions of the related issues. In this paper, we discuss an alternative scenario in which the relaxion corresponds to an exponentially long multi-helical flat direction
in the compact field space spanned by $N$ sub-Planckian periodic axions:
\phi_i\equiv \phi_i+2\pi f_i \quad (i=1,2,..,N)
with $f_i\ll M_{\rm Planck}$. Such a long flat direction
is formed
by a series of mass mixing between nearby axions, producing a multiplicative sequence of helical windings of flat direction, which results in
\frac{\Delta \phi}{f_i} \,=\, {\cal O}(e^{\xi N}) $$
for $\xi={\cal O}(1)$.
Our scenario is inspired by the recent generalization of the axion alignment mechanism for natural inflation
<cit.> to the case of $N$ axions
Although it requires a rather specific form of axion mass mixings, our scheme
does not involve any fine tuning of continuous parameters, nor an unreasonably large discrete parameter.
As we will see, our scheme finds a natural UV completion in high scale or (mini) split supersymmetry (SUSY) scenario with soft SUSY breaking scalar mass $m_{\rm SUSY}\gg v$. In the UV completed model, the axion scales are generated by SUSY breaking <cit.>
$$f_i \sim \sqrt{m_{\rm SUSY}M_*},$$
where $M_*$ can be identified as the Planck scale or the GUT scale.
With the $(N-1)$ hidden Yang-Mills gauge sectors which confine at scales below $f_i$ to generate the desired axion mass mixings, the
canonically normalized relaxion has a field range
$$\Delta \phi \equiv 2\pi f_{\rm eff} \sim 2\pi f_i\left(\prod_{j=1}^{N-1} n_j\right),$$
$n_j>1$ corresponds to the number of flavors of the gauge-charged fermions in the $j$-th hidden sector.
One can then arrange the microscopic parameters in a technically natural way to make
the resulting relaxion potential $V_0(\phi)$ and the $\phi$-dependent Higgs boson mass $\mu_h^2(\phi)$ vary with an exponentially large periodicity of ${\cal O}(f_{\rm eff})$, while the back reaction potential $V_{\rm br}(h,\phi)$ has a periodicity of ${\cal O}(f_i)$. An interesting feature of our model is that the desired $V_0(\phi)$ and $\mu_h^2(\phi)$ arise as a natural consequence of the solution of the MSSM $\mu$-problem advocated in <cit.>.
The outline of the paper is as follows.
In the next section, we describe the basic idea with a simple toy model and discuss the scheme within the framework of an effective theory of $N$ axions. In section 3, we present a UV model with high scale SUSY, realizing our scheme in the low energy limit. Section 4 is the conclusion.
§ EXPONENTIALLY LONG RELAXION FROM MULTIPLE AXIONS
To illustrate the basic idea, let us begin with a simple two axion model.
The lagrangian density of the model is given by
L = 1/2(∂_μϕ_1)^2 +1/2(∂_μϕ_2)^2
-( Ṽ_0 +V_0+ μ_h^2|h|^2 + V_br+...),where $h$ is the
SM Higgs doublet and $\phi_i$ are the periodic axions:
ϕ_i ≡ ϕ_i +2πf_i,with a scalar potential
Ṽ_0 = -Λ^4cos(ϕ_1/f_1+n ϕ_2/f_2),
V_0 =
-ϵf_2^4 cos(ϕ_2/f_2+δ_2),
μ_h^2 = M_h^2-ϵ^'f_2^2 cos(ϕ_2/f_2+δ_2^'),
V_br = -Λ^4_br(h)cos(ϕ_1/f_1+δ_1),
where Λ^4 ≫ ϵf_2^4 ≫ Λ_br^4.
Here $M_h$ is an axion-independent mass parameter which is comparable to the cutoff scale of the above effective lagrangian, and $n>1$ is an integer which will be determined by the underlying UV completion.
We assume
ϵf_2^2 ≳ O(ϵ^'M_h^2), ϵ^'f_2^2 ≳ O(M_h^2),
and therefore the model is stable against the radiative corrections which replace the
Higgs operator $|h|^2$ with the cutoff-square of ${\cal O}(M_h^2)$, while allowing $\mu_h^2=0$ for certain value of $\phi_2$.
As for the back reaction potential, one can consider two different possibilities. One option is to generate it by the coupling of $\phi_1$ to the QCD anomaly, yielding
Λ_br^4(h) ∼ y_u Λ_QCD^3 h,where $y_u$ denotes the up quark Yukawa coupling to the SM Higgs field $h$, and $\Lambda_{\rm QCD}$ is the QCD scale.
This option corresponds to the minimal model, however generically is in conflict with the axion solution to the strong CP problem. Alternative option is to introduce a new hidden gauge interaction which confines around the weak scale and generates
a back reaction potential given by <cit.>
Λ_br^4 = m_1^2 |h|^2 + m_2^4 with
m_2^4 < m_1^2 v^2 ≲ O(v^4).
In order for the model to be technically natural, the underlying dynamics to generate the back reaction potential should be arranged to
make sure that the above conditions on $m_1$ and $m_2$ are stable against radiative corrections.
The above two axion model involves the shift symmetries
U(1)_i: ϕ_i/f_i →ϕ_i/f_i + c_i (i=1,2),
which are broken by $\tilde V_0$ down to the relaxion shift symmetry
U(1)_ϕ: ϕ_1/f_1 →ϕ_1/f_1 +nc, ϕ_2/f_2 →ϕ_2/f_2- c.
The flat direction associated with $U(1)_\phi$
has a helical winding structure in the compact 2-dim field space of $\phi_i$ as depicted in Fig. (<ref>).
Then the periodicity of the flat direction is enlarged as
Δϕ = 2π√(n^2 f_1^2 +f_2^2) ≡2πf_eff,
which is larger than the original axion periodicities $2\pi f_1 \sim\, 2\pi f_2$ by the winding number $n$.
Flat relaxion direction in the two axion model.
The relaxion shift symmetry $U(1)_\phi$ is slightly broken by small nonzero values of $\epsilon, \epsilon^\prime$ and $\Lambda_{\rm br}$. Note that this particular form of $U(1)_\phi$ breaking is technically natural as long as the first condition
of (<ref>) is satisfied.
To find the effective potential of the flat relaxion direction, one can rewrite the model
in terms of the canonically normalized heavy and light axions <cit.>:
ϕ_H = f_2ϕ_1 +nf_1 ϕ_2/f_eff, ϕ = nf_1ϕ_1-f_2 ϕ_2/f_eff,for which
ϕ_1/f_1 = nϕ/f_eff+f_2^2/n^2f_1^2+f_2^2 ϕ_H/f_H
ϕ_2/f_2 = -ϕ/f_eff+nf_1^2/n^2f_1^2+f_2^2 ϕ_H/f_H,where $f_H=f_1f_2/f_{\rm eff}$.
In the limit $\Lambda^4 \gg \epsilon f_2^4 \gg \Lambda^4_{\rm br}$, it is straightforward to integrate out the heavy axion $\phi_H$ to derive the low energy effective lagrangian of the light axion $\phi$. The resulting effective potential
of the canonically normalized $\phi$ is given by
V_eff =
+(M_h^2 -ϵ^'f_2^2 cos(ϕ/f_eff-δ_2^'))|h|^2
- Λ_br^4(h) cos(ϕ/f+δ_1), where
f_eff = √(n^2f_1^2+f_2^2) ≡nf.
We can now generalize the above two axion model
to the case of $N>2$ axions to enlarge the effective axion scale further <cit.>. The lagrangian density is given by
L = 1/2∑_i (∂_μϕ_i)^2 -( Ṽ_0 +V_0+ μ_h^2|h|^2 + V_br+...),
Ṽ_0 = -∑_i=1^N-1Λ_i^4
cos(ϕ_i/f_i+n_i ϕ_i+1/f_i+1)
V_0 =
-ϵf_N^4 cos(ϕ_N/f_N+δ_N),
μ_h^2 = M_h^2-ϵ^'f_N^2 cos(ϕ_N/f_N+δ_N^'),
V_br = -Λ^4_br(h)cos(ϕ_1/f_1+δ_1),
with $\Lambda_i^4 \gg \epsilon f_N^4 \gg \Lambda_{\rm br}^4$. The model involves the $N$ axionic shift symmetries:
U(1)_i: ϕ_i/f_i → ϕ_i/f_i + c_iwhich are broken by $\tilde V_0$ down to the relaxion shift symmetry:
U(1)_ϕ: ϕ_i/f_i → ϕ_i/f_i +γ_i c (γ_i=-n_iγ_i+1),with the corresponding flat direction given by
ϕ ∝ ∑_i=1^N (-1)^i-1(∏_j=i^N-1 n_j) f_i ϕ_i.
Turing on small nonzero values of $\epsilon, \epsilon^\prime$ and $\Lambda_{\rm br}$, the relaxion shift symmetry (<ref>) is slightly broken and nontrivial potential of $\phi$ is developed. Although our way to break $U(1)_\phi$ is rather specific, it is technically natural as the model involves many continuous or discrete axionic
shift symmetries which are distinguishing our particular way of symmetry breaking from other possibilities.
It is again straightforward to integrate out the $(N-1)$ heavy axions which receive a large mass from $\tilde V_0$ <cit.>.
For the canonically normalized $\phi$, the resulting effective potential is given by
V_eff = V_0(ϕ) +μ_h^2(ϕ)|h|^2 +V_br(h,ϕ)
= -ϵf_N^4cos(ϕ/f_eff+(-)^N-1δ_N)
+(M_h^2 -ϵ^'f_N^2 cos(ϕ/f_eff+(-)^N-1δ_N^'))|h|^2
- Λ_br^4(h) cos(ϕ/f+δ_1), where
f_eff = √(∑_i=1^N (∏_j=i^N-1 n_j^2) f_i^2) ∼ (∏_j=1^N-1 n_j) f_1 ∼ e^ξN f_1 (ξ=O(1)),
f = f_eff/(∏_j=1^N-1 n_j) ∼ f_1. For simplicity here we assumed that all $f_i$ are comparable to each other, or $f_1$ is the biggest among $\{f_i\}$.
Obviously, in the limit $N\gg 1$ the above relaxion potential involves two very different axion scales, an exponentially enhanced effective decay constant $f_{\rm eff}$ and another effective decay constant $f$ which
is comparable to the original decay constants $f_i$. Such a big difference between $f_{\rm eff}$ and $f$
can be understood by noting that in order for the $N$-th axion $\phi_N$ to travel one period
along the relaxion direction, i.e. $\Delta \phi_N = 2\pi f_N$, the other axions $\phi_i$ $(i=1,2,..., N-1)$ should experience a multiple windings as $\Delta \phi_i = 2\pi \left(\prod_{j=i}^{N-1} n_j\right) f_i$.
This results in
ϕ_i/f_i = (-1)^i-1 (∏_j=i^N-1 n_j)ϕ/f_eff + ...,where the ellipsis stands for the $(N-1)$ heavy axions receiving a large mass from $\tilde V_0$.
For an illustration of this feature, we depict in Fig. (<ref>) the relaxion field direction for the case of $N=3, n_1=2, n_2=4$.
Flat relaxion direction in the three axion case with $n_1=2$ and $n_2=4$.
The effective potential (<ref>) can easily realize the relaxion mechanism under several consistency conditions.
First of all, like (<ref>) of the two axion model, we need
ϵf_N^2 ≳ O(ϵ^'M_h^2), ϵ^'f_N^2 ≳ O(M_h^2),
in order for the model to be stable against radiative corrections, while allowing $\mu_h = 0$ for certain value of $\phi$.
Without invoking any fine tuning, there is always a certain range of $\delta_N$ and $\delta^\prime_N$ for which the relaxion rolls down toward the minimum of $V_0(\phi)$ starting
from a generic initial value $\phi_0$ with $\mu_h^2(\phi_0) ={\cal O}(M_h^2) > 0$.
After a field excursion $\Delta \phi={\cal O}(f_{\rm eff})$, the relaxion is crossing $\mu_h(\phi)=0$, and
then a nonzero Higgs VEV is developed together with the back reaction potential stabilizing the relaxion at the value giving $\langle h\rangle = v$.
The stabilization condition leads to
ϵf_N^4/f_eff ∼ Λ_br^4(h=v)/f.
From (<ref>), this then yields a lower bound on $f_{\rm eff}$:
f_eff/f ≳ M_h^4/Λ_br^4(h=v) = (M_h/v)^4 v^4/Λ_br^4(h=v),
where $v^4/\Lambda^4_{\rm br}(h=v)\sim 10^{12}$ when $V_{\rm br}$ is generated by the QCD anomaly, or
$v^4/\Lambda^4_{\rm br}(h=v)$ has a model-dependent value not exceeding ${\cal O}(1)$ when $V_{\rm br}$ is generated by the hidden color dynamics which confines around the weak scale.
To summarize,
in our scheme for the relaxion mechanism, $v\ll M_h$ can be technically natural with an exponential hierarchy between
the two effective axion scales:
f_eff/f = O(e^ξN) (ξ=O(1))which is arising as a consequence of a series of mass mixing between nearby axions in the compact field space of $N$ axions.
Although it relies on a rather specific form of axion mass mixings,
the scheme does not involve any fine tuning of continuous parameters, nor an unreasonably large discrete parameter.
§ A UV MODEL WITH HIGH SCALE SUPERSYMMETRY
In this section, we construct an explicit UV completion of the $N$ axion model discussed in the previous section.
We first note that our scheme requires that the axion potential $\tilde V_0$ should dominate over the other part
of the potential in (<ref>) as it determines the key feature of the model, i.e. an exponentially long flat direction in the compact field space of $N$ axions. Specifically we need
f_i^4 ≫ |Ṽ_0| ≫ |V_0| ≳M_h^4.
On the other hand, we wish to have an explicit UV model providing the full part of the axion potential in (<ref>), as well as a mechanism to generate the axion scales $f_i$. This implies that our UV model should allow the natural size of the Higgs boson mass, i.e. $M_h$, to be much lower than its cutoff scale.
As SUSY provides a natural framework for this purpose,
in the following we present a supersymmetric UV completion of the low energy effective potential (<ref>).
First of all, to have $N$ axions with the decay constants $f_i\ll M_{\rm Planck}$, we introduce $N$ global $U(1)$ symmetries
under which
U(1)_i: X_i→e^iβ_iX_i, Y_i→e^-3iβ_iY_i (i=1,2,...,N),where $X_i$ and $Y_i$ are gauge-singlet chiral superfields with the $U(1)_i$-invariant superpotential
W_1 = ∑_i X_i^3Y_i/M_*,where $M_*$ corresponds to the cutoff scale of the model, which might be identified as the GUT scale or the Planck scale.
Here and in the following,
we ignore the dimensionless coefficients of order unity in the lagrangian.
We assume that SUSY is softly broken with SUSY breaking soft masses
m_SUSY ∼ M_h ≪ M_*.In particular, the model involves the soft SUSY breaking terms of $X_i$ and $Y_i$, given by
L_soft = -m_X_i^2|X_i|^2 - m_Y_i^2|Y_i|^2 +(A_iX_i^3Y_i/M_*+h.c),where
m_{X_i}\,\sim\, m_{Y_i}\,\sim \, A_i\,\sim \, m_{\rm SUSY}.
To achieve the $N$ axions in the low energy limit, we need all $m_{X_i}^2$ are tachyonic, which results in
⟨X_i ⟩ ≡x_i ∼ √(m_SUSYM_*), ⟨Y_i⟩ ≡y_i ∼√(m_SUSYM_*).
Then the canonically normalized axion components $\phi_i$ can be identified as
X_i ∝ e^iϕ_i/f_i, Y_i ∝ e^-3iϕ_i/f_iwith
f_i = √(2(x_i^2+9y_i^2)) ∼ √(m_SUSYM_*).
Now we need a dynamics to generate the axion potential $\tilde V_0$ in (<ref>), developing an exponentially long flat direction as described in the previous section. For this purpose, we introduce $(N-1)$ hidden Yang-Mills sectors associated with the gauge group $G=\prod_{i=1}^{N-1}SU(k_i)$, including also the charged matter fields
Ψ_i+Ψ_i^c, Φ_ia+Φ_ia^c (i=1,2,..., N-1; a=1,2,..., n_i),where $\Psi_i$ and $\Phi_{ia}$ are the fundamental representation of $SU(k_i)$, while $\Psi_i^c$ and $\Phi_{ia}^c$ are anti-fundamentals. These gauged charged matter fields
couple to the $U(1)_i$-breaking fields $X_i$ through the superpotential
W_2 = ∑_i=1^N-1( X_i Ψ_iΨ_i^c + X_i+1Φ_iaΦ^c_ia).
Note that $X_i$ couples to a single flavor of the $SU(k_i)$-charged hidden quark $\Psi_i+\Psi_i^c$, while $X_{i+1}$ couples to
$n_i$ flavors of the $SU(k_i)$-charged hidden quarks $\Phi_{ia}+\Phi^c_{ia}$.
With this form of hidden Yang-Mills sectors, the $N$ global $U(1)$ symmetries are explicitly broken down to
a single $U(1)$ by the $U(1)_i\times SU(k_j)\times SU(k_j)$ anomalies.
The charged matter fields $\Psi_i+\Psi_i^c$ and $\Phi_{ia}+\Phi_{ia}$ get a heavy mass of ${\cal O}(f_i)$, so can be integrated out at scales below $f_i$. This yields an axion-dependent threshold correction to the holomorphic gauge kinetic function
$\tau_i$ of $SU(k_i)$
at scales below $f_i$:
τ_i = 1/g_i^2+i/8π^2(ϕ_i/f_i+n_iϕ_i+1/f_i+1)+θ^2M_λ_i,where we ignored the dependence on $|X_i|$, while including the soft SUSY breaking by the gaugino masses $M_{\lambda_i}\sim m_{\rm SUSY}$.
As a consequence, at scales below $f_i$,
the global symmetry breaking by the $U(1)_i\times SU(k_j)\times SU(k_j)$ anomalies
is described by the following axion effective interactions:
∑_i=1^N-11/32π^2(ϕ_i/f_i+n_iϕ_i+1/f_i+1)(FF̃)_SU(k_i),where $(F)_{SU(k_i)}$ denotes the gauge field strength of $SU(k_i)$ and $(\tilde{F})_{SU(k_i)}$ is its dual.
As we wish to generate the axion potential $|\tilde V_0|\gg M_h^4\sim m_{\rm SUSY}^4$ from the above axion couplings,
we assume
Λ̃_i ≫ m_SUSY,where $\tilde\Lambda_i$ denotes the confining scale of the hidden gauge group $SU(k_i)$.
In such case, the resulting axion potential is determined by the non-perturbative effective superpotential describing the formation of the $SU(k_i)$ gaugino condensation <cit.>:
W_np ∼ ⟨λ_iλ_i⟩ ∝(e^-8π^2 τ_i)^1/k_i,
Ṽ_0 = -∑_i=1^N-1Λ_i^4 cos(1/k_i(ϕ_i/f_i+n_iϕ_i+1/f_i+1))with
Λ_i^4 ∼ 8π^2/k_i M_λ_iΛ̃_i^3.
Our next mission is to generate the axion potential $V_0$ and the axion-dependent Higgs mass-square $\mu_h^2$ in (<ref>),
driving the evolution of the relaxion during the inflationary epoch, while scanning the Higgs mass-square
from an initial value of ${\cal O}(m_{\rm SUSY}^2)$ to zero.
This can be done by introducing a superpotential term given by
W_3 = (X_N-1^2/M_* + X_N^2/M_* ) H_u H_d,
together with the associated Kähler potential term:
ΔK = X_N-1^2X_N^*2/M_*^2+h.c.Here we ignore the irrelevant terms such as $|X_N|^4$ or $|X_{N-1}|^4$ in the Kähler potential. Note that
the couplings in $W_3$ leads to a logarithmically divergent radiative correction to $\Delta K$ <cit.>, and our model is stable against such radiative correction as long as the coefficient of $\Delta K$ is of order unity. Note also that $W_3$ provides a
solution to the MSSM $\mu$ problem as it yields naturally the Higgsino mass $\mu_{\rm eff}\,\sim\, m_{\rm SUSY}$ <cit.>.
After integrating out the $(N-1)$ axions which receive a heavy mass from $\tilde V_0$, while leaving the light relaxion $\phi$ as described in the previous section,
the Kähler potential term $\Delta K$ gives rise to
V_0 = -m_0^4 cos( 2(n_N-1+1)ϕ/f_eff +δ),where
m_0^4 ∼ f_N-1^2f_N^2/M_*^2 m_SUSY^2 ∼ m_SUSY^4,
f_eff = √(∑_i=1^N (∏_j=i^N-1 n_j^2) f_i^2) ∼ (∏_j=1^N-1 n_j) f_1, and $\delta$ is a phase angle which is generically of order unity.
In our scheme, the MSSM Higgsino mass $\mu_{\rm eff}$ originates from $W_3$, and therefore is naturally of the order of $m_{\rm SUSY}$ <cit.>.
Again, after integrating out the $(N-1)$ heavy axions,
we find the MSSM Higgs parameters $\mu_{\rm eff}$ and $B\mu_{\rm eff}$ depend on the relaxion $\phi$ as
μ_eff = μ_N-1 exp(- i 2 n_N-1 ϕ/f_eff)+ μ_N exp(i 2 ϕ/f_eff),
Bμ_eff = b_N-1 exp(- i 2 n_N-1 ϕ/f_eff) + b_N exp(i 2 ϕ/f_eff),
|μ_N| ∼ |μ_N-1| ∼ f^2/M_* ∼ m_SUSY, |b_N| ∼ |b_N-1| ∼ m_SUSY^2.Then the determinant of the MSSM Higgs mass matrix
D = (m_H_u^2 + |μ_eff|^2 ) (m_H_d^2 + |μ_eff|^2 ) - |Bμ_eff|^2
also depends on $\phi$ via
|μ_eff|^2 = |μ_N|^2 + |μ_N-1|^2 +2|μ_Nμ_N-1|cos( 2(n_N-1+1)ϕ/f_eff + δ_μ_N - δ_μ_N-1),
|Bμ_eff|^2 = |b_N|^2 + |b_N-1|^2 +2|b_Nb_N-1|cos( 2(n_N-1+1)ϕ/f_eff + δ_b_N - δ_b_N-1),
where $\delta_{\mu}$ and $\delta_{b}$ are the phases of $\mu$ and $b$, respectively.
Obviously, for an appropriate range of $\delta_\mu$ and $\delta_b$, the determinant ${\cal D}$
can flip its sign from positive to negative as the relaxion experiences an excursion of ${\cal O}(f_{\rm eff})$.
Once the relaxion is stabilized near
the point of ${\cal D}=0$, the MSSM Higgs doublets $H_u$ and $H_d$ can be decomposed into the light SM Higgs $h$ with a mass
of ${\cal O}(v)$
and the other heavy Higgs bosons having a mass of the order of $m_{\rm SUSY}\gg v$.
To complete the model, we need to generate the back reaction potential $V_{\rm br}$. In regard to this, we simply adopt the schemes suggested in <cit.>. One option is
to generate $V_{\rm br}$ through the QCD anomaly. For this, one can introduce
W_br = X_1 QQ^c,where $Q+Q^c$ is an exotic quark which receive a heavy mass by $\langle X_1\rangle \sim f_1$.
Once this heavy quark is integrated out, the axion $\phi_1$ couples to the gluons as
After the $(N-1)$ heavy axions are integrated out, this leads to the relaxion-gluon coupling
1/32π^2ϕ/f(F F̃)_QCD,
f = f_eff/(∏_j=1^N-1 n_j) ∼ f_1.Then the resulting back reaction potential is obtained to be
V_br(h,ϕ) ≈ -y_uΛ^3_QCDh cos(ϕ/f +δ_br),
where $y_u$ is the up quark Yukawa coupling to the SM Higgs field $h$, and $\delta_{\rm br}$ is a phase angle of order unity.
Alternatively, we can consider a back reaction potential generated by an $SU(n_{HC})$ hidden color gauge interaction which confines at scales around the weak scale <cit.>. For this, one can introduce the hidden colored matter superfields
L+L^c, N+N^c
with the superpotential couplings
W_br = κ_1 X_1^2/M_* LL^c + κ_uH_u LN^c + κ_d H_d L^c N,
where $L$ is an $SU(n_{HC})$-fundamental and $SU(2)_L$-doublet with the $U(1)_Y$ charge 1/2, $L^c$ is its conjugate representation,
$N$ is an $SU(n_{HC})$-fundamental but $SU(2)_L\times U(1)_Y$ singlet, and $N^c$ is its conjugate representation.
At scales below $m_{\rm SUSY}$, all superpartners can be integrated out, leaving the following Yukawa interactions
between the relevant light degrees of freedom:
L_br = m_L e^2iϕ_1/f_1LL^c +κ_usinβhLN^c +κ_d cosβh^†L^cN + m_NNN^c,where $L+L^c$ and $N+N^c$ denote the fermion components of the original superfields, $\tan\beta =\langle H_u\rangle/\langle H_d\rangle$, and
m_L ∼ κ_1f_1^2/M_* ∼ κ_1 m_SUSYis presumed to be lighter than $m_{\rm SUSY}$.
Note that a nonzero Dirac mass of $N+N^c$ is induced by radiative corrections below $m_{\rm SUSY}$, giving
m_N ∼ 1/16π^2sin(2β)κ_u κ_dm_L^* e^-2iϕ_1/f_1ln(m_SUSY/m_L).
Now this effective theory at scales below $m_{\rm SUSY}$ corresponds to the non-QCD model proposed in <cit.>, yielding
a back reaction potential of the form
V_br = m_1^2 hh^†cos(2ϕ/f+δ_1) + m_2^4cos(2ϕ/f+δ_2) ,where we have expressed the axion component $\phi_1$ in terms of the light relaxion field $\phi$, and
m_1^2 ∼ κ_uκ_dsin(2β)/m_LΛ_HC^3, m_2^4 ∼ m_NΛ_HC^3for the $SU(n_{HC})$ confinement scale $\Lambda_{\rm HC}$.
If $m_2^4 < m_1^2 v^2$ with $m_1^2 \lesssim {\cal O}(v^2)$, which can be achieved for $m_L< 4\pi v$ <cit.>, this back reaction potential can successfully stabilize the relaxion at a value giving $v=\langle h\rangle \ll m_{\rm SUSY}$.
§ CONCLUSION
In this paper, we have addressed the problem of huge scale hierarchy in the relaxion mechanism, i.e. a relaxion excursion
$\Delta \phi \sim 2\pi f_{\rm eff}$
which is bigger than the period $2\pi f$ of the back reaction potential by many orders of magnitudes. We proposed a scheme
to yield an exponentially long relaxion direction within the compact field space of $N$ periodic axions with decay constants well below the Planck scale, giving
$f_{\rm eff}/f\sim e^{\xi N}$ with $\xi={\cal O}(1)$.
Although it relies on a specific form of the mass mixing between nearby axions, our scheme does not involve any fine tuning of continuous parameters, nor an unreasonably large discrete parameter. Furthermore, our scheme finds a natural UV completion in high scale or (mini) split SUSY scenario, in which all decay constants of the $N$ periodic axions are generated by SUSY breaking as
$f_i \sim \sqrt{m_{\rm SUSY} M_*}$, where $m_{\rm SUSY}$ denotes the soft SUSY breaking scalar masses and $M_*$ is the fundamental scale such as the Planck scale or the GUT scale. In our model, the required relaxion potential and the relaxion-dependent Higgs boson mass are generated through a superpotential term providing a natural solution to the MSSM $\mu$-problem.
§ ACKNOWLEDGMENT
We would like to thank Hyungjin Kim for helpful discussions.
This work was supported by IBS under the project code, IBS-R018-D1.
P. W. Graham, D. E. Kaplan and S. Rajendran,
Phys. Rev. Lett. 115, no. 22, 221801 (2015)
[arXiv:1504.07551 [hep-ph]].
G. Dvali and A. Vilenkin,
Phys. Rev. D 70, 063501 (2004)
J. R. Espinosa, C. Grojean, G. Panico, A. Pomarol, O. Pujolŕs and G. Servant,
Phys. Rev. Lett. 115, no. 25, 251803 (2015)
[arXiv:1506.09217 [hep-ph]].
E. Hardy,
JHEP 1511, 077 (2015)
[arXiv:1507.07525 [hep-ph]].
S. P. Patil and P. Schwaller,
arXiv:1507.08649 [hep-ph].
J. Jaeckel, V. M. Mehta and L. T. Witkowski,
arXiv:1508.03321 [hep-ph].
R. S. Gupta, Z. Komargodski, G. Perez and L. Ubaldi,
arXiv:1509.00047 [hep-ph].
B. Batell, G. F. Giudice and M. McCullough,
JHEP 1512, 162 (2015)
[arXiv:1509.00834 [hep-ph]].
O. Matsedonskyi,
JHEP 1601, 063 (2016)
[arXiv:1509.03583 [hep-ph]].
L. Marzola and M. Raidal,
arXiv:1510.00710 [hep-ph].
J. E. Kim, H. P. Nilles and M. Peloso,
JCAP 0501, 005 (2005)
K. Choi, H. Kim and S. Yun,
Phys. Rev. D 90, 023545 (2014)
[arXiv:1404.6209 [hep-th]].
J. E. Kim and H. P. Nilles,
Phys. Lett. B 138, 150 (1984).
H. Murayama, H. Suzuki and T. Yanagida,
Phys. Lett. B 291, 418 (1992).
K. Choi, E. J. Chun and J. E. Kim,
Phys. Lett. B 403, 209 (1997)
O. Antipin and M. Redi,
JHEP 1512, 031 (2015)
[arXiv:1508.01112 [hep-ph]].
K. A. Intriligator and N. Seiberg,
Nucl. Phys. Proc. Suppl. 45BC, 1 (1996)
M. T. Grisaru, M. Rocek and R. von Unge,
Phys. Lett. B 383, 415 (1996)
|
1511.00127
|
Department of Physics, Banaras Hindu University, Varanasi 221 005,India
Base-pockets (non-complementary base-pairs) in a double-stranded DNA
play a crucial role in biological processes. Because of thermal fluctuations,
it can lower the stability of DNA, whereas, in case of DNA aptamer, small
molecules e.g. adenosinemonophosphate(AMP), adenosinetriphosphate(ATP) etc, form
additional hydrogen bonds with base-pockets termed as
“binding-pockets", which enhance the stability. Using the Langevin Dynamics
simulations of coarse grained model of DNA followed by atomistic simulations,
we investigated the influence of base-pocket and binding-pocket on the stability
of DNA aptamer.
Striking differences have been reported here for the separation
induced by temperature and force, which require further investigation by single molecule experiments.
§ INTRODUCTION
Aptamers are Guanine(G)-rich short oligonucleic acids (DNA, RNA), which can perform
specific function <cit.>.
These have been developed in vitro through SELEX (Systematic
Evolution of Ligands by Exponential Enrichment) process
for better understanding of the behavior of antibodies, which are produced
in vivo or in living cells <cit.>.
One of the most extensively characterized
examples of aptamer is found in telomerase at the ends of eukaryotic chromosomes, where it
plays an important role in gene regulation <cit.>.
DNA loops or base-pockets consisting of Guanine can interact with small molecules
and proteins and thus enhance their stability, affinity and specificity <cit.>.
For example, adenosinemonophosphate (AMP) binds to DNA loop (termed as binding-pocket)
with eight hydrogen bonds, that increases the stability <cit.> of
aptamer. It served as the target of drugs for cancer treatment <cit.>.
Their high affinity and selectivity with target
proteins make them ideal and powerful probes in biosensors and potent
pharmaceuticals <cit.>.
Thus, understanding of the conformational stability of DNA aptamers before and after
the release of drug molecule, and the influence of the binding of other molecules, are of crucial importance.
Schematic representations of dsDNA under a shear force applied at
the opposite ends with varying position of (a) binding-pocket (AMP),
(b) base-pocket (without AMP). In the constant velocity simulation, one
end marked by blank circle is kept fixed, while in the constant force simulation,
the force is applied at both ends. The binding-pocket
(AMP within circle)Fig. 1(a) corresponds to additional binding of hydrogen bonds with AMP.
The circle in Fig. 1(b) represents the base-pocket
consisting of $G$ (say), where base-pairing is absent.
A double-stranded DNA (dsDNA) can be separated in two single-stranded DNA
by increasing the temperature and the process is termed as DNA melting.
Traditional spectroscopic techniques used e.g. fluorescence spectroscopy, $UV-$vis
spectroscopy etc. usually provide average responses for molecular interactions <cit.>.
Single molecule force spectroscopy (SMFS) techniques have emerged
as valuable tools for measuring the molecular interactions on a
single molecule level, and thus unprecedented information about the stability of
bio-molecules have been achieved.
For example, DNA rupture induced by SMFS techniques
have been used to understand the strength of hydrogen bonds in nucleic acids,
ligands-nucleic interaction, protein-DNA interactions etc.
<cit.>. In DNA rupture all intact base-pairs break
simultaneously and two strands get separated, when a force is applied either at $3' -3'$ or $5' -5'$ ends.
This force is identified as rupture force.
Motivated by these studies, attempts have recently been
made to understand the enhanced stability of DNA aptamers <cit.>. For example, Nguyen et al. <cit.> measured
the changes in rupture force of a DNA aptamer (that forms binding-pocket)
with AMP (Fig. 1(a)) and without AMP (Fig. 1(b)),
and thereby determined the dissociation constant at single-molecule level.
Papamichael et al. <cit.> used an aptamer-coated probe and an IgE-coated
mica surface to identify specific binding areas. Efforts have also been made to
determine the rupture force of aptamer binding to proteins and cells, and it was
revealed that the binding-pocket enhances the rupture force by many
folds <cit.>.
Despite the progress made, there are noticeable lack of investigations.
For example, the melting of DNA aptamer in presence and absence of AMP and its dependence on the
position in a DNA strand need to be measured and understood correctly.
The aim of this paper is to develop a theoretical model to understand the role
of a base-pocket and a binding-pocket on the rupture of a DNA aptamer.
For this, a coarse grained model of DNA is developed.
Here, dsDNA is made up of two segments. One of the segments
consists of a DNA loop or the base-pocket of eight G type nucleotides. The other segment is stem,
which is made up of twelve G-C base-pairs. We vary the position of the binding-pocket (Fig. 1 (a)) and the
base-pocket (Fig. 1(b)) along the chain, and measure the rupture force and melting temperature
for both cases (with and without AMP). For the first time, we report the profile as a function
of base/binding-pocket position. We find that even though the melting profile has one
minima (U-shape), there are two minima (W-shape) for rupture. This reflects that there are
two symmetric positions, where the system can be more unstable. Extensive atomistic
simulations have been performed to validate these findings, which helped us to
delineate the correct understanding of the role of base-pockets in the stability of DNA aptamer.
In Sec. II, we briefly describe the model and the Langevin dynamics simulation to study the rupture of
DNA aptamer <cit.>. The discussion on the melting and rupture profile of DNA aptamer as
a function of base/binding-pocket position has also been made in this section. In Sec. III,
we briefly explain the atomistic simulation <cit.> and discuss the
model independency of the results. Finally in Sec. IV, we conclude with a discussion on
some future perspectives.
§ MODEL AND METHOD
We first adopt a minimal model introduced in Ref. <cit.> for a homo-sequence of
dsDNA consisting of $N$ base-pairs, where covalent bonds and base-pairing interactions are
modelled by harmonic springs and Lennard-Jones (LJ) potentials, respectively. By
using Langevin dynamics simulation, it was shown that the rupture
force and the melting temperature remain qualitatively similar to
the experiments <cit.>.
Energy of the model system <cit.> is given by.
\begin{eqnarray}
& & E = {\sum_{l=1}^2\sum_{j=1}^N}k({\bf r}_{j+1,j}^{(l)}-d_0)^2
+{\sum_{l=1}^2\sum_{i=1}^{N-2}\sum_{j>i+1}^N}4\left(\frac{C}{{{\bf r}_{i,j}^{(l)}}^{12}}\right) \nonumber \\
& & + {\sum_{i=1}^N\sum_{j=1}^N}4\epsilon\left(\frac{C}{(|{\bf r}_i^{(1)}-{\bf r}_j^{(2)}|)^{12}}-
\frac{A}{(|{\bf r}_i^{(1)}-{\bf r}_j^{(2)}|)^6}\delta_{ij}\right),
\end{eqnarray}
where $N$ is the number of beads in each strand. Here, ${\bf r}_j^{(l)}$ represents the position of bead $j$
on strand $l$. In the present case, $l=1(2)$ corresponds to first (complementary)
strand of dsDNA. The distance between intra-strand beads, ${\bf r}_{i,j}^{(l)}$, is defined as
$|{\bf r}_i^{(l)}-{\bf r}_j^{(l)}|$.
The harmonic (first) term with spring constant $k = 100$ couples the adjacent beads along each
strand. The parameter $d_0 (=1.12)$ corresponds to the equilibrium distance in the harmonic
potential, which is close to the equilibrium position of the LJ potential. The second term
takes care of excluded volume effects i.e., two beads can not occupy the same
space <cit.>. The third term, described by the Lennard-Jones (LJ) potential, takes care
of the mutual interaction between the two strands. The first term of LJ potential
(same as second term of Eq.1) will not allow the overlap of two strands.
The second term of the LJ potential corresponds to the base-pairing between
two strands. The base-pairing interaction is restricted to the native contacts
($\delta_{ij}=1$) only i.e., the $i^{th}$ base of the $1^{st}$ strand forms pair
with the $i^{th}$ base of the $2^{nd}$ strand only. It is to be noted here that $\epsilon$
represents the strength of the LJ potential. In Eq. 1, we use dimensionless distances
and energy parameters and set $\epsilon=1$, $C = 1$ and $A= 1$, which corresponds to a
homosequence dsDNA. The binding-pocket and base-pocket can be modelled by substituting
$A = 1$ $ { \& }$ $ {\epsilon =2}$ (Fig. 1(a)) and $A=0$ ${ \& }$ $ { \epsilon =1}$ (Fig. 1(b)),
respectively, among the bases inside the circle. The equation of motion is obtained
from the following Langevin equation:
(a) Probability distribution of rupture force of DNA aptamer with AMP(solid line)
and without AMP(dashed line) for the same loading rate(0.0205).
(b) Variation of rupture force with loading rate.
\begin{equation}
m\frac{d^2{\bf r}}{dt^2} = -{\zeta}\frac{d{\bf r}}{dt}+{\bf F_c(t)}+\bf{\Gamma(t)},
\end{equation}
where $m ( =1 )$ and $\zeta (=0.4)$ are the mass of a bead and the friction
coefficient, respectively. Here, ${\bf F_c}$ is defined as $-\frac{dE}{d{\bf r}}$ and
the random force ${\bf \Gamma}$ is a white noise <cit.>,
i.e., $<{{\bf\Gamma}(t){\bf\Gamma}(t')}>= 6 \zeta T\delta(t-t')$. The $6^{th}$ order
predictor-corrector algorithm with time step
$\delta t$=0.025 has been used to integrate the equation of motion.
The results are averaged over many trajectories.
First, we have calculated the rupture force of a dsDNA of two different
lengths <cit.>. We have inserted a base-pocket in the interior of the chain
(circle in Fig. 1(b) and calculated the required force for
the rupture at temperature $T = 0.12$. In order to obtain the rupture force
for the DNA aptamer, we switched on the interaction among the base-pocket nucleotides with
AMP (Fig. 1(a)): each of these extra interaction strength is double
of the base-pairing interactions. Following the experimental protocol <cit.>,
we performed constant velocity simulation <cit.> by fixing one end of a DNA strand marked
by a blank circle in Fig. 1. Force $f = K(vt - x)$ is applied on the other end of the strand
marked by filled circle in Fig. 1. Here, $x$ is the displacement of the pulled monomer
from its original position, $v$ is the velocity, $t$ is the time and $K (= 0.8)$
is the spring constant <cit.>. The rupture force is identified as the maximum force, at which two strands separate suddenly. A selection of rupture force distribution of 500 events have been shown in Fig. 2(a).
For a given loading rate, the most probable rupture force $f_c$ is obtained by the Gaussian fit of the distribution
of rupture force <cit.>. In Fig. 2(b), we have shown the variation of $f_c$ (with and without AMP) with
the loading rate ($Kv$). It is apparent from the plots that the rupture force required
for the DNA aptamer is larger than the DNA (without AMP) with base-pocket, and
the qualitative nature remains same as seen in the experiment <cit.>.
(a) Variation of $T_m$ with binding-pocket positions
and (b) with base-pocket positions. (c) Variation of $f_c$ with
binding-pocket positions and (d) with base-pocket positions.
For the better understanding of the role of base-pocket and
binding-pocket, we varied its position along the chain continuously
from one end to the other, and calculated the melting temperature ($T_m$) and
rupture force ($f_c$) for each position in the constant force ensemble (CFE) <cit.>.
The melting temperature is obtained here by monitoring the energy fluctuation ($\Delta E$) or the
specific heat ($C$) with temperature, which are given by the following relations
\begin{eqnarray}
<\Delta E> & = & <E^2>-<E>^2 \\
C & = & \frac{<\Delta E>}{T^2}.
\end{eqnarray}
The peak in the specific heat curve gives the melting temperature.
Since this simulation is in the equilibrium, we have used 10 realizations with different seeds and
reported the mean value of the rupture force.
In Fig.3 (a)$-$(d), we have plotted the variation of $T_m$ (at $f=0$) and $f_c$ ($T=0.06$)
much below $T_m$ as a function of pocket position ($D_i$).
It is interesting to note that for thermal melting, the variation looks
like $``\cap$ shape" for the DNA aptamer (with AMP) shown in Fig.3(a), and for without AMP,
profile has “U- shape" having one minima (Fig. 3(b). The rupture force remains invariant
with binding-pocket positions for AMP (Fig.3(c)), which is consistent with experiment <cit.>.
Interestingly, in the absence of AMP, the profile has “W-shape", i.e.,
with two minima (Fig.3(d)). Although, one may intuitively expect the shape of profiles to be
symmetric for the homosequence. But it is not apparent, why does the profile has one
minimum for the thermal melting and two minima for the DNA rupture in the absence of AMP.
We now confine ourselves to understand these issues. In case of DNA aptamer melting,
one would expect naively that the aptamer is more stable, when the binding-pocket is
in the interior of the chain. Thus, $T_m$ should be high (Fig. 3(a))
compare to the binding pocket at the end resulting in a $``\cap$ shape" profile. In
absence of AMP, the profile looks like “U-shape".
For a short chain, DNA melting is well described by the two state model <cit.> with
$\Delta G= \Delta H - T \Delta S$, where $H$ and $S$ are the enthalapy and entropy, respectively.
The melting temperature $ T_m = \frac{\Delta H}{\Delta S}$ corresponds to the state with $\Delta G =0$
indicating that the system goes from the bound-state to the open-state and change in the free energy of the
system is zero. It is often practically easier to identify $T_m$ as the temperature where $50\%$ hydrogen
bonds are broken.
For a homo-sequence chain (without base-pocket), it was shown that the chain opens from the end
rather than the interior of the chain <cit.>. In such a case, the major contribution
to the entropy comes from the opening of base-pairs near the end of the chain, and
decreases to zero, when one approaches to the interior of the chain.
In this case, there are two contributions to the entropy: entropy associated with opening of the
end base-pairs ($S_E$) and the base-pocket entropy ($S_{BP}$).
When a base-pocket is inserted at the end of DNA chain, there is an additional contribution
to the entropy, which is greater than $\Delta S_E$ or ($\Delta S_{BP}$), but less than the sum of two. Thus, $T_m$ decreases to $\sim 0.171$ from $\sim 0.181$ <cit.>. As the base-pocket moves along
the chain, entropy of interior base-pairs increases. Combined effect of both entropies reduces the
melting temperature further. Once the base-pocket is deep inside, the total entropy of the system
becomes equal to the sum of end-entropy and base-pocket entropy. As a result,
$T_m$ remains constant($\sim 0.161$) (Fig. 3(b)) irrespective of the base-pocket position. As base-pocket
approaches towards the other end, by symmetry we observe “U-"shape profile. The conversion of reduced unit to real unit indicates that it is possible to observe it in vitro <cit.>.
Understanding the decrease in $T_m$ with base-pocket position does not
explain, why the rupture profile of DNA has two minima. It is clear
from the Fig. 3 (b) & (d) that the base-pocket entropy alone is not responsible for this ($W$)
shape. Based on the ladder model of DNA (homo-sequence), de Gennes proposed that the rupture force
$f_c$ is equal to $2 f_1 (\chi^{-1} \tanh(\chi \frac{N}{2}))$ <cit.>.
Here, $f_1 $ is the force required to separate a single base-pair and $\chi^{-1}$ is
the de Gennes length, which is defined as $\sqrt{\frac{Q}{2R}}$. $Q$ and $R$ are the
spring constants characteristic of stretching of the backbone and hydrogen bonds
<cit.>, respectively.
de Gennes length is the length over which differential force is distributed.
Above this length, the differential force approaches to
zero and there is no extension in hydrogen bonds due to the applied force <cit.>.
When the base-pocket is at the end, four bases of one strand is under the tension,
whereas the complementary four bases are free, thereby increasing the entropy of the system.
The four bases act like a tethered length and a force is required to keep it
stretched so that rupture can take place. It is nearly equal to the ruptured force of
a 12 base-pair dsDNA with no defect. As the base-pocket moves towards the center, the entropy of
base-pocket decreases as the ends of the base-pocket is now not free.
Since, the de Gennes length for the base-pocket is infinite ($R =0$), it
implies that the differential force remains constant inside the base-pocket, thereby decreases
the stability of DNA, and as a result, rupture force decreases further. The force
acting at ends penetrates only up to the de Gennes length and within this, the rupture
force keeps on decreasing as base-pocket moves and approaches to its minimum. Above a
certain length, the rupture force starts increasing, which can be seen in Fig.3 (d)
and approaches to the maximum at the middle of the chain. By symmetry, we get profile of two minima of “W-shape".
Variation of $f_c$ with base-pocket positions for different
It should be pointed here, that the de Gennes length in the present model is about ten
<cit.>, but for a DNA of length $16$ base-pairs (12 complementary and 4 non complementary),
the minima is around $D_i = 4$. Note that the penetration depth for the
two ends (say $5'-5'$) is different because of the asymmetry arising due to the presence of base-pocket, and
hence the minima shifts. If we increase the length of DNA, keeping the base-pocket size constant,
one would then expect that the minima will shift towards 10 (above the de Gennes length,
the end-effect vanishes). This indeed we see in Fig. 4, where the variation of $f_c$
with base-pocket position for different chain lengths ($N = 8,10,12,16,20,24$ and $32$)
has been plotted. In all cases, profiles have two minima, whereas minima shifts towards
10 as $N$ increases.
de Gennes equation predicts that $f_c$ increases linearly with length for
small values of $N$, and saturates at the higher values of $N$, which is consistent
with recent experiment <cit.> and simulations <cit.>.
Surprisingly, right side of the profile also appears to follow the de Gennes equation.
This implies that, the base-pocket can reduce
the effective length of the chain, so that the rupture force is less compare
to the bulk value and approaches to a minimum value at a certain position.
As the end-effect decreases, $f_c$ starts increasing and
approaches to its bulk value.
§ ATOMISTIC SIMULATION
In order to rule out the possibility that the above effect may be a consequence of
the adopted model, we performed the atomistic simulations with explicit solvent.
Here, we have taken homo-sequence DNA consisting of 12 G-C bps and a varying base-pocket of
$8$ G-nucleotides <cit.>. The starting structure of the DNA
duplex sequence having the base-pocket is built using make-na server<cit.>.
We used AMBER10 software package <cit.> with all atom (ff99SB)
force field <cit.> to carry the simulation. Using the LEaP module in AMBER,
we add the AMP molecule (Fig. 5(a)) in the base-pocket and then the $Na^{+}$ (counterions) to
neutralize the negative charges on phosphate backbone group of DNA structure (Fig. 5(b)). This
neutralized DNA aptamer structure is immersed in water box using TIP3P model for water <cit.>. We have
chosen the box dimension in such a way that the ruptured DNA aptamer structure remains fully inside the water
box. We have taken the box size of $57 \times 56 \times 183$ $\AA^3$
which contains $15674$ water molecules and $30$ $Na^{+}$ (counterions).
A force routine
has been added in AMBER10 to do simulation at constant force <cit.>.
In this case, the force has been applied at
$5'-5'$ ends . The electrostatic interactions have been calculated
with Particle Mesh Ewald (PME) method <cit.> using a cubic B-spline interpolation of
order 4 and a $10^{-5}$ tolerance is set for the direct space sum cut off. A real space cut off of
10 $\AA$ is used for both the van der Waal and the electrostatic interactions.
The system is
equilibrated at $F =0 $ for 100 ps under a protocol described in Ref. <cit.> and it
has been ensured that AMP has bound with the base-pocket.
We carried out simulations in the isothermal-isobaric (NPT) ensemble using a time step of
1 fs for 10 different realizations. We maintain the constant pressure by isotropic position scaling <cit.> with a
reference pressure of 1 atm and a relaxation time of 2 ps. The constant temperature was
maintained at 300 K using Langevin thermostat with a collision frequency of 1 ps$^{-1}$.
We have used $3D$ periodic boundary conditions during the simulation.
(Color online) (a) Structure of AMP molecule, which has been inserted in the base-pocket.
(b) Snapshots of binding of AMP molecule at three different positions ($D_i$=1, 4 and 7)
in the DNA.
Because of extensive time involved in the
computation, we restricted ourselves at three different (extremum) base-pocket positions
$D_i$=1, 4 and 7 (Fig. 5(b)) and calculated the rupture force with 10 realization of different seeds
as a mean force.
The required rupture forces for these positions are 840 pN $\pm$ 20 pN, 720 pN $\pm$ 20 pN
and 830 pN $\pm$ 20 pN, indicating that the complete profile contains two minima <cit.>.
In presence of AMP, which interacts with the base-pocket, we find $f_c = $925pN $\pm$ 20pN
remains constant irrespective of binding-pocket position as seen
earlier (Fig. 3(c)). This validates the finding of simple coarse grained model, which captured
some essential but unexplored aspects of rupture mechanism of DNA aptamer.
§ CONCLUSIONS
Our numerical studies clearly demonstrate that the force induced rupture
and thermal melting of DNA aptamer will vary quite significantly. In melting, all the nucleotides get almost
equal thermal knock from the solvent molecules, whereas in DNA rupture,
force is applied at the ends and the differential force acts only up to the
de Gennes length <cit.>. As a result, the stability of DNA in presence
of base-pocket is strikingly different (single minima vs double minima).
This may have biological/pharmaceutical significance because after the release of
drug molecule, the stability of carrier DNA depends on the position from which it
is released. Hence, at this stage our studies warrant further investigations most likely
by new experiments to explore the role of base-pockets and its position, which will significantly
enhance our understanding about the stability of DNA aptamer and its suitability in the
development and designing of drugs.
§ ACKNOWLEDGEMENTS
We thank G. Mishra, D. Giri and D. Dhar for many helpful discussions on the subject.
We acknowledge financial supports from the DST, UGC and CSIR, New Delhi, India.
Gold C. Tuerk and L. Gold, Science 249, 505 (1999).
Szostak A. D. Ellington and J. W. Szostak, Nature 346, 818 (1990).
tan M.B. O'Donoghue, Xi. Shi, X. Fang and W. Tan,
Anal. Bioanal. Chem. 402, 3205 (2012).
sarkies P. Sarkies, C. Reams, L. J. Simpson and J. E. Sale, Mol.Cell 40, 703 (2010).
bejugam M. Bejugam et al.. J. Am. Chem. Soc. 129, 12926 (2007).
Keefe A. D. Keefe, S. Pai, and A. Ellington, Nature Reviews Drug Discovery, 9, 537 (2010).
LinC. H. Lin and D. J. Patel, Chem. Biol. 4, 817 (1997).
Teller C. Teller, S. Shimron and I. Willner, Anal. Chem. 81, 9114 (2009).
patel S. Nonin-Lecomte, C. H. Lin and D. J. Patel, Biophys. J. 81, 3422 (2001).
patel1 D. J. Patel, A. T. Phan and V. Kuryavyi, Nucleic Acids Res. 35, 7429 (2007).
Sullenger B. A. Sullenger and E. Gilboa, Nature 418, 252 (2002).
Hoppe F. Hoppe-Seyler and K. Butz, J. Mol. Med. 78, 426 (2000).
Cho E. J. Cho, J. W. Lee and A. D. Ellington, Annu. Rev. Ana. Chem. 2, 241 (2009).
chen J. Chen, Z. Fang, P. Lie and L. Zeng, Anal. Chem. 84, 6321 (2012).
mao P. M. Yangyuoru, S. Dhakal, Z. Yu, D. Koirala, S. M. Mwongela and
H. Mao, Anal. Chem. 84, 5298 (2012).
Zayats M. Zayats, Y. Huang, R. Gill, Chun-an Ma and I. Willner, J. Am. Chem Soc. 128, 13666 (2006).
Zuo X. Zuo, Y. Xiao and K. W. Plaxco, J. Am. Chem Soc. 131, 6944 (2009).
Boyacioglu B. Olcay, H. S. Christopher, K. George and H. G. William, MTNA 2, e107 (2013).
Sun H. Sun, X. Zhu, P. Y. Lu, R. R. Rosato, W. Tan, and Y. Zu, MTNA 3, e182 (2014).
Zhu G. Zhu et al., PNAS 110, 7998 (2013).
Lele L. Li et al., PNAS 111, 17099 (2014).
wartel R. Wartell and A. Benight, Phys. Rep. 126, 67 (1985).
Lee_Science94 G. U. Lee, L. A. Chrisey and R. J. Colton, Science 266, 771 (1994).
Strunge T. Strunz, K. Oroszlan, R. Schäfer and H. J. Güentherodt, PNAS 96, 11277 (1999).
hatch K. Hatch et al., Phys. Rev. E 78, 011920 (2008).
cludia C. Danilowicz, et al., PNAS 106, 13196 (2009).
Nguyen T-H Nguyen, L. J. Steinbock, H-J. Butt, M. Helm and R. Berger, J. Am. Chem Soc. 133, 2025 (2011).
papa I. K. Papamichael, P. M. Kreuzer and G. Guilbault, Sens. Actuators B 121, 178 (2007).
Zlatanova J. Zlatanova, M. S. Lindsay and H. S. Leuba, Prog. Biophys. Mol. Biol. 74, 37 (2000).
Yu J. P. Yu, Y. X. Jiang, X. Y. Ma, Y. Lin and X. H. Fang, Chem. Asian J. 2, 284 (2007).
Jiang Y. Jiang, C. Zhu, L. Ling, L. Wan, X. Fang and C. Bai, Anal. Chem. 75, 2112 (2003).
mishra R. K. Mishra, G. Mishra, M. S. Li and S. Kumar, Phys. Rev. E 84, 032903 (2011).
cg A FORTRAN code is developed to study the dynamics of the system.
Case D. A. Case et al., AMBER 10, University of California, San Francisco (2008).
Duan Y. Duan et al., J. Comput. Chem. 24, 1999 (2003).
nath S. Nath, T. Modi, R. K. Mishra, D. Giri, B. P. Mandal and S. Kumar, J. Chem. Phys. 139, 165101 (2013).
Allen M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids
(Oxford Science, Oxford, UK,) (1987).
Smith D. Frenkel and B. Smit, Understanding Molecular Simulation (Academic Press UK) (2002).
scaling P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University
Press, Ithaca) (1979), S. Kumar and Y. Singh, Phys. Rev. E 48, 734 (1993).
text0 For a chain of 16 base-pairs, the rupture force ($f_c$) and the melting
temperature ($T_m$) are $\sim 16$ and $\sim 0.181$ respectively. Whereas for 12 base-pairs, their values are $\sim 12$ and $\sim 0.171$,
Li1 M. S. Li Biophys. J. 93, 2644 (2007).
Li M. S. Li and M. Cieplak, Phys. Rev E 59, 970 (1999).
sl J. SantaLucia Jr. and D. Hicks, Annul. Rev. Biophys. Biomol. Struct 33 415 (2004).
shikha_jcp S. Srivastava and N. Singh, J. Chem. Phys. 134, 115102 (2011).
text Following relations may be used to convert dimensionless units to real units:
$ T= \frac{{k_{b}}T^{*}}{\epsilon} $, $ t^{*} = (\frac{m\sigma^2}{\epsilon})^{\frac{1}{2}} t $ and
$ r = \frac{r*}{\sigma} $ <cit.>, where $ T^{*}$, $ t^{*}, r^{*}$, and $ \epsilon $ are temperature, time, distance, and characteristic hydrogen bond energy in real units, respectively.
$\sigma$ is the inter-particle distance at which the potential approaches to zero. Now, if one sets effective base paring energy $ \epsilon \sim 0.17 $ $ ev $, which is obtained by comparing the melting temperature in the reduced $T_{m}$($ = 0.161$) to the one obtained from the oligo-calculator <cit.> for the same sequence, one gets $T_{m} \approx 56^{\circ}C$. By
setting average mass of each bead $ \approx 5 \times 10^{-22} g $ and $\sigma = 5.17 \AA$, one can obtained
relation between real time $ t^{*} $ and reduced time $ t $ as $ t^{*} \sim 3 t $ ps.
Kibbe W. A. Kibbe, Nucleic Acids Res. 35, W43 (2007).
degennes P. G. de Gennes, C. R. Acad. Sci. Paris 2, 1505 (2001).
text3 This approximate value can be obtained by putting different elastic constants involved in
the expression of $\chi^{-1}$. In the present simulation, we have used $k = Q$ (=100)
i.e. the spring constant associated with the covalent bonds along the backbone of the chain.
The spring constant $R$ associated with the base-pairs has been obtained by expanding
the LJ potential around its equilibrium value and setting it equal to zero above the
cut off distance, where rupture takes place. The second (harmonic) term of the expansion
gives the elastic constant $R \sim 0.5$ associated with the LJ potential involved
in the base-pairing.
Santosh M. Santosh and P. K. Maiti, J. Phys.: Condens. Matter 21, 034113 (2009).
ser http://structure.usc.edu/make-na/server.html.
Jorgensen W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L.
Klein, J. Chem. Phys. 79 (1983).
text00 We have used the constant force routine developed by P. K. Maiti and his group.
darden T. Darden, D. York and L. J. Pedersen, Chem. Phys. 98, 10089 (1993).
essmann U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee and L. G. Pedersen,
J. Chem. Phys. 103, 8577 (1995).
maiti P. K. Maiti, T. A. Pascal, N. Vaidehi and W. A. Goddard III, Nucleic Acids Res. 32, 6047 (2004).
nanolett P. K. Maiti and B. Bagchi, Nano Lett. 6, 2478 (2006).
text2 Here, the simulation time is about $7ns$, and hence rupture force is found
to be one order higher than AFM.
|
1511.00439
|
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de
Chile, Valdivia, Chile
Faculty of Physics, Department of Astrophysics - Astronomy - Mechanics,
University of Athens, Panepistemiopolis, Athens 157 83, Greece
Academy of Athens, Research Center for Astronomy and Applied Mathematics,
Soranou Efesiou 4, 11527, Athens, Greece
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce
Rd., Cambridge CB3 0WA, UK
98.80.-k, 95.35.+d, 95.36.+x
We consider the application of group invariant transformations in order to
constrain a flat isotropic and homogeneous cosmological model, containing of a
Brans-Dicke scalar field and a perfect fluid with a constant equation of state
parameter $w$, where the latter is not interacting with the scalar field in
the gravitational action integral. The requirement that the Wheeler-DeWitt
equation be invariant under one-parameter point transformations provides us
with two families of power-law potentials for the Brans-Dicke field, in which
the powers are functions of the Brans-Dicke parameter $\omega_{BD}$ and the
parameter $w$. The existence of the Lie symmetry in the Wheeler-DeWitt
equation is equivalent to the existence of a conserved quantity in field
equations and with oscillatory terms in the wavefunction of the universe. This
enables us to solve the field equations. For a specific value of the conserved
quantity, we find a closed-form solution for the Hubble factor, which is
equivalent to a cosmological model in general relativity containing two
perfect fluids. This provides us with different models for specific values of
the parameters $\omega_{BD},$ and $w$. Finally, the results hold for the
specific case where the Brans-Dicke parameter $\omega_{BD}$ is zero, that is,
for the O'Hanlon massive dilaton theory, and consequently for $f\left(
R\right) $ gravity in the metric formalism.
§ INTRODUCTION
The comprehensive analysis of various observational data (Cosmic Microwave
Background, Supernova Type Ia, large-scale structures, etc.) supports a
picture in which the universe is spatially flat with only about $30\%$ of its
total energy in the form of dark or luminous forms of matter. The nature of
the remaining $70\%$, residing in some unknown form of enigmatic 'dark energy'
remains a mystery even though it can be accurately described by a particular
type of anti-gravitating stress. Discovering the physics of this dark energy,
driving the recent accelerated expansion of the universe, is a key goal of
theoretical physics and cosmology. The intense debate among the cosmologists
and theoretical physics has opened up the possibility of various new
cosmological scenarios which offer different sources for the observed
acceleration and even a possibility to link it to the suspected era of
inflationary acceleration in the very early universe. Some of these scenarios
are based on the existence of new fields in nature, while others modify the
classical Einstein-Hilbert action for gravity
In Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies containing
scalar fields, some analytical solutions without matter can be found in
<cit.>. If a matter component is
included in the dynamics then new solutions were also found in
<cit.>. Furthermore, a new class of solutions has
been found from the application of group invariants, namely Lie/Noether
symmetries of the field equations <cit.>. In fact the idea to
use Noether point symmetries in cosmology is not new and indeed there is a lot
of work in the literature (see <cit.>). In some previous papers we have
provided the Lie/Noether symmetries for various cosmological models, including
scalar fields <cit.>, $f(R)$ <cit.>, $f(T)$ <cit.> and
scalar tensor theories <cit.>. Recently, we have used this dynamical
symmetry approach in order to provide solutions to the WdW equation
In general for scalar-tensor theories, including the particular case of
Brans-Dicke gravity <cit.>, various analytical solutions are available
in the literature <cit.>. In Ref.<cit.>, an exact
solution describes a Brans-Dicke scalar field which interacts with a perfect
fluid in the action integral. Moreover, some closed-form solutions for plane
symmetric spacetimes can be found in <cit.>, and some black-hole
solutions in Brans-Dicke gravity are given in <cit.>. Using the
method of group invariants, some new exact solutions without matter source are
found in <cit.>, and with dust in <cit.>.
The purpose of this paper is to extend the method proposed in Ref.<cit.>
to Brans-Dicke gravity when the perfect fluid does not interact with the
Brans-Dicke field in the action integral. The proposed selection rule
determines the potential that defines the Brans-Dicke field in order for the
Wheeler-DeWitt (WdW) equation to be invariant under a group of point
transformations. In <cit.> that proposed method was applied in general
relativistic cosmology for a homogeneous scalar field and a perfect fluid. It
has been shown that when the WdW equation is invariant under one-parameter
point transformation then, the WdW equation can be solved by separation of
variables. The solution
provides oscillatory terms in the wavefunction and, at the same time, the
point transformations give Noetherian conservation laws for the classical
field equations. This latter property can be used to study the integrability
of the field equations and extract closed-form solutions.
The structure of the paper is as follows. In Section <ref> we present
the field equations in Brans-Dicke gravity. In Section <ref> we apply Lie
point symmetries to the WdW equation and, in section <ref>, we
provide the invariant solution of the WdW equation. Also, in the same section,
we use the Hamilton-Jacobi theory to reduce the field equations to a pair of
first-order differential equations and under specific conditions (the
Noetherian conservation law vanishes) we obtain a closed-form solution for the
Hubble parameter in Brans-Dicke gravity. Then we check the performance of this
special Brans-Dicke model against the latest observational data. Finally, in
Section <ref> we discuss our results and we draw our conclusions.
§ FIELD EQUATIONS IN BRANS-DICKE GRAVITY
In Brans-Dicke gravity the gravitational action in the Jordan frame and matter
that is not interacting with the Brans-Dicke field is defined by
\begin{equation}
S=\int dx^{4}\sqrt{-g}\left[ \frac{1}{2}\phi R-\frac{1}{2}\frac{\omega_{BD}%
}{\phi}g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}-V\left( \phi\right) \right] +\int
dx^{4}\sqrt{-g}L_{m}, \label{bd.01}%
\end{equation}
where $L_{m}$ is the Lagrangian of the matter, $\phi$ is the Brans-Dicke
scalar field, and $\omega_{BD}$ is the Brans-Dicke parameter.
Variations of $S$ (<ref>) with respect to the metric tensor and the
field $\phi$ gives the modified Einstein field equations
\begin{equation}
\phi G_{\mu\nu}=\frac{\omega_{BD}}{\phi}\left( \phi_{;\mu}\phi_{;\nu}%
-g_{\mu\nu}V\left( \phi\right) -\left( g_{\mu\nu}g^{\kappa\lambda}%
\phi_{;\kappa\lambda}-\phi_{;\mu}\phi_{;\nu}\right) +kT_{\mu\nu},
\label{bd.02}%
\end{equation}
and the Klein-Gordon equation
\begin{equation}
+\frac{\phi}{2\omega_{BD}}\left( R-2V_{,\phi}\right) =0, \label{bd.04}%
\end{equation}
where $k\equiv8\pi G$. In (<ref>) $T_{\mu\nu}$ is the energy-momentum
tensor of matter. Since we have assumed that the matter is not interacting
with the Brans-Dicke field, we have the conservation law $T_{~~~;\nu}^{\mu\nu
In the following we study the solution of these equations under the following assumptions.
a. Spacetime is spatially flat with FLRW metric line element
\begin{equation}
ds^{2}=-dt^{2}+a^{2}\left( t\right) \left( dx^{2}+dy^{2}+dz^{2}\right) ,
\label{bd.05}%
\end{equation}
whose Ricci scalar is
\begin{equation}
R=6\left[ \frac{\ddot{a}}{a}+\left( \frac{\dot{a}}{a}\right) ^{2}\right] .
\label{bd.06}%
\end{equation}
b. Matter is a perfect fluid for comoving observers $u_{\mu}=\delta_{\mu}^{0}%
$; that is, the energy-momentum tensor is
\begin{equation}
T_{\mu\nu}=\left( \rho_{m}+p_{m}\right) u_{\mu}u_{\nu}+p_{m}g_{\mu\nu},
\label{bd.03}%
\end{equation}
where $\rho_{m},$ is the energy density of the matter and $p_{m}$ is the
isotropic pressure measured by the observers $u_{\mu}.$
c. The perfect fluid has a constant equation of state (EoS) parameter $w$,
i.e., $p_{m}=w\rho_{m}.$ For a barotropic fluid $w\in\left[ 0,1\right] $;
for $w=0$, $T_{\mu\nu}$, describes dust and for $w=1/3$, a radiation fluid. In
what follows we will extend the range of the EoS parameter to be $w\in\left(
-1,1\right) $ and consider effects 'fluids'. Of course, the lower limit,
$w=-1$, corresponds to a cosmological constant which can always be absorbed in
the scalar-field potential.
d. We assume that the scalar field, $\phi$, possesses the same symmetries as
the spacetime, that is $\phi\left( t,x,y,z\right) \equiv\phi\left(
t\right) $.
From the conservation law $T_{~~~;\nu}^{\mu\nu}=0$ we find that $\rho_{m}%
=\rho_{m0}a^{-3\left( 1+w\right) }$, where $\rho_{m0}$ is the energy density
of today, and $\rho_{m0}=3\Omega_{m0}H_{0}^{2}$. Under these assumptions the
Lagrangian of the field equations becomes
\begin{equation}
\mathcal{L}\left( a,\dot{a},\phi,\dot{\phi}\right) =-3a\phi\dot{a}%
{\phi}^{2}-a^{3}V\left( \phi\right) -k\rho_{m0}a^{-3w}, \label{bd.07}%
\end{equation}
and the first Brans-Dicke Friedmann equation is
\begin{equation}
3H^{2}=\frac{\omega_{BD}}{2}\left( \frac{\dot{\phi}}{\phi}\right) ^{2}%
+\frac{V\left( \phi\right) }{\phi}-3H\frac{\dot{\phi}}{\phi}+\frac{k}{\phi
}\rho_{m0}a^{-3\left( 1+w\right) }, \label{bd.08}%
\end{equation}
where $H=\dot{a}/a$, is the Hubble function, and an overdot indicates
differentiation with respect to the comoving proper time coordinate $t$.
In general, the Lagrangian (<ref>) defines the motion of a particle in a
two-dimensional space $\left\{ a,\phi\right\} $ with effective potential
\begin{equation}
V_{eff}=a^{3}V\left( \phi\right) +k\rho_{m0}a^{-3w}. \label{bd.09}%
\end{equation}
For $\omega_{BD}\neq-3/2$, from Lagrangian[In the limit in which
$\omega_{BD}=-3/2$, lagrangian (<ref>) is denerate, i.e. $\left\vert
\frac{\partial^{2}L}{\partial\dot{x}^{i}\partial\dot{x}^{j}}\right\vert =0$,
for a discussion see <cit.>.] (<ref>) we define the momentum,
$p_{a}=\frac{\partial L}{\partial\dot{a}}$, $p_{\phi}=\frac{\partial
L}{\partial\dot{\phi}}$, hence we can write the Hamiltonian
\begin{equation}
\mathcal{E}=\frac{1}{2\omega_{BD}+3}\left[ -\frac{\omega_{BD}}{2a\phi}%
+a^{3}V\left( \phi\right) +k\rho_{m0}a^{-3w}. \label{bd.10}%
\end{equation}
From the first modified Friedmann equation (<ref>) it follows that
Therefore, the second modified Friedmann equation and the Klein Gordon
equation are described by the following Hamiltonian system:
\begin{equation}
\left( 2\omega_{BD}+3\right) \dot{a}=-\frac{2\omega_{BD}}{3a\phi}p_{a}%
-\frac{2}{a^{2}}p_{\phi}, \label{bd.11}%
\end{equation}
\begin{equation}
\left( 2\omega_{BD}+3\right) \dot{\phi}=-\frac{2}{a^{2}}p_{a}+\frac{4\phi
}{a^{3}}p_{\phi}, \label{bd.12}%
\end{equation}
\begin{equation}
\left( 2\omega_{BD}+3\right) \dot{p}_{\phi}=-\frac{\omega_{BD}}{3a\phi^{2}%
}p_{a}^{2}-\frac{2}{a^{3}}p_{\phi}^{2}-\left( 2\omega_{BD}+3\right)
a^{3}V_{,\phi}, \label{bd.13}%
\end{equation}
\begin{align}
\left( 2\omega_{BD}+3\right) \dot{p}_{a} & =-\frac{\omega_{BD}}{3}%
\frac{p_{a}^{2}}{a^{2}\phi}-\frac{4}{a^{3}}p_{a}p_{\phi}+\frac{6\phi}{a^{4}%
& -3\left( 2\omega_{BD}+3\right) \left( a^{2}V\left( \phi\right)
-k\rho_{m0}wa^{-1-3w}\right) . \label{bd.14}%
\end{align}
Under normal quantization, i.e., $p_{i}\simeq i\frac{\partial}{\partial x^{i}%
},$ we can define the WdW equation $W:=\mathcal{E}\left( \Psi\right) =0$,
that is <cit.>,
\begin{align}
0 & =\frac{1}{6\left( 2\omega_{BD}+3\right) }\left[ \frac{\omega}{a\phi
}\right] +\nonumber\\
& +\frac{1}{6\left( 2\omega_{BD}+3\right) }\left[ \frac{6}{a^{3}}%
\Psi_{,\phi}-\frac{\omega}{a^{2}\phi}\Psi_{,a}\right] -\left[ a^{3}V\left(
\phi\right) +k\rho_{m0}a^{-3w}\right] \Psi, \label{bd.15}%
\end{align}
where $\Psi=\Psi\left( a,\phi\right) $ indicates the wavefunction of the
universe. Notice, that we use the following derivatives $\Psi_{,aa}%
=\partial^{2}\Psi/\partial a^{2}$, $\Psi_{\phi\phi}=\partial^{2}\Psi
/\partial\phi^{2}$ and $\Psi_{,a\phi}=\partial^{2}\Psi/\partial a\partial\phi
$. Recall that the dimension of the minisuperspace is two: that is, we do not
introduce the quantum correction term in order for the WdW equation to be
conformally invariant <cit.>.
The WdW equation is defined by the conformally invariant operator
\begin{equation}
\hat{L}_{\gamma}=-\Delta_{\gamma}+\frac{n-2}{4\left( n-1\right) }R_{\gamma}
\label{bd.15a}%
\end{equation}
where $\Delta_{\gamma}$ is the Laplace operator with respect to the
minisuperspace $\gamma_{ij}$, and $R_{\gamma}$, is the Ricci scalar of
$\gamma_{ij}$. The importance of the operator $\hat{L}_{\gamma}$ is that under
a conformal transformation, $\bar{\gamma}_{ij}=e^{2\Omega\left( x^{k}\right)
}\gamma_{ij},$ the scaling $\hat{L}_{\bar{\gamma}}\left( \Psi\right)
=e^{-\frac{n+2}{2}\Omega\left( x^{k}\right) }\hat{L}_{\gamma}\left(
e^{\frac{n-2}{2}\Omega\left( x^{k}\right) }\Psi\right) $ holds, where
$\Omega\left( x^{k}\right) $ is an arbitrary function. Moreover, using the
the kinetic term in the Lagrangian (<ref>), we endow the minisuperspace
$\gamma_{ij}$ (the dimension is $n=2$), with line element:
\begin{equation}
ds_{\left( \gamma\right) }^{2}=-6a\phi da^{2}-6a^{2}dad\phi+\frac
\end{equation}
from which we conform that the associated Ricci scalar vanishes, i.e.
Hence, from (<ref>), it follows that $L_{\gamma}=-\Delta_{\gamma}$,
which is the Laplace operator, and under a conformal transformation we have
$L_{\bar{\gamma}}\left( \Psi\right) =e^{-2\Omega\left( x^{k}\right)
}L_{\gamma}\left( \Psi\right) $. In order to solve (<ref>), we need to
specify the scalar-field potential. This will be done by an ansatz. In the
literature there have been many forms for this potential depending on what one
wants to do. In the present work we adopt the geometric approach to dictate
the physics. The gain from this approach is that it is observer-free and no
conflicts arise between geometry and dynamics. Specifically, we require that
the potential $V(\phi)$ be such that the WdW equation (<ref>) admits Lie
point symmetries.
§ GROUP-INVARIANT TRANSFORMATIONS FOR THE WDW EQUATION
For convenience, we provide below the basic definitions of Lie point
symmetries. Let $W=W\left( x^{i},\Psi,\Psi_{,i},\Psi_{,ij}\right) ~$be a
second-order differential equation, $x^{i}$ are the independent variables, and
$\Psi$ is the dependent variable, where $\Psi_{,i}=\frac{\partial\Psi
}{\partial x^{i}}$ and $\Psi_{,ij}=\frac{\partial^{2}\Psi}{\partial x^{i}
\partial x^{j}}$. The generator $X$ of the infinitesimal one-parameter point
\begin{equation}
\bar{x}^{i}=x^{i}+\varepsilon\xi^{i}\left( x^{i},\Psi\right) ,~\bar{\Psi
}=\Psi+\varepsilon\eta\left( x^{i},\Psi\right) +O(\varepsilon^{2})
\label{bd.16}%
\end{equation}
is defined by
\begin{equation}
{\partial\bar{\Psi}}{\partial\varepsilon}\partial_{\Psi}, \label{bd.17}%
\end{equation}
from which it follows that
\begin{equation}
X=\xi^{i}\left( x^{i},\Psi\right) \partial_{i}+\eta\left( x^{i}%
,\Psi\right) \partial_{\eta}. \label{bd.18}%
\end{equation}
The differential equation, $W$,$~$is invariant under the action of the
one-parameter point transformation (<ref>) if there exists a function
$\kappa$ such that <cit.>
\begin{equation}
X^{\left[ 2\right] }W=\kappa W, \label{bd.20}%
\end{equation}
where $X^{\left[ 2\right] }$ is the second prolongation of $X$ in the jet
space $\left\{ x^{i},\Psi,\Psi_{,i},\Psi_{,ij}\right\} $. When condition
(<ref>) holds, we say that $X$ is a Lie point symmetry of $W$. Notice,
that the Lie point symmetries of a differential equation form a Lie algebra.
For differential equations which follow from a variational principle, i.e.
there exists a Lagrange function, the Lie point symmetries which transform the
action integral in such a way that the Euler-Lagrange equations are invariant
are called Noether point symmetries. The characteristic of Noether point
symmetries is that for each Noether symmetry, $X,$ there corresponds a
conservation law which is called a Noether integral <cit.>. The Noether
point symmetries of a differential equation form a subalgebra of the Lie point
symmetries of that equation which is called the Noether algebra of the
differential equation.
§.§ Lie point symmetries of the WdW equation
The WdW equation (<ref>) is a second-order partial differential equation
defined in the space of the independent variables $\left\{ x^{i}\right\}
\rightarrow\left\{ a,\phi\right\} ,$ where $\Psi$ is the dependent variable.
Hence, the generator (<ref>) of the infinitesimal point transformation
(<ref>) has the following form,
\begin{equation}
X=\xi^{a}\left( a,\phi,\Psi\right) \partial_{\alpha}+\xi^{\phi}\left(
a,\phi,\Psi\right) \partial_{\phi}+\eta\left( a,\phi,\Psi\right)
\partial_{\Psi}. \label{bd.21}%
\end{equation}
From condition (<ref>), and for arbitrary $V\left( \phi\right) $, we
have the following Lie symmetries:
\begin{equation}
X_{\Psi}=\Psi\partial_{\Psi},~X_{b}=b\left( a,\phi\right) \partial_{\Psi},
\label{bd.22}%
\end{equation}
where $b\left( a,\phi\right) $ is a solution of the original equation. The
vector field $X_{\Psi}$ is called a homogeneous symmetry, whereas $X_{b}$
corresponds to the infinite number of solutions. Both these Lie symmetries are
trivial symmetries in the sense that they cannot be used to reduce the
differential equation. However, they indicate that equation (<ref>) is a
linear second-order partial differential equation.
In order that equation (<ref>) admit non trivial Lie symmetries, we must
consider specific forms of the potential $V\left( \phi\right) .$ In
<cit.> it has been shown that the WdW equation (<ref>)
admits nontrivial Lie point symmetries if and only if the potential $V\left(
\phi\right) $ is power law, with
\begin{equation}
V\left( \phi\right) =V_{0}\phi^{\lambda}. \label{bd.23}%
\end{equation}
In our case the power $\lambda=\lambda\left( \omega_{BD},w\right) $ has the
following possible values: (a) $\lambda_{1}=\left( 1+w\right) \left(
1-w\right) ^{-1}$, and, (b) $2\lambda_{2}=\left( \varpi-3\right) \left(
w+1\right) ,$ where
\begin{equation}
\varpi=\sqrt{6\omega_{BD}+9}. \label{bd.23a}%
\end{equation}
The Lie point symmetry vector which corresponds to $\lambda_{1}$
is[Recall that we have considered $w\in\left( -1,1\right) $. In the
limit where $w=1$, only the power law potential with $\lambda_{2}$ exists.]
\begin{equation}
X_{1}=a\partial_{a}+3\left( w-1\right) \phi\partial_{a}, \label{bd.24}%
\end{equation}
and to $\lambda_{2}$ is,
\begin{equation}
X_{2}=A^{\mu_{1}}\phi^{\mu_{2}}\left( a\partial_{a}+\frac{6\phi}{\varpi
-3}\partial_{\phi}\right) , \label{bd.25}%
\end{equation}
where $\mu_{1}=\frac{3}{2\varpi}\left[ \varpi\left( w-1\right)
-3w+1\right] $ and $\mu_{2}=\frac{\varpi+3}{2}\mu_{1}$.
We remark that the vector fields $X_{1}$, $X_{2}$, are conformal symmetries
for the minisuperspace $\gamma$ defined by the kinematic part of the
Lagrangian (<ref>). Therefore, the minisuperspace selects the form of
the potential <cit.>. A special case occurs when the
perfect fluid is radiation, that is, $w=1/3.$ In this case we find another
power-law potential of the form
\begin{equation}
V_{w=1/3}\left( \phi\right) =V_{0}\left[ \left( V_{1}-\phi^{-\frac{1}%
{3}\varpi}\right) ^{2}+V_{1}\left( 2-\phi^{\frac{1}{3}\varpi}\right)
^{2}-V_{1}\right] . \label{bd.26}%
\end{equation}
For this potential the WdW equation admits the Lie point symmetry which is
given by the vector field, $Y=Y_{+}+Y_{-}~,~$where
\begin{equation}
Y_{\pm}=\phi^{-\frac{1}{2}\left( 1\pm\frac{\varpi}{3}\right) }\left(
\partial_{a}+\frac{\varpi\pm3}{\omega_{BD}}\frac{\phi}{a}\partial_{\phi
}\right) . \label{bd.27}%
\end{equation}
The importance of the existence of a nontrivial Lie point symmetry for
equation (<ref>) is the existence of a coordinate system in which
equation (<ref>) is independent of one of the independent variables, and
the solution which corresponds to the zero-order invariants has oscillatory
terms. Furthermore, a Noetherian conservation law for the field equations
(<ref>)-(<ref>) means that they define an integrable Hamiltonian
system <cit.>. In the next section we use the zero-order invariance on
the WdW equation (<ref>) to reduce the field equations to two
first-order ordinary differential equations, which we solve.
§ ANALYTICAL SOLUTIONS
Now we apply the zero-order invariants of the Lie point symmetries for the WdW
equation in order to reduce the equation and find the solution of the
wavefunction $\Psi(a,\phi)$. Moreover, by using the Hamilton-Jacobi equation
we reduce the dimension of the Hamiltonian system for the field equations. We
do that for the potential (<ref>) with $\lambda=\lambda_{1}$.
§.§ Invariant solution for the WdW equation
In order to apply the zero-order invariants of a Lie point symmetry in
equation (<ref>), we prefer to work with the normal coordinates of the
symmetry vector $X_{1}$. We apply the coordinate transformations
\begin{equation}
a=\exp\left( x\right) ~,~\phi=y\exp\left[ 3\left( w-1\right) x\right]
\label{bd.28a}%
\end{equation}
to (<ref>) and the WdW equation takes the following form
\begin{align}
0 & =\left( -\frac{\omega_{BD}}{3y}\Psi_{,xx}+2m_{1}\Psi_{,xy}-ym_{2}%
\Psi_{,yy}\right) +\nonumber\\
& ~~-m_{2}\Psi_{,y}-2\left( \bar{V}_{0}y^{\frac{1+w}{1-w}}+\bar{\rho}%
_{m0}\right) \Psi, \label{bd.29}%
\end{align}
where now we have $\Psi=\Psi\left( x,y\right) $, $m_{1}=3\omega_{BD}\left(
w-1\right) -1,$ $m_{2}=3\omega_{BD}\left( w-1\right) ^{2}-2\left(
3w-2\right) $ and $\left( \bar{V}_{0},\bar{\rho}_{m0}\right) =\left(
2\omega_{BD}+3\right) \left( V_{0},k\rho_{m0}\right) $.
In the new coordinates the Lie point symmetry vector $X_{1}$, takes the simple
form $X_{1}=\partial_{x}$. Since $X_{1}$ is a Lie symmetry of (<ref>)
this means that it transforms solutions to solutions; that is,
\begin{equation}
X_{1}\left( \Psi\right) =\beta\Psi, \label{bd.29a}%
\end{equation}
from where it follows that,[We take the same result when we apply the
zero-order invariants of the symmetry vector $Z=X_{1}-\beta X_{\Psi}$ in
\begin{equation}
\Psi\left( x,y\right) =\sum\limits_{\beta}e^{-\beta x}\Phi\left( y\right)
, \label{bd.30}%
\end{equation}
where the function, $\Phi\left( y\right) $, is given by the following
second-order ordinary differential equation:
\begin{equation}
ym_{2}\Phi_{,yy}+\left( 2\beta m_{1}+m_{2}\right) \Phi_{,y}+2\left( \bar
^{2}\right) \Phi=0. \label{bd.31}%
\end{equation}
In fig. <ref> we present the wavefunction (<ref>), in the case of
the O'Hanlon theory, i.e. with $\omega_{BD}=0;$ which can be seen as an
effective $f(R)$ gravity with $\phi=\frac{df}{dR}$ and $V\left( \phi\right)
=\left( \frac{df}{dR}R-f\right) $ <cit.>. Specifically,
in the left panels of fig. <ref> we show the solutions of $Re\left(
\Psi\left( a,\phi\right) \right) $ and $Im\left( \Psi\left(
a,\phi\right) \right) $ respectively, for $\rho_{m0}=0$, and $w=1/3$. The
latter case is for the the radiation-dominated era. As we have discussed
above, the previously selected dynamical conditions imply $V\left(
\phi\right) =V_{0}\phi^{2}$, hence we can easily show that the
corresponding[In $f\left( R\right) $-theory, in the metric
formalism the gravitational action integral is $S=\int dx^{4}\sqrt{-g}f\left(
R\right) $, where $R$ is the Ricci scalar of the underlying space with metric
$g_{ij}$.] $f\left( R\right) $-theory is $f\left( R\right) \propto R^{2}$.
We would like to remind the reader that the $R^{2}$ term provides a de-Sitter
behavior <cit.> which plays a critical role in the inflationary era. In
fact, if the gravitational Lagrangian is $f(R)$, there exist de Sitter
solutions of the theory with covariantly constant $R_{0}$ if it is a solution
of Barrow and Ottewill's condition $R_{0}f^{\prime}(R_{0})=2f(R_{0}),$
<cit.>, which is satisfied identically for the purely quadratic
Lagrangian. Notice, that in the right panels of fig. <ref>, we plot the
contours of the wavefunction in the $(a,\phi)$ plane.
Left panels: The surface plot of the wavefunction
(<ref>) for $\omega_{BD}=0$, $\bar{\rho}_{m0}=0$, $w=1/3$ and $\beta
=1i$, which corresponds to the quadratic potential $V\left( \phi\right)
=V_{0}\phi^{2}$. Notice that we use $V_{0}=1$ units. The current dynamical
model effectively reduces to $f\left( R\right) \propto R^{2}$ gravity.
Right panels: The contours of the wavefunction in the $(a,\phi)$
Now we focus on Eq.(<ref>). In the special case of $m_{2}=0$, namely
$\omega_{BD}=\frac{2}{3}\frac{\left( 3w-2\right) }{\left( 1-w\right) ^{2}%
},$ the corresponding solution is
\begin{equation}
\Phi\left( y\right) =\Phi_{0}\exp\left[ -\bar{V}_{0}\frac{\left(
1-w\right) }{2\beta m_{1}}y^{\frac{2}{1-w}}-\frac{\bar{\rho}_{m0}}{\beta
m_{1}}y-\frac{\beta}{6m_{1}}\ln y\right] . \label{bd.32}%
\end{equation}
In the next section we continue with the classical solution of the field
equations. As we shall see, the constant $\beta$ is related to the value of
the Noetherian conservation law for the field equations, which corresponds to
the vector field $X_{1}$.
§.§ Classical solution
In the coordinate system (<ref>) the Hamiltonian of the field equations
(<ref>) becomes
\begin{equation}
\mathcal{E}=\frac{e^{-3wx}}{6}\left( -\frac{\omega_{BD}}{y}p_{x}^{2}%
+6m_{1}p_{x}p_{y}-3ym_{2}p_{y}^{2}\right) +e^{-3wx}\left( \bar{V}%
_{0}y^{\frac{1+w}{1-w}}+\bar{\rho}_{m0}\right) , \label{bd.33}%
\end{equation}
and the field equations are given by the following Hamiltonian system
\begin{equation}
e^{3wx}\dot{x}=m_{1}p_{y}-\frac{\omega_{BD}}{3y}p_{x}, \label{bd.34}%
\end{equation}
\begin{equation}
e^{3wx}\dot{y}=m_{1}p_{x}-m_{2}yp_{y}, \label{bd.35}%
\end{equation}
\begin{equation}
e^{3wx}\dot{p}_{x}=3w\mathcal{E}, \label{bd.36}%
\end{equation}
\begin{equation}
_{BD}p_{x}^{2}-V_{0}\frac{\left( 1+w\right) }{\left( 1-w\right) }%
y^{\frac{2}{1-w}}. \label{bd.37}%
\end{equation}
From the first modified Friedmann equation, and equation (<ref>), we
have that
\begin{equation}
p_{x}=I_{0}. \label{bd.38}%
\end{equation}
This is the Noetherian conservation law which corresponds to the symmetry
vector[As we can see we did not use the formulas of Noether theorems
to calculate the conservation law. However we derived it from the Hamilton
equations in the canonical coordinates of the vector field $X_{1}$.] $X_{1}$.
Comparing the last expression with (<ref>) we see that $\left\vert
\beta\right\vert \simeq\left\vert I_{0}\right\vert $.
From (<ref>) we define the (null) Hamilton-Jacobi
equation:[Recall that the action $S\left( x,y\right) $ is related
to the momenta, $p_{x}=\frac{\partial S}{\partial x}$, and $p_{y}%
=\frac{\partial S}{\partial y}$.]
\begin{equation}
\frac{1}{6}\left( -\frac{\omega_{BD}}{y}\left( \frac{\partial S}{\partial
x}\right) ^{2}+6m_{1}\left( \frac{\partial S}{\partial x}\right) \left(
\frac{\partial S}{\partial y}\right) -3ym_{2}\left( \frac{\partial
S}{\partial y}\right) ^{2}\right) +\left( \bar{V}_{0}y^{\frac{1+w}{1-w}%
}+\bar{\rho}_{m0}\right) =0, \label{bd.39a}%
\end{equation}
where $S=S\left( x,y\right) $ is the action. Furthermore, from
(<ref>), we have the constraint $\left( \frac{\partial S}{\partial
x}\right) =I_{0}$. Hence, from this and from (<ref>) we find,
\begin{align}
S\left( x,y\right) & =I_{0}x+\frac{m_{1}}{m_{2}}I_{0}\ln\left( y\right)
& +\int\sqrt{I_{0}^{2}y^{-2}\left[ \left( \frac{m_{1}}{m_{2}}\right)
^{2}-\frac{\omega_{BD}\omega}{m_{2}}\right] +2m_{2}^{-1}\left( \bar{V}%
_{0}y^{\frac{2w}{1-w}}+\bar{\rho}_{m0}y^{-1}\right) }dy, \label{bd.39}%
\end{align}
while for $m_{2}=0,$ the action $S\left( x,y\right) $ becomes
\begin{equation}
S\left( x,y\right) =I_{0}x+\frac{\omega_{BD}I_{0}}{6m_{1}}\ln\left(
y\right) -\frac{1}{2}\left( \bar{V}_{0}\left( 1-w\right) y^{\frac{2}{1-w}%
}+\frac{\bar{\rho}_{m0}}{I_{0}m_{1}}y\right) . \label{bd.40}%
\end{equation}
Hence, the field equations (<ref>)-(<ref>) reduce to the following
two-dimensional system of first-order differential equations:
\begin{equation}
e^{3wx}\dot{x}=m_{1}\left( \frac{\partial S}{\partial y}\right)
-\frac{\omega_{BD}}{3y}\left( \frac{\partial S}{\partial x}\right) ,
\label{bd.40a}%
\end{equation}
\begin{equation}
e^{3wx}\dot{y}=m_{1}\left( \frac{\partial S}{\partial x}\right)
-m_{2}y\left( \frac{\partial S}{\partial y}\right) , \label{bd.40b}%
\end{equation}
where, $S=S\left( x,y\right) $, is given by (<ref>) or (<ref>).
In general, the solution of the system (<ref>)-(<ref>), is not
given in closed form. Below, for the specific case where the Noetherian
conservation law vanishes, we can express the solution in terms of the scale factor.
§.§ A special closed-form solution
We consider now the case where $I_{0}=0$, i.e. $p_{x}=0$ with
$m_{1}\neq0,$ From (<ref>) and (<ref>) we have $\frac{dy}%
{dx}=-\frac{m_{2}}{m_{1}}y,~$so $y=\phi_{0}e^{-\frac{m_{2}}{m_{1}}x}$, hence
with the use of (<ref>) for the field $\phi$, we have
\begin{equation}
\phi\left( a\right) =\phi_{0}a^{M}~,~M=3\left( w-1\right) -\frac{m_{2}%
}{m_{1}} , \label{bd.41}%
\end{equation}
so $\dot{\phi}=\phi_{0}Ma^{M}H.$ Then, from (<ref>), it follows that
\begin{equation}
\left[ 3\left( 1+M\right) -\frac{\omega_{BD}M}{2}\right] H^{2}%
=G_{eff}\left( V_{0}^{\prime}a^{\frac{1+w}{1-w}M}+k\rho_{m0}a^{-3\left(
1+w\right) }\right) , \label{bd.42}%
\end{equation}
where $V_{0}^{\prime}=V_{0}\phi_{0}^{\frac{1+w}{1-w}},$ and $G_{eff}=\left[
\left( 3\left( 1+M\right) -\frac{\omega_{BD}M}{2}\right) \phi\right]
^{-1}$ is the effective gravitational constant. We see that $G_{eff}\left(
a\rightarrow1\right) =\left[ \left( 3\left( 1+M\right) -\frac{\omega
_{BD}M}{2}\right) \phi_{0}\right] ^{-1}$. Therefore, we can say that when
$I_{0}=0$, the scalar field behaves like an effective fluid with constant EoS
parameter. A special solution of the form, $\phi=\phi_{0}a^{\phi_{1}}$, where
$\phi_{1}$ is a constant, has been found in <cit.>, where $V_{0}=0$
and the perfect fluid is interacting with the scalar field in the action
integral. A similar result has been found in <cit.>
. In that paper we applied the same geometric selection rule for the scalar
field, but in general relativity ($G_{eff}=const$)
as a special solution of the field equations[For a different
derivation of the same result see <cit.>.].
By replacing $G_{eff}\left( a\right) $ in (<ref>), we can define the
Hubble function as follows
\begin{equation}
H\left( a\right) ^{2} =H_{0}^{2} \left( \Omega_{\phi0}a^{q_{1}}+\Omega
_{m0}a^{q_{2}} \right) , \label{bd.43}%
\end{equation}
where spatial flatness requires that $\Omega_{\phi0}+\Omega_{m0}=1,$ since
$E\left( a\rightarrow1\right) =1$. Furthermore, the new constants$~q_{1}%
,q_{2}$ are,
\begin{equation}
q_{1}=\frac{2M\left( w,\omega_{BD}\right) w}{1-w}, \label{bd.44}%
\end{equation}
\begin{equation}
q_{2}=-\left[ M\left( w,\omega_{BD}\right) +3\left( 1+w\right) \right] .
\label{bd.45}%
\end{equation}
Hence, the system (<ref>)-(<ref>) can provide us with a Hubble
function for different models of two fluids (<ref>). We study some
special cases:
Case (A): Cosmological constant with dust. This means that $\left(
q_{1},q_{2}\right) =\left( 0,-3\right) $ or $\left( q_{1},q_{2}\right)
=\left( -3,0\right) $, from which we have $\left( w,\omega_{BD}\right)
=\left( 0,\frac{1}{6}\right) $ or $\left( w,\omega_{BD}\right)
\simeq\left( 0.28,-0.77\right) $.
Case (B): Dust with radiation. This requires, $\left( q_{1},q_{2}\right)
=\left( -3,-4\right) $ or $\left( q_{1},q_{2}\right) =\left(
-4,-3\right) ,$ hence $\left( w,\omega_{BD}\right) \simeq\left(
0.63,0\right) $ or $\left( w,\omega_{BD}\right) \simeq\left(
0.55,-1\right) $.
Case (C): Cosmological constant with radiation fluid This requires $\left(
q_{1},q_{2}\right) =\left( 0,-4\right) $ or $\left( q_{1},q_{2}\right)
=\left( -4,0\right) $, hence $\left( w,\omega_{BD}\right) =\left(
0,0\right) ,~\left( w,\omega_{BD}\right) =\left( \frac{1}{3},0\right) $
or,$~\left( w,\omega_{BD}\right) =\left( \frac{1}{3},-\frac{3}{4}\right) $.
Case (D): In the case of $\left( q_{1},q_{2}\right) =\left( 0,0\right) $,
which implies $\left( w,\omega_{BD}\right) =\left( -1,\frac{1}{6}\right) $
or $\left( w,\omega_{BD}\right) =\left( 0,-\frac{4}{3}\right) $, from
(<ref>) we have a de Sitter solution.
It is interesting that when we assume a radiation fluid in (<ref>) we
have a solution of the system (<ref>)-(<ref>) in which
$\omega_{BD}=0$; however, this result is expected since when $\omega_{BD}=0$,
the action (<ref>) reduces to O'Hanlon's massive dilaton gravity
<cit.>, and consequently to $f\left( R\right) $-gravity in the metric
formalism, which provides a radiation term <cit.>.
Before we close this section, we should add that for the power-law potential
(<ref>) with $\lambda=\lambda_{2}$, one may use the same method to
construct the solution of the field equations. We shall not repeat the
calculations but we simply say that in this case the canonical coordinate
transformation $\left\{ a,\phi\right\} \rightarrow\left\{ z,r\right\} $ is
given by the following expression:
\begin{equation}
z=\frac{3-\varpi}{6\mu_{2}+\mu_{1}\left( \varpi-3\right) }a^{-\mu_{1}}%
\phi^{-\mu_{2}}~,~r=\phi a^{-\frac{6}{\varpi-3}}.
\end{equation}
§.§ Observational Constraints
Now we focus on the Hubble parameter (<ref>) in which we have imposed
$w=0$. This means that the perfect fluid in the gravitation action
(<ref>) is dust. Therefore, from (<ref>) and (<ref>) we have
that $q_{2}=-\frac{3\omega_{BD}+4} {3\omega_{BD}+1}$ and $q_{1}=0$. Using the
above conditions the Hubble parameter becomes
\begin{equation}
H\left( a\right) ^{2}=H_{0}^{2}\left[ \left( 1-\Omega_{m0}\right)
+\Omega_{m0}a^{-\frac{3\omega_{BD}+4}{3\omega_{BD}+1}}\right] . \label{bd.46}%
\end{equation}
We mention that from the second term of Eq.(<ref>) one may define an
effective equation of state parameter, namely
\[
w_{m}^{\mathrm{(eff)}}=\frac{1}{3}\frac{1-6\omega_{BD}}{\left( 3\omega
_{BD}+1\right) }.
\]
Obviously, for $\omega_{BD}=1/6$ (or $w_{m}^{\mathrm{(eff)}}=0$) the above
Hubble parameter reduces to that of the concordance $\Lambda$CDM model.
In order to constrain the Brans-Dicke parameter we perform a joint likelihood
analysis using the Type Ia supernova data set of Union 2.1 <cit.>, and
the BAO data <cit.>. Notice, that for the Hubble constant we
utilize $H_{0}=69.6$km/s/Mpc <cit.>. Hence, the overall likelihood
function is defined as follows
\begin{equation}
\mathcal{L}\left( \Omega_{m0},\omega_{BD}\right) \mathcal{=L}_{SNIa}%
\mathcal{\times L}_{BAO}%
\end{equation}
where $\mathcal{L}_{A}\varpropto e^{-\chi_{A}^{2}/2}~$ which means that the
total $\chi^{2}$ is written as
\begin{equation}
\chi^{2}=\chi_{SNIa}^{2}+\chi_{BAO}^{2}.
\end{equation}
Lastly, in order to test the performance of the cosmological models against
the data we use the Akaike information criterion AIC$=\mathrm{min}(\chi
^{2})+2n_{fit}$, where $n_{fit}$ is the number of free parameter
In the case of SNIa the corresponding chi-square parameter is given
by[For the SNIa test we have applied the diagonal covariant matrix
without the systematic errors.]
\begin{equation}
\chi_{SNIa}^{2}=\sum\limits_{i=1}^{N_{SNIa}}\left( \frac{\mu_{obs}\left(
z_{i}\right) -\mu_{th}\left( z_{i};\Omega_{m0},\omega_{BD}\right) }%
{\sigma_{i}}\right) ^{2}%
\end{equation}
where $N_{SNIa}=580$, $z_{i}\in\lbrack0.015,1.414]$ is the observed redshift,
$\mu_{obs}$ is the observed distance modulus and $\mu_{th}=5logD_{L}+25$ with
$D_{L}$ denoting the luminosity distance. Furthermore, for BAOs the
corresponding chi-square parameter has the following form
\begin{equation}
\chi_{BAO}^{2}=\sum\limits_{i=1}^{N_{BAO}}\left( \sum\limits_{j=1}^{N_{BAO}%
}\left[ d_{obs}\left( z_{i}\right) -d_{th}\left( z_{i};\Omega_{m0}%
,\omega_{BD}\right) \right] C_{ij}^{-1}\left[ d_{obs}\left( z_{j}\right)
-d_{th}\left( z_{j};\Omega_{m0},\omega_{BD}\right) \right] \right)
\end{equation}
where $N_{BAO}=6$ and $C_{ij}^{-1}$ is the inverse of the covariant matrix in
terms of $d_{z}=\frac{l_{BAO}}{D_{V}\left( z\right) }$ <cit.>.
Notice, that the quantity $l_{BAO}\left( z_{drag}\right) $ is the BAO scale
at the drag redshift and $D_{V}\left( z\right) $ is the volume distance
In table <ref>, we present the results of the current statistical
analysis while in figure (<ref>) we provide the $1\sigma$, $2\sigma$,
and $3\sigma$ combined likelihood contours for the Brans-Dicke model [see
In particular, we find the following results:
* for the Brans-Dicke model: $\Omega_{m0}=0.29^{+0.032}_{-0.025}$,
$\omega_{BD}=0.19^{+0.075}_{-0.059}$, $\mathrm{min}(\chi^{2})\simeq564.29$,
$n_{fit}=2$ and AIC$\simeq568.29$.
* for the $\Lambda$CDM model: $\Omega_{m0}=0.28^{+0.025}_{-0.024}$,
$\mathrm{min}(\chi_{\Lambda}^{2}) \simeq564.51$, $n_{fit}=1$ and
Since, $\Delta\mathrm{AIC}=|\mathrm{AIC}-\mathrm{AIC}_{\Lambda}| \le2$ we
conclude that the current cosmological models fit equally well the
observational data.
The overall statistical results
(using SNIa+BAO) for the $\Lambda$CDM and Brans-Dicke models
respectively. Notice, that in our analyis we use Eq.(<ref>).
In the last three colums we present the number of
free parameters and the goodness-of-fit statistics.
c]cccccccccModel $\Omega_{m0}$ $\Omega_{\Lambda}$ $w_{\Lambda}~$ $w_{m}$
$n_{fit}$ $\min\left( \chi^{2}\right) $ $\mathrm{AIC}$
$\Lambda$CDM $0.28_{-0.024}^{+0.025}$ $0.72_{-0.024}^{+0.025}$ $-1$
$0$ $1$ $564.51$ $566.51$
$\Omega_{m0}$ $\Omega_{\phi0}$ $w_{\phi}$ $w_{m}^{(\mathrm{eff)}}$
$\omega_{BD}$ $n_{fit}$ $\min(\chi^{2})$ $\mathrm{AIC}$
Brans-Dicke $0.29_{-0.025}^{+0.032}$ $0.71_{-0.025}^{+0.032}$ $-1$
$-0.03_{-0.072}^{+0.091}$ $0.19_{-0.059}^{+0.075}$ $2$ $564.29$
§ CONCLUSIONS
In this paper, we have extended our earlier analysis, which was introduced in
<cit.>, for the case of Brans-Dicke gravity with a perfect fluid in which
the fluid EoS parameter is constant and the underlying geometry is that of a
spatially flat FLRW universe. In particular, in order to select the functional
form of the scalar field potential, $V\left( \phi\right) ,$ in the
gravitational action integral (<ref>), we have used the well known
criterion, namely the existence of group invariant transformations for the WdW
equation <cit.>. The existence of a Lie point symmetry vector for the WdW
equation is related to the existence of oscillatory terms in the solution of
the wavefunction $\Psi$, and to Noetherian conservation laws for the field
equations. The latter can be used to find analytical solutions.
For our model, in which the perfect fluid is not interacting with the scalar
field, we found two families of solutions with power-law potential $V\left(
\phi\right) =V_{0}\phi^{\lambda}$, where the constant $\lambda$ depends on
the EoS parameter, $w$, of the perfect fluid and the Brans-Dicke parameter
$\omega_{BD}$. The results hold for the case where the parameter
$\omega_{BD}$ vanishes, i.e., action (<ref>) is that of O'Hanlon theory
which is equivalent with $f\left( R\right) $-gravity in the metric
formalism. As a special case, a third family of power potentials is found when
the perfect fluid is radiation, i.e. $w=1/3$.
By applying the zero-order invariants of the corresponding Lie symmetry in the
WdW equation, we were able to solve the WdW equation and find the oscillatory
behavior of the wave function. Furthermore, with the use of the
Hamilton-Jacobi theory we reduced the Hamiltonian system, which defines the
classical field equations, to a system of two first-order differential
equations. That is, the field equations for that form of the potential,
$V\left( \phi\right) $, form a Liouville integrable dynamical system.
The combined (SNIa+BAOs) likelihood contours $1\sigma\left(
\Delta\chi^{2}=2.3\right) $, $2\sigma\left( \Delta\chi^{2}=6.18\right) $,
and $3\sigma\left( \Delta\chi^{2}=11.83\right) $, in the $\left(
\Omega_{m0},\omega_{DE}\right) $ plane. In this analysis we use the Hubble
parameter of Eq. (<ref>). The cross corresponds to best fit solution.
As a special case for one of the integrable models, we found a closed-form
solution for the Hubble function from which we saw that the Brans-Dicke field
follows power-law behaviour with respect to the metric scale factor, that is,
$\phi=\phi_{0}a^{M}$. In this context, the Hubble parameter is written as
$H(a)=H_{0}\sqrt{\Omega_{\phi0}a^{q_{1}}+\Omega_{m0}a^{q_{2}}}$, where the
parameters $q_{1,2}$ are given in terms of $(w,\omega_{BD})$. Obviously, one
may recover a similar Hubble parameter to this within the framework of general
relativistic cosmology by using two different perfect fluids with constant
equation of state parameters. This implies that the current Brans-Dicke
gravity model is cosmologically equivalent to that of general relativity as
far as the cosmic expansion is concerned. Therefore, in order to distinguish
Brans-Dicke gravity from GR we need to extend the analysis to the perturbation
level. It is well known that modified gravity affects, via the effective
gravitational constant the growth rate of linear matter perturbations. As an
example, in the case of $f(R)$ gravity models the quantity $G_{eff}$ is given
in terms of the scale factor and of the wave-number (see <cit.>
,<cit.> and references therein). In a forthcoming paper, we attempt to
investigate the impact of the Brans-Dicke gravitational parameter
$G_{eff}=\left[ \left( 3\left( 1+M\right) -\frac{\omega_{BD}M}{2}\right)
\phi\right] ^{-1}$ on the matter perturbations. Lastly, performing a joint
statistical analysis involving the recent SNIa and BAO data, we place
constraints on the main cosmological parameters of the Brans-Dicke model.
It has also been shown that there exists a relation between the value of the
conserved quantity, $I_{0}$, for the classical field equations and the
“frequency”, $\beta$, in the WdW equation.
Hartle proposed that strong peaks in the wavefunction lead to a classical
observable universe <cit.>. Since the WdW equation is a linear
second-order PDE, the general invariant solution is the sum on all possible
values of $\beta$. However, one may relate the “frequency” of the strong peaks of the wavefunction to the
value of the conserved quantity, $I_{0}$, (<ref>). The latter is
included in the solution of the field equations, i.e. $a\left( t\right) $,
and consequently in the Hubble function $H\left( a\right) $. In any case,
the existence of the conservation law indicates a strong relation between the
classical and the quantum solutions and information can be transferred from
among the two systems. However, the physical observable quantities which
correspond to the conservation laws of the field equations are still unknown.
The Brans-Dicke action is defined in the Jordan frame and is conformally
equivalent to the minimally coupled scalar field in the Einstein frame. The
solutions which we have found describe a perfect fluid which is not
interacting with the scalar field in the Jordan frame. However, the solutions
can be transformed into the Einstein frame and will hold for a model in which
there exists an interaction between the perfect fluid and the scalar field of
a particular form. By definition, the two models share the same solution of
the WdW equation.
In a forthcoming work we will study the cosmological evolution of these
integrable models in the Einstein frame.
AP thanks the University of Athens for the hospitality provided there when
this work done. AP is supported by FONDECYT postdoctoral grant no. 3160121. SB
acknowledges support by the Research Center for Astronomy of the Academy of
Athens in the context of the program ”Tracing the
Cosmic Acceleration”. JDB is supported by the STFC.
RatraB. Ratra and P.J.E. Peebles, Phys. Rev. D 37, 3406 (1988)
Barrow93J.D. Barrow and P. Saich, Class. Quant. Grav. 10,
279 (1993)
LinderE.V. Linder, Phys. Rev. Lett. 90, 091301 (2003)
CopelandE.J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys.
D 15, 1753 (2006)
OverduinJ.M. Overduin and F.I. Cooperstock, Phys. Rev. D
58, 043506 (1998)
BasilS. Basilakos, M. Plionis and J. Sola, Phys. Rev. D
80, 083511 (2009)
lamelL. Amendola, M. Baldi and C. Wetterich, Phys. Rev. D
78, 023015 (2008)
caisY.F. Cai, E.N. Saridakis, M.R. Setare and J.Q. Xia, Phys. Rept.
493, 1 (2010)
BentoM.C. Bento, O. Bertolami and A.A. Sen, Phys. Rev. D
66, 043507 (2002)
BransC. Brans and R.H. Dicke, Phys. Rev. 124, 195 (1961)
BudaH.A. Buchdahl, Mon. Not. Roy. Astron. Soc. 150, 1 (1970)
SotiriouT.P. Sotiriou and V. Faraoni Rev. Mod. Phys. 82,
451 (2010)
FerraroR. Ferraro and F. Fiorini, Phys. Rev. D 75, 084031 (2007)
MaartensR. Maartens, Living Rev. Rel. 7, 7 (2004)
faraonibookV. Faraoni, Cosmology in Scalar-Tensor Gravity,
Fundamental Theories of Physics vol. 139, (Kluwer Academic Press: Netherlands, 2004)
Ame10L. Amendola and S. Tsujikawa, Dark Energy Theory and
Observations, (Cambridge University Press: Cambridge, 2010)
jdbJ.D. Barrow, Phys. Lett. B 235, 40 (1990)
ssf01A.G. Muslinov, Class. Quantum Gravit. 13, 3229 (1996)
ssf02G.F.R. Ellis and M.S. Madsen, Class. Quantum Gravit.
8, 667 (1991)
ssf03J.G. Russo, Phys. Lett. B 600, 185 (2004)
basilLukesS. Basilakos and G. Lukes-Gerakopoulos, Phys. Rev. D
78, 083509 (2008)
ssf04J.J. Halliwell, Phys. Lett. B 185, 341 (1987)
ssf05R. Easter, Class. Quantum Gravit. 10 2203 (1993)
ssf06L.P. Chimento and A.S. Jakubi, Int. J. Mod. Phys. D
5, 71 (1996)
BarrowC. Rubano and J. D. Barrow, Phys. Rev. D. 64, 127301 (2001)
UrenaL. A. Urena-Lopez, T. Matos, Phys. Rev. D 62, 081302 (2000)
sahniV. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 9,
373 (2000)
basprd1S. Basilakos, M. Tsamparlis and A. Paliathanasis, Phys. Rev.
D 83, 103512 (2011)
basprd2A. Paliathanasis, M. Tsamparlis and S. Basilakos, Phys. Rev.
D 90, 103524 (2014)
Cap97S. Capozziello, R. de Ritis and A. A. Marino, Class. Quantum
Gravity, 14, 3259 (1997); C. Rubano, and P. Scudellaro, Gen. Relat.
Grav. 34, 307, (2002); A.K. Sanyal, B.Modak, C. Rubano and E.
Piedipalumbo, Gen. Relat. Grav. 37, 407, (2005); M. Szydlowski et
al., Gen. Rel. Grav., 38, 795, (2006); S. Capozziello, A. Stabile,
and A. Troisi, Class. Quant. Grav., 24, 2153, (2007); A. Bonanno, G.
Esposito, C. Rubano and P. Scudellaro, Gen. Rel. Grav., 39, 189,
(2007); S. Capozziello, E. Piedipalumbo, C. Rubano, and, P. Scudellaro, Phys.
Rev. D., 80, 104030, (2009); B. Vakili, Phys. Lett. B., 664,
16, (2008); Yi Zhang, Yun-gui Gong and Zong-Hong Zhu, Phys. Lett. B.,
688, 13, (2010); Hao Wei, Xiao-Jiao Guo and Long-Fei Wang, Phys.
Lett. B., 707, 298, (2010)
TsafrA. Paliathanasis, M. Tsamparlis and S. Basilakos, Phys. Rev. D
84, 123514, (2011).
BASFTS. Basilakos, S. Capozziello, M. De Laurentis, A.
Paliathanasis, M. Tsamparlis, Phys. Rev. D., 88, 103526 (2013)
AnSTA. Paliathanasis, M. Tsamparlis, S. Basilakos and S.
Capozziello, Phys. Rev. D 89, 063532 (2014)
prdBA. Paliathanasis, M. Tsamparlis, S. Basilakos and J.D. Barrow,
Phys. Rev. D 91, 123535 (2015)
VernovA. Yu Kamenshchik, E.O. Pozdeeva, A. Tronconi, G. Venturi and
S. Yu Vernov, Class. Quantum Gravit. 31, 105003 (2014)
riaziE. Ahmadi-Azar and N. Riazi, Astr. Sp. Sci. 226, 1 (1995)
morgR.E. Morganstern, Phys. Rev. D 4, 946 (1971)
tiwariR.N. Tiwari and B.K. Nayak, J. Phys. A: Math. Gen.
9, 369 (1976)
bh01H. Kim, Phys. Rev. D 60, 024001 (1999)
bh02N. Riazi and H.R. Askari, Mon. Not. R. Astron. Soc.
261, 229 (1993)
bh03T. Clifton, J.D. Barrow and R.J. Scherrer Phys. Rev. D
71, 123526 (2005)
KukuY. Kucukakca, U. Camci and I. Semiz, Gen. Rel. Gravit.,
44, 1893 (2012)
terzis1P.A. Terzis, N. Dimakis and T. Christodoulakis, Phys. Rev.
D. 90, 123543 (2014)
KamS. Kamilya and B. Modak, Gen. Rel. Gravit. 36, 674 (2004)
capSTS. Capozziello, S. Nesseris and L. Perivolaropoulos, JCAP
2012 12(9) (2012)
cliftonT. Clifton and J.D. Barrow, Phys. Rev. D 73, 104022 (2006)
sot7T.P. Sotiriou, Gravity and Scalar fields, Proceedings
of the 7th Aegean Summer School: Beyond Einstein's theory of gravity,
Modifications of Einstein's Theory of Gravity at Large Distances, Paros,
Greece, ed. by E. Papantonopoulos, Lect.Notes Phys. 892 (2015)
wd1B.S. DeWitt, Phys. Rev. 160 1113 (1967)
HartleHawJ.B. Hartle and S.W. Hawking, Phys. Rev. D 28
2960 (1983)
WilD. Wiltshire, An introduction to quantum cosmology (2001)
(arXiv: gr-qc/0101003);
HalJ.J. Halliwell, Introductory lectures on quantum cosmology
(2009) (arXiv: 0909.2566 [gr-qc])
OlverP.J. Oliver, Applications of Lie Groups to
Differential Equations, Graduate Texts in Mathematics, Volume 107,
(Springer-Verlag: New York, 1993)
IbragBN.H. Ibragimov, Transformation Groups Applied to
Mathematical Physics Mathematics and Its Applications, Soviet Series, (D
Reidel Publishers: Dordrecht, 1985)
EmmyNE. Noether, Invariante Variationsprobleme, Nachr. v.d. Ges. d.
Wiss. zu Göttingen, 235 (1918)
ijgmmpA. Paliathanasis and M. Tsamparlis, Int. J. Geom. Meth. Mod.
Phys. 11 1450037 (2014)
HanlonJ. O'Hanlon, Phys. Rev. Lett. 29 137 (1972)
bcotJ.D. Barrow and S. Cotsakis, Phys. Lett. B 214, 515 (1988)
nebanA. Paliathanasis, J. Phys.: Conf. Ser. 453 012009 (2013)
bottJ.D. Barrow and A.C. Ottewill, J. Phys. A 16, 2757 (1983)
SuzukiN. Suzuki et.al, Astrophys. J. 746 85 (2012)
PercivalW.J. Percival et al, Mon. Not. R. Astron. Soc.,
401 2148 (2010)
BlakeCC. Blake et al., Mon. Not. R. Astron. Soc. 418 1707 (2011)
BennetH0C.L. Bennet, D. Larson, J.L. Weiland and G. Hinshaw, Ap. J.
794 135 (2014)
BasilNessS. Basilakos, S. Nesseris and L. Perivolaropoulos, Phys.
Rev. D. 87 123529 (2013)
Akaike1974H. Akaike, IEEE Transactions of Automatic Control,
19, 716 (1974); N. Sugiura, Communications in Statistics A, Theory
and Methods, 7, 13 (1978)
hartleJ.B. Hartle, Gravitation in Astrophysics: Cargèse
1986, Proceedings of a Nato Advanced Study Institute on Gravitation in
Astrophysics, Cargèse, France, Ed. by B. Carter and J.B. Hartle, (Plenum:
New York, 1986)
GANNR. Gannouji, B. Moraes and D. Polarski, JCAP, 62, 034
(2009); S. Tsujikawa, R. Gannouji, B. Moraes and D. Polarski, Phys. Rev. D.,
80, 084044 (2009)
|
1511.00475
|
$^1$School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China; [email protected];
$^2$ Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, China;
$^3$ Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA; [email protected];
$^4$ Department of Astronomy, School of Physics, Peking University, Beijing 100871, China;
$^5$ Kavli Institute of Astronomy and Astrophysics, Peking University, Beijing 100871, China
Three giant flares have been detected so far from soft gamma-ray repeaters, each characterized by an initial short hard spike and a pulsating tail. The observed pulsating tails are characterized by a duration of $\sim100\,\unit{s}$, an isotropic energy of $\sim 10^{44}\,\unit{erg}$, and a pulse period of a few seconds. The pulsating tail emission likely originates from the residual energy after the intense energy release during the initial spike, which forms a trapped fireball composed of a photon-pair plasma in a closed field line region of the magnetars. Observationally the spectra of pulsating tails can be fitted by the superposition of a thermal component and a power-law component, with the thermal component dominating the emission in the early and late stages of the pulsating tail observations. In this paper, assuming that the trapped fireball is from a closed field line region in the magnetosphere, we calculate the atmosphere structure of the optically-thick trapped fireball and the polarization properties of the trapped fireball.
By properly treating the photon propagation in a hot, highly magnetized, electron-positron pair plasma, we tally photons in two modes (O mode and E mode) at a certain observational angle through Monte Carlo simulations. Our results suggest that the polarization degree depends on the viewing angle with respect to the magnetic axis of the magnetar, and can be as high as $\Pi\simeq30\%$ in the $1-30\,\unit{keV}$ band, and $\Pi\simeq10\%$ in the $30-100\,\unit{keV}$ band, if the line of sight is perpendicular to the magnetic axis.
§ INTRODUCTION
Magnetars are neutron stars with super-strong magnetic fields. The typical magnetic strength of magnetars is of the order of $(10^{14}-10^{15}) \unit{G}$, which is higher than the critical magnetic field $B_{Q}=m_e^{2}c^{3}/\hbar e\approx 4.4\times 10^{13}\unit{G}$, and is $\sim (10^{2}-10^{3})$ times stronger than that of normal pulsars. Historically, they were discovered in two manifestations, i.e. anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs), even though later observations suggested that there seems no clear distinction between the two sub-categories. SGRs are discovered in the hard X-ray/soft gamma-ray band characterized by repeating bursts. Eleven SGRs have been confirmed so far <cit.>. Three of them each displayed one giant flare (SGR 0526-66 on March 5, 1979; SGR 1900+14 on August 27, 1998; SGR 1806-20 on December 27, 2004). These giant flares are characterized by an initial short-hard spike followed by a pulsating tail <cit.>. The initial spike has a duration of $\sim 0.1\unit{s}$, and an isotropic energy of $(10^{44}-10^{46}) \unit{erg}$. The pulsating tail, on the other hand, shows a decay pattern and is characterized by a typical duration of $\sim 100\,\unit{s}$, an isotropic energy of $\sim 10^{44}\,\unit{erg}$, and a pulsating period of a few seconds (the magnetar spin period). The spectrum of the early spike is characterized by a temperature of $T\sim100\,\unit{keV}$, which is much harder than that of normal SGR bursts. The spectrum of the pulsating tail is softer than that of the initial spike, which may be fitted with the superposition of a thermal component with $T\sim10-30\,\unit{keV}$ and a non-thermal power law component. The decay light curves of the pulsating tail typically show time-evolving, complex pulse profiles.
As pointed out by <cit.>, the tail emission likely originates from the residual energy of the explosion that gives rise to the initial spike. Owing to the large energy density and strong magnetic field pressure in the magnetar magnetosphere, the residual energy is confined in the magnetosphere in the form of an optically thick photon-pair plasma named a “trapped fireball”. Such a trapped fireball would be the source of the thermal emission component observed in the pulsating tail. The power law spectral component may be formed from the jets associated with the open-field lines that are outside the trapped fireballs <cit.>. The structure of the trapped fireball may determine the pulse profile of the pulsating tail. In fact, the magnetars show more complicated pulse structure profiles during the giant flare decaying phase than during the quiescent phase, the latter of which typically shows a single, broad-peak, nearly sinusoidal pulse shape. For example, the decay phase of the giant flare of SGR 0526-66 on March 5, 1979 showed a double-peak pulse profile <cit.>; the giant flare of SGR 1900+14 on August 27, 1998 showed a complex four-peak profile <cit.>; and the giant flare of SGR 1806-20 on December 27, 2004 showed a roughly three-peak profile <cit.>. On the other hand, the temperature of the phase-resolved spectra during the decay phase shows a different evolution pattern from the light curve: For SGR 1900+14, the phase-resolved temperature shows two peaks in each period, with the epoch of the highest thermal flux corresponding to the lowest count rate in the light curve <cit.>; For SGR 1806-20, on the other hand, the evolution of temperature seems to track the photon count in the light curve <cit.>. Therefore, the true structure of the trapped fireball is likely very complex. The effective blackbody radius inferred from the data suggests that the radius of the trapped fireball is a few times of the magnetar radius <cit.>, which suggests that the trapped fireball should be in the magnetosphere rather than being a hot spot on the magnetar surface. A natural idea is that the trapped fireball is a closed-field region in the magnetosphere (and the non-thermal emission comes from the open field line regions). The multi-peak profile of the light curves might be the result of the inhomogeneous distribution of the plasmas within the trapped fireball and the non-thermal jets within the magnetosphere. Due to the complexity of the problem itself, it is not easy to fully understand the mechanism of the giant flares. Nonetheless, a simple model that assumes that the trapped fireball is from a closed field region in the magnetosphere may catch the essence of the problem.
Polarization is a key element to test theoretical models of the physical processes and geometric structure of magnetars. Unfortunately, none of the three giant flares were detected with polarimeters, so that their polarization properties remain a mystery. Thanks to technological development and funding support, a few X-ray/$\gamma$-ray polarimeters are imminent, which would make ground-breaking discoveries to study the underlying physics in many astrophysical objects, such as isolated neutron stars, magnetars, X-ray pulsars, pulsar wind nebulae, rotation-powered pulsars, and millisecond X-ray pulsars <cit.>. For example, in the conventional X-ray band ($2-10\,\unit{keV}$),
polarimeters that are under construction include Imaging X-ray Polarimetry Explorer (IXPE, <cit.>), Polarimeter for Relativistic Astrophysical X-raY Sources (PRAXyS, <cit.>).
For hard X-rays and soft-$\gamma$-rays (above $10\,\unit{keV}$), several detector concepts have been proposed. For example, there has been an extensive study of the POET (Polarimeters for Energetic Transients) mission concept <cit.> as well as its extension LEAP (LargE Area Polarimeter). On the other hand, the POLAR detector <cit.>, a Compton polarimeter designed to measure linear polarization of gamma-rays from gamma ray bursts (GRBs), is scheduled to be launched with the Chinese Space Station Tiangong-2 in 2016.
POLAR has a large effective area of $400\,\unit{cm^2}$ and a field-of-view covering about one third of the sky. Its energy band covers from $50\,\unit{keV}$ to $500\,\unit{keV}$, which can cover part of the energy range of the pulsating tail of SGR giant flares. For a total flux larger than $10^{-5}\,\unit{erg\,cm^{-2}}$, POLAR can perform a measure of polarization degree with a relative systematic error $\sim10\%$ <cit.>. It would be an ideal detector for giant flare polarization studies, should such a giant flare occurs during its mission period.
Among the available detectors, the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) is an instrument designed to study the brightest solar flares, which has the polarization detecting capability <cit.>. Interestingly, the giant flare of SGR 1806-20 on December 27, 2004 was detected with RHESSI <cit.>. However, no polarization study of the pulsating tail emission of this event is published.
Many authors have calculated the polarization properties from neutron stars. <cit.> calculated the polarization of thermal X-rays from isolated neutron stars with strong magnetic fields. <cit.> studied radiation spectra from the atmosphere of neutron stars and the effect of vacuum polarization. <cit.> and <cit.> suggested that the birefringence of the magnetized QED vacuum decouples the polarization modes up to a large radius, so that the surface polarization of neutron stars can be sufficiently large to be observed. <cit.> and <cit.> calculated the soft X-ray polarization in thermal magnetar emission from a hot spot centered around the magnetic dipole axis. <cit.> studied the X-ray polarization signature of the magnetosphere of quiescent magnetars. <cit.> studied how future X-ray polarization measurements by the next generation detectors may be used to probe the magnetar magnetospheric structure.
In this paper, we study the atmospheric structure and polarization properties of trapped fireballs during the decay phase of giant flares. Since the radiation mechanism of the non-thermal component is still uncertain, we mainly consider the thermal emission component from the trapped fireball. Observationally, the thermal radiation component seems to dominate the observed emission during most phases in the decaying tail, and the flux level of the thermal component is higher than that of the power-law component between $10\,\unit{keV}$ and $100\,\unit{keV}$ <cit.>.
The paper is organized as follows. In Sect. 2, we introduce a simple model of the atmospheric structure of the trapped fireball. In Sect. 3, the photon polarization properties are calculated by considering dielectric tensor of the electron-positron pair plasma as well as the effect of vacuum polarization, and the scattering opacities for the two photon modes are presented. In Sect. 4. we calculate the polarization degree of the trapped fireball using Monte Carlo simulations. The results are summarized in Sect. 5 with some discussion.
§ ATMOSPHERIC STRUCTURE OF TRAPPED FIREBALL
In this section, we calculate the atmospheric structure of the trapped fireball in a simplified model. We assume a dipolar configuration for the magnetar magnetic fields, and take the region enclosed by a closed field line (not necessarily the last closed field line) as the trapped fireball (Fig.1). Thanks to the extreme large optical depth of the trapped fireball, we only need to solve the equations in a relatively thin layer (the bisque layer shown in Fig.1), which corresponds to the radius range from an optical depth $\tau \gg 1$ (bottom of the atmosphere of the trapped fireball) to $\tau \ll 1$. The observed photons mostly come from this region. The structure of this thin atmosphere can be solved via a set of equations, including the magnetic dipole field equation, the equations of state, the hydrostatic equilibrium equation, the energy transport equation, and the optical depth equation.
The total energy radiated in the pulsating tail of a giant flare is $E_{tail}\sim10^{44}\unit{erg}$ <cit.>. The requirement for the energy being trapped by magnetic fields demands
\begin{eqnarray}
\frac{B_{R_s+\Delta R}^2}{8\pi}\gtrsim\frac{E_{tail}}{3\Delta R^3},\label{trap}
\end{eqnarray}
where $R_s$ is the radius of the neutron star, and $\Delta R$ is the scale of the trapped fireball (Fig.1). The factor $1/3$ comes from converting the internal energy density (both the thermal energy[Even though at the photosphere the electron temperature is 10s of keV, in the bulk of the trapped fireball, the electron temperature is higher than 511 keV, so that the fireball can be regarded as a relativistic gas. See detailed discussion below.] and the radiation energy) of the trapped fireball to pressure. For a dipolar field, one has $B(r)=B_\ast (r/R_s)^{-3}$, where $B_\ast$ is the characteristic surface magnetic field strength of the neutron star. Observations show that the effective radius of thermal emission is a few times of the magnetar radius, $\Delta R\gtrsim R_s$ <cit.>. We therefore require that the magnetic field strength satisfies
\begin{eqnarray}
B_\ast\gtrsim 2\times10^{14}\unit{G}\left(\frac{\Delta R}{10\unit{km}}\right)^{-3/2}\left(\frac{E_{tail}}{10^{44}\unit{erg}}\right)^{1/2}\left(\frac{1+\Delta R/R_s}{2}\right)^3 > B_Q,
\end{eqnarray}
which suggests that the neutron star is a magnetar.
In the closed field line region, the electron-positron pair plasma is trapped, since transport of charged particles across field lines is suppressed by strong magnetic fields, and since gravity is no longer the dominant confining force. The critical trapping luminosity $L_c$ can be estimated by the balance between the radiative energy density and the magnetic energy density $L/4\pi r^2 c\sim B^2/8\pi$, i.e.
\begin{eqnarray}
L_c\simeq 3\times10^{49}\,\unit{erg\,s^{-1}}\left(\frac{B}{B_Q}\right)^2\left(\frac{r}{R_s}\right)^2,
\end{eqnarray}
which is much larger than the luminosity of the giant flares.
For a magnetic dipole field, the strength of the magnetic field at a point $(r,\theta)$ is
\begin{eqnarray}
\end{eqnarray}
where $B_p$ is the polar cap magnetic field strength at the surface. The border of the closed magnetosphere field lines that enclose the trapped fireball is given by $r=R_{max}\sin^2\theta$, where $R_{max}$ is the maximum radius.
Next, we consider the equation of state in such a trapped fireball. The deposition of $E_{tail}\sim10^{44}\,\unit{erg}$ in the magnetosphere is sufficient to generate a hot photon-pair plasma. For the internal region of the trapped fireball, the electron-positron pairs are mildly-relativistic. The total energy density of the photon-pair plasma is $U=U_\gamma+U_{e^\pm}=(11/4)aT^4$ (see Eq. (<ref>) in Appendix). The temperature in the internal region of the fireball is $T=1.3\,\unit{MeV}(E_{tail}/10^{44}\,\unit{erg})^{1/4} (\Delta R/R_s)^{-3/4}$. For the atmosphere of the trapped fireball, on the other hand, observations show that its effective temperature is $k_BT_{eff}\sim30\,\unit{keV}\ll m_ec^2$, which is much less than the first Landau level energy $\hbar eB/m_ec=508\,\unit{keV}(B/B_Q)$. Therefore, this is a one-dimensional, magnetized, and non-relativistic electron-positron pair plasma, whose density is given by (see Eq. (<ref>) in Appendix)
\begin{eqnarray}
\rho=m_e\frac{(m_ec)^3}{\hbar^3(2\pi^3)^{1/2}}\frac{B}{B_Q}\left(\frac{k_BT}{m_ec^2}\right)^{1/2}\exp(-m_ec^2/k_BT).\label{eos}
\end{eqnarray}
We apply an ideal gas equation of state and assume that the gas is in local thermodynamic equilibrium with the radiation field. The total pressure $P$ is given by the sum of the gas pressure $P_g$, radiation pressure $P_\gamma$, and magnetic pressure $P_B$, i.e.
\begin{eqnarray}
P=P_g+P_r+P_B,\,\,\,P_g=\frac{\rho}{m_e}k_BT,\,\,\,P_r=\frac{4\sigma_{SB}}{3c}T^4,\,\,\,P_B=\frac{B^2}{8\pi}. \label{pressure}
\end{eqnarray}
The giant flare pulsating tails are characterized by a duration of $\sim100\,\unit{s}$. We assume that the trapped fireball is in hydrostatic equilibrium during the pulsating tail phase:
\begin{eqnarray}
\nabla P+\rho \nabla\Phi-\frac{1}{4\pi}(B\cdot\nabla)B=0,
\end{eqnarray}
where $\Phi$ is the gravitational potential. For the dipolar magnetic field, the magnetic stress in radial direction ($r$ direction) is given by
\begin{eqnarray}
\end{eqnarray}
Therefore one has
\begin{eqnarray}
\frac{dP}{dr}=-\frac{GM_s\rho}{r^2}\left(1-\frac{R_g}{r}\right)^{-1}\left[1+\frac{P+1.5P_g+3P_r+P_B}{\rho c^2}\right]+F_B,
\end{eqnarray}
where $M_s$ is the neutron star mass (with the trapped fireball mass neglected), and $R_g=2GM_s/c^2$ is the gravitational radius. Here, we neglect the term $4\pi r^3P/M_sc^2$ in the general Oppenheimer-Volkoff equation, because its contribution is always smaller than $10^{-4}$ in our atmosphere model. In order to solve the atmospheric structure, we also need the temperature gradient equation
\begin{eqnarray}
\frac{dT}{dr}=\frac{T}{P}\frac{dP}{dr}\nabla,
\end{eqnarray}
where $\nabla\equiv(\ln T/\ln P)$ is determined by the energy transport equation. As pointed out by <cit.>, convection is strongly suppressed by strong magnetic fields of magnetars. We therefore only consider radiative transport in this paper.
For simplicity of treatment and without losing generality, we use the result from the radiative transport in spherical symmetry <cit.>, which is given by
\begin{eqnarray}
\nabla\equiv\left(\frac{\ln T}{\ln P}\right)_r&=&\left[\frac{\kappa L}{16\pi cGM_s}\left(\frac{P}{P_r}\right)\left(1-\frac{R_g}{r}\right)^{-1/2}+\frac{P}{\rho c^2}\right]\nonumber\\
&\times&\left[1+\frac{P+1.5P_g+3P_r+P_B}{\rho c^2}\right]^{-1},
\end{eqnarray}
where $L$ is the luminosity of the photosphere (i.e. the observed luminosity of the pulsating tail), and $\kappa$ is the Rosseland mean opacity, which is discussed in the next section. In order to obtain the scale length of the photosphere, one also needs the optical depth equation
\begin{eqnarray}
\frac{d\tau}{dr}=-\kappa\rho \left(1-\frac{R_g}{r} \right)^{-1/2}.\label{optical}
\end{eqnarray}
Finally, we introduce the boundary of the bottom of the atmosphere that is defined by $(R_{max,0},T_0)$, where $R_{max,0}$ is the maximum radius of the closed line of the atmosphere bottom, and $T_0$ is the temperature at the bottom of the atmosphere. Using Eqs.(<ref>)-(<ref>) and Eqs.(<ref>)-(<ref>), we can then solve for the atmospheric structure of the trapped fireball.
We assume $M_s=1.44\,M_{\odot}$, $R_s=10^6\,\unit{cm}$, $B_p=10^{15}\,\unit{G}$, the rotation period of the magnetar $P=5\,\unit{s}$, and the luminosity of the pulsating tail $L=10^{42}\,\unit{erg\,s^{-1}}$. Since observations show that the effective radius of the thermal emission is a few times of the radius of the magnetar, we assume that the maximum radius of the bottom of the atmosphere is $R_{max,0}=2R_s$. The corresponding temperature $T_0$ at $R_{max,0}$ should be less than the pair production temperature $T_\pm\simeq 6\times10^9\,\unit{K}$ in order to reach a quasi-steady state solution.
In our model, we take $T_{0}=10^9\,\unit{K}$. For $T_{0}\ll 10^9\,\unit{K}$, the temperature is so low that the thermal electron-positron pair density decreases exponentially with temperature, leading to a very low optical depth.
In Fig. <ref>-<ref>, we plot the temperature, optical depth, number density of electron-positron pairs and pressure as a function of the radial distance from the bottom of the atmosphere $r-R_{max,0}\sin^2\theta$, where $R_{max,0}\sin^2\theta$ is the radial distance from the neutron star center to the bottom of the atmosphere (see Fig. <ref>, with $\theta=\pi/6,\,\pi/3,\,\pi/2$ denoted by dotted, dashed and solid lines). For the given boundary condition described above, the atmosphere is very thin $\sim 10^3\,\unit{cm}$ and the effective temperature at the photosphere ($\tau=1$) is $T_{eff}=27\,\unit{keV}$ for $\theta=\pi/2$. This is consistent with the observed temperature of the pulsating tail. The temperatures for different $\theta$ have little difference for $\tau>1$, and the temperature near the magnetosphere equator is higher than that near the pole for $\tau<1$. The optical depth from the direction near the pole is lower than that in the direction of the equator, which means that the flux near the pole might be higher. As shown in Fig. <ref>, we find that the electron-positron pair number density decreases rapidly when $\tau<1$. Figure <ref> shows pressure as a function of the radial distance to the atmosphere bottom $r-R_{max,0}\sin^2\theta$. The electron-positron pairs, radiation and magnetic pressures are denoted by black, red and blue lines, respectively. One can see that the total pressure is dominated by the magnetic pressure, so that the pair plasma is trapped in the magnetosphere. Near the equator, one has $P_g<P_\gamma$. However, near the pole, the gas pressure could be larger than the radiation pressure at the atmosphere bottom. The result suggests that the pressure at the pole decreases more rapidly than that of the equator, since the temperature at the pole decreases from a smaller radius $r-R_{max,0}\sin^2\theta$.
§ PHOTON POLARIZATION AND SCATTERING OPACITY OF PAIRS
<cit.> calculated the photon polarization modes in a magnetized electron-ion plasma. In this section, we calculate the photon polarization modes in a magnetized electron-positron pair plasma following the procedure introduced by <cit.>. As shown below, because of the charge symmetry in a pair plasma, some noticeable differences from the case of an electron-ion plasma exist. For example, the photons are completely linearly polarized, and there is no vacuum resonance point and ion cyclotron absorption in the case of the pair plasma.
For the atmosphere of a trapped fireball, the pair plasma is non-relativistic ($k_BT\ll m_ec^2$). In the coordinate system $x^\prime\,y^\prime\,z^\prime$ with $\mathbf{B}$ along $\mathbf{z}^\prime$, the dielectric tensor $\pmb{\epsilon}^{(p)}$ contributed by the plasma is given by <cit.>
\begin{eqnarray}
\left[\pmb{\epsilon}^{(p)}\right]_{\mathbf{z}^\prime=\mathbf{\hat{B}}}=\left(
\begin{array}{ccc}
\varepsilon & ig & 0\\
-ig & \varepsilon & 0\\
0 & 0 & \eta\\
\end{array}
\right).
\end{eqnarray}
\begin{eqnarray}
\varepsilon&=&1-\sum_s\frac{\lambda_s\upsilon_s}{\lambda_s^2-u_s},\\
\eta&=&1-\sum_s\frac{\upsilon_s}{\lambda_s},\\
\end{eqnarray}
where $s$ represents the charged particle species in the plasma, $u_s=\omega_{Bs}^2/\omega^2$, and $\upsilon_{s}=\omega_{ps}^2/\omega^2$. For the charged particle $s$, $\omega_{Bs}=|q_s|B/(m_sc)$ is the cyclotron frequency, and $\omega_{ps}=(4\pi n_sq_s^2/m_s)^{1/2}$ is the plasma frequency. The parameter $\lambda_s=1+i\nu_s/\omega$ delineates damping of the particle motion, where $\nu_s$ is the damping rate. In our calculation, we assume small damping ($\nu_s\ll \omega$). For electron-positron pairs, $s=e^+,e^-$, we have
\begin{eqnarray}
\varepsilon&=&1-\frac{2\upsilon_e}{1-u_e},\\
\eta&=&1-2\upsilon_e,\\
\end{eqnarray}
\begin{eqnarray}
\upsilon_e&=&\left(\frac{E_{pe}}{E}\right)^2=\left(\frac{0.02871\rho_1^{1/2}\,\mathrm{keV}}{E}\right)^2,
\end{eqnarray}
with $B_{14}=B/(10^{14}\,\unit{G})$ and $\rho_1=\rho/(1\,\unit{g\,cm^{-3}})$. We note that the non-diagonal terms in the matrix are zero due to charge symmetry of pairs. On the other hand, in strong magnetic fields, vacuum polarization makes a corrected contribution to the dielectric tensor:
\begin{eqnarray}
\Delta\pmb{\epsilon}^{(v)}=(a-1)\mathbf{I}+q\mathbf{\hat{B}\hat{B}},
\end{eqnarray}
where $\mathbf{I}$ is the unit tensor and $\mathbf{\hat{B}}$ is the unit vector along $\mathbf{B}$. Similarly, vacuum polarization also modifies the magnetic permeability tensor $\pmb{\mu}$ as
\begin{eqnarray}
\mathbf{H}_w=\pmb{\mu}^{-1}\cdot\mathbf{B}_w=(a\mathbf{I}+m\mathbf{\hat{B}\hat{B}})\cdot\mathbf{B}_w,
\end{eqnarray}
where $\mathbf{H}_w$ and $\mathbf{B}_w$ are the magnetizing field and the magnetic field of an electromagnetic wave, respectively.
The vacuum polarization coefficients are given by <cit.>
\begin{eqnarray}
\end{eqnarray}
where $\beta_B\equiv B/B_Q$ is the relevant dimensionless magnetic field parameter, and $\alpha=1/137$ is the fine-structure constant.
If $|\epsilon^{(v)}_{ij}|\ll 1$ ($B\ll5\times10^{16}\mathrm{G}$), the vacuum polarization contribution, $\Delta\pmb{\epsilon}^{(v)}$, could be added linearly to the dielectric tensor $\pmb{\epsilon}^{(p)}$. In the frame with $\mathbf{\hat{B}}$ along $\mathbf{z}^\prime$, one has
\begin{eqnarray}
\left[\pmb{\epsilon}\right]_{\mathbf{z}^\prime=\mathbf{\hat{B}}}&=&\left(
\begin{array}{ccc}
\varepsilon & 0 & 0\\
0 & \varepsilon & 0\\
0 & 0 & \eta\\
\end{array}\right)+\left(
\begin{array}{ccc}
a-1 & 0 & 0\\
0 & a-1 & 0\\
0 & 0 & a+q-1\\
\end{array}
\right)\nonumber\\
\begin{array}{ccc}
\varepsilon^\prime & 0 & 0\\
0 & \varepsilon^\prime & 0\\
0 & 0 & \eta^\prime\\
\end{array}\right),
\end{eqnarray}
where $\varepsilon^\prime=\varepsilon+a-1$ and $\eta^\prime=\eta+a+q-1$.
Using the Maxwell equations, the equation for a plane wave with $\mathbf{E}\propto \mathrm{e}^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ can be expressed as
\begin{eqnarray}
\nabla\times(\pmb{\mu}^{-1}\cdot\nabla\times\mathbf{E})=\frac{\omega^2}{c^2}\pmb{\epsilon}\cdot\mathbf{E},
\end{eqnarray}
which is <cit.>,
\begin{eqnarray}
\left\lbrace\frac{1}{a}\epsilon_{ij}+n^2\left[\hat{k_i}\hat{k_j}-\delta_{ij}-\frac{m}{a}(\hat{k}\times\hat{B})_i(\hat{k}\times\hat{B})_j\right]\right\rbrace E_j=0, \label{wave}
\end{eqnarray}
where $n=ck/\omega$ is the refractive index. In the coordinate system $x\,y\,z$ with $\mathbf{k}$ along $z$-axis and $\mathbf{B}$ in the $x-z$ plane, the dielectric tensor is given by
\begin{eqnarray}
\left[\pmb{\epsilon}\right]_{\mathbf{z}=\mathbf{\hat{k}}}=\left(
\begin{array}{ccc}
\varepsilon^\prime\cos^2\theta_B+\eta^\prime\sin^2\theta_B & 0 & (\varepsilon^\prime-\eta^\prime)\sin\theta_B\cos\theta_B\\
0 & \varepsilon^\prime & 0\\
(\varepsilon^\prime-\eta^\prime)\sin\theta_B\cos\theta_B & 0 & \varepsilon^\prime\sin^2\theta_B+\eta^\prime\cos^2\theta_B\\
\end{array}
\right),
\end{eqnarray}
where $\theta_B$ is the angle between $\mathbf{k}$ and $\mathbf{B}$. According to the $z$-component of Eq.(<ref>), one has
\begin{eqnarray}
\end{eqnarray}
Then Eq.(<ref>) becomes
\begin{eqnarray}
\left(
\begin{array}{cc}
\eta_{xx}-n^2 & 0\\
0 & \eta_{yy}-rn^2\\
\end{array}
\right)\left(
\begin{array}{c}
\end{array}
\right)=0,\label{marix}
\end{eqnarray}
where $r\equiv1+(m/a)\sin^2\theta_B$, and
\begin{eqnarray}
\eta_{xx}&=&\frac{1}{a\epsilon_{zz}}(\epsilon_{zz}\epsilon_{xx}-\epsilon_{xz}\epsilon_{zx})=\frac{1}{a\epsilon_{zz}}\varepsilon^\prime\eta^\prime,\\
\eta_{yy}&=&\frac{1}{a\epsilon_{zz}}\epsilon_{zz}\epsilon_{yy}=\frac{1}{a\epsilon_{zz}}[(\varepsilon^{\prime2}-\varepsilon^\prime\eta^\prime)\sin^2\theta_B+\varepsilon^\prime\eta^\prime].
\end{eqnarray}
There are two solutions to Eq.(<ref>):
(i.) If $n^2=\eta_{xx}$, then $E_y=0$ and $E_x,E_z\neq0$. The electric field vector of the wave is in the $\mathbf{k-B}$ plane. This is the ordinary mode (O mode).
(ii.) If $n^2=\eta_{yy}/r$, then $E_x=E_z=0$ and $E_y\neq0$. The electric field vector of the wave is perpendicular to the $\mathbf{k-B}$ plane. This is the extraordinary mode (E mode).
Different from the electron-ion plasma, we find that for electron-positron pairs, one does not need to define the polarization ellipticity $K_j=-iE_x/E_y$ <cit.>, and both modes are linearly polarized. In this case, there is no “vacuum resonance" point where E mode vs. O mode classification becomes ambiguous and can convert from one mode to the other.
The mode eigenvector in the coordinate system with the wave vector $\mathbf{k}$ along $z$-axis is
\begin{eqnarray}
\pmb{e}^\prime=\frac{1}{(E_x^2+E_y^2+E_z^2)^{1/2}}(E_x,E_y,E_z).
\end{eqnarray}
The scattering opacity depends on the polarization vector through its projection on the coordinate frame with the $z$-axis along the magnetic field $\mathbf{B}$ direction <cit.>. In this new coordinate system with $\mathbf{B}$ along $z$-axis, the eigenvector is given by
\begin{eqnarray}
\pmb{e}=\frac{1}{(E_x^2+E_y^2+E_z^2)^{1/2}}(E_x\cos\theta_B+E_z\sin\theta_B,E_y,-E_x\sin\theta_B+E_z\cos\theta_B).
\end{eqnarray}
The cyclic components of $\pmb{e}$ read[In <cit.>, they defined $iK_j=E_x/E_y$ and $iK_{z,j}=E_z/E_y$, and $K_j$ and $K_{z,j}$ are complex numbers, which are determined by the plasma medium properties. Thus, there is a plus or minus sign in $|{e_\pm^j}|^2$ in their notation. ]
\begin{eqnarray}
|{e_\pm^j}|^2&=&\left|\frac{1}{\sqrt 2}({e_x^j}\pm i {e_y^j})\right|^2=\frac{|E_y|^2+|E_x\cos\theta_B+E_z\sin\theta_B|^2}{2(E_x^2+E_y^2+E_z^2)},\nonumber\\
\end{eqnarray}
where $j=1$ corresponds to the E mode, and $j=2$ corresponds to the O mode. Specifically, for the E mode, the cyclic components are given by
\begin{eqnarray}
|e_\pm^1|^2=\frac{1}{2},\,|e_0^1|^2=0, \label{base}
\end{eqnarray}
and for the O mode, they are given by
\begin{eqnarray}
\end{eqnarray}
where we define $K_{zx}\equiv E_z/E_x=-(\epsilon_{zx}/\epsilon_{zz})$. We note that the eigenvectors of the E mode are constants for the electron-positron plasma. This is different from the case of electron-ion plasma. The scattering opacity from the $j$ mode into the $i$ mode is given by <cit.>
\begin{eqnarray}
\kappa_{ji}^{sc}=\frac{\sigma_T}{m_e}\sum_{\alpha=-1}^1\left[(1+\alpha u_e^{1/2})^2+\gamma_e^2\right]^{-1}|e_\alpha^j|^2A_\alpha^i,
\end{eqnarray}
where $\gamma_e\equiv\nu_e/\omega$ is the damping factor, and $A_\alpha^i=(3/4)\int_0^\pi d\theta \sin\theta |e_\alpha^i|^2$. The electron scattering opacity from the mode $j$ (into both model $i$ and mode $j$) is
\begin{eqnarray}
\kappa_{j}^{sc}=\frac{\sigma_T}{m_e}\sum_{\alpha=-1}^1\left[(1+\alpha u_e^{1/2})^2+\gamma_e^2\right]^{-1}|e_\alpha^j|^2A_\alpha,\label{opacji}
\end{eqnarray}
where $A_\alpha=A_\alpha^1+A_\alpha^2$. The scattering opacities for the two modes can be approximately derived <cit.>. For the E-mode, one has
\begin{eqnarray}
\kappa_1^{sc}=\kappa_{11}^{sc}+\kappa_{12}^{sc},\,\,\kappa_{11}^{sc}\simeq 3\kappa_{12}^{sc},\label{scaE}
\end{eqnarray}
where the second equation is under the condition of $\omega m_ec/eB\sim10^{-2}-10^{-3}$; For the O-mode, one has
\begin{eqnarray}
\kappa_2^{sc}=\kappa_{22}^{sc}+\kappa_{21}^{sc},\,\,\kappa_{21}^{sc}\simeq \left(\frac{\omega}{\omega_{Be}}\right)^2\kappa_{22}^{sc}\ll\kappa_2^{sc}. \label{scaO}
\end{eqnarray}
According to Eq.(<ref>)-Eq.(<ref>), the probability of E-mode photons converting into O-mode photons is $P_{EO}\simeq 1/4$, and the probability of O-mode photons converting into E-mode photons is $P_{OE}\simeq (\omega/\omega_{Be})^2$.
For the transverse-mode approximation ($K_{zx}\ll1$), one has $A_\alpha\simeq1$. Thus, the opacity is approximately
\begin{eqnarray}
\kappa_{1}^{sc}\simeq\frac{\sigma_T}{2m_e}\left[\frac{\omega^2}{(\omega-\omega_{Be})^2+(\gamma_e\omega)^2}+\frac{\omega^2}{(\omega+\omega_{Be})^2+(\gamma_e\omega)^2}\right]
\end{eqnarray}
for the E mode, and
\begin{eqnarray}
\kappa_{2}^{sc}\simeq\frac{\sigma_T}{2m_e}\left[\frac{\omega^2\cos^2\theta_B}{(\omega-\omega_{Be})^2+(\gamma_e\omega)^2}+\frac{\omega^2\cos^2\theta_B}{(\omega+\omega_{Be})^2+(\gamma_e\omega)^2}+\frac{2\sin^2\theta_B}{1+\gamma_e^2}\right]
\end{eqnarray}
for the O model. For radiation frequencies below the cyclotron frequency, i.e. $\omega\ll\omega_{Be}$, and the damping factor $\gamma_e\ll 1$, one has
\begin{eqnarray}
\kappa_1^{sc}\simeq\frac{\omega^2}{\omega_{Be}^2}\kappa_T,\,\,\kappa_2^{sc}\simeq\left(\frac{\omega^2}{\omega_{Be}^2}\cos^2\theta_B+\sin^2\theta_B\right)\kappa_T, \label{opacity}
\end{eqnarray}
where $\kappa_T = \sigma_T/ m_e$ ($\sigma_T$ is the Thomson cross section, and $m_e$ is the electron mass). Due to the absence of the ion cyclotron absorption and vacuum resonance absorption in the case of electron-positron plasma, the above equations could be approximately applied to calculate the opacities for $\omega\ll\omega_{Be}$. We then finally obtain the Rosseland mean opacity $\kappa$ through
\begin{eqnarray}
\frac{1}{{\kappa}}=\left[\int_0^\infty\left(\frac{1}{\kappa_1^{sc}}+\frac{1}{\kappa_2^{sc}}\right)\frac{\partial B_\nu(T)}{\partial T}d\nu\right]\Bigg/\left[\int_0^\infty\frac{\partial B_\nu(T)}{\partial T}d\nu\right].
\end{eqnarray}
For electron energy $\lesssim300\mathrm{keV}$, the contribution from electron-electron bremsstrahlung can be safely ignored <cit.>. We therefore ignore free-free absorption of electron pairs in the atmosphere of the trapped fireball.
We apply the full expressions of $\kappa_1^{sc}$ and $\kappa_2^{sc}$ in Eq. (<ref>) to perform the calculation. The scattering opacities $\kappa_j^{sc}$ of the two modes as a function of energy for various angle $\theta_B$, magnetic fields $B$, and electron-positron pair density $n_e$ are presented in Figures <ref> - <ref>, respectively. Different from the case of electron-ion plasma <cit.>, for electron-positron pair plasma, there are no ion cyclotron absorption and vacuum resonance absorption, so that the opacities satisfy Eq. (<ref>) for $E\ll E_{Be}=\hbar eB/m_ec$. As shown in Fig.<ref>, for magnetic field $B=10^{14}\,\unit{G}$ and electron-positron pair number density $n_e=10^{25}\,\unit{cm^{-3}}$, the O mode opacity exhibits an angle dependence for $\theta_B$, i.e. $\kappa_O\simeq\kappa_T\sin^2\theta_B\,(\omega\ll\omega_{Be})$. The E model opacity, on the other hand, is independent of $\theta_B$ (see Eq. (<ref>) and Eq. (<ref>)). As shown in Fig <ref>, for $n_e=10^{25}\,\unit{cm^{-3}}$ and $\theta_B=\pi/4$, the electron cyclotron peak $E_{Be}$ is proportional to the strength of magnetic field $B$ and the E mode opacity satisfies $\kappa_E\simeq(\omega/\omega_{Be})^2\kappa_T$ for $\omega\ll\omega_{Be}$. As shown in Fig. <ref>, for $B=10^{14}\,\unit{G}$ and $\theta_B=\pi/4$, the plasma peak $E_{pe}$ is proportional to $n_e^{1/2}$.
§ MONTE CARLO SIMULATIONS
In this section, we conduct Monte Carlo simulations to calculate the polarization properties of the X-ray photons emitted from the trapped fireball.
To calculate the polarization signals, we set a coordinate system[Notice that the coordinate systems $XYZ$ and $X'Y'Z'$ introduced in this section are the global coordinate systems for the neutron star, which are very different from the “local" coordinate systems $xyz$ and $x'y'z'$ introduced in Section 3. In the following derivations, the coordinate symbols are still kept in lower case for these global coordinate systems.] $XYZ$ with $Z$-axis along the line of sight $\pmb{e_z}$, as shown in Fig. <ref>. The spin angular velocity vector $\pmb{\Omega}$ is in the $XZ$ plane. We define $\zeta$ as the angle between the magnetic dipole axis $\pmb{\mu}$ and $\pmb{\Omega}$, $\Theta$ as the angle between $\pmb{\mu}$ and the line of sight $\pmb{e_z}$, and $\delta$ as the angle between $\pmb{e_z}$ and $\pmb{\Omega}$.
For a dipolar field, in the coordinate system $X^\prime Y^\prime Z^\prime$ with $\pmb{\mu}$ along the $Z^\prime$-axis, the magnetic field components in a spherical coordinate system are given by
\begin{eqnarray}
\end{eqnarray}
Transforming from the spherical coordinate system to the Cartesian coordinate system, we have
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\tan\theta&=&\sqrt{x^{\prime2}+y^{\prime2}}/z^{\prime},\\
\tan\varphi&=&y^\prime/x^\prime.
\end{eqnarray}
In the coordinate system $XYZ$ with the line of sight along the $Z$-axis, the magnetic field components can be calculated by
\begin{eqnarray}
\left(
\begin{array}{c}
\end{array}
\right)=
\left(
\begin{array}{ccc}
\cos\Theta\cos\phi & \sin\phi & -\sin\Theta\cos\phi\\
-\cos\Theta\sin\phi & \cos\phi & \sin\Theta\sin\phi\\
\sin\Theta & 0 & \cos\Theta\\
\end{array}
\right)
\left(
\begin{array}{c}
\end{array}
\right),
\end{eqnarray}
and the coordinate system transformation is given by
\begin{eqnarray}
\left(
\begin{array}{c}
\end{array}
\right)=
\left(
\begin{array}{ccc}
\cos\Theta\cos\phi & -\cos\Theta\sin\phi & \sin\Theta\\
\sin\phi & \cos\phi & 0\\
-\sin\Theta\cos\phi & \sin\Theta\sin\phi & \cos\Theta\\
\end{array}
\right)
\left(
\begin{array}{c}
\end{array}
\right),
\end{eqnarray}
\begin{eqnarray}
\tan\phi&=&\frac{\sin\zeta\sin\psi}{\cos\zeta\sin\delta+\sin\zeta\cos\delta\cos\psi},\\
\cos\Theta&=&\cos\zeta\cos\delta-\sin\zeta\sin\delta\cos\psi,\\
\sin\Theta&=&\frac{\sin\zeta\sin\psi}{\sin\phi},
\end{eqnarray}
with $\phi$ being the azimuthal angle, and $\psi=\Omega t+\psi_0$ being the rotation phase ($\psi_0$ is the initial rotation phase). We assume that the photon flux is same everywhere at the bottom of the atmosphere. In order to calculate the photon spatial distribution, we need to know the surface area of the magnetosphere $S(\theta)$ with $\theta$. Because the border of the magnetosphere is given by $r=R_{max}\sin^2\theta$, where $R_{max}$ is the maximum radius of the closed field lines, the length of the dipole field line is given by
\begin{eqnarray}
\end{eqnarray}
Thus, the differential area of the magnetosphere is given by
\begin{eqnarray}
dS&=&2\pi r\sin\theta dl\nonumber\\
&=&2\pi R_{max}^2\sin^4\theta\sqrt{1+3\cos^2\theta}d\theta.\label{ds}
\end{eqnarray}
Integrating over $dS$, we obtain the area of the closed magnetic field line within $\theta$, i.e. $S(\theta)=\int_0^\theta dS$. The total area is $S(\theta=\pi)\simeq8.9R_{max}^2$.
Using this information, we can apply the Monte Carlo method to assign the initial spatial distribution of the photons.
As we have discussed above, the observations show that the spectra of the pulsating tails are dominated by the thermal component. Therefore, in this paper, we only consider the thermal component contributed by the photon-pair trapped fireball. Because the optical depth of the trapped fireball is very large, the observed radiation comes from the atmosphere of the trapped fireball. We apply the atmosphere model described in Sec. 2. to calculate the photon polarization properties through Monte Carlo simulations. The procedure of our calculations is the following:
A. We assume that the boundary of the atmosphere of the trapped fireball satisfies the follow conditions:
(i.) The bottom of the atmosphere is a closed magnetic field line surface defined by $r=R_{max,0}\sin^2\theta$ with a same temperature $T_0$.
(ii.) The radiative flux is the same everywhere at the bottom of the atmosphere. As a result, the initial photon spatial distribution can be simulated according to the coordinates $(\theta,\phi)$, i.e. $\xi=\int_0^\theta dS/\int_0^\pi dS$, $dS$ is given by Eq. (<ref>), $\phi=2\pi\xi$, with $\xi$ being a random number between (0,1). The initial directional distribution $(\theta_v,\phi_v)$ of the photons is isotropic, i.e., $\cos\theta_v=1-2\xi$ and $\phi_v=2\pi\xi$.
(iii.) The initial photon number in the E mode is equal to that in the O mode at the bottom of the atmosphere. However, the photons in one mode can be converted to the other mode due to the electron scattering in the atmosphere, leading to the change of the ratio between two modes.
B. We keep track the propagation and transition of each photon through the medium.
(i.) At each time step, we grid the atmosphere of the trapped fireball in 3 dimensions. The properties (e.g. density, temperature, pressure) in each cell are kept constant.
(ii.) The scattering optical depth $\tau_{sc}$ is given by $\tau_{sc}=-\ln(1-\xi)$. Each photon is allowed to travel along its trajectory in each time step. The optical depth in each cell is calculated until $\tau>\tau_{sc}$. Note that we need to use different opacities for different photon modes in this process.
(iii.) Scattering would change the mode of the photons. The probability for E mode converting into O mode is $P_{EO}\simeq1/4$, whereas the probability for O mode converting into E-mode is $P_{OE}\simeq(\omega/\omega_{Be})^2$.
(iv.) Due to the electrons' thermal motion, scattering would be isotropic. The direction $(\theta,\phi)$ of a scattered photon is given by $\cos\theta_v=1-2\xi$ and $\phi_v=2\pi\xi$.
(v.) We assume that the atmosphere satisfies local thermodynamic equilibrium (LTE). The energy distribution of the photons depends on the energy distribution of the electron-positron pairs that have a thermal distribution.
(vi.) We repeat the above procedure for each time step. Based on the atmosphere model discussed above, we find that the photons are almost no longer scattered when $r\gtrsim10\,R_s$, since $\tau(r=10R_s)\ll 1$.
C. We collect the photons along the light of sight within the solid angel $\Delta\Omega=0.15$, and calculate the directional angel $\phi_B$ for both photon modes at $r=10R_s$. For E-mode photons, one has $\phi_{B,E}=\phi_B$, where
\begin{eqnarray}
\phi_B\simeq\left \{
\begin{array}{lll}
\arctan(B_y/B_x), &\,\,& B_x>0,\\
\pi/2+\arctan(B_y/B_x), &\,\,& B_x<0,\\
\end{array}
\right.
\end{eqnarray}
and $B_x$ and $B_y$ are the magnetic field components perpendicular to the light of sight. For O mode photons, one has $\phi_{B,O}=\pi/2+\phi_B$.
The Stokes parameters near the photosphere are given by
\begin{eqnarray}
I&=&\langle E_x^2\rangle+\langle E_y^2\rangle,\\
Q&=&\langle E_x^2\rangle-\langle E_y^2\rangle,\\
U&=&\langle E_a^2\rangle-\langle E_b^2\rangle,\\
V&=&\langle E_l^2\rangle-\langle E_r^2\rangle,
\end{eqnarray}
where the subscripts refer to different bases of the Jones vector space: $(x,y)$ is the Cartesian basis, $(a,b)$ is the Cartesian basis rotated by $\pi/4$, $(l,r)$ is the circular basis, which is defined as $e_{l,r}=(e_x\pm ie_y)/\sqrt{2}$, and $\langle E_x^2\rangle$, $\langle E_y^2\rangle$, $\langle E_a^2\rangle$ and $\langle E_b^2\rangle$ are given by
\begin{eqnarray}
\langle E_x^2\rangle&=&\sum_i\cos^2\phi_{B,\alpha_i},\\
\langle E_y^2\rangle&=&\sum_i\sin^2\phi_{B,\alpha_i},\\
\langle E_a^2\rangle&=&\frac{1}{2}\sum_i(\cos\phi_{B,\alpha_i}-\sin\phi_{B,\alpha_i})^2,\\
\langle E_b^2\rangle&=&\frac{1}{2}\sum_i(\cos\phi_{B,\alpha_i}+\sin\phi_{B,\alpha_i})^2,
\end{eqnarray}
where $i$ presents each observed photon and $\alpha_i$ presents the mode of the photon $i$.
Finally, the polarization degree near the photosphere is given by
\begin{eqnarray}
\Pi=\frac{\sqrt{Q^2+U^2}}{I},
\end{eqnarray}
and the polarization angle is
\begin{eqnarray}
\chi=\frac{1}{2}\arctan\left(\frac{U}{Q}\right).
\end{eqnarray}
After a photon escapes the trapped fireball, it will propagate in the magnetic vacuum. The dielectric and inverse permeability are $\pmb{\epsilon}=a\mathbf{I}+q\mathbf{\hat{B}\hat{B}}$ and $\pmb{\mu}^{-1}=a\mathbf{I}+m\mathbf{\hat{B}\hat{B}}$, respectively. For a transverse electromagnetic wave emitted from a point in the magnetosphere atmosphere, it can be treated as the superposition of E-mode and O-mode photons, i.e.
\begin{eqnarray}
\mathbf{E}=A_O\pmb{e_O}+A_E\pmb{e_E},
\end{eqnarray}
\begin{eqnarray}
\pmb{e_O}=(\cos\phi,\sin\phi),\,\pmb{e_E}=(-\sin\phi,\cos\phi)
\end{eqnarray}
with the refraction indices for different modes being
\begin{eqnarray}
\end{eqnarray}
One therefore has $\Delta n\equiv n_O-n_E=(1/2)(q+m)\sin^2\theta_B$. The evolution of the mode amplitudes is given by <cit.>
\begin{eqnarray}\label{eq:lai-ho}
i\frac{\mathrm{d}}{\mathrm{d} s}\left(
\begin{array}{c}
\end{array}
\right)=
\left(
\begin{array}{cc}
-(\omega/c)\Delta n/2 & i\mathrm{d}\phi/\mathrm{d} s\\
-i\mathrm{d}\phi/\mathrm{d} s & (\omega/c)\Delta n/2\\
\end{array}
\right)
\left(
\begin{array}{c}
\end{array}
\right),
\end{eqnarray}
where $s=ct$ represents the photon trajectory. The condition for adiabatic evolution of the photon modes is
\begin{equation}\label{eq:adiabatic}
(\omega/c)\Delta n\gg2\mathrm{d}\phi/\mathrm{d} s,
\end{equation}
so that the non-diagonal elements in the matrix of Eq.(<ref>) is essentially zero, and the normal modes do not mix <cit.>. As the photon propagates in the magnetosphere, the QED correction to the dielectric tensor falls off as $B^2\sim r^{-6}$, and non-diagonal components would appear. Within the neutron star atmosphere context, <cit.> first defined the QED polarization limiting radius $r_p$ (the distance from the star at which the adiabatic approximation breaks down), and found that the QED limit radius allows the neutron star surface polarization to be sufficiently large to be observed <cit.>. Here we perform a similar treatment for the trapped fireball. We approximately treat the adiabatic region to have a sharp edge at $r_p$. Based on the adiabatic condition (<ref>), we calculate the polarization-limit radius, $r_p$, which is set by the condition $\omega\Delta n/c=2\mathrm{d}\phi/\mathrm{d} s$, i.e.<cit.>
\begin{eqnarray} \label{eq:QED-radius}
r_p=153\,R_s f_\phi\left(\frac{E}{1\unit{keV}}\right)^{1/6}\left(\frac{B}{10^{14}\unit{G}}\right)^{1/3}\left(\frac{2\pi}{\Omega}\right)^{1/6},
\end{eqnarray}
where $R_s$ is the neutron star radius, $f_\phi$ is a slowly varying function of phase and is of the order of unity, $E$ is the photon energy, $B$ is the strength of magnetic filed, and $\Omega$ is the spin angular velocity. One can see that the adiabatic condition is safely satisfied in the trapped fireball and its atmosphere.
Beyond $r_p$, the magnetic field strength has dropped enough so that it no longer affects the state of photon polarization, i.e. the photon polarization state is `frozen'. Because $r_p\gg 10R_s$, the electric vectors of all the photons with a certain energy $E$ emitted from the photosphere would rotate for a same angle $\sim\phi(r_p)$, so that the polarization degree is no longer affected.
Since magnetars are slow rotators, the effect of magnetar spin is not important in the polarization calculations. For simplicity, in the Monte Carlo simulations, we assume an aligned rotator, i.e.
the spin angular velocity vector $\pmb{\Omega}$ is parallel to the magnetic axis $\pmb{\mu}$, so that $\zeta=0$. All the photons with a certain energy have almost the same rotation angle $\phi(r_p)$ under the adiabatic condition. In this case, the Stokes parameter $U=0$, so that the polarization degree is $\Pi=Q/I$ and the polarization angle is $\chi=0$. We consider two energy bands, i.e. a soft band ($1-30\,\unit{keV}$) and a hard band ($30-100\,\unit{keV}$), and collect 5000 photons in each band within a solid angel $\Delta\Omega=0.15$ along the light of sight. The E mode and O mode are defined by the direction of the magnetic field at $r=10R_s$, where the adiabatic condition is satisfied.
Our simulation results are displayed in Table <ref>. As shown in the table, there are more E-mode photons in the soft band than in the hard band, since the opacity of the E-mode photons $\kappa_E\propto\omega^2$. The number ratio between the E-mode and O-mode photons, $N_E/N_O$, is higher for $\Theta=0$ than that for $\Theta=\pi/2$. This is because the opacity of E-mode photons is suppressed in strong magnetic fields, i.e. $\kappa_E\propto B^{-2}$, and $B$ is stronger near the polar. On the other hand, the polarization degree is close to zero for $\Theta=0$ and reaches the maximum value at $\Theta=\pi/2$, i.e. 27.9% in the 1-30 keV band and 10.0% in the 30-100 keV band. The result could be qualitatively explained from the direction of the magnetic field and the photon number ratio between E-mode and O-mode. As shown in the left panel of Fig. <ref>, if the light of sight is parallel to the magnetic axis, the directional angles of E-mode and O-mode are evenly distributed from 0 to $\pi$. Therefore, even though $N_E/N_O$ has the maximum value, the polarization degree is still zero. On the other hand, as shown in the right panel of Fig. <ref>, if the light of sight is perpendicular to the magnetic axis, the projected directions of the magnetic field lines almost have the same direction. In this case, the polarization degree depends on $N_E/N_O$. Since $N_E/N_O$ is of the order of unity, the polarization degree is mainly determined by the direction of the magnetic fields, so that an ordered $B$ field configuration naturally results in a relatively large polarization degree.
§ CONCLUSIONS AND DISCUSSIONS
Polarization observations provide the key information to probe the structure of the magnetar magnetosphere during giant flares. After the initial spike, the residual energy would form an optically thick photon-pair plasma. This photon-pair plasma is trapped in the magnetar magnetosphere, forming a “trapped fireball”. Observations show that the flux of the thermal component is much higher than that of the non-thermal component between $10\,\unit{keV}$ to $100\,\unit{keV}$ in the early and later stages <cit.>. In this paper, we assumed that the trapped fireball is enclosed by a set of closed-field lines in the magnetosphere, and calculated the atmospheric structure and polarization degree of the trapped fireball. We have reached the following conclusions:
A. Regarding the atmosphere of the trapped fireball:
(i.) The atmosphere of the trapped fireball is much thinner than the thickness of the trapped fireball. The size of a trapped fireball is a few times of $R_s$, which is determined by the requirement of the confinement of fireball energy within the magnetosphere, as shown in Eq. (<ref>). Observations show that the effective radius of the thermal component is a few times of the radius of neutron star, which is consistent with this requirement. We calculated the structure of the trapped fireball and its atmosphere. Our results show that the thickness of the atmosphere is $\sim0.001R_s$ with a sharp temperature gradient.
(ii.) In the atmosphere, the electron-positron pairs are non-relativistic, and the thermal pair density decreases with temperature (see Eq. (<ref>)). The pair gas pressure is lower than the radiative pressure at the photosphere, and the total pressure is dominated by the magnetic pressure. The atmosphere could be still in hydrostatic equilibrium due to the magnetic confinement. In the trapped fireball, gravity is not the dominant confining force, and the plasma is trapped due to the suppression of the transport of charged particles across the magnetic field lines.
(iii.) Opacity plays an important role in the structure of the trapped fireball. Different from the surface of a neutron star <cit.>, the trapped fireball is dominated by the electron-positron pairs <cit.>. We assume that the atmosphere is baryon free. For an electron-ion plasma, there is a vacuum resonance point where E-mode and O-mode photons could convert from one mode to another, and there are two peaks in the opacity-energy relation, e.g. the vacuum resonance absorption and the proton cyclotron absorption. However, for electron-positron pairs, both E-mode and O-mode photons are entirely linearly polarized. In this case, there is no vacuum resonance point where they can change mode during their propagation in the medium. Rather, for electron-positron pairs, there is only one electron cyclotron absorption peak with energy $E\simeq1158B_{14}\unit{keV}$, which is much larger than that of the observed effective temperature $k_BT_{\rm eff}\sim 30\,\unit{keV}$.
(iv.) The atmosphere temperature distribution is determined by the structure of the magnetosphere. For a specific angle $\theta$, the longer the distance from the magnetar, the lower the temperature. For a specific distance $r$, the temperature of the equator is higher than that of the pole (Note that the abscissa in Fig. <ref>-<ref> is the logarithm of $r-R_{max,0}\sin^2\theta$ rather than $r$). The pair density and opacity are determined by the temperature, so that the photosphere is no longer a sphere. Observationally one could only infer the effective size of the trapped fireball, but not the shape of the trapped fireball.
(v.) Our results show that the opacity of E mode is independent of $\theta_B$ for electron-positron pairs, which is different from the case of the electron-ion plasma <cit.>.
B. Regarding the polarization properties of the trapped fireball:
(i.) The opacity of the E-mode photons scales as $\kappa_E\propto\omega^2$. As a result, the number of E-mode photons is larger than that of O-mode photons in softer bands. Conversely, O-mode photons become more dominant in harder bands. The number ratio between E-mode and O-mode photons, $N_E/N_O$, is higher for $\Theta=0$ than for $\Theta=\pi/2$, as a result of the suppression of the E-mode photon opacity in strong magnetic fields near the magnetic pole.
(ii.) The polarization degree is zero for $\Theta=0$ and reaches a maximum value for $\Theta=\pi/2$. The maximum polarization degree from the trapped fireball is lower than that ($\sim 100\%$) from a hot spot on the pole of the magnetar surface <cit.> or in the quiescent magnetar magnetosphere <cit.>. This is mostly due to the extended thermal emission from a trapped fireball and the wider collected band, e.g. $1-30\,\unit{keV}$ or $30-100\,\unit{keV}$ considered in our work.
(iii.) During the photon propagation in the magnetic vacuum, due to the adiabatic condition ($r\ll r_p$), the electric vector direction of the electromagnetic wave continues to change according to the direction of the local magnetic field. Because the polarization-limit radius $r_p$ is much larger than that of the trapped fireball radius, the electric vectors of all the photons with a certain energy emitted from the photosphere rotate a same angle $\sim\phi(r_p)$ as they escape the magetosphere.
(iv.) The polarization degree at $\pi/2$ can be as large as $\sim 30\%$ in 1-30 keV, and is $\sim 10\%$ in 30-100 keV. Since giant flares typically have large fluxes, such a polarization degree would be detectable future X-ray/$\gamma$-ray polarimeters such as POLAR and LEAP, and may have been detected by the existing X-ray detectors such as RHESSI. We encourage a re-analysis of the possible polarization signal of SGR 1806-20 giant flare pulsating tail in the RHESSI data <cit.>.
C. Implications for detecting the predicted polarization signal:
Our calculations are based on the magnetar conjecture and robust QED physics. If the future observations confirm the polarization signal at the predicted level, it would suggest the following:
(i.) The magnetar hypothesis is correct. Even if the magnetar model is the leading model to interpret SGRs, some other scenarios have been proposed to interpret SGRs without introducing strong magnetic fields. For example, in the accretion models <cit.>, the strength of the magnetic field is weak, i.e. $B\simeq10^{11}\,\unit{G}$. According to Eq.(<ref>), the QED polarization limiting radius would be $r_p\simeq 15R_s$, which is not much larger than the scale of the trapped fireball. This would give rise to a much smaller polarization degree.
(ii.) QED is right: A high polarization degree suggests that vacuum polarization indeed makes a corrected contribution to the dielectric tensor and the magnetic permeability tensor, and that the QED vacuum is birefringent. Otherwise the polarization degree would be less than $\sim 10\%$ even for $\Theta=\pi/2$.
(iii.) A polarization measurement would reveal the geometry of the system.
Since $N_E/N_O$ is of the order of unity, the polarization degree is mainly determined by the magnetic field line directions. Therefore, a relatively large polarization degree corresponds to an ordered magnetic field configuration, and a relatively large angle between the magnetic pole and line of sight.
(iv.) The idea of the pair-plasma trapped fireball is right: Different from the electron-ion plasma, the pair plasma has no vacuum resonance point and ion cyclotron absorption. Both E-mode and O-mode photons are entirely linearly polarized. The properties of pair opacity are also very different from that of electron-ion plasma. Therefore, an observed polarization degree at the predicted level would prove that the trapped fireball is indeed a photon-pair plasma.
In our Monte Carlo simulations, we have assumed an aligned rotator (the rotation axis and magnetic axis overlap), so that the $\Theta$ angle is constant for a particular observer. In reality, the two axes are mis-aligned. The slow rotation of the magnetars would not affect the $\Theta$-dependent polarization degrees calculated in this paper. However, a certain observer would view the magnetosphere in a range of $\Theta$ values. The average polarization degree would be between the minimum and maximum $\Pi$ values defined the maximum and minimum $\Theta$, which would usually be still $\sim 10\%$ in the 1-30 keV band (or a few percent in 30-100 keV), unless the two axes are nearly aligned and the line of sight is also very close to the pole.
We thank the anonymous referee for valuable and detailed suggestions that have allowed us to improve this manuscript significantly.
This work is partially supported by National Basic Research Program (973 Program) of China under Grant No. 2014CB845800. YPY is supported by China Scholarship Program to conduct research at UNLV.
[Alpar(2001)]alp01 Alpar, M. A., 2001, , 554, 1245
[Boggs et al.(2007)]bog07 Boggs, S. E., Zoglauer, A., Bellm. E.,
Hurley, K., Lin, R. P., Smith, D. M., Wigger, C., & Hajdas, W. 2007, , 661, 458
[Fernández & Davis(2011)]fer11 Fernández, R., & Davis, S. W. 2011, , 730, 131
[Feroci et al.(1999)]fer99 Feroci, M., Frontera, F., Costa, E., Amati, L., Tavani M., Rapisarda, M., & Orlandini, M. 1999, , 515, L9
[Feroci et al.(2001)]fer01 Feroci, M., Hurley, K., Duncan, R. C.,
& Thompson, C. 2001,, 549, 1021
[Feroci & Soffitta(2011)]fer11b Feroci, M. & Soffitta, P., 2011, in High-Energy Emission from Pulsars and their Systems, Springer-Verlag Berlin Heidelberg p585
[Harding & Lai(2006)]har06 Harding, A. K., & Lai, D. 2006, Rep. Prog. Phys., 69, 2631
[Haug(1975)]hau75 Haug, E. 1975, Z. Naturforsch., 30, 1099
[Heyl & Shaviv(2000)]hey00 Heyl, J. S., & Shaviv, N. J. 2000, MNRAS, 311, 555
[Heyl & Shaviv(2002)]hey02 Heyl, J. S., & Shaviv, N., 2002, Phys. Rev. D, 66, 3002
[Heyl et al.(2003)]hey03 Heyl, J. S., Shaviv, N., & Lloyd, D., 2003, MNRAS, 342, 134
[Ho & Lai(2001)]ho01 Ho, W. C. G., & Lai, D. 2001, , 327, 1081
[Ho & Lai(2003)]ho03 Ho, W. C. G., & Lai, D. 2003, , 338, 233
[Hurley et al.(1999)]hur99 Hurley, K., et al. 1999, , 397, 41
[Hurley et al.(2005)]hur05 Hurley, K., et al. 2005, , 434, 1098
[Jahoda et al.(2015)]jah15 Jahoda K. M., Kouveliotou C., Kallman T. R. et al., 2015, American Astronomical Society, AAS Meeting #225, #338.40
[Lai & Ho(2002)]lai02 Lai, D., & Ho, W. C. G. 2002, , 566, 373
[Mazets et al.(1982)]maz82 Mazets, E. P., Golenetskii, S. V.,
Gurian, I. A., & Ilinskii, V. N. 1982, Ap&SS, 84, 173
[McConnell et al.(2007)]mcc07 McConnell, M. L., Ryan, J. M., Smith, D. M. et al., 2007, American Astronomical Society, AAS Meeting #201, #93.01
[McConnell et al.(2014)]mcc14 McConnell, M. L., Baring, M., Bloser, P. et al., 2014, Proceedings of the SPIE, Volume 9144, id. 91440O 8 pp.
[Mereghetti(2008)]mer08 Mereghetti, S., 2008, A&ARv, 15, 225
[Mészáros(1992)]mes92 Mészáros, P., 1992, High-Energy Radiation from Magnetized Neutron Stars. Univ. Chicago Press, Chicago
[Miller(1995)]mil95 Miller, M. C., 1995, , 448, L29
[Olausen & Kaspi(2014)]ola14 Olausen, S. A., & Kaspi, V. M. 2014, , 212, 6
[Paczyński & Anderson(1986)]pac86 Paczyński B., & Anderson N., 1986, ApJ, 302, 1
[Pavlov & Zavlin(2000)]pav00 Pavlov, G. G., & Zavlin, V. E. 2000, , 529, 1011
[Produit et al.(2005)]pro05 Produit, N., Barao, F., Deluit, S., Hajdas, W., Leluc, C., Pohl, M., Rapin, D., Vialle, J. P., Walter, R. & Wigger, C. 2005, NIMPA, 550, 616.
[Rajagopal & Romani(1996)]raj96 Rajagopal, M., & Romani, R. W., 1996, , 461, 327
[Suarez-Garcia et al.(2010)]sua10 Suarez-Garcia, E. et al., 2010, NIMPA, 624, 624
[Taverna et al.(2014)]tav14 Taverna, R., Muleri, F., Turolla, R., Soffitta, P., Fabiani, S., & Nobili, L. 2014, , 438, 1686
[Taverna et al.(2015)]tav15 Taverna, R., Turolla, R., Gonzalez Caniulef, D., et al., 2015, arXiv:1509.05023
[Thompson & Duncan(1995)]tho95 Thompson, C., & Duncan, R. C.
1995, , 275, 255
[Thompson & Duncan(2001)]tho01 Thompson, C., & Duncan, R. C.
2001, , 561, 980
[Thorne(1977)]tho77 Thorne, K. S., 1977, ApJ, 212, 825
[Ulmer(1994)]ulm94 Ulmer, A., 1994, , 437, L111
[van Adelsberg & Lai(2006)]ade06 van Adelsberg, M., & Lai, D., 2006, , 373, 1495
[van Adelsberg & Perna(2009)]ade09 van Adelsberg, M., & Perna, R., 2009, , 399, 1523
[Ventura(1979)]ven79 Ventura, J. 1979, Phys. Rev. D., 12, 1132
[Weisskopf et al.(2014)]wei14 Weisskopf, M. C., Bellazzini, R., Costa, E. et al., 2014, American Astronomical Society, HEAD meeting #14, #116.15
[Xiao et al.(2015)]xia15 Xiao, H., Hajdas, W., Wu, B., & Produit, N., 2015, arXiv:1507.04474
[Xiong et al.(2009)]xio09 Xiong, S., Produit, N.,& Wu, B., 2009, NIMPA, 606, 552
§ FREE ELECTRON GAS IN STRONG MAGNETIC FIELDS
In order to obtain the structure of a trapped fireball, we need to know the relation between the pair energy density and temperature. Here, we summarize the basic properties of a free electron gas in strong magnetic fields. The results are consistent with <cit.>.
Consider an elementary cell in a phase space with a volume
\begin{eqnarray}
\Delta x\Delta y\Delta z\Delta p_x\Delta p_y\Delta p_z=h^3,
\end{eqnarray}
where $(x,y,z)$ and $(p_x,p_y,p_z)$ are the position and momentum vectors of the electrons.
The electron distribution function as a function of energy $E$ is given by
\begin{eqnarray}
\end{eqnarray}
where $g$ is the degeneracy number and $\mu$ is the chemical potential.
For 3-D free electrons, the number density is given by
\begin{eqnarray}
n_e&=&\int_0^\infty \frac{g}{e^{(E-\mu)/k_BT}+1}\frac{4\pi p^2}{h^3}dp.
\end{eqnarray}
In strong magnetic fields, the electron distribution is in 1-D (due to the strong Landau confinement), which is given by <cit.>
\begin{eqnarray}
n_e=\frac{1}{(2\pi r_c)^2\hbar}\sum_{n=0}^\infty g_n\int_{-\infty}^\infty \frac{1}{e^{(E_n-\mu)/k_BT}+1} dp_z,
\end{eqnarray}
where $r_c=(\hbar c/eB)^{1/2}$ is the cyclotron radius, and $g_n$ is the degeneracy at energy level $n$ ($g(n=0)=1$ and $g(n\geq 1)=2$). The Landau level in a strong magnetic field is given by
\begin{eqnarray}
\end{eqnarray}
The first Landau level is
\begin{eqnarray}
\begin{array}{lll}
m_ec^2(B/B_Q), &\,\,& (B\ll B_Q),\\
m_ec^2(2B/B_Q)^{1/2}, &\,\,& (B\gg B_Q).\\
\end{array}
\right.
\end{eqnarray}
i) If $E(n=1)\ll k_BT$, electrons are in the non-magnetic limit, and the higher Landau levels are occupied. Thus one should consider a 3-D Fermi distribution.
a. For $k_BT\ll m_ec^2$ ($B\ll B_Q$) and $\mu\ll m_ec^2$, one has
\begin{eqnarray}
U_{e^\pm}=m_ec^2n_{e^\pm}&\simeq&2m_ec^2\frac{4\pi g}{h^3}e^{(\mu-mc^2)/k_BT}\int_0^\infty e^{-p^2/2m_ek_BT}p^2dp\nonumber\\
\end{eqnarray}
b. For $k_BT\gg m_ec^2$ and $\mu\ll pc$, one has
\begin{eqnarray}
U_{e^\pm}&\simeq&2\frac{4\pi g}{h^3}\int_0^\infty pc \frac{1}{e^{pc/k_BT}+1}p^2dp\nonumber\\
\end{eqnarray}
ii) If $E(n=1)\gg k_BT$, only the ground Landau level is occupied by electrons. One should consider a 1-D magnetized Fermi distribution.
c. For $k_BT\ll m_ec^2$ and $\mu\ll m_ec^2$, one has
\begin{eqnarray}
U_{e^\pm}=m_ec^2n_{e^\pm}&=&2m_ec^2\frac{1}{(2\pi r_c)^2\hbar}\sum_{n=0}^\infty g_n\int_{-\infty}^\infty \frac{1}{e^{(E_n-\mu)/k_BT}+1} dp_z\nonumber\\
&\simeq&2m_ec^2\frac{1}{(2\pi)^2\hbar}\frac{eB}{\hbar c}e^{(\mu-m_ec^2)/{k_BT}}\int_{-\infty}^\infty e^{-p_z^2/2m_ek_BT}dp_z\nonumber\\
&\simeq&m_ec^2\frac{(m_ec)^3}{\hbar^3(2\pi^3)^{1/2}}\frac{B}{B_Q}\left(\frac{k_BT}{m_ec^2}\right)^{1/2}e^{-m_ec^2/k_BT}; \label{a9}
\end{eqnarray}
d. For $k_BT\gg m_ec^2$ ($B\gg B_Q$) and $\mu\ll p_zc$, one has
\begin{eqnarray}
U_{e^\pm}&=&2\frac{1}{(2\pi r_c)^2\hbar}\sum_{n=0}^\infty g_n\int_{-\infty}^\infty p_zc \frac{1}{e^{(p_zc-\mu)/k_BT}+1} dp_z\nonumber\\
&\simeq&2\frac{1}{(2\pi)^2\hbar}\frac{eB}{\hbar c}\frac{(k_BT)^2}{c}2\int_0^\infty \frac{x}{e^x+1} dx\nonumber\\
\end{eqnarray}
The geometric structure of the trapped fireball. The grey circle represents the neutron star. The orange region represents the inner part of the trapped fireball. The thin bisque layer represents the atmosphere of the trapped fireball.
The temperature as a function of the radial distance to the atmosphere bottom ($r-R_{max,0}\sin^2\theta$) for $B_p=10^{15}\,\unit{G}$, $R_{max,0}=2R_s$ and $T_0=10^9\,\unit{K}$. The dotted, dashed, and solid lines denote $\theta=\pi/6,\,\pi/3,\,\pi/2$, respectively.
The optical depth as a function of the radial distance to the atmosphere bottom ($r-R_{max,0}\sin^2\theta$) for $B_p=10^{15}\,\unit{G}$, $R_{max,0}=2R_s$ and $T_0=10^9\,\unit{K}$. The dotted, dashed, and solid lines denote $\theta=\pi/6,\,\pi/3,\,\pi/2$, respectively.
The number density of electron-positron pairs as a function of the radial distance to the atmosphere bottom ($r-R_{max,0}\sin^2\theta$) for $B_p=10^{15}\,\unit{G}$, $R_{max,0}=2R_s$ and $T_0=10^9\,\unit{K}$. The dotted, dashed, and solid lines denote $\theta=\pi/6,\,\pi/3,\,\pi/2$, respectively.
The pressure as a function of the radial distance to the atmosphere bottom ($r-R_{max,0}\sin^2\theta$) for $B_p=10^{15}\,\unit{G}$, $R_{max,0}=2R_s$ and $T_0=10^9\,\unit{K}$. The dotted, dashed, and solid lines denote $\theta=\pi/6,\,\pi/3,\,\pi/2$, respectively. The electron-positron pairs, radiation and magnetic pressures are denoted by black, red and blue lines, respectively.
The scattering opacities $\kappa^{sc}$ as a function of energy for various angles between $\mathbf{k}$ and $\mathbf{B}$, $\theta_B$, for $B=10^{14}\,\unit{G}$ and $n_e=10^{25}\,\unit{cm^{-3}}$. The black curves represent the E model opacity, and the red curves represent the O mode opacity. The dotted, dashed and solid curves denote $\theta_B=\pi/6,\,\pi/4,\,\pi/3$, respectively.
The scattering opacities $\kappa^{sc}$ as a function of energy for different magnetic field strengths $B$ for $n_e=10^{25}\,\unit{cm^{-3}}$ and $\theta_B=\pi/4$. The black curves represent the E mode opacity, and the red curves represent the O mode opacity. The magnetic field strengths $B=10^{14},\,10^{14.5},\,10^{15},\,\unit{G}$ are denoted by dotted, dashed and solid curves, respectively.
The scattering opacities $\kappa^{sc}$ as a function of energy for various electron-positron pair number density $n_e$ for $B=10^{14}\,\unit{G}$ and $\theta_B=\pi/4$. The black curves represent the E mode opacity, and the red curves represent the O mode opacity. The pair number densities $n_e=10^{24},\,10^{25},\,10^{26}\,\unit{cm^{-3}}$ are denoted by dotted, dashed and solid curves, respectively.
The angles and vectors introduced in Section 4.
Schematic pictures of the magnetar magnetosphere. The grey circle represents the neutron star. The black curves represent the magnetic field lines.
The red arrows represent the electric vectors of the E-mode photons, whereas the blue arrows represent the electric vectors of the O-mode photons. The left panel corresponds to the case with the line of sight parallel to the magnetic axis, whereas the right panel corresponds to the case with the line of sight perpendicular to the magnetic axis.
@ ccccc
The polarization degree of the trapped fireball
$\Theta^\mathrm{a}$ $(N_E/N_O)_S^\mathrm{b}$ $\Pi_S^\mathrm{c}$
$(N_E/N_O)_H^\mathrm{d}$ $\Pi_H^\mathrm{e}$
0 2.78 0.5% 1.06 0.9%
$\pi/4$ 2.33 12.7% 0.84 2.7%
$\pi/2$ 1.98 27.9% 0.80 10.0%
a$\Theta$ is the angle between the magnetic axis and the line of sight.
b$(N_E/N_O)_S$ is the rate of the number of E-mode photons and O-mode photons at $1-30\,\unit{keV}$.
c$\Pi_S$ is the polarization degree at $1-30\,\unit{keV}$.
d$(N_E/N_O)_H$ is the rate of the number of E-mode photons and O-mode photons at $30-100\,\unit{keV}$.
e$\Pi_H$ is the polarization degree at $30-100\,\unit{keV}$.
∗Note that E mode and O mode are defined by the direction of the magnetic field at $r=10R_s$ where they are under adiabatic condition.
|
1511.00404
|
September 12, 2025
Yang-Baxter invariance of the Nappi-Witten model
Hideki Kyono[E-mail: [email protected]]
and Kentaroh Yoshida[E-mail: [email protected]]
Department of Physics, Kyoto University,
Kitashirakawa Oiwake-cho, Kyoto 606-8502, Japan
We study Yang-Baxter deformations of the Nappi-Witten model
with a prescription invented by Delduc, Magro and Vicedo.
The deformations are specified by skew-symmetric classical $r$-matrices
satisfying (modified) classical Yang-Baxter equations.
We show that the sigma-model metric is invariant under arbitrary deformations
(while the coefficient of $B$-field is changed) by utilizing the most general classical $r$-matrix.
Furthermore, the coefficient of $B$-field is determined to be the original value from the requirement
that the one-loop $\beta$-function should vanish.
After all, the Nappi-Witten model is the unique conformal theory within the class of
the Yang-Baxter deformations preserving the conformal invariance.
§ INTRODUCTION
The Yang-Baxter sigma-model description, which was originally proposed by Klimcik <cit.>,
is a systematic way to consider integrable deformations of 2D non-linear sigma models.
According to this procedure, the deformations are specified by skew-symmetric classical
$r$-matrices satisfying the modified classical Yang-Baxter equation (mCYBE) .
The original work <cit.> has been generalized to symmetric spaces <cit.>
and the homogeneous CYBE <cit.>.
Yang-Baxter deformations of the AdS$_5\times$S$^5$ superstring can be studied
with the mCYBE <cit.> and the CYBE <cit.>.
For the former case, the metric and $B$-field are derived in <cit.>
and the full background has recently been studied in <cit.>.
For the latter case, classical $r$-matrices are identified with solutions
of type IIB supergravity including $\gamma$-deformations of S$^5$ <cit.>
and gravity duals of non-commutative gauge theories <cit.>,
in a series of works <cit.>
(For a short summary, see <cit.>).
Lately, Yang-Baxter deformations of 4D Minkowski spacetime have been studied <cit.>.
In <cit.>, classical $r$-matrices are identified with exactly-solvable string backgrounds
such as Melvin backgrounds and pp-wave backgrounds. In <cit.>,
Yang-Baxter deformations of 4D Minkowski spacetime are discussed by using classical $r$-matrices
associated with $\kappa$-deformations of the Poincaré algebra <cit.>.
Then the resulting deformed geometries include T-duals of (A)dS$_4$ spaces[T-dual of dS$_4$
can be derived as a scaling limit of $\eta$-deformed AdS$_5$ as well <cit.>.]
and a time-dependent pp-wave background.
Furthermore, the Lax pair is presented for the general $\kappa$-deformations <cit.>.
As a spin off from this progress, it would be interesting to study Yang-Baxter deformations
of the Nappi-Witten model <cit.>. The target space of this model is given
by a centrally extended 2D Poincaré group. Hence the Yang-Baxter deformed
Nappi-Witten models can be regarded as toy models of the previous works <cit.>,
because the structure of the target space is much simpler than that of 4D Minkowski spacetime.
This simplification makes it possible to study the most general Yang-Baxter deformation.
As a matter of course, it is exceedingly complicated in general, hence such an analysis has not been done yet.
In this article, we investigate Yang-Baxter deformations of the Nappi-Witten model
by following a prescription invented by Delduc, Magro and Vicedo <cit.>.
We show that the sigma-model metric is invariant under the deformations
(while the coefficient of $B$-field is changed) by utilizing the most general classical $r$-matrix.
Furthermore, the coefficient of $B$-field is determined to be the original value from the requirement
that the one-loop $\beta$-function should vanish.
After all, the Nappi-Witten model is the unique conformal theory within the class of
the Yang-Baxter deformations preserving the conformal invariance (i. e., Yang-Baxter invariance).
§ NAPPI-WITTEN MODEL
In this section, we shall give a concise review of the Nappi-Witten model <cit.>.
The Nappi-Witten model is a Wess-Zumino-Witten (WZW) model whose target space is given by
a centrally extend 2D Poincaré group.
The associated extended Poincaré algebra $\mathfrak{g}$ is composed of two translations $P_i~(i=1,2)$ ,
a rotation $J$ and the center $T$ . The commutation relations of the generators are given by
\begin{eqnarray}
\label{2D-P}
[J,P_i]=\epsilon_{ij}P_j\,,\quad[P_i,P_j]=\epsilon_{ij}T\,, \quad
[T,J] = [T,P_i] = 0\,,
\end{eqnarray}
where $\epsilon_{ij}$ is an anti-symmetric tensor normalized as $\epsilon_{12}=1$ .
It is convenient to introduce a notation of the generators with the group index $I$
\begin{eqnarray}
T_I=\bigl\{P_1,P_2,J,T\bigr\} \qquad (I=1,2,3,4)\,.
\end{eqnarray}
Let us introduce a group element represented by
\begin{eqnarray}
\end{eqnarray}
By using this group element $g$ , the left-invariant current $A$ can be evaluated as
\begin{eqnarray}
\label{current}
A_\alpha &\equiv& g^{-1}\partial_\alpha g
= A^I\, T_I
\, \nonumber \\
&=&\left(\cos u\, \partial_\alpha a_1+\sin u\, \partial_\alpha a_2\right)\,P_1
+\left(\cos u\, \partial_\alpha a_2-\sin u\, \partial_\alpha a_1\right)\,P_2\nonumber\\
&&+\partial_\alpha u\, J+\left[\partial_\alpha v
+ \frac{1}{2}\,a_2\, \partial_\alpha a_1-\frac{1}{2}\,a_1\,\partial_\alpha a_2\right]\,T\,.
\end{eqnarray}
Here the index $\alpha=\tau$ , $\sigma$ is for the world-sheet coordinates.
It is also helpful to introduce the light-cone expression of $A$ on the world-sheet like
\begin{eqnarray}
A_\pm\equiv A_\tau \pm A_\sigma\,.
\end{eqnarray}
By using $A_{\pm}$ , the classical action of the Nappi-Witten model is given by
\begin{eqnarray}
\label{NWaction}
S[A]=\frac{1}{2}\int_\Sigma\! d^2\sigma ~ \Omega_{IJ}\, A^I_-A^J_+
+ \frac{1}{6}\int_{B_3}\!\! d^3\sigma~\epsilon^{\hat{\alpha}\hat{\beta}\hat{\gamma}}\,
\Omega_{KL}\, {f_{IJ}}^LA^{I}_{\hat{\alpha}}A^{J}_{\hat{\beta}}A^{K}_{\hat{\gamma}}\,.
\end{eqnarray}
This action is basically composed of the two parts, 1) the sigma model part and
2) the Wess-Zumino-Witten (WZW) term.
The sigma model part is defined as usual on the world sheet $\Sigma$ ,
where we assume that $\Sigma$ is compact and the periodic boundary condition
is imposed for the dynamical variables.
A key ingredient contained in this part is
the most general symmetric two-form[The overall factor of $\Omega_{IJ}$
(i.e., the level of the WZW model)
is set to be 1 because it is irrelevant to the deformations we consider later.]
\begin{eqnarray}
\label{symf}
\Omega_{IJ} \equiv
\begin{pmatrix}
\end{pmatrix}\,,
\end{eqnarray}
which satisfies the following condition:
\begin{eqnarray}
\label{invariance}
\end{eqnarray}
Here ${f_{IJ}}^K$ are the structure constants which determine the commutation relations
\[
\]
The WZW term in (<ref>) also contains $\Omega_{IJ}$ , but,
apart from this point, it is the same as the usual.
The symbol $B_3$ denotes a 3D space which has $\Sigma$ as a boundary.
Hence the domain of $A^I$ is implicitly generalized to $B_3$
with $\hat{\alpha}=\tau\,,\sigma$ and $\xi$ in the WZW term[
For the detail of the WZW model, for example, see <cit.>.],
where the extra direction is labeled by $\xi$ .
A remarkable point is that the action (<ref>) can be rewritten
into the following form <cit.>:[In this derivation, we have used the identities (12) and (13) in <cit.>. ]
\begin{eqnarray}
\label{NWaction2}
S=-\frac{1}{2}\int_{\Sigma}\! d^2\sigma\, \Bigl[\,
\gamma^{\alpha\beta}g_{\mu\nu}\,\partial_\alpha X^\mu \partial_\beta X^\nu
-\epsilon^{\alpha\beta}\, B_{\mu\nu}\, \partial_{\alpha} X^\mu\partial_{\beta} X^\nu
\Bigr]\,.
\end{eqnarray}
Here $X^\mu=\{u,v,a_1,a_2\}$ are the dynamical variables.
The metric and anti-symmetric two-form on $\Sigma$ are described by
$\gamma^{\alpha\beta}=\text{diag}(-1,1)$ and $\epsilon^{\alpha\beta}$
normalized as $\epsilon^{\tau\sigma}=1$ .
Then the space-time metric $g_{\mu\nu}$ and two-form field $B$ are given by
\begin{eqnarray}
\label{NW-metric}
ds^2&=&g_{\mu\nu}\,dX^\mu dX^\nu
=2dudv+b\,du^2 +da_1^2+da_2^2+a_1\,da_2du-a_2\,da_1du\,,\nonumber\\
B&=&B_{\mu\nu}\,dX^{\mu}\wedge dX^{\nu} =u~da_1 \wedge da_2\,.
\end{eqnarray}
This is a simple 4D background.
It would be helpful to further rewrite the background (<ref>) .
By performing the following coordinate transformation with a real constant $m$ <cit.>
\begin{eqnarray}
a_1~~&\rightarrow&~~ a_1\,\cos (m\,u) + a_2\,\sin( m\,u)\,,\qquad
a_2~~\rightarrow~~ a_1\,\sin( m\,u) - a_2\, \cos( m\,u)\,,\nonumber\\
u ~~&\rightarrow&~~ 2m\,u\,,\qquad v\rightarrow \frac{1}{2m}\,v-b\,m\,u\,,
\end{eqnarray}
the background (<ref>) can be rewritten as
\begin{eqnarray}
\label{NW pp-wave}
ds^2&=& 2dudv-m^2(a_1^2+a_2^2)\,du^2 + da_1^2 + da_2^2\,,\nonumber\\
B&=&-2m\,u~da_1 \wedge da_2+2m^2a_1\,u~da_1 \wedge du+2m^2a_2\,u~da_2 \wedge du\,.
\end{eqnarray}
This is nothing but a pp-wave background.
Note here that the last two terms of $B$-field in (<ref>) contribute
to the Lagrangian as the total derivatives, which can be ignored in the present setup.
For this background (<ref>), the world-sheet $\beta$-function vanishes
at the one-loop level<cit.>.
§ YANG-BAXTER DEFORMED NAPPI-WITTEN MODEL
In this section, let us consider Yang-Baxter deformations of the Nappi-Witten model.
§.§ A Yang-Baxter deformed classical action
As explained in the previous section, the Nappi-Witten model contains the WZW term.
Hence it is not straightforward to study Yang-Baxter deformations of this model.
Our strategy here is to follow a prescription invented by Delduc, Magro and Vicedo <cit.>.
This is basically a two-parameter deformation. It is an easy task to extend their prescription
to the Nappi-Witten model.
A deformed action we propose is the following:[Here we would like to start
from the WZNW model (non-conformal)
rather than the WZW model (conformal).
Hence, a real constant $k$ has been put
in front of the WZW term to measure the non-conformality.]
\begin{eqnarray}
\label{deformed action}
S &=& \frac{1}{2}\int_\Sigma\! d^2\sigma ~ \Omega_{IJ}\,A^I_-J^J_+
+ \frac{1}{2}k\int_{B_3}\!\! d^3\sigma~\Omega_{KL}\,{f_{IJ}}^LA^{I}_{\xi}A^{J}_{-}A^{K}_{+}\,.
\end{eqnarray}
Here the deformed current $J$ is defined as
\begin{eqnarray}
J_\pm &\equiv& (1+\omega\, \eta^2)\frac{1\pm\tilde{A}R}{1-\eta^2 R^2}A_\pm\,.
\end{eqnarray}
First of all, the classical action (<ref>) includes three constant parameters $\eta$ , $\tilde{A}$ and $k$ .
The deformation is measured by $\eta$ and $\tilde{A}$ . The last parameter $k$
is regarded as the level. When $\eta=\tilde{A}=0$ and $k=1$ ,
the action (<ref>) is reduced to the original Nappi-Witten model.
A key ingredient contained in $J$ is a linear operator $R$: $\alg{g}\rightarrow\alg{g}$ .
In the context of Yang-Baxter deformations, it is supposed that $R$ should be
skew-symmetric and satisfy the (modified) Yang-Baxter equation <cit.>
\begin{eqnarray}
\label{CYBE}
[R(X),R(Y)]-R([R(X),Y]+[X,R(Y)])=\omega\, [X,Y]\qquad (X,\,Y\in \mathfrak{g})\,.
\end{eqnarray}
The constant parameter $\omega$ can be normalized by rescaling $R$ ,
hence it is enough to consider the following three cases: $\omega=\pm1$ and 0 .
In particular, the case with $\omega =0$ is the homogeneous CYBE.
§.§ The general solution of the (m)CYBE
In this subsection, we derive the general solution of the (m)CYBE.
Let us start from the most general expression of a linear $R$-operator:
\begin{eqnarray}
\label{R-matrix2}
R(X) &=& M^{IJ}\Omega_{JK}\,x^K\,T_I\,, \qquad
M^{IJ} \equiv \sum_i \left( a_i^I\,b_i^J - b_i^I\,a_i^J\right)\,.
\end{eqnarray}
Here $M^{IJ}$ is an anti-symmetric $4\times 4$ matrix
which is parametrized as
\begin{eqnarray}
\begin{pmatrix}
\end{pmatrix}\,,
\qquad m_i \in \mathbb{R}\,.
\label{ansatz}
\end{eqnarray}
By using the expression (<ref>) and the defining relation
of $\Omega_{IJ}$ (<ref>) ,
the (m) CYBE (<ref>) can be rewritten into the following form:
\begin{eqnarray}
-\omega {f_{LM}}^K\Omega^{LI}\Omega^{MJ}=0\,. \label{cybe-f}
\end{eqnarray}
Note that we define $\Omega^{IJ}$ as the inverse matrix of $\Omega_{IJ}$ .
Then, by putting the expression (<ref>) into (<ref>) ,
the most general solution can be determined like
\begin{eqnarray}
\begin{pmatrix}
\end{pmatrix}\,.
\label{solution}
\end{eqnarray}
Here the condition (<ref>) has led to the following constraints:
\[
m_1 = \sqrt{\omega}\,, \qquad m_2 = m_4 =0\,.
\]
Then we have also supposed that $\omega \geq 0$
in order to preserve the reality of the background[When we consider $\omega< 0$ ,
we need to multiply the linear $R$-operator by the imaginary unit $i$ .].
After all, $m_3$ , $m_5$ and $m_6$ have survived as free parameters of the $R$-operator
as well as $\omega$ .
§.§ The general deformed background
Let us consider a deformation of the Nappi-Witten model
with the general solution (<ref>) .
The resulting background is given by
\begin{eqnarray}
\nonumber \\
&& + \frac{(1+\omega\,\eta^2)a_2+2\eta^2\{(m_3m_6+m_5\sqrt{\omega})\cos u
-(m_5m_6-m_3\sqrt{\omega})\sin u\}}{1-m_6^2\,\eta^2}da_1du\nonumber\\
&& - \frac{(1+\omega\,\eta^2)a_1-2\eta^2\{(m_5m_6-m_3\sqrt{\omega})\cos u
+(m_3m_6+m_5\sqrt{\omega})\sin u\}}{1-m_6^2\,\eta^2}da_2du\,,\nonumber\\
B &=& k\,u\,da_1\wedge da_2+\frac{\tilde{A}m_6(1+\omega\,\eta^2)}{2(1-m_6^2\eta^2)}
\left(a_2\,da_1\wedge du-a_1\,da_2\wedge du\right)\,. \label{general-bg}
\end{eqnarray}
Here we have ignored the total derivative terms that appeared in the $B$-field part.
This background (<ref>) can be simplified
by performing a coordinate transformation
\begin{eqnarray}
a_1~~&\rightarrow&~~ a_1\,\cos (m\,u) + a_2\,\sin (m\,u) + C_1\cos \left( C_3\,u\right)
+ C_2\,\sin \left( C_3\,u\right)\,, \nonumber \\
a_2 ~~&\rightarrow&~~ -a_2\,\cos ( m\,u) + a_1\,\sin ( m\,u)
- C_2\,\cos\left(C_3\,u\right) + C_1\,\sin \left(C_3\,u\right)\,, \nonumber \\
u ~~&\rightarrow&~~ C_3\,u\,, \nonumber \\
v ~~&\rightarrow&~~ \frac{1}{2\,m}v-\frac{1}{2}b\,C_3\,u-\frac{1}{2}\Bigl[C_2\cos \left(\frac{C_3-C_4}{2}u\right)
&&+\frac{1}{2}\Bigl[C_1\cos\left(\frac{C_3-C_4}{2}u\right) + C_2\sin\left(\frac{C_3-C_4}{2}u\right)\Bigr]a_2\,,
\label{trans}
\end{eqnarray}
where we have introduced the following quantities:
\begin{eqnarray}
C_1 & \equiv &\frac{m_5m_6-m_3\sqrt{\omega}}{m_6^2+\omega}\,,\qquad
C_2 \equiv \frac{m_3m_6+m_5\sqrt{\omega}}{m_6^2+\omega}\,,\nonumber\\
C_3 & \equiv & 2m\frac{1-m_6^2\,\eta^2}{1+\omega\,\eta^2}\,,\qquad
C_4 \equiv 2m\frac{m_6^2+\omega}{1+\omega\,\eta^2}\eta^2\,.
\end{eqnarray}
After performing the transformation (<ref>) , the resulting background is given by
the following pp-wave background equipped with a $B$-field:
\begin{eqnarray}
\label{DBG}
ds^2&=& 2dudv -m^2(a_1^2+a_2^2)du^2 + da_1^2+da_2^2\,,\nonumber\\
B&=&-k\,C_3\,u\,da_1\wedge da_2-m\tilde{A}m_6a_2\,da_1\wedge du+m\tilde{A}m_6 a_1\,da_2\wedge du\,.
\end{eqnarray}
Here we have ignored the total derivative terms again.
Note that the $B$-field in (<ref>) can be rewritten as (up to total derivative terms)
\begin{eqnarray}
\label{DBG2}
B&=&-(k\,C_3-2m\,\tilde{A}\,m_6 )\,u\,da_1\wedge da_2\,.
\end{eqnarray}
Comparing (<ref>) with the $B$-field in (<ref>) ,
one can find that only the difference is the coefficient of $B$-field.
From the viewpoint of the original Nappi-Witten model, the coefficient of the WZW term
has been changed and the resulting theory should be regarded as a WZNW model.
According to this observation, it is obvious that the deformed model is exactly solvable[
In the case of <cit.>, it is necessary to impose a condition for $\tilde{A}$
so as to preserve the integrability (c.f., <cit.>) .
However, such an extra condition is not needed in the present case. ].
§.§ Yang-Baxter deformations and conformal invariance
Finally, let us show that the original Nappi-Witten model is the unique conformal theory
within the class of the Yang-Baxter deformations preserving the conformal invariance.
Due to the requirement of the vanishing $\beta$-function at the one-loop level,
the two $B$-fields should be identical as follows:
\begin{eqnarray}
2m \quad (\mbox{the original}) ~~=~~ k\,C_3-2m\,\tilde{A}\,m_6 \quad (\mbox{the deformed})\,.
\label{equal}
\end{eqnarray}
This condition indicates two interesting results. The first one is the Yang-Baxter invariance of
the Nappi-Witten model.
If we start from the case with $k=1$ ,
then the original system is invariant under the Yang-Baxter deformations preserving the conformal
invariance, which are specified by the parameters satisfying the condition
\[
1 = \frac{1+\omega\,\eta^2}{1-m_6^2\,\eta^2}(1+m_6\tilde{A})\,.
\]
In other words, the Yang-Baxter invariance follows from the conformal invariance.
The second is that the Yang-Baxter deformation may map a non-conformal theory to
the conformal Nappi-Witten model. Suppose that we start from the case with $k \neq 1$ .
Then, by performing a Yang-Baxter deformation with parameters satisfying the condition
\begin{eqnarray}
\end{eqnarray}
the resulting system becomes the Nappi-Witten model.
In other words, the coefficient of $B$-field can be set to the conformal fixed point
by an appropriate Yang-Baxter deformation.
§ CONCLUSION AND DISCUSSION
In this article, we have studied Yang-Baxter deformations of the Nappi-Witten model.
By considering the most general classical $r$-matrix,
we have shown the invariance of the sigma-model metric under arbitrary deformations,
up to two-form $B$-fields. That is, the effect coming from the deformations
is reflected only as the coefficient of $B$-field. Then, the coefficient of $B$-field has been determined
to be the original value from the requirement that the one-loop $\beta$-function should vanish.
After all, it has been shown that the Nappi-Witten model is the unique conformal theory within the class of
the Yang-Baxter deformations preserving the conformal invariance (i. e., Yang-Baxter invariance).
There are many future directions. It would be interesting to consider a supersymmetric
extension of our analysis by following <cit.>. The number of the remaining supersymmetries
should depend on Yang-Baxter deformations because the coefficient of $B$-field is changed.
It is also nice to investigate higher-dimensional cases (e.g.,
the maximally supersymmetric pp-wave background <cit.>). As another direction,
one may consider non-relativistic backgrounds such as Schrödinger spacetimes <cit.>
and Lifshitz spacetimes <cit.>. Although there is a problem
of the degenerate Killing form similarly, it can be resolved by adopting the most general
symmetric two-form <cit.> as in the Nappi-Witten model.
It would be straightforward to apply the techniques
presented in <cit.> to Yang-Baxter deformations by following our present analysis.
It should be remarked that the most interesting indication of this work is
the universal aspect of the dual gauge-theory side. According to our work,
pp-wave backgrounds would have a kind of rigidity against Yang-Baxter deformations.
This result may indicate that the ground state and lower-lying excited states of
the spin chain associated with the $\mathcal{N}=4$ super Yang-Mills theory are invariant.
It is quite significant to extract such a universal characteristic after classifying various examples.
This is the standard strategy in theoretical physics and would be much more important than
identifying the associated dual gauge theory for each of the deformations.
We hope that our result could shed light on a universal aspect of Yang-Baxter deformations
from the viewpoint of the invariant pp-wave geometry.
§.§ Acknowledgments
We are very grateful to Martin Heinze and Jun-ichi Sakamoto for useful discussions.
The work of K.Y. is supported by Supporting Program for Interaction-based Initiative Team Studies
(SPIRITS) from Kyoto University and by the JSPS Grant-in-Aid for Scientific Research (C) No.15K05051.
This work is also supported in part by the JSPS Japan-Russia Research Cooperative Program
and the JSPS Japan-Hungary Research Cooperative Program.
C. Klimcik,
“Yang-Baxter sigma models and dS/AdS T duality,”
JHEP 0212 (2002) 051 [hep-th/0210095];
“On integrability of the Yang-Baxter sigma-model,”
J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518 [hep-th]];
“Integrability of the bi-Yang-Baxter sigma model,” Lett. Math. Phys. 104 (2014) 1095
[arXiv:1402.2105 [math-ph]].
F. Delduc, M. Magro and B. Vicedo,
“On classical q-deformations of integrable sigma-models,”
JHEP 1311 (2013) 192 [arXiv:1308.3581 [hep-th]].
T. Matsumoto and K. Yoshida,
“Yang-Baxter sigma models based on the CYBE,”
Nucl. Phys. B 893 (2015) 287
[arXiv:1501.03665 [hep-th]].
F. Delduc, M. Magro and B. Vicedo,
“An integrable deformation of the AdS$_5\times$S$^5$ superstring action,”
Phys. Rev. Lett. 112 (2014) 051601
[arXiv:1309.5850 [hep-th]];
“Derivation of the action and symmetries of the $q$-deformed AdS$_5\times$S$^5$ superstring,”
JHEP 1410 (2014) 132
[arXiv:1406.6286 [hep-th]].
I. Kawaguchi, T. Matsumoto and K. Yoshida,
“Jordanian deformations of the AdS$_5\times$S$^5$ superstring,”
JHEP 1404 (2014) 153
[arXiv:1401.4855 [hep-th]].
G. Arutyunov, R. Borsato and S. Frolov,
“S-matrix for strings on $\eta$-deformed AdS$_5\times$S$^5$,”
JHEP 1404 (2014) 002 [arXiv:1312.3542 [hep-th]].
G. Arutyunov, R. Borsato and S. Frolov,
“Puzzles of $\eta$-deformed AdS$_5\times$S$^5$ ,”
JHEP 1512 (2015) 049 [arXiv:1507.04239 [hep-th]].
B. Hoare and A. A. Tseytlin,
“Type IIB supergravity solution for the T-dual of the $\eta$-deformed AdS$_{5} \times$ S$^{5}$ superstring,”
JHEP 1510 (2015) 060
[arXiv:1508.01150 [hep-th]].
O. Lunin and J. M. Maldacena,
“Deforming field theories with $U(1) \times U(1)$ global symmetry and their gravity duals,”
JHEP 0505 (2005) 033 [hep-th/0502086].
S. Frolov,
“Lax pair for strings in Lunin-Maldacena background,”
JHEP 0505 (2005) 069
A. Hashimoto and N. Itzhaki,
“Noncommutative Yang-Mills and the AdS / CFT correspondence,”
Phys. Lett. B 465 (1999) 142
J. M. Maldacena and J. G. Russo,
“Large N limit of noncommutative gauge theories,”
JHEP 9909 (1999) 025
T. Matsumoto and K. Yoshida,
“Lunin-Maldacena backgrounds from the classical Yang-Baxter equation
- towards the gravity/CYBE correspondence,”
JHEP 1406 (2014) 135
[arXiv:1404.1838 [hep-th]].
T. Matsumoto and K. Yoshida,
“Integrability of classical strings dual for noncommutative gauge theories,”
JHEP 1406 (2014) 163
[arXiv:1404.3657 [hep-th]].
T. Matsumoto and K. Yoshida,
“Schrödinger geometries arising from Yang-Baxter deformations,”
JHEP 1504 (2015) 180 [arXiv:1502.00740 [hep-th]].
I. Kawaguchi, T. Matsumoto and K. Yoshida,
“A Jordanian deformation of AdS space in type IIB supergravity,”
JHEP 1406 (2014) 146 [arXiv:1402.6147 [hep-th]].
T. Matsumoto and K. Yoshida,
“Yang-Baxter deformations and string dualities,”
JHEP 1503 (2015) 137 [arXiv:1412.3658 [hep-th]].
S. J. van Tongeren,
“On classical Yang-Baxter based deformations of the AdS$_5 \times$S$^5$ superstring,”
JHEP 1506 (2015) 048 [arXiv:1504.05516 [hep-th]];
“Yang-Baxter deformations, AdS/CFT, and twist-noncommutative gauge theory,”
Nucl. Phys. B 904 (2016) 148 [arXiv:1506.01023 [hep-th]].
T. Kameyama, H. Kyono, J. Sakamoto and K. Yoshida,
“Lax pairs on Yang-Baxter deformed backgrounds,”
JHEP 1511 (2015) 043 [arXiv:1509.00173 [hep-th]].
P. M. Crichigno, T. Matsumoto and K. Yoshida,
“Deformations of $T^{1,1}$ as Yang-Baxter sigma models,”
JHEP 1412 (2014) 085
[arXiv:1406.2249 [hep-th]];
“Towards the gravity/CYBE correspondence beyond integrability
– Yang-Baxter deformations of $T^{1,1}$,”
J. Phys. Conf. Ser. 670 (2016) 1, 012019 [arXiv:1510.00835 [hep-th]].
T. Matsumoto and K. Yoshida,
“Integrable deformations of the AdS$_{5} \times$S$^5$ superstring
and the classical Yang-Baxter equation
– Towards the gravity/CYBE correspondence –,”
J. Phys. Conf. Ser. 563 (2014) 1, 012020
[arXiv:1410.0575 [hep-th]];
“Towards the gravity/CYBE correspondence – the current status –,”
J. Phys. Conf. Ser. 670 (2016) 1, 012033.
MORSY T. Matsumoto, D. Orlando, S. Reffert, J. Sakamoto and K. Yoshida,
“Yang-Baxter deformations of Minkowski spacetime,”
JHEP 1510 (2015) 185 [arXiv:1505.04553 [hep-th]].
A. Borowiec, H. Kyono, J. Lukierski, J. Sakamoto and K. Yoshida,
“Yang-Baxter sigma models and Lax pairs arising from $\kappa$-Poincaré $r$-matrices,”
arXiv:1510.03083 [hep-th].
J. Lukierski, H. Ruegg, A. Nowicki and V. N. Tolstoy,
“Q deformation of Poincare algebra,”
Phys. Lett. B 264 (1991) 331;
J. Lukierski and A. Nowicki,
“Quantum deformations of D = 4 Poincare and Weyl algebra from Q deformed D = 4 conformal algebra,”
Phys. Lett. B 279 (1992) 299.
A. Pachol and S. J. van Tongeren,
“Quantum deformations of the flat space superstring,”
Phys. Rev. D 93 (2016) 2, 026008 [arXiv:1510.02389 [hep-th]].
H. Kyono, J. Sakamoto and K. Yoshida,
“Lax pairs for deformed Minkowski spacetimes,”
JHEP 1601 (2016) 143
[arXiv:1512.00208 [hep-th]].
C. R. Nappi and E. Witten,
“A WZW model based on a nonsemisimple group,”
Phys. Rev. Lett. 71 (1993) 3751
F. Delduc, M. Magro and B. Vicedo,
“Integrable double deformation of the principal chiral model,”
Nucl. Phys. B 891 (2015) 312
[arXiv:1410.8066 [hep-th]].
E. Abdalla, M. C. Abdalla and K. Rothe,
“Non-perturbative methods in two-dimensional quantum field theory,”
World Scientific, 1991.
G. W. Gibbons and C. N. Pope,
“Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group,”
Annals Phys. 326 (2011) 1760
[arXiv:1010.2455 [hep-th]].
M. A. Semenov-Tyan-Shanskii, “What is a classical $r$-matrix?”
Funct. Anal. Appl. 17 (1983) 259.
For a nice review, see “Integrable Systems and Factorization Problems,”
I. Kawaguchi, D. Orlando and K. Yoshida,
“Yangian symmetry in deformed WZNW models on squashed spheres,”
Phys. Lett. B 701 (2011) 475.
[arXiv:1104.0738 [hep-th]];
I. Kawaguchi and K. Yoshida,
“A deformation of quantum affine algebra in squashed WZNW models,”
J. Math. Phys. 55 (2014) 062302
[arXiv:1311.4696 [hep-th]].
E. Kiritsis, C. Kounnas and D. Lust,
“Superstring gravitational wave backgrounds with space-time supersymmetry,”
Phys. Lett. B 331 (1994) 321
M. Blau, J. M. Figueroa-O'Farrill, C. Hull and G. Papadopoulos,
“A New maximally supersymmetric background of IIB superstring theory,”
JHEP 0201 (2002) 047
D. T. Son,
“Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrödinger symmetry,”
Phys. Rev. D 78 (2008) 046003
[arXiv:0804.3972 [hep-th]];
K. Balasubramanian and J. McGreevy,
“Gravity duals for non-relativistic CFTs,”
Phys. Rev. Lett. 101 (2008) 061601
[arXiv:0804.4053 [hep-th]].
S. Kachru, X. Liu and M. Mulligan,
“Gravity duals of Lifshitz-like fixed points,”
Phys. Rev. D 78 (2008) 106005
[arXiv:0808.1725 [hep-th]].
S. Schafer-Nameki, M. Yamazaki and K. Yoshida,
“Coset Construction for Duals of Non-relativistic CFTs,”
JHEP 0905 (2009) 038
[arXiv:0903.4245 [hep-th]].
|
1511.00314
|
Primary Facets of Order Polytopes
[Université Libre de Bruxelles, Département de Mathématique c.p. 216, B-1050 Bruxelles, Belgium. {doignon,Selim.Rexhep}@ulb.ac.be]
Jean-Paul Doignon and Selim Rexhep
Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data.
In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type.
The five prominent types of order relations are linear orders, weak orders, semiorders, interval orders and partial orders. For each of them, the problem of finding a complete, workable characterization of the vectors of probabilities is crucial—but it is reputably inaccessible.
Under a geometric reformulation, the problem asks for a linear description of a convex polytope whose vertices are known.
As for any convex polytope, a shortest linear description comprises one linear inequality per facet. Getting all of the facet-defining inequalities of any of the five order polytopes seems presently out of reach. Here we search for the facet-defining inequalities which we call primary because their coefficients take only the values $-1$, $0$ or $1$. We provide a classification of all primary, facet-defining inequalities of three of the five order polytopes.
Moreover, we elaborate on the intricacy of the primary facet-defining inequalities of the linear order and the weak order polytopes.
semiorder, semiorder polytope, order polytope, facet-defining inequality.
MSC Classification: [2010] 06A07, 52B12, 91E99.
§ INTRODUCTION
As it is the case in general for testing a deterministic theory on random sample data <cit.>,
checking whether transitivity is confirmed by
a collection of two-item comparisons raises several interesting issues.
Binary choice data usually consist of the relative frequencies of choice among any two alternatives.
A formal approach to test whether the relative frequencies are consistent with transitivity relies on probabilistic models and derived statistical tests. <cit.> and <cit.> survey models for binary choice (forced or non-forced). We focus here on the characterization problem of the choice probabilities predicted by five of the main models.
A random utility model of binary choice relates the probability of chosing alternative $i$ over $j$ to the probability that the utility of $i$, taken as a random variable, exceeds that of $j$.
As known since a long time for the direct comparison of random utility values[under the assumption that equality of the utilities of two distinct alternatives occur with probability zero],
the model happens to be a mixture model on linear orders <cit.>. In precise terms, vectors of binary choice probabilities coincide with convex combinations of the characteristic vectors of linear orders on the set of alternatives.
Recent work (), extends the traditional setting of linear orders to various types of order relations, principally weak orders, semiorders, interval orders and partial orders (the meaning of the terms will be explained in the next section). The random utility model, based on a specific, modified way of comparing two random utility values, admits a reformulation as a mixture model of order relations.
One of the fundamental problems on probabilistic models is to find out a workable characterization of the (probabilistic) predictions it makes. In the case of the mixture models of order relations, the characterization plays an important role in implementing tests of transitivity on binary choice data.
However, complete characterizations were obtained only when the number of alternatives is small (we give details in Section <ref>).
Even for the particular case of linear orders (in a way the most structured of our relations), the characterization problem raises mathematical difficulties and is even seen as unsolvable.
Section <ref> recalls some explanations for the latter assertion, and moreover indicates why similar difficulties appear for the other four types of order relations.
For mixture models, as the ones we investigate here, a geometric point of view is most useful. Indeed, a characterization of the model is akin to the description of a certain polytope. More precisely, the polytope is given by its vertices (in our cases, the characteristic vectors of the order relations) and the aim is to describe the polytope as the solution set of a system of linear (in)equalities. If such a linear description is moreover a shortest one, then the number of linear equations is equal to the codimension of the polytope, and there is one linear inequality per facet of the polytope. This shows the importance of facet-defining inequalities, or FDI's. We refer the reader to Section <ref> for a short summary of the concepts and results we will need and to <cit.> for more background on general convex polytopes.
The five types of order relations we mentioned before lead to five convex polytopes: the linear ordering polytope, the weak order polytope, the interval order polytope, the semiorder polytope and the partial order polytope. In the past, the first polytope received the most attention (we provide references in the sections dedicated to the respective polytopes but we want to mention here an unpublished manuscript of , the first to promote a common approach to order polytopes).
It is recognized that obtaining a full, linear description of any of these five polytopes is a very difficult problem (cf. Section <ref>). We found it interesting to investigate the facet-defining inequalities with coefficients in the set $\{-1,0,1\}$, with the aim of assessing the relative difficulties in the five cases. A linear inequality is primary when its coefficients (including the independent term) take only the values $-1$, $0$ or $1$.
Here is a summary of our results.
As a warm-up exercice, we provide a complete description of the primary linear inequalities which define facets of the partial order polytope (Section <ref>) or the interval order polytope (Section <ref>). Then we present a rather satisfiable understanding of the FDI's of the semiorder polytope; here, the results turn out to be rather technical (see Sections <ref> to <ref>).
We see the cases of the strict weak order and the linear ordering polytopes to be out of our reach even for primary linear inequalities, as we explain in Sections <ref> and <ref>.
Two directions of possible further research are worth mentioning here. First, techniques from combinatorial optimization could be applied to the primary linear inequalities found here to derive more inequalities (such as those resulting from so-called Chvátal-Gomory cuts; for an introduction to the techniques, see , Section 9.4, or , Chapter 5).
Second, when a polytope $Q$ contains a polytope $P$, new FDI's of one of the two polytopes can sometime be infered from FDI's of the other polytope; we leave for future work the related inspection of the inclusions among our five order polytopes.
The authors thank Samuel Fiorini for helpful discussions at the start of the project.
§ BACKGROUND: TYPES OF ORDER RELATIONS AND THEIR POLYTOPES
In this section we briefly recall some basic facts, first about order relations, then about polytopes.
Throughout the paper, $n$ denotes a natural number with $n \geqslant 2$. We write $\Vset{n}$ for the set $\{1,\ldots,n\}$ of elements (or alternatives).
Moreover, we denote by $\Arcs{n}$ the set of pairs
of distinct elements, that is:
\Arcs{n} = \{(i,j) \st i,j \in \Vset{n}, i \neq j\}.
Let $\RR$ be an irreflexive binary relation on $\Vset{n}$ (that is, $\RR \subseteq \Arcs{n}$); we write $i \RR j$ for $(i,j)\in \RR$. Then $\RR$ is a (strict) partial order if $\RR$ is asymmetric (that is, if $i \RR j$ then not $j \RR i$) and transitive (if $i \RR j$ and $j \RR k$ then $i \RR k$).
A linear order is a partial order which is total (two distinct elements are always comparable).
A strict weak order is a partial order which is negatively transitive (that is, $i \RR k$ implies $i \RR j$ or $j \RR k$). (Notice that strict weak orders are the complements of `complete preorders', as we explain in Section <ref>.)
An interval order $S$ on $\Vset{n}$ is a partial order for which there exist two maps $f$ and $g$ from $\Vset{n}$ to $\R$ such that
i \SS j \text{ if and only if } g(i) < f(j)
(thus $i \SS j$ exactly if the closed interval $[f(i),g(i)]$ of the real line is located entirely below the similar interval $[f(j),g(j)]$).
The pair $(f,g)$ of maps is then called an interval representation of $S$. If $S$ admits an interval representation $(f,g)$ such that $g(i) = f(i) + 1$ for each $i$ in $\Vset{n}$ (every interval has length $1$) then $S$ is also called a semiorder; we then say that $f$ is a unit interval representation. We now state two classical theorems characterizing interval orders and semiorders (for the proofs as well as additional basic terminology, see a textbook as for example <cit.> or <cit.>).
It is easy to check that the partial orders represented by their Hasse diagrams in Figure <ref> are not semiorders; we denote them by $\underline{2} + \underline{2}$ and $\underline{3} + \underline{1}$ respectively. The second one is an interval order, while the first one is not.
vertex=[circle,draw,fill=white, scale=0.3]
(z_1) at (0,0) [vertex] ;
(z_2) at (0,2) [vertex] ;
(z_3) at (1,0) [vertex] ;
(z_4) at (1,2) [vertex] ;
(z_1) – (z_2);
(z_3) – (z_4);
at (0.5,-1) 2 + 2;
(z_5) at (0,0) [vertex] ;
(z_6) at (0,2) [vertex] ;
(z_7) at (0,4) [vertex] ;
(z_8) at (1,2) [vertex] ;
(z_5) – (z_6) – (z_7);
at (0.5,-1) 3 + 1;
The Hasse diagrams of the posets $\underline{2} + \underline{2}$ and $\underline{3} + \underline{1}$.
[Fishburn Theorem]
A partial order is an interval order if and only if it does not induce any $\underline{2} + \underline{2}$.
[Scott-Suppes Theorem]
A partial order is a semiorder if and only if it does not induce any $\underline{2} + \underline{2}$ nor $\underline{3} + \underline{1}$.
Obviously, any strict weak order is a semiorder, any semiorder is an interval order, and any interval order is a partial order.
We now move on to convex polytopes. A detailed treatment of the subject can be found for example in <cit.>.
A convex polytope in some space $\R^d$ is the convex hull of a finite set of points. For $X$ a finite subset of $\R^d$, let $P$ be the polytope
P = \conv \left(\{ x_v \st v \in V \} \right).
The dimension $\dim(P)$ of $P$ is the dimension of its affine hull (notice that, except otherwise mentioned, $\dim$ designates the affine dimension). A linear inequality on $\R^d$,
\sum_{i=1}^d \alpha_i x_i \leqslant \beta,
is valid for $P$ if it is satisfied by all points of $P$. A face of $P$ is the subset of points of $P$ satisfying a given valid inequality with equality (thus $\es$ and $P$ are faces of $P$); then the inequality defines the face. The faces of $P$ are themselves polytopes. The vertices of $P$ are the points $v$ such that $\{v\}$ is a face of dimension $0$; the facets are the faces of dimension $\dim(P)-1$. A valid inequality is facet-defining (or a FDI) if it defines a facet of $P$. The importance of the latter concept is clear from the following result. Assume that the polytope $P$ is full, that is, of dimension $d$. Then $P$ equals the set of solutions to all of its facet-defining inequalities; moreover, any system of linear inequalities on $\R^d$ whose set of solutions equals $P$ necessarily contains all of the facet-defining inequalities.
The five polytopes we consider in the paper are defined in the space $\R^{\Arcs{n}}$, where as before $\Arcs{n} = \{(i,j) \st i,j \in \Vset{n}, i \neq j\}$.
The next lemma recalls, in this setting, a classical result on valid inequalities. Notice how, for a vector $x \in \R^{\Arcs{n}}$ and $(i,j) \in \Arcs{n}$, we abbreviate $x_{(i,j)}$ into $x_{ij}$.
Suppose that the two inequalities
\begin{equation}\label{eq_proper_face}
\sum_{(i,j) \in \Arcs{n}} \alpha_{ij} x_{ij} \leqslant \beta
\end{equation}
\begin{equation}\label{eq_proper_face_bis}
\sum_{(i,j) \in \Arcs{n}} \alpha_{ij}' x_{ij} \leqslant \beta'
\end{equation}
define distinct, proper faces of some polytope $P$ in $\R^{\Arcs{n}}$.
Then their sum
\begin{equation}\label{eq_3}
\sum_{(i,j) \in \Arcs{n}} (\alpha_{ij} + \alpha_{ij}') x_{ij} \leqslant \beta + \beta'
\end{equation}
cannot be facet-defining for $P$.
A relation $\RR$ on $\Vset{n}$ is represented in $\R^{\Arcs{n}}$ by its characteristic vector $\chi^R$, with $\chi^{\RR}_{ij} = 1$ if $i \RR j$, and $\chi^R_{ij} = 0$ otherwise.
The semiorder polytope $\pso n$ on $\Vset{n}$ is defined in
$\R^{\Arcs{n}}$ by
\pso n = \conv \left(\left\{ \chi^{\RR} \st \RR \text{ is a semiorder on } \Vset{n} \right\} \right).
The definitions of the linear ordering polytope $\plo n$, strict weak order polytope $\pswo n$, interval order polytope $\pio n$, and partial order polytope $\ppo n$ are similar. Of course
\plo n \subseteq \pswo n \subseteq \pso n \subseteq \pio n \subseteq \ppo n,
all inclusions being strict when $n \geqslant 4$.
It is easy to see that
\begin{equation}\label{eqn_dim}
\dim(\pswo n) = \dim(\pso n) = \dim(\pio n) = \dim(\ppo n) = n(n-1)
\end{equation}
(so that the four polytopes are full), and it can be shown <cit.> that
\begin{equation}\label{eqn_dim_plo}
\dim(\plo n) = \frac{n(n-1)}{2}.
\end{equation}
Remember from our Introduction that characterizing the binary choice probabilities predicted by five models surveyed in <cit.> amount to listing all of the facet-defining inequalities of the five polytopes $\ppo n$, $\pio n$, $\pso n$, $\pswo n$, and $\plo n$. The next section explains why the task appears to be much difficult.
§ THE DIFFICULTIES IN FINDING ALL FDI'S
We briefly indicate why finding out all the FDI's of any of the five order polytopes may look as a hopeless task. The (traditional) trick is to convert some NP-hard combinatorial problem into the optimization problem of a linear form on the polytope. Among the many possible optimization problems we could select, there are five similar problems which apply in the same way to our five polytopes. All problems require the construction of a `median order' for a given collection of relations. We refer the reader to <cit.> and their references for background on median orders, and also for the results we apply here.
For any family $\RRR$ of relations on $\Vset{n}$ and any relation $P$ on $\Vset{n}$, the $\PPP$-remoteness of $P$ to $\RRR$ equals
\begin{equation}\label{eqn_remoteness}
\rho(P,\RRR) \;=\; \sum_{R \in \RRR} |P \Delta R|,
\end{equation}
where $P \Delta R = (P\setminus R) \cup (R \setminus P)$ is the symmetric difference of $P$ and $R$. Thus $\rho(P,\RRR)$ counts the total number of disagreements between $P$ and the relations in $\RRR$.
In the next problem, the aim is to minimize the remoteness of an order relation of fixed type to a given family $\RRR$ of relations. Let $\PPP$ designate one of the families of linear orders, strict weak orders, semiorders, interval orders of partial orders on $\Vset n$.
Given a family $\RRR$ of relations on $\Vset{n}$, find a relation $P$ in $\PPP$ such that $\rho(P,\RRR) \leqslant \rho(Q,\RRR)$ for all $Q$ in $\PPP$.
Such an order $P$ is $\PPP$-median for the collection $\RRR$ (the notion of a median order plays a role in the aggregation of preferences and in voting theory).
To restate the problem, denote by $\pany{n}$ the corresponding polytope (that is, $\plo n$, $\pwo n$, $\pso n$, $\pio n$ or $\ppo n$) and define from $\RRR$ a vector $c$ in $\R^{\Arcs{n}}$ by
\begin{equation}
c_{ij} \;=\; |\{R\in\RRR \st (i,j)\in R\}| - |\{Q\in\RRR \st (i,j)\notin Q\}|.
\end{equation}
\begin{equation}\label{eqn_rho}
\rho(P,\RRR) \quad=\quad \sum_{ij\in\Arcs{n}} c_{ij} \chi^P_{ij}
\; - \;
\sum_{R\in\RRR} |R|.
\end{equation}
Because the last sum in Equation (<ref>) gives a constant depending only on $\RRR$, we see that Problem is equivalent to the minimization of the linear form $x \mapsto c\,x$ (the scalar product of $c$ and $x$) on the polytope $\pany{n}$ (such a reformulation of the problem is well known).
Now it happens that Problem is NP-hard for each choice of $\PPP$ as one of our five collections of order relations—it is even the case under rather strong restrictions on $\RRR$, for instance about the size of $\RRR$ or the type of relations in $\RRR$; <cit.> and <cit.> provide a wealth of results in this line. It suffices here to record that Problem is NP-hard and moreover reformulable as a linear programming problem having $\pany{n}$ as its feasible set.
The following conclusion follows: if a polynomial-size description of $\pany{n}$ existed, the equality $\texttt{P}= \texttt{NP}$ would follow (answering in a surprising way a famous question in complexity theory, ). Hence, any linear decription of $\pany{n}$ must be intractable in the technical sense (that is, be of non-polynomial size in $n$). Of course, the conclusion we just reached does not preclude the existence of an exponential-size, linear description of $\pany{n}$, even one with a nice mathematical structure.
\begin{equation*}\begin{array}{|c||r|*{6}{r|}}
\hline
n & \multicolumn{2}{c|}{\ppo n} & \multicolumn{2}{c|}{\pio n} & \multicolumn{2}{c|}{\pso n} \\
\hline\hline
2& 3 & 2 & 3 & 2 & 3 & 2 \\
3& 17 & 4 & 17 & 4 & 17 & 4 \\
4& 128 & 8 & 191 & 14 & 563 & 31 \\
5& \geqslant43244 & \geqslant 211 & &&& \\
\hline
\end{array}
\end{equation*}
Numbers of FDI's, in total or up to element relabellings, for three order polytopes when, to our knowledge, they are available.
For `small' values of $n$, computers can produce a linear description of $\pany{n}$ (running for instance the software ). Tables <ref> and <ref> indicate for which values of $n$, to our knowledge, a description was published.
The cells record, respectively, the total number of FDI's, and the number of their equivalence classes under relabellings of the elements in $\Vset n$. The values come from
* <cit.> for the partial order polytope $\ppo n$;
* <cit.> for the interval order polytope $\pio n$ and for the semiorder polytope $\pso n$;
* <cit.>, <cit.> and <cit.> for the (strict) weak order polytope $\pswo n$;
* <cit.> and <cit.> for the linear ordering polytope $\plo n$.
The symbol “$\geqslant$” indicates that the number provided is only a lower bound, the exact value being unknown to us (in many cases, was reported to run out of computer resources).
\begin{equation*}
\begin{array}{|c||r|*{4}{r|}}
\hline
n & \multicolumn{2}{c|}{\pswo n} & \multicolumn{2}{c|}{\quad\plo n}\\[2mm]
\hline\hline
2& 3 & 2 & 2 & 1\\
3& 15 & 3 & 8 & 2\\
4& 106 & 9 & 20 & 2\\
5& 75\,843 & \geqslant318 & 40 & 2\\
6& & & 910 & 5\\
7& & & 87\,472 & 27\\
8& & & \geqslant488\,602\,996 & \geqslant12\,231\\
\hline
\end{array}
\end{equation*}
Numbers of FDI's, in total or up to element relabellings, for the strict weak order polytope and the linear ordering order polytope when, to our knowledge, they are available.
The difficulty of finding a complete linear description of any of the five polytopes led us to investigate their FDI's under some restriction on the coefficients. This is why we focus from now on primary linear inequalities.
§ GENERAL CONDITIONS ON THE VALIDITY OF PRIMARY LINEAR INEQUALITIES
A primary linear inequality on $\R^{\Arcs{n}}$ takes the form
\begin{equation}\label{eqn_rewrite_A_B}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant \beta
\end{equation}
where $\beta\in\{-1,0,1\}$ and $A$, $B$ are two disjoint subsets of $\Arcs{n}$. From now on, we assume that at least one of $A$ and $B$ is nonempty when we consider Equation (<ref>). Throughout the whole paper, the notation $A$, $B$ and $\beta$ asssume the latter condition (additional assumptions come at the end of the section).
It is useful to think of $(\Vset{n}, A)$, $(\Vset{n}, B)$ and $(\Vset{n}, A \cup B)$ as graphs[If not explicitly said otherwise, any graph here is directed, without loops or multiple arcs.]. When convenient, we will represent the graphs (or parts of them) using solid arcs for the pairs in $A$ and dashed arcs for the pairs in $B$.
Figure <ref> displays such graphical representations for the five inequalities in Proposition <ref>, which provide examples of FDI's for our polytopes.
vertex=[circle,draw,fill=white, scale=0.3]
(1) at (0,0) [vertex,label=below:$i$] ;
(2) at (0,1) [vertex,label=above:$j$] ;
[->-=.7,dashed] (1) to (2);
at (0,-1) $\leqslant 0$;
(2-1) at (0,0) [vertex,label=below:$i$] ;
(2-2) at (0,1) [vertex,label=above:$j$] ;
[->-=.7,>= triangle 45] (2-1) to [bend right=25] (2-2);
[->-=.7,>= triangle 45] (2-2) to [bend right=25] (2-1);
at (0,-1) $\leqslant 1$;
(3-i) at (0,0) [vertex,label=below:$i$] ;
(3-j) at (-0.9,1) [vertex,label=left:$j$] ;
(3-k) at (0,2) [vertex,label=above:$k$] ;
[->-=.7,>= open triangle 45] (3-i) to (3-j);
[->-=.7,>= open triangle 45] (3-j) to (3-k);
[->-=.7,>= triangle 45,dashed] (3-i) to (3-k);
at (-0.5,-1) $\leqslant 1$;
(4-i) at (1,0) [vertex,label=below:$i$] ;
(4-j) at (0,2) [vertex,label=above:$j$] ;
(4-k) at (-1,0) [vertex,label=below:$k$] ;
[->-=.7,>= open triangle 45,bend right=15] (4-i) to (4-j);
[->-=.7,>= open triangle 45,bend right=15] (4-j) to (4-k);
[->-=.7,>= triangle 45,bend right=15] (4-k) to (4-i);
[->-=.7,>= open triangle 45,dashed,bend right=15] (4-i) to (4-k);
[->-=.7,>= open triangle 45,dashed,bend right=15] (4-k) to (4-j);
[->-=.7,>= triangle 45,dashed,bend right=15] (4-j) to (4-i);
at (0,-1) $\leqslant 1$;
(5-i) at (0,0) [vertex,label=below:$i$] ;
(5-j) at (0,2) [vertex,label=above:$j$] ;
(5-k) at (2,0) [vertex,label=below:$k$] ;
(5-l) at (2,2) [vertex,label=above:$l$] ;
[->-=.7,>= open triangle 45] (5-i) to (5-j);
[->-=.7,>= open triangle 45] (5-k) to (5-l);
[->-=.7,>= open triangle 45,dashed] (5-i) to (5-l);
[->-=.7,>= open triangle 45,dashed] (5-k) to (5-j);
at (1,-1) $\leqslant 1$;
Graphical representations of the five inequalities in Proposition <ref>. The arcs in $A$ are solid, those in $B$ dashed.
Consider the five primary linear inequalities
\begin{align}
-x_{ij} \;\leqslant&\; 0,\label{eqn_nonneg}
\\ x_{ij} + x_{ji} \;\leqslant&\; 1,\label{eqn_2}
\\ x_{ij} + x_{jk} - x_{ik} \;\leqslant&\; 1,\label{eqn_3}
\\ x_{ij} + x_{jk} + x_{ki} - x_{ji} - x_{kj} - x_{ik} \;\leqslant&\; 1;\label{eqn_4}
\\x_{ij} + x_{kl} - x_{il} - x_{kj} \;\leqslant&\; 1,\label{eqn_5}
\end{align}
where $i$, $j$, $k$ and $l$ are distinct elements in $\Vset{n}$.
Table <ref> indicates for which order polytopes they define facets.
Checkmarks indicate when the five linear inequalities define facets of the five order polytopes.
Equation $\ppo n$ $\pio n$ $\pso n$ $\pswo n$ $\plo n$
(<ref>) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
(<ref>) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
(<ref>) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
(<ref>) $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
(<ref>) $\checkmark$ $\checkmark$ $\checkmark$
<cit.> (unpublished) establishes many of the results—for $\pso n$, we repeat his argument in Section <ref>.
For $\ppo n$, $\pio n$ and $\plo n$, proofs appear in respectively <cit.>, <cit.> and <cit.>.
For $\pso n$ the results are consequences of findings of <cit.> and Theorem <ref> below.
For $\pswo n$, the assertions follow from results in <cit.> on the weak order polytope (the relationship between the latter polytope and the strict weak order polytope is described in Section <ref>). For $\plo n$, see for instance <cit.>.
We call nonnegativity inequality the inequality in Equation (<ref>).
Our main contribution is to characterize the primary linear inequalities that define facets of the polytopes $\ppo n$, $\pio n$ and $\pso n$.
For the rest of this section, let $P_n$ be one of the latter three polytopes (we consider $\pswo n$ and $\plo n$ in Sections <ref> and <ref>, respectively).
The following four conditions on the sets $A$ and $B$ are most useful in our characterization of valid or facet-defining, primary inequalities for $P_n$ in the form of Inequality (<ref>). Figure <ref> illustrates Conditions C2, C3 and C4.
Condition C1: For any distinct pairs $(i,j)$, $(k,l)$ in $A$, there holds $i \neq k$ and $j \neq l$.
Condition C2: For any $(i,j),$ $(k,l)$ in $A$ with $i$, $j$, $k$, $l$ distinct elements, there holds $(i,l) \in B$.
Condition C3: For any pairs $(i,j)$, $(j,k)$ in $A$ with $i$, $j$, $k$ distinct elements, there holds $(i,k)\in B$.
Condition C4: For any pairs $(i,j)$, $(j,k)$ in $A$ with $i$, $j$, $k$ distinct elements, either there holds $(i,k) \in B$ or there exists some element $p$ in $\Vset{n}\setminus\{i,j,k\}$ such that $(i,p),(p,k)\in B$.
vertex=[circle,draw,fill=white, scale=0.3]
(z_l) at (1.5,2) [vertex] ;
(z_k) at (1.5,0) [vertex] ;
(z_j) at (0,2) [vertex] ;
(z_i) at (0,0) [vertex] ;
[->-=.7] (z_i) to (z_j);
[->-=.7] (z_k) to (z_l);
[->-=.8, dashed] (z_i) to (z_l);
[->-=.8, dashed] (z_k) to (z_j);
(z_i.west) node [left] $i$;
(z_j.west) node [left] $j$;
(z_k.east) node [right] $k$;
(z_l.east) node [right] $l$;
(u_k) at (0,4) [vertex] ;
(u_j) at (0,2) [vertex] ;
(u_i) at (0,0) [vertex] ;
[->-=.7] (u_i) to (u_j);
[->-=.7,] (u_j) to (u_k);
[->-=.7, dashed] (u_i) to [bend right] (u_k);
(u_i.west) node [left] $i$;
(u_j.west) node [left] $j$;
(u_k.west) node [left] $k$;
(i) at (0,0) [vertex,label=left:$i$] ;
(j) at (0,2) [vertex,label=left:$j$] ;
(k) at (0,4) [vertex,label=left:$k$] ;
[->-=.7] (i) – (j);
[->-=.7] (j) – (k);
[->-=.7, dashed] (i) to [bend right] (k);
(p) at (2,2) [vertex,label=right:$p$] ;
[->-=.7, dashed] (i) – (p);
[->-=.7, dashed] (p) – (k);
Illustrations of Conditions C2, C3 and C4 on subsets $A$, $B$ of $\Arcs{n}$. The solid arcs are in $A$, the dashed ones in $B$.
Note that Condition C1 exactly means that the graph is a disjoint union of isolated vertices, directed paths and directed cycles (where “disjoint union” means “no two constituents have a vertex in common”). We call PC-graph any graph of this type. In Condition C2, $(k,j) \in B$ also follows from the assumption (by a renaming of the elements).
If Inequality (<ref>) is valid for $P_n$, we have:
* $\beta \geqslant 0$; if $A\neq\es$, then $\beta = 1$;
* the set $A$ satisfies Condition C1;
* the sets $A$ and $B$ satisfy Condition C2;
* if $P_n = \ppo n$ or $P_n = \pio n$, then $A$ and $B$ satisfy Condition C3;
* if $P_n = \pso n$, then $A$ and $B$ satisfy Condition C4.
1) The incidence vector of the empty relation (or antichain) on $\Vset{n}$ gives value $0$ to the left-hand side of Inequality (<ref>). Validity of Inequality (<ref>) then implies $\beta \geqslant 0$.
Next, let $a$ be in $A$. Consider the semiorder consisting only of the pair $\{a\}$; its characteristic vector gives value $1$ to the left-hand side of Inequality (<ref>). Validity of the inequality thus implies $\beta \geqslant 1$. Moreover, we assume $\beta\in\{-1$, $0$, $1\}$.
2) To show that Condition C1 is satisfied, suppose first the existence of pairwise distinct elements $i$, $j$, $k$ in $\Vset{n}$ with $(i,j)$, $(i,k)$ in $A$. Then the vector of $\R^{A_n}$ defined by
\begin{equation*}
y_{uv} =
\begin{cases}
1 & \text{if } (u,v)=(i,j) \text{ or } (u,v)=(i,k),\\
0 & \text{otherwise }
\end{cases}
\end{equation*}
is the incidence vector of a semiorder, and moreover
\sum_{a \in A} y_{a} - \sum_{b \in B}y_b = 2 > \beta.
This contradicts the validity of Inequality (<ref>).
The argument is similar in case $(i,k), (j,k) \in A$.
3) To establish Condition C2, consider $(i,j),$ $(k,l) \in A$ with $i$, $j$, $k$, $l$ pairwise distinct elements. Suppose $(i,l) \notin B$. Then the vector of $\R^{A_n}$ defined by
\begin{equation*}
y_{uv} =
\begin{cases}
1 & \text{if } (u,v)\in \{(i,j),\,(k,l),\,(i,l)\},\\
0 & \text{otherwise}
\end{cases}
\end{equation*}
is the incidence vector of a semiorder, and
\sum_{a \in A} y_{a} - \sum_{b \in B}y_b = 2 > \beta.
4) Let $(i,j), (j,k) \in A$ and suppose $(i,k) \notin B$. Then the vector of $\R^{A_n}$ defined by
\begin{equation*}
y_{uv} =
\begin{cases}
1 & \text{if } (u,v) \in \{(i,j),\,(i,k),\,(j,k)\},\\
0 & \text{otherwise }
\end{cases}
\end{equation*}
is the incidence vector of an interval order such that
\sum_{a \in A} y_{a} - \sum_{b \in B}y_b \in \{2,\,3\},
a contradiction with the validity of Inequality (<ref>) for $\pio n$. Note that the interval order we just constructed is in general not a semiorder.
5) Finally, suppose Inequality (<ref>) is valid for $\pso n$ but does not satisfy Condition C4. Let the pairs $(i,j)$, $(j,k)$ in $A$ invalidate Condition C4. Then $(i,k) \notin B$. Moreover, for evey element $p$ in $\Vset{n}\setminus\{i,j,k\}$ we have $(i,p) \notin B$ or $(p,k)\notin B$; we call $z_p$ the pair not in $B$ (if none of the two pairs happens to be in $B$, we choose $z_p$ arbitrary among them). Then
\{(i,j), (i,k), (j,k), z_p \st p \in \Vset{n}\setminus\{i,j,k\}\}
is a semiorder on $\Vset{n}$ whose characteristic vector invalidates Inequality (<ref>), a contradiction.
Additional conditions obtain when Inequality (<ref>) is moreover facet-defining.
Suppose that Inequality (<ref>) defines a facet of $P_n$. Then:
* if $\beta = 0$, then $A$ is empty and $B$ is a singleton, in other words Inequality (<ref>) is a nonnegativity equality (<ref>);
* if $\beta = 1$, the set $A$ can neither be empty nor be a singleton.
1) If $\beta=0$, we must have $A = \varnothing$ (Theorem <ref>). Now the inequality reads $-\sum_{b \in B}x_b \leqslant 0$, with $B$ nonempty. If $|B|=1$, we get a nonnegativity inequality. If $|B| \geqslant 2$, Lemma <ref> shows that our inequality cannot define a facet.
2) If $A=\es$, equality cannot be reached in (<ref>). Suppose now $A = \{(i,j)\}$. Then if $y$ is a vertex of $P_n$ satisfying equality in Inequality (<ref>), we have $y_{ij} = 1$ and hence $y_{ji} = 0$. So $y$ is contained in the facet defined by $-x_{ij} \leqslant 0$, a contradiction.
Because of Theorem <ref>, when searching for primary FDI's of $P_n$ in the form of Equation (<ref>), we will always suppose $\beta = 1$, $|A| \geqslant 2$ and $B \neq \varnothing$: indeed, the only FDI not satisfying these requirements is $-x_{ij} \leqslant 0$.
In the next two sections, we determine explicitly all the primary FDI's of respectively $\ppo n$ and $\pio n$.
§ THE PRIMARY FACET-DEFINING INEQUALITIES OF THE PARTIAL ORDER POLYTOPE
Among other authors, <cit.> and, later, <cit.> investigate mathematical aspects of the partial order polytope. It happens that their (unsurprisingly) incomplete lists of FDI's nevertheless contain all the primary ones—as we show in Theorem <ref> below. The partial order polytope appears in psychological applications, for instance in <cit.>.
The inequality
\begin{equation}\label{eqn_rerewrite_A_B}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
is valid for the partial order polytope $\ppo n$ if and only if the pair $A$, $B$ satisfies one of the following requirements, for some distinct elements $i$,$j$ and $k$:
* $A=\{(i,j)\}$;
* $A=\{(i,j),\,(j,i)\}$;
* $A=\{(i,j),\,(j,k)\}$ and $(i,k)\in B$;
* $A=\{(i,j),\,(j,k),\, (k,i)\}$ and $(j,i),\, (k,j),\, (i,k)\in B$.
We first show that if Inequality (<ref>) is valid for $\ppo n$, then $A$ satisfies
Condition C0: For any distinct pairs $(i,j)$, $(k,l)$ in $A$, the elements $i$, $j$, $k$ and $l$ are not all distinct.
Indeed, if the elements $i$, $j$, $k$ and $l$ were distinct the characteristic vector of the partial order $\{(i,j)$, $(k,l)\}$ would invalidate (<ref>).
By Theorem <ref>, Condition C1 also holds. The only subsets $A$ of $\Arcs n$ that satisfy both Conditions C0 and C1 are exactly those described in the requirements of the theorem. Condition C3 (which holds by Theorem <ref>.4) then implies the rest of the requirements.
Conversely, any of the inequalities satisfying one of the requirements is the sum of one of the FDI's (<ref>)–(<ref>) with a valid inequality of the form $\sum_{c\in C} -x_c \le 0$ for some $C \subseteq \Arcs{n} $, so it is also valid.
We now describe the FDI's of $\ppo n$.
The primary FDI's of the partial order polytope $\ppo n$ are exactly those in Equations (<ref>), (<ref>), (<ref>) and (<ref>),
that is:
\begin{align}
-x_{ij} \leqslant&\; 0,\label{eqn_PPO_1}
\\ x_{ij} + x_{ji} \leqslant&\; 1, \label{eqn_PPO_2}
\\ x_{ij} + x_{jk} - x_{ik} \leqslant&\; 1,\label{eqn_PPO_3}
\\ x_{ij} + x_{jk} + x_{ki} - x_{ji} - x_{kj} - x_{ik} \leqslant&\; 1\label{eqn_PPO_4}
\end{align}
(where $i,j,k$ are pairwise distinct elements of $\Vset{n}$).
As mentioned in Proposition <ref>, Inequalities (<ref>)–(<ref>) define facets of $\ppo n$. Among the valid inequalities described in Theorem <ref>, only inequalities (<ref>)–(<ref>) are FDI's. This follows by the application of Lemma <ref> to the latter inequalities and inequalities of the form $\sum_{c\in C} -x_c \le 0$ for well-chosen subsets $C$ of $\Arcs{n}$.
§ THE PRIMARY FACET-DEFINING INEQUALITIES OF THE INTERVAL ORDER POLYTOPE
We now turn to the interval order polytope $\pio n$ (; ; ; , for instance).
The primary linear inequality
\begin{equation}\label{eqn_02_schtroumpf}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
is valid for $\pio n$ if and only if $A$ and $B$ satisfy Conditions C1, C2 and C3.
Thus Inequality (<ref>) is valid for $\pio n$ exactly if $(\Vset n,A)$ is a PC-graph and moreover $B$ contains all pairs required to make Conditions C2 and C3 true—and maybe additional pairs.
The necessity of Conditions C1–C3 was established in Proposition <ref>.4.
The sufficiency of Conditions C1–C3 obtains as follows. Let $P$ be an interval order on $\Vset n$ with $(i,j)\in P \cap A$ (if $P \cap A=\es$ validity is obvious). Now for each other pair $(k,l)$ in $P \cap A$, Condition C1 implies $i\neq k$ and $j\neq l$. If moreover $i \neq l$ and $j\neq k$, then Condition C2 implies that $B$ contains both pairs $(i,l)$ and $(k,j)$; at least one of them must be in $P$. If $i = l$, Condition C3 implies $(k,j)\in B$; on the other hand, $(k,j)\in P$. The case $j = k$ similarly gives $(i,l)\in B \cap P$. Notice that distinct pairs $(k,l)$ in $P \cap A$ (all different from the initial $(i,j)$) deliver in this way distinct pairs in $B \cap P$ (because of Condition C1). The validity of (<ref>) follows.
From Theorems <ref>.2 and <ref>, we know that if the primary linear inequality
\begin{equation}\label{eqn_02}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
defines a facet of $\pio n$, then $|A| \geqslant 2$ and Conditions C1, C2, C3 hold.
Let us say that $B$ is $A$[C2–C3]-forced when $B$ consists exactly of the pairs whose existence is required in Conditions C2 and C3, that is when
\begin{equation}\label{eqn_PIO_in_B}
(i,l) \in B \quad\iff
\begin{cases}
&\exists (i,j),(k,l)\in A \text{ with } i,j,k,l \text{ distinct, or}\\
&\exists (i,j),(j,l)\in A \text{ with } i,j,l \text{ distinct}.
\end{cases}
\end{equation}
A result of <cit.> states that when $A$ and $B$ satisfy $|A| \geqslant 2$, Conditions C1, C2, C3 and moreover $B$ is $A$[C2–C3]-forced, then Inequality (<ref>) is facet-defining (in a more general context, call `io-clique inequalities' such inequalities). We now easily establish that no further primary FDI exists.
The primary FDI's of $\pio n$ are exactly the nonnegativity inequalities and the inequalities
\begin{equation}\label{eqn_02_bis}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
for which $|A| \geqslant 2$, Conditions C1, C2, C3 hold, and moreover $B$ is $A$[C2–C3]-forced.
<cit.> establish that Inequality (<ref>) is a FDI when the requirements are satisfied. For the converse, by Theorem <ref>, there remains to prove that, for a primary FDI (<ref>) of $\pio n$ other than a nonnegative inegality, all pairs in $B$ are $A$[C2–C3]-forced. But this follows at once from Lemma <ref>, the result of and the validity of $-x_{ij}\leqslant 0$.
The $n$-fence inequality is just a particular case of the inequalities as in Theorem <ref>, in which any two pairs in $A$ are disjoint (two pairs $(i,j)$ and $(k,l)$ of elements are (vertex) disjoint when $\{i,j\} \cap \{k,l\}=\es$).
As in Figure <ref>, let $A = \{(a_1, b_1)$, $(a_2,b_2)$, …, $(a_m, b_m)\}$ with $m\geqslant 1$ and $|\{a_1$, $a_2$, …, $a_m$, $b_1$, $b_2$, …, $b_m\}|=2m$
(thus $2m \leqslant n$). The resulting set $B$ contains all pairs $(a_i,b_j)$ such that $i \neq j$. Thus the corresponding primary linear inequality reads
\begin{equation}\label{eqn_09}
\sum_{i=1}^{m}x_{a_ib_i} -
\sum_{\substack{i,j=1 \\ i \neq j}}^m x_{a_ib_j}
\;\leqslant\; 1.
\end{equation}
vertex=[circle,draw,fill=white, scale=0.5]
at (3.5,2) ...;
at (4.5,1.3) ...;
at (4.5,0.7) ...;
at (3.5,0) ...;
(z_1) at (0,0) [vertex] ;
(z_2) at (0,2) [vertex] ;
(z_3) at (2,0) [vertex] ;
(z_4) at (2,2) [vertex] ;
(z_5) at (5,0) [vertex] ;
(z_6) at (5,2) [vertex] ;
[->-=.7] (z_1) to (z_2);
[->-=.7] (z_3) to (z_4);
[->-=.7] (z_5) to (z_6);
(z_1.south) node [below] $a_1$;
(z_3.south) node [below] $a_2$;
(z_5.south) node [below] $a_m$;
(z_2.north) node [above] $b_1$;
(z_4.north) node [above] $b_2$;
(z_6.north) node [above] $b_m$;
[->-=.7, dashed] (z_1) to (z_4);
[->-=.7, dashed] (z_1) to (z_6);
[->-=.7, dashed] (z_3) to (z_2);
[->-=.7, dashed] (z_3) to (z_6);
[->-=.7, dashed] (z_5) to (z_2);
[->-=.7, dashed] (z_5) to (z_4);
The $n$-fence inequality first appeared in studies of the linear ordering polytope (see Section <ref>). <cit.> mentions that it defines a facet of $\pio n$.
Here is an immediate corollary of Theorem <ref>.
There exists a bijection between the primary FDI's of $\pio n$ and the PC graphs on $n$ elements with at least one arc.
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant \beta
be a primary FDI of $\pio n$. If $\beta = 0$ then by Theorem <ref>.1, we have a nonnegativity inequality, that is $A = \varnothing$ and there exist $(i,j) \in \Arcs{n}$ such that $B = \{(i,j)\}$. To the inequality we associate the PC graph on $\Vset{n}$ having one single arc $\{(i,j)\}$. If $\beta = 1$ then $A$ contains at least two arcs and to the inequality we associate the PC graph $(\Vset{n},A)$. This association specifies the desired bijection. Indeed, surjectivity is obvious, and injectivity holds because $A$ univocally determines $B$.
§ THE LIFTING LEMMA FOR THE SEMIORDER POLYTOPE
We now turn to the semiorder polytope $\pso n$.
The polytope is mentioned for instance in an unpublished manuscript of <cit.> and in papers by <cit.>.
The following theorem entails that any facet-defining inequality for $\pso n$ remains facet-defining for $\pso k$ for all $k$ such that $k \geqslant n$—it is common to name such a statement the Lifting Lemma. <cit.> states the theorem and gives a sketch of proof. Notice $\dim\left(\pso {n+1}\right) = 2n + \dim\left(\pso {n}\right)$, because Equation (<ref>) reads $\dim \pso n = n(n-1)$.
\begin{equation}\label{eqn_before}
\sum_{(i,j) \in \Arcs{n}} \alpha_{ij}x_{ij} \leqslant \beta
\end{equation}
be an inequality on $\R^\Arcs{n}$ which is valid for $\pso n$ (where $\alpha_{ij}$, $\beta \in \R$). Let $F$ be the face of $\pso n$ defined by (<ref>). Consider then the inequality on $\R^{\Arcs{n+1}}$
\begin{equation}\label{eqn-after}
\sum_{(i,j) \in \Arcs{n+1}} \alpha_{ij}'x_{ij} \leqslant \beta
\end{equation}
with $\alpha_{ij}' = \alpha_{ij}$ if $i,j \in \Vset{n}$ and $\alpha_{ij}' = 0$ if $i = n+1$ or $j = n+1$. Then Equation (<ref>) is valid for $\pso {n+1}$, and it defines a face $F'$ of $\pso{n+1}$ of dimension
\begin{equation}
\dim(F') = 2n + \dim(F).
\end{equation}
In particular, if Equation (<ref>) defines a facet of $\pso n$, then Equation (<ref>) defines a facet of $\pso{n+1}$.
Consider the canonical linear projection $\pi: \R^{\Arcs{n+1}} \to \R^{\Arcs{n}}$ (whose effect is to delete all coordinates attached to a pair of elements one of which is $n+1$).
Then $\pi$ maps the characteristic vector of a semiorder $S$ on $\Vset{n+1}$ to the characteristic vector of the restriction of $S$ to $\Vset{n}$—and this restriction is a semiorder on $\Vset{n}$. Hence Equation (<ref>) is valid for $\pso{n+1}$ and it defines a face $F'$ of $\pso{n+1}$.
From previous sentence there follows $\pi(F') \subseteq F$, and then $F' \subseteq \pi^{-1}(F)$.
Now because the kernel of the linear mapping $\pi$ has dimension $2n$, we derive
\dim(F') \leqslant 2n + \dim(F).
To prove the opposite inequality, let $k=\dim(F)$ and select $1+k$ affinely independent vertices $v_0$, $v_1$, …, $v_k$ in $F$. Thus each $v_i$ is the characteristic vector of some semiorder $R_i$ on $\Vset{n}$. Adding to $R_i$ all pairs $(n+1,i)$ for $i\in \Vset{n}$, we get a semiorder $R_i'$ on $\Vset{n+1}$;
denote by $v'_i$ its characteristic vector in $\R^{\Arcs{n+1}}$. All $v'_i$'s belong to $F'$. Moreover the points $v'_0$, $v'_1$, …, $v'_k$ are affinely independent because $\pi(v'_i)=v_i$.
To prove $\dim(F') \geqslant 2n + \dim(F)$, it suffices now to build $2n$ vertices $w_1$, $w_2$, …, $w_{2n}$ in $F' \cap \pi^{-1}(v_0)$ such that $v'_0$, $w_1$, $w_2$, …, $w_{2n}$ are affinely independent (because then $v'_0$, $w_1$, $w_2$, …, $w_{2n}$, $v'_1$, …, $v'_k$ are together affinely independent).
Consider some unit interval representation of $R_0$. First, by slightly perturbing the real values $f(i)$ if necessary, we make the $2n$ real values $f(i)-1$ and $f(i)$, for $i=1$, $2$, …, $n$, distinct and list them in increasing order as $\gamma_1$, $\gamma_2$, …, $\gamma_{2n}$.
We then form $1+2n$ semiorders on $\Vset{n+1}$ by specifying one of their interval representations: we always leave unchanged the values $f(i)$ (that is, the actual intervals $[f(i),f(i)+1]$) for $1 \leqslant i \leqslant n$, but select for $f(n+1)$ various values: first a value strictly below $\gamma_1$, then a value strictly between $\gamma_1$ and $\gamma_2$, next a value strictly between $\gamma_2$ and $\gamma_3$, …, and finally a value strictly above $\gamma(2n)$. There results $1+2n$ semiorders $R'_0$, $S_1$, $S_2$, …, $S_{2n}$ on $\Vset{n+1}$ whose characteristic vectors we denote by $v'_0$, $w_1$, $w_2$, …, $w_{2n}$. Notice that $\pi$ maps all these vectors to $v_0$. Moreover the points $v'_0$, $w_1$, $w_2$, …, $w_{2n}$ are affinely independent, because either the semiorder $S_j$ lacks a pair $(n+1,i)$ which belongs to all previous semiorders or $S_j$ contains a pair $(i,n+1)$ which was in none of the previous semiorders.
We have thus proved $\dim(F') = 2n + \dim(F)$. The very last assertion directly follows.
Theorem <ref> motivates the following:
When we study an inequality of the form $\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant \beta$ for $\pso n$ , we may restrict ourselves to the smallest $n$ such that $A \cup B \subseteq \Arcs{n}$. If this inequality is valid (respectively, facet-defining) for $\pso n$ it will also be valid (respectively, facet-defining) for $\pso k$ with $k \geqslant n$.
This is how <cit.> establishes several families of primary FDI's for $\pso n$ (some were already mentioned in Proposition <ref>).
vertex=[circle,draw,fill=white, scale=0.3]
(v_i) at (0,0) [vertex] ;
(v_j) at (0,2) [vertex] ;
(v_i.north) node [above] $i$;
(v_j.south) node [below] $j$;
[->-=.7] (v_i) to [bend right] (v_j);
[->-=.7] (v_j) to [bend right] (v_i);
(z_i) at (0,0) [vertex] ;
(z_j) at (0,2) [vertex] ;
(z_k) at (1,0) [vertex] ;
(z_l) at (1,2) [vertex] ;
[->-=.7] (z_i) to (z_j);
[->-=.7] (z_k) to (z_l);
[->-=.8, dashed] (z_i) to (z_l);
[->-=.8, dashed] (z_k) to (z_j);
(z_i.west) node [left] $i$;
(z_j.west) node [left] $j$;
(z_k.east) node [right] $k$;
(z_l.east) node [right] $l$;
(u_k) at (0,4) [vertex] ;
(u_j) at (0,2) [vertex] ;
(u_i) at (0,0) [vertex] ;
[->-=.7] (u_i) to (u_j);
[->-=.7] (u_j) to (u_k);
[->-=.7, dashed] (u_i) to [bend right] (u_k);
(u_i.west) node [left] $i$;
(u_j.west) node [left] $j$;
(u_k.west) node [left] $k$;
(t_k) at (0,4) [vertex] ;
(t_j) at (0,2) [vertex] ;
(t_i) at (0,0) [vertex] ;
(t_l) at (1,2) [vertex] ;
[->-=.7] (t_i) to (t_j);
[->-=.7] (t_j) to (t_k);
[->-=.7, dashed] (t_i) to (t_l);
[->-=.7, dashed] (t_l) to (t_k);
(t_i.west) node [left] $i$;
(t_j.west) node [left] $j$;
(t_k.west) node [left] $k$;
(t_l.east) node [right] $l$;
Graphical representations of the Axiomatic Inequalities (<ref>)–(<ref>) for $\pso n$ (with independent term equal to $+1$).
For $i$, $j$, $k$, $l$ any distinct elements of $\Vset{n}$, the four inequalities (see also Figure <ref>)
\begin{align}
x_{ij} + x_{ji} &\leqslant 1,\label{eqn_axiomatic1}\\
x_{ij} + x_{jk} - x_{ik} &\leqslant 1,\label{eqn_axiomatic2}\\
x_{ij} + x_{kl} - x_{il} - x_{kj} &\leqslant 1,\label{eqn_axiomatic3}\\
x_{ij} + x_{jk} - x_{il} - x_{lk} &\leqslant 1\label{eqn_axiomatic4}
\end{align}
are valid for $\pso n$.
Because Equations (<ref>)–(<ref>) derive from the conditions in the (Scott-Suppes) Theorem <ref> (for the first and second ones, remember that neither $(i,i)$ nor $(j,j)$ appears in any semiorder), we call them the Axiomatic Inequalities.
[Axiomatic FDIs of $\pso n$, ]
For $i,j,k,l \in \Vset{n}$ pairwise distinct, the Axiomatic Inequalities (<ref>)–(<ref>) define facets of $\pso n$.
§ THE PRIMARY VALID INEQUALITIES OF THE SEMIORDER POLYTOPE
Theorem <ref> states that if the primary linear inequality
\begin{equation}\label{eqn_03}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
is valid for the semiorder polytope $\pso n$, then $A$ and $B$ must satisfy Conditions C1, C2 and C4. As we will see in the next proof, the converse does not hold and we now proceed to define an additional necessary condition.
Condition C5: For any pairs $(i,j)$, $(j,k)$ and $(k,l)$ in $A$ with $i$, $j$, $k$ and $l$ distinct elements, at least one the following requirements holds:
* $(i,k)\in B$;
* $(j,l)\in B$;
* there exists some $r$ in $\Vset{n}\setminus\{i,j,k,l\}$ such that $(i,r),(r,k)\in B$;
* there exists some $s$ in $\Vset{n}\setminus\{i,j,k,l\}$ such that $(j,s),(s,l)\in B$;
* there exists some $t$ in $\Vset{n}\setminus\{i,j,k,l\}$ such that $(i,t),(t,l)\in B$;
* there exist some $u$ and $v$ in $\Vset{n}\setminus\{i,j,k,l\}$ such that $(i,u),(u,v)$, $(v,l)\in B$.
An illustration is given in Figure <ref>. Taking into considerations $i$, $j$ and $k$ in Conditions C4 and C5 (cf. Figures <ref> and <ref>), the reader might think that Condition C4 implies Condition C5. However this is not true because it could be that the element $p$ in Condition C4 equals $l$ in Condition C5.
vertex=[circle,draw,fill=white, scale=0.3]
(i) at (0,0) [vertex,label=left:$i$] ;
(j) at (0,1) [vertex,,label=left:$j$] ;
(k) at (0,2) [vertex,label=left:$k$] ;
(l) at (0,3) [vertex,label=left:$l$] ;
[->-=.7] (i) – (j);
[->-=.7] (j) – (k);
[->-=.7] (k) – (l);
[->-=.7, dashed] (i) to [bend right] (k);
[->-=.7, dashed] (j) to [bend right] (l);
(r) at (1,1) [vertex,label=above:$r$] ;
[->-=.7, dashed] (i) – (r);
[->-=.7, dashed] (r) – (k);
(s) at (1,2) [vertex,label=below:$s$] ;
[->-=.7, dashed] (j) – (s);
[->-=.7, dashed] (s) – (l);
(t) at (2,1.5) [vertex,label=right:$t$] ;
[->-=.7, dashed] (i) – (t);
[->-=.7, dashed] (t) – (l);
(u) at (3,1) [vertex,label=right:$u$] ;
(v) at (3,2) [vertex,label=right:$v$] ;
[->-=.7, dashed] (i) – (u);
[->-=.7, dashed] (u) – (v);
[->-=.7, dashed] (v) – (l);
Condition C5 on primary linear inequalities for $\pso n$.
Assume $A$ and $B$ are disjoint subsets of $\Arcs{n}$.
Then the primary linear inequality
\begin{equation}
\label{eqn_general_valid}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
is valid for $\pso n$ if and only if Conditions C1, C2, C4 and C5 are satisfied. Moreover, the four conditions are logically independent.
Before presenting the proof of Theorem <ref> we establish a lemma which provides information on the semiorders $S$ whose characteristic vector $\chi_S$ invalidates Equation (<ref>).
Assume $A$ and $B$ are disjoint subsets of $\Arcs{n}$ which satisfy Conditions C1 and C2. Then for any semiorder $S$ on $\Vset{n}$ such that
\begin{equation}\label{eqn_lemme}
\sum_{a \in A} \chi^{\RR}_{a} - \sum_{b \in B} \chi^{\RR}_b \geqslant 1
\end{equation}
there holds $|S \cap A| \in \{1,2,3,4\}$. Moreover, if
\begin{equation}\label{eqn_lemme_2}
\sum_{a \in A} \chi^{\RR}_{a} - \sum_{b \in B} \chi^{\RR}_b > 1
\end{equation}
$(\alpha$) either $S \cap A$ forms a path of length $2$
and $S \cap B = \es$
$(\beta$) or $S \cap A$ forms a path of length $3$, say $S \cap A =\{(i,j),(j,k),(k,l)\}$,
and moreover $S \cap B = \{(i,l)\}$.
If two pairs $(i,j)$ and $(k,l)$ in $S \cap A$ are disjoint, then by Condition C2 both $(i,l)$ and $(k,j)$ belong to $B$ and because $S$ is a semiorder, at least one of them belongs to $S$. By Condition C1 such a pair $(i,l)$ or $(k,j)$ found to be in $S \cap B$ comes from a unique set of disjoint pairs $(i,j)$, $(k,l)$. Hence
\sum_{a \in A} \chi^{\RR}_{a} - \sum_{b \in B} \chi^{\RR}_b
\quad\leqslant\quad
|S \cap A| - |\{\text{sets of two disjoint pairs in } S \cap A\}|.
Now let $|S \cap A| = m$. By Condition C1, $(\Vset{n},A)$ is a PC-graph, and so also $(\Vset{n},S \cap A)$ is a PC-graph (which moreover has no cycle because $S$ is a semiorder). Hence a pair in $S \cap A$ must be disjoint from at least $m-3$ other pairs in $S \cap A$. We then derive from previous equation
\sum_{a \in A} \chi^{\RR}_{a} - \sum_{b \in B} \chi^{\RR}_b
\quad\leqslant\quad
m - \frac{m(m-3)}{2} \quad=\quad \frac{-m(m-5)}{2}.
The latter expression is positive only for $m \in \{1,2,3,4\}$. This establishes the first assertion.
Next assume Equation (<ref>) holds. Then $m \geqslant 2$. For any two disjoint pairs in $S \cap A$, we have shown in the first part of the proof the existence of some pair in $S \cap B$ (with no two of the latter pairs being equal).
Hence the PC-graph $(\Vset{n},S \cap A)$ has to be a path of length at least $2$, maybe plus isolated vertices.
If the path has length $4$, say $S \cap A =\{(i,j)$, $(j,k)$, $(k,l)$ $(l,p)\}$, then Condition C2 implies that $(i,l)$, $(j,p)$ and $(i,p)$ are in $B$, and because they are also in $S$, Equation (<ref>) cannot hold. There remain two cases: $(\alpha$) $S \cap A$ is a path of length 2, and then $S \cap B=\es$ in order to make Equation (<ref>) valid; $(\beta$) $S \cap A$ is a path of length $3$, say $S \cap A =\{(i,j),(j,k),(k,l)\}$; then by Condition C2 $(i,l)\in B$. No other pair of $S$ can be in $B$.
We now prove Theorem <ref>.
Independence of Conditions C1, C2, C4 and C5. Table <ref> provides, for each of the four conditions, two subsets $A$, $B$ of $\Arcs{n}$ which satisfy all conditions but the one considered. The elements $i$, $j$, $k$, $l$ involved are all distinct.
\begin{array}{c@{\quad}|@{\quad}c@{\quad}|@{\quad}c}
\text{Condition} & A & B \\
\hline
\text{C1} & \{(i,k),\;(i,\;l)\} & \es \\
\text{C2} & \{(i,\;j),\;(k,\;l)\} & \es \\
\text{C4} & \{(i,\;j),\;(j,\;k)\} & \es \\
\text{C5} & \{(i,\;j),\;(j,\;k),\;(k,\;l)\} & \{(i,l),\; (l,\;k),\;(k,\;j),(j,i)\}
\end{array}
The pairs $A$, $B$ used to prove independence of C1, C2, C4 and C5.
Necessity of Conditions C1, C2, C4 and C5. We have already proved in Theorem <ref> that C1, C2 and C4 are necessary, so we no turn to C5. We proceed by contradiction: if C5 does not hold at distinct elements $i$, $j$, $k$, $l$, we produce a semiorder on $\Vset{n}$ whose characteristic vector does not satisfy our primary linear inequality (<ref>). Let
\begin{align}
T &= \{t\in\Vset{n} \setminus \{i,j,k,l\} \st (i,t) \notin B \text{ and } (t,l) \notin B \};\\
U &= \{ u \in\Vset{n} \setminus \{i,j,k,l\} \st (i,u) \in B\};\\
V &= \{v \in\Vset{n} \setminus \{i,j,k,l\} \st (v,l) \in B \}.
\end{align}
Notice that $\Vset{n} = \{i,j,k,l\} \cup T \cup U \cup V$, a disjoint union which we display in Figure <ref>.
Because we assume that C5 does not hold, we have $(i,k)$, $(j,l)\notin B$. For the same reason, $(u,k) \notin B$ and $(u,l) \notin B$ for any $u$ in $U$. Similarly, for any $v$ in $V$, we have $(i,v) \notin B$ and $(j,v) \notin B$. Finally, we have $(u,v) \notin B$ for each $u$ in $U$ and each $v$ in $V$.
Figure <ref> displays the Hasse diagram of a semiorder $\RR$ which does not satisfy Equation (<ref>). To show that $R$ is indeed a semiorder, we indicate an interval representation of constant length $3$ by providing the initial endpoints of the representing intervals:
\begin{array}{c@{\qquad}c@{\qquad}c@{\qquad}c@{\quad}@{\quad}c@{\quad}cc}
i & j & k & l & t\in T & u \in U & v \in V\\[2mm]
0 & 4 & 8 & 12 & 6 & 3 & 9
\end{array}
The value at $\chi_{\RR}$ of the left-hand side of Equation (<ref>) equals at least 2, because the pairs $(i,j)$, $(j,k)$ and $(k,l)$ of $\RR$ contribute a $+1$ while only $(i,l)$ can contribute a $-1$.
vertex=[circle,draw,fill=white, scale=0.3]
(i) at (0,0) [vertex,label=right:$i$] ;
(j) at (0,1) [vertex,,label=left:$j$] ;
(k) at (0,2) [vertex,label=left:$k$] ;
(l) at (0,3) [vertex,label=right:$l$] ;
(i) – (j) – (k) – (l);
(t1) at (-2,1.5) [vertex] ;
at (-1.5,1.5)$\dots$;
(t2) at (-1,1.5) [vertex] ;
(i) – (t1) – (l);
(i) – (t2) – (l);
(u1) at (1,1) [vertex] ;
at (1.5,1)$\dots$;
(u2) at (2,1) [vertex] ;
(v1) at (1,2) [vertex] ;
at (1.5,2)$\dots$;
(v2) at (2,2) [vertex] ;
(u1) – (v1);
(u1) – (v2);
(u1) – (k);
(u2) – (v1);
(u2) – (v2);
(u2) – (k);
(j) – (v1);
(j) – (v2);
[rectangle, rounded corners] (-2.25,1.25) rectangle (-0.75,1.75);
at (-2.5,1.6) $T$;
[rectangle, rounded corners] (0.75,0.75) rectangle (2.25,1.25);
at (2.5,0.8) $U$;
[rectangle, draw, rounded corners] (0.75,1.75) rectangle (2.25,2.25);
at (2.5,2.2) $V$;
The Hasse diagram of the semiorder used in the proof of Theorem <ref>.
Sufficiency of Conditions C1, C2, C4 and C5. Assume that the four conditions are satisfied. We only need to show that any vertex of $\pso{n}$, in other words the characteristic vector $\chi_S$ of any semiorder $S$ on $\Vset{n}$, satisfies Equation (<ref>). Suppose to the contrary that the equation is not satisfied at $\chi_S$. Then we have one of the two cases $(\alpha)$ and $(\beta)$ in Lemma <ref>.
However, Case $(\alpha)$ cannot occur: by C4, we have either $(i,k)$ in $S \cap B$ or, for some $p$ in $\Vset{n}\setminus\{i,j,k\}$, we have $(i,p)$ or $(p,k)$ in $S \cap B$.
Case $(\beta)$ cannot neither occur because each requirement in C5 contradicts $S \cap B = \{(i,l)\}$. Let us verify the assertion for Requirement (v6), leaving (v1)–(v5) to the reader. Because $S$ is a semiorder, $(i,u)$ or $(u,k)$ is in $S$. In the first event, $(i,u)\in S \cap B$. In the second event, $(u,v)\in S \cap B$ or $(v,l)\in S \cap B$. We have reached a contradiction.
For every PC-graph $(\Vset{n},A)$ with $A \neq \Vset{n}$, there is a subset $B$ of $\Arcs{n}$ such that Inequality (<ref>) is valid: it suffices to take $B=\Arcs{n}\setminus A$. Of course, the latter choice gives in general a very “weak” inequality: taking a smaller choice for $B$ gives a “stronger” inequality. As a matter of fact, any “least possible” choice for $B$ produces a facet-defining inequality: this is the core of the next Theorem <ref>.
§ THE PRIMARY FACET-DEFINING INEQUALITIES OF THE SEMIORDER POLYTOPE
Theorem <ref> below provides a characterization of those primary linear inequalities which define facets of the polytope $\pso{n}$. Although unwieldy, the characterization directly leads to a polynomial-time algorithm for deciding whether a given primary linear inequality is facet-defining.
It is moreover a practical tool for explicitly building primary facet-defining inequalities of $\pso{n}$ (see Theorem <ref>) and even, in principle, for listing all of them.
To state Theorem <ref>, we first define in Example <ref> the `exceptional' inequalities. The example makes use of the following simple observation: any permutation of $\Vset n$ induces a permutation of coordinates in $\R^{\Arcs n}$, which leaves $\pso n$ invariant.
vertex=[circle,draw,fill=white, scale=0.3]
(1) at (0,1) [vertex] ;
(2) at (1,1) [vertex] ;
(3) at (1,0) [vertex] ;
(4) at (0,0) [vertex] ;
(1.west) node [above left] $1$;
(2.east) node [above right] $2$;
(3.east) node [below right] $3$;
(4.west) node [below left] $4$;
[->-=.7] (1) to [bend right=15] (2);
[->-=.7] (2) to [bend right=15] (3);
[->-=.7] (3) to [bend right=15] (4);
[->-=.7] (4) to [bend right=15] (1);
[->-=.7,dashed] (3) to [bend right=15] (2);
[->-=.7,dashed] (2) to [bend right=15] (1);
[->-=.7,dashed] (1) to [bend right=15] (4);
[->-=.7,dashed] (4) to [bend right=15] (3);
(5) at (0.25,0.5) [vertex] ;
(6) at (0.75,0.5) [vertex] ;
(5.west) node [left] $5$;
(6.east) node [right] $6$;
[->-=.7,dashed] (5) to (1);
[->-=.7,dashed] (5) to (2);
[->-=.7,dashed] (5) to (3);
[->-=.7,dashed] (5) to (4);
[->-=.5,dashed] (1) to (6);
[->-=.7,dashed] (2) to (6);
[->-=.7,dashed] (3) to (6);
[->-=.5,dashed] (4) to (6);
[->-=.7,dashed] (6) to (5);
(7) at (1.5,0.5) [vertex] ;
at (1.75,0.5) ...;
(n) at (2,0.5) [vertex] ;
(7.south) node [below] $7$;
(n.south) node [below] $n$;
A graphical representation of the exceptional inequality from Example <ref>.
[The exceptional inequality]
Assume $n\geqslant6$ and let (see also Figure <ref>)
\begin{eqnarray}\label{AB_6}
A_6 &=& \{(1,2),\; (2,3),\; (3,4),\; (4,1)\}, \\
B_6 &=& \{(2,1),\; (3,2),\; (4,3),\; (1,4), \\
&& \;\;(5,1),\; (5,2),\; (5,3),\; (5,4), \;(6,5), \\
&& \;\;(1,6),\; (2,6),\; (3,6),\; (4,6)\}.
\end{eqnarray}
A pair $A$, $B$ of subsets of $\Arcs n$ is exceptional when, possibly after a relabelling of the elements of $\Vset n$,
there hold $A=A_6$ and $B=B_6$. A primary linear inequality on $\R^{\Arcs n}$ is exceptional when, possibly after a relabelling of the elements of $\Vset n$,
it takes the form
\begin{equation}\label{eqn_6}
\sum_{a \in A_6} x_{a} - \sum_{b \in B_6}x_b \leqslant 1.
\end{equation}
All exceptional inequalities are valid for $\pso n$ (this results from Theorem <ref> or can be checked directly). None of them defines a facet, because the characteristic vector of any semiorder on $\Vset{n}$ containing the pair $(5,6)$ cannot give equality in Equation (<ref>), thus all vertices of $\pso n$ satisfying Equation (<ref>) with equality lie also in the face defined by $-x_{(5,6)} \leqslant 0$.
Let $A$ and $B$ be disjoint, nonempty subsets of $\Arcs{n}$. The primary linear inequality
\begin{equation}\label{eqn_general}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
defines a facet of $\pso n$ if and only if
* it is not exceptional;
* $A$, $B$ satisfy Conditions C1, C2, C4 and C5;
* for any $c$ in $B$, replacing $B$ with $B\setminus\{c\}$ makes at least one of the Conditions C1, C2, C4 and C5 becomes false.
When $A$ and $B$ are disjoint, nonempty subsets of $\Arcs{n}$, and moreover the condition after the “if and only if” in Theorem <ref> holds, we say that the set $B$ is $A$-minimal.
In view of Theorem <ref>, here is a rephrasing of the theorem (again assuming $A\neq \es \neq B$ and $A \cap B =\es$): Inequality (<ref>) is facet-defining for $\pso n$ if and only it is valid and not exceptional but, for any $c $ in $B$, the following inequality is not valid:
\begin{equation}
\label{eqn_general_A_B_minus_c}
\sum_{a \in A} x_{a} - \sum_{b \in B\setminus\{c\}}x_b \leqslant 1.
\end{equation}
Remember from Theorem <ref>.2 that for $A=\es$ or $A$ a singleton, Equation (<ref>) never gives a facet-defining inequality. For $B=\es$, it gives a facet-defining inequality exactly if $A=\{(i,j),(j,i)\}$ for some distinct elements $i$ and $j$. According to Theorem <ref>, the $n$-fence inequality from Example <ref> defines a facet of $\pso n$.
Why do we need to mention the exceptional inequalities in Theorem <ref>? Because they satisfy all the other conditions after the “if and only if”, but they do not define facets (as we saw in Example <ref>).
Necessity. When inequality (<ref>) defines a facet, Conditions C1, C2, C4 and C5 necessarily hold (Theorem <ref>). Working now by contradiction, assume that moreover (<ref>) is valid for some $c$ in $B$. By adding to the last inequality the valid inequality $-x_c \leqslant 0$, we get Inequality (<ref>). Hence the latter cannot be facet-defining, a contradiction.
Sufficiency. If Conditions C1, C2, C4 and C5 hold, then by Theorem <ref> inequality (<ref>) is valid for $\pso n$. To prove moreover that (<ref>) defines a facet, we also assume that (<ref>) is not exceptional and that $B$ is $A$-minimal. For (<ref>) to be non-valid, $|A|$ must be at least $2$. If $|A| = 2$, our assumptions imply that (<ref>) must be one of the four Axiomatic Inequalities (<ref>)–(<ref>), which we know to be facet-defining (Theorem <ref>).
From now on we suppose $|A| \geqslant 3$. By the Lifting Lemma (Theorem <ref>), we may moreover assume that any element from $\Vset{n}$ appears in at least one pair in $A \cup B$.
Remember $\dim \pso{n}=n(n-1)$. To show that the Inequality (<ref>) is facet-defining, we will produce $n(n-1)$ semiorders $S_{ij}$, one for each pair $(i,j)$ in $\Arcs{n}$, in such a way that their characteristic vectors are affinely independent and satisfy (<ref>) with equality.
Any pair $(i,j)$ in $\Arcs{n}$ is of exactly one of the following six types (the first type covers the situation where $(i,j)\in A$, the second, third and fifth $(i,j)\notin A \cup B$, the fourth and the sixth $(i,j)\in B$). Figure <ref> illustrates the four first types.
vertex=[circle,draw,fill=white, scale=0.3]
(i1) at (0,0) [vertex,label=left:$i$] ;
(j1) at (1,1) [vertex,,label=left:$j$] ;
[->-=.7] (i1) – (j1);
(i2) at (0,0) [vertex,label=left:$i$] ;
(j2) at (-1,1) [vertex,,label=left:$j$] ;
(l2) at (1,1) [vertex,label=right:$l$] ;
[->-=.7] (i2) – (l2);
(i3) at (-1,0) [vertex,label=left:$i$] ;
(j3) at (0,1) [vertex,,label=left:$j$] ;
(k3) at (1,0) [vertex,label=right:$k$] ;
[->-=.7] (k3) – (j3);
(i4) at (0,0) [vertex,label=left:$i$] ;
(j4) at (1,1) [vertex,label=right:$j$] ;
(l4) at (0,1) [vertex,,label=left:$l$] ;
(k4) at (1,0) [vertex,label=right:$k$] ;
[->-=.7,dashed] (i4) – (j4);
[->-=.7] (k4) – (j4);
[->-=.7] (i4) – (l4);
of pairs $(i,j)$ in the proof of Theorem <ref>.
$(i,j) \in A$. We then take the semiorder $S_{ij}=\{(i,j)\}$.
$(i,j) \notin A \cup B$ but $(i,l) \in A$ for some $l$ in $\Vset{n}$. By Condition C1, there can be only one such element $l$. We let $S_{ij}=\{(i,j), (i,l)\}$.
$(i,j) \notin A \cup B$, $(i,l)\in A$ for no element $l$ and $(k,j) \in A$ for some element $k$ in $\Vset{n}$ (such a $k$ is unique by Condition C1). We let $S_{ij}=\{(i,j), (k,j)\}$.
$(i,j) \in B$ and there exist some pairs $(i,l)$, $(k,j)$ in $A$ with $k\neq l$. Notice that the latter pairs are unique. We then let $S_{ij}=\{(i,l), (k,j), (i,j)\}$.
$(i,j) \notin A \cup B$ and $(i,j)$ is not of . This implies $(i,u), (v,j)\notin A$ for all $u$, $v$ in $\Vset{n}$. Because of $|A| \geqslant 3$ and of Conditions C1, there must exist $k$ and $l$ in $\Vset{n}\setminus\{i,j\}$ such that $(k,l) \in A$.
Suppose first that it is possible to select such a pair $(k,l)$ with moreover $(i,l) \notin B$ or $(k,j) \notin B$. We then take the semiorder $S_{ij}=\{(i,j), (k,l), z\}$, the pair $z$ being $(i,l)$ in the first case and $(k,j)$ in the second one.
Suppose next no such choice of $(k,l)$ exists, that is $(k,j), (i,l) \in B$ for each pair $(k,l)$ in $A$ with $k,l \notin \{i,j\}$.
In this case, we claim that $A$ equals $\{(j,k),$ $(k,l),$ $(l,i)\}$ and that $B$ is either equal to $\{(j,i),$ $(i,l),$ $(l,k),$ $(k,j),$ $(j,t),$ $(t,i)\}$, for some $t \in \Vset{n} \setminus \{i,j,k,l\}$,
or to $\{(j,i)$, $(i,l)$, $(l,k)$, $(k,j)$, $(j,u)$, $(u,v)$, $(v,i)\}$, for some $u$, $v \in \Vset{n} \setminus \{i,j,k,l\}$, with $u$ and $v$ distinct; moreover, in both cases, the resulting inequality is facet-defining. The proof of the latter assertions being long, we defer them to Lemmas <ref> and <ref>.
$(i,j)$ is of none of the previous . Then necessarily $(i,j) \in B$.
By our basic assumption, $A$ and $B \setminus \{(i,j)\}$ do not satisfy Conditions C1, C2, C4 or C5. Notice that Conditions C1 remains true (because $A$ is not modified) and Condition C2 also because $(i,j)$ is not of .
Now the inequality
\begin{equation}
\label{eqn_general_A_B_minus_(i,j)}
\sum_{a \in A} x_{a} - \sum_{b \in B\setminus\{(i,j)\}}x_b \leqslant 1
\end{equation}
is not valid for $\pso n$, and so by Lemma <ref> there exists some semiorder $S$ on $\Vset{n}$ such that one of the two following holds:
$\alpha$) $S \cap A$ is a path of length $2$, and $S \cap (B \setminus\{(i,j)\}) = \es$. Then we must have $S \cap B = \{(i,j)\}$, so we attach the semiorder $S$ to the pair $(i,j)$.
$\beta$) $S \cap A$ is a path of length 3, say $S \cap A =\{(u,v),(v,k),(k,l)\}$, and moreover $S \cap (B \setminus\{(i,j)\}) = \{(u,l)\}$. Notice that the pair $(u,l)$ is of . Also, $S \cap B = \{(u,l),(i,j)\}$. We attach the semiorder $S$ to the pair $(i,j)$.
At this point, we have attached to any pair $(i,j)$ in $\Arcs{n}$ some semiorder $S_{ij}$ on $\Vset{n}$—except in the singular situation as in the second part of , for which the conclusion results from the following two lemmas. The characteristic vectors of all the $n(n-1)$ semiorders $S_{ij}$ satisfy Equation (<ref>) with equality.
They are moreover affinely independent. Indeed, the semiorder $S_{ij}$ contains the pair $(i,j)$ while all previously constructed semiorders do not contain that particular pair $(i,j)$; in other words, the characteristic vector of $S_{ij}$ has a $1$ in component $(i,j)$ while all the previous characteristic vectors have a $0$.
The two lemmas below complete the handling of in the previous proof.
Assume that $A$ and $B$ are disjoint subsets of $\Arcs{n}$ which satisfy Conditions C1 and C2. If $A$ contains pairs $(i,j)$, $(j,k)$ and $(u,v)$, $(v,w)$ such that
\{(i,j), (j,k)\} \cap \{(u,v), (v,w)\} = \varnothing
then $A$ and $B$ satisfy also Condition C4 at the given elements $i$, $j$ and $k$. Similarly, if $A$ contains pairs $(i,j)$, $(j,k)$, $(k,l)$ and $(u,v)$, $(v,w)$ such that
\{(i,j), (j,k), (k,l)\} \cap \{(u,v), (v,w)\} = \varnothing
then $A$ and $B$ satisfy also Condition C5 at the given elements $i$, $j$, $k$ and $l$.
By Condition C1, $(\Vset{n},A)$ is a PC-graph. So $(i,j)$ and $(u,v)$ are disjoint, as well as $(j,k)$ and $(v,w)$. Condition C3 then implies that $(i,v)$ and $(v,k)$ are in $B$; this establishes Condition C4 at $i$, $j$ and $k$. The proof of the second assertion is similar.
Consider nonempty, disjoint subsets $A$ and $B$ of $\Arcs{n}$ which satisfy Conditions C1, C2, C4 and C5, with moreover $|A|\geqslant 3$, $B$ being $A$-minimal and $A$, $B$ not exceptional (in the sense of Example <ref>). Suppose that there exists $(i,j)$ in $\Arcs{n}\setminus(A \cup B)$ such that
$(i,p)\notin A$ for all $p$ in $\Vset{n}$;
$(q,j)\notin A$ for all $q$ in $\Vset{n}$;
$(k,j), (i,l) \in B$ for each pair $(k,l)$ in $A$ disjoint from $(i,j)$.
Then there exist $k$, $l \in \Vset{n}$ with $i$, $j$, $k$, $l$ pairwise distinct such that
A = \{(j,k), (k,l), (l,i)\}
and either of two cases:
* $B = \{(j,i),$ $(i,l),$ $(l,k),$ $(k,j),$ $(j,t),$ $(t,i)\}$ for some $t \in \Vset{n} \setminus \{i,j,k,l\}$,
* $B = \{(j,i)$, $(i,l)$, $(l,k)$, $(k,j)$, $(j,u),$ $(u,v),$ $(v,i)\}$ for some distinct $u$, $v$ in $\Vset{n} \setminus \{i,j,k,l\}$.
In both cases, Inequality (<ref>) defines a facet of $\pso n$.
The specific inequalities in the two cases are valid in view of Theorem <ref>. To prove that they define facets, it suffices to exhibit respectively $5\cdot4 = 20$ and $6\cdot5 = 30$ semiorders with affinely independent characteristic vectors satisfying (<ref>) with equality: in view of Theorem <ref>, it suffices to work with $\Vset{n}$ equal to $\{i$, $j$, $k$, $l$, $t\}$ or $\{i$, $j$, $k$, $l$, $u$, $v\}$, respectively. We leave this to the reader.
We now show that if $A$ and $B$ satisfy the assumptions, then they are of one of the two latter forms. Because of $|A| \geqslant 3$ and Conditions C1 in Theorem <ref>, there must exist $k$ and $l$ in $\Vset{n}\setminus\{i,j\}$ such that $(k,l) \in A$.
First, let us infer the existence of some element $s$ in $\Vset{n}\setminus\{j,k,l\}$ such that $(l,s)\in A$.
If no such element $s$ exists in $A$ we derive, from the present Assumption (II) together with our $A$-minimality assumption (applied to the pair $(k,j)$ in $B$), that $(k,j)$ cannot be anything else than a pair as $(u,v)$ in Condition C5: as shown in Figure <ref>, there exist distinct elements $w$, $x$, $y$, $z$ in $\Vset{n}\setminus\{k,j\}$ such that the pairs $(w,x)$, $(x,y)$, $(y,z)$ are in $A$ and $(w,k)$, $(j,z)$ in $B$.
vertex=[circle,draw,fill=white, scale=0.3]
(i) at (-1,0) [vertex,label=left:$i$] ;
(j) at (0,1) [vertex,label=left:$j$] ;
(k) at (0,0) [vertex,label=left:$k$] ;
(l) at (-1,1) [vertex,label=left:$l$] ;
(w) at (1,-1) [vertex,label=right:$w$] ;
(x) at (2,0) [vertex,label=right:$x$] ;
(y) at (2,1) [vertex,label=right:$y$] ;
(z) at (1,2) [vertex,label=right:$z$] ;
[->-=.7] (k) – (l);
[->-=.7,dashed] (k) – (j);
[->-=.7,dashed] (i) – (l);
[->-=.7,dashed] (w) – (k);
[->-=.7,dashed] (j) – (z);
[->-=.7] (w) – (x);
[->-=.7] (x) – (y);
[->-=.7] (y) – (z);
First illustration for the proof of Lemma <ref>.
Moreover, we must have $(k,l)$ and $(x,y)$ disjoint. Then by Condition C1, $(k,y) \in B$. This shows that the deletion of $(k,j)$ never invalidates Condition C5 (whatever the choices of $w$, $x$, $y$ and $z$), a contradiction with our present assumption in the statement. We conclude that for some element $s$ in $\Vset{n}\setminus \{j,k,l\}$ the pair $(l,s)$ is in $A$.
Similarly (this time because $B\setminus\{(i,l)\}$ is $A$-minimal), there is some $r$ in $\Vset{n}\setminus \{i,k,l\}$ such that $(r,k) \in A$ (see Figure <ref>).
vertex=[circle,draw,fill=white, scale=0.3]
(i) at (-1,0) [vertex,label=left:$i$] ;
(j) at (0,1) [vertex,label=right:$j$] ;
(k) at (0,0) [vertex,label=right:$k$] ;
(l) at (-1,1) [vertex,label=left:$l$] ;
(r) at (0,-1) [vertex,label=right:$r$] ;
(s) at (-1,2) [vertex,label=left:$s$] ;
[->-=.7] (k) – (l);
[->-=.7,dashed] (k) – (j);
[->-=.7,dashed] (i) – (l);
[->-=.7] (r) – (k);
[->-=.7] (l) – (s);
Second illustration for the proof of Lemma <ref>. Here the following equalities may occur: $i=s$, $j=r$, $r=s$.
Again by our present assumptions on the pair $(i,j)$, we know $r \neq i$ and $s \neq j$. However $r$ might be equal to $j$ and $s$ might be equal to $i$. We could also have $r=s$ in which case the two previous equalities cannot hold together.
Suppose first that both equalities $r=j$ and $s=i$ hold (as in Figure <ref>).
vertex=[circle,draw,fill=white, scale=0.3]
(j) at (0,0) [vertex,label=left:$j=r$] ;
(k) at (0,1) [vertex,label=left:$k$] ;
(l) at (0,2) [vertex,label=left:$l$] ;
(i) at (0,3) [vertex,label=left:$i=s$] ;
[->-=.7] (j) – (k);
[->-=.7] (k) – (l);
[->-=.7] (l) – (i);
[->-=.7, dashed] (i) to [bend left] (l);
[->-=.7, dashed] (k) to [bend left] (j);
Third illustration for the proof of Lemma <ref>: when $i=s$ and $j=r$.
We prove that the Lemma holds by first etablishing $A = \{(j,k), (k,l), (l,i)\}$. Suppose that $(x,y)$ is another pair of $A$. Then $(i,y) \in B$ by Assumption (III) and we will prove that replacing $B$ with $B \setminus \{(i,y)\}$ leaves a valid inequality, a contradiction with the $A$-minimality of $B$. If replacing $B$ with $B \setminus \{(i,y)\}$ gives a nonvalid inequality, then either Condition C4 is not satisfied or Condition C5 is not satisfied by $A$ and $B \setminus \{(i,y)\}$.
If Condition C4 is not, then $(w,x) \in A$ for some $w$ distinct from $x,$ $y,$ $j,$ $k,$ $l,$ $i$. But this is impossible since then $(w,l)$ and $(l,y)$ are in $B$ by Condition C2, and hence Condition (C4) is satisfied by $A$ and $B \setminus \{(i,y)\}$..
So if $B \setminus \{(i,y)\}$ gives a nonvalid inequality, then Condition C5 is not satisfied. But then the three consecutive pairs of this condition must be all disjoint from $(j,k)$, $(k,l)$ and $(l,i)$, a contradiction by Lemma <ref>. This proves $A = \{(j,k)$, $(k,l)$, $(l,i)\}$.
Next, remember $(i,l)$, $(k,j)\in B$. Moreover, by Condition C2, $(j,i)$, $(l,k) \in B$. We now apply Condition C5 to the pairs $(j,k),$ $(k,l),$ $(l,i)$ to deduce that we are in one of the two cases of the statement of the lemma we are proving. Actually, these two cases correspond to Requirements (v5) and (v6) in Condition C5. Hence, we only have to show that if Requirements (v1) to (v4) are satisfied we have a contradiction with the $A$-minimality of $B$. Suppose Requirement (v1) or Requirement (v3) is satisfied. Then $B$ is not $A$-minimal, because replacing $B$ with $B \setminus \{(i,l)\}$ still preserves Conditions C1, C2, C4, C5. The argument is similar for Requirements (v2) and (v4), with the pair $(k,j)$ of $B$. By the $A$-minimality of $B$, there is no further pair in $B$. This concludes the proof when $r=j$ and $s=i$.
If $r \neq j$ or $s \neq i$, the situation must be as in one of the cases of Figure <ref>. We deduce a contradiction between Assumption (III) and the $A$-minimality of $B$ in each case.
vertex=[circle,draw,fill=white, scale=0.3]
at (-0.2,3.7) Case 1;
(z_r) at (0,0) [vertex] ;
(z_k) at (0,1) [vertex] ;
(z_l) at (0,2) [vertex] ;
(z_i) at (0,3) [vertex] ;
(z_j) at (1,2) [vertex] ;
[->-=.7] (z_r) to (z_k);
[->-=.7] (z_k) to (z_l);
[->-=.7] (z_l) to (z_i);
[->-=.7, dashed] (z_k) to (z_j);
[->-=.7, dashed, bend left] (z_i) to (z_l);
(z_r.west) node [left] $r$;
(z_k.west) node [left] $k$;
(z_l.west) node [left] $l$;
(z_i.west) node [left] $s=i$;
(z_j.north) node [right] $j$;
at (-0.4,3.7) Case 2;
(z_r) at (-1,0) [vertex] ;
(z_k) at (-1,2) [vertex] ;
(z_l) at (0,2) [vertex] ;
(z_i) at (1,0) [vertex] ;
(z_j) at (0,0) [vertex] ;
[->-=.7] (z_r) to (z_k);
[->-=.7] (z_k) to (z_l);
[->-=.7] (z_l) to (z_r);
[->-=.7, dashed] (z_k) to (z_j);
[->-=.7, dashed] (z_i) to (z_l);
(z_r) node [below] $r = s$;
(z_k) node [above] $k$;
(z_l) node [above] $l$;
(z_i) node [below] $i$;
(z_j) node [below] $j$;
at (0,3.7) Case 3;
(z_s) at (0,3) [vertex] ;
(z_l) at (0,2) [vertex] ;
(z_k) at (0,1) [vertex] ;
(z_r) at (0,0) [vertex] ;
(z_i) at (-1,1) [vertex] ;
(z_j) at (1,2) [vertex] ;
[->-=.7] (z_r) to (z_k);
[->-=.7] (z_k) to (z_l);
[->-=.7] (z_l) to (z_s);
[->-=.7, dashed] (z_k) to (z_j);
[->-=.7, dashed] (z_i) to (z_l);
(z_r.west) node [left] $r$;
(z_k.west) node [left] $k$;
(z_l.west) node [left] $l$;
(z_s.west) node [left] $s$;
(z_j.east) node [right] $j$;
(z_i.east) node [left] $i$;
at (-0.5,3.7) Case 4;
(z_s) at (0,3) [vertex] ;
(z_l) at (0,2) [vertex] ;
(z_k) at (0,1) [vertex] ;
(z_r) at (0,0) [vertex] ;
(z_i) at (-1,1) [vertex] ;
[->-=.7] (z_r) to (z_k);
[->-=.7] (z_k) to (z_l);
[->-=.7] (z_l) to (z_s);
[->-=.7, dashed, bend left] (z_k) to (z_r);
[->-=.7, dashed] (z_i) to (z_l);
(z_r.west) node [left] $r=j$;
(z_k.west) node [left] $k$;
(z_l.east) node [right] $l$;
(z_s.east) node [right] $s$;
(z_j.east) node [right] $j$;
(z_i.west) node [left] $i$;
Fourth illustration for the proof of Lemma <ref>: the four cases where $s \neq i$ or $r \neq j$.
By Assumption (III), the pair $(i,l)$ belongs to $B$; when $r \neq j$, the same assumption (after replacement of $(k,l)$ with $(r,k)$) gives $(r,j)\in B$.
Consider first Case 2 of Figure <ref>. We know that $B \setminus \{(r,j)\}$ is not valid. Either Condition C4 or Condition C5 is not satisfied by $B \setminus \{(r,j)\}$. By Lemma <ref>, $A$ does not contain two consecutive pairs $(u,v), (v,w)$ that are disjoint from $(r,k)$ and $(k,l)$
(this would imply that $B \setminus \{(r,j)\}$ still satisfies Conditions C1, C2, C4, C5, a contradiction). Hence $A$ does not contain three consecutive pairs $(u,v), (v,w), (w,z)$ with $u,v,w,z$ pairwise distinct, and we know that it is Condition C4 that is not satisfied by $B \setminus \{(r,j)\}$. This implies then that $(j,l) \in B$ and then $B \setminus \{(i,l)\}$ is still valid. This gives the desired contradiction.
Cases 1 and 4 are similar, so we only treat Case 4. We claim that $B \setminus \{(l,j)\}$ is still valid (here, $(l,j)\in B$ because of Assumption (III) and $(l,s)\in A$). If not, because $A$ and $B \setminus \{(l,j)\}$ satisfy Condition C4, it must be Condition C5 which is not satisfied and there exist pairs $(u,v)$, $(v,w)$, $(w,x)$ in $A$. These pairs must be different from $(j,k)$ and $(k,l)$ and hence Condition C5 must in fact be satisfied for $\{(u,v)$, $(v,w)$, $(w,x)\}$, a contradiction.
Finally, consider Case 3. Exactly as we derived in the beginning of the present proof the existence of the element $s$ from the pair $(k,l)$ in $A$, we derive the existence of an element $t$ with $(s,t)\in A$ from the pair $(l,s)$ in $A$.
Here is a general argument which we will be using several times. Assume $A'$ and $B'$ satisfy Conditions C1 and C2, where we write primes to make the distinction with our present notation. Consider in Condition C5 the given, distinct elements $i'$, $j'$, $k'$ and $l'$ and also a pair $(u',v')$ whose existence is asserted in Requirement (v6). If for some element $p'$ in $\Vset{n}\setminus\{j',k'\}$ we have $(u',p')$ in $A'$, then Requirement (v3) holds (because $(u',p')$ and $(j',k')$ are disjoint pairs in $A'$). Similarly, if for some element $q'$ in $\Vset{n}\setminus\{j',k'\}$ we have $(q',v')$ in $A'$, then Requirement (v4) is met.
If $t\neq r$ we derive that $A$ and $B\setminus\{(i,l)\}$ satisfy Conditions C1, C2, C4 and C5, in contradiction with our assumptions. Indeed, $A$ and $B$ satisfy Conditions C1, C2, C4 and C5; by Assumption (I) and $(r,s)$, $(s,l)\in B$, we see that $A$ and $B\setminus\{(i,l)\}$ can only invalidate Condition C5 with $(i,l)$ as $(u,v)$ in Requirement (v6). But the general argument from previous paragraph with $(k,l)\in A$ rules this out.
If $t=r$, the pairs $(r,k)$, $(k,l)$, $(l,s)$, $(s,r)$ form a cycle of length $4$. Assumptions (I) and (II) imply $(i,p)$, $(p,j)\in B$ for any $p$ in $\{r$, $k$, $l$, $s\}$. Thus we may take advantage of the cyclic symmetry w.r.t. $r$, $k$, $l$, $s$.
Notice also that $(l,k)$, $(k,r)$, $(r,s)$ and $(s,l)$ must be in $B$ because of Condition C2.
We may not have $(p,i)$ in $B$ (for any $p$ in $\{r$, $k$, $l$, $s\}$), because otherwise $A$ and $B \setminus \{(p,j)\}$ still satisfy Conditions C1, C2, C4 and C5 (use the general argument just above to check that $(p,j)$ is not as $(u,v)$ in Requirement (v6); then, if for instance $p=k$, use Assumption (II) and $(k,i)$, $(i,s)\in B$ to check that all Conditions C1, C2, C4, C5 still hold). Similarly, we may not have $(j,p)$ in $B$.
A similar argument shows $(s,k) \notin B$ (and by symmetry, $(r,l)$, $(k,s)$, $(l,r)\notin B$).
Next, consider the pair $(i,l)$. By assumption $B$ is $A$-minimal, thus $A$ and $B\setminus\{(i,l)\}$ must invalidate Conditions C1, C2, C4 or C5. However, from Assumption (I) and all the pairs we have obtained in $A$ and $B$ this is impossible except if $(i,l)$ is as $(v,l)$ in Requirement (v6). Thus there exists $u$ in $\Vset{n}\setminus\{s$, $r$, $k$, $l\}$ such that $(s,u)$, $(u,i) \in B$. Similarly, $A$ and $B\setminus\{(s,j)\}$ must invalidate Conditions C1, C2, C4 or C5, and this can only occur with the existence of $v$ in $\Vset{n}\setminus\{s$, $r$, $k$, $l\}$ such that $(j,v)$, $(v,l) \in B$. Now if $v\neq i$ or $u \neq j$, we see that we cannot invalidate Requirement (v6) both times,
a contradiction which completes Case 3.
So we are left with $u=j$ and $v=i$. Notice that because of Assumption (III), there cannot be any further pair $(x,y)$ in $A$ (because $A$ and $B\setminus\{(i,y)\}$ would not invalidate Conditions C1, C2, C4, C5). So we arrive at the exceptional example, (there may be isolated elements, not appearing in any pair of $A$ or of $B$).
Condition C1 implies that for a valid inequality
\begin{equation}\label{eqn_general_valid_again}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1
\end{equation}
the graph $(\Vset{n}, A)$ is a PC-graph, that is, its components are isolated vertices, directed paths or directed cycles. For many of the structural forms of the PC-graph $(\Vset{n}, A)$, there is only one set $B$ such that Equation (<ref>) provides a facet-defining inequality. We say that a pair $(i,l)$ in $\Vset{n}$ is $A$[C2]-forced if there exist elements $j$ and $k$ such that $i$, $j$, $k$ and $l$ are distinct and moreover $(i,j)$ and $(k,l)$ are in $A$.
Thus the $A$[C2]-forced pairs $(i,l)$ are exactly those that Condition C2 forces to be in $B$.
Consider for disjoint, nonempty subsets $A$ and $B$ of $\Arcs{n}$, the inequality
\begin{equation}\label{eqn_general_once_more}
\sum_{a \in A} x_{a} - \sum_{b \in B}x_b \leqslant 1,
\end{equation}
and suppose that the graph $(\Vset{n}, A)$ is a PC-graph whose components meet at least one of the following six assumptions:
* at least one component contains two opposite pairs (of elements) in $A$ (and thus no other pair);
* at least two components contain at least two pairs in $A$;
* at least one component contains at least six pairs in $A$;
* all nontrivial components contain exactly one pair of $A$.
Then Equation (<ref>) describes a facet-defining inequality for $\pso n$ if and only if $B$ consists exactly of the $A$[C2]-forced pairs.
First, remember that $(\Vset{n}, A)$ is a PC-graph exactly if Condition C1. Next, we note that if the PC-graph $(\Vset{n}, A)$ satisfies one of the assumptions 1–6, then the validity of Condition C2 implies the validity of Conditions C4 and C5 (this is easily established). Then in the present situation Theorem <ref> entails that Equation (<ref>) gives a facet-defining inequality if and only if $B$ contains the $A$[C2]-forced pairs but no other pair.
To obtain a full list of all the primary facet-defining inequalities of $\pso{n}$, it remains to investigate the pairs $A$ and $B$ as in Theorem <ref> for which the PC-graph $(\Vset{n}, A)$ does not satisfy any of the six assumptions of Theorem <ref>. Such a PC-graph has exactly one nontrivial component which contains more than one pair in $A$, and the number of its pairs equals $2$, $3$, $4$ or $5$ (moreover, in the first case the two pairs are not opposite). Besides the $A$[C2]-forced pairs, $B$ must contain at least one other pair in order to make Conditions C4 and C5 valid. For any given value of $n$, it is in principle possible to list all such possible pairs $A$ and $B$ (say up to relabelling of elements). However, even for small values of $n$, the task becomes quite tedious and from $n=6$ the number of examples is daunting.
§ SEARCHING FOR THE PRIMARY FACET-DEFINING INEQUALITIES OF THE STRICT WEAK ORDER POLYTOPE
The strict weak order polytope $\pswo n$ <cit.> is closely linked to still another polytope, the weak order polytope $\pwo n$ <cit.>. A weak order (or complete preorder) on $\Vset{n}$ is a relation which is reflexive, transitive and total. Thus the asymmetric part of a weak order is a strict weak order. The weak order polytope is defined (again in $\R^{\Arcs{n}}$) by
{\pwo n} = \conv \left(\left\{ \chi^{W} \st W \text{ is a weak order on } \Vset n \right\}\right).
In fact, the polytopes ${\pwo n}$ and ${\pswo n}$ are mutual images by the symmetry in the point $(\frac{1}{2}, \ldots, \frac{1}{2})$ (because the complement of a strict weak order is a weak order, and reciprocally). Thus the inequality
\begin{equation}\label{eqn_preorder_bis}
\sum_{(i,j) \in \Arcs{n}} \alpha_{ij}x_{ij} \leqslant \beta
\end{equation}
is facet-defining (resp. valid) for ${\pswo n}$ if and only if
\begin{equation}\label{eqn_preorder}
\sum_{(i,j) \in \Arcs{n}} -\alpha_{ij}x_{ij} \leqslant \beta - \sum_{(i,j) \in \Arcs{n}} \alpha_{ij}
\end{equation}
is facet-defining (resp. valid) for ${\pwo n}$.
Our focus here is on primary FDIs for $\pswo n$. Written as Equation (<ref>), such a primary FDI arises from an FDI of $\pwo n$ as in (<ref>) with $\alpha_{ij} \in \{-1,0,1\}$ and $\beta - \sum_{(i,j) \in \Arcs{n}} \alpha_{ij}\in\{-1,0,1\}$.
A search in <cit.> and <cit.> led to only nine primary FDIs of $\pswo n$, which we represent in Figure <ref> (adhering to our usual conventions for such figures, also adding the value of the independent term).
vertex=[circle,draw,fill=white, scale=0.3]
(1) at (0,0) [vertex] ;
(2) at (0,1) [vertex] ;
[->-=.7,dashed] (1) to (2);
at (-0.5,0.75) $F_1$;
at (0.5,0.25) $\leqslant 0$;
(2-1) at (0,0) [vertex] ;
(2-2) at (0,1) [vertex] ;
[->-=.7,>= triangle 45] (2-1) to [bend right=15] (2-2);
[->-=.7,>= triangle 45] (2-2) to [bend right=15] (2-1);
at (-0.5,0.75) $F_2$;
at (0.5,0.25) $\leqslant 1$;
(3-1) at (0.45,0) [vertex] ;
(3-2) at (-0.45,0) [vertex] ;
(3-3) at (0,1) [vertex] ;
[->-=.7,>= open triangle 45,dashed] (3-1) to (3-2);
[->-=.7,>= open triangle 45,dashed] (3-2) to (3-3);
[->-=.7,>= triangle 45] (3-1) to (3-3);
at (-0.5,0.5) $F_3$;
at (0.75,0.25) $\leqslant 0$;
(4-1) at (1,0) [vertex] ;
(4-2) at (0,1) [vertex] ;
(4-3) at (-1,0) [vertex] ;
(4-4) at (0,-1) [vertex] ;
[->-=.7,>= open triangle 45,dashed] (4-2) to [bend right=15] (4-3);
[->-=.7,>= open triangle 45,dashed] (4-3) to [bend right=15] (4-2);
[->-=.7,>= open triangle 45,dashed] (4-1) to [bend right=15] (4-3);
[->-=.7,>= open triangle 45,dashed] (4-3) to [bend right=15] (4-1);
[->-=.7,>= open triangle 45,dashed] (4-3) to [bend right=15] (4-4);
[->-=.7,>= open triangle 45,dashed] (4-4) to [bend right=15] (4-3);
[->-=.7] (4-1) to [bend right=15] (4-2);
[->-=.7] (4-2) to [bend right=15] (4-1);
[->-=.7] (4-2) to [bend right=15] (4-4);
[->-=.7] (4-4) to [bend right=15] (4-2);
[->-=.7] (4-1) to [bend right=15] (4-4);
[->-=.7] (4-4) to [bend right=15] (4-1);
at (-1.25,0.75) $F_4$;
at (1,-0.75) $\leqslant 0$;
(5-1) at (1,0) [vertex] ;
(5-2) at (0,1) [vertex] ;
(5-3) at (-1,0) [vertex] ;
(5-4) at (0,-1) [vertex] ;
[->-=.7,>= open triangle 45,dashed] (5-1) to (5-2);
[->-=.7,>= open triangle 45,dashed] (5-3) to (5-2);
[->-=.7,>= open triangle 45,dashed] (5-4) to (5-1);
[->-=.7,>= open triangle 45,dashed] (5-4) to (5-3);
[->-=.7,>= open triangle 45,dashed] (5-4) to [bend right=15] (5-2);
[->-=.7] (5-1) to [bend right=15] (5-3);
[->-=.7] (5-3) to [bend right=15] (5-1);
[->-=.7] (5-2) to [bend right=15] (5-4);
at (-1.25,0.75) $F_5$;
at (1,-0.75) $\leqslant 0$;
(6-1) at (1,0) [vertex] ;
(6-2) at (0,1) [vertex] ;
(6-3) at (-1,0) [vertex] ;
(6-4) at (0,-1) [vertex] ;
[->-=.7,>= open triangle 45,dashed] (6-1) to [bend right=15] (6-2);
[->-=.7,>= open triangle 45,dashed] (6-2) to [bend right=15] (6-1);
[->-=.7,>= open triangle 45,dashed] (6-1) to [bend right=15] (6-4);
[->-=.7,>= open triangle 45,dashed] (6-4) to [bend right=15] (6-1);
[->-=.7,>= open triangle 45,dashed] (6-1) to (6-3);
[->-=.7] (6-2) to [bend right=15] (6-4);
[->-=.7] (6-4) to [bend right=15] (6-2);
[->-=.7] (6-2) to (6-3);
[->-=.7] (6-4) to (6-3);
at (-1.25,0.75) $F_6$;
at (1,-0.75) $\leqslant 1$;
(7-1) at (1,0) [vertex] ;
(7-2) at (0,1) [vertex] ;
(7-3) at (-1,0) [vertex] ;
(7-4) at (0,-1) [vertex] ;
[->-=.7,>= open triangle 45,dashed] (7-1) to [bend right=15] (7-2);
[->-=.7,>= open triangle 45,dashed] (7-2) to [bend right=15] (7-1);
[->-=.7,>= open triangle 45,dashed] (7-1) to [bend right=15] (7-4);
[->-=.7,>= open triangle 45,dashed] (7-4) to [bend right=15] (7-1);
[->-=.7,>= open triangle 45,dashed] (7-3) to (7-2);
[->-=.7,>= open triangle 45,dashed] (7-3) to (7-4);
[->-=.7] (7-2) to [bend right=15] (7-4);
[->-=.7] (7-4) to [bend right=15] (7-2);
[->-=.7] (7-3) to (7-1);
at (-1.25,0.75) $F_7$;
at (1,-0.75) $\leqslant 0$;
(8-1) at (1,0) [vertex] ;
(8-2) at (0,1) [vertex] ;
(8-3) at (-1,0) [vertex] ;
(8-4) at (0,-1) [vertex] ;
[->-=.7,>= open triangle 45,dashed] (8-2) to [bend left=15] (8-1);
[->-=.7,>= open triangle 45,dashed] (8-1) to [bend left=15] (8-2);
[->-=.7,>= open triangle 45,dashed] (8-4) to [bend left=15] (8-1);
[->-=.7,>= open triangle 45,dashed] (8-1) to [bend left=15] (8-4);
[->-=.7,>= open triangle 45,dashed] (8-3) to (8-1);
[->-=.7] (8-2) to [bend left=15] (8-4);
[->-=.7] (8-4) to [bend left=15] (8-2);
[->-=.7] (8-3) to (8-2);
[->-=.7] (8-3) to (8-4);
at (-1.25,0.75) $F_8$;
at (1,-0.75) $\leqslant 1$;
(9-1) at (1,0) [vertex] ;
(9-2) at (0,1) [vertex] ;
(9-3) at (-1,0) [vertex] ;
(9-4) at (0,-1) [vertex] ;
[->-=.7,>= open triangle 45,dashed] (9-1) to [bend left=15] (9-2);
[->-=.7,>= open triangle 45,dashed] (9-2) to [bend left=15] (9-1);
[->-=.7,>= open triangle 45,dashed] (9-1) to [bend left=15] (9-4);
[->-=.7,>= open triangle 45,dashed] (9-4) to [bend left=15] (9-1);
[->-=.7,>= open triangle 45,dashed] (9-2) to (9-3);
[->-=.7,>= open triangle 45,dashed] (9-4) to (9-3);
[->-=.7] (9-4) to [bend left=15] (9-2);
[->-=.7] (9-2) to [bend left=15] (9-4);
[->-=.7] (9-1) to (9-3);
at (-1.25,0.75) $F_9$;
at (1,-0.75) $\leqslant 0$;
Graphical representations of nine primary FDIs of $\pswo n$ built from FDI's of $\pwo n$ from <cit.> and <cit.> (the independent term appears after the symbol “$\leqslant$”).
In several primary FDI's the independent term vanishes (in contrast to Theorem <ref>.1).
Moreover, Example $F_4$ in Figure <ref> invalidates Conditions C1 and C3 from Section <ref>, while Example $F_5$ invalidates Condition C2. In particular, $A$ does not always form a PC-graph—which complicates a lot the search for a classification of the primary FDIs of $\pswo n$. We leave unsettled the problem of characterizing the primary FDIs of $\pswo n$.
§ SEARCHING FOR THE PRIMARY FACET-DEFINING INEQUALITIES OF THE LINEAR ORDERING POLYTOPE
The linear ordering polytope $\plo n$ has a richer history than the other order polytopes, including in psychology where it is often called the binary choice polytope (see for instance ).
According to Equation (<ref>), it is a polytope of dimension only $n(n-1)/2$.
Its affine hull is minimally described by all linear equations, for distinct elements $i$, $j$,
\begin{equation}\label{eq_equality_PLO}
x_{ij} + x_{ji} = 1.
\end{equation}
Moreover, for any two distinct elements $i$, $j$ and $k$, the two following inequalities define facets of $\plo n$:
\begin{align}
\label{eq_trivial_PLO}
-x_{ij} &\leqslant 0,\\
\end{align}
\begin{align}
\label{eq_trans_PLO}
x_{ij} + x_{jk} - x_{ik} &\leqslant 0;
\end{align}
they are respectively the trivial inequality and the transitive inequality.
Equations (<ref>), (<ref>) and (<ref>) form a linear description of the linear ordering polytope $\plo n$ if and only if $n \leqslant 5$.
Hence we already know all the FDI's of $\plo n$ for $2 \leqslant n \leqslant 5$; moreover, for such values of $n$, there is a linear description of $\plo n$ which consists only of primary linear equations. The same holds for $n=6$, by results of <cit.>.
Notice that the $3$-fence inequality defines a facet of $\plo 6$, and thus of $\plo n$. When $3 \leqslant m$ and $2m \leqslant n$, <cit.> and <cit.> <cit.> independently established the same assertion for the $m$-fence and $\plo n$.
When searching for primary FDI's of $\plo n$ we have to take into account that a facet admits several descriptions (for instance, Equations (<ref>) and (<ref>) define the same facet).
Among the many descriptions of a facet, some might be primary. Let us check this on a particular example extracted from a family in <cit.>.
vertex=[circle,draw,fill=white, scale=0.3]
(1) at (26:2) [vertex] ;
(2) at (77:2) [vertex] ;
(3) at (128:2) [vertex] ;
(4) at (180:2) [vertex] ;
(5) at (-128:2) [vertex] ;
(6) at (-77:2) [vertex] ;
(7) at (-26:2) [vertex] ;
at (26:2.15) $1$;
at (77:2.15) $2$;
at (128:2.15) $3$;
at (180:2.15) $4$;
at (-128:2.15) $5$;
at (-77:2.15) $6$;
at (-26:2.15) $7$;
(8) at (26:1) [vertex] ;
(9) at (77:1) [vertex] ;
(10) at (128:1) [vertex] ;
(11) at (180:1) [vertex] ;
(12) at (-128:1) [vertex] ;
(13) at (-77:1) [vertex] ;
(14) at (-26:1) [vertex] ;
at (26:0.8) $8$;
at (77:0.8) $9$;
at (128:0.8) $10$;
at (180:0.8) $11$;
at (-128:0.8) $12$;
at (-77:0.8) $13$;
at (-26:0.8) $14$;
at (2.5,-1.8) $\leqslant 17$;
[->-=.6] (1) – (2);
[->-=.6] (2) – (9);
[->-=.6] (9) – (8);
[->-=.6] (8) – (1);
[->-=.6] (3) – (2);
[->-=.6] (10) – (3);
[->-=.6] (9) – (10);
[->-=.6] (3) – (4);
[->-=.6] (4) – (11);
[->-=.6] (11) – (10);
[->-=.6] (5) – (4);
[->-=.6] (12) – (5);
[->-=.6] (11) – (12);
[->-=.6] (5) – (6);
[->-=.6] (6) – (13);
[->-=.6] (13) – (12);
[->-=.6] (7) – (6);
[->-=.6] (14) – (7);
[->-=.6] (13) – (14);
[->-=.6] (7) – (8);
[->-=.6] (1) – (14);
The graphical description of a Möbius inequality as in Example <ref>.
[A Möbius inequality]
Assume $n=14$ and let $A$ consist of the 21 pairs shown in Figure <ref>. Taking $B=\es$, we form the inequality
\begin{equation}\label{eqn_Mobius}
\sum_{a\in A} x_a \;\leqslant\; 17.
\end{equation}
The latter inequality defines a facet of $\plo n$ when $n \geqslant 14$ <cit.>. It fails to be primary only because of the independent term. However we may remedy this by subtracting $16$, $17$ or $18$ equations (<ref>) from (<ref>) (selecting as many distinct, unordered pairs $\{i$, $j\}$).
In the last example, a whole menagerie of primary FDI's of $\plo n$ results from the various possible choices of the pairs $\{i$, $j\}$. Notice also that most of the resulting primary FDI's invalidate Conditions C1 and C2; moreover, some of them invalidate Conditions C3, C4 and C5.
Transforming a FDI of $\plo n$ into a primary FDI for the same facet can be done, of course, starting from many other equations than Equation (<ref>); for instance, all equations whose coefficients in front of variables take value $0$ or $+1$ apply (their independent term must be nonnegative and at most the number of $+1$, as seen by evaluating the left-hand side at a vertex in the facet), and even more equations do. Consequently, the process produces a huge number of primary FDI's of the linear ordering polytope starting from known FDI's such as
* Möbius ladder inequalities, a family generalizing Equation (<ref>) <cit.>,
* $Z_k$-inequalities <cit.>,
* Paley inequalities <cit.>,
* graphical inequalities <cit.>,
* inequalities derived from nonorientable surfaces<cit.>,
* a vast family containing further inequalities <cit.>.
In addition to these well-known classes, additional primary FDI's for $\plo n$ were obtained for $n=7$ <cit.>.
For a summary of FDI families known in year 2000, see <cit.>. The profusion and diversity of primary FDI's explain why we do not enter the classification enterprise for the linear ordering polytope.
|
1511.00141
|
Department of Mathematics, National
University of Singapore, Singapore 119076
Beijing Computational Science Research Center, Beijing 100094,
P. R. China
Department of Mathematics, Purdue University, West Lafayette,
IN 47907, USA
Department of Mathematics, National
University of Singapore, Singapore 119076
We derive rigorously one- and two-dimensional mean-field equations for
cigar- and pancake-shaped Bose-Einstein condensates (BEC) with higher order interactions (HOI).
We show how the higher order interaction modifies the contact interaction of
the strongly confined particles. Surprisingly, we find that the usual Gaussian profile assumption for the strongly confining direction
is inappropriate for
the cigar-shaped BEC case, and a Thomas-Fermi type profile should be adopted instead.
Based on the derived mean field equations, the Thomas-Fermi densities are analyzed in presence of the contact interaction and HOI.
For both box and harmonic traps in one, two and three dimensions, we identify the analytical Thomas-Fermi densities, which depend on the competition between
the contact interaction and the HOI.
03.75.Hh, 67.85.-d
§ INTRODUCTION
Quantum-degenerate gases have been extensively explored since the
remarkable realizations of Bose-Einstein Condensate (BEC) in 1995 <cit.>.
In the typical experiments of BEC, the ultra-cold bosonic gases
are dilute and weakly interacting, and yet the major properties of the system are governed
by these weak two-body interactions <cit.>. Though the atomic interaction
potentials are rather complicated, they can be effectively described by
the two-body Fermi contact interaction – interaction kernel taken as the
Dirac delta function – in the ultra cold dilute regime,
with a single parameter, the zero energy $s$-wave scattering
length $a_s$. This is the heart of the mean field Gross-Pitaevskii equation (GPE)
theory for BEC <cit.>. Based on GPE, various aspects of
BEC have been extensively studied, including the static
properties <cit.> and dynamical properties <cit.>.
The treatment of effective two-body contact interactions has been proven to be successful,
but it is limited due to the low energy or low density assumption <cit.>.
In the case of high particle densities or strong confinement, there will be a wider
range of possible momentum states and correction terms should be
included in the GPE for better description <cit.>. Within the
perturbative framework,
higher order interaction (HOI) (or effective range expansion) as a
correction to the delta function, has to be taken into account, resulting
in a modified Gross-Pitaveskii equation (MGPE), e.g. Eq. (<ref>) <cit.>.
Based on the MGPE (<ref>), <cit.>
have shown the stability conditions and collective excitations of a
harmonically trapped BEC. In the Thomas-Fermi(TF) limit regime, <cit.> has shown the
approximate density profile for BEC with HOI in a radial trap. On the other hand,
in most experiments, a strong harmonic trap along one or two directions confine
(or suppress) the condensate into pancake or cigar shape, respectively.
In such cases, the usual TF approximation for the full three-dimensional (3D) case
becomes invalid. It is then desirable to derive the effective one- (1D) and two-dimensional (2D) models,
which offers compelling advantage for numerical computations compared to the 3D case.
In this paper, we present effective mean-field equations for trapped
BECs with HOI in one and two dimensions. Our equations are based on
a mathematically rigorous dimension reduction of the 3D
MGPE (<ref>) to lower dimensions. Such dimension reduction has been formally derived in <cit.> and rigorously
analyzed in <cit.>, for the conventional GPE, i.e. without HOI.
While for the MGPE, to our knowledge, this result has not been obtained, except for
some initial work <cit.>, where the Gaussian profile is assumed in the strongly confining direction following the conventional GPE case.
Surprisingly, our findings suggest that the Gaussian profile assumption is inappropriate for the quasi-1D BEC.
In the derivation of
the quasi-1D (2D) model for the BEC with HOI, we assume that the leading order (in terms of aspect ratio)
of the full 3D energy is from the radial (longitudinal) wave function,
such that the BEC can only be excited in the non-confining directions,
resulting in effective 1D (2D)
condensates. Based on this principle, we show that the longitudinal wave function
can be taken as the ground state of the longitudinal harmonic trap in quasi-2D BEC, and the radial
wave function has to be taken as the Thomas-Fermi (TF) type (see (<ref>)) in
quasi-1D BEC, which is totally different from the conventional GPE case <cit.>. Furthermore,
we derive simple TF densities in 1D, 2D and 3D from our effective
equations, with different HOI and contact interaction parameters, for both harmonic potentials and box potentials.
These results demonstrate very interesting phase diagrams of the TF ground state densities regarding the contact interaction and HOI.
We compare the ground states of the quasi-1D and quasi-2D
BEC with the ground states of the full 3D BEC and find good
agreement. In particular, our ground states are good approximations
to those of the full 3D MGPE in regimes where the TF
approximation fails.
The paper is organized as follows. In Sec. <ref>, we introduce the modified GPE in presence of
HOI that will be considered in this paper.
As the first main
result, we present in Sec. <ref> a mean-field equation for a
quasi-1D cigar-shaped BEC. We compare the ground state solutions of this
1D equation with the full 3D computation.
In Sec. <ref>, we present the second main result, a mean-field equation for a quasi-2D pancake-shaped BEC. Comparisons are
made between ground state density profiles of the 2D equation and ground state density profiles calculated from the full 3D model.
In Sec. <ref>, we provide a complete summary of the TF approximation in 1D, 2D and 3D cases, with harmonic potential or box potential separately.
Depending on the HOI strength and contact interaction strength, TF approximate solutions are surprisingly different, which are compared with the corresponding
ground state solutions obtained via mean field equations. Finally, we present our conclusions in Sec. <ref>. Appendix <ref> provides the
details of the dimension reduction from full 3D MGPE to our 1D mean-field equation, and Appendix <ref> provides the details for the reduction to 2D case.
§ 3D MODIFIED GPE
At the temperature $T$ much smaller than the critical temperature $T_c$, the mean-field Gross-Pitaevskii equation (GPE) of a BEC
can be combined with the HOI effect <cit.>. Inserting the HOI corrections to the two-body interaction potential,
we can obtain the modified Gross-Pitaevskii equation (MGPE) <cit.> for the wave function $\psi:=\psi(\bx,t)$ as
\begin{equation}\label{mgpe}
\end{equation}
where $\bx=(x,y,z)^T\in \mathbb R^3$ is the Cartesian coordinate
vector, $\hbar$ is the reduced Planck constant, $m$ is the mass of the particle,
$g_0=\frac{4\pi \hbar^2 a_{s}}{m}$ is the
contact interaction strength with $a_s$ being the $s$-wave scattering
lengths, HOI correction is given by the parameter
$g_1=\frac{a_s^2}{3}-\frac{a_sr_e}{2}$ with $r_e$ being the effective range of the two-body interactions and
$r_e=\frac{2}{3}a_s$ for hard sphere potential,
$V:=V(\bx)$ is the given
real-valued external trapping potential.
As in typical current experiments, we assume BEC is confined in the following harmonic potential
\begin{equation}\label{trap}
\end{equation}
where $\omega_x>0$, $\omega_y>0$ and $\omega_z>0$ are trapping frequencies in $x$-, $y$- and $z$-direction,
The wave function $\psi$ is normalized as
\begin{equation} \label{norm}
\|\psi(\cdot,t)\|^2:=\int_{{\mathbb R}^3}|\psi(\bx,t)|^2d\bx=N,
\end{equation}
where $N$ is the total number of particles in BEC.
We introduce the dimensionless quantities by rescaling length, time, energy and
wave function as
$\bx\to \bx x_s$, $t\to t/\omega_0$, $E\to E \hbar\omega_0$ and $\psi\to\psi\sqrt{N/x_s^3}$, respectively,
where $x_s=\sqrt{\frac{\hbar}{m\omega_0}}$ with $\omega_0=\min\{\omega_x,\omega_y,\omega_z\}$, $E$ is the energy.
After rescaling, the dimensionless form of the MGPE (<ref>) reads
\begin{equation}\label{eq:mgpe}
\end{equation}
\begin{equation}
\quad\beta=4\pi{N}\frac{a_s}{x_s},\quad\delta=-\frac{4\pi{N}}{x_s^3}\left(\frac{a_s^3}{3}-\frac{a_s^2r_e}{2}\right),
\end{equation}
and the dimensionless trapping potential is $V(\bx)=\gm_x^2x^2/2+\gm_y^2y^2/2+\gm_z^2z^2/2$
with $\gm_x=\omega_x/\omega_0,\quad\gm_y=\omega_y/\omega_0,\quad\gm_z=\omega_z/\omega_0$. The normalization condition becomes
When $\delta=0$, the MGPE (<ref>) collapses to the conventional GPE
and the corresponding dimension reduction problem has been studied in
<cit.> and references therein.
When $\delta<0$, there is no ground state of (<ref>),
and when $\delta>0$ and the trapping potential is a confinement,
there exist ground states of (<ref>) and the positive ground state is unique if $\beta\ge0$.
Thus hereafter we assume $\delta>0$.
§ QUASI-1D BEC WITH HOI
With a sufficiently large radial trapping frequency, it is possible to freeze the radial motion of BEC <cit.>, which becomes a quasi-1D system. Intuitively, the energy separation between stationary states is much larger in the radial direction than in the axial direction, and the dynamics is then freezed in radial direction.
As a consequence, the wave function of the system is in the variable separated form, i.e., it is the multiplication of the axial direction function and the radial direction function. In
this section, we present an effective mean-field equation for the
axial wave function of such a strongly confined BEC with HOI, by assuming a strong radial confinement.
§.§ 1D mean-field equation
In order to derive the mean-field equation for the axial wave function, we start with the 3D MGPE (<ref>) and assume a harmonic potential with
$\omega_r=\omega_x=\omega_y\gg\omega_z$. Choosing rescaling parameters used in (<ref>) as $\omega_0=\omega_z$, $x_s=\sqrt{\hbar/m\omega_z}$, we now work
with the dimensionless equation (<ref>).
In the quasi-1D BEC with HOI, the 3D wavefunction can be factorized as
\begin{equation}\label{3to1factor}
\psi(\bx,t)=e^{-i\mu_{2D}t}\omega_{2D}(x,y)\psi_{1D}(z,t),
\end{equation}
with appropriate radial state function $\omega_{2D}$ and $\mu_{2D}\in\Bbb R$. Once the radial state $\omega_{2D}$ is known, we could project the MGPE (<ref>)
onto the axial direction to derive the quasi-1D equation. The key to find such $\omega_{2D}$ is the criteria that, the energy separation between stationary states should be much larger in the radial direction than in the axial direction, i.e., there is energy scale separation between the radial state $\omega_{2D}$ and axial wavefunction.
We denote the aspect ratio of the harmonic trap as
\begin{equation}
\gamma=\omega_r/\omega_z.
\end{equation}
For conventional GPE, i.e., $\delta=0$, a good choice for $\omega_{2D}$ is the Gaussian function <cit.>, which is the ground state of the radial harmonic trap, as $\omega_{2D}(r)=\sqrt{\frac{\gm}{\pi}}e^{-\frac{\gm{r}^2}{2}}$. The reason is that the order of energy separation between states of conventional BEC is dominated in the radial direction by the radial harmonic oscillator part, which is $O(\gm)$, much larger than the interaction energy part if $\beta=O(1)$ by a similar computation shown in Appendix <ref>. Alternatively, it would be possible to use variational Gaussian profile approach to find $\omega_{2D}(r)$ <cit.>. However, for the BEC with HOI case, the extra HOI term contributes to the energy. Thus, a more careful comparison between the kinetic energy part and the HOI energy part is demanded.
By a detailed computation (see Appendix <ref>), we identify the energy contribution from the HOI term (<ref>) in transverse direction, is dominant as $\gamma\gg1$. It shows a completely different scenario compared to the conventional GPE, in which the transverse harmonic oscillator terms are dominant. The explicit form for the transverse radial state function $\omega_{2D}(r)$ for quasi-1D BEC with HOI is determined as
\begin{equation}
\omega_{2D}(x,y)\approx\frac{\gm(R^2-r^2)_+}{4\sqrt{2\delta_{r}}},\quad r=\sqrt{x^2+y^2},\label{eq:dt2}
\end{equation}
where $R=2\left(\frac{3\delta_r}{2\pi\gm^2}\right)^{\frac{1}{6}}$,
$\delta_{r}=\frac{2\cdot 3^{\frac{5}{7}}\pi^{\frac{1}{7}}\delta^{\frac{6}{7}}}{5^{\frac{9}{7}}\gm^{\frac{4}{7}}}$, $\mu_{2D}{\approx\frac{3^{\frac{4}{7}}\delta^{\frac{2}{7}}\gm^{\frac{8}{7}}}{\pi^{\frac{2}{7}}5^{\frac{3}{7}}}}$
and $(f)_+=\max\{f,0\}$.
It is worth pointing out that the determination of the radial state $\omega_{2D}(r)$
is coupled with the axial direction state (see (<ref>)). Therefore, a coupled system of the radial and axial states is necessary to get refined approximate density profiles for ground states, instead of using the above approximate $\omega_{2D}(r)$.
In the axial $z$ direction, multiplying (<ref>) by $\omega_{2D}$ and integrating the $x,y$ variables out, we obtain the mean-field equation for quasi-1D BEC with HOI as
\begin{equation}\label{3to1model}
\end{equation}
where $V_{1D}(z)=\frac12\gamma_z^2z^2=\frac12z^2$, and
\begin{align}
\beta_1&=\frac{5^{\frac{6}{7}}}{3^{\frac{1}{7}}\cdot4\pi^{\frac{3}{7}}}\delta^{\frac{3}{7}}\gm^{\frac{12}{7}}+
\frac{3^{\frac{10}{7}}}{4\cdot5^{\frac{4}{7}}\pi^{\frac{5}{7}}}
\frac{\beta\gm^{\frac{6}{7}}}{\delta^{\frac{2}{7}}},\label{eq:beta1}\\
\delta_1&=\frac{3^{\frac{10}{7}}}{4\cdot5^{\frac{4}{7}}\pi^{\frac{5}{7}}}\delta^{\frac{5}{7}}\gm^{\frac{6}{7}}.
\end{align}
From Eq. (<ref>), it is observed that the HOI provides extra repulsive contact interactions in the quasi-1D BEC. More interestingly, the first term in $\beta_1$ suggests that
the contact interactions is dominated by HOI part.
If the repulsive contact interaction dominates the dynamics in (<ref>), we could neglect the kinetic and HOI parts to obtain an analytical expression for
the quasi-1D BEC with HOI. This agrees with the usual Thomas-Fermi approximation for conventional quasi-1D BEC, and its validity is shown in Sec. <ref> (referred as region I). In such situation, the approximate density profile is given as:
\begin{equation}\label{eq:denstiy1d}
\end{equation}
where $z^*=\left(\frac{3\beta_1}{2}\right)^{\frac{1}{3}}$.
(quasi-1D ground state) Red line: approximation (<ref>) in radical direction and numerical solution of (<ref>) in axial direction. Blue dash line: Thomas-Fermi approximation of (<ref>) in axial direction. Shaded area: numerical solution from the original 3D model (<ref>).
The corresponding $\gm$'s are given in the plots. For other parameters, we choose $\beta=1,\delta=20$.
In Fig. <ref>, we compare the ground state densities of quasi-1D BEC with HOI determined via (<ref>), analytical predication (<ref>) and the numerical results from 3D MGPE in (<ref>) by integrating over the transversal directions. Noticing that HOI term produces effective repulsive potential, the BEC is broadened compared to the analytically predicated profile. As a consequence, in the regime of small or moderate interaction energy $\beta_{1}$,
we predict that the usual approach to BECs with HOI via
conventional TF approximation fails. On the other hand, our proposed 1D equation, Eq. (<ref>), describes the BEC accurately in the mean-
field regime at experimentally relevant trap aspect ratios $\gamma$.
§ QUASI-2D BEC WITH HOI
In this section, we consider the BEC being strongly confined in $z$ axis, which corresponds to $0<\gm\ll1$. Accordingly, we choose rescaling parameters used in (<ref>) as $\omega_0=\omega_r$, $x_s=\sqrt{\hbar/m\omega_r}$, and we work with the dimensionless equation (<ref>).
Similarly to the case of quasi-1D BEC, we assume that the wave function can be factorized in the quasi-2D case, as
\begin{equation}\label{3to2factor}
\psi(\bx,t)=e^{-i\mu_{1D}t}\psi_{2D}(x,y,t)\omega_{1D}(z),
\end{equation}
for appropriate longitudinal state $\omega_{1D}(z)$ and $\mu_{1D}\in\mathbb{R}$.
Following the same procedure as that for the quasi-1D BEC case, we find that, the leading order energy separations in $z$ direction is due to the longitudinal harmonic oscillator, while the cubic interaction and HOI parts are less important (see Appendix <ref> for details). This fact suggests that the ground mode of the longitudinal harmonic oscillator is a suitable choice for $\omega_{1D}(z)$, i.e., a Gaussian type function as
\begin{equation}\label{eq:w1d}
\omega_{1D}(z)\approx\left(\frac{1}{\pi\gm}\right)^{\frac{1}{4}}e^{-\frac{z^2}{2\gm}},
\end{equation}
and $\mu_{1D}\approx 1/2\gamma$.
Substituting (<ref>) with (<ref>) into the MGPE (<ref>), then multiplying (<ref>) by $\omega_{1D}$ and integrating the longitudinal $z$ out, we obtain a mean-field equation for quasi-2D BEC with HOI as
\begin{equation}\label{3to2model}
\end{equation}
where $V_{2D}(x,y)=\frac{1}{2}(x^2+y^2)$ and
\begin{align}
\beta_2&=\frac{\beta}{\sqrt{2\pi\gm}}+\frac{\delta}{\sqrt{2\pi\gm^3}},
\quad\delta_2=\frac{\delta}{\sqrt{2\pi\gm}}.\label{beta_2}
\end{align}
Similarly to the quasi-1D BEC case, HOI induces effective contact interactions in the quasi-2D regime, which dominates the contact interaction ($\beta$ part). We then conclude that
even for small HOI $\delta$, the contribution of HOI could be significant in the high particle density regime of quasi-2D BEC.
Analogous to the quasi-1D BEC case, we can derive the usual Thomas-Fermi (TF) approximation when the repulsive interaction $\beta_2$ dominates the dynamics, and
the analytical densities for the quasi-2D BEC with HOI reads
\begin{equation}\label{eq:tf2d}
n_{2D}(r)=|\psi_{2D}|^2=\frac{\left(R^2-r^2\right)_+}{2\beta_2},\quad r=\sqrt{x^2+y^2},
\end{equation}
where $R=\left(\frac{4\beta_2}{\pi}\right)^{\frac{1}{4}}$.
In order to verify our findings in this section, we compare the quasi-2D ground state densities obtained via Eq. (<ref>), TF density (<ref>) and
the numerical results from 3D MGPE (<ref>) by integrating the longitudinal $z$ axis out. The results are displayed in Fig. <ref>. Similarly to the quasi-1D case, the BEC is broadened compared to the analytically predicated profile because of the effective repulsive interaction from the HOI. Thus, in the regime of small or moderate interaction energy $\beta_{2}$,
the usual approach to BECs with HOI via conventional Thomas-Fermi approximation fails. On the other hand, it turns out that our proposed 2D equation, Eq. (<ref>), is accurate for quasi-2D BEC in the mean-field regime at experimentally relevant trap aspect ratios $\gamma$.
(quasi-2D ground state) Red line: approximation (<ref>) in axial direction and numerical solution of (<ref>) in radical direction. Blue dash line: Thomas-Fermi approximation of (<ref>) in radical direction. Shaded area: numerical solution from the original 3D model (<ref>).
The corresponding $\gm$'s are given in the plots. For other parameters, we choose $\beta=5,\delta=1$.
§ THOMAS-FERMI (TF) APPROXIMATION
In the previous sections, we have derived 1D (<ref>) and 2D (<ref>) equations for the quasi-1D and quasi-2D BECs, respectively. Indeed, all the 1D (<ref>), 2D (<ref>) and 3D (<ref>) equations can be written in a unified form as
\begin{equation}\label{eq:mgpe:d}
\end{equation}
where $\bx\in\mathbb{R}^d$, $d=3, 2, 1$, $\beta$ and $\delta$ are treated as parameters ($\delta$ is positive). Though $V(\bx)$ is assumed to be harmonic potential in the previous derivation, it is not necessary to restrict ourselves for the harmonic potential case. Thus, we treat $V(\bx)$ as a general real-valued potential in this section. In particular, we will address the cases when $V(\bx)$
is a radially symmetric harmonic potential as
\begin{equation}
V(\bx)=\frac12\gamma_{0}^2r^2, \qquad r=|\bx|,
\end{equation}
where $\gamma_0>0$ is a dimensionless constant,
or a radial box potential as
\begin{equation}
V_{\rm box}(\bx)=\begin{cases}0,&0\le r< R,\\
\infty, &r\ge R.
\end{cases}
\end{equation}
As pointed out in the quasi-1D, 2D cases, a dominant repulsive contact interaction will lead to an analytical TF densities, analogous to the conventional BEC case. However, with HOI (<ref>), the system is characterized by two interactions, contact interaction strength $\beta$ and HOI strength
$\delta$, which is totally different from the classical GPE theory that the BEC is purely characterized by the contact interaction $\beta$. Hence, for BEC with HOI (<ref>), it is possible that HOI interaction competes with contact interaction, and may be the
major effect determining the properties of BEC. In this section, we will discuss how the competition between $\beta$ and $\delta$ leads to different density profiles for the strong interactions, for which we refer such analytical density approximations as the TF approximations. We notice that it might not be physical to consider HOI as the key factor of BEC
in three dimensions in current BEC experiments, but we treat $\delta$ and $\beta$ in (<ref>) as arbitrary parameters and the result presented here may find its application in future and/or in the other fields.
In previous sections on quasi-1D and 2D BECs, we have given the analytical TF densities for $\beta$ dominant system. For the general consideration of the large $\beta$ and $\delta$
interactions, we show in Fig. <ref> the phase diagram of the different parameter regimes for $\beta$ and $\delta$, in which the TF approximation are totally different. Intuitively, there are three of them: $\beta$ term is more important (regime I in Fig. <ref>), $\delta$ term is more important (regime III), and $\beta$ term is comparable to the $\delta$ term (regimes II & IV). Detailed computations and arguments for the results shown in Fig. <ref> can be found in the Appendix <ref>. Based on Fig. <ref>, we will discuss the harmonic potential and the box potential cases separately.
§.§ TF approximation with harmonic potential
From Fig. <ref>(a), the curve $\beta=O(\delta^{\frac{d+2}{d+4}})$ is the boundary that divides the regimes for harmonic potential case. To be more specific, if $\beta\gg\delta^{\frac{d+2}{d+4}}$, the cubic nonlinear term is more important, and vise versa. If $\beta= O(\delta^{\frac{d+2}{d+4}})$, both of the two nonlinear terms are important, and have to be taken care of in the TF approximation. The resulting analytical TF density profiles in different regimes, are listed below:
Phase diagram for extreme regimes: (a) is for harmonic potential case and (b) is for box potential case.
In the figure, we choose $\beta_0\gg1$ and $\delta_0\gg1$.
Regime I, i.e. $\beta\gg\delta^{\frac{d+2}{d+4}}$, the $\delta$ term and the kinetic energy term are dropped, and the density profile is determined as
\begin{equation}\label{ground_sol_har_lb}
n_{\rm TF}(r)=|\psi_{\rm TF}|^2=\frac{\gm_0^2(R^2-r^2)_+}{2\beta},
\end{equation}
where $R=\left(\frac{(d+2)C_d\beta}{\gm_0^2}\right)^{\frac{1}{d+2}}$, and the constant $C_d$ is defined as
\begin{equation}\label{eq:cd}
\begin{cases}
\frac{1}{2},& d=1,\\
\frac{1}{\pi},& d=2,\\
\frac{3}{4\pi},& d=3.
\end{cases}
\end{equation}
With the above TF densities, the leading order approximations for chemical potential $\mu$ and energy $E$ of the ground state are: $\mu_{\rm TF}=\frac{1}{2}\left((d+2)C_d\beta\right)^{\frac{2}{d+2}}\gm_0^{\frac{2d}{d+2}}$, $E_{\rm TF}=\frac{d+2}{d+4}\mu_{\rm TF}$
for $d$ ($d=3,2,1$) dimensional case.
Regime II, i.e. $\beta=C_0\delta^{\frac{d+2}{d+4}}$ with $C_0>0$, neglecting the kinetic term in the time-independent MGPE, we have
\begin{equation}\label{eq:har_bd}
\mu\psi=\frac{\gm_0^2|\bx|^2}{2}\psi+C_0\delta^{\frac{d+2}{d+4}}|\psi|^2\psi-\delta\nabla^2(|\psi|^2)\psi.
\end{equation}
Formally, Eq. (<ref>) degenerates at position $\bx$ if $\psi(\bx)=0$ and it is indeed a free boundary problem (boundary of the zero level set of $\psi$), which
requires careful consideration. Motivated by <cit.> for the 3D case,
besides the condition that $n(R)=0$ along the free boundary $|\bx|=R$, we impose $n^{\prime}(R)=0$;
and assume $n(r)=0$ for $r>R$.
The TF density profile in regime II is self similar under appropriate scalings.
To be more specific, the analytical TF density takes the form
where $n_0(r)$ is the function can be calculated exactly as below.
Plugging (<ref>) into (<ref>), we obtain the equation for $n_0(r)$ by imposing the aforementioned conditions at the free boundary,
=_0^2r^2/2+C_0n_0-∂_rrn_0(r)-d-1/r∂_r n_0(r),
for $r\leq R$ and $n_0(s)=0$ for $s\ge R$, and $n_0(R)=0$, $n_0^\prime(R)=0$, where $R$ is the free boundary that has to be determined and $\tmu=\delta^{-\frac{2}{d+4}}\mu$.
In addition, we assign the boundary condition
at $r=0$ as $n_0^\prime(0)=0$, because of the symmetry.
Note that $C_0$ can be negative as $\delta$ term can bound the negative
cubic interaction, which corresponds to Regime IV. In fact in Regime IV, we will repeat the above procedure.
Denote $a=\sqrt{C_0}$ and
the ordinary differential equation (<ref>) in $d$ dimensions can be solved analytically. Denote
\begin{gather}\label{eq:fdr}
\begin{cases}
e^{ar}+e^{-ar},& \text{for } d=1,\\
I_0(ar),& \text{for } d=2,\\
(e^{ar}-e^{-ar})/r, &\text{for } d=3,
\end{cases}
\end{gather}
where $I_0(r)$ is the standard modified Bessel function $I_{\alpha}$ with $\alpha=0$.
Then the solution of Eq .(<ref>) with prescribed Neumann boundary conditions reads as
\begin{equation}\label{eq:TF_har_pc}
\end{equation}
Inserting the above expression to the normalization condition that $\int_{\mathbb{R}^d}n_0(\bx)\,d\bx=1$, we
find chemical potential,
\begin{equation}\label{eq:TF_har_mu_pc}
\tmu=\frac{C_da^2}{R^d}+\frac{d\gm_0^2R^2}{2(d+2)}.
\end{equation}
Combining (<ref>) and (<ref>), noticing the Dirichlet condition $n(R)=0$,
we have the equation for $R$,
Thus, the free boundary $R$ can be calculated and $n_0(r)$ is then determined.
Regime III, i.e. $\beta\ll\delta^{\frac{d+2}{d+4}}$, the $\beta$ term and the kinetic energy term are dropped, and the TF density profile is
\begin{equation}\label{ground_sol_har_ld}
n_{\rm TF}(r)=|\psi_{\rm TF}|^2=\frac{\gm_0^2(R^2-r^2)^2_+}{8(d+2)\delta},
\end{equation}
where $R=\left(\frac{(d+2)^2(d+4)C_d\delta}{\gm_0^2}\right)^{\frac{1}{d+4}}$. Again, the leading order approximations for chemical potential and energy,
with the above TF densities, are $\mu_{\rm TF}=\frac{d}{2(d+2)}\left((d+2)^2(d+4)C_d\delta\gm_0^{d+2}\right)^{\frac{2}{d+4}}$ , $E_{\rm TF}=\frac{d+4}{d+6}\mu_{\rm TF}$ in $d$ dimensions.
Regime IV, i.e. $\beta=-C_0\delta^{\frac{d+2}{d+4}}$ with $C_0>0$.
By a similar procedure as in Regime II, we'll get (<ref>) and
=_0^2r^2/2-C_0n_0-∂_rrn_0(r)-d-1/r∂_r n_0(r),
for $r\leq R$ and $n_0(s)=0$ for $s\ge R$, and $n_0^\prime(0)=0$, $n_0(R)=0$, $n_0^{\prime}(R)=0$, where $R$ is the free boundary that has to be determined and $\tmu=\delta^{-\frac{2}{d+4}}\mu$.
Again, let $a=\sqrt{C_0}$ and denote
\begin{gather}\label{eq:gddef}
\begin{cases}
\cos(ar),& \text{for } d=1,\\
J_0(ar),& \text{for } d=2,\\
\sin(ar)/r,& \text{for } d=3,
\end{cases}
\end{gather}
where $J_0(r)$ is the Bessel function of the first kind $J_{\alpha}(r)$ with $\alpha=0$.
The solution of Eq. (<ref>) with the assigned Neumann boundary conditions can be written as:
\begin{equation}\label{eq:TF_har_nc}
\end{equation}
The chemical potential is then calculated from normalization condition as
\begin{equation}\label{eq:TF_har_mu_nc}
\tmu=-\frac{C_da^2}{R^d}+\frac{d\gm_0^2R^2}{2(d+2)}.
\end{equation}
Finally, the free boundary $R$ is determined from the Dirichlet condition $n_0(R)=0$,
After $R$ is computed, we then find $n_0(r)$.
Comparisons of 3D numerical ground states with TF densities, the harmonic potential case in region I, II, III and IV, which are define in Fig. <ref>(a).
Red line: Thomas-Fermi approximation, and shaded area: numerical solution from the equation (<ref>). The parameters are chosen to be
$\gm=2$ and
(I) $\beta=1280$, $\delta=1$; (II) $\beta=828.7$, $\delta=1280$; (III) $\beta=1$, $\delta=1280$; (IV) $\beta=-828.7$, $\delta=1280$; respectively.
[comparison of energy (harmonic potential case)]
[comparison of chemical potential (harmonic potential case)]
Comparisons of numerical energies and chemical potentials with TF approximations, the harmonic potential case. 3D problem is considered here.
Blue line: Thomas-Fermi approximation, and red circles: numerical results obtained from the equation (<ref>). The parameters are chosen to be
$\gm=2$ and
(I) $\delta=1$, (II) $\beta=5\delta^{\frac{5}{7}}$, (III) $\beta=1$, (IV) $\beta=-5\delta^{\frac{5}{7}}$, respectively.
In Fig. <ref>, we compare the analytical TF densities (<ref>), (<ref>) and (<ref>) with
the numerical results computed via full equation (<ref>)
by the background Euler finite difference (BEFD) method <cit.>.
We can observe that in all the extreme regions, the analytical TF densities agree very well with
the full equation simulations. As a byproduct, we show the comparisons of the corresponding chemical potentials and energies in Fig. <ref>.
It has been shown that the usual TF densities provide accurate approximations for the density profiles for quasi-1D an 2D BECs. Indeed, we can check that for fixed three dimensional parameter $\beta$ and $\delta$, the effective contact interaction $\beta_1$ and HOI $\delta_1$ (or $\beta_2$ and $\delta_2$) for quasi-1D (2D) condensate, are
in the TF regime I, in the quasi-1D (2D) limit, i.e. $\gamma\to\infty$ ($\gamma\to0^+$). This justifies that the effective contact interactions is dominant for the dynamics in the quasi-1D
(2D) limit.
For instance, we know $\beta_1\sim O(\gamma^{\frac{12}{7}})$ and $\delta_1\sim O(\gamma^{\frac{6}{7}})$ in quasi-1D limit, and it implies that
$\beta_1\gg \delta_1^{3/5}$ as $\gamma\gg1$. This immediately suggests that the TF density (<ref>) is a good approximation for the density profiles in quasi-1D limit regime,
which has been shown in Fig. <ref>. In the quasi-2D limit, i.e. $\gamma\to 0^+$, we find $\beta_{2}\gg \delta_2^{2/3}$ in view of $\beta_2\sim O(\gamma^{-3/2})$ and $\delta_2\sim O(\gamma^{-1/2})$, which again confirms that TF density (<ref>) is a good approximations for the density profiles, as observed in Fig. <ref>.
§.§ TF approximation with box potential
In this section, we consider the box potential case, which confines the BEC in a bounded domain $\{|\bx|\leq R\}$. Using similar method for the harmonic potential case, we could obtain the analytical TF densities as the contact interaction and/or HOI dominates the ground state in Eq. (<ref>).
In detail, we have the analytical TF densities for different regimes shown in Fig. <ref>(b). Different from the harmonic potential case,
the borderline of the three regimes is $\beta= O(\delta)$.
Regime I, $\beta$ term is dominant, i.e. $\beta\gg1$ and $\delta=o(\beta)$. The kinetic term and the HOI term are dropped and the time independent MGPE equation in the radial variable $r$ becomes
\begin{equation}
\mu\psi(r)=\beta|\psi|^2\psi,\quad 0\le r=|\bx|< R,
\end{equation}
with boundary condition $\psi(R)=0$. Thus, the TF density is a constant, which can be uniquely determined by the normalization condition $\|\psi\|=1$. Explicitly, TF density is given by $n_{\rm TF}(r)=|\psi|^2=\frac{C_d}{R^d}$, and $\mu_{\rm TF}=\frac{C_d\beta}{R^d}$, where $C_d$ is defined in previous subsection.
It is obvious that the TF density is inconsistent with zero boundary condition, thus a boundary layer appears in the ground state density profiles <cit.>. In fact, as in <cit.>, if $\delta\sim{o}(1)$, for $d=1$, to match the boundary layers at $x=\pm R$, an asymptotic analysis leads to the
following matched density as $\beta\gg1$ for $0\le r=x\le R$,
\begin{equation}
n_{\rm as}(r)=|\psi_{\rm as}|^2=\frac{1}{2R}\left(\tanh(\sqrt{\mu_{\rm as}}(R-r))\right)^2,
\end{equation}
with the chemical potential $\mu_{\rm as}=\frac{1}{2R}\beta+\frac{1}{R}\sqrt{\frac{\beta}{2R}}$, and the energy $E_{\rm as}=\frac{1}{4R}\beta+\frac{2}{3R}\sqrt{\frac{\beta}{2R}}$.
For $d=2,3$, similar matched densities can be derived.
From our numerical experience, the matched asymptotic density $n_{\rm as}$ provides much more accurate approximation to the ground state of Eq. (<ref>), than the TF density $n_{\rm TF}$, in the parameter regimes $\beta\gg1$ and $\delta=O(1)$.
Regime II, both $\beta$ and $\delta$ are important, i.e. $\beta=O(\delta)$ as $\delta\to\infty$. We assume that $\beta=C_0\delta$, with $\delta\gg1$ for some constant $C_0>0$.
Omitting the less important kinetic part, the radially symmetric time independent MGPE reads
μψ(r)=C_0δ|ψ|^2ψ-δ∇^2(|ψ|^2)ψ, r<R,
with $\psi(R)=0$.
The above equation can be simplified for
density $n(r)=|\psi|^2$ in $d$ dimensions as
with $n(R)=0$, and at $r=0$ with $n^{\prime}(0)=0$.
Eq. (<ref>) can be solved analytically. Again, we introduce $a=\sqrt{C_0}$ and recall
function $f_{a,d}$ defined in (<ref>).
The TF density, or solution of the boundary value problem (<ref>), is given explicitly as
with $\mu_{\rm TF}=C_da^2\delta/(R^d-d\frac{\int_0^Rf_{a,d}(r)r^{d-1}dr}{f_{a,d}(R)})$
and $E_{\rm TF}=\mu_{\rm TF}/2$, where $C_d$ is defined in Eq. (<ref>).
Regime III, $\delta$ term is dominant, i.e. $\delta\gg1$, $\beta=o(\delta)$. The kinetic term and the $\beta$ term are dropped. The corresponding stationary MGPE
for the ground state reads
with boundary condition $\psi(R)=0$.
Solving the equation and using the normalization condition, we obtain the TF density as
with chemical potential $\mu_{\rm TF}={C_d}d(d+2)\delta/R^{d+2}$ and energy $ E_{\rm TF}=\mu_{\rm TF}/2$.
Regime IV, i.e. $\beta=-C_0\delta$, with $\delta\gg1$ for some constant $C_0>0$.
Intuitively, if $C_0$ is small, the repulsive
HOI $\delta$ term is dominant and the particle density will still occupy the entire domain; if $C_0$ is sufficiently large, the attractive $\beta$ interaction becomes
the major effect, where the particles will be self trapped and the density profile will concentrate in a small portion of the domain.
Therefore, unlike the corresponding whole space case with harmonic potential, we have two different situations here.
By a similar procedure as in Regime II, we get
with $n(R^\prime)=0$ and $R^\prime$ to be determined.
In the first situation, the density spreads over the whole domain and thus $R^\prime=R$;
in the second situation, the density would concentrate and $0<R^\prime<R$.
Case I, i.e. $C_0\leq C_{\rm cr}$, where $C_{\rm cr}=\hat{R}^2/R^2$ and $\hat{R}$ is the
first positive root of $g_{a,d}^\prime(r/a)=0$ defined in Eq. (<ref>) with $a=\sqrt{C_0}$ .
As mentioned before, because of the relatively weak attractive interaction, we still have the following boundary conditions at the boundary:
$n(R)=0$, $n^{\prime}(0)=0$.
The TF density, or solution of Eq. (<ref>), can be expressed as:
with $\mu_{\rm TF}=C_da^2\delta/(d\frac{\int_0^Rg_{a,d}(r)r^{d-1}dr}{g_{a,d}(R)}-R^d)
$ and
$E_{\rm TF}=\mu_{\rm TF}/2$, where $C_d$ is given in (<ref>).
If $aR>\hat{R}$, we know from the properties of $g_{a,d}(r)$ that $g_{a,d}(r)$ ($r\in[0,\hat{R}]$) would take any value between the maximum (positive) and minimum (negative) of
$g_{a,d}(r)$ ($r\ge0$). Then $1-g_{a,d}(r)/g_{a,d}(R)$ would change sign for $r\in[0,\hat{R}]$, when $g_{a,d}(r)$ takes value around $r_0\in(0,\hat{R})$ such that $g_{a,d}(r_0)=g_{a,d}(R)$.
On the other hand, since the density must be nonnegative, $1-g_{a,d}(r)/g_{a,d}(R)$ can not change sign in $[0,R]$. So we conclude that $aR\leq\hat{R}$, i.e. the condition
$C_0\leq C_{\rm cr}$ is necessary.
$g_{a,d}^\prime$ at $r/a$ can be computed as
\begin{gather}\label{def:box_nbd_df}
\begin{cases}
-a\sin(r),& d=1,\\
-aJ_1(r),& d=2,\\
a^2(r\cos(r)-\sin(r))/r^2,& d=3,
\end{cases}
\end{gather}
and we have
for 1D case, $\hat{R}=\pi$;
for 2D case, $\hat{R}=3.8317\cdots$;
for 3D case, $\hat{R}=4.4934\cdots$.
Case II, $C_0>C_{\rm cr}$.
As observed above, the density profiles may be away from the boundaries of the domain. Thus, free boundary conditions should be used as
$n(\tilde{R})=0$, $n^{\prime}(\tilde{R})=0$, $n^{\prime}(0)=0$, where $\tilde{R}<R$ is the boundary for the TF density that we want to find.
Hence domain $[0,\tilde{R}]$ replaces the domain $[0,R]$ in Case I, and we have extra boundary condition $n^{\prime}(\tilde{R})=0$.
Denoting $a=\sqrt{C_0}$ and using the solution in Case I, we get $g^\prime_{a,d}(\tilde{R})=0$, and $a\tilde{R}\leq \hat{R}$, where
both conditions can only be satisfied if and only if $a\tilde{R}=\hat{R}$. So, we identify that $\tilde{R}=\hat{R}/a<R$.
Comparisons of 1D numerical ground states with TF densities, the box potential case in region I, II, III and IV, which are define in Fig. <ref>(b). Red line: analytical TF approximation, and shaded area: numerical solution obtained from (<ref>).
Domain is $\{r|0\le r<2\}$ and the corresponding $\beta$'s and $\delta$'s are (I) $\beta=1280$, $\delta=1$; (II) $\beta=320$, $\delta=160$; (III) $\beta=1$, $\delta=160$; (IV) $\beta=-400$, $\delta=80$.
[comparison of energy]
[comparison of chemical potential]
Comparisons of numerical energies and chemical potentials with TF approximations, the box potential case. 1D problem is considered here. Blue line: analytical TF approximation, and red
circles: numerical results obtained from (<ref>).
The parameters are chosen to be (I) $\delta=1$, (II) $\beta=2\delta$, (III) $\beta=1$, (IV) $\beta=-5\delta$, respectively, and domain is $\{r|0\le r<2\}$.
Replacing $R$ with $\hat{R}/a$ in TF solution of Case I, we obtain the analytical TF density as
with $\mu_{\rm TF}=-C_da^{d+2}\delta/\hat{R}^d$ and $E_{\rm TF}=\mu_{\rm TF}/2$,
where $\hat{R}$ is defined in Case I.
We compare in Fig. <ref> the analytical TF densities listed above with the ground state obtained from numerical results via Eq. (<ref>) computed by the BEFD method <cit.> in various
parameter regimes discussed above.
Fig. <ref> shows our analytical TF densities are very good approximations for the ground states. We also make comparisons for chemical potentials and energies between the TF approximations and the numerical values by solving Eq. (<ref>) in Fig. <ref>.
§ CONCLUSION
We have presented the mean-field modified Gross-Pitaevskii equations for quasi-1D, Eq. (<ref>), and quasi-2D, Eq. (<ref>), BECs with higher-order interaction (HOI) term.
These equations are based on a rigorous
dimension reduction from the full 3D MGPE with the assumptions that the energy separations
in radial and longitudinal directions scales differently in the strongly anisotropic aspect ratio limit,
and the wave function can be separated into radial and longitudinal variables. By carefully studying
the energy separation, we obtain the correct radial or longitudinal states used in the dimension reduction. In particular, it is quite interesting
that the radial states has to be taken in the form different from the ground state of radial harmonic potential in the quasi-1D BEC, which
is counterintuitive compared with the conventional GPE.
Our result shows that quasi-1D and quasi-2D
BECs with HOI are governed by a modified contact interaction term
and a modified HOI term, and all the equations for quasi-1D and quasi-2D BECs have the same form as the 3D MGPE.
We have computed the ground states of our 1D and 2D equations
numerically and compared them with the ground states of the 3D MGPE, and we find excellent agreements.
We have also completely determined Thomas-Fermi approximation in various parameter regimes with both box potential and harmonic potential, for the 1D, 2D and 3D cases.
In presence of HOI, TF approximations become very complicated as HOI competes with contact interaction.
§ DERIVATION OF THE QUASI-1D EQUATION
Under the assumption in Sec. <ref>,
we take the ansatz
\begin{equation}\label{factorization}
\psi(x,y,z,t)=e^{-i\mu_{2D}t}\omega_{2D}(x,y)\psi_{1D}(z,t),
\end{equation}
where the transverse state is frozen, i.e. $\omega_{2D}$ is the radial minimum energy state and the energy separation is much larger in the radial direction than the longitudinal $z$ direction.
Substitute (<ref>) into Eq. (<ref>), we can get the equations for $\psi_{1D}$ for appropriate $\mu_{2D}$ as
where $V_{1D}(z)=\frac12z^2$,
\begin{align}
\beta_1&=\beta\iint|\omega_{2D}|^4dxdy+\delta\iint|\nabla_{\perp}|\omega_{2D}|^2|^2dxdy,\label{beta_z}\\
\delta_1&=\delta\iint|\omega_{2D}|^4dxdy,\label{delta_z}
\end{align}
and $\nabla_{\perp}=(\p_x,\p_y)^T$.
It remains to determine $\omega_{2D}$ and we are going to use the criteria that the energy separations scale differently in different directions. In order to
do this, we need calculate the energy scale in $z$ direction. Hence, we take the stationary states (ground states) of (<ref>) as
Combining Eqs. (<ref>) and (<ref>), following the way to find Eq. (<ref>),
we can derive the equations for $\omega_{2D}(x,y)$ as
\begin{equation}
\mu_{2D}\omega_{2D}=-\frac{1}{2}\nabla^2_{\perp}\omega_{2D}+V_{2D}(r)\omega_{2D}+\beta_2|\omega_{2D}|^2\omega_{2D}-\delta_2(\nabla^2_{\perp}|\omega_{2D}|^2)\omega_{2D},\label{rmodel}
\end{equation}
where $\nabla^2_\perp=\p_{xx}+\p_{yy}$, the radially symmetric potential $V_{2D}=\frac{\gamma^2}{2}(x^2+y^2)$,
\begin{align}
\beta_2&=\beta\int|\phi_{1D}|^4dz+\delta\int|\partial_{z}|\phi_{1D}|^2|^2dz,\label{beta_r}\\
\delta_2&=\delta\int|\phi_{1D}|^4dz.\label{delta_r}
\end{align}
To determine the frozen state $\omega_{2D}$, we need minimize the energy of Eq. (<ref>), while parameters $\beta_2$ and $\delta_2$ depends on $\phi_{1D}$. So actually, we need solve a coupled system together for $\omega_{2D}$ and $\phi_{1D}$. To this purpose, we will consider the problem in the quasi-1D limit $\gamma\to\infty$. Intuitively, transverse direction is almost compressed to a Dirac function as $\gamma\to\infty$, so that a proper scaling is needed to obtain the correct form of $\omega_{2D}$.
We will determine $\omega_{2D}$ via a self consistent iteration as follows:
given some $\beta_2$ and $\delta_2$, under proper scaling as $\gamma\to\infty$, (i)
drop the less important part to get approximate $\omega_{2D}$, (ii)
put $\omega_{2D}$ into Eq. (<ref>) to determine the longitudinal ground state $\phi_{1D}$,
(iii) use $\phi_{1D}$ to compute $\beta_2$ and $\delta_2$, and then (iv) check if it is consistent.
In the quasi-1D regime, $\gamma\to\infty$, similarly to the conventional GPE case, due to the strong confinement in transverse direction, the ground state solution $\phi_{1D}$ is very flat in $z$ direction, as both nonlinear terms exhibit repulsive interactions. It is easy to get the scalings of $\int|\partial_z|\phi_{1D}|^2|^2dz=O(L^{-3})$, $\int|\phi_{1D}|^4dz=O(L^{-1})$, where $L$ indicates the correct length scale of $\phi_{1D}$. Therefore $\beta_2$ and $\delta_2$ are of the same order by definition, since $L\to \infty$ in the quasi-1D limit.
For mathematical convenience, we introduce $\vep=1/\sqrt{\gm}$ such that $\vep\to0^+$. In the radial variable, introduce the new scale $\tr=r/\vep^{\alpha}$ and $\tilde{\omega}(\tr)=\vep^{\alpha}\omega_{2D}(r)$ such that $\tr\sim O(1)$ and $\|\tilde{\omega}\|=1$, then (<ref>) becomes
\begin{equation}\label{eq:r_eig_eq_m}
\mu_{2D}\tilde{\omega}=-\frac{\nabla^2_{\perp}\tilde{\omega}}{2\vep^{2\alpha}}
\end{equation}
Noticing that the term $\beta_2/\vep^{2\alpha}\tilde{\omega}^3$ can be always neglected compared to the last term since $\beta_2\sim\delta_2$ and $\vep^{-\alpha}\ll\vep^{-3\alpha}$ as $\vep\to0^{+}$. On the other hand, $\beta_2$ and $\delta_2$ are both repulsive interactions while only the potential term confines the condensate. Thus, the correct leading effects (HOI or kinetic term)
should be balanced with the potential term. Now, we are only left with two possibilities:
Case I, $-\frac{1}{2\vep^{2\alpha}}\tilde{\nabla^2}_{\perp}\tilde{\omega}$ is balanced with term $\frac{\tr^2}{2\vep^{4-2\alpha}}\tilde{\omega}$, and $\frac{\delta_2}{\vep^{4\alpha}}\tilde{\nabla^2}_{\perp}(|\tilde{\omega}|^2)\tpsi$ is smaller. In this case, $\vep^{2\alpha}\sim\vep^{4-2\alpha}$. So we get $\alpha=1$. Besides, we also need $\vep^{-2\alpha}\gg\frac{\delta_2}{\vep^{4\alpha}}$, i.e. $\delta_2\ll\vep^2$.
Case II, $\frac{\delta_2}{\vep^{4\alpha}}\tilde{\nabla^2}_{\perp}(|\tilde{\omega}|^2)\tilde{\omega}$ is balanced with term $\frac{\tr^2}{2\vep^{4-2\alpha}}\tilde{\omega}$, and $-\frac{1}{2\vep^{2\alpha}}\tilde{\nabla^2}_{\perp}\tilde{\omega}$ is much smaller. In this case, $\frac{\delta_2}{\vep^{4\alpha}}\sim\frac{1}{\vep^{4-2\alpha}}$ and $\vep^{-2\alpha}\ll\frac{1}{\vep^{4-2\alpha}}$, i.e. $\alpha<1$ and $\delta_2\sim\vep^{6\alpha-4}$.
We will check if the scaling is consistent for each case.
Case I. Since $\alpha=1$ , we have $\omega_{2D}$ as the ground state of radial harmonic
\begin{equation}\label{r_sol1}
\omega_{2D}(r)=\frac{1}{\sqrt{\pi\vep^2}}e^{-\frac{r^2}{2\vep^2}},
\end{equation}
\begin{equation*}
\iint|\omega_{2D}|^4dxdy=\frac{1}{2\pi\vep^2},\quad
\iint|\nabla_{\perp}(|\omega_{2D}|^2)|^2dxdy=\frac{1}{\pi\vep^4}.
\end{equation*}
Recalling $\beta_1$ and $\delta_1$ in Eq. (<ref>), the parameters are in TF regime I (cf. Sec. <ref>), so in $z$ direction we can get the approximate solution from Sec. <ref> as:
\begin{equation}\label{z_sol}
\phi_{1D}\approx\sqrt{\frac{\left((z^{*})^2-z^2\right)_+}{2\beta_1}},\quad z^*=\left(\frac{3\beta_1}{2}\right)^{\frac{1}{3}},
\end{equation}
By definition of $\delta_2$ (<ref>), we obtain
\begin{equation}\label{db1_3to1}
\delta_2=\delta\int|\phi_{1D}|^4dz=\frac{3\delta}{5}\left(\frac{2}{3\beta_1}\right)^{\frac{1}{3}},
\end{equation}
\begin{equation}\label{bd1_3to1}
\beta_1\sim\delta\iint|\nabla_{\perp}(|\omega_{2D}|^2)|^2dxdy=\frac{\delta}{\pi\vep^4}.
\end{equation}
Combining (<ref>)) and (<ref>), we get $\delta_2=O(\vep^{\frac{4}{3}})$. But it contradicts with the requirement that $\delta_2\ll\vep^2$.
Thus Case I is inconsistent.
Case II. As $\delta_2$ term is more significant than the kinetic term, we solve $\mu_{2D}=r^2/2\vep^4-\delta_2\nabla_{\perp}^2|\omega_{2D}|^2$
within the support of $\omega_{2D}(r)$ and get
\begin{equation}\label{r_sol2}
\omega_{2D}(r)=\frac{(R^2-r^2)_+}{\sqrt{32\vep^4\delta_2}},\, R=2a\vep,\,a=\left(\frac{3\delta_2}{2\pi\vep^2}\right)^{\frac{1}{6}}.
\end{equation}
Hence, we know
Again, recalling $\beta_1$ and $\delta_1$ in Eq. (<ref>), the parameters are in TF regime I (cf. Sec. <ref>), so in $z$ direction we can get the approximate solution from Sec. <ref> as Eq. (<ref>). Having $\phi_{1D}(z)$ Eq. (<ref>), we can compute
\begin{equation}\label{db2_3to1}
\delta_2=\delta\int|\phi_{1D}|^4dz=\frac{3\delta}{5}\left(\frac{2}{3\beta_1}\right)^{\frac{1}{3}},
\end{equation}
\begin{equation}\label{bd2_3to1}
\beta_1\sim\delta\int|\nabla_{\perp}|\omega_{2D}|^2|^2dxdy=\left(\frac{3}{2\pi}
\right)^{\frac{1}{3}}\frac{\delta}{2\vep^{\frac{8}{3}}\delta_2^{\frac{2}{3}}}.
\end{equation}
Combining (<ref>) and (<ref>), we find $\delta_2=\frac{2\cdot 3^{\frac{5}{7}}\pi^{\frac{1}{7}}\delta^{\frac{6}{7}}\vep^{\frac{8}{7}}}{5^{\frac{9}{7}}}$,
Noticing the requirement that $\delta_2\sim\vep^{6\alpha-4}$, we get $\alpha=6/7$, and it satisfies the other constraint $\alpha<1$. Thus, Case II is self consistent.
Therefore, for the quasi-1D BEC, this is the case that we should choose to derive the mean field equation and $\beta_1$, $\delta_1$ can be obtained as in Eq. (<ref>).
To summarize, we identify that $\omega_{2D}$ should be taken as Eq. (<ref>) and the mean field equation Eq. (<ref>) for quasi-1D BEC is derived.
With this explicit form of the approximate solutions, we can further get the leading order approximations for chemical potential and energy for the original 3D problem. It turns out that $\mu_g^{3D}\approx\frac{9}{8}\mu_{2D} \text{ and } E_g^{3D}\approx\frac{7}{8}\mu_{2D},$ where $\mu_{2D}$ is computed approximately as before.
§ DERIVATION OF THE QUASI-2D EQUATION
Under the assumption in Sec. <ref>,
we take the ansatz
\begin{equation}\label{factorization2}
\psi(x,y,z,t)=e^{-i\mu_{1D}t}\omega_{1D}(z)\psi_{2D}(x,y,t),
\end{equation}
where the longitudinal state is frozen, i.e. $\omega_{1D}$ is the minimum energy state and the energy separation is much larger in the longitudinal $z$ direction than the radial direction.
Plugging Eq. (<ref>) into Eq. (<ref>), we can get the equations for $\psi_{2D}$ with appropriate $\mu_{1D}$ as
where the radially symmetric potential $V_{2D}(r)=\frac12r^2$ and
\begin{align}
\beta_2&=\beta\int|\omega_{1D}|^4dxdy+\delta\int|\partial_{z}|\omega_{1D}|^2|^2dz,\label{beta_2:r}\\
\delta_2&=\delta\int|\omega_{1D}|^4dxdy,\label{delta_2:r}
\end{align}
with $\nabla_{\perp}=(\p_x,\p_y)^T$ and $\nabla^2_\perp=\partial_{xx}+\partial_{yy}$.
It remains to determine $\omega_{1D}$ and we are going to use the same idea as that in the quasi-1D BEC. In order to
do this, we need calculate the energy scale in $r$ direction. Hence, we take the stationary states (ground states) of Eq. (<ref>) as
Combining Eq. (<ref>) with Eq. (<ref>),
we can derive the equations for $\omega_{1D}(z)$ as
\begin{equation}
\mu_{1D}\omega_{1D}=-\frac{1}{2}\partial_{zz}\omega_{1D}+V_{1D}(z)\omega_{1D}+\beta_1|\omega_{1D}|^2\omega_{1D}-\delta_1(\partial_{zz}|\omega_{1D}|^2)\omega_{1D},\label{rmodel:2d}
\end{equation}
where $V_{1D}(z)=\frac{z^2}{2\gamma^2}$,
\begin{align}
\beta_1&=\beta\int|\phi_{2D}|^4dz+\delta\int|\nabla_{\perp}|\phi_{2D}|^2|^2dz,\label{beta_r:2d}\\
\delta_1&=\delta\int|\phi_{2D}|^4dz.\label{delta_r:2d}
\end{align}
We proceed similarly to the quasi-1D case. For mathematical convenience, denote $\vep=\sqrt{\gm}$ such that $\vep\rightarrow{0^+}$.
Rescale $z$ variable as $\tilde{z}=z/\vep^{\alpha}$, $\tilde{\omega}(\tilde{z})=\vep^{\frac{\alpha}{2}}\omega_{1D}(z)$ for some $\alpha>0$.
By removing the tildes, Eq. (<ref>) becomes
\begin{equation}\label{eq:z_eig_eq_m}
\mu_{1D}\omega=-\frac{1}{2\vep^{2\alpha}}\partial_{zz}\omega+\frac{z^2}{2\vep^{4-2\alpha}}\omega
\end{equation}
Assuming that the scale is correct, then $\omega$ will be a regular function, independent of $\vep$ so that its norm will be $O(1)$.
Now, we will determine the scale similarly to the quasi-1D BEC. Intuitively, by the same reason in the quasi-1D case, the term $\frac{\beta_1}{\vep^{\alpha}}\omega^3$ can always be neglected compared to the HOI term. In addition, potential term is the only effects that confine the condensate, which can not be neglected. Then, there are two possibilities:
Case I. $-\frac{1}{2\vep^{2\alpha}}\partial_{zz}\omega$ is balanced with term $\frac{z^2}{2\vep^{4-2\alpha}}\omega$, and $\frac{\delta_1}{\vep^{3\alpha}}(\partial_{zz}|\omega|^2)\omega$ is much smaller. In this case, $\vep^{2\alpha}\sim\vep^{4-2\alpha}$. So we get $\alpha=1$. Besides, we also need $\vep^{-2\alpha}\gg\frac{\delta_1}{\vep^{3\alpha}}$, i.e. $\delta_1\ll\vep$.
Case II. $\frac{\delta_1}{\vep^{3\alpha}}(\partial_{zz}|\omega|^2)\omega$ is balanced with term $\frac{z^2}{2\vep^{4-2\alpha}}\omega$, and $-\frac{1}{2\vep^{2\alpha}}\partial_{zz}\omega$ is much smaller. In this case, $\frac{\delta_1}{\vep^{3\alpha}}\sim\frac{1}{\vep^{4-2\alpha}}$ and $\vep^{-2\alpha}\ll\frac{1}{\vep^{4-2\alpha}}$, i.e. $\alpha<1$ and $\delta_1\sim\vep^{5\alpha-4}$.
Now, we check the consistency of each case.
Case I. Since $\alpha=1$, we can obtain $\omega_{1D}(z)$ as the ground state
of longitudinal harmonic oscillator as
\begin{equation}\label{z_sol1}
\omega_{1D}(z)=\left(\frac{1}{\pi\vep^2}\right)^{\frac{1}{4}}e^{-\frac{z^2}{2\vep^2}},
\end{equation}
and the following quantities can be calculated:
\begin{equation}\label{z_order1}
\int|\omega_{1D}|^4dz=\frac{1}{\sqrt{2\pi}\vep} \text{ , } \int|(|\omega_{1D}|^2)^{\prime}|^2dz=\frac{1}{\sqrt{2\pi}\vep^3}.
\end{equation}
By examining $\beta_2$ and $\delta_2$ in Eq. (<ref>), we find $\beta_2$ is dominant as $\vep\to0^+$ and the ground state
$\phi_{2D}(r)$ can be obtained as TF approximation in the parameter regime I as shown in Sec. <ref>,
\begin{equation}\label{r_sol}
\phi_{2D}(r)=\sqrt{\frac{(R^2-r^2)_+}{2\beta_2}},\text{ where } R=\left(\frac{4\beta_2}{\pi}\right)^{\frac{1}{4}}.
\end{equation}
Then we can compute
\begin{equation}\label{r1_order}
\iint|\phi_{2D}|^4dxdy=\frac{2}{3\sqrt{\pi\beta_2}},
\end{equation}
\begin{equation}\label{r2_order}
\iint|\nabla_{\perp}(|\phi_{2D}|^2)|^2dxdy=\frac{2}{\beta_2}.
\end{equation}
Having $\phi_{2D}$, we can check the consistency of Case I.
By definition of $\delta_1$ in Eq. (<ref>), we get
\begin{equation}\label{db1_3to2}
\delta_1=\delta\iint|\phi_{2D}|^4dxdy=\frac{2\delta}{3\sqrt{\pi\beta_2}},
\end{equation}
while it follows from the definition of $\beta_2$ in Eq. (<ref>),
\begin{equation}\label{bd1_3to2}
\beta_2\sim\delta\int|(|\omega_{1D}|^2)^{\prime}|^2dz=\frac{\delta}{\sqrt{2\pi}\vep^3}.
\end{equation}
Combining Eqs. (<ref>) and (<ref>), we obtain
\vep^{\frac{3}{2}}=O(\vep^{\frac{3}{2}})=
o(\vep)$, which satisfies the requirement for $\delta_1$.
Thus, Case I is self consistent.
Case II. In this case, we solve equation $\mu_{1D}=\frac{z^2}{2\vep^4}-\delta_1\partial_{zz}|\omega_{1D}|^2$ within the support
of $\omega_{1D}$ and get
\begin{equation*}
\omega_{1D}(z)=\frac{\left((z^{*})^2-z^2\right)_+}{2\vep^2\sqrt{6\delta_1}},\quad z^{*}=\left(\frac{45\delta_1\vep^4}{2}\right)^{\frac{1}{5}}.
\end{equation*}
Then we have the identities as
\begin{equation}\label{z_order2}
\int|\omega_{1D}|^4dz=\frac{2}{63}\left(\frac{45}{2}\right)^{\frac{4}{5}}\left(\vep^4\delta_1\right)^{-\frac{1}{5}},
\end{equation}
\begin{equation}
\int|(|\omega_{1D}|^2)^{\prime}|^2dz=\frac{2}{21}\left(\frac{45}{2}\right)^{\frac{2}{5}}\left(\vep^4\delta_1\right)^{-\frac{3}{5}}.
\end{equation}
In the quasi-2D limit regime, i.e. $0<\vep\ll1$, by the definitions of $\beta_2$ and $\delta_2$ in Eq. (<ref>), we find $\beta_2$
is dominant and $\phi_{2D}$ can be obtained as the TF density in parameter regime I shown in Sec. <ref>, which is exactly the same
as Eq. (<ref>).
Similarly to the previous case, we can calculate
\begin{equation}\label{db2_3to2}
\delta_1=\delta\iint|\phi_{2D}|^4dxdy=\frac{2\delta}{3\sqrt{\pi\beta_2}},
\end{equation}
\begin{equation}\label{bd2_3to2}
\beta_2\sim\delta\int|(|\omega_{1D}|^2)^{\prime}|^2dz=\frac{2\delta}{21}\left(\frac{45}{2}\right)^{\frac{2}{5}}\left(\vep^4\delta_1\right)^{-\frac{3}{5}}.
\end{equation}
Combining Eqs. (<ref>) and (<ref>), we can get $\delta_1\approx\frac{2}{45}\left(\frac{105\delta}{\pi}\right)^{\frac{5}{7}}\vep^{\frac{12}{7}}$.
But the requirement is $\delta_1\sim\vep^{5\alpha-4}$ and we get $\alpha=8/7$. This contradicts with the other requirement that $\alpha<1$.
In other words, Case II is inconsistent.
In summary, Case I is true and $\omega_{1D}$ should be chosen as Eq. (<ref>). Thus, mean-field equation for quasi-2D BEC is derived in Eq. (<ref>)
with given constants in Eq. (<ref>).
§ RESCALING WITH HARMONIC POTENTIAL
In this section, we show how to distinguish the four extreme regions in the TF approximations for Eq. (<ref>).
In $d$ ($d=3,2,1$) dimensions, introduce $\tilde{\bx}=\frac{\bx}{x_s}$, and
$\tilde \psi(\tilde{\bx})=x_s^{d/2}\psi(\bx)$ such that $x_s$ is the Thomas-Fermi radius of
the wave function and then the Thomas-Fermi radius in the new scaling is at $O(1)$. It's easy to check that such scaling conserves the normalization condition Eq. (<ref>). Substituting
$\tilde{\bx}$ and $\tilde\psi$ into the time-independent version of (<ref>) and then removing
all $\tilde{\ }$, we get
\begin{equation*}
\frac{\mu}{x_s^2}\psi=-\frac{1}{2x_s^4}\nabla^2\psi+\frac{\gm_0^2|\bx|^2}{2}\psi
\end{equation*}
Since it is assumed $x_s$ is the length scale and the potential term would be $O(1)$. To balance the confinement with repulsive interactions, we need $\frac{\beta}{x_s^{2+d}}\sim O(1)$ and/or $\frac{\delta}{x_s^{4+d}}\sim O(1)$. For simplicity, we require $\frac{\delta}{x_s^{4+d}}=1$, then $x_s=\delta^{\frac{1}{4+d}}$, and further $\beta\sim O(x_s^{2+d})\sim O(\delta^{\frac{2+d}{4+d}})$. So the borderline case is
$\beta=C_0\delta^{\frac{2+d}{4+d}}$. If $C_0\gg1$, $\beta$ term is much more significant than the $\delta$ term; if $|C_0|\ll 1$, $\delta$ term is much more significant than the $\beta$ term.
M. H. Anderson, J. R. Ensher, M. R. Matthewa, C. E.
Wieman and E. A. Cornell,
Science, 269 (1995), 198–201.
K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten,
D. S. Durfee, D. M. Kurn and W. Ketterle,
Phys. Rev. Lett., 75 (1995), 3969–3973.
C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet,
Phys. Rev. Lett., 75 (1995), 1687–1690.
F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari,
Rev. Mod. Phys., 71 (1999), 463.
C. J. Pethick and H. Smith,
Bose-Einstein Condensation in Dilute Gases, (Cambridge University Press, London, 2002).
L. P. Pitaevskii and S. Stringari,
Bose-Einstein Condensation,
(Oxford University Press, New York, 2003).
E. H. Lieb, R. Seiringer and J. Yngvason,
Phy. Rev. A, 61 (2000), 043602.
L. V. Hau, B. D. Busch, C. Liu, Z. Dutton, M. M. Burns, and
J. A. Golovchenko,
Phys. Rev. A, 58 (1998), R54.
W. Bao and Y. Cai,
Kinet. Relat. Mod., 6 (2013), pp. 1–135.
M. J. Holland, D. Jin, M. L. Chiofalo, and J. Cooper,
Phys. Rev. Lett., 78 (1997), 3801.
D. M. Stamper-Kurn,, M. R. Andrews, A. P. Chikkatur, S.
Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle,
Phys. Rev. Lett., 80 (1998), 2027.
W. Bao, D. Jaksch and P. A. Markowich,
J. Comput. Phys., 187 (2003), 318–342.
B. D. Esry and C. H. Greene,
Phys. Rev. A, 60 (1999), 1451–1462.
J. O. Andersen,
Rev. Mod. Phys., 76, 599-639 (2004).
H. Fu, Y. Wang, and B. Gao,
Phys. Rev. A, 67 (2003), 053612.
A. Collin, P. Massignan and C. J. Pethick,
Phys. Rev. A, 75 (2007), 013615.
H. Veksler, S. Fishman and W. Ketterle,
Phys. Rev. A, 90 (2014), 023620.
W. Qi, Z. Liang and Z. Zhang,
J. Phys. B: At. Mol. Opt. Phys., 46 (2013), 175301.
X. Qi and X. Zhang,
Phys. Rev. E, 86 (2012), 017601.
E. Wamba, S. Sabari, K. Porsezian, A. Mohamadou, and T. C. Kofané,
Phys. Rev. E, 89 (2014), 052917.
M. Thøgersen, N. T. Zinner and A. S. Jensen,
Phys. Rev. A, 80 (2009), 043625.
W. Bao, Y. Ge, D. Jaksch, P. A. Markowich and R. M. Weishaeupl,
Comput. Phys. Commun., Vol. 177(2007), 832–850.
W. Bao, F. Y. Lim and Y. Zhang,
Bulletin of the Institute of Mathematics, Vol.2 (2007), pp. 495-532.
L. Salasnich,
J. Phys. A: Math. Theor., 42 (2009), 335205.
L. E. Young-S., L. Salasnich and S. K. Adhikari,
Phys. Rev. A 82 (2010), 053601.
Y. Cai, M. Rosenkranz, Z. Lei and W. Bao,
Phys. Rev. A, 82 (2010), 043623.
N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl
SIAM J. Math. Anal., 47 (2005), pp. 189–199.
W. Bao, L. Le Treust and F. Mehats,
Nonlinearity, 28 (2015), 755-772.
A. Görlitz et al.,
Phys. Rev. Lett., 87 (2001), 130402.
W. Bao and Q. Du,
SIAM J. Sci. Comput., 25 (2004), 1674–1697.
|
1511.00369
|
Branislav Vlahovic, Maxim Eingorn & Cosmin Ilie Uniformity of CMB as a non-inflationary geometrical effect
North Carolina Central University, CREST and NASA Research Centers,
Fayetteville st. 1801, Durham, North Carolina 27707, U.S.A.
North Carolina Central University, CREST and NASA Research Centers,
Fayetteville st. 1801, Durham, North Carolina 27707, U.S.A.
Department of Physics and Astronomy, University of North Carolina at Chapel Hill,
CB#3255, Phillips Hall, Chapel Hill, NC 27599, U.S.A.
Received (Day Month Year)Revised (Day Month Year)
The conventional $\Lambda$CDM cosmological model supplemented by the inflation concept describes the Universe very well. However, there are still a few concerns:
new Planck data impose constraints on the shape of the inflaton potential, which exclude a lot of inflationary models; dark matter is not detected directly, and
dark energy is not understood theoretically on a satisfactory level. In this brief sketch we investigate an alternative cosmological model with spherical spatial
geometry and an additional perfect fluid with the constant parameter $\omega=-1/3$ in the linear equation of state. It is demonstrated explicitly that in the
framework of such a model it is possible to satisfy the supernovae data at the same level of accuracy as within the $\Lambda$CDM model and at the same time
suppose that the observed cosmic microwave background (CMB) radiation originates from a very limited space region. This is ensured by introducing an additional
condition of light propagation between the antipodal points during the age of the Universe. Consequently, the CMB uniformity can be explained without the
inflation scenario.
The corresponding drawbacks of the model with respect to its comparison with the CMB data are also discussed.
PACS Nos.: 04.20.-q; 98.80.-k
§ INTRODUCTION
Even if it was a very successful theory in its own right, predicting for instance the primordial abundance of light elements, Big Bang (BB) cosmology suffers from
serious theoretical drawbacks such as the horizon, flatness, and the grand unified theory (GUT) magnetic monopoles problems. Cosmic
inflation <cit.> can solve all three of those shortcomings, and, moreover, provides a natural mechanism for causally generating
the observed nearly scale invariant spectrum of primordial adiabatic density fluctuations[Alternatives are the
ekpyrotic <cit.> and cyclic models <cit.>.]. It is a theory
that has gained enormous popularity since being introduced in the early 1980s. However, as we shall discuss in some detail in Sec.<ref>, inflation
faces some theoretical challenges that motivate the search for alternative solutions to the problems of BB cosmology.
Moreover there is the puzzle of the initial singularity and the question it raises immediately: what happened prior to the Big Bang? Historically, the first model
of cosmic inflation was discovered by Alexei Starobinsky in 1980, when he suggested the possibility of a non-singular cosmological model <cit.>
as a consequence of including one-loop quantum vacuum polarization effects due to conformally covariant matter fields on a Friedmann-Lemaitre-Robertson-Walker
(FLRW) background. In his model the early universe goes through a maximally symmetric, de Sitter phase, corresponding to an exponential growth of the scale
factor. This accelerated nearly exponential expansion is typical in any inflationary theory ever proposed. Mukhanov & Chibisov <cit.> have
considered the quantum fluctuations of the metric during this intermediate de Sitter phase and shown that the spectrum is nearly scale invariant, with an
amplitude consistent with generating the observed large scale structure of the Universe. All subsequent successful models of inflation replicate those two
In the remainder of this section we will give a brief review of inflationary cosmology, structured as follows: in Sec. <ref> we expand on the
discussion of the mechanisms and generic predictions of inflationary cosmology; in Sec. <ref> we argue how current Cosmic Microwave Background (CMB)
data can be used as a testbed for inflationary model building. We will end this introduction in Sec. <ref> with a discussion of the theoretical
drawbacks that any inflationary scenario faces, thus motivating us to search for alternative solutions to the problems of Big Bang cosmology. In the reminder of
the paper we present such an alternative, based on a curved geometry.
§.§ Mechanisms and Generic Predictions of Inflationary Models
Inflation is made possible whenever the universe is dominated by states of high energy density that do not dilute significantly with the cosmic expansion. As
alluded before, one possibility for the high energy density state is due to curved space corrections to the energy-momentum tensor of a scalar field. This is the
case for what is currently known as $R^2$ or Starobinsky inflation <cit.>. In 1981 Alan Guth <cit.> showed
how inflation can solve the horizon, flatness, and magnetic monopole problems of BB cosmology. In contrast to Starobinsky's model, in Guth's original proposal
inflation was driven by a scalar field (inflaton) trapped in a high energy, false vacuum, state. This mechanism, commonly referred to as “old inflation”, was
soon shown to be unviable, as the phase transition to the true vacuum via bubble collisions cannot be completed efficiently and the universe continues to inflate
forever in most patches thus generating large inhomogeneities <cit.>.
A solution to this “graceful exit” problem was found in 1982 by Linde <cit.> and Albrecht & Steinhardt <cit.>, who proposed that
inflation is driven by a scalar field starting initially perched on the plateau of its effective potential. Subsequently the field “slow rolls” towards the
minimum (true vacuum state), with no quantum tunneling necessary. This “new inflation” scenario has its own theoretical drawbacks, especially when one attempts
to implement realizations of this mechanism based on the theory of high temperature phase transitions. One then has to postulate that the universe started
relatively homogeneous prior to inflation, therefore rendering one of the merits of the theory itself into a possible problem. Moreover, this scenario requires
existence of a pre-inflationary thermal state of the universe. This motivated introduction by Andrei Linde, in 1983, of “chaotic
inflation” <cit.>; in this scenario inflation is driven by a scalar field being initially trapped, due to Hubble friction, at a high value of its
potential. In contrast to new inflation, chaotic inflation does not require assuming existence of an initial false vacuum state. This is replaced by the
assumption that the scalar field initial conditions are chaotic, and that at least in some patches of the universe the inflaton starts at high values
(“large-field”). Chaotic initial conditions can be useful even when inflation occurs near the maximum of the potential, such as in new inflation, “low-field”
models. In fact they are more natural initial conditions, as opposed to the thermal equilibrium state required in the original version of new inflation. This
scenario is commonly referred to as “hilltop inflation” in the literature.
The next theoretical development was the realization that once started, inflation never completely ends everywhere in the universe, at least in most models. In
the case of plateau-like potentials of new inflation there is always a non-zero probability of finding the inflaton field in the decaying metastable false vacuum
state (see, e.g., Refs. StLin,Vilenkin:1983). For chaotic inflation, large quantum fluctuations of the scalar field may end up kicking it back up the
potential, leading to a process of eternal self-reproduction of the universes, i.e. “eternal inflation” <cit.>. On Hubble sized
patches the eternal inflationary universe appears homogeneous, however on much larger scales it has a fractal like structure, consisting of many bubble universes,
i.e. the “multiverse”! This scenario raises yet unsolved problem of defining probabilities in an infinite multiverse, as the distinction between rare and
common events becomes ambiguous without some regularization scheme. We will discuss this in more detail in Sec. <ref>. For recent reviews and
current status and perspectives on inflationary theory written by some of its original proponents see Refs. <cit.>
Measurements of the temperature of the CMB are uniform to one part in $10^5$, indicating that the Universe is homogeneous and isotropic to a high degree prior to
recombination. In standard BB cosmology this becomes problematic, since extrapolating back to the BB, assuming radiation domination, would lead to the universe at
last scattering being comprised of an extremely large number of causally disconnected patches. This is the “horizon problem”. The accelerated expansion phase of
inflationary cosmology solves it, as now our observable universe can be easily generated from one single, causally connected patch that will be stretched out
during inflation. To get a sense of numbers, if the Universe starts to undergo inflation when it had a size corresponding to the Planck length, $l_P\sim10^{-33}$
cm, in $10^{-30}~\mathrm{s}$ of inflation it will become many orders of magnitude larger than our current observable universe ($l~\sim 10^{28}$ cm)! By the same
argument inflation solves all the other cosmological problems of BB. It leads to a Universe that is homogeneous on large scales and spatially flat, i.e.
$\Omega_K\equiv 1-\Omega\ll1$, since any initial amount inhomogeneity/curvature or any other unwanted relics is stretched out/diluted to unobservable levels
during the vacuum dominated expansion, if inflation lasts long enough.
Single field inflationary models typically predict Gaussian spectrum of scalar perturbations, in agreement with CMB data. Quantum sized fluctuations in the
inflaton field are stretched by the exponential expansion and become classical prior to sourcing the CMB temperature anisotropies. Therefore the Gaussian nature
of perturbations, as inferred from CMB temperature anisotropies, is due to the Gaussian statistics in the case of a single quantum field. The power spectrum
contains all the statistical information needed in that case, as odd n-point correlation functions are identically zero and all the others can be related to
products of the two point correlation, i.e. to the power spectrum. In multi-field or single field models with non-trivial kinetic terms, or whenever the slow roll
conditions are violated one expects a significant amount of non-Gaussianity, even for the scalar modes.
In principle there are two different possible types of primordial perturbations: “curvature” (or adiabatic) and “isocurvature”(or entropy). In the former case
each species has an equal perturbation in the number density, $\delta n_f/n_f$. For entropy perturbations one has $\delta\rho=0$, for the total fluid, therefore
the total density, or local curvature, remains homogeneous. Any generic primordial perturbation can be decomposed in a combination of those two orthogonal types.
Requiring that the primordial perturbations have an amplitude that allows them grow via gravitational instability to become the bound structures we observe today
on small scales (clusters, etc.) and matching this with the amplitude measured by CMB experiments at larger scales can place constraints on the type of primordial
perturbations, assuming scale invariance. If isocurvature perturbations were the sources for gravitational structures, then the anisotropy in the CMB would be
about 6 times larger than the one measured <cit.>. COBE was the first experiment to fix this amplitude and since then adiabatic perturbations are
favored. As shown in Refs. Hawking:Irreg,Guth:Fluctuations,Bardeen:1983, assuming radiation domination at the end of inflation, the comoving curvature
perturbation can be related to the inflationary potential $V$ and its derivative, $V_{\phi}=dV/d\phi$, in the following way:
\begin{equation}\label{Eq:CurvPert}
\mathcal{R}=-H\frac{\delta\phi}{\dot{\phi}}\sim\frac{H^2}{2\pi\dot{\phi}}=\frac{V^{3/2}}{2\sqrt{3}\pi V_{\phi}}
\end{equation}
During inflation both $H$ (the Hubble parameter) and $\dot{\phi}$ change very slowly, therefore one generic prediction is that the spectrum is nearly flat, i.e.
scale invariant[A careful analysis shows that there is a mild, logarithmic deviation from scale invariance. This general result was first discovered in
the context of Starobinsky inflation by Mukhanov & Chibisov <cit.>]. Scalar perturbations generated during inflation become super-horizon and no
longer evolve whenever $k\sim aH$. At a later stage, after inflation ends, modes re-enter the horizon and evolve again, starting with the lower wavelength ones,
in a last-out first-in fashion. Gravitational waves, also known as tensor modes, can also be generated during inflation, as discussed in
Ref. Rubakov:GravWaves for example. For massless graviton there are two independent polarizations $(h^+,h^\times)$ of the transverse and traceless parts
of the metric. It is convenient to introduce the power spectrum of perturbations in the following way:
⟨R(k_1)R(k_2)⟩ = (2π)^32π^2/k^3P_R(k)δ^3(k_1+k_2)
⟨h^+,×(k_1)h^+,×(k_2)⟩ = (2π)^32π^2/k^3P_h^+,×(k)δ^3(k_1+k_2), with
$\langle...\rangle$ denoting ensemble average fluctuations. For the gravitational waves, we consider the sum of the two independent polarization and define the
tensor power spectrum as $\mc{P}_t=\mc{P}_{h^+}+\mc{P}_{h^\times}$. The scale dependence of the power spectrum is parametrized in the following way:
P_R(k) = A_s(k_*)(k/k_*)^n_s-1+1/2α_s(k_*)ln(k/k_*)+...
P_t(k) = A_t(k_*)(k/k_*)^n_t+1/2α_t(k_*)ln(k/k_*)+..., where $A_{s,t}$ is the amplitude and $n_{s,t}$ is the spectral
index for the scalar and tensor modes respectively. Running of the spectral index is quantified by $\alpha_{s,t}\equiv dn_{s,t}/d\ln k$. In both cases $k_*$ is an
arbitrary reference or pivot scale where the normalization can in principle be fixed experimentally. In the slow roll approximation one can express the amplitudes
and spectral indices of the potential $V$ and its derivatives ($V_{\phi}=dV/d\phi$, $V_{\phi\phi}=d^2V/d\phi^2$) in the following way:
\begin{align}\label{Eq:SRoll}
A_s &\approx \frac{V}{24\pi^2M_{pl}^4\epsilon} & A_t &\approx \frac{2V}{3\pi^2M_{pl}^4}\\
n_s-1 &\approx 2\eta-6\epsilon & n_t &\approx -2\epsilon,
\end{align}
with $\epsilon\equiv M_{pl}^2V_{\phi}^2/2V^2$ and $\eta=M_{pl}^2V_{\phi\phi}/V$ being parameters that in the slow roll regime are $\ll 1$. There are similar,
higher order expressions for the running of the spectral indices (see, e.g., Eqs. (17)-(19) in Ref. Planck:xxii). One can see from Eq. Eq:SRoll
that the scalar modes generated during a slow roll phase have a nearly scale invariant power spectrum, with deviations sensitive to features in the potential. The
value for the ratio of the tensor to scalar power spectra at the pivot scale can be obtained in the slow roll approximation by combining
Eqs. Eq:Ns-Eq:SRoll: Eq:ScalarTensor r=P_t(k_*)/P_ℛ(k_*)≈16ϵ≈-8n_t This important result is
known as the consistency relation, and in multi-field models becomes an inequality.
Last, but not least, inflaton perturbations naturally lead to a specific pattern in the spectrum of CMB radiation which has been confirmed experimentally by WMAP
and Planck satellites. This provides an extremely powerful link that spans over $20$ orders of magnitude between the low energy Universe during recombination
($T\sim$ eV) and the high energy universe at the end of inflation ($T\sim 10^{15}$ GeV), made possible by the existence of large scale, super-horizon, modes that
did not evolve much after exiting the horizon during inflation. For a theoretical computation of the CMB spectrum of perturbations, that allows a transparent
understanding of the connections between the CMB features and basic cosmological parameters such as the spectral index and amplitude of the primordial
perturbations generated during inflation, see Ref. Mukhanov:CMBSlow.
It is worth mentioning that sufficiently complicated models of inflation can deviate from any of the predictions mentioned in this section, even the ones taken
almost for granted such as the flatness of the universe!
§.§ Tests and Current Status
CMB is the best available probe we have of the early Universe. Its temperature anisotropies and polarization measured with accuracy by the Planck satellite, the
Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT) can be used to test predictions of inflation. First they fix the amplitude of the scalar
primordial fluctuations to $A_s=2.196^{+0.051}_{-0.06}\times10^{-9}$ and the scalar spectral index $n_s=0.9603\pm0.0073$ at the pivot scale $k_*=0.05$
Mpc$^{-1}$. This rules out exact scale invariance, and indicates a red tilt ($n_s<1$) of the spectrum. The amplitude measured from the CMB anisotropy has a value
that is compatible to the one required in order to generate the observed structures in the late Universe such as galaxy clusters from adiabatic initial
perturbations! Moreover, Planck data does not indicate any non-Gaussianity, or isocurvature perturbation, favoring simple, single field models.
The strongest constraints to date placed on inflationary models come from likelihood contours in the $n-r$ plane placed using a combination of datasets. As one
can see from Fig. <ref> concave, plateau like potentials are favored in addition to Hilltop, natural, Starobinsky and Higgs inflation models, which fit the
data well. More importantly, one can rule out a number of models, such as power-law (exponential potential), simplest hybrid models, chaotic models with monomial
potentials (i.e. $V\propto\phi^{p}$), if $p>2$. For an in depth analysis of many of the surviving models see Refs. Encyclopedia,Best.
Marginalized $68\%$ and $95\%$ CL regions for $n_s$ and $r$ from Planck and
other data sets: WP refers to the WMAP large-scale polarization likelihood, highL is high l data from ACT and SPT telescopes. Superimposed are predictions from
various inflationary models for $50-60$ e-foldings. [From Ref. <cit.>]
In 2014 the BICEP2 experiment claimed detection of B-mode polarization in the CMB <cit.>, a pattern that can be generated by the tensor perturbations of
the metric produced during inflation. Their result implied that $r=0.2$, in strong tension with the bound from CMB data by Planck in combination with WMAP
polarization results: $r\lesssim 0.12$! Recently this issue has been settled, when a combined analysis of the data by the Planck and BICEP teams in
Ref. PlanckBICEP lead to the conclusion that the signal can be attributed to large degree to galactic dust and placed an upper bound $r<0.12$. A
detection of tensor modes would have important consequences for inflationary theories. First that would fix the scale of the inflationary potential, as from
Eqs. Eq:SRoll-Eq:ScalarTensor one gets:
Eq:Potential V^1/4=(r/0.12)^1/41.9×10^16GeV Additionally a value of the tensor to scalar ratio $r\gtrsim 0.01$
would favor models with large, super Planckian, inflaton field excursions, as can be seen from the Lyth bound, a relation that relates the inflaton field
evolution to the number of e-folds, N, and $r$: Lyth Δϕ/M_pl≈1/√(8)∫_0^N dN√(r) $N$ is usually assumed to
be an integer between $50-60$ in most inflationary models.
In summary, basic predictions of single field inflation such as a spectrum of adiabatic, nearly scale invariant, Gaussian primordial perturbations have been
confirmed by the current CMB data. This has been interpreted by many as a very strong case for inflation. However, as we shall see in the next section, there are
severe theoretical unsolved problems that the inflationary paradigm faces.
§.§ Theoretical Problems and alternatives to Inflation
In this section we briefly discuss some of the open questions of inflationary cosmology. For simplicity we will omit the trans-Planckian, $\eta$, and Higgs
instability problems, focusing on the more generic, model independent issues, that are most difficult to overcome.
§.§.§ Reheating and subsequent expansion history
At the end of inflation the inflaton field must decay into standard model particles, in a process called reheating. This has important observable consequences,
because in order to relate predictions of slow roll inflations to observables in the CMB one needs to know the time $t_*$ when the observable pivot scale has
exited the horizon during inflation. Obviously this depends on the details of reheating and the subsequent expansion of the Universe. Neither of those two are yet
fully constrained. Reheating can proceed in a number of different ways, and one has to take this into account when deriving constraints in parameter space, as
done for instance in Ref. Reheating. The expansion history of the universe is constrained by the requirement that radiation domination starts prior to
the Big Bang Nucleosynthesis era, or when the temperature of the universe was of the order of $\sim$MeV. It is natural to assume that radiation domination starts
as soon as inflation ends, as the inflaton will decay into light species. However, during reheating in a quadratic potential, the coherent oscillations of the
inflaton field make it act as pressureless dust. Depending on how slow the inflaton field decays into radiation, this “early matter dominated era” (EMDA) can
have important consequences on the expansion history, and thus on relating observables measured at CMB scales to inflationary theory.
§.§.§ Initial Conditions
In order for inflation to start one needs a Universe homogeneous on scales larger than the Hubble radius, prior to inflation, as large kinetic and gradient terms
inhibit inflation. Thus, in order to solve the BB cosmology homogeneity and flatness problem one needs to start from a very homogeneous patch. One “natural”
solution to this problem, proposed by Linde in Ref. Linde:Chaotic, is to assume chaotic initial conditions: by the time the Universe reached the Planck
energy scale, all different energy densities are of the same order. In the case of single field inflation this amounts to:
$\frac{1}{2}\dot{\phi}^2\sim\half(\partial_i\phi)^2\sim V(\phi)\sim M_{pl}^4$. The potential term will quickly dominate the energy density and therefore inflation
can proceed. Patches which start with large kinetic or gradient terms compared with the potential do not inflate, and therefore are disfavored at a classical
level. However, CMB data, and the upper bound it places on the tensor modes: $r\lesssim 0.12$, implies that the scale of inflaton potential is much smaller than
the Planck energy: $V\lesssim 10^{-12}M_{pl}^4$, as one can see from Eq. Eq:Potential. As argued in Ref. Ijjas:2013, for the case of featureless,
plateau-like potentials, such as the ones favored by the data, this becomes problematic. First, this class of potentials require an amount of fine tuning much
larger than the now-disfavored power-law potentials. More importantly, extrapolating back to the Planck scale one finds initial conditions where gradient,
inflation prohibitive, terms dominate over the potential. In order for inflation to proceed one needs to start from a patch that is homogeneous on scales $1000$
times larger than the Hubble horizon! A possible solution to this problem, postulating that inflation starts from a region of negative spatial curvature, has been
proposed in Ref. Guth:2013. This assumption can reduce the required homogeneity length at Planck scale from $\sim 1000$ to $\sim 1-15$ Hubble lengths.
Features in the potential at scales not probed via CMB anisotropies, and hence of no observational consequence, could also help to solve this initial homogeneity
problem. One possibility is to assume an initial stage of inflation that proceeds just as in the case of old inflation, via tunneling from a false vacuum and
bubble nucleation. Symmetry of the true vacuum bubbles guarantee homogeneity prior to the onset of the last stage of inflation. A very intricate possibility that
seems outside of experimental reach, and that is introduced only to reduce the homogeneity scale.
§.§.§ Multiverse and the measure problem
For inflationary models with plateau-like potentials, such as the ones favored by CMB data, there are regions where the quantum fluctuations over a Hubble time,
$\Delta\phi_{qu}\sim H/2\pi$ dominate over the classical evolution, $\Delta\phi_{cl}=\dot{\phi}/H$. As explained in Sec. <ref>, this leads to
an eternally self-reproducing regime, where inflation never ends globally and the universe has a fractal-like structure on scales much larger than the Hubble
length. Actually, eternal inflation and the multiverse is a very generic picture, and there are very few models that do not have a self-reproducing regime. This
leads to a loss of predictability, as any cosmological possibilities are realized in at least some of the bubble universes that are part of the multiverse.
Probabilities cannot be defined unless a regularization scheme is used, since in an infinite multiverse they involve comparison on infinities[For a
review of the measure problem in the multiverse see e.g. Ref. <cit.>]. For example Refs. Bousso:2006,Bousso:2008,Nomura:2011 propose such
regularization schemes; however the predicted probabilities are scheme dependent, and there is no consensus yet as to which one is correct. A naïve, weight by
volume, measure will actually exponentially favor a much younger patch than our Universe <cit.> or Boltzmann brains <cit.>. Those are
known as the youngness paradox and Boltzmann brain problem, respectively.
§.§.§ Large scale correlations and structure problems
Even if CMB data seems to confirm the vanilla $\Lambda$CDM paradigm, there are certain anomalies at large scales, indicative of an incomplete understanding of the
physics in the early universe. For instance, inflation is not consistent with observed large scale angular correlations in CMB data. Inflation models require
angular correlation at all angles, not only at angles up to $\sim 60^{\circ}$, because inflation occurred at all scales. The discrepancy in angular correlations
between CMB and $\Lambda$CDM model that is presented in Fig. <ref> was first noticed in Ref. CMBCOBE and confirmed later in
Refs. CMBWMAP1,SevenWMAP, and NineWMAP.
Angular correlation function of the best fit
$\Lambda$CDM model, a finite size universe model, and WMAP data on large angular scales (adopted from Ref. <cit.>).
There is an obvious difference between the CMB spectrum and predictions of the standard model. The figure also includes a curve that shows very good agreement
between the observable data and a finite size universe model (similar to the model proposed here). Please note that the finite size model gives not only a better
match to the observed correlation function than the $\Lambda$CDM model, but also predicts the distinctive signature in the temperature polarization $(TE)$
spectrum; see Fig. <ref>.
The comparison of the data to the predicted TE power spectrum in a
finite universe model (solid line) and the $\Lambda$CDM model (dashed line), adopted from Ref. <cit.>.
§.§.§ Alternatives to inflation
Motivated by the thorny theoretical problems of inflationary cosmology some authors have proposed alternative scenarios such as string gas cosmology, matter
bounces, ekpyrotic/cyclic scenarios (for a review, see e.g. Ref. Alternatives). As shown in Ref. Gratton:2003 there are two robust cases for
the effective equation of state parameter $w$ required for a scalar field to produce a nearly scale-invariant spectrum of density perturbations: $w\approx -1$
(inflation) and $w\gg 1$ (ekpyrotic/cyclic). The latter scenario predicts almost no tensor perturbations and a degree of non-Gaussianity larger than simple,
single field inflation. If B-modes are detected in the CMB with future experiments, one could use the consistency relation in Eq. Eq:ScalarTensor to test the
slow roll hypothesis vs alternatives. A very interesting review of the ekpyrotic/cyclic scenario and its problems can be found in the Appendix of
Ref. Linde:2014.
Of the three problems of BB cosmology the horizon problem is the most serious one, as “there are possible solutions of the flatness and monopole problems that do
not rely on inflation”<cit.>. The explanation for flatness may be the anthropic principle<cit.>, that intelligent life would only arise
in those patches of universe with $\Omega$ very close to 1; another explanation could be that space is precisely flat at time of BB. Guth's monopoles may be
explained by inflation, or the physics may be such that they never existed in appreciable abundances. An explanation may be that there is no simple gauge group
that is spontaneously broken to the gauge group $SU(3)\times SU(2)\times U(1)$ of the Standard Model. In the remainder of this paper we will discuss a novel,
geometric, approach that could be used to solve the horizon problem without the need for postulating an inflationary epoch.
§ POSITIVE CURVATURE AND $\OMEGA=-1/3$ QUINTESSENCE
Inflation explains uniformity of CMB and solves horizon problem by generating our observable universe from one single causally connected patch.
Homogeneity in the CMB on the level of $10^{-5}$ is explained by inflation era. However, arguments in favor of inflation only exist if space was already
homogeneous before inflation. If the pre inflationary universe was not already homogenous, inflation will not lead to homogeneity <cit.>.
So, the homogeneity problem is pushed only back in time, because the Big Bang itself is taken to be inherently free of correlations.
In addition, after the time when inflation ended to the moment of the last recombination, when CMB was emitted, densities changed from $10^{38} ~\mathrm{kg/m}^3$
to $10^{-17} ~\mathrm{kg/m}^3$, and temperature from $10^{29}$ K to $3000$ K. The high degree of isotropy observed in the microwave background indicates that any
density variations from one region of space to another at the time of decoupling must have been small, at most a few parts in $10^{-5}$. This requires that
changes in density at any part of the universe is the same to the 60 orders of magnitude. After inflation ended, during the time period of 380k years some parts
of the universe are not anymore causally connected and there is no reason that they will have the same density and the same temperature at the time of decoupling.
Taking in account that during that period we also have acoustic oscillations (heat of photon-matter interactions creates a large amount of outward pressure that
counteract gravity) it is statistically unlikely that the CMB will have observed uniformity. This is similar to the horizon problem, but after inflation,
inflation does not help to solve it.
It is important to note that the transfer of the quantum fluctuations in the inflaton field to density perturbations that lead to the CMB anisotropies is not well
For instance, at the end of inflation during reheating the energy of vacuum was transferred to ordinary matter and radiation, but we have no clear idea how the
energy transfer took place and which particles are first created. Later, we can only guess when and how some particles effectively stopped interacting with the
rest of matter and radiation and become cold dark matter. These uncertainties do not allow us to make predictions that will be accurate on 60 decimal places. So,
how do we explain the CMB uniformity?
We argue that the observed uniformity in the CMB does not mean that space was uniform at the time of decoupling. We propose a cosmological model that allows for a
different interpretation of the CMB data and for inhomogeneity of the universe at the early stage. A large-scale homogeneity and isotropy is not required by
classical GR theory. It is well known that in the Big Bang models homogeneity of space cannot be explained, it is simply assumed in initial conditions.
In our positive curvature closed universe model the CMB is always coming from a very small vicinity of the antipodal point. Therefore measuring the same CMB by
looking in the opposite directions of the universe does not represent or reflect the uniformity of the universe at the time of decoupling, because we always
measure CMB originating from approximately the same antipodal point regardless of the direction of observation. For that reason we always must obtain the same
result. Small variations for the CMB are possible and observed, but these variations are the result of measuring CMB from a small region and not exactly from a
single point, and because of the interaction between matter and light during its travel. For instance, depending on the direction we choose to measure the CMB,
light will travel through different galaxies and will interact with different amounts of matter, which will result in small observed variations of the CMB at
large angular scales (as the photons pass through large scale structures) by the integrated Sachs – Wolfe effect <cit.>.
To establish a connection between the uniformity of the earlier universe at the time of decoupling and the CMB we will need to make a completely different kind of
measurement of the CMB. We can see the CMB in any direction we can look in the sky. However, we must keep in mind that the CMB emitted from over-dense regions that would
ultimately form, for instance the Milky Way, is long gone. It left our part of the universe at the speed of light billions of years ago and now forms the CMB for
observers in remote parts of the universe, for an observer located at the antipodal point. To measure the uniformity of the universe at the time of decoupling we
will need to measure the CMB in at least two different points. If, that measurements give the same result, then and only then may we speak about the uniformity of
the CMB and uniformity of the universe at the time of decoupling. However, such measurements are not possible at the present time since we cannot move to
different place to perform such measurement.
Following the above and ideas expressed in the recent papers <cit.>, let us try to find a possibility for an affirmative answer to the
following question: “Is there a chance from the mathematical point of view that the last scattering surface is approximately point-like, or, in other words, that
the CMB radiation originates from a very limited space region in the vicinity of the only one point?" For this purpose, taking into account that the answer is
definitely negative in the framework of the conventional $\Lambda$CDM cosmological model with flat spatial geometry, we consider its extension with the positive
curvature, i.e. the closed space. The corresponding FLRW metric reads:
1 ds^2=c^2dt^2-a^2(t)[dχ^2+sin^2χ(dθ^2+sin^2θdφ^2)] ,
where the hyperspherical coordinates $\chi\in[0,\pi]$, $\theta\in[0,\pi]$, $\varphi\in[0,2\pi)$; $c$ represents the speed of light, while $a(t)$ stands for the
scale factor. This function satisfies the well-known first Friedmann equation:
2 H^2 = (ȧ/a)^2=κε_radc^2/3+κρ̅c^4/3a^3+Λc^2/3-c^2/a^2
= H_0^2[Ω_rad(a_0/a)^4+Ω_mat(a_0/a)^3+Ω_Λ+Ω_K(a_0/a)^2]
where the dot denotes the derivative with respect to time $t$; $\kappa=8\pi G_N/c^4$, with $G_N$ being the Newtonian gravitational constant; $a_0$ is the current
scale factor value, and $H_0\approx67.4$ km/s/Mpc is the current value of the Hubble parameter $H(t)=\dot a/a$. Further, $\varepsilon_{rad}$ and $\bar\rho
c^2/a^3$ represent the energy densities of radiation and nonrelativistic matter (with the average rest mass density $\bar\rho$ in the comoving coordinates),
respectively, while $\Lambda$ stands for the cosmological constant. Finally, the following well-known energy fractions are introduced:
3 Ω_rad=κε_rad(0)c^2/3H_0^2≈0.000055, Ω_mat=κρ̅c^4/3H_0^2a_0^3≈0.31 ,
Ω_Λ=Λc^2/3H_0^2≈0.69, Ω_K=-c^2/H_0^2a_0^2∼-0.005 ,
where $\varepsilon_{rad(0)}$ denotes the today's radiation energy density. Here the numerical values approximately correspond to what is observed according to the
recent Planck results (see Ref. Planck). It should be noted that the value $0.005$ may be considered as an approximate upper limit for the quantity
$|\Omega_K|$ in the considered case of the positive curvature space.
Introducing the dimensionless quantities $\tilde a=a/a_0$ and $\tilde t=H_0t$ (so $\tilde a(0)=1$ and $\tilde a(-\tilde t_0)=0$, where $\tilde t_0$ represents the
dimensionless age of the Universe), we get
4 t̃_0=∫_0^1ãdã/√(Ω_rad+Ω_matã+Ω_Λã^4+Ω_Kã^2)≈0.96
and this value corresponds to $t_0\approx13.9$ billions of years of the Universe evolution. In addition, it is worth mentioning that the today's value of the
deceleration parameter $q=-\ddot a/\left(aH^2\right)$ reads $q_0\approx-0.535$, being in complete agreement with the supernovae data.
Now, let us demand that
5 ∫_-t_0^0cdt/a(t)≈π .
The physical interpretation of this condition is clear: if it holds true, than light travels between the antipodal points during the age of the Universe. One can
easily verify that it is impossible to reach the approximate equality 5 within the standard pure $\Lambda$CDM model. Really, with the help of the values
3 for the left-hand side we immediately obtain
6 √(-Ω_K)∫_0^1dã/√(Ω_rad+Ω_matã+Ω_Λã^4+Ω_Kã^2)≈0.2≠π
However, without the crucial condition 5 the proposed elegant geometrical solution of the horizon problem is not valid. Therefore, in what follows we
continue demanding its fulfilment. Of course, this is apparently forbidden if the composition of the Universe remains
unchanged. In this connection we supplement the positive spatial curvature extension of the conventional cosmological model with an additional perfect fluid with
the constant parameter $\omega$ in the linear equation of state $p_Q=\omega \varepsilon_Q$, where $\varepsilon_Q$ and $p_Q$ represent its energy density and
pressure, respectively. If $-1<\omega<0$, such a perfect fluid may be called quintessence <cit.>.
According to Ref. BUZ1, only two negative values of the constant parameter $\omega$ are admissible from the point of view of the cosmological
perturbations theory: $\omega=-1$ (this possibility is already completely exhausted by introducing the nonzero cosmological constant $\Lambda$, which, as it is
known, can be interpreted as a perfect fluid with the vacuum equation of state $p_{\Lambda}=-\varepsilon_{\Lambda}$) and $\omega=-1/3$ (for the foundations of the
mechanical/discrete approach to cosmological problems inside the cell of uniformity, leading directly to these severe theoretical restrictions, see
Refs. EZcosm1,EKZ2,EZcosm2). Consequently, we make the choice $\omega=-1/3$. It is interesting that the frustrated network of such topological defects
as cosmic strings <cit.> is characterized by exactly the same value of the parameter $\omega$. Such a constituent is also used in
Refs. Melia1,Melia2 within another alternative cosmological model.
It should be noted that the extension of the standard $\Lambda$CDM model with respect to the positive curvature space without quintessence has an important
problematic aspect: the gravitational potential $\phi$ in the comoving coordinates produced by a point-like mass $m$ diverges at the antipodal point where this
mass is actually absent, and this may be considered as a disadvantage of the investigated spherical topology. As shown in Ref. EZcosm1, if
the considered gravitating mass is situated at the point $\chi=0$, then
7 ϕ=2Ccosχ-G_Nm(1/sinχ-2sinχ) ,
where $C$ is some integration constant. It immediately follows from 7 that the function $\phi$ diverges not only at the point $\chi=0$ where it has the
correct Newtonian limit $\phi\rightarrow-G_Nm/\chi$, as it certainly should be, but also at the antipodal point $\chi=\pi$ where there is no any mass! Let us note
as well that the inevitably coming to mind conclusion that this result is nonphysical may be drawn in the case of the infinite-range gravitational interaction
analyzed in Ref. EZcosm1. At the same time, in the opposite case of the finite-range gravitational interaction studied in detail in Ref. EBV
the situation may improve drastically. However, this chance lies beyond the scope of the present manuscript.
The situation with the gravitational potential improves as well, if one introduces the above-mentioned $\omega=-1/3$ quintessence Ref. BUZ1. Now under
the same problem statement
8 ϕ=-G_Nmsin[(π-χ)√(μ^2+1)]/sin(π√(μ^2+1))sinχ ,
where $\mu^2=(3-\kappa\varepsilon_{Q(0)}a_0^2)>0$. Here, in its turn, $\varepsilon_{Q(0)}$ denotes the today's quintessence energy density. Obviously, for
$\sqrt{\mu^2+1}\neq2,3,\ldots$ the function $\phi$ 8 is finite at any point $\chi\in(0,\pi]$, including the suspect antipodal point $\chi=\pi$. The
finiteness remains in force also in the opposite case $(3-\kappa\varepsilon_{Q(0)}a_0^2)<0$.
Avoidance of the gravitational potential divergency in the presence of quintessence in the closed Universe may serve as a valid reason to incorporate such a
constituent into the cosmological model under consideration. The other reason consists in allowing the condition 5 of light traveling between the antipodal
points during the age of the Universe to be satisfied. The next section is entirely devoted to determination of numerical values of the corresponding energy
§ COMPARISON WITH SUPERNOVAE DATA
The contribution of quintessence to the right-hand side of the first Friedmann equation 2 reads:
9 κε_Qc^2/3=H_0^2Ω_Q(a_0/a)^2, Ω_Q=κε_Q(0)c^2/3H_0^2 .
Neglecting the radiation contribution, we get
10 Ω_mat+Ω_Λ+Ω_K+Ω_Q=1 .
Demanding that the acceleration parameter $q$ has approximately the same current value as in the standard $\Lambda$CDM model, that is $q_0\approx-0.535$, we
obtain one more equation for the energy fractions:
11 -1/2Ω_mat+Ω_Λ≈0.535 .
Besides, instead of 6 we have now
12 √(-Ω_K)∫_0^1dã/√(Ω_matã+Ω_Λã^4+Ω_Kã^2+Ω_Qã^2)≈π
For illustrative purposes we single out two concrete examples. First, it is the so-called “exact compensation" case when the positive contribution $\Omega_Q$ of
quintessence exactly compensates the negative one of the spatial curvature $\Omega_K=-|\Omega_K|$: $\Omega_Q=-\Omega_K\approx0.93$, so the rest two fractions
exactly coincide with those from the conventional model, namely $\Omega_{mat}\approx0.31$, $\Omega_{\Lambda}\approx0.69$. In this case the gravitational potential
in the comoving coordinates produced by a point-like mass reads:
13 ϕ=G_Nm/2π-G_Nmcosχ/sinχ(1-χ/π) ,
again being a finite function of $\chi\in(0,\pi]$.
Second, it is the so-called “visible matter" case when the nonrelativistic matter contribution equals $\Omega_{mat}\approx0.040$ (which allows interpreting these
$4\%$ as approximately corresponding to the visible matter only, without dark matter). Then $\Omega_{\Lambda}\approx0.555$, $\Omega_Q\approx 0.721$ and
$\Omega_K\approx-0.316$. Both found sets of energy fractions satisfy the equations 10 and 11 as well as the condition 12. In the following two
tables we present some useful calculations for the “exact compensation" and “visible matter" cases, respectively.
Numerical values for the “exact compensation" case.
$\ z\ $ $\ a/a_0\ $ $\ -t\ $ $\ -ct\ $ $\ d_L\ $
$\ d_{L(\Lambda CDM)}\ $
(Gyr) (Gpc) (Gpc) (Gpc)
0.1 0.91 1.34 0.41 0.48 0.48
0.5 0.67 5.17 1.59 2.85 2.94
1.0 0.50 7.99 2.45 6.23 6.84
1.5 0.40 9.58 2.94 9.57 11.27
2.0 0.33 10.63 3.26 12.69 16.04
1000 0.001 13.89 4.26 489.4 14046
Numerical values for the “visible matter" case.
$\ z\ $ $\ a/a_0\ $ $\ -t\ $ $\ -ct\ $ $\ d_L\ $
$\ d_{L(\Lambda CDM)}\ $
(Gyr) (Gpc) (Gpc) (Gpc)
0.1 0.91 1.34 0.41 0.48 0.48
0.5 0.67 5.23 1.61 2.95 2.94
1.0 0.50 8.26 2.54 6.89 6.84
1.5 0.40 10.13 3.11 11.36 11.27
2.0 0.33 11.45 3.51 16.11 16.04
1000 0.001 16.89 5.17 1385.7 14046
The last two columns of these tables show the numerical values of the luminosity distance
14 d_L(z)=c/H_0(1+z)√(-1/Ω_K)sin[√(-Ω_K)∫_1/1+z^1dã/√(Ω_matã+Ω_Λã^4+(Ω_Q+Ω_K)ã^2)]
within the cosmological model under consideration and the luminosity distance
15 d_L(ΛCDM)(z)=c/H_0(1+z)∫_1/1+z^1dã/√(Ω_matã+Ω_Λã^4)
in the framework of the standard $\Lambda$CDM one, respectively, as functions of the cosmological redshift $z=a_0/a-1$. These dependences are also depicted in
Fig. <ref>. We see that both examples agree with the supernovae data quite well for sufficiently small $z$. Nevertheless, for large $z$ the discrepancy
between the considered model and the conventional one with respect to the luminosity distance becomes wide. Indeed, for $z_*=1100$ (this redshift corresponds
approximately to the recombination time) we have $d_{L(\Lambda CDM)}\approx15400$ Gpc while for two above-mentioned sets of energy fractions the luminosity
distance is about $510$ Gpc or $1450$ Gpc, respectively. This leads to drawbacks discussed briefly in the next section.
The dependence of the luminosity distance $d_L$ on the cosmological redshift $z$ for the standard $\Lambda$CDM model (a black solid line)
and both the “exact compensation" (a blue dashed line) and “visible matter" (a red dotted line) cases.
§ COSMIC MICROWAVE BACKGROUND ANISOTROPY
Certainly, it is not enough to reach agreement only with the supernovae data. Any novel cosmological scenario must take a lot of other tests in order to be called
successful and viable. Among them here we single out consistency with the CMB data. The simplest possible check consists in estimation of the location of the
first acoustic peak directly related to the so-called acoustic scale (see, e.g., Refs. PlanckCMB,Ruth,Rubakov):
theta θ_*=r_s/D_A ,
representing the ratio of the comoving size of the sound horizon at the time of recombination and the corresponding angular diameter distance, which is directly
proportional to the luminosity distance $d_L(z_*)$ divided by $(1+z_*)^2$. Let us restrict ourselves, for example, to the “visible matter" case, which matches
the luminosity distance redshift relation of $\Lambda$CDM to within $1\%$ up to high redshifts, as one can see from Fig. <ref>. Then from the quite natural
requirement $\theta_*/\theta_{*(\Lambda CDM)}\sim 1$ we obtain:
16 r_s/r_s(ΛCDM)∼D_A/D_A(ΛCDM)=d_L/d_L(ΛCDM)≈1450/15400≈0.1 .
Unfortunately, it can hardly be so. Indeed, since the comoving size of the sound horizon $r_s$ is determined by the average sound speed before the recombination,
the requirement 16 means that in the framework of the alternative cosmological model under consideration the average sound speed during this earliest epoch
of the Universe evolution should be decreased in about $10$ times as compared to the standard $\Lambda$CDM one. However, this quantity in its turn is determined
mainly by the radiation contribution and thus is approximately equal to $1/\sqrt{3}$ if defined as $\sqrt{\partial p/\partial\varepsilon}$. The physical mechanism
of such decreasing of the sound speed is actually completely unknown to us. The reasoning may be as follows. First, we cannot simply compensate the radiation
contribution to the sound speed squared by some other contribution of the opposite (negative) sign, because all popular exotic perfect fluids including
quintessence become relativistic at the earliest evolution stage if their initial peculiar velocities are nonzero, and therefore are described by the same
parameter $1/3$ in the equation of state as the radiation. Second, even if there is no thermal motion of some assumed additional fluid at the very beginning of
the Universe evolution, which is hard to imagine, then this fluid should affect not only the sound propagation making it difficult, but also possess the energy
density and pressure compared to those of radiation. This would evidently have grave consequences not only for the epoch before the recombination but also for the
epoch after it, and from the experimental point of view this is also hard to imagine with respect to what is observed or supposed to be known about the early
evolution stages.
The other possibility to save the situation lies in the following. One can reject comparing the acoustic scales and change the initial power spectrum instead in
such a way that its new shape already contains the observed acoustic peaks. Then their positions may be adjusted to the experimental data. According,
e.g., to Ref. Ruth, the conversion from the flat geometry to the spherical space (with the positive spatial curvature) may be implemented, in
particular, by replacing the today's conformal time $t_0$ by the comoving angular diameter distance $\chi(t_0)$ to the last scattering surface, where we have
switched for convenience to the corresponding designations adopted in Ref. Ruth, see, e.g., the formulas (2.254) and (2.255), respectively, for the
positions of the peaks:
l_n∼nπ√(3)t_0/t_* (Ω_K=0) → l_n∼nπ√(3)χ(t_0)/t_* (Ω_K<0) .
The difference between these quantities (see our estimation 16) leads to a conclusion that the repetition frequency for the peaks within the alternative
cosmological model under consideration would be theoretically much higher than in the conventional one, in disagreement with the experimental data, if the same
initial power spectrum is assumed, namely
17 k^3P(k)∼(kt_0)^n-1 ,
where, as usual, $n$ denotes the spectral index Ref. Ruth. Note that, up to numerical factors, $k^3P(k)$ is what we have labeled
$\mathcal{P}_{\mathcal{R}}$ in Eq. Eq:PSpec. Moreover, we neglect running of the spectral index, as Planck data suggest no evidence for it, and assume here a
pivot scale equal to the horizon scale today, i.e. $k_*=1/t_0$. Now, in order to adjust the acoustic peaks of the CMB anisotropy, we are forced to assume the
other form of the spectrum, namely
18 P̃(k)=(χ(t_0)/t_0)^3P(kχ(t_0)/t_0) ,
so that
19 k^3P̃(k)=(kχ(t_0)/t_0)^3P(kχ(t_0)/t_0)∼(kχ(t_0)/t_0t_0)^n-1=[kχ(t_0)]^n-1 .
In other words, we absorb the difference between the angular diameter distance and the comoving distance for a curved universe in the amplitude of the
primordial power spectrum. Let us estimate the change needed in order to match the location of the acoustic peaks in the CMB. From Eqs. 16, 17
and 19 we get:
Therefore, if one assumes $n=0.96$, the best fit value of the spectral index of scalar perturbations from CMB, then the change in the amplitude of the power
spectrum needs to be of the order of $10\%$, to compensate for a $90\%$ difference in the location of the first acoustic peak. This drastic reduction is made
possible by the near scale invariance! CMB fixes the amplitude of the scalar modes to $A_s=2.196^{+0.051}_{-0.06}\times10^{-9}$ at the pivot scale $k_*=0.05$
Mpc$^{-1}$. A value $\tilde A_s\approx2.4\times 10^{-9}$, as required by Eq. Eq:Ratio, is about $4\sigma$ away from the central value. Albeit not excluded,
this seems statistically problematic. However, the pivot scale assumed in Eq. Eq:Ratio is the horizon scale today, corresponding to $k_*=2.2\times 10^{-4}$
Mpc$^{-1}$, a scale $\sim 200$ times larger than the pivot assumed in the CMB analysis! At such large scales the primordial power spectrum is barely constrained
by CMB or other cosmological probes[See, e.g., Fig. 6 in Ref. Bringmann:2011.], therefore a modification of the order of $10\%$ should be well
within experimental bounds.
One has to keep in mind however that the conventional (geometric) estimation of the acoustic peak position can be influenced by the presence of giant voids, and
one of the CMB anomalies, the Cold Spot, could be indicative of a great void <cit.>. This could further relax constrains and make our model even
more plausible.
§ CONCLUSION
In this paper we demonstrate explicitly that in the framework of the $\Lambda$CDM model supplemented in the spherical space with an additional perfect fluid
(namely, quintessence with the constant parameter $w = -1/3$ in the linear equation of state) there is an elegant solution of the horizon problem without
inflation: under the proper choice of the parameters light travels between the antipodal points during the age of the Universe. Consequently, one may suppose that
the observed CMB radiation originates from a very limited space region, which explains its uniformity. Then there seems no need for various inflation scenarios.
In addition this removes any constraints on the uniformity of the universe at the early stage and opens a possibility that the universe was not uniform and that
creation of galaxies and large structures is due to the inhomogeneities that originated in the Big Bang. Besides, in the constructed model the gravitational
potential of any single mass is convergent at any point except for the point of its location, and agreement with the supernovae data is reached. There are certain
serious difficulties when one tries to adjust the proposed concept to the CMB anisotropy arriving at the necessity to change the amplitude of the initial power
spectrum. However, the changes that should be done are well inside experimentally allowed constrains.
§ ACKNOWLEDGEMENTS
This work is supported by NSF CREST award HRD-1345219 and NASA grant NNX09AV07A. The authors thank Yu. Shtanov for useful discussions and valuable comments.
A. H. Guth,
Phys. Rev. D 23, 347 (1981).
A. D. Linde,
Phys. Lett. B 108, 389 (1982).
A. Albrecht and P. J. Steinhardt,
Phys. Rev. Lett. 48, 1220 (1982).
J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,
Phys. Rev. D 64, 123522 (2001)
J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok,
Phys. Rev. D 65, 086007 (2002)
J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,
Phys. Rev. D 66, 046005 (2002)
P. J. Steinhardt and N. Turok,
Science 296, 1436 (2002).
P. J. Steinhardt and N. Turok,
Phys. Rev. D 65, 126003 (2002)
J. Khoury, P. J. Steinhardt and N. Turok,
Phys. Rev. Lett. 92, 031302 (2004)
A. A. Starobinsky,
Phys. Lett. B 91, 99 (1980).
V. F. Mukhanov and G. V. Chibisov,
JETP Lett. 33, 532 (1981)
[Pisma Zh. Eksp. Teor. Fiz. 33, 549 (1981)].
A. A. Starobinsky,
Sov. Astron. Lett. 9, 302 (1983).
L. A. Kofman, A. D. Linde and A. A. Starobinsky,
Phys. Lett. B 157, 361 (1985).
S. W. Hawking, I. G. Moss and J. M. Stewart,
Phys. Rev. D 26, 2681 (1982).
A. H. Guth and E. J. Weinberg,
Nucl. Phys. B 212, 321 (1983).
A. D. Linde,
Phys. Lett. B 129, 177 (1983).
P. J. Steinhardt, “Natural Inflation,” In: The Very Early
Universe, ed. G.W. Gibbons, S.W. Hawking and S.Siklos, Cambridge
University Press, (1983).
A. Vilenkin,
Phys. Rev. D 27, 2848 (1983).
A. D. Linde,
Phys. Lett. B 175, 395 (1986).
A. S. Goncharov, A. D. Linde and V. F. Mukhanov,
Int. J. Mod. Phys. A 2, 561 (1987).
A. H. Guth,
Phys. Rept. 333, 555 (2000)
A. Ijjas, P. J. Steinhardt and A. Loeb,
Phys. Lett. B 723, 261 (2013)
[arXiv:1304.2785 [astro-ph.CO]].
A. H. Guth, D. I. Kaiser and Y. Nomura,
Phys. Lett. B 733, 112 (2014)
[arXiv:1312.7619 [astro-ph.CO]].
A. Linde, Inflationary Cosmology after Planck 2013 (2014); arXiv:1402.0526 [hep-th].
A. R. Liddle and D. H. Lyth,
Phys. Rept. 231, 1 (1993)
S. W. Hawking,
Phys. Lett. B 115, 295 (1982).
A. H. Guth and S. Y. Pi,
Phys. Rev. Lett. 49, 1110 (1982).
J. M. Bardeen, P. J. Steinhardt and M. S. Turner,
Phys. Rev. D 28, 679 (1983).
V. A. Rubakov, M. V. Sazhin and A. V. Veryaskin,
Phys. Lett. B 115, 189 (1982).
P. A. R. Ade et al. [Planck Collaboration],
Astron. Astrophys. 571, A22 (2014)
[arXiv:1303.5082 [astro-ph.CO]].
V. F. Mukhanov,
Int. J. Theor. Phys. 43, 623 (2004)
J. Martin, C. Ringeval and V. Vennin,
Phys. Dark Univ. 5-6, 75 (2014)
[arXiv:1303.3787 [astro-ph.CO]].
J. Martin, C. Ringeval, R. Trotta and V. Vennin,
JCAP 1403, 039 (2014)
[arXiv:1312.3529 [astro-ph.CO]].
P. A. R. Ade et al. [BICEP2 Collaboration],
Phys. Rev. Lett. 112, no. 24, 241101 (2014)
[arXiv:1403.3985 [astro-ph.CO]].
P. A. R. Ade et al. [BICEP2 and Planck Collaborations],
Phys. Rev. Lett. 114, 101301 (2015)
[arXiv:1502.00612 [astro-ph.CO]].
J. Martin, C. Ringeval and V. Vennin,
Phys. Rev. Lett. 114, no. 8, 081303 (2015)
[arXiv:1410.7958 [astro-ph.CO]].
R. Bousso,
Phys. Rev. Lett. 97, 191302 (2006)
R. Bousso, B. Freivogel and I. S. Yang,
Phys. Rev. D 79, 063513 (2009)
[arXiv:0808.3770 [hep-th]].
Y. Nomura,
JHEP 1111, 063 (2011)
[arXiv:1104.2324 [hep-th]].
A. Albrecht and L. Sorbo,
Phys. Rev. D 70, 063528 (2004)
CMBCOBEG. Hinshaw et al., Astrophys. J. Lett. 464 (1996)L25.
CMBWMAP1 D. N. Spergel et al., Astrophys. J. Supp. 148 (2003)175.
SevenWMAP Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Are There Cosmic Microwave Background Anomalies?
Bennett, C., et.al., 2011, ApJS, 192, 17
NineWMAP Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results
Bennett, C.L., et.al., 2013, ApJS., 208, 20B
R. H. Brandenberger,
Int. J. Mod. Phys. Conf. Ser. 01, 67 (2011)
[arXiv:0902.4731 [hep-th]].
S. Gratton, J. Khoury, P. J. Steinhardt and N. Turok,
Phys. Rev. D 69, 103505 (2004)
S. Weinberg, Cosmology, Oxford University Press, Oxford (2008).
Barrow J.D. Barrow, F.J.Tipler, The antropic Cosmological Principle, Oxford, 1986.
S.D. Goldwirth and T. Piran, in the Sixth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and
Relativistic Field Theories, eds. H. Sato and T. Nakamura (World Scientific, 1992), p. 1211.
4a C. Ho, C, Hirata, N. Padmanabhan, U. Seljak, N. Bahcall,
Phys. Rev. D 78 (4) (2008)043519.
4b T. Giannantonio, et al.,
Phys. Rev. D 77 (12) (2008)123520.
4c A. Raccanelli, et al.,
Monthly Notices of the Royal Astronomical Society 386 (4) (2008)2161.
4d B.R. Granett, M.C. Neyrinck, and I. Szapudi,
Astrophys. J. 683 (2) (2008)L99.
B. Vlahovic, New cosmological model and its implications on observational data interpretation, in Proc. of the Time Machine Factory Conf. (Turin, Italy, 14-20
October, 2012), EPJ Web of Conferences 58, 263 (2013); arXiv:1303.3203 [physics.gen-ph].
B. Vlahovic, Spherical shell cosmological model and uniformity of cosmic microwave background radiation, in Proc. of the Low Dimensional Physics and Gauge
Principles Conf. (Yerevan, Armenia, 21-26 September, 2011), eds. V.G. Gurzadyan, A. Klumper, and A.G. Sedrakyan (World Scientific, 2012), p. 241; arXiv:1207.1720
P.A.R. Ade et al. [Planck Collaboration],
Astron. Astrophys. 571, A16 (2014);
arXiv:astro-ph/1303.5076; P.A.R. Ade et al. [Planck Collaboration],
P.G. Ferreira and M. Joyce,
Phys. Rev. Lett. 79, 4740 (1997); arXiv:astro-ph/9707286.
L. Wang and P.J. Steinhardt,
Astrophys. J. 508, 483 (1998); arXiv:astro-ph/9804015.
I. Zlatev, L. Wang and P.J. Steinhardt,
Phys. Rev. Lett. 82, 896 (1999);
A. Burgazli, M. Eingorn and A. Zhuk,
Eur. Phys. J. C 75, 118 (2015); arXiv:astro-ph/1301.0418.
M. Eingorn and A. Zhuk,
JCAP 09, 026 (2012); arXiv:astro-ph/1205.2384.
M. Eingorn, A. Kudinova and A. Zhuk,
JCAP 04, 010 (2013);
M. Eingorn and A. Zhuk,
JCAP 05, 024 (2014); arXiv:astro-ph/1309.4924.
A. Vilenkin and E.P.S. Shellard, Cosmic strings and other topological defects, Cambridge University Press, Cambridge (1994).
S. Kumar, A. Nautiyal and A.A. Sen,
Eur. Phys. J. C 73, 2562 (2013);
F. Melia,
Astrophysics and Space Science 356, 393 (2015); arXiv:astro-ph/1411.5771.
F. Melia,
MNRAS 446, 1191 (2015); arXiv:astro-ph/1406.4918.
M. Eingorn, M. Brilenkov and B. Vlahovic,
Eur. Phys. J. C 75, 381 (2015); arXiv:astro-ph/1407.3244.
P.A.R. Ade et al. [Planck Collaboration],
Astron. Astrophys. 571, A15 (2014);
R. Durrer, The Cosmic Microwave Background, Cambridge University Press, Cambridge (2008).
D.S. Gorbunov and V.A. Rubakov,
Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory, World Scientific,
Singapore (2011).
T. Bringmann, P. Scott and Y. Akrami,
Phys. Rev. D 85, 125027 (2012)
[arXiv:1110.2484 [astro-ph.CO]].
V. G. Gurzadyan, A. L. Kashin, H. Khachatryan, E. Poghosian, S. Sargsyan and G. Yegorian,
Astron. Astrophys. 566, A135 (2014)
[arXiv:1404.6347 [astro-ph.CO]].
I. Szapudi et al.,
Mon. Not. Roy. Astron. Soc. 450, no. 1, 288 (2015)
[arXiv:1405.1566 [astro-ph.CO]].
|
1511.00347
|
Control synthesis from temporal logic specifications has gained popularity in recent years.
In this paper, we use a model predictive approach to control discrete time linear systems with additive bounded disturbances subject to constraints given as formulas of signal temporal logic (STL).
We introduce a (conservative) computationally efficient framework to synthesize control strategies based on mixed integer programs. The designed controllers satisfy the temporal logic requirements, are robust to all possible realizations of the disturbances, and optimal with respect to a cost function. In case the temporal logic constraint is infeasible, the controller satisfies a relaxed, minimally violating constraint. An illustrative case study is included.
§ INTRODUCTION
Model predictive control (MPC), also known as receding horizon control (RHC), is a popular approach to generate
(sub)optimal control strategies for systems with constraints <cit.>,<cit.>. During the last decades, many theoretical aspects of MPC, such as stability and robustness, have been investigated <cit.>,<cit.>. However, most works in the MPC literature focus on simple classes of performance objectives and constraints such as closeness to a reference point or trajectory. Recently, there has been a growing trend in control theory in considering a richer class of constraints that are described using rules and symbolism from formal methods such as temporal logics.
Temporal logics <cit.>, such as linear temporal logic (LTL), computational tree logic (CTL), and signal temporal logic (STL), are able to describe a wide range of specifications. For example, satisfying disjoint sets of constraints infinitely often with specific deadlines for the satisfaction of each of them, (i.e., oscillatory behavior), can be naturally expressed in a temporal logic such as STL.
Temporal logic control based on finite abstractions <cit.>,<cit.>,<cit.> is a correct by construction control synthesis method that involves a finite state machine representation of the control system that is generally expensive to compute. To address this limitation, some works proposed receding horizon approaches to temporal logic control <cit.>, <cit.>. Inspired by <cit.>, the authors in <cit.>,<cit.> have developed methods for translation of LTL constraints into mixed-integer constraints that are used in the controller synthesis algorithm. More recently, <cit.>, <cit.> have extended this methodology to MPC from STL specifications by developing mixed-integer encodings of bounded time model checking.
In this paper, we use STL formulas over predicates in the state of the system to specify correctness requirements. We focus on discrete-time linear systems with additive bounded uncertainties. We propose an MPC approach to the synthesis problem with the goal of satisfying the STL specification globally (i.e., for all times) by characterizing the bounded propagation of uncertainties into STL constraints. We take a conservative approach that is computationally as tractable as STL control of deterministic systems. Furthermore, the notions introduced in this paper enable to treat STL constraints as soft constraints that may be violated if a feasible control policy does not exist. We are thus able to find minimally violating solutions in the presence of uncertainties and limited control actuation.
In the closest related work, the authors in <cit.> use a counter example guided approach to receding horizon control of disturbed systems. A major disadvantage of this approach is that the generation of counter examples may never terminate. Also, taking into account a large number of counter examples is computationally intractable. Furthermore, since there does not exist a global feasibility guarantee for STL MPC, the control algorithm in <cit.> may encounter infeasibility, an issue that we address in this paper.
This paper is organized as follows. First, we informally state the problem in Section <ref>. Next, the receding horizon mechanism of STL control is explained in Section <ref>. After providing technical details about robust prediction in Section <ref>, we formalize the problem as an optimization problem in Section <ref>. Finally, a case study is presented in Section <ref>.
§ PROBLEM STATEMENT
We consider discrete-time dynamical systems of the form
\begin{equation}
\label{eq:dynamics}
\end{equation}
\begin{equation}
\label{eq:secondary_signals}
\end{equation}
where $x \in \mathbb{R}^n $ is the state, $u \in \mathcal{U} \subseteq \mathbb{R}^m$ is the control restricted to an admissible set
$\mathcal{U}$ and $t \in \mathbb{Z}_{\ge 0}$ is time. The exogenous inputs, or additive disturbances, $w[t] \in \mathcal{W} \subset \mathbb{R}^n$ are unknown but restricted to a known bounded set $\mathcal{W}$, which is assumed to be a polytope. Matrices $A,B$ are fully known with appropriate dimensions. We also consider an associated stage cost function $J(x[t],u[t]), J:\mathbb{R}^n \times \mathcal{U} \rightarrow \mathbb{R}$, defined for each time step. The output $y$, and the corresponding $C$, $D$ and $e$ matrices, are defined from the predicates present in the temporal logic specification, as it will be shown below.
STL is a formal framework for describing a wide range of specifications in a convenient and compact form. The formal definition of STL is not presented in this paper, and the interested reader is referred to <cit.>. Informally, STL formulas consist of boolean connectives $\neg$ (negation), $\wedge$ (conjunction), $\vee$ (disjunction), and bounded-time temporal operators $\mathcal{U}_{[a,b]}$ (until between $a$ and $b$), $\Diamond_{[a,b]}$ (eventually between $a$ and $b$) and $\Box_{[a,b]}$ (always between $a$ and $b$) that operate on a finite set of numerical predicates over the underlying signals. In discrete time setting, we assume that interval bounds $a$ and $b$ are nonnegative integers, $b > a$.
STL was originally developed for monitoring dense-time (continuous-time) signals. The reason for restriction of control synthesis to discrete time systems is translating bounded time model checking to mixed integer constraints by using a finite number of integers, a method that has been developed in <cit.>. It should be noted that while discrete time approximations may be used to resemble a continuous time system, validity of a STL formula in discretized time, in general, does not indicate validity of the formula in the original continuous system.
Basically, an STL specification for a control system is intended to require certain behavior from the system. The predicates determine thresholds for functions of state and control values. For example, $(x_1+x_2 \ge 1)$, is a predicate over state values $x_1$ and $x_2$.
In this paper, each predicate, $\mu_i$, is written in the form of:
\begin{equation}
\label{eq:predicates}
\mu_i:=\left(y_i[t] \geq 0 \right)~~~~~ i=1,\cdots,p,
\end{equation}
where the set of outputs $y=(y_1,\cdots, y_p)^T \in \mathbb{R}^p$ are required to be linear functions over the state and control in the form of (<ref>), and $p$ is the number of predicates. Matrices $C,D$ and vector $e$ are appropriately defined with respect to the specification predicates.
For instance, $(x_1+x_2 \ge 1)$ corresponds to output $y=x_1+x_2-1$ where the predicate appearing in the STL formula is $(y\ge0)$. In this case, $C=(1~1), D=0$ and $e=-1$.
Following the terminology from <cit.>, we refer to the state and control as primary signals and to the scalars $y_i$ as secondary signals. Throughout the terminology used in this paper, the specification, which naturally is over the state and control, is written over the secondary signals. The secondary signals notion provides a consistent and convenient format for analyzing the STL constraints later in the paper.
The validity of an STL formula is a function of the secondary signals. STL quantitive semantics defines a function called robustness that associates a scalar for the quality of satisfaction. The robustness function is recursively defined as follows <cit.>:
\begin{equation}
\label{eq:semantics}
\begin{array}{l}
\rho_y^\mu[t]=y[t],
\\
\rho_y^{\neg \varphi} [t]=-\rho_y^\varphi[t], \\
\rho_y^{\varphi \vee \psi}[t]=\max (\rho_y^\varphi[t],\rho_y^\psi[t] ),
\\
\rho_y^{\varphi \wedge \psi} [t]=\min (\rho_y^\varphi[t],\rho_y^\psi[t] ),
\\
\rho_y^{\Diamond_{[a,b]} \varphi}[t]=\underset{t^\prime \in [t+a,t+b]} \max \rho_y^\varphi[t^\prime],
\\
\rho_y^{\Box_{[a,b]} \varphi}[t]=\underset{t^\prime \in [t+a,t+b]} \min \rho_y^\varphi[t^\prime],
\\
\rho_y^{\varphi~\mathcal{U}_{[a,b]} \psi}[t]= \underset{t^\prime \in [t+a,t+b]} \max \left( \min (\rho_y^\varphi[t^\prime], \underset{t^{\prime\prime} \in [t,t^\prime]} \min \rho_y^\psi[t^{\prime\prime}])\right ),
\end{array}
\end{equation}
where $\mu=(y \geq 0)$ is a predicate over scalar $y$ and $\varphi$ and $\psi$ are STL formulas. Basically, $\rho^\varphi_y[t] > 0$ indicates that the formula $\varphi$ is satisfied by secondary signals starting at time $t$ whereas negative robustness indicates violation.
Zero robustness, which has measure zero in the continuous domain, does not indicate satisfaction nor violation. In this paper, however, algorithms are developed such that zero robustness is considered as satisfaction.
Consider the STL formula $\varphi=\Box_{[0,2]} \mu_1 \wedge \Diamond_{[0,3]} \mu_2$ where $\mu_i=(y_i \geq 0), i=1,2$. In words, this formula requires that $\mu_1$ is always true between time zero and two, and, $\mu_2$ is eventually true between time zero and three. By applying the quantitative semantics in (<ref>), the robustness function becomes:
\begin{equation*}
\begin{split}
\rho_y^\varphi[t]=\min & (\min (y_1[t],y_1[t+1],y_1[t+2]), \\ & \max (y_2[t],y_2[t+1],y_2[t+2],y_2[t+3])).
\end{split}
\end{equation*}
Suppose the values of $y_i[t],t=0, \cdots, 5$ are given in Table <ref>:
$y_1[t]$ and $y_2[t]$ for Example <ref>
t 0 1 2 3 4 5
$y_1[t]$ -0.5 1.5 1 1 0.8 -0.5
$y_2[t]$ 3 2 0.5 -1 -1.5 -1
The robustness values are $\rho_y^\varphi[0]=-0.5$, $\rho_y^\varphi[1]=1$, $\rho_y^\varphi[2]=0.5$. Note that the values of $\rho_y^\varphi[t],t \geq 3$ are not computable, in general, without knowing the values of $y_i[t], t \geq 6$.
Not that the robustness function depends on the future values of secondary signals. In this paper, we are interested in finding controls in a receding horizon manner that evolve the system such that an STL formula $\varphi$ is globally satisfied, in the sense that the robustness function is always nonnegative, $\rho_y^\varphi[t] \geq 0 ~ \forall t \in \mathbb{Z}_{\geq 0}$. Global satisfaction can also be viewed as equivalent to satisfying ${\Box}_{[0,\infty]} \varphi$.
Now we are ready to informally state the problem. Our primary aim is to develop a receding horizon controller that is correct, robust and optimal.
Note that, like other finite horizon controllers, suboptimal solutions are sought since the control synthesis algorithm depends on the size of the horizon. Furthermore, an issue in constrained controllers is infeasibility that often arises in highly disturbed or under-actuated systems. In this paper, when encountering infeasibility, we soften the STL predicates using slack variables. For example, $(-2 \geq 0)$ is softened as $(-2+\zeta \geq 0)$, which is satisfied by $\zeta \ge 2$. Details on optimality criterion and predicate softening are given in Section <ref>.
Given a discrete-time system (<ref>),(<ref>), an STL formula $\varphi$ over the predicates in the form of (<ref>), a stage cost function $J(x[t],u[t])$, find a receding horizon controller that is:
* Correct and Robust: The STL specification $\varphi$ is globally satisfied for all realizations of disturbances,
* Optimal: cost $J(x[t],u[t])$ cumulated over the receding horizon is optimized, and
* Minimally Violating: in case the global (finite horizon) satisfaction of $\varphi$ is infeasible, find controls such that the constraints are minimally violated.
After explaining necessary details in Section <ref>, the problem is formulated as an optimization problem in Section <ref>.
§ STL RECEDING HORIZON CONTROL
In this section, we describe the receding horizon mechanism of STL control.
As mentioned earlier, the robustness function depends on the future values of the secondary signals. The time bounds of the temporal operators determine the future times of the secondary signals that are necessary to compute robustness.
The horizon length of STL formula $\varphi$, denoted by $h^\varphi$, is defined as the largest $\tau$ such that that robustness $\rho_y^\varphi[t]$ depends on $y[t+\tau]$.
The formula horizon can be recursively computed as <cit.>:
\begin{equation}
\label{eq:horizon}
\begin{array}{l}
h^\mu=0, \\
h^{\neg \varphi}=h^\varphi, \\
h^{\varphi \wedge \psi}=h^{\varphi \vee \psi}=\max(h^{\varphi},h^{\psi}), \\
h^{\varphi \mathcal{U}_{[a,b]} \psi}=b+\max(h^{\varphi},h^{\psi}), \\
\end{array}
\end{equation}
where $\mu$ is a numerical predicate and $\varphi,\psi$ are STL formulas. In discrete time, robustness at a given time is a function of $h^\varphi$ steps of secondary signals in future, i.e. $\rho_y^\varphi[t]$ is a function of $y[t],y[t+1],\cdots,y[t+h^\varphi]$.
For instance, the horizon length of the specification in Example <ref> is 3. It should be noted that the horizon length of an STL formula, which follows from the terminology used in <cit.>, should not be confused with the horizon length of the control sequence that is used in receding horizon control.
At a given time $t$, the latest robustness that is computable is $\rho_y^\varphi[t-h^\varphi-1]$ given the history of values of secondary signals $y^{his}[t]=\{y[t-h^\varphi-1], \cdots,y[t-1] \}$. The robustness values starting from $\rho_y^\varphi[t-h^\varphi]$ depend on the values of secondary signals starting from time $t$ that are determined by the evolution of the system. We use the system model to predict robustness values and maintain satisfaction, i.e. nonnegative robustness, of the STL specification. Let $\rho_y^\varphi[t+h_p]$ be the most distant in future robustness value that is computed using the predicted values of $y[t],\cdots,y[t+h_p+h^\varphi]$, where $h_p$ is the prediction horizon that is a user chosen integer. Consecutively, at time $t$, we enforce the following constraints to maintain the global STL satisfaction:
\begin{equation}
\label{eq:constraints}
\left \{
\begin{array}{cc}
\rho_y^\varphi[t-h^\varphi] & \geq 0, \\
\rho_y^\varphi[t-h^\varphi+1] & \geq 0, \\
\vdots & \\
\rho_y^\varphi[t+h_p] & \geq 0.
\end{array}
\right.
\end{equation}
Note that when starting the control software, while $t\le h^\varphi$, the set of constraints begin from $\rho_y^\varphi[0]$.
The horizon of predictions for secondary signals is $H=h^\varphi+h_p$ and at each time, the control sequence that is searched for is:
\begin{equation}
\label{eq:control}
u^H[t]=\{ u^t[t],u^{t}[t+1],\cdots,u^t[t+H] \}.
\end{equation}
Note that only the current time control is applied to the system and according to the new measurements, a new control sequence is found at next time step.
In case of $D=0$ in (<ref>), i.e. secondary signals only functions of state, we can reduce the number of constraints and variables for faster computation. At time $t$, the value of $y[t]$ is already determined hence $\rho^\varphi_y[t-h^\varphi]$ is fixed. Therefore, the set of constraints can be written as $\rho^\varphi[t-h^\varphi+1] \geq 0, \cdots, \rho^\varphi[t+h_p] \geq 0$ and since $y[t+H]$ is not dependent on $u^t[t+H]$, the finite horizon control sequence (<ref>) is $u^H[t]=\{u^t[t],u^t[t+1],\cdots,u^t[t+H-1]\}$.
The following theorem establishes closed-loop soundness of the receding horizon controller.
If ${u}^H[t]$ is found for all $t\in \mathbb{Z}_{\geq 0}$ such that (<ref>) is satisfied, then the evolution of the system from applying the closed loop control sequence
\begin{equation*}
u^0[0],u^1[1], \cdots, u^t[t],\cdots
\end{equation*}
globally satisfies $\varphi$.
At time $t$, constraint $\rho_y^\varphi[t-h_p] \geq 0$ is satisfied and $\rho_y^\varphi[t-h_p]$ is fixed since it is not dependent on the further evolution of the system.
By applying $u^t[t]$, $\rho_y^\varphi[t-h_p]+1 \geq 0$ is guaranteed since $u^t[t]$ was found such that all constraints in (<ref>) were satisfied. By induction, $\rho_y^\varphi[t] \geq 0 ~ \forall t \in \mathbb{Z}_{\geq 0}$ and the specification $\varphi$ is hence globally satisfied.
The theorem above does not state that $u^H[t]$ satisfying (<ref>) always exist. Intuitively, larger values of prediction $h_p$ may result in better performance due to longer horizon feasibility. However, in uncertain systems, larger horizon robust controllers are usually excessively conservative, and even infeasible since worst case predictions are made for a longer time. We address infeasibility issues in Sec. <ref>.
The closed-loop synthesis approach of this paper differs from the approach in <cit.>. In the mentioned work, at each time step the control sequence $u^{2h^\varphi}[t]$ is found such that the set of constraints $\rho_y^\varphi[t] \geq 0, \rho_y^\varphi[t+1] \geq 0,\cdots \rho_y^\varphi[t+h^\varphi] \geq 0$ are satisfied and controls $u^t[t],\cdots,u^t[t+h^\varphi-1]$ are fixed by the values found in the previous iterations. Therefore, the synthesis algorithm is based on maintaining STL satisfaction in future whereas the past robustness values are fixed since the control values $u^t[t],\cdots,u^t[t+H-1]$ are not updated anymore, even though they are not yet applied to the system. The control strategy also involves a transient phase where control inputs for initial steps are computed by a slightly different algorithm.
The receding horizon scheme in this paper, on the other hand, is based on both past and future satisfaction maintenance where by measuring the current state and storing the history of the system, the control sequence starting at the current time is updated at each time step. Therefore, our approach is more appropriate for online implementation since in the presence of uncertainties, given online measurements, the controller is able to find solutions that the control sequence stitching approach in <cit.> may not.
§ ROBUST STL SATISFACTION
In this section, we explain our approach to construct a robust MPC mechanism to find controls such that the system evolution satisfies the STL constraints for all possible realizations of disturbances. First, we introduce the notion of positive normal form STL which is important to characterize the behavior of STL constrains with respect to the changes in the secondary signals values. Next we provide the technical details on our approach to STL robust control.
Notation For two same-length vectors $a$ and $b$, inequalities such as $a\ge b$ are interpreted element-wise. The notation $\underline{1}$ stands for appropriately sized vector of all ones. The notation $\otimes$ denotes the Kronecker product.
§.§ Positive Normal Form STL
The robustness function constructed from the quantitative semantics in (<ref>) is a $\min/\max$ function that is piecewise linear and, in general, non-convex. The only scaling operation that appears in the quantitative semantics is multiplication by $-1$ from applying the semantics for negation operator.
An STL formula $\varphi$ is in positive normal form if its robustness is a non-decreasing function with respect to the secondary signals, i.e. if $y_1$ and $y_2$ are two secondary signals such that $y_1[\tau] \geq y_2[\tau] ~\forall \tau \in [t,t+h^\varphi]$, then $\rho_{y_1}^\varphi[t] \geq \rho_{y_2}^\varphi[t]$.
By writing the STL specification in positive normal form, the constraints become monotonic with respect to the values of secondary signals, which enables us to characterize the bounds of propagation of uncertainties into STL constraints. Furthermore, in Section <ref>, we exploit the positive normal form structure to soften the STL constraints by the means of addition of slack variables to the secondary signals.
An STL specification that does not contain negation, i.e. only consisting of $\wedge,\vee, \mathcal{U}_\mathcal{I}, \Diamond_\mathcal{I}, \Box_\mathcal{I}$ where $\mathcal{I}$ is a time interval, is in positive normal form.
By excluding the negation operator, the robustness function consists only of $\min$ and $\max$ operators over the secondary signal values without sign change. Therefore, an increase in secondary signal values will either increase robustness, or does not alter robustness.
Every STL formula can be transformed into positive normal form.
We explain how to recursively transform a general STL formula to positive normal form using negation propagation into numerical predicates and modifying the sign of secondary signals:
* Negation recursively propagates into numerical predicates:
\begin{equation*}
\begin{array}{l}
\neg (\varphi_1 \wedge \varphi_2) = \neg \varphi_1 \vee \neg\varphi_2,
\\
\neg (\varphi_1 \vee \varphi_2) = \neg \varphi_1 \wedge \neg\varphi_2, \\
\neg (\varphi_1 \mathcal{U}_{[a,b]} \varphi_2) = \neg\varphi_1 \mathcal{R}_{[a,b]} \neg\varphi_2, \\
%\footnote{$\mathcal{R}$ is the \emph{release} operator defined as $\varphi_1 \mathcal{R}_{[a,b]} \varphi_2=\Box_{[a,b]} \varphi_2 \vee (\varphi_2 \mathcal{U}_{[a,b]} (\varphi_1\wedge \varphi_2)) $}
\neg \Diamond_{[a,b]} \varphi = \Box_{[a,b]} \neg\varphi, \\
\neg \Box_{[a,b]} \varphi= \Diamond_{[a,b]} \neg\varphi,
\end{array}
\end{equation*}
where $\mathcal{R}$ is the release operator defined as $\varphi_1 \mathcal{R}_{[a,b]} \varphi_2=\Box_{[a,b]} \varphi_2 \vee (\varphi_2 \mathcal{U}_{[a,b]} (\varphi_1\wedge \varphi_2))$.
* Negation of predicates: If $\neg \mu_i$ appears in the STL formula where $\mu_i=(y_i\ge 0)$ that $y_i=c_i x+d_i u+e_i$ is a secondary signal:
* If only $\neg \mu_i$ appears in the STL formula: Redefine $y_i=-c_i x-d_i u-e_i$, thus $\neg$ is removed.
* If both $\mu_i$ and $\neg \mu_i$ appear: Introduce new secondary signal $y_{j}=-c_i x-d_i u - e_i$, thus $\neg \mu_i$ is replaced by the new predicate $\mu_j=(y_j \geq 0)$.
Note that, in the worst case, the number of secondary signals is doubled during the construction of the positive normal form STL formula. As explained in Section <ref>, the computational complexity of control synthesis is exponential with respect to the number of secondary signals.
§.§ Robust Prediction System
At time $t$, given the control sequence $u^H[t]=(u^t[t]^T,\cdots,u^t[t+H]^T)^T$ and the uncertain input sequence $w^H[t]=(w^t[t]^T,\cdots,w^t[t+H-1]^T)^T$, the prediction for secondary signals $y^H[t]=(y^t[t]^T, \cdots,y^t[t+H]^T )^T$ is:
[With a slight abuse of notation, $w^H[t]$,$u^H[t]$ and $y^H[t]$ are interchangeably used for the described sequences and their corresponding representations as column vectors.]
\begin{equation}
\label{eq:matrix}
\Phi_0^H x[t]
\Phi_1^H u^H[t]
\Phi_2^H w^H[t]
\underbar{1} \otimes e,
\end{equation}
where the flow matrices $\Phi_0,\Phi_1,\Phi_2$ are given by:
\begin{equation*}
\label{eq:phi}
\Phi_0^H=
\left(
\begin{array}{c}
C \\ CA \\ \vdots \\ CA^H
\end{array}
\right ),
\end{equation*}
\begin{equation*}
\Phi_1^H
\left(
\begin{array}{ccccc}
D & 0 && \cdots & 0 \\
CB & D && \cdots & 0 \\
CAB & CB && \cdots &0 \\
\vdots & \vdots & \ddots &\ddots& \vdots \\
CA^{H-1}B & CA^{H-2}B & \cdots & CB & D
\end{array}
\right),
\end{equation*}
\begin{equation*}
\Phi_2^H=
\left(
\begin{array}{cccc}
0 & 0 & \cdots & 0 \\
C & 0 & \cdots & \vdots \\
CA & C & \cdots &\vdots \\
\vdots & \vdots & \ddots & \vdots \\
CA^{H-1} & CA^{H-2} & \cdots & C
\end{array}
\right).
\end{equation*}
Since the uncertain inputs belong to the polytope set $\mathcal{W}$, the admissible set of $y^H[t]$ is also a polytope in the finite horizon secondary signals space:
\begin{equation}
\mathcal{Y}_{u^H[t]}=\left \{ y^H[t] \middle| w^t[\tau] \in \mathcal{W}, \tau=t,\cdots,t+H-1 \right\},
\end{equation}
which consists of uncertainty image set, $\left\{ \Phi_2 w^H[t] \middle| w^t[\tau] \in \mathcal{W} \right\} $, which can be computed beforehand, plus an affine map of the control sequence $\Phi_0^H x[t]+\Phi_1^Hu^H[t]+\underline{1}\otimes e$. The finite horizon robust prediction problem is finding controls $u^H[t]$ such that the set of constraints (<ref>) is satisfied for all points in $\mathcal{Y}_{u^H[t]}$.
[fill=cyan] (4,2) – (3,3) – (5,3) – (6,2) – cycle;
[] at (3.5,4.4) $y[t+2]$;
[] at (7.6,2.5) $y[t+1]$;
[->] (2,2.5) – (6.4,2.5);
[->] (3.8,1.5) – (3.8,4);
(4.0,1.2) node[] a);
[dashed] (3,2) – (6,2) – (6,3) – (3,3) – cycle;
[fill=cyan] (4,2) – (3,3) – (5,3) – (6,2) – cycle;
[] at (3.5,4.4) $y[t+2]$;
[] at (7.6,2.5) $y[t+1]$;
[->] (2,2.5) – (6.4,2.5);
[->] (3.8,1.5) – (3.8,4);
(3,2) node[] •;
(4.0,1.2) node[] b);
a) An example representation of the admissible set of $y^H[t]$ in the finite horizon secondary signals space. b) The lower left corner of the axis-aligned minimum bounding box of the set.
Since the robustness function is non-convex, its extreme values do not necessarily lie on the vertices of $\mathcal{Y}_{u^H[t]}$. For example,
consider a simple robustness function $\rho_y^\varphi=\max(y[t+1],y[t+2])$ and assume that for some $u^H[t]$, the set $\mathcal{Y}_{u^H[t]}$ is the shaded parallelogram illustrated in Fig. <ref> a). It is observed that even though all the vertices of the parallelogram are in the positive robustness region, i.e. the first, second and fourth quadrants in Fig. <ref> a), a small section lies in the third quadrant which corresponds to negative robustness.
In order to maintain robust satisfaction, we enforce the constraints (<ref>) at a single point that is the lower left corner of the axis-aligned minimum bounding box of the uncertainty image set (See Fig. <ref> b) for an illustration).
As stated in Theorem <ref> later in the paper, with STL formula being in positive normal form, the robustness is guaranteed to be greater or equal to the robustness at the lower left corner of the box since the secondary signals of the image are greater element-wise. Therefore, the robust prediction system is constructed based on the mentioned lower left corner. Note that this approach is, in general, conservative yet computationally manageable as the set of constraints (<ref>) are imposed for a single point.
The lower left corner of the axis-aligned minimum bounding box of a bounded set $\mathcal{S} \subset \mathbb{R}^n$ is denoted by $\Omega(\mathcal{S})$, where the $i$'th element is given by:
\begin{equation}
\Omega_i(\mathcal{S})=\underset{s \in \mathcal{S} }\inf \quad s_i, ~i=1,\cdots,n.
\end{equation}
A polytope set $\mathcal{P} \subset \mathbb{R}^n$ can be represented by the convex hull of its vertices. For a given polytope $\mathcal{P}$, we define the matrix $P$ whose columns contain its vertices.
The function $ \omega: \mathbb{R}^{n\times m} \rightarrow \mathbb{R}^n$ maps a $n\times m$ matrix to a $n$-dimensional vector where the $i$'th element is:
\begin{equation}
\omega_i(P)=\underset{j} \min \quad P_{ij}.
\end{equation}
In words, the $i$'th element of the vector is the minimum value in the $i$'th row of the matrix.
Let $\mathcal{P}$ be a polytope and matrix $P$ whose columns contain its vertices. Then:
\begin{equation}
\label{eq:omega_equal}
\Omega(\mathcal{P})=\omega(P).
\end{equation}
The value of $i$'th element is the solution to the following linear program:
\begin{equation*}
\begin{array}{cl}
\Omega_i(\mathcal{P})=\min & \sum\limits_{j} P_{ij} \lambda_{jk}, \\
s.t.&0\le\lambda_{jk} \le1,\\
& \sum\limits_{k} \lambda_{jk} =1.
\end{array}
\end{equation*}
It is straightforward to verify that the solution is:
\begin{equation*}
\lambda_{jk}= \left \{
\begin{array}{ll}
1 & j= argmin~ P_{ij}, \\
0 & otherwise.
\end{array}
\right.
\end{equation*}
Therefore, $\Omega_i(\mathcal{P})=\omega_i(P)$ holds element-wise.
If the open-loop control sequence $u^H[t]$ is applied to the system (<ref>),(<ref>), the following relation holds:
\begin{equation}
y^{\omega,H}[t] \le y^H[t],
\end{equation}
\begin{equation}
\label{eq:predicted}
y^{\omega,H}[t]= \Phi_0 x[t]+\Phi_1 u^H[t]+ \omega\left(\Phi_2 (\underline{1}\otimes W)\right)+\underline{1}\otimes e,
\end{equation}
where $W$ is the matrix whose columns are given by the polytope $\mathcal{W}$'s vertices.
The proof follows from the definition of $\Omega$ function where
\begin{equation}
y^{\omega,H}[t]= \Omega \left (\left\{ \mathcal{Y}_{u^H[t]} \right\} \right) \le y^H[t].
\end{equation}
By applying Lemma <ref> to (<ref>) we arrive at (<ref>).
Note that $\omega\left(\Phi_2 (\underline{1}\otimes W)\right)$ is computed prior to starting the control synthesis optimization problem.
Given a linear system (<ref>), (<ref>), STL specification $\varphi$ in positive normal form and secondary signals history $y^{his}[t]$, for any control sequence $u^H[t]$ the following relations hold:
\begin{equation}
\label{eq:robust_predict}
\begin{array}{lcl}
\rho_y^\varphi[t-h^\varphi] & \geq & \rho_{y^{pre}}^\varphi[t-h^\varphi], \\
\rho_y^\varphi[t-h^\varphi+1] & \geq & \rho_{y^{pre}}^\varphi[t-h^\varphi+1], \\
& \vdots & \\
\rho_y^\varphi[t+h_p] & \geq & \rho_{y^{pre}}^\varphi[t+h_p],
\end{array}
\end{equation}
where $y^{pre}$ is the prediction secondary signal that is composed from the stored values of $y^{his}$ and the robust prediction values from (<ref>).
The proof follows immediately from Lemma <ref> and the definition of positive normal form STL.
If the control sequence $u^H[t]$ is found such that the set of constraints
\begin{equation}
\label{eq:constraints_robust}
\left \{
\begin{array}{cc}
\rho_{y^{pre}}^\varphi[t-h^\varphi] & \geq 0, \\
\rho_{y^{pre}}^\varphi[t-h^\varphi+1] & \geq 0, \\
\vdots & \\
\rho_{y^{pre}}^\varphi[t+h_p] & \geq 0,
\end{array}
\right.
\end{equation}
are satisfied, then the open-loop system response of $u^H[t]$ satisfies the set of original constraints (<ref>).
The methodology of this paper can be easily extended to linear time variant (LTV) systems. The required modification is generalizing the flow matrices in (<ref>) for time dependent matrices. In this case, the necessary assumption is that the time dependencies of the system matrices are known.
We have not assumed any restriction on the plant matrix $A$. In principle, an unstable $A$ results in large entries in matrices in (<ref>) that causes control decisions to be very conservative and may even cause infeasibility in longer horizon predictions. A well known technique in MPC literature is designing a control law in the form of $u[t]=Kx[t]+v[t]$, where $K$ is a fixed state feedback gain. The closed loop matrix $A^{cl}=A+BK$ can be designed to possess some important properties such as stability and nil-potency (if the pair $(A,B)$ is controllable). We have not investigated this approach since STL constraints, in general, are different from stability. We also remark that an analogous investigation of robust invariant sets <cit.> for STL MPC is an open problem.
§ OPTIMIZATION BASED CONTROL
In the previous section, we explained our approach to the first objective of Problem <ref>. In this section, after explaining our approach to the formalization of the second and third objectives, i.e. optimality and minimality of violations, we formulate Problem <ref> as an optimization problem. Finally we explain how to express the optimization problem as a mixed integer programming (MIP) problem that is solvable using standard solvers.
§.§ Optimization Problem
A performance criterion is required for selecting a control sequence from the robust open-loop control candidates $u^H[t]$. In principle, there exist two different approaches to define a cost criteria for a nondeterministic system. First, one can optimize the cost using predictions from the nominal system, where the disturbances are assumed to take a known nominal value. A more complicated alternative is optimizing the worst case cost that is admissible by the disturbance realizations. In this paper, we choose the nominal system cost since it is found to perform better in many classical MPC problems <cit.>. Furthermore, worst case cost approaches lead to optimization problems that are computationally more expensive. It should be noted that if the cost is only dependent on controls, the two approaches are identical.
We define $\hat{w}$ as the nominal disturbance, that may be given or may be chosen by some means such as finding the centroid of the polytope $\mathcal{W}$. Given the current state $x[t]$, the nominal system prediction is given by
\begin{equation}
\label{eq:nominal}
\begin{split}
\hat{x}[\tau+1]=& A\hat{x}[\tau]+Bu[\tau]+\hat{w}, \\ & t \le \tau \le t+H-1, \\ & \hat{x}[t]=x[t].
\end{split}
\end{equation}
At time $t$, the finite horizon cost function is:
\begin{equation}
\sum \limits_{\tau=t}^{t+H} J(\hat{x}[\tau],u[\tau]).
\end{equation}
Effectively, within all control sequences that robustly satisfy STL constraints, we choose the one with the least finite horizon nominal evolution cost.
On the other hand, in systems with large disturbances or limited control actuation, it is possible that the STL constraints may be inevitably violated.
If encountered with infeasibility, instead of terminating the control software, we find minimally violating solutions. With STL formula $\varphi$ in positive normal form, a counterfeit increase in all the secondary signals values eventually restores the satisfaction of STL constraints. This method is similar to constraint softening method in <cit.>. We introduce softened secondary signals values as:
\begin{equation}
\label{eq:soft}
y^{his}_{soft}=y^{his}+\underline{1} \zeta,
y^{\omega,H}_{soft}=y^{\omega,H}+\underline{1} \zeta,
\end{equation}
where $\zeta \geq 0$ is the softening slack variable. Note that both robust prediction values and history values are softened to recover feasibility of (<ref>). The artificial secondary signal composed from $y^{his}_{soft}$ and $y^{\omega,H}_{soft}$ is denoted by $y^{soft}$.
We desire that if the STL constraints are infeasible, $\zeta$, the amount of softening, is minimized without optimizing the cost function. Finally, Problem <ref> is formulated as the following optimization problem:
\begin{equation}
\label{eq:optimization}
\begin{array}{ccl}
u^H[t]= & argmin & \sum \limits_{\tau=t}^{t+H} J(\hat{x}[\tau],u[\tau]) + M \zeta \\
& s.t.& \rho_{y^{soft}}^\varphi[t-h^\varphi] \geq0 \\
& & \vdots \\
& & \rho_{y^{soft}}^\varphi[t+h_p] \geq 0, \\
& & \text {Eqn. \eqref{eq:predicted}, \eqref{eq:nominal}, \eqref{eq:soft}}, \\
& & \zeta \geq 0,
\end{array}
\end{equation}
where $M$ is a large penalizing positive number that unifies the separate optimization problems for cost optimality and violation minimality. In case the STL constraints are feasible, large $M$ enforces $\zeta=0$ and the cumulated cost is optimized. In case of infeasibility, effectively, $\zeta$ is minimized without optimization of the cumulated cost.
The smallest $\zeta$ such that the constraints set in (<ref>) is feasible is:
\begin{equation}
\zeta_{min}= \max\{0,-\underset{\tau=t-h^\varphi,\cdots,t+h_p}\min \rho_{y^{pre}}^\varphi[\tau] \}.
\end{equation}
The detailed proof is not included as it basically follows from the monotonicity of the robustness function of a positive normal form STL formula.
Eqn. (<ref>) can be modified by using weights for softening different secondary signals. Multiple softening values for different secondary signals may also be used. In these cases, a practically efficient controller may require a tuning procedure to find the best softening strategy.
Removing the constraint $\zeta \geq 0$ results in a STL robustness maximization receding horizon policy. A negative $\zeta$ value can be viewed as constraint tightening, i.e. how much constraints can be tightened while keeping feasibility.
§.§ Mixed Integer Formulation
STL constraints can be written as mixed integer constraints. One can encode the robustness function by representing $\max$ and $\min$ operations in the quantitative semantics, Eqn. (<ref>), by a set of mixed integer constraints. This method typically introduces a large number of integer variables as the number of $\max/\min$ arguments may be large. An alternative approach, which is computationally more efficient, is binary encoding of quantitative semantics, which has been first introduced by the authors in <cit.>. In this section, we briefly explain this method.
The binary encoding is recursively executed. For a single predicate $\mu=(y \geq 0)$, a binary $z^\mu[t] \in \{0,1\}$ indicates whether the predicate at time $t$ is true, $z^\mu[t]=1$, or false, $z^\mu[t]=0$. The corresponding mixed integer constraints are:
\begin{equation}
\left \{
\begin{array}{ll}
y[t]-Kz^\mu[t] & \le 0, \\
y[t]+K(1-z^\mu[t]) & \geq 0,
\end{array}
\right.
%\left \{
%1 & y[t] \ge 0 \\
%0 & y[t] \le 0
\end{equation}
where $K$ is a sufficiently large positive number. Overall, $p\times(H+h^\varphi+1)$ number of binary variables is required to binary encode the secondary signals in (<ref>).
For encoding the STL formula, an additional number of variables are recursively defined as <cit.>:
* Conjunction: $\psi=\bigwedge_{i=1}^m \varphi_i~$:
\begin{equation}
\left\{
\begin{array}{c}
z^\psi[t] \le z^{\varphi_i} \\
z^\psi[t] \ge 1-m+\sum \limits_{i=1}^m z^{\varphi_i}[t]
\end{array}
\right.
\end{equation}
* Disjunction: $\psi=\bigvee_{i=1}^m \varphi_i~$:
\begin{equation}
\left\{
\begin{array}{c}
z^\psi[t] \ge z^{\varphi_i}[t] \\
z^\psi[t] \le \sum \limits_{i=1}^m z^{\varphi_i}[t]
\end{array}
\right.
\end{equation}
* Eventually $\psi=\Diamond_{[a,b]} \varphi$
\begin{equation}
z^\psi[t] = \bigvee_{\tau=t+a}^{t+b} z^{\varphi_i}[\tau]
\end{equation}
* Always $\psi=\Box_{[a,b]} \varphi$
\begin{equation}
z^\psi[t] = \bigwedge_{\tau=t+a}^{t+b} z^{\varphi_i}[\tau]
\end{equation}
* Until $\psi=\varphi_1\mathcal{U}_{[a,b]} \varphi_2$
\begin{equation}
z^\psi[t] = \bigvee_{\tau=t+a}^{t+b} (z^{\varphi_2}[\tau] \wedge \bigwedge_{\tau^\prime=t}^{\tau} z^{\varphi_1}[\tau^\prime])
\end{equation}
Note that each $z^\psi \in [0,1]$ is not required to be declared as an integer since is automatically enforced to take values from $0$ or $1$. Finally, the set of constraints (<ref>) becomes binary encoded as:
\begin{equation}
\left \{
\begin{array}{cc}
z^\varphi[t-h^\varphi] & = 1, \\
z^\varphi[t-h^\varphi+1] & = 1, \\
\vdots & \\
z^\varphi[t+h_p] & = 1.
\end{array}
\right.
\end{equation}
Depending on the cost function $J$, the optimization problem is a mixed integer linear programming (MILP) (in case of linear $J$), a mixed integer quadratic programming (MIQP) (in case of quadratic $J$) or a mixed integer nonlinear programming (MINLP) (in case of nonlinear $J$). Mixed integer programs are exponentially expensive with respect to the number of integer variables, therefore the real time applications of STL MPC are restricted to small systems.
§ CASE STUDY
We consider a linear system in the form (<ref>), with
\begin{equation*}
%x_1[t+1] \\ x_2[t+1]
\left( \begin{array}{cc} 1 & 0.5 \\ 0 & 0.8 \end{array} \right),
\left( \begin{array}{c} 0 \\ 1 \end{array} \right).
\end{equation*}
The two dimensional state is $x[t]=(x_1[t],x_2[t])^T$ and control input is a scalar. This system represents a double integrator with energy loss, a type of system which is encountered in many engineering problems.
The disturbance $w[t]$ is bounded to the two dimensional box $\left \|w[t] \right \|_\infty \le w_0$, where $w_0$ is assigned multiple values as explained further. The stage cost function is $J(u[t])=\left | u[t] \right |$, which penalizes the control effort.
We desire a STL specification that enforces $x_1$ to oscillate between $2\le x_1\le 4$ and $-4\le x_1\le -2$, with each interval being visited at least once within any five consecutive time steps. The specification, written in positive normal form is:
\begin{equation*}
\begin{split}
\varphi=&\left(\Diamond_{[0,4]} ((y_1\ge 0) \wedge (y_2 \ge 0))\right) \wedge \\ & \left(\Diamond_{[0,4]} ((y_3 \ge 0) \wedge (y_4 \ge 0))\right),
\end{split}
\end{equation*}
for which the corresponding matrices from (<ref>) are:
\begin{equation*}
%y[t]=\left( \begin{array}{cc} 1 & 0 \\ -1 & 0 \\ -1 & 0 \\ 1 & 0 \end{array} \right) x[t] + \left( \begin{array}{c} -2 \\ 4 \\ -2 \\ 4
C=\left( \begin{array}{cc} 1 & 0 \\ -1 & 0 \\ -1 & 0 \\ 1 & 0 \end{array} \right), D=0, e= \left( \begin{array}{c} -2 \\ 4 \\ -2 \\ 4
\end{array} \right).
\end{equation*}
We chose $h_p=2$, which makes $H=h^\varphi+2=6$. The initial state values are $x_1[0]=x_2[0]=0$. The control admissible set is initially assigned as $\left | u \right | \le 20$. We formulate the optimization problem given in (<ref>) as a MILP and find solutions using the MATLAB standard optimization toolbox MILP solver. The assigned value of $M$ in (<ref>) is $10^5$. We simulate the system for 30 time steps. The solution of each step takes less than 0.1s using a 2.8 GHz core i5 processor on an iMac computer. The uncertain values $w[t]$ are drawn randomly from a uniform distribution on $\mathcal{W}$. We investigated the following scenarios:
* We observe that if the system was fully deterministic, $w_0=0$, the optimal-correct solution oscillates between $x_1=2$ and $x_1=4$ as illustrated in Fig. <ref> a). The solution does not enter the mentioned regions as it is unnecessary and is associated with a larger control effort. The robustness function is always zero for this solution.
* We consider a disturbed system $w_0 = 0.2$. We observe that the nominal MPC, i.e. neglecting the presence of disturbances, fails to satisfy the specification. The trajectory is shown in Fig. <ref> b) and the robustness function values in this case are occasionally negative.
* We now implement the robust MPC introduced in this paper. It is observed that the robust solution enters the regions to conservatively maintain STL satisfaction (See Fig. <ref> c)). The robustness function is always above zero for this case.
* We broaden the disturbances set to $\left \|w[t] \right \|_\infty \le 0.5$. The controller fails to find a robust solution to this scenario, see Fig. <ref> d), thus constraint softening is required. This is particularly due to the long horizon considered where the worst case predictions cause infeasibility. We observed that by decreasing the horizon length to $H=5$ and $H=4$, better solutions were found (results not shown).
* We now limit the admissible control set to $\left | u \right | \le 2$ with disturbances from the set $\left \|w[t] \right \|_\infty \le 0.2$. It is impossible to satisfy STL constraints with such a limited control set. The implementation result, shown in Fig. <ref> e), does not satisfy the STL specification, but nevertheless, the observed trajectory is maximally oscillating between the two regions in order to minimally violate the specification.
a) $w[t]=0$ b) $\left \|w[t] \right \|_\infty \le 0.2$
MPC nominal MPC
c) $\left \|w[t] \right \|_\infty \le 0.2$ d) $\left \|w[t] \right \|_\infty \le 0.5$
robust MPC soft constrained MPC
e) $\left |u[t] \right | \le 2$ f) STL robustness function
minimally violating MPC values
Case study results
The robustness values as a function of time are shown in Fig. <ref> f) for the five different scenarios.
§ CONCLUSION AND FUTURE WORK
In this paper, we focused on the connection between optimality and correctness for discrete time linear systems with additive bounded disturbances. Specifically, we proposed a model predictive control approach for the case when the correctness specifications are given as signal temporal logic formulas. We plan to extend these results to classes of discrete-time piecewise affine systems for which the monotonicity property stated in this paper holds, and apply them to controlling traffic networks. We are also working on extending this work to distributed model predictive control, where the additive disturbances of the component subsystems are used in assume-guarantee reasoning schemes. We believe that such techniques have the potential to impact temporal logic optimal control of large networks.
|
1511.00233
|
The convergence and optimality theory of adaptive Galerkin methods is
almost exclusively based on the Dörfler marking. This entails a fixed
parameter and leads to a contraction constant bounded below away from
zero. For spectral
Galerkin methods this is a severe limitation which affects performance. We
present a dynamic marking strategy that allows for a super-linear
relation between consecutive discretization errors, and show
exponential convergence with linear computational
complexity whenever the solution belongs to a Gevrey
approximation class.
§ INTRODUCTION
The modern analysis of adaptive discretizations of partial
differential equations aims at establishing rigorous results of
convergence and optimality. The former results
concern the convergence of the approximate solutions produced by the
successive iterations of the adaptive algorithm towards the exact
solution $u$, with an estimate of the error decay rate measured
in an appropriate norm. On the other hand, optimality results
compare the cardinality of the active set of basis functions used to expand
the discrete solution to the minimal cardinality
needed to approximate the exact solution with similar accuracy;
this endeavor borrows ideas from Nonlinear Approximation
theory. Confining ourselves in the sequel to second-order elliptic
boundary value problems, such kind of analysis has been carried out
first for wavelet discretizations <cit.>, then for $h$-type finite elements
<cit.>, <cit.> dealing just
with convergence, and more recently for spectral-type methods
<cit.>; we refer to the surveys
In contrast, the state of the art for
$hp$-type finite elements is still in evolution; see
<cit.> and the more recent
paper <cit.> which includes optimality estimates.
For all these cases, convergence is proven to be linear, i.e., a certain expression controlling the error (a norm, or a combination of norm and estimator) contracts with some fixed parameter $\rho<1$ from one iteration to the next one, e.g., $\Vert u - u_{k+1} \Vert \leq \rho \Vert u - u_{k} \Vert $. This is typically achieved if the adaptation strategy is based on some form of Dörfler marking (or bulk chasing) with fixed parameter $\theta<1$: assuming that $\sum_{i \in {\cal I}} \eta_i^2$ is some additive error estimator at iteration $k$, one identifies a minimal subset ${\cal I}' \subset {\cal I}$ such that
\sum_{i \in {\cal I}'} \eta_i^2 \geq \theta^2 \sum_{i \in {\cal I}} \eta_i^2
and utilizes ${\cal I}'$ for the construction of the new
discretization at iteration $k+1$.
For wavelet or $h$-type fem discretizations, optimality is guaranteed by performing cautious successive adaptations, i.e., by choosing a moderate value of
$\theta$, say $0 < \theta \leq \theta_{\max } <1$ <cit.>.
This avoids the need of cleaning-up the discrete solution from time to
time, by subjecting it to a coarsening stage.
On the
other hand, the resulting contraction factor $\rho=\rho(\theta)$ turns
out to be bounded from below by a positive constant, say $0 <
\rho_{\min } \leq \rho<1$ (related to the `condition number' of the
exact problem), regardless of the choice of $\theta$.
This entails a limitation on the speed of
convergence for infinite-order methods <cit.>,
but is not restrictive for fixed-order methods <cit.>.
It has been shown in <cit.> that such an obstruction can be
avoided if a specific property of the differential operator
holds, namely the so-called quasi-sparsity of the inverse of the associated stiffness matrix. Upon exploiting this information, a more aggressive marking strategy can be adopted, which judiciously enlarges the set ${\cal I}'$ coming out of Dörfler's stage. The resulting contraction factor $\rho$ can be now made arbitrarily close to $0$ by choosing $\theta$ arbitrarily close to $1$.
When a method of spectral type is used, one expects a fast (possibly,
exponentially fast) decay of the discretization error for smooth
solutions. In such a situation, a slow convergence of the iterations
of the adaptive algorithm would endanger the overall performance of the
method; from this perspective, it is useful to be able to make the contraction factor as close to 0 as desired. Yet, linear convergence of the adaptive iterations is not enough to guarantee the optimality of the method. Let us explain why this occurs, and why a super-linear convergence is preferable, using the following idealized setting.
As customary in Nonlinear Approximation, we consider the
best $N$-term approximation error $E_N(u)$ of the exact
solution $u$, in a suitable norm, using combinations of at most $N$
functions taken from a chosen basis. We prescribe a decay law of
$E_N(u)$ as $N$ increases, which classically for
fixed-order approximations is algebraic and reads
\begin{equation}\label{alg-decay}
\sup_N N^s E_N(u) <\infty,
\end{equation}
for some positive $s$.
However, for infinite-order methods such as spectral approximations an
exponential law is relevant that reads
\begin{equation}\label{exp-decay}
\sup_N {\rm e}^{\eta N^\alpha} E_N(u) <\infty
\end{equation}
for some $\eta>0$ and $\alpha \in (0,1]$, where $\alpha<1$ accommodates the inclusion of $C^\infty$-functions that are not analytic.
This defines corresponding algebraic and exponential
sparsity classes for the exact solution $u$.
These classes are related to Besov and Gevrey regularity of $u$ respectively.
We now assume the
ideal situation that at each iteration of our adaptive algorithm
Throughout the paper, we write $A_k \lesssim B_k$ to indicate that
$A_k$ can be bounded by a multiple of $B_k$, independently of the iteration counter $k$
and other parameters which $A_k$ and $B_k$ may depend on; $A_k\eqsim B_k$
means $A_k\lesssim B_k$ and $B_k \lesssim A_k$. ]
\begin{equation}\label{ideal-decay}
\Vert u - u_{k} \Vert \eqsim N_k^{-s}
\qquad\textrm{or}\qquad
\Vert u - u_{k} \Vert \eqsim e^{-\eta N_k^\alpha},
\end{equation}
where $N_k$ is the cardinality of
the discrete solution $u_k$, i.e., the dimension of the
approximation space activated at iteration $k$. We assume
in addition that the error decays
linearly from one iteration to the next, i.e., it satisfies precisely
\begin{equation}\label{ideal-contraction}
\Vert u - u_{k+1} \Vert = \rho \, \Vert u - u_{k} \Vert.
\end{equation}
If $u$ belongs to a sparsity class of algebraic type, then one easily
gets $N_k \eqsim \rho^{-k/s}$, i.e., cardinalities grow exponentially fast and
\[
\Delta N_k := N_{k+1}-N_k \eqsim N_k
\eqsim \|u-u_k\|^{-1/s},
\]
i.e., the increment of cardinality between consecutive iterations is
proportional to the current cardinality as well as to the error raised to the power $-1/s$. The important message stemming from this ideal setting is
that for a practical adaptive algorithm
one should be able to derive the estimates
$\Vert u - u_{k+1} \Vert \leq \rho \, \Vert u - u_{k} \Vert$ and
$\Delta N_k \lesssim \|u-u_k\|^{-1/s}$, because they yield
\[
N_n=\sum_{k=0}^{n-1}\Delta N_k \lesssim \sum_{k=0}^{n-1}\|u-u_k\|^{-1/s}
\le \|u-u_n\|^{-1/s} \sum_{k=0}^{n-1} \rho^{(n-k)/s}
\lesssim \|u-u_n\|^{-1/s}.
\]
This geometric-series argument is precisely the strategy used in
<cit.> and gives an estimate similar to (<ref>).
The performance of a practical adaptive algorithm is thus quasi-optimal.
If $u$ belongs to a sparsity class of exponential type, instead, the
situation changes radically. In fact, assuming
(<ref>) and (<ref>), one has $e^{-\eta N_k^\alpha} \eqsim \rho^k$, and so
\[
\lim_{k \rightarrow \infty} k^{-1/\alpha} N_k=
\big(|\log \rho|/\eta\big)^{1/\alpha},
\]
i.e., the cardinality $N_k$ grows polynomially.
For a practical adaptive algorithm, proving such a growth
is very hard if not impossible. This obstruction has
motivated the insertion of a coarsening stage in the adaptive
presented in <cit.>.
Coarsening removes the negligible components of the discrete solution
possibly activated by the marking strategy and guarantees that the
final cardinality is nearly optimal <cit.>,
but it does not account for the workload to create $u_k$.
One of the key points of the present contribution is the observation
that if the convergence of the adaptive algorithm is super-linear,
then one is back to the
simpler case of exponential growth of cardinalities which
is ameanable to a sharper performance analysis. To see this,
let us assume a super-linear
relation between consecutive errors:
\begin{equation}\label{ideal-superlinear}
\Vert u - u_{k+1} \Vert = \Vert u - u_{k} \Vert^q
\end{equation}
for some $q>1$. If additionally
$u_k$ satisfies (<ref>), then one infers that
${\rm e}^{-\eta N_{k+1}^\alpha} \eqsim {\rm e}^{-\eta q N_{k}^\alpha}$, whence
\[
\lim_{k \rightarrow \infty} \frac{\Delta N_k}{N_k} =q^{1/\alpha}-1,
\qquad
\lim_{k \rightarrow \infty} \frac{|\log\|u-u_k\||^{1/\alpha}}{N_k} =\eta^{1/\alpha},
\]
the latter being just a consequence of (<ref>).
This suggests that the geometric-series argument may be invoked again in the
optimality analysis of the adaptive algorithm.
This ideal setting does not apply directly to our
practical adaptive algorithm. We will be able to prove
estimates that are consistent with
the preceding derivation to some extend, namely
\[
\Vert u - u_{k+1} \Vert \leq \Vert u - u_{k} \Vert^q,
\qquad \Delta N_k\le Q |\log\|u-u_k\||^{1/\bar{\alpha}},
\]
with constants $Q>0$ and $\bar{\alpha} \in (0,\alpha]$.
Invoking $\|u-u_n\| \leq \|u-u_k\|^{q^{n-k}}$, we then realize that
N_n =
\sum_{k=0}^{n-1} \Delta N_k \le
Q \sum_{k=0}^{n-1}
\big|\log\|u-u_k\| \, \big|^{1/\bar{\alpha}}
\leq
\frac{Q q^{1/\bar{\alpha}}}{q^{1/\bar{\alpha}}-1}
\big|\log\|u-u_n\|\, \big|^{1/\bar{\alpha}}.
Setting $\bar{\eta}:=\big(\frac{Q
we deduce the estimate
\[
\sup_n {\rm e}^{\bar{\eta} N_n^{\bar{\alpha}}} \|u-u_n\| \leq 1,
\]
which is similar to (<ref>), albeit with different class parameters.
The most important parameter is $\bar{\alpha}$. Its possible
degradation relative to $\alpha$
is mainly caused by the fact that the residual, the only
computable quantity accessible to our practical algorithm, belongs to
a sparsity class with a main parameter generally smaller than
that of the solution $u$.
This perhaps unexpected property is
typical of the exponential class and has
been elucidated in <cit.>.
In order for the marking strategy to guarantee super-linear
convergence, one needs to adopt a dynamic choice of Dörfler's
parameter $\theta$, which pushes its value towards $1$ as the
iterations proceed. We accomplish this requirement by equating the
quantity $1-\theta_k^2$ to some function of the dual norm of the
residual $r_k$, which is monotonically increasing and
vanishing at the origin.
This defines our dynamic marking strategy. The order of the
root at the origin dictates the exponent $q$ in the super-linear
convergence estimate of our adaptive algorithm.
The paper is organized as follows. In Sect. 2 we introduce the model
elliptic problem and its spectral Galerkin approximation based on
either multi-dimensional Fourier or (modified) Legendre expansions.
In particular, we highlight properties of the resulting
stiffness matrix that will be fundamental
in the sequel. We present the adaptive algorithm in
Sect. 3, first for the static marking ($\theta$ fixed) and
later for the dynamic marking ($\theta$ tending towards 1); super-linear
convergence is proven. With the optimality analysis in mind, we next
recall in Sect. 4 the definition and crucial properties of a family of
sparsity classes of exponential type, related to Gevrey
regularity of the solution, and we investigate how the sparsity class
of the Galerkin residual deteriorates relative to that of the exact
solution. Finally, in Sect. 5 we relate the cardinality of the
adaptive discrete solutions, as well as the workload needed to compute
them, to the expected accuracy of the approximation. Our analysis
confirms that the proposed dynamic marking strategy avoids any form of
coarsening, while providing exponential convergence
with linear computational complexity, assuming optimal linear solvers.
§ MODEL ELLIPTIC PROBLEM AND GALERKIN METHODS
Let $d \geq 1$ and consider the following elliptic PDE in a $d$-dimensional rectangular domain $\Omega$
with periodic or homogeneous Dirichlet boundary conditions:
\begin{equation}\label{eq:four03}
{ L}u=-\nabla \cdot (\nu \nabla u)+ \sigma u = f \qquad \text{in } \Omega ,
\end{equation}
where $\nu$ and $\sigma$ are sufficiently smooth real coefficients satisfying
$0 < \nu_* \leq \nu(x) \leq \nu^* < \infty$ and $0 < \sigma_* \leq \sigma(x) \leq \sigma^* < \infty$
in $\Omega$; let us set
\alpha_* = \min(\nu_*, \sigma_*) \qquad \text{and} \qquad \alpha^* = \max(\nu^*, \sigma^*) \;.
Let $V$ be equal to $H^1_0(\Omega)$ or $H^1_p(\Omega)$ depending on the boundary conditions and denote by $V^*$ its dual space. We formulate (<ref>) variationally as
\begin{equation}\label{weak}
u \in V \ \ : \quad a(u,v)= \langle f,v \rangle \qquad \forall v \in V \;,
\end{equation}
where $a(u,v)=\int_\Omega \nu \nabla u \cdot \nabla \bar{v} + \int_\Omega \sigma u \bar{v}$ (bar
indicating as usual complex conjugate). We denote by
$\tvert v \tvert = \sqrt{a(v,v)}$
the energy norm of any $v \in V$, which satisfies
\begin{equation}\label{eq:four.1bis}
\sqrt{\alpha_*} \Vert v \Vert_V \leq \tvert v \tvert \leq
\sqrt{\alpha^*} \Vert v \Vert_V \;.
\end{equation}
§.§ Riesz Basis
We start with an abstract formulation which encompasses the two
examples of interest: trigonometric functions and Legendre polynomials.
Let $\phi=\{\phi_k \, : \, k \in {\cal K}\}$ be a Riesz basis of $V$. Thus, we assume
the following relation between a function
$v = \sum_{k \in {\cal K}} \hat{v}_k \phi_k\in V$ and its coefficients:
\begin{equation}\label{eq:propNOBS.3}
\Vert v \Vert_{V}^2 \simeq \sum_{k \in {\cal K}}
|\hat{v}_k|^2 d_k=:\Vert v \Vert_\phi^2 \; ,
\end{equation}
for suitable weights $d_k>0$.
Correspondingly, any element $f \in V^*$ can be expanded along the
dual basis $\phi^*=\{\phi_k^*\}$ as
$f = \sum_{k \in {\cal K}} \hat{f}_k \phi_k^*$, with $\hat{f}_k = \langle f,\phi_k \rangle$,
yielding the dual norm representation
\begin{equation}\label{eq:propNOBS.4}
\Vert f \Vert_{V^*}^2 \ \simeq \ \sum_{k \in {\cal K}} |\hat{f}_k|^2 d_k^{-1}=:\Vert v \Vert_{\phi*}^2\;.
\end{equation}
For future reference, we introduce the vectors $\bv = (\hat v_k d_k^{1/2})_{k\in {\cal K}}$
and $\bF = (\hat f_k d_k^{-1/2})_{k\in {\cal K}}$ as well as the
constants $\beta_*\le\beta^*$ of the norm equivalence in (<ref>)
\begin{equation}\label{eq:propNOBS.7}
\beta_* \Vert v \Vert_{V} \leq \Vert v \Vert_\phi = \|\bv\|_{\ell^2} \leq \beta^* \Vert v \Vert_{V}
\qquad \forall v \in V \;.
\end{equation}
This implies
\begin{equation}\label{eq:propNOBS.8}
\frac1{\beta^*} \Vert f \Vert_{V^*} \leq \Vert f \Vert_{\phi^*} =
\|\bF\|_{\ell^2} \leq \frac1{\beta_*} \Vert f \Vert_{V^*}
\qquad \forall f \in V^* \;.
\end{equation}
The two key examples
to keep in mind are trigonometric basis and tensor products of
Babuška-Shen basis. We discuss them briefly below.
Trigonometric basis.
Let $\Omega=(0,2\pi)^d$ and the trigonometric basis be
\phi_k(x)=\frac1{(2\pi)^{d/2}} \, {\rm e}^{i k \cdot x}
for any $k \in \mathcal{K}=\mathbb{Z}^d$ and $x \in \Omega$.
Any function $v\in L^2(\Omega)$ can be expanded in terms of
$\{\phi_k\}_{k\in\mathbb{Z}^d}$ as follows:
\begin{equation}\label{fourier-exp}
v = \sum_k \hat{v}_k \phi_k \;, \qquad \hat{v}_k=\langle v,\phi_k\rangle \;,
\qquad \ \Vert v \Vert_{L^2(\Omega)}^2= \sum_k |\hat{v}_k|^2 \;.
\end{equation}
The space $V := H^1_p(\Omega)$ of periodic functions with square
integrable weak gradient
can now be easily characterized as the subspace of those $v \in L^2(\Omega)$ for which
\Vert v \Vert_V^2 =
\Vert v \Vert_{H^1_p(\Omega)}^2 = \sum_k |\hat{V}_k|^2 <\infty
\qquad (\text{where }\hat{V}_k := \hat{v}_k d_k^{1/2}, \text{ with }{d_k}:=1+|k|^2).
This induces an isomorphism between $H^1_p(\Omega)$ and $\ell^2(\mathbb{Z}^d)$:
for each $v \in H^1_p(\Omega)$ let $\bv=(\hat{V}_k)_{k \in {\cal K}} \in\ell^2(\mathbb{Z}^d)$ and
note that $\|v\|_{H^1_p(\Omega)} = \|\bv\|_{\ell^2}$.
Likewise, the dual space
$H^{-1}_p(\Omega)=(H^1_p(\Omega))'$ is characterized as the space of those functionals $f$ for which
\Vert f \Vert_{V^*}^2 =
\Vert f \Vert_{H^{-1}_p(\Omega)}^2 = \sum_k |\hat{F}_k|^2
\qquad\text{with}\quad
\hat{F}_k := \hat{f}_k d_k^{-1/2}.
We also have an isomorphism between $H^{-1}_p(\Omega)$ and $\ell^2(\mathbb{Z}^d)$
upon setting $\bF=(\hat{F}_k)_{k \in {\cal K}}$ for $f\in H^{-1}_p(\Omega)$ and
realizing that $\|f\|_{H^{-1}_p(\Omega)} = \|\bF\|_{\ell^2}$.
Babuška-Shen basis.
Let us start with the one-dimensional case $d=1$. Set $I=(-1,1)$, $V:=H^1_0(I)$, and let
$L_k({x})$, $k \geq 0$, stand for the $k$-th Legendre orthogonal polynomial in $I$,
which satisfies ${\rm deg}\, L_k = k$, $L_k(1)=1$ and
\begin{equation}\label{eq:Leg-ort}
\int_I L_k({x}) L_m({x}) \, d{x} = \frac2{2k+1}\, \delta_{km}\;, \qquad m \geq 0 \;.
\end{equation}
The natural modal basis in $H^1_0(I)$ is the Babuška-Shen basis (BS basis), whose elements are defined as
\begin{equation}\label{eq:defBS}
\eta_k({x})=\sqrt{\frac{2k-1}2}\int_{{x}}^1 L_{k-1}(s)\,{d}s =
\frac1{\sqrt{4k-2}}\big(L_{k-2}({x})-L_{k}({x})\big)\ ,
\qquad k \geq 2\;.
\end{equation}
The basis elements satisfy ${\rm deg}\, \eta_k = k$ and
\begin{equation}\label{eq:propBS.1}
(\eta_k,\eta_m)_{H^1_0({I})} =
\int_I \eta_k'({x}) \eta_m'({x}) \, d{x} = \delta_{km}\;, \qquad k,m \geq 2 \;,
\end{equation}
i.e., they form an orthonormal basis for the ${H^1_0({I})}$-inner product. Equivalently,
the (semi-infinite) stiffness matrix ${S}_\eta$ of the Babuška-Shen basis with respect to this inner product is the identity matrix.
We now consider, for simplicity, the two-dimensional case $d=2$ since
the case $d>2$ is similar. Let $\Omega=(-1,1)^2$, $V=H^1_0(\Omega)$,
and consider the tensorized Babuška-Shen basis,
whose elements are defined as
\begin{equation}\label{eq:defBS.2}
\eta_k(x) = \eta_{k_1}(x_1) \eta_{k_2}(x_2)\;, \qquad k_1, k_2 \geq 2 \;,
\end{equation}
where we set $k=(k_1,k_2)$ and $x=(x_1, x_2)$; indices vary in the set
${\cal K}=\{k \in \mathbb{N}^2 \, : \, k_i \geq 2 \text{ for }i=1,2 \}$,
which is ordered `a la Cantor' by increasing total degree $k_\text{tot}=k_1+k_2$ and, for the same total degree, by increasing $k_1$.
The tensorized BS basis is no longer orthogonal, since
\begin{equation}\label{eq:orthogonal}
(\eta_k,\eta_m)_{H^1_0(\Omega)} = (\eta_{k_1},\eta_{m_1})_{H^1_0({I})}(\eta_{k_2},\eta_{m_2})_{L^2({I})}+
(\eta_{k_1},\eta_{m_1})_{L^2({I})}(\eta_{k_2},\eta_{m_2})_{H^1_0({I})} \;,
\end{equation}
whence $ (\eta_k,\eta_m)_{H^1_0(\Omega)} \not = 0$ if and only if $k_1=m_1$ and $k_2-m_2 \in \{-2,0,2\}$, or
$k_2=m_2$ and $k_1-m_1 \in \{-2,0,2\}$. Obviously, we cannot have a Parseval representation of the $H^1_0(\Omega)$-norm of $v = \sum_{k \in {\cal K}} \hat{v}_k \eta_k$ in terms of the coefficients $\hat{v}_k$.
With the aim of getting (<ref>), we follow
<cit.> and we first perform the orthonormalization of the BS
basis via a Gram-Schmidt procedure. This allows us to build a sequence of functions
\begin{equation}\label{eq:defOBS}
\Phi_k = \sum_{m \leq k} g_{mk} \eta_m \;,
\end{equation}
such that $g_{kk}\not = 0$ and
(\Phi_k,\Phi_m)_{H^1_0(\Omega)} = \delta_{km} \qquad \forall \ k, m \in {\cal K}\;.
We will refer to the collection $\Phi:=\{\Phi_k:\ k \in\mathcal{K}\}$
as the orthonormal Babuška-Shen basis (OBS basis), for which
the associated stiffness matrix ${S}_\Phi$ with respect to the $H^1_0(\Omega)$-inner product is the identity matrix.
Equivalently, if ${G}=(g_{mk})$ is the upper triangular matrix which collects the coefficients generated by the
Gram-Schmidt algorithm above, one has
\begin{equation}\label{eq:propOBS.1}
{G}^T {S}_\eta {G} = {S}_\Phi = {I} \;,
\end{equation}
that is the validity of (<ref>) with $d_k=1$.
However, unlike ${S}_\eta$, which is very sparse, the upper
triangular matrix ${G}$ is full; in view of this, we next apply a
thresholding procedure to wipe-out a significant portion of the non-zero entries sitting in the leftmost columns of ${G}$.
This leads to a modified basis whose computational efficiency is quantitatively improved,
without significantly deteriorating the properties of the OBS basis. To be more precise, we
use the following notation: $G_t$ indicates the matrix obtained from
$G$ by setting to zero a certain finite set of off-diagonal entries, so that in particular
${\rm diag}(G_t)= {\rm diag}(G)$; correspondingly, $E:=G_t-G$
is the matrix measuring the truncation quality, for which ${\rm diag}(E)=0$.
Finally, we introduce the matrix
\begin{equation}\label{eq:defNOBS:1}
{S}_\phi = {G}^T_t {S}_\eta {G}_t
\end{equation}
which we interpret as the stiffness matrix associated to the modified BS basis defined in analogy to (<ref>) as
\begin{equation}\label{eq:defNOBS}
\phi_k = \sum_{m \in \mathcal{M}_t(k)} g_{mk} \eta_m \;
\end{equation}
where $\mathcal{M}_t(k)=\{m \leq k: E_{mk}=0\}$. This forms a new basis in
$H^1_0(\Omega)$ (because $k \in \mathcal{M}_t(k)$ and $g_{kk}\not =0$).
We will term it a nearly-orthonormal Babuška-Shen basis (NOBS basis).
Note that only the basis functions $\phi_k$ having total degree not exceeding a certain value, say $p$, may be affected by the compression, while all the others coincide with the corresponding orthonormal basis functions $\Phi_k$ defined in (<ref>).
If $D_{\phi} = {\rm diag}\,S_\phi$, then for any value of $p$ there are strategies to build $G_t$ (depending on $p$)such that the eigenvalues $\lambda$ of
\begin{eqnarray}\label{eqn:maineigpb}
S_\phi x = \lambda D_{\phi} x
\end{eqnarray}
are close to one and bounded from above and away from 0, independently of $p$ <cit.>.
This guarantees the validity for NOBS basis of (<ref>) with $d_k$ equal to the diagonal elements of the matrix $D_\phi$ and (<ref>) for suitable choice of constants $\beta_*,\beta^*$ depending on the eigenvalues of (<ref>) (see <cit.> for more details).
§.§ Infinite Dimensional Algebraic Problem
Let us identify the solution $u = \sum_k \hat{u}_k \phi_k$ of Problem (<ref>)
with the vector $\bu=(\hat{u}_k)_{k\in {\cal K} }$
of its coefficients w.r.t. the basis $\{\phi_k\}_{k\in\mathcal{K}}$. Similarly, let us identify
the right-hand side $f$ with the vector $\bF=(\hat{f}_\ell)_{\ell \in {\cal K}}$ of its dual coefficients.
Finally, let us introduce the bi-infinite, symmetric and positive-definite stiffness matrix
\begin{equation}\label{eq:four100}
{{\bf A}}=(a_{\ell, k})_{\ell,k \in {\cal K}} \qquad \text{with} \qquad
a_{\ell, k}= a(\phi_k,\phi_\ell)\;.
\end{equation}
Then, Problem (<ref>) can be equivalently written as
\begin{equation}\label{eq:four110}
{{\bf A}} \bu = \bF \;,
\end{equation}
where, thanks to (<ref>) and the norm equivalences (<ref>)-(<ref>), ${\bf A}$ defines a bounded invertible operator
in $\ell^2({\cal{K}})$.
Decay Properties of $\bA$ and $\bA^{-1}$.
The decay of the entries of $\bA$ away from the diagonal depends on the
regularity of the coefficients $\nu$ and $\sigma$ of $L$.
If $\nu$ and $\sigma$ are real analytic in a neighborhood of $\Omega$, then
$a_{k,m}$ decays exponentially away from the diagonal <cit.>:
there exist parameters $c_L,\eta_L>0$ such that
\begin{equation}\label{eq:decay-entries-A}
|a_{k,m}| \leq \ c_L {\rm exp}(-\eta_L |k-m|) \quad\forall k,m \in \mathcal{K};
\end{equation}
we then say that $\bA$ belongs to the exponential class
$\cD_e(\eta_L)$, in particular $\bA$ is quasi-sparse.
This justifies the symmetric truncation $\bA_J$ of $\bA$ with parameter $J$, defined as
$(\mathbf{A}_J)_{\ell,k}=a_{\ell,k}$ if $|\ell-k| \leq J$
and $(\mathbf{A}_J)_{\ell,k}=0$ otherwise, which satisfies <cit.>
\begin{equation}\label{decay-A}
\Vert \mathbf{A}-\mathbf{A}_J \Vert \leq
C_{\mathbf{A}} (J+1)^{d-1}{\rm e}^{-\eta_L J}
\end{equation}
for some $C_{\mathbf{A}}>0$ depending only on $c_L$.
Most notably, the inverse matrix $\mathbf{A}^{-1}$ is also
quasi-sparse <cit.>. Precisely $\bA^{-1} \in \cD_e(\bar\eta_L)$ for some $\bar{\eta}_L \in (0,\eta_L]$ and $\bar{c}_{L}$ only dependent on $c_L$ and $\eta_L$.
Thus, there exists an explicit constant $C_{\mathbf{A}^{-1}}$ (depending only on $c_L$ and $\eta_L$) such that the symmetric truncation $(\mathbf{A}^{-1})_J$ of $\mathbf{A}^{-1}$ satisfies
\begin{equation}\label{decayA-1}
\Vert \mathbf{A}^{-1}-(\mathbf{A}^{-1})_J \Vert \leq
C_{\mathbf{A}^{-1}} (J+1)^{d-1} {\rm e}^{-\bar{\eta}_L J} \leq
C_{\mathbf{A}^{-1}} {\rm e}^{-\tilde{\eta}_L J} \; ,
\end{equation}
for a suitable exponent $\wt\eta_L < \bar\eta_L$.
Galerkin Method.
Given any finite index set $\Lambda \subset {\cal K}$, we define the subspace $V_{\Lambda} = {\rm span}\,\{\phi_k\, | \, k \in \Lambda \}$ of $V$; we set $|\Lambda|= \rm{card}\, \Lambda$, so that $\rm{dim}\, V_{\Lambda}=|\Lambda|$. If $v\in V$ admits the
expansion $v = \sum_{k \in {\cal K}} \hat{v}_k \phi_k $, then we define its
projection $P_\Lambda v$ upon $V_\Lambda$ by setting
$P_\Lambda v := \sum_{k \in \Lambda} \hat{v}_k \phi_k$.
Similarly, we define the subspace $V_{\Lambda}^* = {\rm
span}\,\{\phi^*_k\, | \, k \in \Lambda \}$ of $V^*$. If $f$ admits an expansion $f = \sum_{k \in {\cal K}} \hat{f}_k \phi^*_k $, then we define its projection $P^*_\Lambda f$ onto $V^*_\Lambda$ upon setting
$P^*_\Lambda f := \sum_{k \in \Lambda} \hat{f}_k \phi^*_k $.
Given any finite $\Lambda \subset {\cal K}$, the Galerkin approximation of (<ref>) is defined as
\begin{equation}\label{eq:four.2}
u_\Lambda \in V_\Lambda \ \ : \quad a(u_\Lambda,v_\Lambda)=
\langle f,v_\Lambda \rangle \qquad \forall v_\Lambda \in V_\Lambda \;.
\end{equation}
Let $\mathbf{u}_\Lambda$ be the vector collecting the coefficients
of $u_\Lambda$ indexed in $\Lambda$;
let $\mathbf{f}_\Lambda$ be the analogous restriction for
the vector of the coefficients of $f$. Finally, denote by $\mathbf{R}_\Lambda$ the
matrix that restricts a vector indexed in ${\cal K}$ to the portion indexed in $\Lambda$, so that
$\mathbf{R}_\Lambda^H$ is the corresponding extension matrix.
\begin{equation}\label{eq:four120}
\mathbf{A}_\Lambda := \mathbf{R}_\Lambda \mathbf{A} \mathbf{R}_\Lambda^H \;,
\end{equation}
then problem (<ref>) can be equivalently written as
\begin{equation}\label{eq:four130}
\mathbf{A}_\Lambda \mathbf{u}_\Lambda = \mathbf{f}_\Lambda \;.
\end{equation}
For any $w \in V_\Lambda$, we define the residual $r(w) \in V^*$ as
r(w)=f-{L}w = \sum_{k \in {\cal K}} \hat{r}_k(w) \phi^*_k \;, \qquad \text{where} \qquad
\hat{r}_k(w) = \langle f - {L}w, \phi_k \rangle = \langle f,\phi_k \rangle -a(w,\phi_k) \;.
The definition (<ref>) of $u_\Lambda$ is equivalent to the condition $P^*_\Lambda r(u_\Lambda) = 0$, i.e., $\hat{r}_k(u_\Lambda)=0$ for every $k \in \Lambda$.
By the continuity and coercivity of the bilinear form, one has
\begin{equation}\label{eq:four.2.1}
\frac1{\alpha^*} \Vert r(u_\Lambda) \Vert_{V^*} \leq
\Vert u - u_\Lambda \Vert_{V} \leq
\frac1{\alpha_*} \Vert r(u_\Lambda) \Vert_{V^*} \;,
\end{equation}
which in view of (<ref>) and (<ref>) can be rephrased as
\begin{equation}\label{eq:four.2.1bis}
\frac{\beta_*}{\sqrt{{\alpha^*}}} \Vert r(u_\Lambda) \Vert_{\phi^*} \leq
\tvert u - u_\Lambda \tvert \leq
\frac {\beta^*}{\sqrt{\alpha_*}} \Vert r(u_\Lambda) \Vert_{\phi^*} \;.
\end{equation}
Therefore, if $(\hat{r}_k(u_\Lambda))_{k \in {\cal K}}$ are the coefficients of ${r}(u_\Lambda)$ with respect to the dual basis $\phi^*$, the quantity
\begin{equation*}\label{eq:four.2bis}
\Vert r(u_\Lambda) \Vert_{\phi^*}
=\left( \sum_{k \not \in \Lambda} |\hat{R}_k(u_\Lambda)|^2 \right)^{1/2} \qquad\text{with}\quad \hat{R}_k(u_\Lambda) = \hat{r}_k(u_\Lambda) d_k^{-1/2}
\end{equation*}
is an error estimator from above and from below.
However, this quantity is not computable because it involves infinitely
many terms. We discuss feasible versions in
<cit.> but not here.
Equivalent Formulation of the Galerkin Problem.
For future reference, we now rewrite the Galerkin problem (<ref>) in an equivalent (infinite-dimensional) manner.
$$\mathbf{P}_\Lambda: \ell^2(\mathcal{K}) \to \ell^2(\mathcal{K})$$
be the projector operator defined as
\[
(\mathbf{P}_\Lambda \mathbf{v})_\lambda :=
\begin{cases}
v_\lambda & \text{\rm if } \lambda\in\Lambda \;, \\
0 & \text{\rm if } \lambda\notin\Lambda \;.
\end{cases}
\]
Note that $\mathbf{P}_\Lambda$ can be represented as a diagonal bi-infinite matrix whose diagonal elements
are $1$ for indexes belonging to $\Lambda$, and zero otherwise.
We set $\mathbf{Q}_\Lambda := \mathbf{I}-\mathbf{P}_\Lambda$ and
introduce the bi-infinite matrix $\widehat{\mathbf{A}}_\Lambda:=
\mathbf{P}_\Lambda \mathbf{A} \mathbf{P}_\Lambda + \mathbf{Q}_\Lambda$ which
is equal to $\mathbf{A}_\Lambda$ for indexes in $\Lambda$ and to the identity matrix, otherwise.
The definitions of the projectors $\mathbf{P}_\Lambda$ and
$\mathbf{Q}_\Lambda$ yield the following property:
\begin{equation}\label{prop:inf-matrix}
\textit{If $\mathbf{A}$ is invertible with
$\mathbf{A}\in\mathcal{D}_e(\eta_L)$, then the same holds for
\end{equation}
Furthermore, the constants $C_{\widehat{\mathbf{A}}_\Lambda}$ and $C_{(\widehat{\mathbf{A}}_\Lambda)^{-1}}$ which appear in the inequalities (<ref>) and (<ref>) for $\widehat{\mathbf{A}}_\Lambda$ can be bounded uniformly in $\Lambda$, since in turn they can be bounded in terms of $\eta_L$ and $c_L$, respectively.
Now, let us consider the following extended Galerkin problem:
find $\hat{\mathbf{u}}\in\ell^2$ such that
\begin{equation}\label{eq:inf-pb-galerkin}
\widehat{\mathbf{A}}_\Lambda \hat{\mathbf{u}}
= \mathbf{P}_\Lambda \mathbf{f}\ .
\end{equation}
Let $\mathbf{u}_\Lambda$ be the Galerkin solution to (<ref>);
then, it is easy to check that $\hat{\mathbf{u}}={\mathbf{R}}_\Lambda^H \mathbf{u}_\Lambda$.
§ ADAPTIVE SPECTRAL GALERKIN METHOD
In this section we present our adaptive spectral Galerkin method,
named DYN-GAL, that is based on a new notion of marking
strategy, namely a dynamic marking. In Section
<ref> we recall the enriched Dörfler marking
strategy, introduced in <cit.>, which represents an enhancement of the classic Dörfler marking strategy. In Section <ref> we introduce the dynamic marking strategy, present DYN-GAL and prove its quadratic convergence.
§.§ Static Dörfler Marking
Fix any $\theta \in (0,1)$ and set
$\Lambda_0= \emptyset$, $u_{\Lambda_0}=0$.
For $n=0,1, \dots$, assume that $\Lambda_n$ and $u_n := u_{\Lambda_n} \in V_{\Lambda_n}$
and $r_n := r(u_n) = Lu_n-f$
are already computed and choose
$\Lambda_{n+1} := \Lambda_n \cup \partial\Lambda_n$ where the set $\partial\Lambda_n$ is built by a two-step procedure that we call $\textbf{E-D\"ORFLER}$ for enriched Dörfler:
$\partial\Lambda_n = \textbf{E-D\"ORFLER} \, (\Lambda_n,\theta)$
$\widetilde{\partial\Lambda}_n={\bf{DORFLER}} \, (r_n,\theta)$
${\partial\Lambda}_n={\bf ENRICH} \, (\widetilde{\partial\Lambda}_n,J)$
The first step is the usual Dörfler's marking
with parameter $\theta$:
\begin{equation}\label{doerfler}
\Vert P_{\wt{\partial \Lambda}_{n}}^* r_n \Vert_\ps
= \Vert P_{\wt\Lambda_{n+1}}^* r_n \Vert_\ps \geq \theta
\Vert r_n \Vert_\ps
\quad\textrm{or}\quad
\sum_{k \in \wt{\partial\Lambda}_{n}} |\hat{R}_k(u_n)|^2
\ \geq \ \theta^2 \sum_{k \in \mathcal{K}} |\hat{R}_k(u_n)|^2 \;,
\end{equation}
with $\wt\Lambda_{n+1}=\Lambda_{n}\cup \wt\partial\Lambda_{n}$.
This also reads
\begin{equation}\label{doerfler-equiv}
\Vert r_n-P_{\wt\Lambda_{n+1}}^* r_n \Vert_\ps \leq \sqrt{1-\theta^2}
\Vert r_n \Vert_\ps \;
\end{equation}
and can be implemented by rearranging the coefficients
$\hat{R}_k(u_n)$ in decreasing order of modulus
and picking the largest ones (greedy approach).
However, this is only an idealized algorithm because the number of coefficients
$\hat{R}_k(u_n)$ is infinite. This marking is known to yield a
contraction property between $u_n$ and the Galerkin solution $\wt u_{n+1}\in V_{\wt\Lambda_{n+1}}$ of the form
\[
\tvert u-\wt u_{n+1} \tvert \leq
\rho(\theta) \tvert u-u_n \tvert \;,
\]
with $\rho(\theta) = \sqrt{1-\frac{\alpha_*}{\alpha^*}\theta^2}$
<cit.>. When $\alpha_* < \alpha^*$ we see that
in contrast to (<ref>), $\rho(\theta)$ is bounded below
away from $0$ by $\sqrt{1-\frac{\alpha_*}{\alpha^*}}$.
The second step of $\textbf{E-D\"ORFLER}$ is meant to remedy this
situation and hinges on the a priori structure of $\bA^{-1}$ already
alluded to in <ref>. The goal is to augment the set
$\wt{\partial\Lambda}_{n}$ to $\partial\Lambda_{n}$
judiciously. This is contained in the following proposition (see <cit.>) whose proof is reported here for completeness.
Let $\widetilde{\partial\Lambda}_n={\bf{DORFLER}} \, (r_n,\theta)$, and let $J=J(\theta)>0$ satisfy
\begin{equation}\label{eq:aggr2}
C_{\bA^{-1}} \textrm{e}^{-\wt\eta_L J} \leq \sqrt{\frac{1-\theta^2}{\alpha_* \alpha^*}} \;,
\end{equation}
where $C_{\bA^{-1}}$ and $\wt\eta_L$ are defined in (<ref>).
Let ${\partial\Lambda}_n={\bf ENRICH} \, (\widetilde{\partial\Lambda}_n,J)$ be built as follows
\partial\Lambda_{n} := \big \{ k\in\mathcal{K}: \quad\textrm{there exists } \ell\in\wt{\partial\Lambda}_{n}
\textrm{ such that } |k-\ell|\le J \big\}.
Then for $\Lambda_{n+1}=\Lambda_n\cup \partial\Lambda_n$, the Galerkin solution $u_{n+1}\in V_{\Lambda_{n+1}}$ satisfies
\begin{equation}
\tvert u - u_{n+1} \tvert \leq \bar{\rho}(\theta) \tvert u - u_{n} \tvert \;
\end{equation}
\begin{equation}\label{eq:aggr3}
\bar{\rho}(\theta)=2
\frac{\beta^*\sqrt{\alpha^*}}{\beta_*\sqrt{\alpha_*}}\sqrt{1-\theta^2}.
\end{equation}
Let $g_n := P^*_{\wt{\partial\Lambda}_n} r_n = P^*_{\wt\Lambda_{n+1}} r_n$ which, according to
(<ref>), satisfies
\Vert r_n- g_n \Vert_\ps \leq \sqrt{1-\theta^2} \Vert r_n \Vert_\ps \; .
Let $w_n \in V$ be the solution of $L w_n = g_n$, which in general
will have infinitely many components, and let us split it as
w_n= P_{\Lambda_{n+1}} w_n + P_{\Lambda_{n+1}^c} w_n
=: y_n + z_n \in V_{\Lambda_{n+1}} \oplus
V_{\Lambda_{n+1}^c} \;.
The minimality property in the energy norm
of the Galerkin solution $u_{n+1}$ over the set $\Lambda_{n+1}$ yet to be defined,
in conjunction with (<ref>)
and (<ref>), implies
\begin{align*}
\tvert u - u_{n+1} \tvert &\leq \tvert u - (u_{n}+y_{n}) \tvert \leq
\tvert u- u_n - w_{n} + z_{n} \tvert \\
&\leq \frac1{\sqrt{\alpha_*}} \Vert L(u- u_n - w_{n}) \Vert + \sqrt{\alpha^*}\Vert z_{n} \Vert
= \frac{\beta^*}{\sqrt{\alpha_*}} \Vert r_n - g_{n} \Vert_\ps + \sqrt{\alpha^*} \Vert z_{n} \Vert \;,
\end{align*}
\tvert u - u_{n+1} \tvert \leq
\frac{\beta^*}{\sqrt{\alpha_*}}\sqrt{1-\theta^2} \, \Vert r_n \Vert_\ps
+ \sqrt{\alpha^*} \Vert z_{n}\Vert \;.
$z_n= \big( P_{\Lambda_{n+1}^c} L^{-1}P^*_{\widetilde{\partial\Lambda}_n} \big) r_n $,
we now construct $\Lambda_{n+1}^c$ to control $\|z_n\|$. If
k \in \Lambda_{n+1}^c \quad \text{and} \quad \ell \in \widetilde{\partial\Lambda}_n\qquad \Rightarrow \qquad
|k - \ell | > J \;,
then we have
\Vert P_{\Lambda_{n+1}^c} L^{-1} P^*_{\widetilde{\partial\Lambda}_n} \Vert \leq
\Vert \mathbf{A}^{-1}-(\mathbf{A}^{-1})_J \Vert \leq
C_{\bA^{-1}} \textrm{e}^{-\wt\eta_L J} \;,
where we have used (<ref>). We now choose $J=J(\theta)>0$ to satisfy (<ref>),
and we exploit that $\|z_n\|\le C_{\bA^{-1}} e^{-\tilde\eta_L J} \|r_n\|$ to obtain
\begin{equation}\label{eq:aggr_error_reduct}
\tvert u - u_{n+1} \tvert \leq 2\frac{\beta^*}{\sqrt{\alpha_*}}
\sqrt{1-\theta^2} \, \Vert r_n \Vert_\ps
\leq 2\frac{\beta^*\sqrt{\alpha^*}}{\beta_*\sqrt{\alpha_*}}
\sqrt{1-\theta^2} \, \tvert u - u_{n} \tvert \; ,
\end{equation}
as asserted.
We observe that, as desired, the new error reduction rate
\begin{equation}
\bar{\rho}(\theta)=2
\frac{\beta^*\sqrt{\alpha^*}}{\beta_*\sqrt{\alpha_*}}\sqrt{1-\theta^2}
\end{equation}
can be made arbitrarily small by choosing $\theta$ suitably close to $1$.
This observation was already made in <cit.>, but
we improve it in Section <ref> upon choosing $\theta$ dynamically.
Since we add a ball of radius $J$ around each point of
$\wt{\partial\Lambda}_{n}$ we get a crude estimate
\begin{equation}\label{card-Lambda}
|\partial\Lambda_{n}| \le |B_d(0,J) \cap \Z^d| ~ |\widetilde{\partial\Lambda}_{n}| \approx
\omega_d J^d |\widetilde{\partial\Lambda}_{n}|,
\end{equation}
where $\omega_d$ is the measure of the $d$-dimensional Euclidean unit
ball $B(0,1)$ centered at the origin.
§.§ Dynamic Dörfler Marking and Adaptive Spectral Algorithm
In this section we improve on the above marking strategy upon making the choice of $\theta$ dynamic.
At each iteration $n$ let us select the Dörfler parameter $\theta_n$ such that
\begin{equation}\label{aux:1}
\sqrt{1-\theta_n^2}=C_0\frac{\| r_n \|_\ps}{\| r_0 \|_\ps}
\end{equation}
for a proper choice of the positive constant $C_0$ that will be made precise later.
This implies
\begin{equation}\label{eq:Jk}
J(\theta_n) = - \frac{1}{\wt{\eta}_L} \log \frac{\| r_n \|_\ps}{\| r_0 \|_\ps} + K_1
\end{equation}
according to (<ref>), where
$K_1:= - \frac{1}{\wt\eta_L}
\log \big( \frac{1}{\sqrt{\alpha_*\alpha^*}}\frac{C_0}{C_{\bA^{-1}}} \big) +
\delta_n$ and $\delta_n \in [0,1)$.
We thus have the following adaptive spectral Galerkin method with
dynamic choice (<ref>) of the marking parameter
$\theta_n=(1-C_0^2 \|r_n\|_\ps^2/ \|r_0\|_\ps^2)^{1/2}$:
set $r_0:=f$, $\Lambda_0:=\emptyset$, $n=-1$
$n \leftarrow n+1$
$\partial\Lambda_{n}:= \textbf{E-D\"ORFLER} \, \big(\Lambda_n, (1-C_0^2
\|r_n\|_\ps^2/\| r_0 \|^2_\ps)^{1/2} \big)$
$\Lambda_{n+1}:=\Lambda_{n} \cup \partial\Lambda_{n}$
$u_{n+1}:= \textbf{ GAL} \, (\Lambda_{n+1})$
$r_{n+1}:= \textbf{ RES} \, (u_{n+1})$
while $\Vert r_{n+1} \Vert_\ps > \tol ~\Vert r_0 \Vert_\ps$
where GAL computes the Galerkin solution and RES the residual.
The following result shows the quadratic convergence of DYN-GAL.
Let the constant $C_0$ of (<ref>) satisfy
$C_0\le \frac 1 4 \sqrt{\frac{\alpha_*}{\alpha^*}}\frac{\beta_*}{\beta^*}$ and
$C_1 := \frac{\sqrt{\alpha^*}}{2\beta_*\|f\|_\ps}$.
Then the residual $r_n$ of DYN-GAL satisfies
\begin{equation}\label{quadratic-residual-1}
\frac{\|r_{n+1} \|_\ps}{2\|r_{0} \|_\ps} \leq \left(\frac{\|r_n
\|_\ps}{2\|r_{0} \|_\ps}\right)^2 \quad\forall n\ge0,
\end{equation}
and the algorithm terminates in finite steps for any
tolerance $\tol$. In addition, two consecutive solutions of DYN-GAL satisfy
\begin{equation}\label{quadratic}
\tvert u - u _{n+1} \tvert \leq C_1 \tvert u-u_n \tvert^2
\quad\forall n\ge0.
\end{equation}
Invoke (<ref>) and (<ref>) to
figure out that
\begin{equation}\label{quadratic-residual}
\begin{aligned}
\frac{\|r_{n+1} \|_\ps}{\|r_{0} \|_\ps}
\frac{\sqrt{\alpha^*}}{\beta_*} \frac{\tvert u- u_{n+1}\tvert}{\|r_{0} \|_\ps}
\leq
2 \sqrt{\frac{\alpha^*}{\alpha_*}}\frac{\beta^*}{\beta_*}
\sqrt{1-\theta^2_n} \frac{ \| r_n\|_\ps}{\| r_0\|_\ps}
\\
2 C_0 \sqrt{\frac{\alpha^*}{\alpha_*}}\frac{\beta^*}{\beta_*} \left(\frac{ \| r_n\|_\ps}{ \| r_0\|_\ps} \right)^2
\leq
\frac 1 2 \left(\frac{ \| r_n\|_\ps}{ \| r_0\|_\ps} \right)^2,
\end{aligned}
\end{equation}
which implies (<ref>). We thus
realize that DYN-GAL converges quadratically and terminates in
finite steps for any tolerance $\tol$. Finally, combining (<ref>)
with (<ref>), we readily obtain (<ref>)
upon using $r_0=f$.
If the dynamic marking parameter $\theta_n$ is chosen so that
$\sqrt{1-\theta_n^2}=C_0\left(\frac{\| r_n \|_\ps}{\| r_0
\|_\ps}\right)^\sigma$ for some
$\sigma >0$, then we arrive at the rate
$\tvert u - u _{n+1} \tvert \leq C_1 \tvert u-u_n \tvert^{1+\sigma}$.
It seems to us that
the quadratic rate (<ref>) is the first one in adaptivity theory.
The relation (<ref>) reads equivalently
\begin{equation}\label{quadratic-residual-2}
\frac{\|r_{n+1} \|_\ps}{2\|r_{0} \|_\ps} \leq \left(\frac{\|r_{n+1-k} \|_\ps}{2\|r_{0} \|_\ps}\right)^{2^k}\qquad 0\leq k \leq n+1,
\end{equation}
and implies that $\|r_n\|_\ps/\|r_0\|_\ps$ is within machine precision
in about $n=6$ iterations. This fast decay is consistent with spectral
methods. Upon termination,
we obtain the relative error
\[
\tvert u - u_n \tvert \le \frac{\beta^*\sqrt{\alpha^*}}{\beta_*
\sqrt{\alpha_*}} \tvert u \tvert ~\tol,
\]
because $\|f\|_\ps \le \frac{\sqrt{\alpha^*}}{\beta_*}\tvert u \tvert$.
The algorithm DYN-GAL entails exact computation of the
residual $r_n$, which in general has infinitely many terms. We do not
dwell here with inexact or feasible versions of DYN-GAL and
refer to <cit.> for a full
discussion which extends to our present setting.
§ NONLINEAR APPROXIMATION AND GEVREY SPARSITY CLASSES
Given any $v\in V$ we define its best $N$-term approximation error as
E_N(v)= \inf_{\Lambda \subset {\cal K} ,
\ |\Lambda|=N} \Vert v - P_{\Lambda} v \Vert_\phi \;.
We are interested in classifying functions $v$ according to the decay
law of $E_N(v)$ as $N\to\infty$, i.e., according to the “sparsity” of their expansions in terms of the basis
$\{\phi_k\}_{k\in \mathcal{K}}$.
Of special interest to us is the following exponential Gevrey class.
For $\eta >0$ and $0 < t \leq d$,
we denote by ${\mathcal A}^{\eta,t}_G$ the subset of $V$ defined as
{\mathcal A}^{\eta,t}_G { := \Big\{ v \in V \ :
\ \Vert v \Vert_{{\mathcal A}^{\eta,t}_G}:=
\sup_{N \geq 0} \, \left( E_N(v) \, {\rm exp}\left(\eta \omega_d^{-t/d}
N^{t/d} \right) \right) < +\infty \Big\} \;}
where $\omega_d$ is the measure of the $d$-dimensional Euclidean
unit ball $B_d(0,1)$ centered at the origin.
Let $\ell_G^{\eta,t}({\mathcal{K}})$ be the subset of sequences
${\bv} \in \ell^{2}({\mathcal{K}})$ so that =-1
\Vert {\bv} \Vert_{\ell_G^{\eta,t}({\mathcal{K}})} := \sup_{n \geq 1}
\Big( n^{(1-t/d)/2} {\rm exp}\left(\eta \omega_d^{-t/d} n^{t/d} \right)
|v_n^*| \Big) < +\infty \;,
where ${\bv}^*=(v_n^*)_{n=1}^\infty$ is the non-increasing rearrangement of ${\bv}$.
The relationship between ${\mathcal A}^{\eta,t}_G$ and $\ell_G^{\eta,t}({\mathcal{K}})$ is stated
in the following <cit.>.
Given a function $v \in V$ and the sequence ${\bv}={(\hat{v}_k \sqrt{d_k})_{k \in {\mathcal{K}}}}$ of its coefficients,
one has $v \in {\mathcal A}^{\eta,t}_G$ if and only if
${\bv} \in \ell_G^{\eta,t}({\mathcal{K}})$, with
\|v \|_{{\mathcal A}^{\eta,t}_G} \lesssim
\Vert {\bv} \Vert_{\ell_G^{\eta,t}({\mathcal{K}})}
\lesssim \| v \|_{{\mathcal A}^{\eta,t}_G}\,.
For functions $v$ in ${\mathcal A}^{\eta,t}_G$ one can estimate the
minimal cardinality of a set $\Lambda$ such that
$\|v-P_\Lambda v\|_\phi \le\varepsilon$ as follows:
since $\|v-P_{\wt\Lambda} v\|_\phi > \varepsilon$ for any set $\wt\Lambda$ with cardinality
$|\wt\Lambda|=|\Lambda|-1$, we deduce
\begin{equation}\label{bound:optimal}
\vert \Lambda \vert \leq \omega_d \left( \frac{1}{\eta} \log
\frac{\Vert v \Vert_{{\mathcal A}^{\eta,t}_G}}{\varepsilon} \right)^{d/t} +1.
\end{equation}
For the analysis of the optimality of our algorithm it is important to investigate
the sparsity class of the image ${L}v$ for the operator ${L}$ defined
in (<ref>), when
the function $v$ belongs to the sparsity class
$\mathcal{A}^{\eta,t}_G$. Sparsity classes of exponential type for functionals $f \in V^*$ can be defined analogously as above,
using now the best $N$-term approximation error in $V^*$
E^*_N(f)= \inf_{\Lambda \subset {\cal K} ,
\ |\Lambda|=N} \Vert f - P^*_{\Lambda} f \Vert_{\phi^*} \;.
The following result is based on
Let $L$ be such that the associated stiffness matrix $\mathbf{A}$ satisfies the decay condition (<ref>).
Given $\eta>0$ and $t \in (0,d]$, there exist $\bar{\eta}>0$, $\bar{t} \in (0,t]$ and a constant $C_L\ge1$ such that
\begin{equation}\label{eq:spars11bis}
\Vert Lv \Vert_{{\mathcal A}_G^{\bar{\eta},\bar{t}}} \le {C}_L
\Vert v \Vert_{{\mathcal A}_G^{\eta,t}} \qquad \forall v \in {\mathcal A}_G^{\eta,t}\;.
\end{equation}
Let $\bA$ be the stiffness matrix associated with the operator $L$. In <cit.> it is proven
that if $\mathbf{A}$ is banded with $2p+1$ non-zero diagonals, then the result holds with
$\bar{\eta}= \frac{\eta}{(2p+1)^{t/d}}$ and $\bar{t}= t$; on the other hand, if $\mathbf{A}\in\mathcal{D}_e(\eta_L)$ is dense, but the coefficients $\eta_L$ and $\eta$ satisfy the inequality $\eta< \eta_L \omega_d^{t/d}$, then the result
holds with $\bar{\eta}= \zeta(t)\eta$ and $\bar{t}= \frac{t}{1+t}$, where ${\zeta(t) = \left( \frac{1+t}{2^d \, \omega_d^{1+t}} \right)^{\frac{t}{d(1+t)}}}$. Finally, if $\eta \geq \eta_L \omega_d^{t/d}$, we introduce an arbitrary $\hat{\eta}>0$ satisfying $\hat{\eta}< \eta_L \omega_d^{t/d}$; then the result holds with $\bar{\eta}= \zeta(t)\hat{\eta}$ and $\bar{t}= \frac{t}{1+t}$, since $\Vert v \Vert_{{\mathcal A}_G^{\hat{\eta},t}} \leq \Vert v \Vert_{{\mathcal A}_G^{\eta,t}} $.
Keeping into account that $\zeta(t) \leq 1$ for $1 \leq d \leq 10$ (see again <cit.>), this result indicates that the residual is expected to belong to a
less favorable sparsity class than the one of the
solution. Counterexamples in <cit.> show that
(<ref>) cannot be improved.
Finally, we discuss the sparsity class of the residual
$r=r(u_\Lambda)$ for any Galerkin solution $u_\Lambda$.
Let $\mathbf{A}\in\mathcal{D}_e(\eta_L)$ and
$\bA^{-1} \in\mathcal{D}_e(\bar\eta_L)$, for constants $\eta_L>0$
and $\bar\eta_L\in(0,\eta_L]$ so that (<ref>) and
(<ref>) hold.
If $u \in {\mathcal A}^{\eta,t}_G$ for some $\eta>0$ and $t \in
then there exist suitable positive constants $\tilde{\eta} \leq \eta$ and
$\tilde{t} \leq t$ such that
$r(u_\Lambda) \in {\mathcal A}_G^{\tilde{\eta},\tilde{t}}$ for any index
set $\Lambda$ and
\Vert r(u_\Lambda) \Vert_{{\mathcal A}_G^{\tilde{\eta},\tilde{t}}} \lesssim
\Vert u \Vert_{{\mathcal A}^{\eta,t}_G} \;.
Proposition <ref> yields the existence of $\bar{\eta}>0$ and $\bar{t} \in (0,t]$ such that
\begin{equation}\label{eq:spars11ter}
\Vert f \Vert_{{\mathcal A}_G^{\bar{\eta},\bar{t}}}= \Vert Lu \Vert_{{\mathcal A}_G^{\bar{\eta},\bar{t}}}
\lesssim \Vert u \Vert_{{\mathcal A}_G^{\eta,t}} \;.
\end{equation}
In order to bound $\Vert r(u_\Lambda) \Vert_{{\mathcal A}_G^{\tilde{\eta},\tilde{t}}}$ in terms of $\Vert f \Vert_{{\mathcal A}_G^{\bar{\eta},\bar{t}}}$,
let us write
{\bf r}_\Lambda = \mathbf{A}({\bf u} - {\bf u}_\Lambda ) = {\bf f} - \mathbf{A} {\bf u}_\Lambda \;,
then use $ {\bf u}_\Lambda = ({\widehat{\mathbf{A}}_\Lambda})^{-1} (\mathbf{P}_\Lambda{\bf f})$ from
(<ref>) to get
{\bf r}_\Lambda = {\bf f} - \mathbf{A} ({\widehat{\mathbf{A}}_\Lambda})^{-1} (\mathbf{P}_\Lambda{\bf f}).
Now, assuming just for simplicity that the indices in $\Lambda$ come first (this can be realized by a permutation),
we have
\mathbf{A} = \left(
\begin{matrix}
\mathbf{A}_\Lambda & \mathbf{B} \\[5pt]
\mathbf{B}^T & \mathbf{C}
\end{matrix}
\right)
\qquad
\text{and}
\qquad
\widehat{\mathbf{A}}_\Lambda = \left(
\begin{matrix}
\mathbf{A}_\Lambda & \mathbf{O} \\[5pt]
\mathbf{O}^T & \mathbf{I}
\end{matrix}
\right),
\qquad
\text{whence}
\qquad
(\widehat{\mathbf{A}}_\Lambda)^{-1} = \left(
\begin{matrix}
(\mathbf{A}_\Lambda)^{-1} & \mathbf{O} \\[5pt]
\mathbf{O}^T & \mathbf{I}
\end{matrix}
\right)
Setting $\mathbf{f}=(\mathbf{f}_\Lambda \ \ \mathbf{f}_{\Lambda^c} )^T$, so that $\mathbf{P}_\Lambda{\bf f} = (\mathbf{f}_\Lambda \ \ \mathbf{0} )^T$, we have
\begin{eqnarray*}
\mathbf{A} ({\widehat{\mathbf{A}}_\Lambda})^{-1} (\mathbf{P}_\Lambda{\bf f}) &=&
\left(
\begin{matrix}
\mathbf{A}_\Lambda & \mathbf{B} \\[5pt]
\mathbf{B}^T & \mathbf{C}
\end{matrix}
\right)
\left(
\begin{matrix}
(\mathbf{A}_\Lambda)^{-1} & \mathbf{O} \\[5pt]
\mathbf{O}^T & \mathbf{I}
\end{matrix}
\right)
\left(
\begin{matrix}
\mathbf{f}_\Lambda \\[5pt]
\mathbf{0}
\end{matrix}
\right) \\[5pt]
\left(
\begin{matrix}
\mathbf{A}_\Lambda & \mathbf{B} \\[5pt]
\mathbf{B}^T & \mathbf{C}
\end{matrix}
\right)
\left(
\begin{matrix}
(\mathbf{A}_\Lambda)^{-1} \mathbf{f}_\Lambda \\[5pt]
\mathbf{0}
\end{matrix}
\right)
\ = \
\left(
\begin{matrix}
\mathbf{f}_\Lambda \\[5pt]
\mathbf{B}^T (\mathbf{A}_\Lambda)^{-1} \mathbf{f}_\Lambda
\end{matrix}
\right)
\end{eqnarray*}
{\bf r}_\Lambda = \left(
\begin{matrix}
\mathbf{0} \\[5pt]
\mathbf{f}_{\Lambda^c} - \mathbf{B^T} (\mathbf{A}_\Lambda)^{-1} \mathbf{f}_\Lambda
\end{matrix}
\right)
\left(
\begin{matrix}
\mathbf{O} & \mathbf{O} \\[5pt]
-\mathbf{B}^T (\mathbf{A}_\Lambda)^{-1} & \mathbf{I}
\end{matrix}
\right)
\left(
\begin{matrix}
\mathbf{f}_\Lambda \\[5pt]
\mathbf{f}_{\Lambda^c}
\end{matrix}
\right)
=: \mathbf{R} \mathbf{f} \;.
Now, since $\mathbf{A}\in\mathcal{D}_e(\eta_L)$ and
$(\bA_\Lambda)^{-1} \in\mathcal{D}_e(\bar\eta_L)$, it is easily seen that $\mathbf{R}\in\mathcal{D}_e(\tilde{\eta}_L)$ with $\tilde{\eta}_L = \bar{\eta}_L$ if $\bar{\eta}_L< \eta_L$, or $\tilde{\eta}_L < \eta_L$ arbitrary if $\bar{\eta}_L= \eta_L$.
Finally, we apply Proposition <ref> to the operator $R$ defined by the matrix $\mathbf{R}$,
obtaining the existence of constants $\tilde{\eta}>0$ and $\tilde{t} \in (0,\bar{t}\,]$ such that
\Vert r(u_\Lambda) \Vert_{{\mathcal A}_G^{\tilde{\eta},\tilde{t}}} \lesssim
\Vert f \Vert_{{\mathcal A}_G^{\bar{\eta},\bar{t}}} \;,
whence the result.
§ OPTIMALITY PROPERTIES OF DYN-GAL
In this section we derive an exponential rate of convergence for
$\tvert u -u_n\tvert $ in terms of the number of degrees of freedom
$\Lambda_n$ activated by DYN-GAL and assess the computational
work necessary to achieve this rate. This is made precise in the
following theorem.
Let $u\in \mathcal{A}_G^{\eta,t}$ where the Gevrey class
$\mathcal{A}_G^{\eta,t}$ is introduced in Definition <ref>.
Upon termination of DYN-GAL, the iterate $u_{n+1}\in V_{\Lambda_{n+1}}$
and set of active coefficients $\Lambda_{n+1}$
satisfy $\tvert u-u_{n+1}\tvert \le \frac{\beta^*}{\sqrt{\alpha_*}} \|f\|_{\phi^*}\tol$ and
\begin{equation}\label{exp-rate}
|\Lambda_{n+1}| \le \omega_d
\left( \frac{1}{\eta_*}
\log \frac{C_* \frac{\|u\|_{\mathcal{A}_G^{\eta,t}}}{\|f\|_{\phi^*}}}{\tol}
\right)^{d/t_*},
\end{equation}
with parameters $C_*>0$, $\eta_*<\eta$ and $t_*<t$. Moreover, if
the number of arithmetic operations needed to solve a
linear system scales linearly with its dimension, then the workload
$\mathcal{W}_\tol$ of DYN-GAL upon completion satisfies
\begin{equation}\label{workload}
\mathcal{W}_\tol \le \omega_d
\left(
\frac{1}{\eta^*} \log
\frac{C^* \frac{\|u\|_{\mathcal{A}_G^{\eta,t}}}{\|f\|_{\phi^*}}}{\tol^4|\log|\log\tol||^{-1}},
\right)^{d/t_*}
\end{equation}
where $\eta^*<\eta_*$ and $C^*>C_*$ but $t_*$ remains the same as in (<ref>).
We proceed in several steps.
1. Expression of $J(\theta_k)$:
Our first task is to simplify the expression (<ref>) for
$J(\theta_k)$, namely to absorb the term $K_1$:
there is $C_2>1$ such that
\begin{equation}\label{J_k}
J(\theta_k)\leq \frac{C_2}{{\wt{\eta}_L}} \Big\vert \log
\frac{\| r_k\|_\ps}{2\|r_0\|_\ps} \Big\vert .
\end{equation}
In fact, if $C_2$ is given by $C_2 = 1+ \frac{\wt\eta_L \max(0,K_1)}{\log 2}$,
\[
K_1 \leq \frac{C_2 - 1}{\wt\eta_L} \log 2 \le
\frac{C_2 - 1}{\wt\eta_L} \left\vert \log \frac{\|r_k\|_\ps}{2\|r_0\|_\ps}\right\vert
\]
because $\|r_k\|_\ps \le \|r_0\|_\ps$ for all $k\ge0$ according to
(<ref>). This in turn implies (<ref>)
\[
J(\theta_k) \le\frac{1}{\wt\eta_L} \left\vert \log \frac{\|r_k\|_\ps}{2\|r_0\|_\ps} \right\vert
+ \frac{C_2 - 1}{\wt\eta_L} \left\vert \log \frac{\|r_k\|_\ps}{2\|r_0\|_\ps}\right\vert
= \frac{C_2}{{\wt{\eta}}_L} \left\vert \log
\frac{\| r_k\|_\ps}{2\|r_0\|_\ps} \right\vert .
\]
2. Active set $\Lambda_k$:
We now examine the output $\Lambda_k$ of E-DÖRFLER.
Employing the minimality of Dörfler marking and (<ref>), we deduce
\begin{equation}\label{non-lin:dual}
E^*_{\vert\widetilde{\partial\Lambda_k} \vert} (r_k)=\Vert r_k-P_{\widetilde{\partial\Lambda_k}}^* r_k \Vert_\ps \leq \sqrt{1-\theta_k^2}\Vert r_k \Vert_\ps ,
\end{equation}
which clearly implies $E^*_{\vert\widetilde{\partial\Lambda_k}\vert-1}> \sqrt{1-\theta_k^2}\Vert r_k \Vert_\ps$.
The latter inequality, together with the Definition <ref>
of $\|r_k\|_{\cA_G^{\bar{\eta},\bar{t}}}$, yields
\begin{equation*}
\Vert r_k \Vert_{{\mathcal A}^{\bar\eta,\bar t}_G} >
\sqrt{1-\theta_k^2}\Vert r_k \Vert_\ps\, {\rm exp}\left(\bar{\eta}
\omega_d^{-\bar{t}/d} (\vert\widetilde{\partial\Lambda}_k\vert-1)^{\bar{t}/d} \right).
\end{equation*}
We note that this, along with $\vert\widetilde{\partial\Lambda}_k\vert\geq 1$,
$\frac{\|r_k\|_{\cA_G^{\bar{\eta},\bar{t}}}}{\sqrt{1-\theta_k^2}\|r_k\|_\ps}>1$ whence
\[
|\widetilde{\partial\Lambda_k}| \leq \omega_d \left(
\frac{1}{\bar{\eta}} \log
\frac{\|r_k\|_{\cA_G^{\bar{\eta},\bar{t}}}}{\sqrt{1-\theta_k^2}\|r_k\|_\ps}\right)^{d/\bar{t}}+1.
\]
We now recall the membership of the residual $r_k := r(u_k)$ to the Gevrey
class $\mathcal{A}_G^{\bar \eta, \bar t}$ for all $k\ge 0$,
established in Proposition <ref>: there exists
$C_3>0$ independent of $k$ and $u$ such that
\begin{equation*}
\|r_k\|_{\mathcal{A}_G^{\bar \eta, \bar t}} \le C_3 \|u\|_{\mathcal{A}_G^{\eta, t}}.
\end{equation*}
Combining this with the dynamic marking (<ref>) implies
\[
\frac{\|r_k\|_{\mathcal{A}_G^{\bar \eta, \bar t}}}{\sqrt{1-\theta_k^2} \|r_k\|_\ps}
= \frac{\|f\|_\ps \|r_k\|_{\mathcal{A}_G^{\bar \eta, \bar t}}}{C_0\|r_k\|_\ps^2}
\le \frac{C_4\|f\|_\ps \|u\|_{\mathcal{A}_G^{\eta, t}}}{\|r_k\|_\ps^2},
\]
with $C_4=C_3/C_0$, whence
\[
|\widetilde{\partial\Lambda_k}| \leq \omega_d \left(
\frac{1}{\bar{\eta}} \log
\frac{C_4 \|f\|_\ps \|u\|_{\mathcal{A}_G^{\eta, t}}}{\|r_k\|_\ps^2}\right)^{d/\bar{t}}+1.
\]
We let $C_5$ satisfy
$1=\omega_d \big(\frac{1}{\bar\eta} \log C_5 \big)^{d/\bar t}$, and
use that $\bar t < t \le d$, to obtain the simpler expression
\begin{equation}\label{aux-card:1}
|\widetilde{\partial\Lambda}_k| \leq
\omega_d \left(
\frac{1}{\bar{\eta}} \log
\frac{C_4 C_5 \|f\|_\ps \|u\|_{\mathcal{A}_G^{\eta, t}}}{\|r_k\|_\ps^2}\right)^{d/\bar{t}}.
\end{equation}
On the other hand, in view of (<ref>),
the enrichment step (<ref>) of E-DÖRFLER yields
\begin{equation}\label{aux-card:2}
|\partial\Lambda_k| \le C_6 \omega_d J(\theta_k)^d |\widetilde{\partial\Lambda}_k|
\le \frac{C_6C_2^d \omega_d^2}{\wt\eta_L^{d}\bar\eta^{d/\bar t}}
\left\vert \log \frac{\| r_k\|_\ps}{2\|r_0\|_\ps} \right\vert^d
\left (
\log \frac{ C_4 C_5\|f\|_\ps \|u\|_{\cA_G^{\eta,t}}}{\|r_k\|_\ps^2}\right)^{d/\bar{t}}.
\end{equation}
We exploit the quadratic convergence (<ref>) to
write for $k\le n$
\begin{equation}\label{aux-card:4}
\left| \log\frac{\|r_k\|_\ps}{2\|r_0\|_\ps}\right|^d \le
2^{d(k-n)} \left| \log \frac{\|r_n\|_\ps}{2\|r_0\|_\ps} \right|^d
=2^{d(k-n)} \left( \log \frac{2\| f\|_\ps}{\|r_n\|_\ps} \right)^d.
\end{equation}
Using the bound $\|f\|_{\phi^*}\le \|f\|_{\cA_G^{\bar\eta,\bar{t}}}
= \|r_0\|_{\cA_G^{\bar\eta,\bar{t}}} \leq C_3 \|u\|_{\cA_G^{\eta,t}}$ in the two previous inequalities
\[
\log \frac{ C_4 C_5\|f\|_\ps \|u\|_{\cA_G^{\eta,t}}}{\|r_k\|_\ps^2} \leq
2 \log \frac{ C_7 \|u\|_{\cA_G^{\eta,t}}}{\|r_k\|_\ps} \qquad \text{and} \qquad
\log \frac{2\| f\|_\ps}{\|r_n\|_\ps} \leq \log \frac{2C_3 \|u\|_{\cA_G^{\eta,t}}}{\|r_n\|_\ps},
\]
with $C_7=(C_3C_4C_5)^{1/2}$. Introducing the constants
\[
C_8 = \max (2C_3, C_7),
\qquad
\frac1{\hat \eta} = \frac{2^{d/\bar t}C_6C_2^d \omega_d}{\wt\eta_L^{d}\bar\eta^{d/\bar t}},
\qquad
t_* = \frac{\bar t}{1+\bar t},
\]
the derived upper bound for $|\partial\Lambda_k|$ can be simplified as follows:
\[
|\partial\Lambda_k| \le 2^{d(k-n)}\frac{\omega_d}{\hat\eta}
\left( \log \frac{C_8 \|u\|_{\cA_G^{\eta,t}}}{\|r_n\|_\ps} \right)^{d/t_*}.
\]
Recalling now that $|\Lambda_0|=0$ and for $n\ge0$
\begin{equation*}
\vert \Lambda_{n+1}\vert = \sum_{k=0}^n \vert \partial \Lambda_k \vert
\end{equation*}
we have
\begin{equation*}
\vert \Lambda_{n+1}\vert \leq \frac{\omega_d}{\hat\eta}
\left( \sum_{k=0}^{n} 2^{d(k-n)}\right)
\left ( \log\frac{C_8 \|u\|_{\cA_G^{\eta,t}}}{\|r_n\|_\ps} \right)^{d/t_*}.
\end{equation*}
This can be written equivalently as
\begin{equation}\label{card:Lambda_n}
\vert \Lambda_{n+1}\vert \leq \omega_d \left( \frac1{{\eta_*}} \log \frac{C_8 \|u\|_{\cA_G^{\eta,t}}}{\Vert r_n \Vert_\ps} \right)^{d/t_*}
\end{equation}
with $\eta_*=\left(\frac{\hat\eta}{\sum_{k=1}^\infty 2^{-dk}}\right)^{t_*/d}$. At last, we make use of
$\|r_n\|_{\phi^*} > \tol \|f\|_{\phi^*}$ to
get the desired estimate (<ref>) with $C_*=C_8$.
3. Computational work:
Let us finally discuss the total computational work $\mathcal{W}_\tol$
of DYN-GAL. We start with some useful notation.
We set $\delta_n:=\frac{\|r_n\|_{\ps}}{\|r_0\|_\ps}$ for
$n\ge0$ and $\varepsilon_{\ell+1}=\varepsilon_\ell^2$ for $\ell\geq 1$
with $\varepsilon_1=\frac 1 2$. We note that there exists an integer $L>0$
such that $\varepsilon_{L+1} < \varepsilon \le \varepsilon_{L}$ with
$\varepsilon$ being the tolerance of DYN-GAL.
In addition, we observe that for every iteration $n>0$ of DYN-GAL there exists $\ell>0$ such that
$\delta_n \in I_\ell:=(\varepsilon_{\ell+1}, \varepsilon_\ell]$ and that for
each interval $I_\ell$
there exists at most one $\delta_n \in I_\ell$ because
$\delta_{n+1}\le\frac12 \delta_n^2$ according to (<ref>).
Finally, to each interval $I_\ell$ we associate the following computational work
$W_\ell$ to find and store $u_{n+1}$
\[
\begin{cases}
C_\# \vert \Lambda_{n+1}\vert & \text{\rm if there exists $n$ such that }
\delta_n\in I_\ell \\
0 & \text{\rm otherwise}\;.
\end{cases}
\]
This assumes that the number of arithmetic operations needed to solve the
linear system for $u_n$ scales linearly with its dimension and $C_\#$
is an absolute constant that may depend on the specific solver.
The total computational work of DYN-GAL is bounded by
\[
\mathcal{W}_\tol = \sum_{\ell=1}^L W_\ell.
\]
We now get a bound for $\mathcal{W}_\tol$.
In view of (<ref>) we have
\[
W_\ell \leq C_\# \omega_d \left( \frac1{{\eta_*}} \log \frac{C_*
\frac{\|u\|_{\cA_G^{\eta,t}}}{\|f\|_{\phi^*}}}{\varepsilon_{\ell+1} } \right)^{d/t_*} =
C_\# \omega_d \left( \frac1{{\eta_*}} \log \frac{C_*
\frac{\|u\|_{\cA_G^{\eta,t}}}{\|f\|_{\phi^*}}}{\varepsilon_1^{2^{\ell}}} \right)^{d/t_*}.
\]
Therefore, upon adding over $\ell$ and using that $d/t_*\ge1$, we
\begin{align*}
\mathcal{W}_\tol
\leq \frac{C_\#\omega_d}{\eta_*^{d/t_*}}
\left( \sum_{\ell=1}^L \log C_* \frac{\|u\|_{\cA_G^{\eta,t}}}{\|f\|_{\phi^*}}
+ \sum_{\ell=1}^L \log \varepsilon_1^{-2^{\ell}} \right)^{d/t_*}
\leq \frac{C_\#\omega_d}{\eta_*^{d/t_*}}
\left( \log \frac{L C_* \frac{\|u\|_{\cA_G^{\eta,t}}}{\|f\|_{\phi^*}}}{\varepsilon_1^{2^{L+1}}} \right)^{d/t_*}.
\end{align*}
Since $\tol\le \varepsilon_{L}=\varepsilon_1^{2^{L-1}}$, we deduce
$\tol^4\leq \varepsilon_1^{2^{L+1}}$ and
$L \le \frac{\log \frac{|\log \tol|}{\log 2}}{\log 2} +1
\le C_9 \log|\log\tol|$. Inserting this bound in the preceding
expression yields
\begin{equation}
\mathcal{W}_\tol \leq \omega_d \left( \frac1{\eta^*} \log
\frac{C^* \frac{\| u\|_{\cA_G^{\eta,t}}}{\|f\|_{\phi^*}}}{\tol^4 |\log \vert\log
\tol \vert \vert^{-1}} \right)^{d/t_*}.
\end{equation}
\[
\eta^* = \frac{\eta_*}{C_\#^{t_*/d}},
\quad
C^* = C_9 C_*,
\]
which is the asserted estimate (<ref>). The proof is thus complete.
Note that the bound on the workload, given in (<ref>), is at most an absolute multiple of the bound, given in (<ref>), on the number of active coefficients.
If $\sqrt{1-\theta_n^2}=C_0\left(\frac{\| r_n \|_\ps}{\| r_0
\|_\ps}\right)^\sigma$ with $\sigma >0$, then (<ref>) still holds with the same parameters $\eta_*,t_*$.
Let us consider the case when $u$ belongs to the algebraic class
\[
\mathcal{A}_B^s:= { \Big\{ v \in V \ : \ \Vert v \Vert_{{\mathcal A}^{s}_B}:=
\sup_{N \geq 0} \, E_N(v) \, \big(N+1\big)^{s/d} < +\infty \Big\}},
\]
which is related to Besov regularity. We can distinguish two cases:
* $\mathbf{A}$ belongs to an exponential class but the residuals belong to an algebraic class;
* $\mathbf{A}$ belongs to an algebraic class $\mathcal{D}_a(\eta_L)$, i.e. there exists a constant $c_L>0$
such that its elements satisfy $| a_{\ell,k} | \leq c_L (1+ \vert \ell - k \vert )^{-\eta_L}$, and the residual belongs to an algebraic class.
We now study the optimality properties of DYN-GAL for
these two cases. Let us first
observe that whenever the residuals belong to the algebraic class $\mathcal{A}_B^s$, the bound
(<ref>) becomes
\begin{equation}\label{aux-card:3}
|\widetilde{\partial\Lambda}_k| \lesssim \| u -u_k\|^{-2d/s}.
\end{equation}
This results from the dynamic marking (<ref>) together
with (<ref>) in the algebraic case.
Let us start with Case $1.$ Using (<ref>) and
(<ref>), which is still valid here as we assume that $\mathbf{A}$
belongs to an exponential class, the bound (<ref>) is replaced by
\begin{equation}
|{\partial\Lambda}_k| \lesssim \| u -u_k\|^{-2d/s}\Big\vert \log
\frac{\| r_k\|_\ps}{2\|r_0\|_\ps} \Big\vert^d \lesssim \| u
-u_k\|^{-2d/s}2^{d(k-n)} \left( \log \frac{2\| f\|_\ps}{\|r_n\|_\ps}
\right)^d ,
\end{equation}
where in the last inequality we have employed (<ref>). Hence, we have
\begin{eqnarray}
\vert \Lambda_{n+1}\vert &=& \sum_{k=0}^n \vert \partial \Lambda_k \vert
\lesssim \left( \log \frac{2\| f\|_\ps}{\|r_n\|_\ps} \right)^d \sum_{k=0}^n \| u- u_n\|^{-2\frac d s 2^{k-n}} 2^{d(k-n)}\nonumber\\
&\lesssim& \left( \log \frac{2\| f\|_\ps}{\|r_n\|_\ps} \right)^d \| u- u_n\|^{-2\frac d s} \sum_{k=0}^n 2^{d(k-n)}\nonumber\\
&\lesssim& \left( \log \frac{2\| f\|_\ps}{\|r_n\|_\ps} \right)^d\| u- u_n\|^{-2\frac d s} \lesssim \left( \log \frac{2\| f\|_\ps}{\|u-u_n\|} \right)^d \|u-u_{n+1}\|^{- \frac d s }\nonumber
\end{eqnarray}
where in the last inequality we have employed the quadratic convergence of DYN-GAL.
The above result implies that DYN-GAL is optimal
for Case 1 (up to a logarithmic factor).
Let us now consider Case $2.$ Since $\mathbf{A}$ belongs to an algebraic class, (<ref>) is replaced by
\begin{equation*}\label{aux-card:5}
J(\theta_k)\eqsim \| u-u_k\|^{-1/s}.
\end{equation*}
Thus, the bound (<ref>) is replaced by
\begin{equation*}
|{\partial\Lambda}_k| \lesssim \| u -u_k\|^{-3d/s},
\end{equation*}
which implies
\begin{equation*}
\vert \Lambda_{n+1}\vert = \sum_{k=0}^n \vert \partial \Lambda_k \vert
\lesssim \|u-u_n\|^{-3 \frac d s }
\lesssim \|u-u_{n+1}\|^{-\frac 3 2 \frac d s }
\end{equation*}
where in the last inequality we have again employed the
quadratic convergence of DYN-GAL.
This result is not optimal for Case 2,
due to the factor 2/3 multiplying $s$ in the exponent.
We recall that the algorithm FA-ADFOUR of
<cit.> is similar to DYN-GAL
but with static marking parameter $\theta$. The theory of FA-ADFOUR
requires neither a restriction on $\theta$ nor coarsening and is proven to
be optimal in the algebraic case; see <cit.>.
The first and the fourth authors are partially supported by the Italian research grant Prin 2012 2012HBLYE4 “Metodologie innovative nella modellistica differenziale numerica”.
The first author is also partially supported by Progetto INdAM-GNCS 2015 “Tecniche di riduzione computazionale per problemi di fluidodinamica e interazione fluido-struttura".
The second author is partially supported by NSF grants DMS-1109325 and DMS-1411808.
The fourth author is also partially supported by Progetto INdAM - GNCS 2015 “Non-standard numerical methods for geophysics".
P. Binev, W. Dahmen, and R. DeVore.
Adaptive finite element methods with convergence rates.
Numer. Math., 97(2):219–268, 2004.
P. Binev and R. DeVore.
Fast computation in adaptive tree approximation.
Numer. Math., 97(2):193–217, 2004.
M. Bürg and W. Dörfler.
Convergence of an adaptive $hp$ finite element strategy in higher
Appl. Numer. Math., 61(11):1132–1146, 2011.
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang.
Spectral methods. Fundamentals in single domains.
Scientific Computation. Springer-Verlag, Berlin, 2006.
C. Canuto, R. H. Nochetto, R. Stevenson, and M. Verani.
High-order adaptive Galerkin methods.
Proceedings of ICOSAHOM 2014 (to appear).
C. Canuto, R. H. Nochetto, and M. Verani.
Adaptive Fourier-Galerkin methods.
Math. Comp., 83(288):1645–1687, 2014.
C. Canuto, R. H. Nochetto, and M. Verani.
Contraction and optimality properties of adaptive
Legendre-Galerkin methods: the one-dimensional case.
Comput. Math. Appl., 67(4):752–770, 2014.
C. Canuto, R. H. Nochetto, R. Stevenson, and M. Verani.
Convergence and Optimality of $hp$-AFEM.
C. Canuto, V. Simoncini, and M. Verani.
On the decay of the inverse of matrices that are sum of Kronecker
Linear Algebra Appl., 452:21–39, 2014.
C. Canuto, V. Simoncini, and M. Verani.
Contraction and optimality properties of an adaptive
Legendre-Galerkin method: the multi-dimensional case.
J. Sci. Comput., doi: 10.1007/s10915-014-9912-3.
C. Canuto and M. Verani.
On the numerical analysis of adaptive spectral/$hp$ methods for
elliptic problems.
In Analysis and numerics of partial differential equations,
volume 4 of Springer INdAM Ser., pages 165–192. Springer, Milan, 2013.
C. Carstensen, M. Feischl, M. Page, and D. Praetorius.
Axioms of adaptivity.
Comput. Math. Appl.
67(6):1195–-1253, 2014.
J. M. Cascón, C. Kreuzer, R. H. Nochetto, and K. G. Siebert.
Quasi-optimal convergence rate for an adaptive finite element method.
SIAM J. Numer. Anal., 46(5):2524–2550, 2008.
A. Cohen, W. Dahmen, and R. DeVore.
Adaptive wavelet methods for elliptic operator equations –
convergence rates.
Math. Comp, 70:27–75, 1998.
L. Diening, Ch. Kreuzer, and R. Stevenson,
Instance optimality of the adaptive maximum strategy.
Found. Comp. Math., 1–36, 2015.
W. Dörfler and V. Heuveline.
Convergence of an adaptive $hp$ finite element strategy in one
space dimension.
Appl. Numer. Math., 57(10):1108–1124, 2007.
G. B. Folland, Real analysis, second ed., John Wiley & Sons Inc., New
York, 1999.
T. Gantumur, H. Harbrecht, and R. Stevenson.
An optimal adaptive wavelet method without coarsening of the
Math. Comp., 76(258):615–629, 2007.
P. Morin, R.H. Nochetto, and K. G. Siebert.
Data oscillation and convergence of adaptive FEM.
SIAM J. Numer. Anal., 38(2):466–488 (electronic), 2000.
R. H. Nochetto, K. G. Siebert, and A. Veeser.
Theory of adaptive finite element methods: an introduction.
In Multiscale, nonlinear and adaptive approximation, pages
409–542. Springer, Berlin, 2009.
Ch. Schwab.
$p$- and $hp$-finite element methods.
Numerical Mathematics and Scientific Computation. The Clarendon
Press, Oxford University Press, New York, 1998.
Theory and applications in solid and fluid mechanics.
R. Stevenson.
Optimality of a standard adaptive finite element method.
Found. Comput. Math., 7(2):245–269, 2007.
R. Stevenson.
The completion of locally refined simplicial partitions created by
Math. Comp., 77:227–241, 2008.
R. Stevenson.
Adaptive wavelet methods for solving operator equations: An overview.
In Multiscale, nonlinear and adaptive approximation, pages
543-597. Springer, Berlin, 2009.
|
1511.00105
|
=6.0in =8.25in
=-0.3in =-0.20in
Sezione INFN di Ferrara, Ferrara, Italy,
European Organization for Nuclear Research (CERN), Geneva, Switzerland, Università di Ferrara, Ferrara, Italy
[On behalf of the LHCb Collaboration.]
Submitted to #1
PRESENTED AT
|
1511.00105
|
[-.7ex] $#1$
0.18em-0.18em K
0.18em-0.18em D
0.18em-0.18em B
^+ '^-
ℓ^+ ℓ^-
1 TELL1
1 UKL1
|
1511.00220
|
Constraints on top quark flavor changing neutral currents using diphoton events at the LHC
Sara Khatibi and Mojtaba Mohammadi Najafabadi
School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran
In this paper we show that the diphoton mass spectrum in proton-proton collisions at the LHC is sensitive to the top quark flavor changing neutral
current in the vertices of $tu\gamma$ and $tc\gamma$.
The diphoton mass spectrum measured by the CMS experiment at the LHC at a
center-of-mass energy of 8 TeV and an integrated luminosity of 19.5 fb$^{-1}$ is used as an example
to set limits on these FCNC couplings.
It is also shown that the angular distribution of the diphotons is sensitive to
anomalous $tu\gamma$ and $tc\gamma$ couplings and it is a powerful tool to probe any value of the
branching fraction of top quark rare decay to an up-type quark plus a
photon down to the order of $10^{-4}$. We also show that the $tu\gamma$ FCNC coupling has a significant
contribution to the neutron electric dipole
moment (EDM) and the upper bound on neutron EDM can be used to constrain the $tu\gamma$ FCNC coupling.
PACS Numbers: 13.66.-a, 14.65.Ha
Keywords: Top quark, photon, flavor changing neutral current.
§ INTRODUCTION
The top quark with a mass of $173.34 \pm 0.75$ GeV <cit.> is the heaviest particle of the Standard Model (SM).
With such a mass, the top quark has the largest Yukawa coupling to the Higgs boson and therefore
measurement of its properties would provide a promising way to probe the electroweak
symmetry breaking mechanism and new physics beyond the SM.
New physics can
show up either through direct production of new particles or indirectly
via higher order effects. Observing indirect evidences is important as
it provides hints to look for new physics before direct discovery.
In the Standard Model (SM), the branching fractions of top quark rare
decays $t\rightarrow q V$, with $q = u,c$ and $V=\gamma,Z,g$, are
at the order of $10^{-14}-10^{-12}$ <cit.>. Such branching fractions are
extremely small and are out of the ability of the current and future
collider experiments to be measured. Within the SM, such
Flavor Changing Neutral Current (FCNC) transitions only occur at loop level and are strongly suppressed due
to the Glashow-Iliopoulos-Maiani (GIM) mechanism <cit.>. On the other hand, it has been shown that
several extensions of the SM are able to relax the GIM suppression of
the top quark FCNC transitions due
to additional loop diagrams mediated by new particles. Models,
such as supersymmetry, two Higgs doublet models, predict significant
enhancements for the FCNC top quark rare decays
<cit.>. As a result, the
observation of any excess for these rare decays would be indicative
of indirect effects of new physics. Many studies on searches for the
top quark FCNC and other anomalous couplings have been already done
In this paper, a direct search for the top quark FCNC interactions in the vertex
of $tq\gamma$ is discussed. Such interactions can be described in a model-independent
way using the effective Lagrangian approach, which has the
following form <cit.>:
\begin{equation}\label{eff}
\mathcal{L}_{\rm FCNC} = - eQ_{t} \sum_{q=u,c} \kappa_{tq\gamma} \bar{q} (\lambda^v_{tq\gamma}+\lambda^a_{tq\gamma} \gamma_{5}) \frac{i \sigma_{\mu \nu} q^{ \nu}}{\Lambda} t A^{ \mu} +h.c.,
\end{equation}
where the electric charges of the electron and top quark are denoted
by $e$ and $eQ_{t}$, respectively and $q^{\nu}$ is the
four momentum of the involved photon, $\Lambda$ is the cutoff of the
effective theory, which is conventionally assumed to be equal to the top
quark mass, unless we mention. In the FCNC Lagrangian in Eq.<ref>, $\sigma_{\mu \nu} = \frac{1}{2}[\gamma_{\mu},\gamma_{\nu}]$ and
the anomalous couplings strength is denoted by $\kappa_{tq\gamma}$. Throughout this paper, no specific chirality is assumed for the
$tq\gamma$ FCNC couplings, i.e. $\lambda^v_{tq\gamma} = 1$ and $
\lambda^a_{tq\gamma}=0$.
Within the SM framework, the values of $\kappa_{tq\gamma}$, $q=u,c$, vanish at tree level.
The leading order (LO) partial width of the top quark FCNC decay $t\rightarrow q\gamma$,
neglecting the masses of the up and charm quarks, has the
following form <cit.>:
\begin{eqnarray}
\Gamma (t \rightarrow q \gamma) = \frac{\alpha}{2} Q^{2}_{t} m_{t} |\kappa_{tq\gamma}|^{2},
\end{eqnarray}
and the LO width of $t\rightarrow b W^{+}$ can be written as <cit.>:
\begin{eqnarray}
\Gamma(t \rightarrow b W^{+}) = \frac{\alpha |V_{tb}|^{2}}{16
s_{W}^{2}}\frac{m_{t}^{3}}{m_{W}^{2}} \left(1-\frac{3m_{W}^{4}}{m_{t}^{4}}+\frac{2m_{W}^{6}}{m_{t}^{6}} \right),
\end{eqnarray}
where $\alpha$ and $V_{tb}$ are the fine structure constant and the CKM
matrix element, respectively. The sine of the Weinberg angle is
denoted by $s_{W}$ and $m_{t},m_{W}$ are the top quark and $W$ boson mass,
respectively. The branching fraction of $t\rightarrow q \gamma$ is estimated as the ratio of
$\Gamma (t \rightarrow q \gamma)$ to the width of $t\rightarrow bW^{+}$
which takes the following form <cit.>:
\begin{eqnarray}\label{branchingratio}
Br(t\rightarrow q \gamma) = 0.2058\times |\kappa_{tq\gamma}|^{2}.
\end{eqnarray}
To obtain the above branching fraction, we set $m_{t}$=172.5 GeV,
$\alpha=1/128.92$, $m_{W}$= 80.419 GeV and $s_{W}^{2} = 0.234$ in
$t\rightarrow q\gamma$ and $t\rightarrow bW^{+}$ widths.
The $tu\gamma$ and $tc\gamma$ FCNC couplings have been studied in different experiments with
no observation of any excess above the SM expectation up to now.
In $p\bar{p}$ collisions at the Tevatron, the CDF experiment has set
the following upper bounds on the branching fraction at the $95\%$
confidence level (CL) <cit.>:
\begin{eqnarray}
Br(t \rightarrow q \gamma) < 3.2 \times 10^{-2}~,~ \text{with}~q =
u, c.
\end{eqnarray}
This upper bound has been obtained using the study of the top quark
decays in top quark pair production.
Further searches for the anomalous $tq\gamma$ couplings in
electron-positron and electron-proton colliders (LEP and HERA) have provided
the following limits on the anomalous couplings at the $95\%$ CL <cit.>:
\begin{eqnarray}
\kappa_{tc\gamma} < 0.486~\text{(DELPHI)}~,~\kappa_{tu\gamma} <
0.174~\text{(ZEUS)}~,~\kappa_{tu\gamma} < 0.18~\text{(H1)}.
\end{eqnarray}
The ZEUS limit has been
obtained under the assumption of $m_{t} = 175$ GeV.
The most stringent bounds on
the $tq\gamma$ FCNC interactions have been obtained by the CMS experiment
at the LHC, using proton-proton collisions
at a center-of-mass energy of 8 TeV, by studying
the final state of single top quark production in association with a
photon. The following upper bounds have been obtained on the anomalous
couplings and the corresponding branching fractions at the $95\%$ CL <cit.>:
\begin{eqnarray}\label{cms}
\kappa_{tu\gamma} < 0.028 ~,\text{corresponding to}~Br(t\rightarrow u\gamma) < 1.61 \times
10^{-4}, \nonumber \\
\kappa_{tc\gamma} < 0.094 ~,\text{corresponding to}~Br(t\rightarrow c\gamma) < 1.82 \times
\end{eqnarray}
These limits has been obtained based on 19.1 fb$^{-1}$ of integrated
luminosity of data using only the muonic decay mode of the W boson in the
top quark decay.
All the above searches are based on final states containing at
least a top quark. As the top quark
has a short lifetime, it decays immediately (before hadronization). Therefore one has to
reconstruct top quark from its decay products to be able to probe the $tq\gamma$ couplings. This needs
a careful attention to correctly select the final state objects,
i.e. top quark decay products, and
consider several sources of systematic uncertainties associated to each final
state object in the detector. In this work, we propose instead to use diphoton events to probe the $tq\gamma$
FCNC couplings which have less difficulties and challenges with respect to
the events with top quarks in the final state.
The measurement of the diphoton invariant mass spectrum is one of the particular interests at
the LHC as it is sensitive to several new physics models beyond the SM
being one of the most sensitive channels to the Higgs boson production at the LHC.
On the other hand, the excellent mass resolution of the diphoton spectrum in the ATLAS
and CMS detectors at the LHC provides the possibility for precise
measurement of new signals above the SM expectation. Randall-Sundrum
model <cit.> and large extra dimensions <cit.>
are of the examples of the models which affect the diphoton
differential cross sections. In this paper, we show that the presence
of the FCNC anomalous coupling
$tq\gamma$ leads to significant change in the diphoton mass spectrum
and the diphoton angular distribution. Using a mass spectrum
measurement by the CMS experiment <cit.>, we obtain bounds on the anomalous
couplings $\kappa_{tq\gamma}$. In addition it is shown that the diphoton angular
distribution would be able to constrain the $tq\gamma$ FCNC couplings
As an indirect way to probe the FCNC couplings, we calculate
the effect of $tu\gamma$ coupling to the neutron electric dipole
moment (EDM) and show that the neutron EDM can receive significant
contribution from the FCNC couplings.
This paper is organized as follows. In Section <ref>, the
details of the calculations and methods to constrain the $tq\gamma$
FCNC couplings using the diphoton mass spectrum are
presented. Section <ref> is dedicated to present the
the potential of the LHC to study the $tq\gamma$
FCNC couplings using the angular distribution of the diphoton events.
Finally, Section <ref> concludes the paper. In the appendix <ref>,
using the upper bound on the neutron EDM, an upper limit on the anomalous $tu\gamma$ is derived.
§ DIPHOTON: MASS SPECTRUM
In this section, we calculate the contribution of $tq\gamma$ FCNC
couplings to diphoton production at the LHC. Then, based on the
measured diphoton mass spectrum by the CMS experiment <cit.>, constraints on
the anomalous couplings are derived.
Within the SM, the LO diphoton production proceeds through
quark-antiquark annihilation.
The $tq\gamma$ FCNC couplings affect the diphoton production through
the scattering of $u,c,\bar{u}$, and $\bar{c}$ quarks which proceed through
$t$-channel as shown in Fig. <ref>.
The representative Feynman diagrams of the $tq\gamma$ FCNC
contributions to the diphoton production at the LHC. The right
diagram represents the lowest order SM contribution to
diphoton production which interferes with diagrams from $tq\gamma$.
We calculate the leading order matrix
element of diphoton production analytically for the Feynman diagrams shown in
Fig.<ref>. After averaging over the color and spin indices of the
initial state partons and summing over the polarizations of the final
state photons, the amplitude takes the following form:
\begin{eqnarray}
\overline{|\mathcal{M}|}^2 \propto \frac{2 e^{4}u}{t}+\frac{ Q_{t}^{2}e^{4}\kappa^{2}t (u-4
t)}{m^2_{t}(t - m^2_{t})} +\frac{2Q_{t}^{4} e^{4} \kappa^{4}t^2 (m_{t}^2 s+t u)}{m^4_{t}(t - m^2_{t})^{2}} ,
\end{eqnarray}
where for simplicity, we have assumed
$\kappa_{tu\gamma}=\kappa_{tc\gamma}=\kappa$ and $s,t,u$ are the
Mandelstam variables which can be written in terms of the scattering
angle $\theta^{*}$ in the center-of-mass frame as: $t =
-\frac{s}{2}(1-\cos\theta^{*})$ and $u= -\frac{s}{2}(1+\cos\theta^{*})$. The first term in the
above expression is the leading order amplitude describing the SM
diphoton production, the
second term is the interference between the SM and FCNC diagram, and
the last term is the contribution of pure $tq\gamma$ FCNC diphoton
production. The interference term (SM+FCNC) is found to be constructive and the contribution of the third term,
which is purely coming from FCNC, is smaller than the
the interference term by a factor of $\approx 10^{-3}$.
One of the characteristics of the LO SM is the
enhancement of diphoton production at small angles as the production
proceed through a $t$-channel virtual exchange.
In order to perform the signal simulation, the $tq\gamma$ effective
Lagrangian, Eq.<ref>, is implemented into the
FeynRules package <cit.> and then the model is exported to a UFO
module <cit.> which is linked to MadGraph 5 <cit.>. Events are generated, describing
the diphoton production at the LHC with the center-of-mass energy of
$\sqrt{s} = 8$ TeV. The LO parton distribution functions (PDFs)
of CTEQ6L1 <cit.> are used as the input for the calculations and events
generation. The renormalization and factorization scales are chosen to be equal and
set to the default dynamic scales of the MadGraph generator.
Pythia 8 <cit.> is used for parton showering and
hadronization of the parton-level events.
Finally, the detector-level effects are emulated by
Delphes-3.3.2 package <cit.>. It includes
a reasonable modeling of the CMS detector performances as
described in <cit.>.
In <cit.>, the CMS collaboration has performed a
search for diphoton resonances in high mass in proton-proton
collisions at the center-of-mass energy of 8 TeV using an integrated
luminosity of 19.5 fb$^{-1}$ of data. The analysis searches for resonant diphoton production via
gravitons in the Randall-Sundrum scenario with a
warped extra dimension. According to the calculations presented above, the
$tq\gamma$ FCNC couplings affect the production of diphotons at the
LHC. In this work, we follow the quite similar strategy to the CMS
collaboration and use their result to probe the $tq\gamma$ FCNC
anomalous couplings.
In the CMS experiment analysis, two isolated photons with transverse energy ($E_{T}$) greater
than 80 GeV within the pseudorapidity range of $|\eta_{\gamma}| <
1.4442$, and with a diphoton-system invariant mass greater than 300 GeV
are selected. In this region of the pseudorapidity, an excellent
resolution for the photon energy is experimentally achieved. For the photons with
$E_{T} \sim 60$ GeV and $|\eta_{\gamma}| < 1.4442$, the energy resolution
varies between $1\%-3\%$ <cit.>.
The used cuts for isolation and identification of the
photons by the CMS collaboration lead to an efficiency of $86\%$ for
the photons with $E_{T} > 80$ GeV and $|\eta_{\gamma}|
<1.4442$. Small changes are seen in this efficiency when the $E_{T}$ and
$\eta$ of the photons change. In the current work, quite similar
selection is employed for the analysis <cit.>.
The background to the diphoton final state originates from SM diphoton
production, $\gamma$+jet, and from dijet productions where one or two
jets are misidentified as photons in the detector for the latter two
background processes. Table
<ref> shows the number of observed events in data and the
background prediction for several ranges of the diphoton mass
spectrum <cit.>. The uncertainties presented in the Table <ref>
include both the statistical and systematic sources.
The data and SM background expectation
are found to be in agreement, considering the uncertainties on the
predicted background and no significant excess over the SM background
is found.
The values reported in Table <ref> are used to probe $tq\gamma$ anomalous
couplings. As the measurement is compatible with the SM prediction, we
set upper limit on the diphoton production cross section in the
presence of anomalous couplings.
Figure <ref> shows the diphoton mass distribution at LO for the SM and
SM+FCNC signal assuming $\kappa_{tu\gamma} = \kappa_{tc\gamma}=\kappa
= 0.1$ obtained from the MadGraph simulation. The diphoton mass
distribution at NNLO estimated based on the Monte Carlo program 2gNNLO <cit.> is also shown in Fig.<ref> for
2gNNLO program calculates the production
cross section of diphoton in hadron collisions to the accuracy of next-to-next-to-leading-order.
As depicted, the presence of $tq\gamma$ FCNC couplings lead
to increase the diphoton cross section in the high invariant mass
region. According to Table <ref>, the total number of
observed data events above $m_{\gamma\gamma} > 500$ GeV is 333 events
with the SM background prediction of $375.8 \pm 29.9$ <cit.>, where the SM diphoton production has been
estimated based on the Monte Carlo program 2gNNLO.
Assuming $\kappa_{tu\gamma} = \kappa_{tc\gamma}=\kappa
= 0.15$, $99.4 \pm 8.5$ FCNC events are expected in this region for an
integrated luminosity of 19.5
fb$^{-1}$ of data. The uncertainty on the number of FCNC events includes the contributions coming from
the choice PDFs, variations of renormalization and factorization scales, and the statistical uncertainty.
The PDF uncertainty is obtained according to the PDF4LHC recommendation <cit.> using PDF sets
CTEQ6L1 <cit.>, NNPDF 3.0 <cit.>, and MSTW 2008 <cit.>. The uncertainty originating from the
variations of the scales has been estimated by varying the renormalization and factorization scales simultaneously by factors of 0.5 and 2.
Number of observed events in data and the SM background
prediction in different ranges of diphoton invariant mass with 19.5
fb$^{-1}$ of data <cit.>.
$ m_{\gamma\gamma} $ [GeV] Data Total expected (SM)
500-750 265 310.8 $\pm 29.9$
750-1000 46 48.6 $\pm 5.4$
1000-1250 16 11.4 $\pm 1.5$
1250-1500 3 3.3 $\pm 0.5$
1500-1750 2 1.1 $\pm 0.2$
1750-$\infty$ 1 0.6 $\pm 0.1$
Diphoton invariant mass distribution for SM
and SM+FCNC with $\kappa = 0.1$ obtained from
LO MadGraph simulation at the
center-of-mass energy of 8 TeV. The SM prediction at
NNLO obtained from 2gNNLO is also depicted for comparison.
We proceed to set an upper limit on the diphoton cross section in the
presence of FCNC couplings. We compare the number of
observed events in data and the expected events from SM in the region
$m_{\gamma\gamma} > 500$ GeV. The limit at the $95\%$ CL is set on the
quantity $\sigma_{s} = (\sigma_{\rm Total} - \sigma_{\rm SM})\times\epsilon_{A}$, where the
whole diphoton production cross section (SM and FCNC signal) is
denoted by $\sigma_{\rm Total}$ and $\sigma_{\rm SM}$ is the SM diphoton
cross section. The FCNC signal acceptance is taken into account by the $\epsilon_{A}$ term.
The CL$_{s}$ technique <cit.> is used to calculate the upper limit on the cross
section. An efficiency of $77.45 \%$ with an
uncertainty of $10\%$ is found for the FCNC signal. The observed and
expected $95\%$ CL upper limit on $\sigma_{s}$ are found to be 3.2 fb
and 5.0 fb. The observed limit at the $95\%$ CL and the FCNC
signal cross section, $\sigma_{s}$ are shown in Fig.
Parameterization of the signal cross section
versus the anomalous FCNC coupling $\kappa$ and the
observed $95\%$ CL upper limit on the cross section.
The $95\%$ CL upper limit on the
anomalous coupling parameter $\kappa$ is the intersection of the
observed limit on cross section with the theoretical cross section
curve. The upper limit on $\sigma_{s}$ (3.2 fb) is corresponding to
the upper limit of 0.153 on the anomalous coupling $\kappa$. This
limit can be expressed to the upper limit on the branching fraction
using Eq.<ref>:
\begin{eqnarray}
Br(t\rightarrow q\gamma) < 4.81 \times 10^{-3}, ~\text{with} ~q = u,c.
\end{eqnarray}
The value obtained is comparable to the most stringent limits which has been
obtained from the anomalous single top quark production in association
with a photon by the CMS experiment (Eq.<ref>) <cit.>.
This provides a motivation for using this channel as a complementary technique
for studying the $tq\gamma$ FCNC interactions at the LHC experiments. A combination of
this result with the results of other channels can lead to an improvement of the best limit.
§ ANGULAR DISTRIBUTION OF THE DIPHOTON SYSTEM
In this section, we propose and use a diphoton angular variable to
probe the $tq\gamma$ anomalous couplings. In the SM, as the diphoton
production proceeds through a $t$-channel exchange, the angular distribution
peaks at $\cos\theta^{*} = 1$, where $\theta^{*}$ is the
scattering angle in the center-of-mass frame of two partons.
The scattering angle between two photons
can also be expressed by the variable $\chi =
e^{|\eta_{\gamma_{1}}-\eta_{\gamma_{2}}|}$. This variable has been used widely
in searches for new physics such as searches for contact interactions,
large extra dimensions, and excited quarks in dijet events in the
Tevatron and LHC experiments <cit.>.
It has been found that new phenomena affect this angular variable
considerably and consequently is used to probe beyond SM.
In order to produce the SM diphoton events, including the QCD next-to-leading order corrections and
the contributions from the fragmentation processes, the Diphox (v 1.3.3) program <cit.> is used.
The CT10 PDF set <cit.> is used as the input of the parton distribution functions and all the scales are set to
$m_{\gamma\gamma}$. Figure <ref>
shows the distribution of the angular variable $\chi$ for the SM diphoton events
with an invariant mass above 500 GeV and $|\eta| < 1.442$. The error bars
represent the systematic uncertainties coming from the parton distribution functions and strong coupling constant $\alpha_{S}$.
The shaded bars show the uncertainty from the theoretical scales in each bin of the angular distribution.
The uncertainties arising from the scales variations are calculated by varying the factorization, renormalization, and fragmentation scales simultaneously
by factors of 0.5 and 2. In each bin of the $\chi$ angular distribution, the uncertainty is calculated as the maximum difference between the new angular distributions and the
distribution with the reference inputs.
The uncertainty coming from limited knowledge on
the choice of PDF is obtained using PDF4LHC recommendation <cit.>.
The PDF sets CT10 <cit.>, MSTW08 <cit.>, and NNPDF 3.0 <cit.> are used to estimate the uncertainty on
from PDFs and strong coupling constant $\alpha_{S}$ is varied by 0.012 similar to the prescription adopted in <cit.>.
Distribution of the angular variable $\chi$ for the SM diphoton events
with an invariant mass above 500 GeV and $|\eta| < 1.442$. The error bars
represent the systematic uncertainties coming from PDFs and strong coupling constant $\alpha_{S}$.
The shaded bars show the uncertainty from the theoretical scales in each bin of the angular distribution.
The left plot in Fig. <ref> shows the distributions of
$\chi=e^{|\eta_{\gamma_{1}}-\eta_{\gamma_{2}}|}$ as a function of the
anomalous coupling $\kappa$. The distribution is normalized to
unity since the sensitivity to FCNC couplings affects the angular
distribution rather than normalization. This figure depicts the
predicted SM distribution as well as the SM+FCNC with $\kappa =
0.2$ and 0.5. These distributions are after all selection cuts
described previously requiring in addition that $m_{\gamma\gamma} > 500$ GeV. As seen, the presence of the anomalous FCNC
couplings of $tq\gamma$ changes the shape of the angular variable
$\chi$. Increasing the value of the anomalous coupling $\kappa$ causes
more events to be concentrated at small values of $\chi$. It is
notable that due to the detector acceptance cut applied on the photon
pseudorapidity, $\chi $ varies from 0 to $e^{2\times 1.442} = 17.96$.
Left: Normalized distributions of diphoton angular
variable $\chi$ for the SM and SM+FCNC with
anomalous couplings $\kappa$ = 0.2,0.5. Right:
$\mathcal{R}$ versus the anomalous coupling
$\kappa$ for various choices of $\chi_{0}$.
In order to quantify the difference in the shape of $\chi$ for SM and FCNC, a ratio is
defined as:
\begin{eqnarray} \label{ratioc}
\mathcal{R}_{\chi_{0}} (\kappa)=\frac{\int_{0}^{\chi_{0}}\frac{1}{N}\frac{dN}{d\chi}}{\int_{\chi_{0}}^{\infty}\frac{1}{N}\frac{dN}{d\chi}},
\end{eqnarray}
where $\chi_{0}$ is an arbitrary cut which is chosen in such a way
that the best sensitivity to the FCNC couplings is achieved. The right plot in
Fig. <ref> shows the behavior of $\mathcal{R}$ versus the
anomalous coupling $\kappa$ for different choices of $\chi_{0}$.
As the normalized
distribution of $\chi$ depends on the cut on the diphoton invariant
mass, the value of $\mathcal{R}$ varies with the cut on
$m_{\gamma\gamma}$. Figure <ref> shows the behavior of
$\mathcal{R}$ for the SM and SM+FCNC
for the
$\chi_{0} = 8$ choice and different cuts on the minimum $m_{\gamma\gamma}$.
The uncertainties in this plot includes both the statistical and theoretical uncertainties for the SM
and only the statistical uncertainty for the SM+FCNC.
The cut on $m_{\gamma\gamma}$ can be chosen to optimize the expected sensitivity
to $\kappa$.
We define the statistical significance of the observable $\mathcal{R}$ as:
\begin{eqnarray}
\mathcal{S}_{\chi_{0}} (\kappa) = \frac{\mathcal{R}_{\chi_{0}}^{\rm FCNC+SM}(\kappa)-\mathcal{R}_{\chi_{0}}^{\rm SM}}{\Delta \mathcal{R}_{\chi_{0}}^{SM}},
\end{eqnarray}
where $\mathcal{R}_{\chi_{0}}^{\rm SM}$ and
$\mathcal{R}_{\chi_{0}}^{\rm FCNC+SM}$ are the values of the ratio defined
in Eq.<ref> with a choice of $\chi_{0}$ for the SM and for the case of the presence of
FCNC. The uncertainty on $\mathcal{R}_{\chi_{0}}^{\rm SM}$ is denoted
by $\Delta \mathcal{R}_{\chi_{0}}^{SM}$. Considering the theoretical
and statistical uncertainties in the region of $m_{\gamma\gamma} > 500$, the value
of $\chi_{0} =8$ is found to provide the best sensitivity.
The upper bounds at the $68\%$ CL and at the $95\%$ CL on the FCNC anomalous couplings including only statistical uncertainties are found to be:
\begin{eqnarray}
68\%~\text{CL}:~ \kappa < 2.75\times 10^{-2}~\text{corresponding to}~Br(t\rightarrow
q\gamma) < 1.56\times 10^{-4}, \nonumber \\
95\%~\text{CL}:~ \kappa < 3.91\times 10^{-2}~\text{corresponding to}~Br(t\rightarrow
q\gamma) < 3.15\times 10^{-4},
\end{eqnarray}
and the $68\%$ and $95\%$ CL limits after including both the statistical and systematic
uncertainties are:
\begin{eqnarray}
68\%~\text{CL}:~ \kappa < 4.46\times 10^{-2}~\text{corresponding to}~Br(t\rightarrow
q\gamma) < 4.10\times 10^{-4}, \nonumber \\
95\%~\text{CL}:~ \kappa < 6.26\times 10^{-2}~\text{corresponding to}~Br(t\rightarrow
q\gamma) < 8.06\times 10^{-4}.
\end{eqnarray}
From these results it can be concluded that the angular variable $\chi$ is able to provide additional
sensitivity to $tq\gamma$ anomalous coupling with respect to the diphoton
mass spectrum.
Further optimization on both $m_{\gamma\gamma}$ and $\chi_{0}$ is expected to improve
the sensitivity to possible $tq\gamma$ contribution to diphoton production at the LHC.
The behavior of $\mathcal{R}$ in terms of
the cut on diphoton mass for the SM and SM+FCNC at
$\chi_{0} = 8$ and with the choice of $\kappa = 0.2$.
§ SUMMARY AND CONCLUSIONS
Rare top quark decays through flavor changing neutral currents in the
vertices of $tq\gamma$, $tqZ$, and $tqg$ are particularly interesting
as they are significantly sensitive to many extensions of the SM.
The SM predictions for the branching ratios of these rare decay modes are expected to be unobservable at the LHC
( $ < 10^{-12}$) while new physics models are able to enhance the
branching fractions by several order of magnitudes. As a consequence, any
observation of such processes would indicate new physics beyond the
SM. In this paper, we propose a new indirect way to search for the $tu\gamma$ and
$tc\gamma$ FCNC interactions. So far, these couplings have been directly
studied by CDF, DELPHI, H1, ZEUS, and CMS experiments at
colliders with at least a top quark in the final state of the
collisions. In this work, we propose to use the diphoton
invariant mass and angular differential distributions to probe $tq\gamma$
FCNC couplings. Using a measured mass spectrum of diphoton
at the LHC with the CMS experiment, an upper limit of $4.81 \times
10^{-3}$ is set on the branching fraction of $t\rightarrow
q\gamma$. Furthermore, we show that the angular variable
$\chi=e^{|\eta_{\gamma_{1}}-\eta_{\gamma_{2}|}}$ would allow us to
probe this branching fraction down to $8.2\times 10^{-4}$. These
limits have been obtained based on the LO prediction of the $tq\gamma$ FCNC contribution in diphoton productions at the LHC and are
comparable with the ones recently obtained from the search for anomalous single top
events by the CMS experiment.
§ ELECTRIC DIPOLE MOMENT ANALYSIS
In this section, we obtain upper limit on the $tu\gamma$ FCNC coupling using the present upper bound on the neutron electric dipole moment.
Such an approach has been used in constraining the $W$ boson electric
dipole moment <cit.>, top-Higgs non-standard interactions <cit.>,
and probing heavy charged gauge boson mass and couplings <cit.>.
We calculate the contribution of the $tu\gamma$
coupling to the neutron EDM using the effective interaction for the
$q\bar{q}\gamma$ vertex. The most general effective vertex
describing the interaction of a photon with two on-shell quarks can be
written as <cit.>:
\begin{eqnarray}\label{edmlag}
\Gamma_{\mu}(q^{2})= -ie
\left(\gamma_{\mu} F_{1v}(q^{2}) +
\frac{\sigma_{\mu\nu}}{2m_{q}}q^{\nu}[iF_{2v}(q^{2})+F_{2a}(q^{2})\gamma_{5}]
\right),
\end{eqnarray}
where $q$ is the four-momentum of
the off-shell photon. The functions $F_{1v}(q^{2})$ and
$F_{2v,2a}(q^{2})$ are called form factors which in the low energy
limit $q^{2} \rightarrow 0$, they are physical parameters and be
related to the static physical quantities according to the following
\begin{eqnarray}
F_{1v}(0) = Q_{q}~,~F_{2v}(0) = a_{q}~,~F_{2a} = d_{q}\frac{2m_{q}}{e},
\end{eqnarray}
where $Q_{q}$ is the electric charge of a quark $q$, $a_{q}$ and
$q_{q}$ are the magnetic dipole moment and electric dipole moment of a quark.
The electric dipole moment term violates the P and CP
invariance. Within the SM at tree level, $d_{q}$ and $a_{q}$ are zero
and however non-zero values for $d_{q}$ and $a_{q}$ arise from higher
order corrections. The SM prediction for the electric dipole moments
of the quarks are extremely small and expected to be smaller
$10^{-30}$ e.cm <cit.>.
Using the interactions described by Eq.<ref> at low energy, the $tu\gamma$
FCNC coupling introduced by Eq.<ref>, the induced CP violating
amplitude coming from the $tu\gamma$ FCNC interaction in
Fig. <ref> can be expressed as:
\begin{eqnarray}
\Gamma_{\mu}= \bar{u}(p_{2}) [\int \frac{d^4 k}{(2 \pi)^4} (\frac{Q_{t}e \kappa_{tu\gamma}}{\Lambda} \sigma_{\beta \beta'} k^{\beta'})\frac{\imath (k\!\!\!/-p\!\!\!/_{2}+m_{t})}{(k-p_{2})^2-m^{2}_{t}} (-\imath d_{t} \gamma_{5} \sigma_{\mu \nu} q^{\nu}) \frac{\imath (k\!\!\!/-p\!\!\!/_{1}+m_{t})}{(k-p_{1})^2-m^{2}_{t}}(\frac{Q_{t}e \kappa_{tu\gamma}}{\Lambda} \sigma_{\alpha \alpha'} k^{\alpha'})\frac{-\imath g^{\alpha \beta}}{k^2} ] u(p_{1}).
\end{eqnarray}
Feynman diagram contributing to the on shell $u\bar{u}\gamma$
vertex originating from the $tuH$ interaction.
After employing Dirac equation and Gordon identity, the above expression can be
simplified to find the up quark EDM arising from $tu\gamma$ FCNC coupling, which is the coefficient of
This integral on $k$ is divergent, therefore a mass scale of $\Lambda_{cut}$
is introduced as an ultraviolet cutoff scale.
After performing some algebraic
manipulations and integration over $k$, the amplitude is found to be:
\begin{eqnarray}
\Gamma_{\mu} &=& \frac{d_{t} (\frac{Q_{t}e \kappa_{tu\gamma}}{\Lambda})^2 m^2_{t}}{(4 \pi)^2} [\bar{u}(p_{2}) \imath\gamma_{5} \sigma_{\mu \nu} q^{\nu} u(p_{1})] \int 2 dx dy \nonumber \\
&\times & \lbrace \frac{- \Lambda_{\rm cut} ^2}{m^2_{t} }+ \ln \frac{\Lambda_{\rm cut} ^2}{\Delta}[6 x_{u} (x+y)(x+y-1)+3(x+y)+x_{u}-1]+\frac{x_{u}(x+y)}{2(1+x_{u}(x+y-1))} \rbrace,
\end{eqnarray}
where $d_{t}$ is the top quark EDM and
\begin{eqnarray}
\Delta = m^2_{t}(x+y)[1+ x_{u}(x+y-1)]~,
~x_{u} = \frac{m^2_{u}}{m^2_{t}}. \nonumber
\end{eqnarray}
As $x_{u}\sim 10^{-5}$, we take the limit of the amplitude for the
case of $x_{u} \rightarrow 0$. We find the following form for the amplitude:
\begin{eqnarray}
\Gamma_{\mu} &=& \frac{d_{t} e^2 Q_{t}^{2}\kappa^2_{tu\gamma} m^2_{t}}{(4 \pi)^2 \Lambda ^2} [\bar{u}(p_{2}) \imath\gamma_{5} \sigma_{\mu \nu} q^{\nu} u(p_{1})] \lbrace \frac{1}{6} - \frac{ \Lambda _{\rm cut} ^2}{m^2_{t} }+ \ln \frac{\Lambda_{\rm cut} ^2}{m^2_{t} } \rbrace.
\end{eqnarray}
Employing the effective coupling for $u\bar{u}\gamma$ coupling, we find
the $tu\gamma$ contribution to the up quark EDM:
\begin{equation}
d_{u}= \frac{d_{t} e^2Q_{t}^{2} \kappa^2_{tu\gamma} }{(4 \pi)^2 } \frac{m^2_{t}}{\Lambda ^2} \lbrace \frac{1}{6} - \frac{ \Lambda _{\rm cut} ^2}{m^2_{t} }+ \ln \frac{\Lambda_{\rm cut} ^2}{m^2_{t} } \rbrace.
\end{equation}
As seen, there are quadratic and logarithmic divergences to the up
quark EDM from $tu\gamma$ FCNC. However, there is a factor
$\Lambda^{2}$ in the denominator which comes form the $tu\gamma$
effective coupling and is the scale at which new physics effects is
expected to appear. It is natural to assume that the cutoff scale of
the loop divergences ($\Lambda_{\rm cut}$) is equal to $\Lambda$ which
is the scale at which new physics effects are expected to show
up, i.e. $\Lambda_{\rm cut} = \Lambda$.
Using the non-relativistic SU(6) wave functions, the neutron EDM can
be related to the up and down quark EDMs.
The neutron EDM in terms of EDMs of quarks is written as <cit.>:
\begin{eqnarray}
d_{n} = \eta (\frac{4}{3}d_{d} - \frac{1}{3}d_{u}) \,,
\end{eqnarray}
where the up and down quarks EDMs are denoted by $d_{u}$ and $d_{d}$
and $\eta$ describes the QCD higher order
corrections and is equal to 0.61.
The current experimental bound on the neutron EDM is $d_{n} < 2.9\times 10^{-26}$ e.cm. <cit.>.
The measured upper limit on the top quark EDM has been found to be $d_{t} < 10^{-16}$
e.cm <cit.>. For example, by setting $d_{t} = 10^{-16}, \Lambda = m_{t}$ the upper bound of 0.0026 on $\kappa_{tu\gamma}$ is obtained.
This is corresponding to an upper limit of the order of $10^{-6}$ on
$Br(t\rightarrow u\gamma) $.
K. A. Olive et al. [Particle Data Group Collaboration],
Chin. Phys. C 38, 090001 (2014).
J. A. Aguilar-Saavedra and B. M. Nobre,
Phys. Lett. B 553, 251 (2003)
S. L. Glashow, J. Iliopoulos and L. Maiani,
Phys. Rev. D 2 (1970) 1285.
G. Eilam, J. L. Hewett and A. Soni,
Phys. Rev. D 44, 1473 (1991)
Erratum: [Phys. Rev. D 59, 039901 (1999)].
doi:10.1103/PhysRevD.44.1473, 10.1103/PhysRevD.59.039901
F. del Aguila, J. A. Aguilar-Saavedra and R. Miquel,
Phys. Rev. Lett. 82, 1628 (1999)
D. Atwood, L. Reina and A. Soni,
Phys. Rev. D 55, 3156 (1997)
S. Bejar,
R. Gaitán, R. Martinez and J. H. M. de Oca,
arXiv:1503.04391 [hep-ph].
G. Abbas, A. Celis, X. Q. Li, J. Lu and A. Pich,
JHEP 1506, 005 (2015)
[arXiv:1503.06423 [hep-ph]].
B. Altunkaynak, W. S. Hou, C. Kao, M. Kohda and B. McCoy,
Phys. Lett. B 751, 135 (2015)
[arXiv:1506.00651 [hep-ph]].
J. M. Yang, B. L. Young and X. Zhang,
Phys. Rev. D 58, 055001 (1998)
G. M. de Divitiis, R. Petronzio and L. Silvestrini,
Nucl. Phys. B 504, 45 (1997)
J. L. Lopez, D. V. Nanopoulos and R. Rangarajan,
Phys. Rev. D 56, 3100 (1997)
J. Guasch and J. Sola,
Nucl. Phys. B 562, 3 (1999)
J. J. Liu, C. S. Li, L. L. Yang and L. G. Jin,
Phys. Lett. B 599, 92 (2004)
J. J. Cao, G. Eilam, M. Frank, K. Hikasa, G. L. Liu, I. Turan and J. M. Yang,
Phys. Rev. D 75, 075021 (2007)
K. Agashe, G. Perez and A. Soni,
Phys. Rev. D 75, 015002 (2007)
K. Agashe and R. Contino,
Phys. Rev. D 80, 075016 (2009)
[arXiv:0906.1542 [hep-ph]].
R. Goldouzian,
Phys. Rev. D 91, no. 1, 014022 (2015)
[arXiv:1408.0493 [hep-ph]].
J. A. Aguilar-Saavedra and G. C. Branco,
Phys. Lett. B 495, 347 (2000)
G. Durieux, F. Maltoni and C. Zhang,
Phys. Rev. D 91, no. 7, 074017 (2015)
[arXiv:1412.7166 [hep-ph]].
S. Khatibi and M. M. Najafabadi,
Phys. Rev. D 89, no. 5, 054011 (2014)
[arXiv:1402.3073 [hep-ph]].
J. L. Agram, J. Andrea, E. Conte, B. Fuks, D. Gelé and P. Lansonneur,
Phys. Lett. B 725, 123 (2013)
[arXiv:1304.5551 [hep-ph]].
J. A. Aguilar-Saavedra,
Nucl. Phys. B 837, 122 (2010)
[arXiv:1003.3173 [hep-ph]].
F. Larios, R. Martinez and M. A. Perez,
Phys. Rev. D 72, 057504 (2005)
H. Hesari, H. Khanpour, M. K. Yanehsari and M. M. Najafabadi,
Adv. High Energy Phys. 2014, 476490 (2014)
[arXiv:1412.8572 [hep-ex]].
H. Khanpour, S. Khatibi, M. K. Yanehsari and M. M. Najafabadi,
arXiv:1408.2090 [hep-ph].
N. Craig, J. A. Evans, R. Gray, M. Park, S. Somalwar, S. Thomas and M. Walker,
Phys. Rev. D 86, 075002 (2012)
[arXiv:1207.6794 [hep-ph]].
S. M. Etesami and M. Mohammadi Najafabadi,
Phys. Rev. D 81, 117502 (2010)
[arXiv:1006.1717 [hep-ph]].
D. Kim and M. Park,
arXiv:1507.03990 [hep-ph].
J. Cao, C. Han, L. Wu, J. M. Yang and M. Zhang,
Eur. Phys. J. C 74, no. 9, 3058 (2014)
[arXiv:1404.1241 [hep-ph]].
A. Arhrib, R. Benbrik, C. H. Chen, M. Gomez-Bock and S. Semlali,
arXiv:1508.06490 [hep-ph].
A. Greljo, J. F. Kamenik and J. Kopp,
JHEP 1407, 046 (2014)
[arXiv:1404.1278 [hep-ph]].
C. Degrande, F. Maltoni, J. Wang and C. Zhang,
Phys. Rev. D 91, 034024 (2015)
[arXiv:1412.5594 [hep-ph]].
S. Khatibi and M. M. Najafabadi,
Phys. Rev. D 90, no. 7, 074014 (2014)
[arXiv:1409.6553 [hep-ph]].
S. Y. Ayazi, S. Khatibi and M. Mohammadi Najafabadi,
JHEP 1210, 103 (2012)
[arXiv:1205.3311 [hep-ph]].
J. A. Aguilar-Saavedra,
Nucl. Phys. B 812, 181 (2009)
[arXiv:0811.3842 [hep-ph]].
J. A. Aguilar-Saavedra,
Acta Phys. Polon. B 35, 2695 (2004)
M. Mohammadi Najafabadi,
J. Phys. G 34, 39 (2007)
F. Abe et al. [CDF Collaboration],
Phys. Rev. Lett. 80, 2525 (1998).
J. Abdallah et al. [DELPHI Collaboration],
Phys. Lett. B 590, 21 (2004)
S. Chekanov et al. [ZEUS Collaboration],
Phys. Lett. B 559, 153 (2003)
F. D. Aaron et al. [H1 Collaboration],
Phys. Lett. B 678, 450 (2009)
[arXiv:0904.3876 [hep-ex]].
CMS Collaboration [CMS Collaboration],
CMS Collaboration [CMS Collaboration],
S. Chatrchyan et al. [CMS Collaboration],
Phys. Rev. Lett. 108, 111801 (2012)
[arXiv:1112.0688 [hep-ex]].
G. Aad et al. [ATLAS Collaboration],
Phys. Rev. D 92, no. 3, 032004 (2015)
[arXiv:1504.05511 [hep-ex]].
G. Aad et al. [ATLAS Collaboration],
New J. Phys. 15, 043007 (2013)
[arXiv:1210.8389 [hep-ex]].
L. Randall and R. Sundrum,
Phys. Rev. Lett. 83, 3370 (1999)
N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali,
Phys. Rev. D 59, 086004 (1999)
A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks,
Comput. Phys. Commun. 185, 2250 (2014)
[arXiv:1310.1921 [hep-ph]].
C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter,
Comput. Phys. Commun. 183, 1201 (2012)
[arXiv:1108.2040 [hep-ph]].
J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer,
JHEP 1106, 128 (2011)
[arXiv:1106.0522 [hep-ph]].
J. Alwall et al.,
JHEP 1407, 079 (2014)
[arXiv:1405.0301 [hep-ph]].
J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung,
JHEP 0207, 012 (2002)
T. Sjostrand, S. Mrenna and P. Z. Skands,
Comput. Phys. Commun. 178, 852 (2008)
[arXiv:0710.3820 [hep-ph]].
J. de Favereau et al. [DELPHES 3 Collaboration],
JHEP 1402, 057 (2014)
[arXiv:1307.6346 [hep-ex]].
S. Chatrchyan et al. [CMS Collaboration],
JINST 3, S08004 (2008).
V. Khachatryan et al. [CMS Collaboration],
JINST 10, no. 08, P08010 (2015)
[arXiv:1502.02702 [physics.ins-det]].
S. Catani and M. Grazzini,
Phys. Rev. Lett. 98, 222002 (2007)
S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini,
Phys. Rev. Lett. 108, 072001 (2012)
[arXiv:1110.2375 [hep-ph]].
S. Alekhin et al.,
arXiv:1101.0536 [hep-ph].
M. Botje et al.,
arXiv:1101.0538 [hep-ph].
A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt,
Eur. Phys. J. C 63, 189 (2009)
[arXiv:0901.0002 [hep-ph]].
R. D. Ball et al. [NNPDF Collaboration],
JHEP 1504, 040 (2015)
[arXiv:1410.8849 [hep-ph]].
A. L. Read,
J. Phys. G 28, 2693 (2002).
G. Aad et al. [ATLAS Collaboration],
Phys. Rev. Lett. 114, no. 22, 221802 (2015)
[arXiv:1504.00357 [hep-ex]].
P. Meade and L. Randall,
JHEP 0805, 003 (2008)
[arXiv:0708.3017 [hep-ph]].
L. A. Anchordoqui, J. L. Feng, H. Goldberg and A. D. Shapere,
Phys. Lett. B 594, 363 (2004)
U. Baur, I. Hinchliffe and D. Zeppenfeld,
Int. J. Mod. Phys. A 2, 1285 (1987).
T. Binoth, J. P. Guillet, E. Pilon and M. Werlen,
Eur. Phys. J. C 16, 311 (2000)
H. L. Lai, M. Guzzi, J. Huston, Z. Li, P. M. Nadolsky, J. Pumplin and C.-P. Yuan,
Phys. Rev. D 82, 074024 (2010)
[arXiv:1007.2241 [hep-ph]].
S. Chatrchyan et al. [CMS Collaboration],
JHEP 1201, 133 (2012)
[arXiv:1110.6461 [hep-ex]].
W. J. Marciano and A. Queijeiro,
Phys. Rev. D 33, 3449 (1986).
A. Soni and R. M. Xu,
Phys. Rev. Lett. 69, 33 (1992).
W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia and E. Remiddi,
Phys. Rev. Lett. 95, 261802 (2005)
W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber and E. Remiddi,
Nucl. Phys. B 723, 91 (2005)
M. Pospelov and A. Ritz,
Annals Phys. 318, 119 (2005)
I. S. Altarev et al.,
Phys. Lett. B 276, 242 (1992).
I. S. Altarev et al.,
Phys. Atom. Nucl. 59, 1152 (1996)
[Yad. Fiz. 59N7, 1204 (1996)].
J. L. Hewett and T. G. Rizzo,
Phys. Rev. D 49, 319 (1994)
|
1511.00333
|
Department of Physics and Astronomy, Johns Hopkins University,
Baltimore, MD 21218, USA
Radiative outflow and support in AGN tori
Chan & Krolik
Radiation-driven outflows from and radiative support in dusty tori of
active galactic nuclei
Chi-Ho Chan, Julian H. Krolik
Substantial evidence points to dusty, geometrically thick tori obscuring the
central engines of AGN, but so far no mechanism satisfactorily explains
why cool dust in the torus remains in a puffy geometry. Near-Eddington IR
and UV luminosities coupled with high dust opacities at these frequencies
suggest that radiation pressure on dust can play a significant role in shaping
the torus. To explore the possible effects of radiation pressure, we perform
three-dimensional RHD simulations of an initially smooth torus. Our code
solves the hydrodynamics equations, the time-dependent multi–angle group
IR RT equation, and the time-independent UV RT equation. We
find a highly dynamic situation. IR radiation is anisotropic, leaving
primarily through the central hole. The torus inner surface exhibits a break in
axisymmetry under the influence of radiation and differential rotation;
clumping follows. In addition, UV radiation pressure on dust launches a
strong wind along the inner surface; when scaled to realistic AGN
parameters, this outflow travels at
E)]^{1/4}\,\si{\kilo\meter\per\second}$ and carries
E)]^{3/4}\,\si{\solarmass\per\year}$, where $M$, $L_\su{UV}$, and $L_\su E$ are
the mass, UV luminosity, and Eddington luminosity of the central object
§ INTRODUCTION
The discovery of reflected broad emission lines hidden in polarized light of
type-2 AGN <cit.> came as a revelation to AGN research in that it can
only be reasonably explained by a geometrically and optically thick structure
surrounding the central source. Further observations established the properties
of the obscurer. The ratio of type-2 to type-1 objects
implies a high torus covering fraction, although the exact value of the ratio,
as well as its dependence on luminosity and redshift, is still under debate
<cit.>. There is unequivocal proof for dust
<cit.>; in particular, the broad
∼1∼100 bump in the SED is attributed to
thermal radiation from warm dust <cit.>, and the cutoff
at ≲2 is indicative of dust close to sublimation
<cit.>. Finally, the existence of ionization
<cit.> and scattering
cones also signifies a small, geometrically and optically thick, toroidal
structure with an opening spanning a fraction of the solid angle around the
central source. IR interferometry has provided the first direct
observation of the obscuring torus in the form of warm dust within several
parsecs from the center in NGC 1068 <cit.>, NGC 4151
Centaurus A <cit.>, Circinus <cit.>, and other nearby AGN
<cit.>. A sample of 29 AGN have
thus far been studied in this way <cit.>. The
preponderance of evidence in favor of the torus inspires the idea that
observational variations between AGN types 1 and 2 can be attributed to
orientation <cit.>.
A crucial missing piece to this AGN unification picture is an
understanding of torus dynamics. The torus has an aspect ratio of unity if its
velocity dispersion is comparable to its orbital velocity. If the velocity
dispersion were entirely due to thermal motion, hydrogen atoms at a distance
$r$ from SMBH of mass $M$ would have temperature $\num{\gtrsim1.7e6}\,
(M/\SI{e7}{\solarmass})|(r/\si{\parsec})^{-1}\,\si{\kelvin}$, hot enough to
destroy dust by sputtering <cit.>. Many models of
angle-dependent obscuration in AGN have been put forward over the past
decades in an effort to solve this problem. They fall into five general
categories, but as we shall show below, none of them is entirely satisfactory.
Some proposed intrinsically warped structures. For example,
<cit.> and <cit.>
<cit.> advanced the notion that in
lieu of a torus, obscuration could be provided by a geometrically thin warped
disk. The disk must stretch from ∼1∼e4 to reproduce
the observed IR spectrum, at odds with the presence of well-defined
ionization cones on ∼100 scales, with IR interferometric
observations, and with optical variability on a timescale of years
<cit.>. Worse still, the covering fraction is
less than half except for the most severe warps and twists, and twists are
imperative if one must obstruct more than half of the lines of sight at high
inclinations. Parsec-scale warps and twists have garnered recent attention,
with proponents arguing that they can be sustained by stochastic accretion of
clumps from random directions <cit.>,
or that they are bending modes excited by radial flows caused by a lopsided
disk <cit.>. However, the torus advocated by
<cit.> still suffers from the same shortcomings above,
whereas the aspect ratio of the <cit.> torus is only
Another option to partially avoid the dynamical problem is dust clumping.
Collisions between clumps can convert orbital shear to bulk velocity dispersion
<cit.>. The collision rate must be almost once per orbit
for the mid-plane to be completely covered. Should these supersonic encounters
be inelastic, the resulting shocks would quickly turn the velocity dispersion
of clumps into internal energy; a torus that cools efficiently would settle to
the mid-plane, and one that does not would be geometrically thick, but so hot
that dust is burnt away. Clumps threaded with magnetic fields could be
sufficiently elastic, but the conditions are rather unusual, and one would ask
how adequate field strength could be sustained.
Other workers turn to large-scale magnetic fields for an answer. Dusty
molecular material lifted up from the surface of the accretion disk around the
central mass could be entrained in a magnetocentrifugal wind; in this scenario,
the torus is merely the parts of the wind which happen to be optically thick
enough <cit.>. The dust perhaps takes the form of
optically thick clumps embedded in the wind
. Alternatively, magnetic fields could directly
support a static torus against gravity <cit.>. Magnetic
models, however, require strong, ordered fields on large scales, which are
difficult to justify.
Still another alternative is to invoke the nuclear starbursts seen in some
Seyfert 2s <cit.>. They prompted
<cit.> to suggest turbulence stirred up by supernovae as a
means of creating a quasi-stable torus, but its size needs to be
≳30, and even then the covering fraction is
≲0.2. The obscuring gas disk of <cit.> has
similar drawbacks in that its size and aspect ratio are ≳10
and ≲0.3. Stellar feedback is in fact too weak to keep the torus
geometrically thick on parsec scales <cit.>. Attacking the
problem from a different perspective, <cit.> considered
mass and energy injection by stars in a spherical and isotropic nuclear
cluster. Filaments in that scheme are formed by shock waves from supernovae and
planetary nebulae interacting with one another, while cold clumps come from
cooling. An analogous proposal by <cit.> looked at
supernova ejecta and stellar winds released with some angular momentum. The gas
cools and is compressed to filaments, which then flows inward and accumulates
at the centrifugal barrier, forming a torus made geometrically thick by
X-ray heating. Both models attempt to circumvent the weakness of
stellar feedback by injecting gas at the positions of the stars of a spatially
extended cluster, hence the fate of the torus is unclear once the starburst
ends. Moreover, the specific mass injection rate in the latter model is
∼6e3 times the galactic specific star formation rate.
Last but not least, <cit.> realized that since dust
opacity in the IR is ≳10 times Thomson opacity, even
sub-Eddington AGN luminosities could dramatically affect the torus through
radiation pressure. In their picture, UV radiation from the central source
is converted to IR on the inner surface of a smooth cylindrical torus
<cit.>; part of the IR radiation diffuses through the
torus and supports it. revisited the problem and
constructed an analytic solution of a smooth axisymmetric torus under the
combined influence of gravity and radiation; <cit.> later
extended his work by incorporating the effects of hard X-ray and
stellar heating. Unfortunately, both models are overly simplistic in that they
assume a hydrostatic torus and the diffusion approximation for the IR
radiative flux.
Others have developed ideas along a similar vein. For example,
<cit.> considered radiation pressure
from both AGN and a nuclear starburst ring, yet their obscuring structure
is stable near the mid-plane only for specific parameters.
studied a magnetocentrifugal wind accelerated by
radiation from an accretion disk; the wind again depends on the existence of
some postulated large-scale magnetic field. An alternative model from
<cit.> focuses instead on turbulence
generated when gas streams lifted up by radiation fall back to the mid-plane
and intersect. Its conclusions can only be tentative because UV heating
and radiative cooling in this model assume ionization by starlight while
X-ray heating is based on stellar-mass black hole X-ray
spectra, entirely ignoring AGN radiation. The model also does not treat
dust destruction by sputtering at temperatures ≳e5. In
addition, the omission of reprocessed IR radiation in these three schemes
renders their applicability to optically thick tori doubtful. Less directly
related is the suggestion from <cit.> that a starburst
disk with Eddington luminosity in the IR possesses a tenuous, dusty, and
geometrically thick atmosphere.
In a series of articles,
<cit.> investigated the effects of IR radiation pressure on
dusty tori using simulations that couple hydrodynamics and radiation.
Encouragingly, they found that gas evolves naturally to a geometrically thick
obscuring wind. However, there are two limitations to this work. They neglected
momentum deposition from direct UV illumination. More worrisome is their
use of the FLD approximation, which can yield radiative fluxes in
completely wrong directions wherever the optical depth is comparable to or
smaller than unity. This problem is especially troubling when the dynamical
effect of radiation is important <cit.>, as it is
here. took the complementary direction of
performing Monte Carlo RT on dusty gas and calculating the radiative
acceleration. The fact that they find accelerations exceeding gravity
emphasizes that hydrodynamics and RT should be treated together.
We adopt a different approach in this article. Our program is to conduct a
series of numerical experiments designed to yield physical insight into each of
the most prominent mechanisms affecting torus dynamics; by adding mechanisms
one at a time, we hope to be able to distinguish their effects. Only toward the
end of this process will it be appropriate to draw specific relations between
our results and observable quantities. We begin in this article by presenting
three-dimensional, time-dependent RHD simulations of a dusty torus that
experiences radiative acceleration on dust due to UV radiation from the
central source and diffuse IR radiation in the torus. Our simulations used
the finite-volume hydrodynamics code Athena <cit.>
augmented by its time-dependent RT module <cit.> for
IR radiation and a new long-characteristics RT module for UV
radiation (<ref>). The code simultaneously solves
the time-dependent hydrodynamics and RT equations; most notably, it solves
the RT equations on a large number of grid rays rather than adopting
ad hoc closure prescriptions. We leave other ingredients, such as
magnetic fields, realistic atomic and molecular heating and cooling rates, and
dust destruction by sputtering in high-temperature regions, to future
In interpreting these results, it must be remembered that since the character
of the system demands mass loss from the inner surface, realistic tori
must be resupplied externally. Our simulation, and indeed any other
simulation beginning with a finite amount of mass, cannot portray
steady-state tori. The common device of putting a large gas reservoir at large
distances would impose a misleading radiative boundary condition. For this
reason, any connection between simulated and real tori must be posed in terms
of the rate of mass resupply necessary to secure stationarity.
We dedicate <ref> to our equations and simulation parameters.
Results are presented in <ref>, while discussion can be found in
§ METHODS
We consider a cold, dusty, and optically thick torus orbiting a point mass $M$
at the origin. Isotropic UV radiation of luminosity $L_\su{UV}$ emerges
from the origin. UV radiation impinging on the inner surface is absorbed
by dust and re-emitted in the IR; radiation pressure from both the IR
and the UV, in concert with rotation, supports the torus against gravity.
Cylindrical coordinates $(R,\phi,z)$ are a natural choice for describing this
system, although we do occasionally refer to the spherical radius
$r\eqdef(R^2+z^2)^{1/2}$. From now on, the adjective `radial' shall implicitly
refer to the cylindrically radial direction. We also call the section of the
inner surface near the mid-plane the `inner edge.'
§.§ Hydrodynamics
We begin by examining the equations governing the dynamics of the torus. The
hydrodynamics equations are
\begin{align}
\label{eq:gas mass}
\pd\rho t+\divg(\rho|\vec v) &= 0, \\
\label{eq:gas momentum}
\pd{}t(\rho|\vec v)+\divg(\rho|\vec v|\vec v+p|\tsr I) &=
-\rho|\grad\Phi+\momsrc{IR}+\momsrc{UV}, \\
\label{eq:gas energy}
\pd Et+\divg[(E+p)|\vec v] &=
-\rho|\vec v\cdot\grad\Phi+\ergsrc{IR}+\ergsrc{UV}.
\end{align}
Here $\rho$, $\vec v$, and $p$ are gas density, velocity, and pressure. Gas
temperature and total energy density are $T=p/(\rho R_\su{ideal})$ and
$E=\ifaastex{\rho v^2/2}{\tfrac12|\rho v^2}+p/(\gamma-1)$, where $R_\su{ideal}$
and $\gamma$ are the specific ideal gas constant and the ratio of specific
heats. The gravitational potential of the central mass is $\Phi(\vec r)=-GM/r$.
The energy and momentum source terms due to radiation are $\ergsrc{IR,UV}$ and
$\momsrc{IR,UV}$; we shall define the IR source terms in <ref>, and the UV source terms in <ref>. Finally, the isotropic rank-two tensor is denoted by $\tsr I$.
The presence of dust means that gas temperature is ≲e5,
otherwise dust would be rapidly destroyed by sputtering. This temperature is
much smaller than the virial temperature, hence the gas sound speed is also a
tiny fraction of the orbital velocity, or $c_\su s/v_\phi\ll1$. Because gas
pressure alone falls far short of maintaining the geometrical thickness of the
torus, it is dynamically unimportant compared to whatever pressure that
actually provides support against gravity, such as IR radiation pressure,
so an approximate equation of state for the gas is entirely satisfactory. This
approximation breaks down outside the torus body, particularly in the central
hole where photoionization heating and Compton recoil can strongly heat the gas
<cit.>. In the interest of focusing
attention on radiation-driven dynamics, in the simulations presented here we do
not change the equation of state between the body and the central hole. We plan
in future work to incorporate photoionization heating and related processes;
the increased gas pressure in the central hole could potentially alter the
shape of the inner surface.
We treat dust and gas as a single fluid with common velocity and temperature.
The fact that dust contributes significantly to IR emission implies a dust
temperature below sublimation. We expect hydrogen at such temperature to remain
molecular and the vibrational modes of the molecule to be weakly excited; we
therefore set $R_\su{ideal}=\kB/(2|m_\element H)$ and
§.§ Radiative transfer
Dust has ∼e2∼e3 times greater opacity to UV radiation
than to IR radiation <cit.>; such a large
contrast compels us to treat radiation at the two frequencies separately.
UV radiation comes from the innermost regions of an accretion disk at the
origin, but the angular distribution of its radiative flux is poorly known. The
classical picture of a limb-darkened disk only holds for a Newtonian,
scattering-dominated, geometrically thin, and optically thick disk; disk
turbulence, thermal instabilities, coronal scattering, as well as relativistic
boosting, beaming, and ray-bending, could all skew the angular profile of
emergent radiation. The axis of the disk also need not be aligned with that of
the torus. Because we lack a detailed disk model, and because our desire is to
understand physical principles rather than to provide observables, we simply
allow our UV radiative flux to be isotropic instead of giving it a more
complicated and more model-dependent angular distribution.
Several RT modules have already been developed for Athena. The
time-independent module <cit.>
performs RT on a snapshot of the simulation, computes the Eddington
tensor, and uses it to close the angular moments of the RT equation. In
comparison, the time-dependent module <cit.> tracks the
propagation of radiation by solving the multi–angle group RT equation
directly. Both modules are suited to handling diffusive IR radiation
inside the torus, but we are restricted to the time-dependent module because it
is the only one available for cylindrical coordinates.
None of these modules is appropriate for point-source radiation crossing the
optically thin region between the central source and the torus because they
concentrate radiation along directions defined by the angle grid. Contours of
constant radiation energy density, instead of being spherically symmetric, show
prominent spherically radial spikes coincident with the grid rays. We therefore
reserve the time-dependent module for reprocessed IR radiation inside the
torus. UV radiation emitted by the central source is handled with the
method of long characteristics, as described in <ref>.
§.§.§ Time-dependent IR radiative
To first order in $v/c$, where $c$ is the speed of light, the mixed-frame
time-dependent RT equation for IR radiation interacting with gray
material reads <cit.>
\begin{multline}\label{eq:radiative transfer}
\frac1c|\pd{I_\su{IR}}t+\uvec n\cdot\grad I_\su{IR}=
\Bigl(-1+\uvec n\cdot\frac{\vec v}c\Bigr)|
\rho|(\kappa_\su{IR}+\sigma_\su{IR})|I_\su{IR} \\
+\Bigl(1+3\,\uvec n\cdot\frac{\vec v}c\Bigr)|
\rho|(\kappa_\su{IR}|B+\sigma_\su{IR}|J_\su{IR})
-2|\rho|\sigma_\su{IR}|\frac{\vec v}c\cdot\vec H_\su{IR} \ifaastex{}\\
+\rho|(\kappa_\su{IR}-\sigma_\su{IR})|\frac{\vec v}c\cdot
(\vec H^0_\su{IR}-\vec H_\su{IR}).
\end{multline}
The specific intensity integrated over the IR in the observer frame is
$I_\su{IR}(\uvec n)$; its lowest three angular moments are $J_\su{IR}$, $\vec
H_\su{IR}$, and $\tsr K_\su{IR}$, from which the IR radiation energy
density and flux follow as $E_\su{IR}=(4\pi/c)|J_\su{IR}$ and $\vec
F_\su{IR}=4\pi|\vec H_\su{IR}$. The frequency-integrated blackbody mean
intensity is $B(T)=c|\aSB|T^4/(4\pi)$, where $\aSB$ is the radiation constant.
The coupling between gas and radiation is mediated by $\kappa_\su{IR}$ and
$\sigma_\su{IR}$, the comoving absorption and scattering cross sections per
mass in the IR. If we take the zeroth and first angular moments of
<ref>, we get
1/c|J_IRt+H⃗_IR =
1/c|H⃗_IRt+K_IR =
The remaining piece to specify in <ref> is $\vec H^0_\su{IR}$, the first
angular moment of the IR specific intensity in the fluid frame. It is
related to the angular moments in the observer frame by a Lorentz
transformation <cit.>:
\begin{equation}
\vec H^0_\su{IR}=\vec H_\su{IR}
-\frac{\vec v}c|J_\su{IR}-\frac{\vec v}c\cdot\tsr K_\su{IR}+\bigO(v^2/c^2).
\end{equation}
An unreasonably small time step is needed for the time-dependent RT module
if $v\ll c$, but the fact that radiation relaxes to equilibrium much faster
than the hydrodynamic timescale means we can circumvent the problem with the
reduced speed of light approximation <cit.>. The details are in <ref>; for
now, it suffices to know that the approximation replaces the physical light
speed $c$ attached to the time derivatives in <ref> with the reduced
light speed $\hat c$ subject to the requirement $v<\hat c\ll c$. An improvement
to how the time-dependent RT module treats scattering is set forth in
§.§.§ IR and
UV opacities
The chief sources of opacity in our system are dust absorption and electron
scattering, which we model as
\begin{align}
\kappa_\su{IR}(T) &\eqdef \bar\kappa_\su{IR}\times
\frac12|\left[1-\tanh\frac{\log_{10}(T/T_\su{ds})}{\Delta_\su{ds}}\right], \\
\kappa_\su{UV}(T) &\eqdef \bar\kappa_\su{UV}\times
\frac12|\left[1-\tanh\frac{\log_{10}(T/T_\su{ds})}{\Delta_\su{ds}}\right], \\
\sigma_\su{IR}(T) &\eqdef \kappaT\times
\frac12|\left[1+\tanh\frac{\log_{10}(T/T_\su{hi})}{\Delta_\su{hi}}\right].
\end{align}
In these fitting formulae, $T_\su{ds}\approx\SI{1500}{\kelvin}$ is the dust
sublimation temperature <cit.>, $T_\su{hi}\approx\SI{4013}{\kelvin}$
is the temperature at which hydrogen atoms in LTE at a number density of
e4 are collisionally half-ionized, and
$\kappaT\approx\SI{0.397}{\centi\meter\squared\per\gram}$ is the Thomson
scattering cross section per mass. The dust opacities are normalized to Thomson
as $\bar\kappa_\su{IR}/\kappaT=20$ and $\bar\kappa_\su{UV}/\kappaT=80$, a
choice we shall justify in <ref>; the parameters
governing the transition between opacity regimes are $\Delta_\su{ds}=0.05$ and
§.§ Simulation setup
We now spell out in detail the initial and boundary conditions, as well as
various tricks to keep the simulation stable.
§.§.§ Initial condition
The initial condition is based on the analytic solution of an axisymmetric
hydrostatic torus by <cit.>. To summarize, the radiation
energy density inside the torus is determined along the mid-plane by
\begin{equation}
\frac1{1+\xi}|\biggl[\left(\frac R{R_\su{in}}\right)^{-(1+\xi)}-1\biggr]
\biggl[\left(\frac R{R_\su{in}}\right)^{-\xi}-1\biggr]\right\},
\end{equation}
\begin{multline}
+\frac{3GM|\rho_\su{in}}{R_\su{in}}\times{} \\
\left\{\frac1{1+\xi}|
\biggl[\left(\frac R{R_\su{in}}\right)^{-(1+\xi)}-1\biggr]
\biggl[\left(\frac R{R_\su{in}}\right)^{-\xi}-1\biggr]\right\},
\end{multline}
and everywhere else by the constant-$E^0_\su{IR}$ contours
\begin{equation}
\frac12|\left(\frac z{R_\su{in}}\right)^2
+\frac12|\left(\frac R{R_\su{in}}\right)^2
-\frac13|j_\su{in}^2|\left(\frac R{R_\su{in}}\right)^3=\const.
\end{equation}
Of the five free parameters, four pertain to quantities measured at the inner
edge: radial coordinate $R_\su{in}$, gas density $\rho_\su{in}$, comoving
IR radiation energy density $(E^0_\su{IR})_\su{in\vphantom0}\nosup$, and
ratio of gas orbital to Keplerian velocity $j_\su{in}$. The remaining free
parameter is the radial power-law exponent $\xi$ of gas density along the
mid-plane. We distinguish between $E_\su{IR}$ and $E^0_\su{IR}$, the IR
radiation energy density in the observer and fluid frames respectively.
Although the radiative initial condition inside the torus is fully specified by
$E^0_\su{IR}$, the procedure for assigning $I_\su{IR}$ to individual grid rays
is somewhat elaborate, and is therefore relegated to <ref>. Gas density inside the torus is given by
\begin{equation}
\rho(R,z)\eqdef-\left[\frac{3GM}{R^2}|
\left(1-j_\su{in}^2|\frac R{R_\su{in}}\right)\right]^{-1}|\pd{E^0_\su{IR}}R;
\end{equation}
in particular, $\rho(R,0)=\rho_\su{in}|(R/R_\su{in})^{-\xi}$. Gas temperature
and pressure are established by thermal equilibrium between gas and radiation,
to wit, $E^0_\su{IR}=\aSB|T^4$. Lastly, gas inside the torus has orbital
\begin{equation}\label{eq:IC velocity}
\vec v\eqdef j_\su{in}|\left(\frac{GM}{R_\su{in}}\right)^{1/2}\,\uvec e_\phi;
\end{equation}
in other words, $j/j_\su{in}=(R/R_\su{in})^{1/2}$, where $j\eqdef
v_\phi|(R/GM)^{1/2}$. This velocity profile in fact applies to all hydrostatic
radiation-supported tori in point-mass potentials (<ref>). The torus has extent $1<R/R_\su{in}<j_\su{in}^{-2}$ and
The free parameters are selected in a similar fashion to
<cit.>. We pick $j_\su{in}=\ifaastex{1/2}{\tfrac12}$ such
that the inner edge is not supported by rotation alone, and that its vertical
extent is comparable to its radial coordinate. Having
\begin{equation}\label{eq:IC inner edge energy ratio}
\end{equation}
ensures IR and gravitational accelerations are comparable, that is,
$\norm{\grad E_\su{IR}}/\rho\sim GM/r^2$, although
$(E^0_\su{IR})_\su{in\vphantom0}\nosup$ could take on any value as long as
$E^0_\su{IR}\ge0$ inside the torus. The only deviation from
<cit.> is in our choice that $\xi=1$, which results in a
less massive torus.
This initial condition is not an exact equilibrium since the central source may
not be able to maintain the initial distribution of IR radiation energy
density along the inner surface. It is not even intended to resemble the
quasi-steady state of a realistic, axisymmetric, radiation-supported torus
since its properties, such as its radial and vertical extent, can be
arbitrarily altered by manipulating, say, the parameter $j_\su{in}$. The
initial condition is merely an approximate analytic solution of a hydrostatic
radiation-supported torus; as such, it is a convenient initial condition to
The exterior of the torus is filled with isothermal and hydrostatic ambient
material; we grant it nonzero orbital velocity because static ambient material
is found to be numerically unstable. To determine the properties of the ambient
material, we build upon the method used by <cit.> for
slender tori. The gravitational term of the force equation is clearly the
gradient of some scalar field; if we stipulate polytropic gas and a power-law
orbital velocity profile, then both pressure and centrifugal terms are
gradients as well, and the force equation becomes an easily solvable algebraic
equation. Unlike the solution of <cit.>, which is an
expansion around some $(R,z)$, our solution is exact. The density, pressure,
and velocity given by our method assuming a polytropic index of unity are
\begin{align}
\rho_\su{amb}(R,z) &\eqdef \bar\rho_\su{amb}\exp\biggl[
\frac{GM}{r|(c_\su s^2)_{\smash{\su{amb}}}\nosup}
+\frac{(v_\phi^2/c_\su s^2)_{\smash{\su{amb}}}\nosup}
{2-2|q_{\smash{\su{amb}}}}\biggr], \\
p_\su{amb}(R,z) &\eqdef
\rho_\su{amb}(R,z)\times(c_\su s^2)_{\smash{\su{amb}}}\nosup, \\
\vec v_\su{amb}(R) &\eqdef
\left(\frac{GM}{R_\su{amb}}\right)^{1/2}|
\biggl(\frac R{R_\su{amb}}\biggr)^{1-q_\su{amb}}\,\uvec e_\phi.
\end{align}
The center of the simulation domain $R_\su{amb}$ sets the length scale of the
ambient material, while the other parameters are
$\bar\rho_\su{amb}=\num{2e-8}\,\rho_\su{in}$, $(c_\su
s^2)_{\smash{\su{amb}}}\nosup=GM/R_\su{amb}$, and $q_\su{amb}=1.75$. The shear
parameter must satisfy $1.5<q_\su{amb}<2$ in order that the ambient material
have finite height and be stable. Since it is preferable that density and
pressure vary monotonically across the torus boundary, we additionally require
$\rho\ge\rho_\su{amb}$ and $p\ge p_\su{amb}$ everywhere in the initial
§.§.§ Central mass and reduced speed of light
The astute reader will notice that we have evaded any mention of the value of
the central mass $M$. This is because its choice is by far the most complicated
consideration in our simulations.
Sharp discontinuities in numerical calculations are flanked by ringing
artifacts, which resemble wiggles associated with the Gibbs phenomenon. These
artifacts usually damp out over time; however, in the case of a cylindrical
discontinuity in a gas partially supported against gravity, such as the inner
edge, the artifact grows rapidly at any spatial resolution. Experimentation
with different values of $c_\su s/v_\phi$ shows that the artifact can be
suppressed by demanding $c_\su s/v_\phi\gtrsim\bigO(0.1)$. If $c_\su s/v_\phi$
is kept at the low end of the numerically permitted range, gas pressure should
always be weak compared to gravity; as long as gas pressure is a minor
influence, it should not matter if it is not as tiny as in realistic
astrophysical circumstances.
These constraints determine $M$. The gas equilibrium temperature $T_\su{in}$ at
the inner edge is set by $\kappa_\su{UV}|L_\su{UV}/(4\pi|R_\su{in}^2)=
\kappa_\su{IR}|c|\aSB|T_\su{in}^4$; the stability requirement
$R_\su{ideal}|T_\su{in}/(GM/R_\su{in})\gtrsim\bigO(0.1)^2$ then becomes
\begin{align}
\nonumber
M &\lesssim \frac{R_\su{ideal}^2|(\kappa_\su{UV}/\kappa_\su{IR})}
{G|\kappaT|\aSB|T_\su{ds}^2}|\left(\frac{L_\su{UV}}{L_\su E}\right)|
\left(\frac{T_\su{in}}{T_\su{ds}}\right)^{-2}\bigO(0.1)^{-4} \\
&\approx \num{7.58e-4}\,\left(\frac{L_\su{UV}/L_\su E}{0.1}\right)|
\left(\frac{T_\su{in}}{T_\su{ds}}\right)^{-2}\bigO(0.1)^{-4}\,
\si{\solarmass},
\end{align}
with $L_\su E$ being the Eddington luminosity. We use
$M\approx\SI{0.758}{\solarmass}$ in practice. We shall argue in
<ref> that our failure to simulate a torus around a genuine
supermassive black hole is completely superficial.
We now consider how $M$ affects our choice of $\hat c$. The dynamical timescale
is $[R_\su{in}^3/(GM)]^{1/2}$, whereas the IR radiation diffusion
timescale in the reduced speed of light approximation is
c$. Clean separation of dynamical evolution from IR radiation diffusion
\begin{equation}
\frac{\hat c}{(GM/R_\su{in})^{1/2}}\gg
\rho_\su{in}|\bar\kappa_\su{IR}|R_\su{in}\times\frac14|(j_\su{in}^{-2}-1)^2;
\end{equation}
the right-hand side is an overestimate by a factor of a couple because density
falls off away from the inner edge. We settle on $\hat
c\sim50\,(GM/R_\su{in})^{1/2}$ as a trade-off between accuracy and
computational time (see <ref> for the actual value),
although we find little qualitative difference even at $\hat
c\approx8.94\,(GM/R_\su{in})^{1/2}$ as long as $v<\hat c$ everywhere.
§.§.§ Normalization and parameters
Physical quantities are hereafter normalized to their respective fiducial
values. The fundamental fiducial quantities are the central mass $M$, the dust
sublimation temperature $T_\su{ds}$, and the Thomson scattering cross section
per mass $\kappaT$; all other fiducial quantities, listed in
<ref>, are derived from them. In particular, $L_\su E$ is the
Eddington luminosity, and $r_0$ is the distance where the effective temperature
of the radiative flux in vacuum from a source with Eddington luminosity equals
$\smash{\sqrt2}$ times the dust sublimation temperature. Note that a system in
which rotational support is provided by diffusive radiation must have
$\rho_0|v_0^2/r_0\sim E_0/r_0$.
Derived fiducial quantities.
Fiducial quantity Symbol Definition
luminosity $L_\su E$ $4\pi GMc/\kappaT$
length $r_0$ $[L_\su E/(4\pi c|\aSB|T_\su{ds}^4)]^{1/2}$
velocity $v_0$ $(GM/r_0)^{1/2}$
time $t_0$ $(GM/r_0^3)^{-1/2}$
gas density $\rho_0$ $(\kappaT|r_0)^{-1}$
gas pressure $p_0$ $\rho_0|v_0^2=\aSB|T_\su{ds}^4$
radiation energy density $E_0$ $L_\su E/(4\pi|r_0^2|c)=p_0$
radiative flux $F_0$ $c|E_0$
One virtue of our normalization is that, because the characteristic length
scale is $r_0\propto M^{1/2}$, the gravitational acceleration at $r=r_0$ does
not depend on $M$. We can guarantee accelerations due to gas pressure and
radiation are likewise independent of $M$ by fixing $c_\su s/v_\phi$ and
$L_\su{UV}/L_\su E$ for each simulation. These invariances ensure that the
character of the dynamics simulated differs from that for more astrophysically
relevant values of $M$ only in the magnitude of the timescale $t_0\propto
M^{1/4}$. The normalizations of other quantities, such as momentum density,
could nevertheless vary with $M$.
Now that we have a system of normalization in place, we can translate our
choice $M\approx\SI{0.758}{\solarmass}$ in <ref> to
dimensionless parameters that the simulation actually accepts, namely,
$R_\su{ideal}=0.05\,p_0/(\rho_0|T_\su{ds})$ and $c\approx\num{2.70e4}\,v_0$.
It remains to pick the appropriate parameters for the simulations. To start
with, we choose $0.10\le L_\su{UV}/L_\su E\le0.15$ in steps of 0.01 because
these luminosities are high enough to hold back the infall of the torus, but
low enough not to push it away too briskly. The simulation at each $L_\su{UV}$
is run for about two orbits at the inner edge, at which point the radial
component of velocity is positive throughout the torus body.
Three of the five parameters governing the initial condition have already been
picked in <ref>; the remaining two will be given here.
The inner edge $R_\su{in}$ should be just outside the dust sublimation radius
<cit.>, that is,
Our initial condition puts $R_\su{in}=0.8\,r_0$, so that $R_\su{in}$ goes from
$1.26\,r_\su{ds}$ to $1.03\,r_\su{ds}$ as $L_\su{UV}/L_\su E$ varies from 0.10
to 0.15. The reduced light speed introduced in <ref> can be
recast in terms of fiducial values as $\hat c=50\,v_0$.
The density at the inner edge is selected to be $\rho_\su{in}=\rho_0$. The
radial Thomson optical depth of our initial condition along the mid-plane is
\begin{equation}
\int_{R_\su{in}}^\infty dR\,\rho|\kappaT=
\rho_\su{in}|R_\su{in}|\kappaT\times
\begin{cases}
[j_\su{in}^{-2|(1-\xi)}-1]/(1-\xi), & \xi\ne1, \\
2\ln j_\su{in}^{-1}, & \xi=1,
\end{cases}
\end{equation}
while the vertical Thomson optical depth at $R=R_\su{in}$ is
\begin{equation}
\int_{-\infty}^\infty dz\,\rho|\kappaT=
\int_1^{j_\su{in}^{-2}}dx\,\frac{x^{-(2+\xi)}|(1-j_\su{in}^2|x)}
\end{equation}
\begin{multline}
\int_{-\infty}^\infty dz\,\rho|\kappaT= \\
\int_1^{j_\su{in}^{-2}}dx\,\frac{x^{-(2+\xi)}|(1-j_\su{in}^2|x)}
\end{multline}
Our parameters yield Thomson optical depths of ≈1.11 and
≈1.01 respectively, consistent with the observed range of values
<cit.>. The corresponding IR optical depths
are established by numerical integration to be ≈19.9 and
≈10.9. The ratios of Thomson to IR optical depths are not
$\kappaT/\bar\kappa_\su{IR}$ due to the higher temperature and lower IR
opacity near the inner edge.
The simulation domain spans
in $(R,\phi,z)$. The vertical direction is made as tall as possible to capture
escaping material, while not so tall that the centrifugal barrier would cause
numerical problems at the inner-radial boundary. The number of grid cells is
$188\times33\times320$ in $(R,\phi,z)$, and the number of grid rays per cell is
§.§.§ Boundary conditions and numerical limits
Periodic hydrodynamic and radiative boundary conditions are adopted for the
azimuthal direction, with the understanding that grid rays at one boundary must
be rotated through $\pm\ifaastex{\pi/2}{\tfrac12|\pi}$ before they can be
copied to the ghost zones at the opposite boundary, to account for the fact
that the simulation domain covers only a quarter of a circle
Outflow hydrodynamic boundary conditions are applied at both boundaries in the
radial and vertical directions. The value of $\vec v$ in the ghost zones is
duplicated from the last physical cell, and components pointing into the
simulation domain are zeroed. We then adjust $\rho$ and $p$ in the ghost zones
at constant $c_\su s^2\eqdef p/\rho$ so that the pressure gradient exactly
cancels the gravitational and centrifugal forces. The value of $c_\su s^2$ is
the greater of $p/\rho$ of the last physical cell and $(c_\su
s^2)_{\smash{\su{amb}}}\nosup$; bounding $c_\su s^2$ from below protects $\rho$
and $p$ from numerical underflow.
Outflow radiative boundary conditions are used for the outer-radial and both
vertical boundaries. For grid rays pointing away from the simulation domain, we
copy their values of specific intensity from the last physical cell to the
ghost zones; for all other grid rays, we set their values of specific intensity
to zero. We also implement a cutout boundary condition for the inner-radial
boundary. Ghost zones are filled out in the same way as an outflow boundary; on
top of that, for every radially inward grid ray intersecting this boundary, we
trace its trajectory across the cylindrical cutout to where it re-enters the
domain, and add the specific intensity of the exiting grid ray to the
corresponding grid ray in the ghost zone at the re-entry point without allowing
for any time delay. Since the angle grid does not vary with coordinates, the
matching of exiting to re-entering grid rays is exact. Grid rays re-entering at
azimuthal coordinates outside the simulation domain are wrapped back after a
suitable rotation.
Limits on gas density and temperature are enforced for the sake of numerical
stability. We require that $\rho$ satisfy $\rho\ge\rho_\su{amb}$, and that $T$
satisfy $\num{e-3}\,T_\su{ds}\le T\le 10\,(c_\su
s^2)_{\smash{\su{amb}}}\nosup/R_\su{ideal}$; if at any time $\rho$ and $T$
violate these conditions, we reset them to the nearest value within the
acceptable range. The density floor guarantees a stable vacuum. A static
pressure floor is unsatisfactory because pressure could hit the floor before
density; any further drop in density would result in erroneous heating of the
gas, making the overall time step unreasonably small. A better approach is to
restrict temperature to within a generous range. Because
$\kappa_\su{IR,UV}\approx0$ when $T\gg T_\su{ds}$ (<ref>) and
$R_\su{ideal}|T_\su{ds}\ll(c_\su s^2)_{\smash{\su{amb}}}\nosup$ if the torus is
supported by radiation pressure, radiative heating cannot bring the gas to the
temperature ceiling.
§ RESULTS
We now present the results of our simulations. It is important to remember that
the simulations do not reach steady state; therefore, what we report below is
the transient response of the <cit.> hydrostatic torus to
UV irradiation. We already see within the first two orbits that the torus
will never be hydrostatic for any $L_\su{UV}$. This is because the degree of
radiative support varies strongly with time and location (<ref>), because the inner surface is corrugated radially by radiation and
sheared azimuthally by differential rotation (<ref>), and because mass is continually lost in the form of a
radiation-driven wind from the inner surface (<ref>). We do not claim our simulations represent the only possible
configuration of a torus; instead, we wish to draw qualitative conclusions that
apply to any flavor of smooth, radiation-supported torus, and to let this
information guide us toward constructing a more realistic torus.
§.§ Qualitative description of gas motion
For the purpose of orientation, we begin by examining the evolution of the
torus in general terms, using <ref> as reference. Parts of
the simulation domain with $z<0$ are discarded from our figures on the grounds
that we observe no breaking of symmetry about the mid-plane.
Zoom-in of the azimuthally averaged poloidal plane at times $t=n|t_0$,
where $n$ is the number in the top-left corner of each panel. Gas density is
presented on a logarithmic scale as blue intensities (see color bar along the
right edge). The dust sublimation surface $r=r_\su{ds}$
(<ref>) is the dashed black curve around the origin, and the
red contour traces the surface on which $\tau_\su{UV}=1$. Momentum density is
shown by arrows with lengths $\mathrelp\propto\rho v$; the arrow in the
bottom-right corner of the first panel has length $0.5\,\rho_0|v_0$. All
quantities are normalized to fiducial units (<ref>).
Top grid: Plot of the $L_\su{UV}/L_\su E=0.11$ simulation.
Bottom grid: Plot of the $L_\su{UV}/L_\su E=0.14$ simulation; note
that the abscissa shifts at a constant rate toward the right from panel to
UV radiation creates two immediate effects on the inner surface. Gas at
the inner surface is swept up in the radially outward direction, forming a
density concentration along it. The inner surface recedes as a result, first
supersonically, then subsonically; this excites a transient in the form of an
acoustic density perturbation peeling away from the density concentration and
propagating outward through the torus, discernible at times $t\gtrsim1$. The
perturbation is shaped like a chevron bending outward when viewed with the full
range of $z$.
More notably, UV radiation shaves off gas at high latitudes and creates a
wind, while the central hole opens up from a cylindrical to a flaring shape.
There are two reasons why this gas is the most vulnerable to UV stripping.
First, we designate the UV optical depth from the central source by
$\tau_\su{UV}$. Only gas at $\tau_\su{UV}\lesssim1$ experiences substantial
UV acceleration, and the $\tau_\su{UV}=1$ surface slants radially outward
with increasing $\abs z$ in the initial condition since $\rho$ diminishes
monotonically with $\abs z$. Second, let us mentally divide the solid angle as
seen from the origin into infinitely many sectors, and let us study the
dynamics of the gas column contained within each sector with the proviso that
neighboring columns do not interact. This is akin to the approach used by
<cit.> to calculate accelerations in their simulations.
The acceleration of a column of thickness $\Delta r$ due to point-source
UV radiation is $\mathrelp\propto(1-e^{-\rho|\kappa_\su{UV}|\Delta
r})/(\rho|\Delta r)$, an expression that drops with increasing $\rho|\Delta r$,
while for any plausible initial condition of the torus, including ours
(<ref>), $\rho|\Delta r$ rises with inclination, defined
as the angle from the polar axis. An intuitive way to think about the second
argument is that UV radiative flux is spherically symmetric, and if it is
capable of supporting an optically thick column against gravity at low
latitudes, then it is fully equipped to expel an optically thin column at high
One might think that only gas at high latitudes participates in the wind, while
gas at low latitudes accelerated by UV radiation is simply rammed against
the inner surface. This is untrue because gas pressure along the flaring inner
surface is virtually constant. At any height above the mid-plane, UV
acceleration has a component parallel to the inner surface; unchecked by
pressure gradients, this component is free to peel off gas into a wind gliding
outward along the inner surface. Also note that, according to <ref>, gas starting out from smaller $R$ has smaller $R|v_\phi$, so the
wind preferentially removes gas with lower specific angular momentum.
Care must be exercised in reading <ref> after this initial
phase. We shall see in <ref> that the initially
axisymmetric inner surface becomes radially corrugated at $t\gtrsim4\,t_0$. For
$L_\su{UV}/L_\su E=0.11$, averaging this undulating structure in the azimuthal
direction produces the illusion that the inner surface at $t\gtrsim6\,t_0$
resembles a thick shell while in fact the density concentration remains thin in
any single poloidal slice. For $L_\su{UV}/L_\su E=0.14$, the inner surface
stays relatively axisymmetric; however, the fact that it moves radially outward
almost as quickly as the transients excited along it gives it the appearance of
multiple shells.
The radial motion subsequent to the initial phase depends on $L_\su{UV}$, which
determines the IR radiative flux across the torus. For $L_\su{UV}/L_\su
E\ge0.13$, IR radiative flux is strong enough that gas velocity is
radially outward in the torus body almost all the time, hence there is little
doubt the torus will be driven outward. In contrast, for $L_\su{UV}/L_\su
E\le0.12$, a region develops above and below the mid-plane at greater radial
coordinates than the inner edge in which the sum of the radial components of
IR and centrifugal accelerations falls slightly short of counteracting
gravity, and thus the radial component of velocity is negative. The size of
this region decreases with $L_\su{UV}$. Gas outside the region continues to be
propelled outward, but gas inside slides slowly toward the mid-plane and
inward; as it reaches the $\tau_\su{UV}=1$ surface, it is flung away by UV
radiation. This kind of inflow–outflow is essentially a balance between the
infall of gas toward the inner edge and the ability of UV radiation to
clear out the pileup. Because there is only a finite amount of gas in the
simulated torus, the inflow–outflow in our simulations cannot last forever.
The density distribution at times $t\gtrsim4\,t_0$ bears little resemblance to
the initial condition. Gas continues to be removed in the wind, but the
detailed shape of the body depends on whether vertical support due to IR
radiation is stronger or weaker than gravity. For $L_\su{UV}/L_\su E\ge0.14$,
IR radiative flux is sufficiently strong to inflate the body in the
vertical direction. But for $L_\su{UV}/L_\su E\le0.13$, the body falls toward
the mid-plane, reaching a thickness comparable to the gas pressure scale
height, and then expands back vertically. The density concentration along the
inner surface is shaped like another chevron and is taller than the body thanks
to UV radiation constantly accelerating the gas upward and outward.
Although the IR covering fraction drops steadily with time, the vertically
extended inner surface and the wind keep it at a value higher than would be due
to the body alone.
The degree of IR radiative support differs from place to place at these
late times. For all $L_\su{UV}$, IR vertical support in the chevron-shaped
inner surface is generally insufficient to counteract gravity; as we move
radially outward, we encounter a wedge-shaped, lower-density region in which
marginal IR vertical support prevails, followed by another region of even
lower density in which IR vertical support again falls short of gravity.
As $L_\su{UV}$ increases, IR vertical support becomes stronger more
rapidly at the inner surface than further outward in the torus, such that the
inner surface is completely supported against gravity at $L_\su{UV}/L_\su
E=0.15$ even when other parts of the torus are not.
Significant mass loss in the wind leads to a substantial drop in radial IR
optical depth along the mid-plane over time: By $t=10\,t_0$, the optical depth
is less than half its initial value for $L_\su{UV}/L_\su E=0.10$, and down to
∼0.05 times its initial value for $L_\su{UV}/L_\su E=0.15$. This
diminution in optical depth can be quite uneven as a function of azimuthal
coordinate for $L_\su{UV}/L_\su E$ at the low end of the simulated range
because, as we shall discuss in <ref>, those
are the conditions in which the non-axisymmetric radial perturbation at the
inner surface grows the most; at the high end of $L_\su{UV}/L_\su E$,
axisymmetry is maintained much more closely.
The rate at which UV radiation deposits momentum in the torus is
proportional to the UV covering fraction, whereas the mass of the torus is
roughly proportional to the covering fraction times the optical depth; hence,
the sharp plunge in IR optical depth explains why the body experiences
progressively stronger radially outward acceleration. For $L_\su{UV}/L_\su
E\le0.12$, this means the inflow–outflow eventually ceases, the radial
component of velocity turns positive throughout the body, and the body slides
outward more and more quickly as further mass loss accompanies its outward
§.§ Radial perturbation of the torus inner surface
Another intriguing complication at $t\gtrsim4\,t_0$ is the breaking of
axisymmetry along the inner surface.
The three-dimensional structure of the inner surface stays remarkably vertical
throughout the simulation for all $L_\su{UV}$. Isosurfaces of constant density
extend almost perpendicularly upward and downward from the mid-plane until they
are cut off at some height. This height depends on the density at the
isosurface, and typically increases with radial coordinate due to the flaring
shape of the inner surface (<ref>).
The verticality of the inner surface allows us to focus our attention on the
mid-plane, as we do in <ref>. Non-axisymmetry along the mid-plane
assumes the form of a slight radial perturbation of the inner edge going
through three oscillations per quarter circle at $t\sim4\,t_0$. The
perturbation grows in amplitude afterward, and its behavior in the nonlinear
regime depends on $L_\su{UV}$.
Mid-plane at times $t=n|t_0$, where $n$ is the number along the rim of
each quadrant of the circles. Gas density is presented on a linear scale as
blue intensities (see color bar above each circle), the red contour traces the
surface on which $\tau_\su{UV}=1$, and the black contours display $R^2\,\uvec
e_R\cdot\vec F_\su{IR}$ at levels indicated in the legend. Orbital motion is
counter-clockwise, all quantities are normalized to fiducial units
(<ref>), and all circles have different scales. Top
circles: Plots of the $L_\su{UV}/L_\su E=0.11$ simulation. Bottom-left
circle: Plot of the $L_\su{UV}/L_\su E=0.12$ simulation. Bottom-right
circle: Plot of the $L_\su{UV}/L_\su E=0.13$ simulation.
The top circles illustrate how, for $L_\su{UV}/L_\su E\le0.11$, the originally
smooth inner surface breaks up into dense, thin sheets overlapping in the
azimuthal direction; seen along the mid-plane, the dominant sheets resemble
trailing spiral density waves. For $L_\su{UV}/L_\su E=0.11$, the three original
oscillations combine into one at $t\sim9\,t_0$; for $L_\su{UV}/L_\su E=0.10$,
the three oscillations merge into two at $t\sim5\,t_0$ and break apart into
three again at $t\sim12\,t_0$.
In comparison, the bottom circles show that for $L_\su{UV}/L_\su E\ge0.12$, the
inner edge is characterized by a series of fingers pointing radially inward,
connected at the outward end by arcs which are convex outward. The fingers are
better described in three dimensions as vertical inward protrusions of the
inner surface shaped like rounded flaps in poloidal section. The tips of the
fingers and the middle portions of the arcs are slightly denser than other
parts of the inner edge. The tips of the fingers are also sheared azimuthally
into hooks by differential rotation. At any given time, the amplitude of the
perturbation, as well as the azimuthal distortion of the fingers due to
shearing, both decrease with $L_\su{UV}$.
There is nothing physical about the number three in the number of oscillations
at $t\sim 4\,t_0$. The initial perturbation is seeded by a small numerical
artifact associated with the angle grid whose influence is the strongest at six
azimuthal coordinates; the six originally tiny oscillations then merge to three
easily discernible ones. Since the artifact is fixed in space while the orbital
motion of the gas takes it across azimuthal coordinates, the artifact is not
expected to act on the same gas packet continually; therefore, we believe the
growing perturbation is a real effect.
§.§ Anisotropy of IR radiation
We now discuss the properties of IR radiation with the aid of
<ref>. Although the figure pertains to one snapshot of a single
simulation, it is representative of the configuration of the torus at earlier
times for all $L_\su{UV}$.
The first thing we notice in the top panel is that gas and IR radiation
temperature contours coincide in the torus body, and diverge only in
low-density regions outside the body. This confirms our expectation that
thermal equilibrium holds deep inside the torus but not outside.
A more significant observation, verifiable by a quick inspection of the bottom
panel, is that IR radiative flux streaming vertically through the central
hole is stronger by a factor of a few than its nearly horizontal counterpart
diffusing through the torus. This is explained by the conversion of UV
radiation to IR taking place in a thin layer of thickness
$\mathrelp\sim(\rho_\su{in}|\kappa_\su{UV})^{-1}$ centered at $\tau_\su{UV}=1$.
The IR optical depth is ≫1 from there to the outer surface, but
merely $\mathrelp\sim\kappa_\su{IR}/\kappa_\su{UV}\ll1$ to the central hole;
consequently, it is much easier for the freshly created IR radiation to
head back into the central hole than to penetrate the body.
In a geometrically and optically thick torus, some of the IR radiation
emitted by the inner edge can cross the central hole, reach the far side, and
be absorbed again, giving IR radiation multiple chances at breaking into
the torus. However, owing to the high optical depth of the torus, the
probability per attempt that IR radiation can cross the entire torus is
very small, so most of the IR radiation eventually leaves in the vertical
direction after a few ricochets off the inner surface. Through this process,
IR radiation transfers its momentum several times to a thin layer of gas
at the inner surface.
This focusing of IR radiative flux into the vertical direction means
$F_\su{IR}/F_\su{UV}$ rises gradually with $r$ in the central hole, as seen in
the bottom panel of <ref>. A consequence is that although the
wind is launched by UV radiation, IR radiation also contributes to
its acceleration once it reaches altitudes comparable to the vertical extent of
the torus.
We investigate $\vec F_\su{IR}$ more quantitatively with <ref>. The top panel displays $[L_\su{UV}/(4\pi r^2)]^{-1}\,\uvec e_r\cdot\vec
F_\su{IR}$ for $L_\su{UV}/L_\su E=0.11$ along lines emanating from the central
source at various inclinations; this quantity would be unity if the IR
radiative flux were spherically symmetric. The solid portions of the curves
highlight the parts of the lines belonging to the torus proper. The lines are
divided into two classes. Lines at high inclinations pass through the torus and
have flux magnitudes below the spherically symmetric value. Conversely, lines
at low inclinations lie completely within the central hole and have flux
magnitudes above the spherically symmetric value; in fact, the curves appear to
converge to $\mathrelp\sim C_\su{IR}/(1-C_\su{IR})$ at large $r$, where
$C_\su{IR}$ is the IR covering fraction (<ref>). The increasing discrepancy from spherical symmetry toward the
mid-plane illustrates the high degree of flux anisotropy.
Azimuthally averaged poloidal plane of the $L_\su{UV}/L_\su E=0.11$
simulation at time $t=2\,t_0$, but extending farther than in the top grid of
<ref>. All quantities are normalized to fiducial units
(<ref>). Top panel: Gas density is presented on a
logarithmic scale as blue intensities (see color bar along the right edge). The
dust sublimation surface $r=r_\su{ds}$ (<ref>) is the dashed
black curve around the origin. Purple and green contours respectively show gas
and IR radiation temperatures, both going from $0.3\,T_\su{ds}$ to
$0.7\,T_\su{ds}$ in steps of $0.1\,T_\su{ds}$ as one moves from the right to
the left; to avoid confusion, contours not passing through the torus body are
hidden. Bottom panel: The background colors display $[L_\su{UV}/(4\pi
r^2)]^{-1}|F_\su{IR}$ (see color bar along the right edge), which is unity for
spherically symmetric radiation. The gray contours plot density rising from
$0.1\,\rho_0$ on the outside to $0.5\,\rho_0$ on the inside in steps of
$0.1\,\rho_0$. The $\tau_\su{UV}=1$ surface is traced by a red contour. The
white and black arrows graph $\vec F_\su{IR}/F_\su{IR}$ and $\vec v$
respectively; the arrow in the bottom-right corner has length $5\,v_0$.
Azimuthally averaged IR radiative flux in the $L_\su{UV}/L_\su
E=0.11$ simulation at time $t=2\,t_0$ measured along lines with inclinations
indicated in the legend. The solid portion of each curve terminates at
$\tau_\su{UV}=1$ on the left and $\rho=(\bar\kappa_\su{IR}|r_0)^{-1}$ on the
right. All quantities are normalized to fiducial units
(<ref>). Top panel: Plot of $[L_\su{UV}/(4\pi
r^2)]^{-1}\,\uvec e_r\cdot\vec F_\su{IR}$; the upper and lower horizontal
dotted lines are drawn at $C_\su{IR}/(1-C_\su{IR})$ and 1. Bottom
panel: Plot of $\arccos(\uvec e_r\cdot\vec F_\su{IR}/F_\su{IR})$.
The bottom panel of <ref> shows the angle between $\vec
F_\su{IR}$ and $\uvec e_r$. For lines at low inclinations, $\vec F_\su{IR}$ is
roughly parallel to $\uvec e_r$ everywhere; for lines at high inclinations, it
is intriguing that the IR radiative flux snaps immediately to $\uvec e_r$
past the $\tau_\su{UV}=1$ surface. The fact that $\vec F_\su{IR}$ is nearly
aligned with $\uvec e_r$ in the body is all the more striking considering that
the IR optical depth from the inner edge to the outer surface in the
vertical direction is a quarter that in the radial direction
Similar conclusions were also reached by <cit.>, who found
that, for a smooth torus with geometrical thickness under a certain threshold,
most of the bolometric radiative flux exits through the central hole while only
a small fraction traverses the body. In addition, because $\vec
F_\su{UV}\propto\uvec e_r$ by definition, the bolometric radiative flux is
likewise spherically radial except where $\vec F_\su{IR}$ deviates most from
spherically radial, that is, just inside the uppermost parts of the inner
also stated that $F_\su{IR}\propto r^{-2}$ at large
$r$. The top panel of <ref> certainly suggests such a
trend, especially for IR radiation beyond the outer surface. Nevertheless,
since the radial coordinate ratio of the outer to inner edge is small, we
cannot say with confidence if the inverse-square law holds inside the body. The
situation is also complicated by the torus not being in a quasi-steady state.
§.§ Mass, momentum, and kinetic energy loss rates
It is natural to ask how much mass, momentum, and kinetic energy are carried
away by the UV-launched wind mentioned in <ref>. The rate at which mass is evacuated allows us to determine the
ultimate fate of the torus by balancing it against possible mass resupply.
Moreover, we can connect the loss rates in our simulations with observations of
AGN outflows.
We emphasize that the chevron-shaped transient (<ref>) is not
the wind, and that the density concentration along the inner surface
(<ref>) does not trace the trajectory of individual gas
packets. Since the wind encompasses a large solid angle and density range, we
have no reliable way of separating it from the torus body, which is moving
radially outward at the same time along the mid-plane. In practice, we define
the mass loss rate as
\begin{equation}
\dot M\eqdef\int_{R=R_\su{max},\,\abs z>r_0}
R\,d\phi\,dz\,\uvec e_R\cdot(\rho|\vec v)
-\int_{z=z_\su{min}}R\,dR\,d\phi\,\uvec e_z\cdot(\rho|\vec v)
+\int_{z=z_\su{max}}R\,dR\,d\phi\,\uvec e_z\cdot(\rho|\vec v),
\end{equation}
\begin{align}
\nonumber
\dot M &\eqdef \int_{R=R_\su{max},\,\abs z>r_0}
R\,d\phi\,dz\,\uvec e_R\cdot(\rho|\vec v) \\
\nonumber
&\noeq -\int_{z=z_\su{min}}R\,dR\,d\phi\,\uvec e_z\cdot(\rho|\vec v) \\
&\noeq +\int_{z=z_\su{max}}R\,dR\,d\phi\,\uvec e_z\cdot(\rho|\vec v),
\end{align}
and the momentum and kinetic energy loss rates in a similar fashion; here
$R_\su{max}$ and $z_{\su{min},\su{max}}$ denote the coordinates of the
boundaries of the simulation domain. We must be mindful to terminate our
analysis before the IR half–opening angle becomes too large and the wind
drops below $\abs z=r_0$ at the outer-radial boundary, at $t\gtrsim6\,t_0$. All
loss rates derived from the simulations are implicitly quadrupled to account
for our limited azimuthal extent (<ref>).
We begin with an analytic estimate of the mass loss rate. Supposing that the
wind is propelled by UV momentum and reaches into $\tau_\su{UV}\sim1$, the
mass loss rate may be estimated by either $\dot M\sim L_\su{UV}/(c|v_\infty)$
\begin{equation}
\dot M\sim2|\left(\frac{2\pi|R_\su{in}}{\rho_\su{in}|\kappa_\su{UV}}\right)|
\end{equation}
These two estimates agree if the wind terminal speed is
\begin{equation}\label{eq:escape velocity}
v_\infty\eqdef\biggl(\frac{GM}{R_\su{in}}|\frac{L_\su{UV}}{L_\su E}|
\frac{\kappa_\su{UV}}\kappaT\biggr)^{1/2}.
\end{equation}
It follows that
\begin{equation}\label{eq:mass loss rate}
\dot M\sim4\pi|
\biggl(\frac{GM|R_\su{in}}{\kappaT^2}|\frac{L_\su{UV}}{L_\su E}\biggr)^{1/2}|
\biggl(\frac{\kappa_\su{UV}}\kappaT\biggr)^{-1/2}
\end{equation}
\begin{equation}\label{eq:kinetic luminosity}
\frac{\dot M|v_\infty^2}{L_\su{UV}}=\frac{v_\infty}c=
\biggl(\frac{GM}{c^2|R_\su{in}}|
\frac{L_\su{UV}}{L_\su E}|\frac{\kappa_\su{UV}}\kappaT\biggr)^{1/2}.
\end{equation}
When appropriately rewritten, <ref> will
also serve as the basis of our scaling relations for extrapolating our
simulation results to more astrophysically relevant values of $M$ and
$\kappa_\su{UV}$ (<ref>).
The top panel of <ref> demonstrates that, in keeping with this
simple picture, the mass loss rates in our simulations normalized by
$L_\su{UV}/(c|v_\infty)$ are of order unity and nearly the same for all
$L_\su{UV}/L_\su E$ until $t\sim 4\,t_0$.
Top panel: Plot of the mass loss rate divided by
$L_\su{UV}/(c|v_\infty)$, with $v_\infty$ from <ref>, for
each value of $L_\su{UV}/L_\su E$ indicated in the legend. The dotted line
shows the mass loss rate required to deplete an isolated torus within five
orbits if $L_\su{UV}/L_\su E=0.11$. Middle panel: Plot of the
spherically radial gas momentum loss rate divided by $L_\su{UV}/c$.
Bottom panel: Plot of the ratio of kinetic to UV luminosity. The
dotted line shows the value of
$\ifaastex{v_\infty/(3c)}{\tfrac13|(v_\infty/c)}$ for $L_\su{UV}/L_\su E=0.11$;
the factor $\ifaastex{1/3}{\tfrac13}$ merely brings the line into the plot
range and has no physical meaning. All quantities are normalized to fiducial
units (<ref>).
The middle panel traces the rate at which $\uvec e_r\cdot(\rho|\vec v)$, the
spherically radial component of gas momentum, leaves the simulation domain; the
normalization is $L_\su{UV}/c$, the rate of momentum injection in the form of
UV radiation. This quantity is about half for $t_0\lesssim
t\lesssim4\,t_0$, suggesting that a sizable fraction of the radiation momentum
is not transferred to the gas.
We show in the bottom panel the ratio of kinetic to UV luminosity, where
the kinetic luminosity is the loss rate of kinetic energy. Because
$R_\su{in}\propto M^{1/2}|(L_\su{UV}/L_\su E)^{1/2}$
(<ref>), <ref> predicts $\dot
M|v_\infty^2/L_\su{UV}\propto M^{1/4}|(L_\su{UV}/L_\su E)^{1/4}$. The
$L_\su{UV}/L_\su E$ scaling is undetectable in our results since our range of
$L_\su{UV}/L_\su E$ spans a mere factor of 1.5; in fact, our ratio of kinetic
to UV luminosity is effectively constant for all $L_\su{UV}$ simulated,
contrary to the $\mathrelp\propto L_\su{UV}^{1.8}$ scaling offered by
<cit.>. Moreover, our explicit value is ∼4e-3
times that of <cit.>, but this is largely because our $M$
is ∼e-8 theirs and $\dot M|v_\infty^2/L_\su{UV}\propto M^{1/4}$.
§ DISCUSSION
We now interpret our simulation results and generalize them to
radiation-supported tori with different parameters.
§.§ Estimation of IR radiation energy
density at the torus inner edge
The maximum of $E_\su{IR}$ is attained at the inner edge because that is where
UV radiation is reprocessed (<ref>). We can estimate the
magnitude of the peak $(E_\su{IR})_\su{in\vphantom0}$ by considering the
radiation energy balance at the inner edge:
\begin{equation}\label{eq:inner edge energy balance}
\approx (F_\su{IR}^+)_\su{in\vphantom0}\nosup.
\end{equation}
We denote by $C_\su{IR,UV}$ the IR and UV covering fractions. Similar
to the two-stream approximation, we divide the radial component of the IR
radiative flux into outward and inward parts, and we assign them to
$F_\su{IR}^\pm$ respectively. The second term on the left-hand side represents
the part of the IR radiative flux leaking from the torus through the inner
edge into the central hole, and then absorbed at the far side after crossing
the hole.
<Ref> relates five variables at fixed $L_\su{UV}$
and is therefore difficult to verify against our simulations; two assumptions
simplify it. The first one is $C_\su{UV}\approx C_\su{IR}$. The second one
comes from observing that, for IR optical depth $\Delta\tau_\su{IR}\gg1$
and covering fraction $C_\su{IR}\lesssim1$, IR radiation propagates
diffusively at $R>R_\su{in}$, that is, $F_\su{IR}^++F_\su{IR}^-\approx
c|E_\su{IR}\gg F_\su{IR}^+-F_\su{IR}^-$, or
$F_\su{IR}^\pm\approx\ifaastex{c|E_\su{IR}/2}{\tfrac12|c|E_\su{IR}}$; we
suppose this holds at $R=R_\su{in}$ as well. <Ref> then turns into
\begin{equation}\label{eq:peak IR energy estimate}
\frac{L_\su{UV}}{4\pi|R_\su{in}^2|c}|\frac{2|C_\su{IR}}{1-C_\su{IR}}.
\end{equation}
Our assumptions are not strictly correct because $C_\su{UV}>C_\su{IR}$, but
their errors act in opposite directions in such a way that <ref> is still an excellent description of our simulations. The
factor $C_\su{IR}/(1-C_\su{IR})$ is the number of scatterings IR radiation
suffers at the inner surface prior to exit; $(E_\su{IR})_\su{in\vphantom0}$
goes up with $C_\su{IR}$ because the torus traps IR more efficiently as
$C_\su{IR}$ approaches unity.
<Ref> is a verification that, despite numerous
simplifications, <ref> captures the physics well.
In preparation of this figure, we construct the radial profile of $E_\su{IR}$
by azimuthally averaging its mid-plane value; we then assign
$(E_\su{IR})_\su{in\vphantom0}$ and $R_\su{in}$ to the peak of the radial
profile and its radial coordinate respectively. We measure $C_\su{IR}$ by
considering a tight cylindrical envelope of the simulation domain, and
measuring the solid angle subtended at the origin by the parts of this envelope
for which the IR optical depth toward the origin is greater than unity.
The success of <ref> confirms the inner surface
does act like a mirror to IR radiation.
Plot of the left- and right-hand sides of <ref> in blue and green curves respectively for each value of
$L_\su{UV}/L_\su E$ indicated in the top-right corner of each panel. All
quantities are normalized to fiducial units (<ref>).
Our study of the IR covering fraction leads to another useful result: We
can predict the value of $L_\su{UV}$ that marginally balances gravity in our
initial condition (<ref>). Using <ref>, we get
\begin{equation}
\frac{L_\su{UV}}{L_\su E}\approx
\frac{\rho_\su{in}}{\rho_0}|\frac{R_\su{in}}{r_0}|
\frac{1-C_\su{IR}}{2|C_\su{IR}}.
\end{equation}
The IR half–opening angle at $t=0$ is ≈0.727; the
estimate $L_\su{UV}/L_\su E\approx0.135$ agrees with what we have found in
§.§ Variation of simulation parameters
It is useful to extend beyond the tiny parameter space explored by our
simulations. At fixed $M$, the principal parameters of the system are
$L_\su{UV}/L_\su E$, as well as $\rho_\su{in}$ and $j_\su{in}$ in the initial
condition (<ref>). We ignore detailed mass and angular
momentum distributions, although interesting local effects may arise if we
consider them fully. We also suppose the inner edge has temperatures near dust
sublimation <cit.>, so $R_\su{in}$ is not a free parameter once $M$ and
$L_\su{UV}/L_\su E$ are given. All these parameters enter into the net
\begin{equation}
\vec a\eqdef-\frac{GM}{r^2}\,\uvec e_r+j^2|\frac{GM}{R^2}\,\uvec e_R+
\frac{\kappa_\su{IR}}c|\vec F_\su{IR}+\frac{\kappa_\su{UV}}c|\vec F_\su{UV},
\end{equation}
which is a crucial factor governing torus dynamics. As far as global dynamics
are concerned, it is essentially a poloidal vector with radial and vertical
components $a_R\eqdef\uvec e_R\cdot\vec a$ and $a_z\eqdef(\sign z)\,\uvec
e_z\cdot\vec a$.
Consider how each parameter affects $a_R$ and $a_z$ in the torus body. Clearly
$a_R$ increases with $j_\su{in}$, while $a_R$ and $a_z$ increase with
$L_\su{UV}/L_\su E$ through $\vec F_\su{IR}$ and $\vec F_\su{UV}$. The
influence of $\rho_\su{in}$ on $a_R$ and $a_z$ is subtler as it simultaneously
controls $\Delta\tau_\su{IR}$ and $C_\su{IR}$, which play a role when
$\Delta\tau_\su{IR}\gtrsim1$. On the one hand, greater $\Delta\tau_\su{IR}$
reduces $F_\su{IR}$ in the body according to $F_\su{IR}\sim
c|(E_\su{IR})_\su{in\vphantom0}/\Delta\tau_\su{IR}$; on the other hand, greater
$C_\su{IR}$ better traps IR radiation within the central hole, which at
constant $L_\su{UV}/L_\su E$ raises $(E_\su{IR})_\su{in\vphantom0}$
(<ref>) and thus $F_\su{IR}$ in the body. Both
$\Delta\tau_\su{IR}$ and $C_\su{IR}$ rise with $\rho_\su{in}$, so it is
difficult to determine which effect dominates. In short, raising $j_\su{in}$
increases radial support, raising $L_\su{UV}/L_\su E$ increases both radial and
vertical support, whereas raising $\rho_\su{in}$ has an indeterminate effect on
We now turn to local effects that can appear at the inner surface. First,
consider two tori with different $L_\su{UV}/L_\su E$ and $\rho_\su{in}$ tuned
so that they share $\vec F_\su{IR}$ and $\vec a$ in the body. Dynamics in the
body may be identical, but $\vec F_\su{IR}$ in the central hole of the torus
with greater $L_\su{UV}/L_\su E$ is necessarily stronger. Since <ref> show that $v_\infty$ and $\dot M$ depend on
$L_\su{UV}/L_\su E$ but not $\rho_\su{in}$, this torus must host a faster wind
than the other, as well as more severe losses of mass, momentum, and kinetic
energy. Second, a sufficiently large increase in either $L_\su{UV}/L_\su E$ or
$C_\su{IR}$ could make $(E_\su{IR})_\su{in\vphantom0}\gtrsim\aSB|T_\su{ds}^4$
and $\kappa_\su{IR}\approx0$ at the inner edge.
§.§ Scaling simulation results to more realistic parameters
As already remarked, for numerical reasons we have adopted artificially reduced
values of $M$ and $\kappa_\su{UV}/\kappa_\su{IR}$. Our tiny $M$ is the result
of requiring $c_\su s/v_\phi$ to be small, but not nearly as small as it would
be in a real system (<ref>). The thickness of the UV
radiation absorption layer at the inner surface is
times the radial extent of the torus, so a large opacity ratio would entail the
use of a grid size small enough to resolve an extremely thin absorption layer.
Moreover, because all of the momentum in UV radiation is delivered within
the layer, gas in the layer experiences an acceleration
$\mathrelp\propto\kappa_\su{UV}$. With greater $\kappa_\su{UV}$, tracking the
development of the inner surface would necessitate high temporal resolution,
and the value of $\hat c$ would also need to be revised upward to keep $v<\hat
It is of course desirable to explore how the properties of our simulations
might change if those two parameters were pushed to astrophysically realistic
values. The true opacity ratio should be ∼e2∼e3
<cit.>, but since the essential requirement for
capturing the physics is that the correct ordering of $\kappa_\su{IR}$ and
$\kappa_\su{UV}$ be kept, we argue our simulations are undamaged by our reduced
opacity ratio. To explore the effect of altering $\bar\kappa_\su{UV}$, we have
experimented with two simulations at twice the normal spatial resolution, one
with the usual value of $\bar\kappa_\su{UV}$, the other with twice the value.
The inner surface recedes slightly faster and is sharper at higher
$\bar\kappa_\su{UV}$, but otherwise the overall evolution of the torus and its
qualitative features are unaffected.
Nevertheless, quantitative results do vary with $\kappa_\su{UV}$; in
particular, care must be taken when scaling the wind terminal speed and mass
loss rate found in <ref>. A higher value of $\kappa_\su{UV}$
means the optically thin wind is faster but restricted to a thinner layer.
Rewriting <ref> in terms of
$R_\su{in}/r_\su{ds}$ highlights how this scaling should be performed:
\begin{equation}
\biggl(\frac{L_\su{UV}}{L_\su E}\biggr)^{1/4}|
\biggl(\frac{\kappa_\su{IR}|\kappa_\su{UV}}{\kappaT^2}\biggr)^{1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{-1/2}
\end{equation}
\begin{multline}
v_\infty\sim(GM|\kappaT|\aSB|T_\su{ds}^4)^{1/4}\times{} \\
\biggl(\frac{L_\su{UV}}{L_\su E}\biggr)^{1/4}|
\biggl(\frac{\kappa_\su{IR}|\kappa_\su{UV}}{\kappaT^2}\biggr)^{1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{-1/2}
\end{multline}
\begin{equation}
\dot M\sim4\pi|\left[\frac{(GM)^3}{\kappaT^5|\aSB|T_\su{ds}^4}\right]^{1/4}|
\biggl(\frac{L_\su{UV}}{L_\su E}\biggr)^{3/4}|
\biggl(\frac{\kappa_\su{IR}|\kappa_\su{UV}}{\kappaT^2}\biggr)^{-1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{1/2}.
\end{equation}
\begin{multline}
\dot M\sim4\pi|\left[\frac{(GM)^3}{\kappaT^5|\aSB|T_\su{ds}^4}\right]^{1/4}
\times{} \\
\biggl(\frac{L_\su{UV}}{L_\su E}\biggr)^{3/4}|
\biggl(\frac{\kappa_\su{IR}|\kappa_\su{UV}}{\kappaT^2}\biggr)^{-1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{1/2}.
\end{multline}
These forms cleanly separate the dependence on $M$ and $\kappa_\su{UV}$ from
everything else.
Shifting the fiducial values of these parameters from those used in our
simulations to more astrophysical numbers changes the wind terminal speed and
mass loss rate found in our simulations to
\begin{equation}
\biggl(\frac M{\SI{e7}{\solarmass}}\biggr)^{1/4}|
\biggl(\frac{L_\su{UV}/L_\su E}{0.1}\biggr)^{1/4}|
\biggr(\frac{\kappa_\su{IR}/\kappaT}{20}\biggr)^{1/4}|
\biggr(\frac{\kappa_\su{UV}/\kappaT}{2000}\biggr)^{1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{-1/2}\,
\si{\kilo\meter\per\second}
\end{equation}
\begin{multline}
\biggl(\frac M{\SI{e7}{\solarmass}}\biggr)^{1/4}|
\biggl(\frac{L_\su{UV}/L_\su E}{0.1}\biggr)^{1/4}\times{} \\
\biggr(\frac{\kappa_\su{IR}/\kappaT}{20}\biggr)^{1/4}|
\biggr(\frac{\kappa_\su{UV}/\kappaT}{2000}\biggr)^{1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{-1/2}\,
\si{\kilo\meter\per\second}
\end{multline}
\begin{equation}\label{eq:explicit mass loss rate}
\dot M\sim0.1\,
\biggl(\frac M{\SI{e7}{\solarmass}}\biggr)^{3/4}|
\biggl(\frac{L_\su{UV}/L_\su E}{0.1}\biggr)^{3/4}|
\biggr(\frac{\kappa_\su{IR}/\kappaT}{20}\biggr)^{-1/4}|
\biggr(\frac{\kappa_\su{UV}/\kappaT}{2000}\biggr)^{-1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{1/2}\,
\si{\solarmass\per\year}.
\end{equation}
\begin{multline}\label{eq:explicit mass loss rate}
\dot M\sim0.1\,
\biggl(\frac M{\SI{e7}{\solarmass}}\biggr)^{3/4}|
\biggl(\frac{L_\su{UV}/L_\su E}{0.1}\biggr)^{3/4}\times{} \\
\biggr(\frac{\kappa_\su{IR}/\kappaT}{20}\biggr)^{-1/4}|
\biggr(\frac{\kappa_\su{UV}/\kappaT}{2000}\biggr)^{-1/4}|
\biggl(\frac{R_\su{in}}{r_\su{ds}}\biggr)^{1/2}\,
\si{\solarmass\per\year}.
\end{multline}
Outflows with speeds from ∼100∼2000
have been identified in observations of X-ray warm absorbers
<cit.> and UV absorbers
<cit.> in Seyfert 1s. Mass
loss rates inferred from X-ray warm absorbers go from
<cit.>, whereas studies of
UV absorbers suggest a wider range of ∼e-4∼10 <cit.>. These empirical
results are roughly consistent with our fiducial values of $v_\infty$ and $\dot
The mass loss rate can be understood in a more intuitive fashion. The initial
mass of the torus is $M_\su{tor}\eqdef
C_\su{tor}\times2\pi|\rho_\su{in}|R_\su{in}^3$, where $C_\su{tor}\approx4$ for
our initial condition (<ref>). We define the lifetime of
the torus against mass loss in the radiation-driven wind as $t_\su{tor}\eqdef
M_\su{tor}/\dot M$; from <ref>, we have
\begin{equation}\label{eq:torus lifetime}
(\tau_\su T\nosup|\tau_\su{UV}\nosup)_\su{in\vphantom0}^{1/2}|
\left(\frac{L_\su{UV}}{L_\su E}\right)^{-1/2}.
\end{equation}
In this equation, $\Omega_\su{in}=(GM/R_\su{in}^3)^{1/2}$ is the orbital
frequency at the inner edge, and $(\tau_\su{T,UV})_\su{in\vphantom0}\eqdef
\rho_\su{in}|R_\su{in}|\kappa_\su{T,UV}$ stand for Thomson and UV optical
depths evaluated with inner-edge values. Our simulations have $(\tau_\su
T)_\su{in\vphantom0}\sim0.8$, $(\tau_\su{UV})_\su{in\vphantom0}\sim64$, and
$0.10\le L_\su{UV}/L_\su E\le0.15$, so $t_\su{tor}|\Omega_\su{in}\sim42$. Our
torus remains inside the simulation domain for a shorter amount of time because
the torus body moves radially outward at late times (<ref>).
Our Thomson optical depth may be reasonable for real AGN, but our UV
optical depth is too small by a factor of ≳10, so we expect the
lifetime of realistic tori against mass loss to be ≳3 times longer.
§.§ Balance between radiation-driven mass loss and mass resupply
A salient feature of simulations for all $L_\su{UV}$ is a radiation-driven wind
from the inner surface (<ref>). The wind always has
temperatures below $T_\su{ds}$ since it lies outside of the dust sublimation
radius. Depending on the geometry of the inner edge, the wind can be found at
higher latitudes than the torus body; this enhances the covering fraction, and
hints at a connection between the wind and dust observed in the polar regions
of NGC 424 <cit.> and NGC 3783
The radiation-driven wind is distinct from the thermally driven wind
<cit.> commonly discussed in the context of the torus
<cit.>. The
latter refers to gas lifted from the inner surface, exposed to ionizing
radiation from the central source, and heated to the Compton temperature soon
after its ionization parameter exceeds unity <cit.>. The
mass loss rate due to the thermally driven wind is
∼0.4 <cit.>, similar to that
of the radiation-driven wind found in <ref>. The
two winds could consequently augment each other despite their different
physical properties.
The mass lost to these winds could be resupplied from the outside. A steady
state could also obtain in which the IR optical depth across the body is
approximately constant, so that the IR radiative flux does not become
powerful enough to shove the body collectively outward (<ref>). A combined molecular and ionized gas inflow rate of
∼0.2 has been observed down to
∼40 in NGC 1097 <cit.>. Inflows of
this magnitude at the outskirts of the torus suffice to replenish the mass loss
given by <ref>.
Magnetic effects can strongly influence the resupply rate. MHD turbulence
stirred up by the MRI could lead to outward angular momentum transport
through the torus and subsequent accretion toward the inner edge. The ideal
MHD condition holds even at extremely low ionization fractions
<cit.>, which can be maintained by
X-rays <cit.> if they carry a sizeable
fraction of the energy in the UV <cit.>.
Indeed, magnetic fields have been detected on ≲30 scales
in the nucleus of NGC 1068 <cit.>.
Recall that the steady-state mass inflow timescale in a disk is
$\mathrelp\sim[\alpha|(H/R)^2|\Omega]^{-1}$, where $H/R$ and $\Omega$ are the
aspect ratio and orbital frequency of the disk. Accretion driven by MHD
stresses has $0.01\lesssim\alpha\lesssim0.1$, so the inflow timescale is quite
close to the torus lifetime calculated in <ref> if, as here,
$H/R\sim 1$. The relatively mild dependence of $t_\su{tor}|\Omega_\su{in}$ on
$L_\su{UV}/L_\su E$ suggests that equilibrium between inflow and outflow could
be attained over a wide range of luminosities.
The presence of MHD stresses can redistribute angular momentum in the
torus, altering the distribution of IR radiation needed to achieve radial
force balance against gravity; this change could in turn affect whether the
torus is vertically supported. Magnetic fields could also remove angular
momentum altogether from the torus through a magnetized wind
§.§ Radial perturbation of the torus inner surface
The nonlinear development of the radial perturbation of the inner surface
(<ref>) is reminiscent of the Rayleigh–Taylor
instability <cit.>. For
$L_\su{UV}/L_\su E\ge0.12$, the emergence of fingers and arcs from an
originally smooth inner surface is a hallmark of the instability. For
$L_\su{UV}/L_\su E=0.11$, the azimuthal wavenumber of the most prominent mode
of the perturbation decreases as the development of the perturbation becomes
nonlinear; this mirrors the classical picture in which the fastest-growing mode
of the perturbation of the interface separating the two fluids shifts from high
wavenumbers in the linear regime to low wavenumbers in the nonlinear regime
Since radiation and not a physical fluid is supporting the gas against gravity,
it is more accurate to compare our simulations with the radiative
Rayleigh–Taylor instability <cit.>. <Ref> shows that in both
linear and nonlinear regimes, wherever a part of the $\tau_\su{UV}=1$ surface
is farther from the origin, the region immediately radially outward of it has
greater $R^2\,\uvec e_R\cdot\vec F_\su{IR}$ because the optical depth to the
outer surface is smaller; the perturbation grows as a consequence. This
mechanism is similar to what <cit.> described.
Nonetheless, the cylindrical geometry of our simulations, as well as the
presence of an acceleration gradient and differential rotation, complicates
direct comparison with these previous analyses.
The amplification of the perturbation turns a smooth density distribution
inhomogeneous; this kind of fragmentation process could provide a physical
mechanism for the formation of dusty clumps often invoked to explain the
observed broad ∼1∼100 bump in the SED of
AGN <cit.>, the weak
9.7 silicate emission or absorption feature
<cit.>, and the
gentle radial temperature profile of dust within the central parsec of Circinus
Radiation-driven clump formation has already been reported in FLD
simulations of super-Eddington outflows from axisymmetric accretion disks
<cit.> and from two-dimensional planar atmospheres
<cit.> where the dominant source of opacity is electron
scattering. Clumps in these simulations are irregular and typically one optical
depth across. Anisotropic structures are likewise observed in our
three-dimensional simulations employing genuine RT, but they have multiple
characteristic length scales. Magnetic fields certainly exist in the torus
<cit.> and could change how fragments are formed and
destroyed, but we must leave its study to future work.
§ CONCLUSIONS
We have conducted three-dimensional, time-dependent RHD simulations of
AGN tori in which gas and radiation are evolved simultaneously, and
IR and UV radiative fluxes are not approximated using arbitrary
closure prescriptions. The simulations reveal that a smooth, geometrically and
Compton thick torus is not very permeable to IR radiation, whereas the
optically thin central hole allows IR radiation to escape immediately;
therefore, the IR radiative flux is much stronger through the central hole
than across the torus, and IR radiative support inside the torus is weaker
than if the torus body were optically thin (<ref>). Meanwhile,
IR radiation undergoing several reflections at the inner surface before
leaving the central hole enhances the IR radiation energy density at the
inner edge (<ref>) and reduces the luminosity
needed to achieve marginal IR radiative support.
The inner surface experiences a spontaneous breaking of axisymmetry under
radiation and differential rotation; the consequent radial perturbation
amplifies rapidly with time (<ref>). The growth
of the perturbation conjures up the picture of the radiative Rayleigh–Taylor
instability, but with critical differences. The fragmentation of the inner
surface alludes to a physical mechanism for the creation of clumps; however,
the steady-state configuration of the fragments is not probed by our
simulations and is likely affected by magnetic fields (<ref>).
Most importantly, a dusty wind can be launched from the inner surface by
UV radiation and propelled outward by a combination of IR and UV
radiation. The appearance of this wind is inevitable in a torus with vertical
density stratification (<ref>). High dust opacity in the
UV, along with the concentration of IR radiative flux into the
vertical direction (<ref>), means the wind likely experiences
an acceleration well above gravity. The radiation-driven wind carries momentum
comparable to that in UV radiation (<ref>). It is also a
powerful mechanism of mass loss with the capacity to remove an isolated torus
within ∼20 orbital periods at the inner edge (<ref>).
Our study calls attention for the first time to the possibility that UV
radiation pressure acting on dust can drive a wind with speed and mass loss
rate of the same order as values inferred from observations
(<ref>), and with mass loss rate similar to the
better-known thermally driven wind (<ref>). In order to achieve
an approximate steady state against mass loss through both kinds of winds, any
such torus must be furnished with a new inventory of mass every ∼20
orbital periods. The strong variation of radiative support throughout the body
(<ref>), the existence of a radiation-driven wind
(<ref>), and the growth of perturbations along
the inner surface (<ref>), demonstrate that the
internal structures of tori are unlikely ever to achieve strict hydrostatic
The authors thank Jim Stone, Yanfei Jiang, and Shane Davis for generously
allowing Athena and its time-dependent RT module to be used for this
project and for providing technical support. This research was partially
supported by NASA/ATP grants NNX11AF49G and NNX14AB43G. The simulations were
performed on the Johns Hopkins Homewood High-Performance Cluster.
§ TIME-INDEPENDENT LONG-CHARACTERISTICS UV RT
We have developed a time-independent long-characteristics RT module to
deal with UV radiation from a point source at the origin in cylindrical
coordinates. Our ray-casting algorithm is similar to that of
<cit.>. We construct a ray from the source to the center of
every cell in the simulation domain, extend it so that it reaches the far side
of the destination cell, and then chop it up into segments, one for each cell
the ray passes through. This ray-casting is done once, before the simulation
starts. Our adoption of cylindrical coordinates means that we only need to
solve the ray-casting problem in two dimensions. A subtlety of our algorithm is
that, whenever a ray passes very close to a cell corner, we allow the ray to
pass diagonally through it.
At the beginning of a time step, we compute the UV radiation energy
density in the destination cell by
\begin{equation}
\frac{4\pi}c|J_\su{UV}\eqdef\frac{L_\su{UV}}{4\pi r^2c}|e^{-\tau_\su{UV}}|
\frac{\exp(\tfrac12\tau^*_\su{UV})-\exp(-\tfrac12\tau^*_\su{UV})}
\end{equation}
where $L_\su{UV}$ is the luminosity of the source in the UV and $\vec r$
is the displacement from the source to the destination cell. We determine the
UV optical depth $\tau_\su{UV}$ by accumulating the products of the length
of each segment and $\rho|\kappa_\su{UV}$ averaged over the cell in which the
segment lies; note that we consider only half of the length of the last segment
in this exercise. The last factor in the equation comes from averaging
$J_\su{UV}$ over the entire last segment, which has UV optical depth
$\tau^*_\su{UV}$; its inclusion improves the agreement of the UV energy
and momentum absorption rate between runs at different resolutions,
particularly at locations where $\tau_\su{UV}\lesssim1$.
To arrive at the energy and momentum source terms of gas due to UV
radiation, we remind ourselves of the RT equation in the form derived by
\begin{multline}\label{eq:MK1982 radiative transfer}
\frac1c|\pd{I_\su{UV}}t+\uvec n\cdot\grad I_\su{UV}=
\Bigl(-1+\uvec n\cdot\frac{\vec v}c\Bigr)|
\rho|(\kappa_\su{UV}+\sigma_\su{UV})|I_\su{UV} \\
+\Bigl(1+3\,\uvec n\cdot\frac{\vec v}c\Bigr)|
\rho|(\kappa_\su{UV}|B+\sigma_\su{UV}|J_\su{UV})
-2|\rho|\sigma_\su{UV}|\frac{\vec v}c\cdot\vec H_\su{UV}.
\end{multline}
The zeroth and first angular moments of <ref> are
1/c|J_UVt+H⃗_UV =
1/c|H⃗_UVt+K_UV =
pointed out that <ref> do
not give the correct equilibrium in moving fluids. To overcome this problem,
the time-dependent RT module of Athena solves the modified
<ref>, but <ref> are identical to first order in $v/c$ save for the
For time-independent RT, which applies to the UV, we drop the time
derivatives from <ref>. In the special case of point-source
UV radiation interacting with purely absorbing material that does not
re-radiate in the UV, we set $\sigma_\su{UV}=0$, $B=0$, $\vec
H_\su{UV}=\uvec e_r\,J_\su{UV}$, and $\tsr K_\su{UV}=\uvec e_r\,\uvec
e_r\,J_\su{UV}$ in <ref>; the source terms we seek can be skimmed off as
\begin{align}
-\frac1{4\pi}|\ergsrc{UV} &\eqdef
-\rho|\kappa_\su{UV}|J_\su{UV}|\Bigl(1-\uvec e_r\cdot\frac{\vec v}c\Bigr), \\
-\frac c{4\pi}|\momsrc{UV} &\eqdef
-\rho|\kappa_\su{UV}|J_\su{UV}\,\uvec e_r\,
\Bigl(1-\uvec e_r\cdot\frac{\vec v}c\Bigr).
\end{align}
Observe that a consistent solution cannot be reached with <ref>.
As implied by <ref>, the source terms are added
directly to the gas at the beginning of the time step. The energy source term
is rather large compared to the other terms of <ref>, so the gas
is temporarily overheated. The IR radiative sub-step is then carried out
as described by <cit.>, during which the gas releases
almost all of the energy it gained from the UV into the IR. Although
the source terms are added using the explicit Euler method, the IR
radiative sub-step proceeds by the implicit Euler method, hence a large energy
source term does not pose a problem.
Despite the sharp rise in gas temperature after the UV
long-characteristics sub-step, we must not change the IR and UV
opacities until the IR radiative sub-step is finished; otherwise, gas
exposed to UV radiation would be absorbing UV and emitting IR at
two unrelated opacities, which would generate specious temperature fluctuations
with a period equal to two or three time steps around the true equilibrium
§ REDUCED SPEED OF LIGHT APPROXIMATION
The radiative timescale governing <ref> is shorter than the hydrodynamic
timescale of <ref> by a factor of
$c/v\gg1$. Our primary concern is the hydrodynamic timescale, whereas the fast
variation of $I_\su{IR}$ relative to $\rho$, $\vec v$, and $p$ is uninteresting
since radiation merely equilibriates with the gas in between hydrodynamic time
steps. To avoid following the system on the radiative timescale, we adopt the
method of reduced speed of light <cit.>.
The physical light speed $c$ attached to the time derivatives in
<ref> is substituted with the reduced light speed $\hat c$. This allows the
use of coarser temporal resolution since the rate of change of $I_\su{IR}$ in
<ref>, including thermalization by absorption,
isotropization by scattering, propagation in vacuum, and advection in optically
thick gas, is slowed down by a factor of $\hat c/c$. The source terms
$\ergsrc{IR}$ and $\momsrc{IR}$ are not altered, only the rate at which they
change $J_\su{IR}$ and $\vec H_\su{IR}$ in <ref>; in fact, they must not be touched in
<ref> if we are to preserve gas dynamics.
Our approximation does not stop radiation from reaching equilibrium with the
gas inasmuch as $v<\hat c\ll c$. However, it is critical that we not replace
$c$ attached to $\vec v/c$ in <ref>, otherwise $\vec H_\su{IR}$ could be beamed
in the direction of $\vec v$ even when $v\lesssim\hat c\ll c$.
Because the rate of change of energy and momentum of gas is $c/\hat c$ times
that of radiation, <ref> taken together do not conserve the physical
values of energy and momentum, but $E+4\pi|J_\su{IR}/\hat c$ and $\rho|\vec
v+4\pi|\vec H_\su{IR}/(c\hat c)$ instead. Granted that spurious transients may
manifest themselves on approach to energy and momentum equilibrium between gas
and radiation, we nevertheless expect time-averaged values of $\ergsrc{IR}$ and
$\momsrc{IR}$ to vanish once equilibrium prevails.
A technical point to bear in mind is that the time-dependent RT module of
Athena evaluates $\ergsrc{IR}\,\Delta t$ and $\momsrc{IR}\,\Delta t$ not from
the right-hand sides of <ref>, but directly from $\Delta I_\su{IR}$ as computed by the IR
radiative sub-step; the conversion from $\Delta I_\su{IR}$ to
$\ergsrc{IR}\,\Delta t$ and $\momsrc{IR}\,\Delta t$ therefore necessitates a
factor of $c/\hat c$.
§ IMPROVEMENT TO TREATMENT OF SCATTERING IN ATHENA
We consider the treatment of scattering opacity by the time-dependent RT
module of Athena. The notation follows <cit.>, except that
here $c$ and $\hat c$ are the physical and reduced light speeds. We define
$\sigma_\su s$ as the scattering cross section per volume, $\Delta t$ as the
time step, and $\zeta\eqdef\tau^*_\su s|v/c$, where $\tau^*_\su
s\eqdef\sigma_\su s|\hat c|\Delta t$.
The module handles scattering by solving equations (29) and (30) listed in the
reference. We repeat the equations below, minus a couple typos:
\begin{equation}
\ifaastex{}{\arraycolsep.03em\medmuskip-.2\medmuskip}
\begin{pmatrix}
a_1+b_1+c_1 & b_2+c_1 & b_3+c_1 & \cdots & b_N+c_1 \\
b_1+c_2 & a_2+b_2+c_2 & b_3+c_2 & \cdots & b_N+c_2 \\
b_1+c_3 & b_2+c_3 & a_3+b_3+c_3 & \cdots & b_N+c_3 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
b_1+c_N & b_2+c_N & b_3+c_N & \cdots & a_N+b_N+c_N
\end{pmatrix}
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_N \end{pmatrix}=
\begin{pmatrix} r_1 \\ r_2 \\ r_3 \\ \vdots \\ r_N \end{pmatrix},
\end{equation}
\begin{align}
a_l &\eqdef W_l^{-1}|[1+\tau^*_\su s|(1-\vec n_l\cdot\vec v/c)], \\
b_l &\eqdef \tau^*_\su s|
[2|(\vec n_l\cdot\vec v/c)-(\vec n_l\cdot\vec v/c)^2-v^2/c^2], \\
c_l &\eqdef -\tau^*_\su s|[1+3|(\vec n_l\cdot\vec v/c)], \\
x_l &\eqdef W_l|I^{n+1}_l, \\
r_l &\eqdef I^n_l.
\end{align}
In the process of computing the $LU$ decomposition of the matrix,
$c_l-c_N\sim\zeta$ appears multiple times in the denominator. If $\zeta\ll1$,
some elements of the resultant matrices are $\num{\sim1}$ while others are
$\mathrelp\sim\zeta^{-1}$; put differently, the matrices are ill-conditioned,
with the ratio of the greatest to smallest singular values of either matrix
being $\mathrelp\sim\zeta$. Because the solution is computed using
back-substitution as $x_i=U\inv_{ij}|L\inv_{jk}|r'_k$, it could be highly
inaccurate. The code already includes checks to avoid this kind of situation,
but it is easy to construct triples of $\tau^*_\su s$, $v$, and $c$ that bypass
When $\zeta$ is below a certain threshold, it is preferable to regard the
scattering equation, written in the form
\begin{equation}
\end{equation}
\begin{align}
\lambda_{lm} &\eqdef (1+\tau^*_\su s)|\krons{lm}-\tau^*_\su s|W_m, \\
\epsilon_{lm} &\eqdef \tau^*_\su s|(a'_l|\krons{lm}+b'_m|W_m+c'_l|W_m), \\
a'_l &\eqdef -\vec n_l\cdot\vec v/c, \\
b'_l &\eqdef 2|(\vec n_l\cdot\vec v/c)-(\vec n_l\cdot\vec v/c)^2, \\
c'_l &\eqdef -[3|(\vec n_l\cdot\vec v/c)+v^2/c^2],
\end{align}
and $\krons{lm}$ is the Kronecker delta, as a perturbative equation and solve
it iteratively. The procedure starts with $I^{n+1}_l\leftarrow I^n_l$; each
iterative step updates the solution as
\begin{equation}
\left[I^n_m-\sum_{p=1}^N\epsilon_{mp}|I^{n+1}_p\right].
\end{equation}
The special structure of the matrix allows the inner multiplication to be
accomplished with time expenditure $\bigO(N)$, while the multiplication of any
vector $v_m$ by the inverse matrix $\lambda\inv_{lm}$ is simply
\begin{equation}
\sum_{m=1}^N\lambda_{lm}^{-1}|v_m
=\frac1{1+\tau^*_\su s}|\left(v_l+\tau^*_\su s\sum_{m=1}^NW_m|v_m\right)
\end{equation}
since $\mathop{\smash{\sum_{m=1}^N}}W_m=1$. The chief aim of this modification
is not to obtain a more accurate solution when $\zeta\sim\num{e-15}$; rather,
it prevents the numerical instability that the standard algorithm exhibits in
the static or extremely optically thin limit. Furthermore, the new solution is
not to replace, but to complement, the old solution.
The threshold at which we switch between solution strategies is somewhat
arbitrary; our choice is $\zeta=\num{e-5}$ as the standard algorithm has not
yet shown instability above it. If double-precision floating-point numbers are
used, the machine epsilon is $2^{-52}\approx\num{2.22e-16}$, so the iterative
step should be performed at least four times.
§ IR INITIAL CONDITION
Because the time-dependent RT module of Athena operates on $I_\su{IR}$
rather than $J_\su{IR}$ and $\vec H_\su{IR}$, we must convert $E^0_\su{IR}$
provided by the initial condition to $I_\su{IR}$. Inside the optically thick
torus body, the IR specific intensity in the fluid frame can be found in
the FLD approximation as <cit.>
\begin{equation}
I^0_\su{IR}(\uvec n^0)\eqdef\frac c{4\pi}|E^0_\su{IR}|\mathcal R^{-1}|
(\coth\mathcal R-\uvec m^0\cdot\uvec n^0)^{-1};
\end{equation}
it follows that
\begin{equation}
\vec H^0_\su{IR}=
\frac c{4\pi}|E^0_\su{IR}|(\coth\mathcal R-\mathcal R^{-1})\,\uvec m^0.
\end{equation}
Here $\mathcal R\eqdef\norm{\grad
E^0_\su{IR}}/(\rho|\kappa_\su{IR}|E^0_\su{IR})$ is the Knudsen number for
radiation diffusion, and $\uvec m^0\eqdef-\grad E^0_\su{IR}/\norm{\grad
E^0_\su{IR}}$. Geometrically speaking, if we draw arrows $\uvec n^0$ from the
origin with lengths proportional to $I^0_\su{IR}(\uvec n^0)$, the envelope is a
prolate ellipsoid with ellipticity $\tanh\mathcal R$ and one focus at the
origin. We impose the additional constraint that $0\le\tanh\mathcal R\le0.95$;
the ceiling makes radiation less unidirectional in optically thin regions so
that at least a few grid rays carry finite specific intensity. The specific
intensity is then boosted to the observer frame by
\begin{equation}
I_\su{IR}(\uvec n)=I^0_\su{IR}(\uvec n^0)|
\left[\frac{(1-v^2/c^2)^{1/2}}{1-\uvec n\cdot\vec v/c}\right]^4.
\end{equation}
We remarked after <ref> that $I_\su{IR}$ is a
frequency-integrated quantity, which explains why the exponent is four, not
three. Note that the FLD approximation is used merely to define the
initial condition; it is not used to solve <ref>.
§ ORBITAL VELOCITY PROFILE OF A HYDROSTATIC, RADIATION-SUPPORTED TORUS
The force balance of a hydrostatic torus supported by IR radiation against
the gravity of a point mass is expressed in
\begin{equation}
-\grad\left(-\frac{GM}r\right)+\frac{\kappa_\su{IR}}c|\vec F_\su{IR}+
\frac{v_\phi^2}R\,\uvec e_R=\vec0.
\end{equation}
The equation is solved together with the constraint of IR radiation energy
conservation, $\divg\vec F_\su{IR}=0$, and the assumption that $\kappa_\su{IR}$
is not a strong function of position. A similar equation has been solved by
<cit.> under axisymmetry; here we present a more intuitive
approach. Because the gravitational and radiative terms are both
divergence-free, the same must also be true for $(\smash{v_\phi^2}/R)\,\uvec
e_R$. The only radial and divergence-free vector field is
$C(\phi,z)|R^{-1}\,\uvec e_R$ for some function $C(\phi,z)$, hence $v_\phi$ is
a constant over $R$. If we further restrict $v_\phi$ to be axisymmetric, we can
\begin{equation}
\end{equation}
Here $j_\su{in}(z)$ is some dimensionless function that measures the shortfall
of orbital velocity at $R=R_\su{in}$ from Keplerian as a consequence of
radiative support, so we have $0\le j_\su{in}\le 1$.
|
1511.00345
|
A stratification on the moduli of K3 surfaces]A stratification on the moduli of K3 surfaces in positive characteristic
Korteweg-de Vries Instituut, Universiteit van
Postbus 94248, 1090 GE Amsterdam, The Netherlands.
We review the results on the cycle classes of the strata
defined by the height and the Artin invariant
on the moduli of K3 surfaces in positive characteristic
obtained in joint work with Katsura and Ekedahl.
In addition we prove a new irreducibility result
for these strata.
Dedicated to the memory of Fritz Hirzebruch
§ INTRODUCTION
Moduli spaces in positive characteristic often possess stratifications for which we do not know
characteristic $0$ analogues. A good example is the moduli space of elliptic curves in characteristic $p>0$.
If $E$ is an elliptic curve over an algebraically closed field $k$
of characteristic $p$, then multiplication
by $p$ on $E$ factors as
\times p : E \stackrel{F} {\longrightarrow} E^{(p)}
\stackrel{V}{\longrightarrow} E ,
where Frobenius $F$ is inseparable and Verschiebung $V$ can be
separable or inseparable. If $V$ is separable, then $E$ is called ordinary,
while if $V$ is inseparable $E$ is called supersingular.
In the moduli space $\A_{1} \otimes k$ of elliptic
curves over $k$ there are finitely
many points corresponding to supersingular elliptic curves,
and a well-known formula by Deuring, dating from 1941,
gives their (weighted) number:
\sum_{E/k\, {\rm supers.}\\ /\cong_k} \frac{1}{\# {\rm Aut}_k(E)} = \frac{p-1}{24}\, ,
where the sum is over the isomorphism classes of supersingular elliptic curves and each curve is counted
with a weight. We thus find a stratification of the moduli ${\A}_1\otimes {\FF}_p$
of elliptic curves with two strata: the ordinary stratum and the supersingular stratum.
This stratification generalizes to the moduli of principally polarized abelian varieties of
dimension $g$ in positive characteristic where it leads to two stratifications, the Ekedahl-Oort
stratification with $2^g$ strata and the Newton-polygon stratification.
These stratifications have been the
focus of much study in recent years (see for example
The dimension of these strata are known and in the case of
the Ekedahl-Oort stratification we also know by [E-vdG1]
the cycle classes of these strata
in the Chow groups of a suitable compactification.
The formulas for such cycle classes can be seen as a
generalization of the formula of Deuring.
Besides abelian varieties, K3 surfaces form another
generalization of elliptic curves. The stratification
on the moduli of elliptic curves in positive characteristic
generalizes to a stratification of the moduli ${\F}_g$ of primitively
polarized K3 surfaces of degree $2g-2$ in positive
characteristic. In fact, in the 1970s Artin and Mazur
obtained in <cit.>
an invariant of K3 surfaces by looking at the formal Brauer group
of a K3 surface.
For an elliptic curve the distinction between ordinary and supersingular can be formulated
by looking at the formal group, that is,
the infinite infinitesimal neighborhood of the origin with the
inherited group law. If $t$ is a local parameter at the origin, then multiplication by $p$ is given by
[p] \, t = a\, t^{p^h}+ \text{higher order terms} \eqno(1)
with $a \neq 0$. Since multiplication by $p$ on $E$ is of degree $p^2$
and inseparable, we have $1\leq h \leq 2$
and $h=1$ if $E$ is ordinary and $h=2$ if $E$ is supersingular.
The formal group allows a functorial description as the functor on spectra $S$
of local Artin $k$-algebras with residue field $k$
given by
S \mapsto \ker \{ H^1_{\rm et} (E\times S, {\GG}_m) \to H^1_{\rm et} (E,{\GG}_m)\},
where $H^1_{\rm et}(E,{\GG}_m)\cong H^1(E,{\mathcal O}_E^*)$ classifies line bundles
on $E$. The invariant of Artin and Mazur generalizes this. For a K3 surface $X$ they looked
at the functor of local Artinian schemes over $k$
with residue field $k$ given by
S \mapsto \ker \{
H^2_{\rm et} (X\times S, {\GG}_m) \to H^2_{\rm et} (X,{\GG}_m)\},
and showed that it is representable by a formal Lie group,
called the formal Brauer group. Its tangent space is $H^2(X,{\mathcal O}_X)$, so we
have a $1$-dimensional formal group.
Now over an algebraically closed field $1$-dimensional formal groups are classified by their height:
in terms of a local coordinate $t$ multiplication by $p$ is either zero
or takes the form
$[p]\, t = a \, t^{p^h}+ $ higher order terms, with $a \neq 0$.
If multiplication by $p$ vanishes we say $h=\infty$,
and then we have
the formal additive group $\hat{\GG}_a$ and if $h<\infty$
we have a $p$-divisible
formal group.
Artin and Mazur connected this invariant $h(X)$ to the geometry
of the K3 surface
by proving that if $h(X) \neq \infty$ then
\rho(X) \leq 22-2h(X), \eqno(2)
where $\rho(X)$ is the Picard number of $X$. In particular,
either we have $\rho=22$ (and then necessarily $h=\infty$), or $\rho \leq 20$.
The case that $\rho=22$ can occur in positive characteristic as Tate observed:
for example, in characteristic $p\equiv 3 \, (\bmod 4)$
the Fermat surface $x^4+y^4+z^4+w^4=0$ has $\rho=22$ (see <cit.>).
If $h(X)=\infty$ then the K3 surface $X$ is called supersingular.
By the result of Artin and Mazur a K3 surface $X$ with $\rho(X)=22$
must be supersingular. In 1974 Artin conjectured the converse (<cit.>):
a supersingular K3 surface has $\rho(X)=22$.
This has now been proved by Maulik,
Charles and Madapusi Pera for $p\geq 3$, see <cit.>.
In the 1980s Rudakov, Shafarevich and Zink proved that supersingular K3 surfaces
with a polarization of degree $2$ have $\rho=22$ in characteristic
$p\geq 5$, see
The height is upper semi-continuous in families.
The case $h=1$ is the generic case; in particular the K3 surfaces with
$h=1$ form an open set. By the inequality (2) we have
1 \leq h \leq 10 \qquad {\rm or} \quad h=\infty.
In the moduli space
${\F}_g$ of primitively polarized K3 surfaces of genus $g$ (or equivalently, of
degree $2g-2$) with $2g-2$ prime to $p$, the locus of K3
surfaces with height $\geq h$ is locally closed and
has codimension $h-1$ and we thus have
$11$ strata in the $19$-dimensional moduli space ${\F}_g$.
The supersingular locus has dimension $9$. Artin showed that it is
further stratified by the Artin invariant $\sigma_0$: assuming that $\rho=22$
one looks at
the Néron-Severi group ${\rm NS}(X)$ with its intersection pairing;
it turns out that the discriminant group
${\rm NS}(X)^{\vee}/{\rm NS(X)}$ is an elementary $p$-group isomorphic to
$({\ZZ}/p{\ZZ})^{2\sigma_0}$ and one thus
obtains another invariant.
The idea behind this
is that, though $\rho=22$ stays fixed, divisor classes in the limit
might become divisible by $p$, thus changing the Néron-Severi lattice and
$\sigma_0$. The invariant $\sigma_0$ is lower semi-continuous.
The generic case (supersingular) is where
$\sigma_0=10$ and the most degenerate case is the so-called superspecial
case $\sigma_0=1$.
In total one obtains a stratification on the moduli space ${\mathcal F}_g$
of K3 surfaces with a
primitive polarization of genus $g$
with $20$ strata $V_j$ with $1 \leq j \leq 20$
\overline{V}_j=
\{ [X]\in {\mathcal F}_g : h(X) \geq j\} \qquad 1\leq j \leq 10,
\overline{V}_j=
\{ [X] \in {\mathcal F}_g : h(X)=\infty, \, \sigma_0(X) \leq 21-j\} \qquad
11\leq j \leq 20 \, ,
the closures $\overline{V}_j$ of which
are linearly ordered by inclusion.
In joint work with Katsura <cit.> we determined the
cycle classes of the strata
$\overline{V}_j$ ($j=1,\ldots,10$) of height $h\geq j$
[V_j]= (p-1)(p^2-1)\cdots (p^{j-1}-1)\, \lambda_1^{j-1}, \qquad 1\leq j \leq 10
\eqno(3)
$\lambda_1= c_1(\pi_* (\Omega^2_{\mathcal X / {\F}_g}))$
is the Hodge class
with $\pi: {\mathcal X} \to {\F}_g$ the universal family.
We also determined the class of the supersingular locus $\overline{V}_{11}$.
Moreover, we proved that the singular locus of $\overline{V}_j$ is
contained in the stratum of the supersingular
locus where the Artin invariant is at most $j-1$, see <cit.>. Ogus made this more
precise in <cit.>. For more on the moduli of supersingular K3 surfaces
we also refer to <cit.>.
But the
cycle classes of the other strata $\overline{V}_j$
for $j=12,\ldots, 20$ given by the Artin
invariant turned out to be elusive.
In joint work with Ekedahl <cit.> we developed a
uniform approach by applying the philosophy of <cit.>
of interpreting
these stratifications in terms of flags on the cohomology and
eventually were able to determine all cycle classes.
All these cycle classes
are multiples of powers of the Hodge class
Our approach uses flags on the de Rham cohomology,
here on $H^2_{\rm dR}$ as opposed to $H^1_{\rm dR}$
for abelian varieties. The space $H^2_{\rm dR}(X)$ is
provided with a non-degenerate intersection form
and it carries a filtration, the Hodge filtration.
But in positive characteristic it carries a second filtration
deriving from the fact that we do not have a Poincaré lemma,
or in other words, it derives from the Leray spectral sequence
applied to the relative Frobenius morphism $X \to X^{(p)}$.
See later for more on this so-called conjugate filtration.
We thus find two filtrations on $H^2_{\rm dR}(X)$ and these are not
necessarily transversal. We say that $X$ is ordinary if the
two filtrations are transversal and that $X$ is superspecial
if the two filtrations coincide.
These are two extremal cases, but by considering the relative position
of flags refining the two flags one obtains a further discrete invariant
and one
retrieves in a uniform way the invariants encountered above,
the height $h$ and the Artin invariant $\sigma_0$.
For applications it is important that we consider
moduli of K3 surfaces together with an embedding of a
non-degenerate lattice $N$ in the Néron-Severi group of $X$
such that it contains a semi-ample class of degree prime
to the characteristic $p$,
and then look at the primitive part of the de Rham cohomology.
If the dimension $n$ of this primitive cohomology is even,
this forces us to deal with very subtle questions related to the
distinction of orthogonal group ${\rm O}(n)$ versus the special orthogonal
group ${\rm SO}(n)$.
Instead of working directly on the
moduli spaces of K3 surfaces, we work on the space of flags on the primitive
part of $H^2_{\rm dR}$, that is, we work on a flag bundle over the moduli space.
The reason is that the strata that are defined on this
space are much better behaved than the strata on the moduli
of K3 surfaces itself.
In fact, up to infinitesimal order $p$ the strata on the flag space over ${\F}_g$
look like the strata (the Schubert cycles) on the flag space for the orthogonal group.
These strata are indexed by elements of a Weyl group.
In order to get the cycle classes of the strata on the moduli of K3 surfaces
we note that these latter strata are linearly ordered. This allows us to
apply fruitfully a Pieri type formula which expresses the intersection product
of a cycle class with a first Chern class (the Hodge class in our case) as
a sum of cycle classes of one dimension less.
We apply this on the flag space and
then project it down. There are many more strata on the flag space of the
primitive cohomology than on the moduli space. Some of these strata, the so-called
final ones, map in an étale way to their image in the moduli space; for the
non-final ones, either the image is lower-dimensional, and hence its cycle class
can be ignored, or the map is inseparable and factors through a final stratum
and the degree of the inseparable map can be calculated. In this way one arrives
at closed formulas for the cycle classes of the strata on the moduli space.
We give an example of the formula for the cycle classes from
<cit.> in the following special case.
Let $p>2$ and $\pi: {\mathcal X} \to {\F}_g$ be the universal family of
K3 surfaces with a primitive polarization
of degree $d=2g-2$ with $d$ prime to $p$. Then there are $20$
strata on the $19$-dimensional
moduli space ${\F}_g$ parametrized by so-called final elements
$w_i$ with $1 \leq i \leq 20$ in the Weyl group of ${\rm SO}(21)$.
These are ordered by their length $\ell(w_i)$ (in the sense of length in
Weyl groups)
starting with the
longest one.
The strata ${\mathcal V}_{w_i}$ for $i=1,\ldots 10$ are the strata
of height $h= i$, the stratum ${\mathcal V}_{w_{11}}$ is the supersingular stratum,
while the strata ${\mathcal V}_{w_i}$ for $i=11,\ldots,20$ are the strata where
the Artin invariant satisfies $\sigma_0 = 21-i$.
The cycle class of the closed stratum $\overline{\mathcal V}_{w_i}$ on the moduli space ${\F}_g$
is given by
\begin{eqnarray*}
{\rm i)} \quad
[\Vc_{w_k}] &=& (p-1)(p^2-1)\cdots(p^{k-1}-1) \lambda_1^{k-1} \quad
\hbox{\rm if $1\leq k\leq 10$,}\\
{\rm ii)} \quad [\Vc_{w_{11}}] &=&\frac{1}{2} (p-1)(p^2-1)\cdots(p^{10}-1)
\lambda_1^{10},\\
{\rm iii)} \quad
[\Vc_{w_{10+k}}] &=&\frac{1}{2}
\frac{(p^{2k}-1)(p^{2(k+1)}-1)\cdots(p^{20}-1)}{(p+1)\cdots(p^{11-k}+1)}
\lambda_1^{9+k} \quad
\hbox{\rm if $2\leq k\leq 10$.}
\end{eqnarray*}
Here $\lambda_1=c_1(L)$ with $L=\pi_* (\Omega^2_{{\mathcal X}/{\mathcal F}_g})$
is the Hodge class. Sections of $L^{\otimes r}$ correspond to modular forms
of weight $r$.
It is known (cf. <cit.> ) that the class
$\lambda_1^{18}\in {\rm CH}_{\QQ}^{18}({\mathcal F}_g)$
vanishes on ${\mathcal F}_g$. But the formulas
can be made to work also on the closure of the image ${\mathcal F}_g$
embedded in projective
space by the sections of a sufficiently high power of $L$,
so that the last two formulas (involving $\lambda_1^{18}$ and $\lambda_1^{19}$)
are non-trivial and still make sense.
Note here that $\lambda_1$ is an ample class;
this is well-known by Baily-Borel in characteristic $0$, but now
we know it too in characteristic $p\geq 3$ by work
of Madapusi-Pera <cit.> and Maulik <cit.>.
In particular, we can give an explicit formula for
the weighted number of superspecial K3 surfaces of genus $g$ by using a
formula for $\deg(\lambda_1^{19})$ from <cit.>.
We consider the situation where we have a primitive polarization of degree
$d=2d'$ inside $N$ with with
N^{\bot}= 2 \, U \bot m \, E_8(-1) \bot \langle -d \rangle \, , \eqno(4)
where $U$ is a hyperbolic plane and $m=0$ or $m=2$.
The weighted number
\sum_{X \, {\rm superspecial}/ \cong} \frac{1}{\# {\rm Aut}_k(X)}
of superspecial K3 surfaces with a primitive polarization $N$
of degree $d=2d^{\prime}$ prime to the characteristic $p$ with $N^{\bot}$
as in (1),
is given by
\frac{-1}{2^{4m+1}}\frac{p^{8m+4}-1}{p+1} \,
\left( (d^{\prime})^{10} \prod_{\ell|d^{\prime}} (1+\ell^{-4m-2}) \right)
\, \zeta(-1)\zeta(-3) \cdots \zeta(-8m-3)
where $\zeta$ denotes the Riemann zeta function and $\ell$ runs over the primes
dividing $d'$.
For $m=0$ this formula can be applied to count the number of Kummer surfaces
coming from superspecial principally polarized abelian surfaces and the
formula then agrees with the formulas of <cit.>.
Formulas like those given in (3) and in Theorem 1.1 for the classes
of the height strata were obtained in joint work with Katsura
<cit.> by different (ad hoc) methods using formal groups and Witt vector
cohomology; but these methods did not suffice to calculate the cycle classes
of the Artin invariant strata.
A simple corollary is (see <cit.>).
A supersingular (quasi-)elliptic K3 surface with a section
cannot have Artin invariant $\sigma_0=10$.
This result was obtained independently by Kondo and Shimada using a
different method in <cit.>.
In addition to reviewing the results from <cit.>
we prove irreducibility results for the strata; about half
of the strata on the moduli space ${\mathcal F}_g$ are
shown to be irreducible. Here we use the local structure of the
strata on the flag space.
Let $p\geq 3$ prime to the degree $d=2g-2$.
For a final element $w \in W_m^B$ (resp. $w \in W_m^D)$ of length $\ell(w)\geq m$ the stratum
$\overline{\mathcal V}_w$ in ${\mathcal F}_g$ is irreducible.
So the strata above the supersingular locus are all irreducible.
We have a similar result in ${\mathcal F}_N$.
The formulas we derived deal with the group ${\rm SO}(n)$;
for K3 surfaces we can restrict $n \leq 21$, but the formulas for larger $n$
might find applications to the moduli of hyperkähler varieties in positive
characteristic (by looking at the middle dimensional de Rham cohomology
or at $H^2_{\rm dR}$ equiped with the Beauville-Bogomolov form).
§ FILTRATIONS ON THE DE RHAM COHOMOLOGY OF A K3 SURFACE
Let $X$ be a K3 surface over an algebraically closed field of
characteristic $p>2$ and let $N \hookrightarrow {\rm NS}(X)$ be an isometric embedding of a non-degenerate
lattice in the Néron-Severi group ${\rm NS}(X)$ (equal to the Picard group for a
K3 surface)
and assume that $N$ contains a semi-ample line bundle and that the discriminant
of $N$ is coprime with $p$ (that is,
$p$ does not divide $\# N^{\vee}/N$).
We let $N^{\bot}$ be the primitive cohomology,
that is, the orthogonal complement of
the image of $c_1(N)$ of $N$ in $H^2_{\rm dR}(X)$.
It carries a Hodge filtration
0 = U_{-1} \subset U_0 \subset U_1 \subset U_2=N^{\bot}
of dimensions $0,1,n-1,n$ and comes with a non-degenerate intersection form
for which the Hodge filtration is self-dual: $U_0^{\bot}=U_1$. Now in
positive characteristic we have another filtration
0 = U_{-1}^c \subset U_0^c \subset U_1^c \subset U_2^c =N^{\bot} \, ,
the conjugate filtration; it is self-dual too.
The reason for its existence
is that the Poincaré lemma does not hold in positive
characteristic. If $F: X \to X^{(p)}$ is the relative Frobenius
morphism then we have a canonical (Cartier) isomorphism
C: {\mathcal H}^j(F_*\Omega_{X/k}^{\bullet}) \cong \Omega^j_{X^{(p)}/k}
and we get a non-trivial spectral sequence from this:
the second spectral sequence of hypercohomology
with $E_2$-term
$E^{ij}_2=H^i(X^{(p)},{\mathcal H}^j(\Omega^{\bullet}))$
which by the inverse Cartier isomorphism $C^{-1}: \Omega^j_{X^{(p)}} \simeq
{\mathcal H}^j(F_*(\Omega^{\bullet}_{X/k}))$
can be rewritten as
degenerating at the $E_2$-term and abutting to $H^{i+j}_{\rm dR}(X/k)$.
This leads to a second filtration on the de Rham cohomology.
The inverse Cartier operator gives an isomorphism
F^*(U_i/U_{i-1})\cong U_{2-i}^c/U^c_{1-i}\, .
As a result we have two (incomplete) flags
forming a so-called F-zip in the sense of <cit.>.
Unlike the characteristic zero situation
where the Hodge flag and its complex conjugate are
transversal, the two flags in our situation are not
necessarily transversal. In fact, the
K3 surface $X$ is called ordinary if these flags are transversal
and superspecial if they coincide. These are just two extremal
cases among more possibilities.
Before we deal with these further possibilities,
we recall some facts about isotropic flags in a
non-degenerate orthogonal space. Let $V$ be a non-degenerate orthogonal
space of dimension $n$ over a field of characteristic $p>2$. We have
to distinguish the cases $n$ odd and $n$ even, the latter being more subtle.
We look at isotropic flags
(0) =V_0 \subset V_1 \subset \cdots \subset V_r
with $\dim V_i=i$ in $V$, that is,
we require that the intersection form vanishes on $V_r$.
We call the flag maximal if $r=[n/2]$. We can complete a maximal flag by
putting $V_{n-j}=V_j^{\bot}$. Now if $n=2m$ is even, a complete isotropic
flag $V_{\bullet}$ determines another complete isotropic flag by putting
$V_i^{\prime}=V_i$ for $i < n/2$
and by taking for $V_m^{\prime}$ the unique maximal isotropic space
containing $V_{m-1}$ but different from $V_m$. We call this flag
$V_{\bullet}^{\prime}$ the twist of $V_{\bullet}$.
In fact, if $n$ is even the group ${\rm SO}(n)$ does not act transitively
on complete flags.
Given two complete isotropic flags their relative position is given by
an element of a Weyl group. If $n$ is odd we let $W_m^B$ be the Weyl group
of ${\rm SO}(2m+1)$. It can be identified with the following
subgroup of the symmetric group $\frak{S}_{2m+1}$
\{ \sigma \in \mathfrak{S}_{2m+1} : \sigma(i)+\sigma(2m+2-i)=2m+2
\text{ for all $1\leq i \leq 2m+1$} \}.
The fact is now that the ${\rm SO}(2m+1)$-orbits of pairs of totally
isotropic complete flags are in 1-1 correspondence
with the elements of $W_m^B$ given by
w \longleftrightarrow \big(\sum_{j\leq i} k\cdot e_j, \sum_{j\leq i}
k\cdot e_{w^{-1}(j)} \big)
with the $e_i$ a fixed orthogonal basis with $\langle e_i, e_j \rangle
=\delta_{i,2m+2-j}$. The simple reflections $s_i\in W_m^B$
for $i=1,\ldots,m$ are given by $s_i=(i, i+1)(2m+1-i, 2m+2-i)$ if $i<m$
and by $s_m=(m, m+2)$, and will play an important role here.
But in the case that $n=2m$ is even we have to replace the Weyl group $W_m^C$
(of ${\rm O}(2m)$) given by
\big\{ \sigma \in \mathfrak{S}_{2m} : \sigma(i)+\sigma(2m+1-i)=2m+1
\text{ for all $1\leq i \leq 2m$}\big\}
by the index $2$ subgroup $W_m^D$ given by the extra parity condition
\# \{ 1 \leq i \leq m : \sigma(i)>m\} \equiv \, 0 \, (\bmod 2) .
The simple reflections $s_i\in W_m^D$
are given by $s_i=(i, i+1)(2m-i, 2m+1-i)$
for $i<m$ and by $s_m=(m-1,m+1)(m, m+2)$.
In the larger group $W_m^C$ we have the simple reflections $s_i$ with $1\leq i \leq m-1$ and $s_m^{\prime}=(m, m+1)$.
Note that $s_m^{\prime}$ commutes with the $s_i$ for $i=1,\ldots,m-2$
and conjugation by it interchanges $s_{m-1}$ and $s_m$.
The ${\rm SO}(2m)$-orbits of pairs of totally isotropic complete flags
are in bijection with the elements of $W_m^C$ given by
w \longleftrightarrow \big(\sum_{j\leq i} k\cdot e_j, \sum_{j\leq i}
k\cdot e_{w^{-1}(j)} \big)
with basis $e_i$ with $\langle e_i,e_j\rangle =\delta_{i,2m+1-j}$.
Twisting the first (resp. second) flag corresponds to changing $w$
to $ws_m'$ (resp. $s_m'w$).
Back to K3 surfaces.
We can refine the conjugate flag on $H^2_{\rm dR}(X)$
to a full (increasing) flag $D^{\bullet}$
and use the Cartier operator to transfer it back to a decreasing flag
$C^{\bullet}$ on the Hodge filtration $U_{\bullet}$.
We thus get two full flags.
A full flag refining the conjugate filtration is called
stable if $D_j\cap U_i + U_{i-1}$ is an element of the
$C^{\bullet}$ filtration or of its twist.
A flag is called final if it is stable and complete.
Final flags correspond to so-called final elements in the Weyl group
defined as follows.
Elements in the Weyl group $W_m^B$ (resp. $W_{m}^D$)
which are reduced with respect to the set of
roots obtained after removing the first root (so that the remaining roots form
a root system of type $B_{m-1}$ (resp. $D_{m-1}$)) are called
final elements.
If $n=2m+1$ is odd we have $2m$ final elements in $W_m^B$. These are
the elements $\sigma$ given by $[\sigma(1),\sigma(2),\ldots,\sigma(m)]$
and we can list these as $w_1=[2m+1,2,3,\ldots]$, $w_2=[2m,1,3,\ldots], \ldots,
w_{2m}=[1,2,\ldots,m]$. These final elements $w_j$ are linearly ordered
by their length $\ell(w_j)=2m-j$, with $w_1$ being the longest element
and $w_{2m}$ equal to the identity element.
If $n=2m$ then we also have $2m$ final elements in $W_m^C$,
but these are no longer linearly ordered by their length,
but the picture is rather
w_m+1 [dr]
w_1[r] w_2[r] ⋯[r] w_m-1 [dr][ur] w_m+2
[r] ⋯[r] w_2m
w_m [ur]
Conjugation by $s_m^{\prime}$ interchanges the
two final elements of length $m-1$. This corresponds to twisting a complete
isotropic flag. We shall denote $w_js_m^{\prime}$ by $w_j^{\prime}$ and
we shall call these twisted final elements.
The following theorem shows that we can read off the height $h(X)$ of the
formal Brauer group and the Artin invariant $\sigma_0(X)$ from these final
filtrations on $H^2_{\rm dR}$.
The following theorem is proven in <cit.>.
Recall that the discriminant of $N$ is assumed to be prime to $p$.
Let $X$ be a K3 surface with an
embedding $N \hookrightarrow {\rm NS}(X)$ and
let $H\subset H^2_{\rm dR}$ be the primitive part of the
cohomology with $n=\dim(H)$ and $m=[n/2]$.
Then $H$ possesses a final filtration; all final filtrations
are of the same combinatorial type $w$. Moreover,
i) $X$ has finite height $h < n/2$ if and only if $w=w_h$ or $w_h'$.
ii) $X$ has finite height $h=n/2$ if and only if $w=w_m'$.
iii) $X$ has Artin invariant $\sigma_0< n/2$ if and only
if $w=w_{2m+1-\sigma_0}$ or $w= w_{2m+1-\sigma_0}'$.
iv) $X$ has Artin invariant $\sigma_0=n/2$
if and only if $w=w_{m+1}$.
In case i) (resp. in case iii))
we can distinguish these cases $w=w_h$ or $w=w_h'$
(resp. $w=w_{n-\sigma_0}$ or $w=w_{n-\sigma_0}'$)
for even $n$ by looking whether the
so-called middle part of the cohomology is split,
or equivalently, by the sign of the permutation $w$. We get $w_h'$ in
case i) exactly if $w$ is an odd permutation and in case
iii) we get $w=w_{n-\sigma_0}$ exactly if $w$ is an even permutation.
Looking at the diagram above one sees that the theorem excludes one of
the two possibilities corresponding to the two final elements
$w_m$ and $w_{m+1}$ of length $m$.
This is analyzed in detail in <cit.>.
It then agrees with the fact that the (closed) strata defined by the
height and the Artin invariant are linearly ordered by inclusion,
whereas the final $w_i$ in the above diagram are not.
§ STRATA ON THE FLAG SPACE
Suppose that we have a family $f: {\mathcal X} \to S$ of
$N$-marked K3 surfaces over a smooth base $S$.
We shall make a versality assumption. At a geometric point $s$ of $S$
we have the Kodaira-Spencer map
$T_sS \to H^1(X_s,T_{X_s}^1)$. We have a natural map
{\rm Hom}(H^0(X_s,\Omega^2_{X_s}), H^1(X_s,\Omega^1_{X_s}))$
and we can project $H^1(X_s,\Omega^1_{X_s})$
to the orthogonal complement $P$ of the image of
$N\hookrightarrow {\rm NS}(X_s)$
in $H^1(X_s,\Omega^1_{X_s})$.
The versality assumption is the requirement that the resulting map
T_sS \to {\rm Hom}(H^0(X_s,\Omega^2_{X_s}),P)
is surjective.
The primitive cohomology forms a vector bundle ${\mathcal H}$
of rank $n$ over $S$. It comes with two partial orthogonal isotropic
flags: the conjugate flag and the Hodge flag.
If we choose a complete orthogonal flag refining the conjugate filtration and
transfer it to the Hodge filtration by the Cartier operator we get
two flags and we can measure the relative position. This defines strata on $S$.
This implies that we have to choose a flag and we are thus forced to work
on the flag space ${\mathcal B}$ over $S$ (or ${\mathcal B}_N$ over
${\mathcal F}_N$)
of complete isotropic flags refining the Hodge filtration.
(Since we are using ${\F}_N$ for the moduli space of $N$-polarized K3 surfaces
we use another letter for the flag space; say ${\mathcal B}_N$ for banner).
To define the strata
scheme-theoretically we consider the general case of a semi-simple Lie group
$G$ and a Borel subgroup $B$ and a $G/B$-bundle $R\to Y$ over some scheme $Y$
with $G$ as structure group. Let $r_i: Y\to R$ ($i=1,2$) be two sections.
If $w$ is an element of the Weyl group $W$ of $G$ we define a locally closed
subscheme ${\mathcal U}_w$ of $Y$ as follows. We choose locally (possibly
in the étale topology) a trivialization of $R$ such that $r_1$ is a constant
section. Then $r_2$ corresponds to a map $Y \to G/B$ and we define
${\mathcal U}_w$ (resp. $\overline{\mathcal U}_w$)
to be the inverse image of the
$B$-orbit $BwB$ (resp. of its closure).
We thus find strata ${\mathcal U}_w$ and $\overline{\mathcal U}_w$ of
${\mathcal B}_N$;
it turns out that $\overline{\mathcal U}_w$
is the closure of ${\mathcal U}_w$.
It might seem that working on the flag space brings us farther
from the goal of defining and studying strata on the base space $S$ or on
the moduli spaces. However, working with the strata on the flag space
has the advantage that the strata are much better behaved there.
The space ${\mathcal B}_N$ together with the strata ${\mathcal U}_w$ is
a stratified space. The space ${\mathcal Fl}_n$ of complete self-dual flags
on an orthogonal space $V$ also carries a stratification, namely
by Schubert cells.
It is fiber space
over the space of maximal isotropic subspaces ${\mathcal I}_n$.
The main idea is now that our space ${\mathcal B}_N$ over ${\mathcal F}_N$
locally at a point up to the $(p-1)$st infinitesimal neighborhood looks like ${\mathcal Fl}_n
\to {\mathcal I}_n$
at a suitable point as stratified spaces.
This idea was developed in <cit.> and here it profitably can be used too.
If $(R,m)$ is a local ring the height $1$ hull of $R$
(resp. of $S={\rm Spec}(R)$) is $R/m^{(p)}$ (resp. ${\rm Spec}(R/m^{(p)})$)
with $m^{(p)}$ the ideal
generated by the $p$th powers of elements of $m$. It defines the
height $1$ neigborhood of the point given by $m$. We call two local rings
height $1$-isomorphic if their height $1$ hulls are isomorphic.
Let $k$ be a perfect field of characteristic $p$. For each $k$-point $x$ of
${\mathcal B}_N$ there exists a $k$-point $y$ of ${\mathcal Fl}_n$ such that
the height $1$ neighborhood of $x$ is isomorphic (as stratified spaces)
to the height $1$ neighborhood of $y$.
Indeed, we can trivialize the de Rham cohomology with its Gauss-Manin
connection on the height $1$ neighborhood of $x$ (because the ideal of $x$
has a divided power structure for which divided powers of degree $\geq p$
are zero). This has strong consequences for our strata, cf. the
following result from <cit.>.
The strata ${\mathcal U}_w$ on the flag space ${\mathcal B}_N$ satisfy the following properties:
* The stratum ${\mathcal U}_w$ is smooth of dimension equal to the
length $\ell(w)$ of $w$.
* The closed stratum $\overline{\mathcal U}_w$ is reduced, Cohen-Macaulay and normal of dimension $\ell(w)$ and equals the closure of ${\mathcal U}_w$.
* If $w$ is a final element then the restriction of ${\mathcal B}_N
\to {\mathcal F}_N$ to ${\mathcal U}_w$ is a finite surjective étale
covering from ${\mathcal U}_w$ to its image ${\mathcal V}_w$.
The degrees of the maps $\pi_w: {\mathcal U}_w \to {\mathcal V}_w$ for final $w$
coincide with the number of final filtrations of type $w$ and these numbers
can be calculated explicitly. For example,
for $w_i\in W_m^B$ with $1 \leq i <m $ we have
\deg \pi_{w_i}/\deg \pi_{w_{i+1}} = p^{2m-2i-1}+p^{2m-2i-2}+\ldots + 1.
§ THE CYCLE CLASSES
We consider a family of $N$-polarized K3 surfaces ${\mathcal X} \to S$
with $S$ smooth and satisfying the versality assumption <ref>.
Our strategy in <cit.> is to apply inductively a Pieri formula
to the final strata, which expresses the intersection
$\lambda_1 \cdot [\overline{\mathcal U}_w]$ as a sum over the classes
$[\overline{\mathcal U}_v]$, where $v$ is running through the elements of the
Weyl group of the form
$v=ws_{\alpha}$ with $s_{\alpha}$ simple and
$\ell(ws_{\alpha})=\ell(w)-1$. In fact,
we use a Pieri formula due to Pittie and Ram <cit.>.
In general these elements $v$ of colength $1$ are not final and this
forces us to analyze what happens with the strata ${\mathcal U}_v$
under the projection ${\mathcal B}_N \to {\mathcal F}_N$.
It turns out that for a non-final stratum either the projection is
to a lower-dimensional stratum or factors through an inseparable
map to a final stratum. The degree of these inseparable maps can be calculated.
In the case of a map to a lower dimensional stratum we can neglect these
for the cycle class calculation.
So suppose that for an
element $w$ in the Weyl group we have $\ell(ws_i)=\ell(w)-1$
for some $1<i\leq m$.
This means that if $A_{\bullet}$ and $B_{\bullet}$ denote
the two flags, that the image of $B_{w(i+1)}\cap A_{i+1}$ in $A_{i+1}/A_i$
is $1$-dimensional and thus we can change the flag $A_{\bullet}$ to a flag $A'_{\bullet}$ by setting
$A'_j=A_j$ for $j\neq i$ and $A_i'/A_{i-1}$ equal to the image of
$B_{w(i+1)}\cap A_{i+1}$. This gives us a map
\sigma_{w,i} : {\mathcal U}_w \to {\mathcal F}_N, \qquad
(A_{\bullet}, B_{\bullet}) \mapsto (A'_{\bullet},B_{\bullet}).
In this situation the image depends on the length $\ell(s_iws_i)$:
If $\ell(s_iws_i)=\ell(w)$ then the image of $\sigma_{w,i}$ is equal to ${\mathcal U}_{s_iws_i}$ and the map is purely inseparable of degree $p$. If
$\ell(s_iws_i)=\ell(w)-2$ then $\sigma_{w,i}$ maps onto ${\mathcal U}_{ws_i}
\cup {\mathcal U}_{s_iws_i}$ and the map $\sigma_{w,i}$ is not generically finite.
We then analyze in detail the colength $1$ elements occuring and whether they give rise to projections that loose dimension or are inseparable to final strata.
This is carried out in detail in <cit.>.
In this way the Pieri formula enables us to calculate the cycle classes.
The result for the cycle classes of the strata $\overline{\mathcal V}_{w_i}$
in the case that $n$ is odd (and with $m=[n/2]$) reads (cf. <cit.>):
The cycle classes of the final strata $\Vc_w$ on the base $S$ are polynomials
in $\lambda_1$ with coefficients that are polynomials in $\frac{1}{2}\ZZ[p]$
given by
\begin{eqnarray*}
{\rm i)} \quad
[\Vc_{w_k}] &=& (p-1)(p^2-1)\cdots(p^{k-1}-1) \lambda_1^{k-1} \quad
\hbox{\rm if $1\leq k\leq m$,}\\
{\rm ii)} \quad [\Vc_{w_{m+1}}] &=&\frac{1}{2} (p-1)(p^2-1)\cdots(p^{m}-1)
\lambda_1^{m},\\
{\rm iii)} \quad
[\Vc_{w_{m+k}}] &=&\frac{1}{2}
\frac{(p^{2k}-1)(p^{2(k+1)}-1)\cdots(p^{2m}-1)}{(p+1)\cdots(p^{m-k+1}+1)}
\lambda_1^{m+k-1} \quad
\hbox{\rm if $2\leq k\leq m$.}
\end{eqnarray*}
In the case where $n$ is even the result for the untwisted final
elements is the following:
The cycle classes of the final strata $\Vc_w$ for final
elements $w=w_j \in W_m^D$ on the base $S$ are
powers of $\lambda_1$ times polynomials
in $\frac{1}{2}\ZZ[p]$ given by
\begin{eqnarray*}
{\rm i)} \quad
[\Vc_{w_k}] &=& (p-1)(p^2-1)\cdots(p^{k-1}-1) \lambda_1^{k-1} \quad
\hbox{\rm if $k\leq m-1$, } \\
{\rm ii)} \quad [\Vc_{w_{m+1}}]
&=& (p-1)(p^2-1)\cdots(p^{m-1}-1)
\lambda_1^{m-1},\\
{\rm iii)} \quad
[\Vc_{w_{m+k}}] &=&
\frac{1}{2}
\frac{\prod_{i=1}^{m-1}(p^i-1)\prod_{i=m-k+2}^m(p^i+1)}
\lambda_1^{m+k-2} \quad
\hbox{\rm if $2\leq k\leq m$. }
\end{eqnarray*}
Furthermore, we have that $\Vc_{w_m}=\emptyset$.
Finally in the twisted even case we have:
The cycle classes of the final strata $\Vc_w$ for twisted final elements
$w=w_j \in W_m^D s_m^{\prime}$ on the base $S$ are
powers in $\lambda_1$ with coefficients that are polynomials
in $\frac{1}{2}\ZZ[p]$ given by
\begin{eqnarray*}
{\rm i)} \quad
[\Vc_{w_k}] &=& (p-1)(p^2-1)\cdots(p^{k-1}-1) \lambda_1^{k-1} \quad
\hbox{\rm if $k\leq m-1$, } \\
{\rm ii)} \quad [\Vc_{w_m}]
&=& (p-1)(p^2-1)\cdots(p^{m}-1)
\lambda_1^{m-1},\\
{\rm iii)} \quad
[\Vc_{w_{m+k}}] &=&
\frac{1}{2}
\frac{\prod_{i=1}^{m}(p^i-1)\prod_{i=m-k+2}^{m-1}(p^i+1)}
\lambda_1^{m+k-2} \quad
\hbox{\rm if $2\leq k\leq m$.}
\end{eqnarray*}
Furthermore, we have $\Vc_{w_{m+1}}=\emptyset$.
§ IRREDUCIBILITY
In this section we shall show that about half of the $2m$
strata ${\mathcal V}_{w_i}$ on our moduli space ${\mathcal F}_N$ of
$N$-polarized K3 surfaces are irreducible ($m$ strata in the B-case, $m-1$
in the D-case).
Let $p\geq 3$ and assume that ${\mathcal F}_N$
is the moduli space of primitively $N$-polarized K3 surfaces where
$N^{\vee}/N$ has order prime to $p$.
If $w \in W_m^B$ (resp. $w \in W^D_{m}$ or $w \in W_m^Ds^{\prime}_m$) is a (twisted) final element with
length $\ell(w)\geq m$, then the locus $\overline{\mathcal V}_{w}$
in ${\mathcal F}_N$ is irreducible.
(We do the $B$-case, leaving the other case to the reader.)
The idea behind the proof is to show that for $1 \leq i \leq m$ the stratum
$\overline{\mathcal U}_{w_i}$ is connected in the flag space ${\mathcal B}_N$.
Note that ${\mathcal F}_N$ is connected by our assumptions.
By Theorem <ref> the stratum $\overline{\mathcal U}_{w_i}$
is normal, so if it is connected it must be irreducible.
But then its image
$\overline{\mathcal V}_{w_i}$ in ${\mathcal F}_N$ is irreducible as well.
This shows the
advantage of working on the flag space.
To show that $\overline{\mathcal U}_{w_i}$ is connected in ${\mathcal B}_N$
we use that its $1$-skeleton is connected, that is, that the union of the
$1$-dimensional strata that it contains, is connected and that every
irreducible component of $\overline{\mathcal U}_{w_i}$ intersects the
$1$-skeleton. To do that we
prove the following facts:
* The loci $\overline{\mathcal V}_{w_i}$ in ${\mathcal F}_N$ are
connected for $i<2m$ (that is, for $w_i\neq 1$).
* Any irreducible component of any $\overline{\mathcal U}_{w}$
contains a point of ${\mathcal U}_1$.
* The union $\cup_{i=2}^m \, \overline{\mathcal U}_{s_i}$ intersected
with a fibre of ${\mathcal B}_N \to {\mathcal F}_N$ over a point of the
superspecial locus ${\mathcal V}_{1}$ is connected.
* For $1 \leq i \leq m$ the locus $\overline{\mathcal U}_{w_i}$
contains $\cup_{i=1}^m \overline{\mathcal U}_{s_i}$.
In the proof we use the fact that the closure of strata on the flag
space is given by the Bruhat order in the Weyl group: ${\mathcal U}_v$
occurs in the closure of ${\mathcal U}_w$ if $v\geq w$ in the Bruhat order.
Furthermore, we observe that one knows by <cit.>, <cit.>
that $\lambda_1$ is an ample class.
Together (1) and (3) will prove that the locus
$\cup_{i=1}^m \overline{\mathcal U}_{s_i}$ (whose image in ${\mathcal F}_N$
is $\overline{\mathcal V}_{2m-1}$) is connected. We begin by proving (1).
Sections of a sufficiently high multiple of $\lambda_1$ embed ${\mathcal F}_N$
into projective space and we take its closure $\overline{\mathcal F}_N$.
By the result of Theorem <ref>
(resp. <ref> and <ref>) we know that the cycle
class $[\overline{\mathcal V}_{w_i}]$ is a multiple of $\lambda_1^{i-1}$,
so these loci are connected in $\overline{\mathcal F}_N$ for $i-1 <
\dim{\mathcal F}_N$.
In particular, the $1$-dimensional
locus $\overline{\mathcal V}_{w_{2m-1}}$ in ${\mathcal F}_N$
(which equals its closure in $\overline{\mathcal F}_N$) is connected.
On any irreducible component of $Y$ of $\overline{\mathcal V}_{w_i}$
in $\overline{\mathcal F}_N$
with $i <2m-1$ the intersection with $\overline{\mathcal V}_{w_{2m-1}}$
is cut out by a multiple of a positive power of
$\lambda_1$, hence it intersects this locus (in ${\mathcal F}_N$).
Since ${\mathcal V}_{w_i}$ contains $\overline{\mathcal V}_{w_{2m-1}}$
for $i<2m-1$ the connectedness follows.
To prove (4) consider the reduced expression for $w_i$ for $i\leq m$:
it is $s_is_{i+1}\cdots s_m s_{m-1}\cdots s_1$, see <cit.>.
This shows that all
the $s_i$ occur in it and we see that the $\overline{\mathcal U}_{s_i}$
for $i=1,\ldots,m$
occur in the closure of $\overline{\mathcal U}_{w_i}$.
The proof of (2) is similar to the
proof of <cit.> and uses induction on the Bruhat order.
If $\ell(w) \leq 2m-2$ then $\overline{\mathcal U}_w$ is proper in
${\mathcal B}_N$. If an irreducible component has
a non-empty intersection with a $\overline{\mathcal U}_{w'}$
with $w'>w$, then induction provides a point of ${\mathcal U}_1$;
otherwise we can apply a version of the Raynaud trick as in
<cit.> and conclude that $w=1$.
If $\ell(w)=2m-1$ then the image of
any irreducible component $Y$ of $\overline{\mathcal U}_w$
in ${\mathcal F}_N$ is either contained in $\overline{\mathcal V}_{w_3}$
and then $Y$ is proper in ${\mathcal B}_N$
or the image coincides with $\overline{\mathcal V}_{w_2}$.
In the latter case it maps in a generically finite way to it and therefore
any irreducible component $Y$ of $\overline{\mathcal U}_w$
intersects the fibres over the superspecial points, hence by induction
contains a point of ${\mathcal U}_1$.
For (3) we now look in the fibre $Z$ of the flag space over the
image of a point of ${\mathcal U}_{1}$.
This corresponds to a K3 surfaces for which the Hodge
filtration $U_{-1}\subset U_0 \subset U_1 \subset U_2=H$
coincides with the conjugate filtration
$U_{-1}^c\subset U_0^c \subset U_1^c \subset U_2^c=H$.
Moreover, we have the identifications
F^*(U_0)\cong (U_2^c/U^c_1)=(U_0^c)^{\vee}= U_0^{\vee},
given by Cartier and the intersection pairing and similarly
F^*(U_1/U_0)\cong (U_1/U_0)^{\vee},
giving $U_0$ and $U_1/U_0$ (and also $U_2/U_1)$) the structure of a $p$-unitary space. Indeed,
if $S$ is an ${\FF}_p$-scheme then a $p$-unitary vector bundle ${\mathcal E}$ over $S$ is a
vector bundle together with an isomorphism $F^*({\mathcal E})\cong {\mathcal E}^*$ with $F$ the absolute Frobenius.
This gives rise to a bi-additive map
$\langle \, , \, \rangle: {\mathcal E}\times {\mathcal E} \to {\mathcal O}_S$
$\langle f x, y\rangle = f^p\langle x, y\rangle$
$\langle x,fy\rangle = f \langle x, y \rangle$ for $f$ a section
of ${\mathcal O}_S$.
In the étale topology this notion is equivalent
to a local system of ${\FF}_{p^2}$-vector spaces,
cf. <cit.>.
In case that $S={\rm Spec}({\FF}_{p^2})$ we can then consider the
flag space $Z$ of complete $p$-unitary flags on $U_1/U_0$.
The smallest $p$-unitary stratum there is the stratum of flags that
coincide with their $p$-unitary dual.
Such flags are defined over ${\FF}_{p^2}$
as one sees by taking the dual once more.
So we look now at self-dual flags $A_{\bullet}= \{ A_1 \subset A_2 \subset
\cdots \subset A_{2m-1}\}$
on $A=U_1/U_0$ and we compare
the flag $A_{\bullet}^{(p)}$ with the flag $A_{\bullet}$.
(Note that the indices $i$ now run from $1$ to $m-1$ instead of from $1$ to $m$
since we leave $U_0$ fixed.)
For an element $s=(i, i+1)$ with $1 \leq i \leq m-2$ we look at
the intersection of the stratum $\overline{\mathcal U}_{s}$ with the fibre $Z$.
It consists of those flags $A_{\bullet}$ with the property that the
steps $A_j$ for $j\neq i$ and $j\neq 2m-1-i$ are ${\FF}_{p^2}$-rational
and that for all $j$ we have $A_{j}= A_{2m-1-j}^{\bot}$.
We see that we can choose $A_i$ freely by prescribing its image in
$A_{i+1}/A_{i-1}$, hence this locus is a ${\PP}^1$.
In case $s=(m-1, m+1)$ we have to choose a space $A_{m-1}$ and its orthogonal
in $A_{m-1} \subset A_{m-1}\subset A_m \subset A_{m+1}$. Now all
non-degenerate $p$-unitary forms are equivalent, so we may choose the
form $x^{p+1}+y^{p+1}+z^{p+1}$ on the $3$-dimensional space $A_{m+1}/A_{m-2}$.
So in this case $\overline{\mathcal U}_s$ is isomorphic to the Fermat curve.
The points of ${\mathcal U}_1$ are the ${\FF}_{p^2}$-rational points on it.
In case the space $A=U_1/U_0$ is even-dimensional, say $\dim A=2m-2$
the same argument works for $s_i$ with $1\leq i \leq m-2$.
For $s=(m-2, m)(m-1, m+1)$ (resp. for $s=(m-1,m)$)
we remark that it equals $s' s_{m-2}s'$ with $s'=(m-1, m)$,
hence we find a ${\PP}^1$ by picking a flag $A_{m-2} \subset
A_{m-1} \subset A_{m}$.
This shows that we remain in the same connected component of
$\overline{\mathcal U}^{(1)} \cap Z$,
with ${\mathcal U}^{(1)}$ the union of the $1$-dimensional strata ${\mathcal U}_v$, if we change the flag $A_{\bullet}$ at
place $i$ and $2m-1-i$ compatibly. By Lemma 7.6 of <cit.>
this implies that $\overline{\mathcal U}^{(1)} \cap Z$ is connected.
M. Artin, Supersingular $K3$ surfaces: Ann. Sci. École Norm. Sup.
7 (1974), 543–567.
[A-M]A-M M. Artin and B. Mazur: Formal groups
arising from algebraic varieties. Ann. Sci. Ec. Norm. Sup.
bf 10, (1977), 87–132.
M. Artin and H. P. F. Swinnerton-Dyer: The Shafarevich-Tate
conjecture for pencils of elliptic curves on $K3$ surfaces.
Invent. Math.
20 (1973), 249–266.
S. Billey and V. Lakshmibai: Singular loci of Schubert varieties.
Progr. in Math., vol. 182, Birkhäuser, Boston, MA, 2000.
[C]C F. Charles:
The Tate conjecture for K3 surfaces over finite fields.
Invent. Math. 194 (2013), no. 1, 119–-145.
[E]E T. Ekedahl:
On supersingular curves and abelian varieties.
Math. Scand. 60 (1987), 151–178.
T. Ekedahl and G. van der Geer: Cycle classes of the E-O
stratification on the moduli of Abelian varieties. In:
Algebra, Arithmetic and Geometry, Progress in Mathematics, vol. 269-270, Birkhäuser,
[E-vdG2]E-vdG2 T. Ekedahl, G. van der Geer:
Cycle Classes on the Moduli of K3 surfaces in positive characteristic.
Selecta Math. (N.S.), (2015) 21, 245–-291 , arXiv:1104.3024 (2011).
http://dx.doi.org/10. 1007/s00029-014-0156-8
[vdG]vdG G. van der Geer:
Cycles on the moduli space of abelian varieties.
Moduli of curves and abelian varieties, 65–89, Aspects Math., E33,
Vieweg, Braunschweig, 1999.
G. van der Geer and T. Katsura: On a stratification of the moduli of
$K3$ surfaces. J. Eur. Math. Soc. (JEMS) 2 (2000), no. 3,
G. van der Geer and T. Katsura:
Note on tautological classes of moduli of K3 surfaces.
Mosc. Math. J. 5 (2005), 775–-779.
[G-H-S]G-H-S V. Gritsenko, K. Hulek, G. Sankaran:
The Hirzebruch-Mumford volume for the
orthognal group and applications.
Documenta Math. 12 (2007), 215–241
[H]H D. Huybrechts:
Lectures on K3 surfaces. Lecture Notes.
[K-S]K-S S. Kondo, I. Shimada:
On certain duality of Néron-Severi lattices of
supersingular K3 surfaces and its application to generic
supersingular K3 surfaces.
Algebraic Geometry 1 (2014), no 3, 311–333.
[L]L C. Liedtke:
Supersingular k3 surfaces are unirational.
: arXiv:1304.5623
[M]M D. Maulik:
Supersingular K3 surfaces for large primes.
[Mo]Mo B. Moonen:
A dimension formula for Ekedahl-Oort strata.
Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 666-–698.
[M-W]M-W B. Moonen, T. Wedhorn:
Discrete invariants of varieties in positive characteristic.
Int. Math. Res. Notices (2004) 2004 (72): 3855-3903.
A. Ogus: Supersingular K3 crystals. Journées de
Géométrie Algébrique de Rennes (II), Astérisque,
vol. 64, Soc. Math. Fr., 1979, pp. 3–86.
[Og2]Og2 A. Ogus:
Singularities of the height strata in the moduli of K3 surfaces.
In: Moduli of Abelian Varieties. (Texel Island, 1999), 325–-343,
Progr. Math., 195, Birkhäuser, Basel, 2001.
[O]O F. Oort:
A stratification of a moduli space of abelian varieties.
Moduli of abelian varieties (Texel Island, 1999), 345–-416, Progr. Math., 195, Birkhäuser, Basel, 2001.
C-L. Chai, F. Oort:
Monodromy and irreducibility of leaves.
Ann. of Math. (2) 173 (2011), 1359–-1396.
[P1]P1 K. Madapusi Pera:
Toroidal compactifications of integral models of
Shimura varieties of Hodge type.
: arXiv:1211.1731
[P2]P2 K. Madapusi Pera:
The Tate conjecture for K3 surfaces in odd characteristic.
: arXiv:1301.6326
H. Pittie and A. Ram: A Pieri-Chevalley formula in the $K$-theory
of a $G/B$-bundle. Electron. Res. Announc. Amer. Math. Soc. 5
(1999), 102–107.
[R]R M. Rapoport:
On the Newton stratification. Séminaire Bourbaki. Vol. 2001/2002.
Astérisque No. 290 (2003), Exp. No. 903, viii, 207–224.
A.N. Rudakov, I.R. Shafarevich:
Supersingular K3 surfaces over fields of characteristic $2$.
Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 848–869.
Also in:
I.R. Shafarevich: Collected Mathematical Papers. Springer Verlag.
A.N. Rudakov, I.R. Shafarevich:
Surfaces of type K3 over fields of finite characteristic.
Current problems in mathematics, Vol. 18, pp. 115–-207,
Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn.
Informatsii, Moscow, 1981.
Also in:
I.R. Shafarevich: Collected Mathematical Papers. Springer Verlag.
A.N. Rudakov, I.R. Shafarevich, Th. Zink:
The effect of height on degenerations of K3 surfaces.
Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 1, 117–-134, 192.
Also in:
I.R. Shafarevich: Collected Mathematical Papers. Springer Verlag.
[S]S T. Shioda:
Algebraic cycles on certain K3 surfaces in characteristic $p$.
Manifolds–Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973),
pp. 357–364. Univ. Tokyo Press, Tokyo, 1975.
[T]T J. Tate: Algebraic cycles and poles of zeta functions.
Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963)
pp. 93–110 Harper & Row, New York 1965.
|
1511.00038
|
[Corresponding author: ][email protected]
Institute for Quantum Science and Technology,
Department of Physics and Astronomy,
University of Calgary, Calgary, Alberta, Canada T2N 1N4
The ability to transfer quantum information from one location to another with
high fidelity is of central importance to quantum information science.
Unfortunately for the simplest system of a uniform chain (a spin chain or a
particle in a one-dimensional lattice), the state transfer time grows
exponentially in the chain length $N$ at fixed fidelity. In this work we show
that the addition of an impurity near each endpoint, coupled to the uniform
chain with strength $w$, is sufficient to ensure efficient and high-fidelity
state transfer. An eigenstate localized in the vicinity of the impurity can be
tuned into resonance with chain extended states by tuning
$w(N)\propto N^{1/2}$; the resulting avoided crossing yields resonant
eigenstates with large amplitudes on the chain endpoints and approximately
equidistant eigenvalues. The state transfer time scales as $t\propto N^{3/2}$
and its fidelity $F$ approaches unity in the thermodynamic limit $N\to\infty$;
the error scales as $1-F\propto N^{-1}$. Thus, with the addition of two
impurities, asymptotically perfect state transfer with a uniform chain is
possible even in the absence of external control.
§ INTRODUCTION
The ability to transfer information is crucial for digital communications.
Likewise in quantum computation and communication, the ability to efficiently
and reliably transfer quantum information is central to both current and
future quantum technologies <cit.>. In the standard circuit model of
quantum computation, the quantum information is encoded on localized spins with
two or more accessible distinguishable quantum states (qubits or qudits), and
an algorithm is effected by manipulating individual spins, performing two-spin
operations, and making measurements <cit.>. The physical objects
encoding the spins, for example spins in semiconductors <cit.> or
nitrogen-vacancy centers in diamond <cit.>, may be widely
separated, requiring the development of robust quantum state transfer protocols
for spin networks <cit.>.
Schemes for perfect quantum state transfer (PST) in spin networks were
developed over a decade ago <cit.>. Quantum
information encoded on a spin at one end of a linear chain of nearest-neighbor
coupled spins was found to propagate perfectly to the opposite end, as long as
the non-uniform spin coupling coefficients could be carefully chosen.
Subsequent work showed how to find all possible coupling constants consistent
with PST for a spin chain of arbitrary length <cit.>. The
ability to adjust a large number of coupling constants is expected to be
experimentally challenging, and unfortunately PST is not possible in uniform
spin chains (i.e. where all coupling constants are identical) longer than
three sites. This has prompted the investigation of different network
topologies with uniform (but possibly signed) couplings that can support
PST <cit.>.
An alternative strategy is to relax the assumption of perfect quantum state
transfer, replacing it with `pretty good state transfer'
(PGST) <cit.> or equivalently `almost perfect state
transfer' (APST) <cit.>. In these cases, one sets the desired
fidelity $F$ of the output state, and determines the time required (if any) to
achieve it. While these equivalent concepts in the literature are frequently
referred to as APST, we prefer to employ the term `imperfect quantum state
transfer' (IST) to clearly distinguish the behavior from PST. In IST there
exists a time at which the initial state at some site transfers to a different
site with probability approaching unity to within some error.
For uniform spin chains, IST is only possible in principle for particular
values of the number of spins $N$ <cit.>. For fixed
minimum fidelity, however, the transfer time $t$ increases exponentially with
$N$ <cit.>. A linear scaling of $t$ with $N$ can be achieved by
coupling the initial and final spins only weakly to the uniform
chains <cit.>. Because the quantum information
is strongly
localized in the vicinity of the chain ends at all times, this model is also
more robust against noise than the bare uniform chain. Unfortunately, in this
model after optimizing the strength of the weak coupling parameter (which is
found to decrease like $N^{-1/6}$) and initializing the channel, the maximum
output amplitude decreases with $N$, attaining an asymptotic value of $0.8469$
for $N\to\infty$ <cit.>. Adding additional weak links improves the
results, but the asymptotic error in fidelity remains
finite <cit.>. Instead applying a magnetic field in the vicinity
of the chain ends yields fidelities that approach unity in the large-$N$ limit,
but at the cost of $t$ growing exponentially with
$N$ <cit.>.
In this work we consider a uniform spin chain of length $N$, with sites labeled
from 1 through $N$, with the minor modification of an additional impurity spin
coupling to the spin at site $3$ and another to site $N-2$, both with coupling
constant $w$. The value of $w$ is left as a variable to be optimized. It is
found that there exists a value of $w\sim\sqrt{N}$ that yields a resonance
between the quantum state localized near the impurity and extended states of
the chain. The resulting avoided crossing yields strongly mixed eigenvectors
with equally spaced eigenvalues and large overlaps with the chain endpoints.
Under these conditions, the fidelity for quantum state transfer from site 1 to
site $N$ approaches unity in the thermodynamic (large-$N$) limit, with error
$1-F\propto N^{-1}$. The system therefore exhibits `asymptotically perfect
quantum state transfer,' a behavior previously unobserved for a spin network.
The time is also found to scale efficiently with the chain length,
$t_{\rm IST}\propto N^{3/2}$. In concrete terms, the output fidelity surpasses
$F=0.9$ when $N>99$ and exceeds $F=0.99$ for $N\geq 900$.
Section <ref> briefly reviews the essential characteristics of IST,
with an emphasis on mirror-symmetric networks such as spin chains. The model of
interest in the present work is introduced in Sec. <ref>. The minimum
IST time found by optimizing the impurity coupling strength $w$ is found to be
a strongly non-monotonic function of $N$. To find a functional form for $w(N)$,
the model is investigated analytically in Sec. <ref>;
high-fidelity IST in time $t_{\rm IST}\propto N^{3/2}$ is found to occur if
$w(N)\propto\sqrt{N}$. Section <ref> is devoted to a full
numerical simulation of the evolution under the model Hamiltonian, validating
and clarifying the analytical predictions. A brief discussion of the results is
found in Sec. <ref>.
§ IMPERFECT QUANTUM STATE TRANSFER
The time evolution of a quantum state $|\phi(t)\rangle$ under the action of a
governing Hamiltonian $H$ is given by
\begin{equation}
\label{eq:evol1}
\end{equation}
Suppose that $|\phi(t)\rangle$ is only defined at discrete sites
$j=1,2,\ldots,N$. Defining states $|i\rangle$ corresponding to unit basis
vectors associated with site $i$, one obtains the probability amplitudes
$\langle i|\phi(t)\rangle=\phi_i(t)$. Given $N$ eigenvalues $\lambda^{(n)}$ and
orthonormal eigenvectors $|\psi^{(n)}\rangle$ of $H$, Eq. (<ref>)
\begin{equation}
\label{eq:evol2}
\end{equation}
Suppose furthermore that the initial state is completely localized at a
particular site $j$, so that $\langle i|\phi(0)\rangle=\delta_{ij}$.
Eq. (<ref>) can then be rewritten as
\begin{equation}
\lambda^{(n)}-\lambda^{(m)}\right)/\hbar}|\psi^{(n)}\rangle
\langle\psi^{(n)}|j\rangle.
\label{eq:evol3}
\end{equation}
Evidently if $(\lambda^{(n)}-\lambda^{(m)})t/\hbar=2\pi s$ ($s\in\mathbb{Z}$)
for any $n$ and $m$, then $|\phi(t)\rangle=|j\rangle$ for all $j$ up to an
unimportant overall phase; that is, the discrete Hamiltonian is site periodic.
This immediately implies that the Hamiltonian is site periodic if all the
eigenvalues of $H$ satisfy the ratio condition
\begin{equation}
\frac{\lambda^{(n)}-\lambda^{(m)}}
{\lambda^{(p)}-\lambda^{(q)}}\in \mathbb{Q}
\end{equation}
for all possible indices $\{n,m,p,q\}$ (except $\lambda^{(p)}=\lambda^{(q)}$).
For large systems in practice, condition (<ref>) can only be satisfied
if the spectrum is linear, i.e. the gap between successive non-degenerate
eigenvalues is constant.
In perfect quantum state transfer (PST) between sites $i$ and $j$, a state
initially localized in state $|i\rangle$ ends up in state $|j\rangle$,
$j\neq i$ (and vice versa) after some elapsed time $t$; these sites are
therefore periodic in time $2t$. PST therefore implies site periodicity, but
the converse is not generally true. For space-symmetric Hamiltonians that
commute with the parity operator, however, site periodicity implies PST.
Suppose that the smallest difference in eigenvalues is
$\Delta=\lambda^{(n)}-\lambda^{(m)}$ for some $m$ and $n$; then site
periodicity occurs in time $t_P=2\pi\hbar/\Delta$. Consider an initial state
in a superposition state of sites $i$ and $j$ equidistant from an axis of
symmetry, $|\phi(0)\rangle=\frac{1}{\sqrt{2}}\left(|i\rangle+|j\rangle\right)$.
Because $|\phi(0)\rangle$ has even parity, only even-parity eigenvectors
will contribute to the sum in Eq. (<ref>). Site periodicity must be
maintained, but the absence of odd eigenvalues implies that the gap has now
doubled to $\Delta'=2\Delta$ (assuming the spectrum is linear); now
$t_P=4\pi\hbar/\Delta'$. At half this time $t=t_P/2=2\pi\hbar/\Delta'$ the sum
over $n$ in Eq. (<ref>) still resolves to the identity. Thus,
even-parity states such as the superposition state
$\frac{1}{\sqrt{2}}\left(|i\rangle+|j\rangle\right)$ evolve to themselves in
half the site periodicity time $t=\pi\hbar/\Delta$, which is only possible if
after this time $|i\rangle\leftrightarrow|j\rangle$, i.e. each site undergoes
PST to its mirror-symmetric counterpart. To summarize: mirror-symmetry and a
linear spectrum are together sufficient to ensure that the Hamiltonian supports
Unfortunately, most Hamiltonians do not possess a perfectly linear spectrum,
even if they are mirror symmetric. One may instead probe for IST, where the
initial state at some site transfers to a different site with high probability.
A naïve approach is to approximate
all eigenvalues in the spectrum by rationals with the same common denominator,
so that in principle the ratio condition (<ref>) is automatically
satisfied. Then in principle there should exist some time at which the
Hamiltonian is almost site periodic, where initial states localized at a given
site return with probability proportional to the accuracy of the rational
approximation. By extension, if parity is a commuting operator then at half
this time the probability at the mirror-symmetric vertex (the fidelity) should
approach unity to within a similar error.
In fact the criteria for IST are slightly more involved than the naïve
approach discussed above <cit.>. Recall that for a site-periodic
Hamiltonian there exists a time $\lambda^{(n)}t/\hbar=2\pi M_n$,
$M_n\in\mathbb{Z}$ for all $n$ so that the sum over $n$ in Eq. (<ref>)
resolves to the identity. For almost site periodicity, and allowing for an
arbitrary $n$-independent phase, this criterion would become
$\left|\lambda^{(n)}t_P/\hbar+\varphi-2\pi M_n\right|<\delta$, where
$\delta\ll 1$.
For reflection-symmetric Hamiltonians the eigenvalues of even and odd parity
eigenstates interleave. At the IST time $t=t_P/2$, odd-$n$ eigenstates map
$|i\rangle$ to $|N-i+1\rangle$ but with an additional $\pi$ phase. The IST
condition then reads
\begin{equation}
-\delta<\frac{\lambda^{(n)}t}{\hbar}-\alpha_n-2\pi M_n<\delta,
\quad n=1,2,\dots,N,
\label{eq:inequality}
\end{equation}
where $\alpha_n=\pi n-\varphi$. The first task would be to find integers $M_n$,
the phase $\varphi$, and time $t$ to satisfy this set of inequalities for each
$n$ at fixed $\delta$. Once this is accomplished, one would calculate the
output fidelity. Given initial occupation of vertex $1$, the desired output
vertex is $N$. The output fidelity is therefore
\begin{equation}
F(t)=|\langle N|e^{-iHt}|1\rangle|^2.
\label{eq:fidelity}
\end{equation}
IST is said to occur at time $t$ if $F(t)$ exceeds some minimum threshold, for
example $F_{\rm min}=0.9$.
§ MODEL AND BEHAVIOR
Geometry of the model. Chain sites are labeled 1 through $N$, and
the tunneling amplitude between adjacent sites is constant $J$ (not labeled in
the figure). The impurities consist of additional leaves at sites 3 and $N-2$
labeled by $N+1$ and $N+2$, respectively. The amplitude to tunnel between chain
and impurity sites is $w$.
Consider an array of spin-$\frac{1}{2}$ particles, each confined to its own
lattice site. As shown in Fig. <ref>, $N$ sites are arranged in a
one-dimensional chain with an additional spin connected to the third site from
each end. Given a spin-spin coupling constant $J$ along the chain (not
explicitly labeled in the figure) and coupling constant $w$ between the
additional spins and their counterparts on the chain, the XY Hamiltonian reads
\begin{equation}
H=H_{\rm 1D}+H',
\label{eq:spinHam}
\end{equation}
\begin{equation}
H_{\rm 1D}=\frac{J}{2}\sum_{i=1}^{N-1}\left(X_iX_{i+1}+Y_iY_{i+1}\right)
\end{equation}
corresponds to the one-dimensional uniform chain Hamiltonian and the
Hamiltonian for the two additional impurity sites is
\begin{eqnarray}
H'&=&\frac{w}{2}\big(X_3X_{N+1}+Y_3Y_{N+1}\nonumber \\
\end{eqnarray}
here $X=\sigma_x$ and $Y=\sigma_y$ are two-dimensional Pauli matrices. The
total spin projection is a good quantum number and the Hamiltonian diagonalizes
into blocks with a fixed number of excitations. Because only one excitation is
required in order to effect state transfer, it is conventional to work in the
single-excitation subspace <cit.>. In this case,
the spin Hamiltonian (<ref>) is equivalent to a single particle
hopping via a tight-binding Hamiltonian on an array with the same geometry,
but with spin-coupling constants replaced by hopping
amplitudes <cit.>:
\begin{equation}
H_{\rm 1D}=J\sum_{i=1}^{N-1}\left(|i\rangle\langle i+1|+|i+1\rangle\langle i|
\right);
\label{eq:H1D}
\end{equation}
\begin{equation}
H'=w\left(|3\rangle\langle N+1|+|N-2\rangle\langle N+2|+\mbox{H.c.}\right).
\label{eq:Hprime}
\end{equation}
Here, states $|i\rangle$ are unit vectors associated with site $i$, and H.c.stands for the Hermitian conjugate.
Determination of the conditions under which IST can occur (if any) for the
geometry shown in Fig. <ref> hinges on the diagonalization of the
spin Hamiltonian (<ref>) or alternatively (and more simply) the
equivalent hopping Hamiltonian, Eqs. (<ref>) and (<ref>).
To probe numerically for IST,
we explicitly obtain the time-dependence of the probability on the output
vertex using Eq. (<ref>), and calculate the time-dependent fidelity
$F(t)$, Eq. (<ref>).
The minimum time, in units of $\hbar/J$, for IST on the
impurity-modified chain is shown as a function of the chain length for $N$
between 6 and 200 sites. The inset shows the value of the impurity hopping
parameter (impurity spin coupling constant) $w$ associated with the minimum
time, in units of the chain parameter $J$.
Figure <ref> shows the minimum time for which IST is possible as a
function of chain length $N$, for $6\leq N\leq 200$. The results were obtained
by looping over values of $\tilde{w}=w/J$ in the range $0\leq\tilde{w}\leq 5$
in $0.1$ increments for each value of $N$. For each $\tilde{w}$, the value of
$tJ/\hbar$ was increased in $0.1$ increments until $F(t)$ was found to exceed
$F_{\rm min}=0.9$. Though IST is found to occur for many choices of $t$, only
the lowest
value of $t$ for each $\tilde{w}$ is shown in Fig. <ref>; the value
of $\tilde{w}$ that minimizes $t$ is shown in the inset. The data clearly show
that the minimum time (and the impurity coupling constants associated with
these) are nonmonotonic in $N$. There are intervals where the minimum time
appears to scale linearly with $N$, but these are interrupted and interspersed
with different trends. Likewise, the impurity coupling constants seem to scale
roughly as $\sqrt{N}$, but the data are not clean.
In principle, one could try to determine the conditions on $\tilde{w}$ that
yield the absolute lowest-slope curve. This would yield a slightly modified
spin chain where the IST time would scale linearly with length, albeit with
restrictions on the values of $N$ for which IST is possible. In this work,
however, we pursue a different tack; namely, determining values of $w$ valid
for any instance of $N$ and for which the IST time scales efficiently with $N$
(i.e. as a power-law with $N$ preferably with a low exponent). The analytical
treatment discussed in the next section addresses this strategy.
§ ANALYTICAL TREATMENT
While a complete analytical solution for arbitrary $w$ (or for arbitrary $w/J$
choosing $J$ as the characteristic energy scale) appears difficult to obtain,
approximate solutions may be obtained by solving the problem in the vicinity of
the additional site(s) and matching to the bulk solution. First consider the
left block in Fig. <ref>, consisting only of sites
$i=\{1,2,3,4,5,N+1\}$ (the block with sites $i\to N-i+1$, $i=1,2,\ldots,N$, and
$N+1\to N+2$ is wholly equivalent).
Expressing the left block solution as
\begin{equation}
|\psi_{\rm L}\rangle=\sum_{i=1}^4a_i|i\rangle+a_{N+1}|N+1\rangle,
\end{equation}
and operating with the left-block hopping Hamiltonian
\begin{eqnarray}
H_{\rm L}&=&J\sum_{i=1}^4\left(|i\rangle\langle i+1|+|i+1\rangle\langle i|
\right)
\nonumber \\
&+&w\left(|3\rangle\langle N+1|+|N+1\rangle\langle 3|\right)
\label{eq:hopHamL}
\end{eqnarray}
such that $H_{\rm L}|\psi_{\rm L}\rangle=\lambda|\psi_{\rm L}\rangle$, one
\begin{eqnarray}
&&\{a_1,a_2,a_3,a_4,a_{N+1}\}\nonumber \\
\frac{\tilde{\lambda}^2-1+\gamma}{\tilde{\lambda}},
\tilde{w}\frac{\tilde{\lambda}^2-1}{\tilde{\lambda}}\right\},
\label{eq:leftsol}
\end{eqnarray}
where $\gamma=1+\tilde{w}^2-(3+\tilde{w}^2)\tilde{\lambda}^2
+\tilde{\lambda}^4$, and $\tilde{w}=w/J$ and $\tilde{\lambda}=\lambda/J$ are
the rescaled values of $w$ and $\lambda$ relative to the characteristic energy
scale $J$.
The most important eigenvectors for
IST are those with large amplitudes on the first and last sites of the chain,
and by extension small amplitudes elsewhere. Small amplitude on the third site
can be ensured if $\tilde{\lambda}\sim 1$. Likewise, for the amplitude on the
fourth site to be small one requires
$\gamma/\tilde{\lambda}\approx\gamma\ll 1$; this
is possible if one chooses $\tilde{w}\gg 1$. More concretely, suppose one sets
$a_1=\alpha a_4$ assuming $\alpha\gg 1$; using the explicit coefficients in
Eq. (<ref>) one obtains
\begin{equation}
\tilde{w}=\sqrt{\tilde{\lambda}_c(\alpha\tilde{\lambda}_c^3-2\alpha
\tilde{\lambda}_c-1)\over\alpha(\tilde{\lambda}_c^2-1)}.
\end{equation}
Setting $\tilde{\lambda}_c=1-\epsilon$ and expanding to lowest order in
$\epsilon\sim 0$ gives
\begin{equation}
\tilde{w}\approx\pm\sqrt{1+\alpha\over 2\alpha\epsilon}
\approx\pm\frac{1}{\sqrt{2\epsilon}}.
\label{eq:wapprox}
\end{equation}
Note that necessarily $\epsilon>0$ which ensures that
$\tilde{\lambda}_c\lesssim 1$. Thus, the eigenvalue that ensures that the
amplitude on the endpoint(s) is resonantly enhanced is
\begin{equation}
\tilde{\lambda}_c\approx 1-\epsilon\approx 1-\frac{1}{2\tilde{w}^2}.
\label{eq:lambdaapprox}
\end{equation}
The value of $\tilde{w}$
can be positive or negative. Inserting this expression for $\tilde{\lambda}$
into Eq. (<ref>) and again assuming $\tilde{w}\gg 1$ gives
\begin{equation}
\{a_1,a_2,a_3,a_4,a_{N+1}\}\propto
\left\{\tilde{w}^2,\tilde{w}^2,-1,-\frac{1}{4},-\tilde{w}\right\}.
\label{eq:leftsolbigw}
\end{equation}
Note that the amplitudes on both the first and second sites of the chain are
much larger than those elsewhere, which implies that any possible IST could
equally originate at either of these sites. Thus, $\tilde{\lambda}_c$ is the
eigenvalue for a state strongly localized in the vicinity of the impurity.
Next consider the solutions $|\psi_{\rm 1D}\rangle$ for the bulk
one-dimensional chain $H_{\rm 1D}$, Eq. (<ref>):
\begin{equation}
\langle i|\psi_{\rm 1D}^{(n)}\rangle=\sin(k_ni);\;
\tilde{\lambda}_{\rm 1D}^{(n)}=2\cos(k_n).
\label{eq:bulksol}
\end{equation}
The reflection symmetry imposes the constraint that
$\langle i|\psi_{\rm 1D}^{(n)}\rangle
=\pm\langle N-i+1|\psi_{\rm 1D}^{(n)}\rangle$, i.e. that the solution is an
eigenstate of the parity operator. Setting
$\sin(k_ni)=-\sin(k_ni)\cos[k_n(N+1)]+\cos(k_ni)\sin[k_n(N+1)]$ for the
even-parity case gives the conditions $\cos[k_n(N+1)]=-1$ and
$\sin[k_n(N+1)]=0$ which is satisfied by $k_n=\frac{\pi(2n+1)}{N+1}$,
$n=1,2,\ldots$. Likewise, the odd-parity solution requires
$k_n=\frac{2\pi n}{N+1}$, so that overall $k_n=\frac{\pi n}{N+1}$,
$n=0,1,\ldots,N-1$. The invariance of the probability density under parity
implies that all eigenvalues appear in plus/minus pairs, as
$2\cos(k_n)\to-2\cos(k_n)$ for $n\to N+1-n$ ($k_n\to\pi-k_n$).
The bulk eigenvectors should automatically match the left-block
solution (<ref>) when $\tilde{w}=0$. Whether using the bulk solution
$\sin\left[\cos^{-1}\left(\tilde{\lambda}/2\right)i\right]$ from
Eq. (<ref>) or setting $\tilde{w}=0$ in Eq. (<ref>), one
obtains the same result:
\begin{equation}
\{a_1,a_2,a_3,a_4\}\propto\left\{1,\tilde{\lambda},
\tilde{\lambda}^2-1,\tilde{\lambda}\left(\tilde{\lambda}^2-2\right)\right\}.
\label{eq:solwzero}
\end{equation}
Alternatively, one can require that the bulk solution matches the amplitude on
the last site of the left block:
\begin{equation}
\sin(4k_n)=\frac{\sin(5k_n)(\tilde{\lambda}^2-1+\gamma)}{\gamma\tilde{\lambda}}
\end{equation}
Using trigonometric identities it is straightforward to verify that this
condition is (non-uniquely) satisfied by choosing $\tilde{\lambda}
=\tilde{\lambda}^{(n)}=2\cos(k_n)$, as expected for the one-dimensional chain.
Return again to the $\tilde{w}\gg 1$ case, but now for generic
$\tilde{\lambda}$ (keeping in mind that the $\tilde{\lambda}\sim 1$ are of
particular interest, and that the reflection symmetry of the site array implies
that all eigenvalues have negative counterparts
$\tilde{\lambda}\to -\tilde{\lambda}$). The parameter $\gamma$ in
Eq. (<ref>) becomes
$\gamma\to\tilde{w}^2\left(1-\tilde{\lambda}^2\right)$. The left-block
solution for $\tilde{w}\gg 1$ therefore becomes
\begin{eqnarray}
\nonumber \\
\frac{\tilde{w}(\tilde{\lambda}^2-1)}{\tilde{\lambda}}\Bigg\},\hphantom{aaa}
\label{eq:leftsolbigw2}
\end{eqnarray}
neglecting unimportant prefactors.
Comparison of Eqs. (<ref>) and (<ref>) reveals that
the $\tilde{w}=0$ and $\tilde{w}\gg 1$ wavefunctions match at all but the
fourth site in the chain: if $\tilde{\lambda}\sim 1$ then $a_{N+1}\to 0$. This
implies that the $\tilde{w}\gg 1$
eigenvalues almost coincide with those of the impurity-free chain. On the face
of it, this is disappointing, as the bare chain cannot support efficient IST.
Yet Eq. (<ref>) clearly states that there always exists one
eigenvalue for the localized state near the impurity that varies as
$1-1/2\tilde{w}^2$, approaching unity asymptotically. The implication is that
there must be (avoided) level crossings at finite $\tilde{w}$ for every
$\tilde{\lambda}^{(n)}$ that comes into resonance with the eigenvalue
$\tilde{\lambda}_c$ of the localized state. It turns out that efficient IST
hinges on the first of these avoided crossings.
Target eigenvalues are those in the vicinity of (but just below) unity,
$\tilde{\lambda}^{(n)}=2\cos(k_n)-\epsilon\sim 1^-$, so that $k_n\gtrsim\pi/3$.
That said, the $k_n=\pi n/(N+1)$ are discrete even in the limit of large $N$.
There are therefore three different cases to consider: $N=3m$ and $3m+1$
with $m=2,3,\ldots$, and $3m-1$ with $m=3,4,\ldots$. Consider first the $N=3m$
case. The wave vectors of interest are indexed by $n=m+r$:
\begin{equation}
\end{equation}
where $r=1,2,\ldots$ for $k_{m+r}>\pi/3$ to ensure that the associated
eigenvalues $\tilde{\lambda}_r<1$ (subscripts are now used to remind the reader
that $r=1$ corresponds to $n=3m+r$). The $r=1$ eigenvalue
$\tilde{\lambda}_1\approx 2\cos(k_{m+1})$ must cross the critical eigenvalue
$\tilde{\lambda}_c$ in Eq. (<ref>) at some value of
$\tilde{w}\gg 1$. The behavior
of the system for large chains is of particular interest; expanding around
$N=3m\gg 1$ and large $\tilde{w}$, the levels cross when
\begin{equation}
1-\frac{2\pi}{\sqrt{3}N}\approx 1-\frac{1}{2\tilde{w}^2},
\end{equation}
which yields the critical impurity coupling constant
\begin{equation}
\tilde{w}_c\approx\pm\frac{3^{1/4}}{2\sqrt{\pi}}\sqrt{N}\approx\pm 0.37\sqrt{N}.
\label{eq:weight0}
\end{equation}
Alternatively, consider the left-block eigenvector (<ref>), which
\begin{eqnarray}
\nonumber \\
+\mathcal{O}\left(\frac{\tilde{w}^2}{N^2}\right),\nonumber \\
\end{eqnarray}
Clearly, the expansions above are analytic only if $\tilde{w}$ varies with $N$
more slowly than $\sqrt{N}$ (or one could obtain a convergent series by
expanding the solution (<ref>) in $N/\tilde{w}$ for $\tilde{w}$ a
polynomial in $N$ with exponent greater than $1/2$). The critical case
$\tilde{w}_c=\alpha\sqrt{N}$ is therefore of particular interest. For large
$N$ one obtains
\begin{eqnarray}
\frac{\sqrt{3}}{2\left(1-\frac{4\pi\alpha^2}{\sqrt{3}}\right)},
\frac{\sqrt{3}}{2\left(1-\frac{4\pi\alpha^2}{\sqrt{3}}\right)},
\nonumber \\
\label{eq:resonance}
\end{eqnarray}
which matches the bulk solution at the fourth site. The amplitudes on the first
and second sites are strongly enhanced relative to the others if one sets
$1-\frac{4\pi\alpha^2}{\sqrt{3}}=0$ or $\alpha=3^{1/4}/2\sqrt{\pi}$, consistent
with Eq. (<ref>). Equivalently, the state (<ref>) has
maximal overlap with the state (<ref>) when the first two
amplitudes above equal $\tilde{w}^2=\alpha^2N$, which also occurs for
The $N=3m+1$ and $N=3m-1$ cases proceed analogously. For $N=3m+1$ for large
$N$ one obtains the critical impurity coupling constant
\begin{equation}
\tilde{w}_c\approx\pm\frac{3^{1/4}}{\sqrt{2\pi}}\sqrt{N}\approx 0.53\sqrt{N},
\label{eq:weight1}
\end{equation}
while for $N=3m-1$ in the same limit one obtains
\begin{equation}
\tilde{w}_c\approx\frac{1}{3^{1/4}\sqrt{2\pi}}\approx 0.30\sqrt{N}.
\label{eq:weight2}
\end{equation}
While all critical impurity coupling constants scale as $\sqrt{N}$, the
prefactor depends on the particular choice of $N$ (mod 3), $N+1$ (mod 3), or
$N-1$ (mod 3).
Return again to the $N=3m$ case. As $\tilde{w}$ is increased through the
critical value (<ref>), the $2\cos(k_{m+1})$ eigenvalue must exhibit
and an avoided crossing with $\tilde{\lambda}_c$, while its associated
eigenvector strongly mixes with the next odd-parity state with eigenvalue
$\tilde{\lambda}_3$. Likewise, the eigenvalue
$\tilde{\lambda}_2$ of the first relevant even-parity state should also follow
Eq. (<ref>) for large $\tilde{w}$, while strongly mixing with
the $\tilde{\lambda}_4$ state for intermediate $\tilde{w}$ at the second
avoided crossing in the vicinity of $2\cos(k_{m+3})$, etc. That said,
presumably the $\tilde{\lambda}_2$ state only mixes weakly with the
$\tilde{\lambda}_4$ state near the first avoided crossing (and of course not at
all with the $\tilde{\lambda}_1$ and $\tilde{\lambda}_3$ states due to parity),
which should occur for much larger values of $\tilde{w}$ than the second
avoided crossing. With this assumption, $\tilde{\lambda}_2$ follows
Eq. (<ref>) throughout the first level crossing.
By inference, therefore, only three states are relevant to the first avoided
crossing, corresponding to eigenvalues indexed by $r=1,2,3$. The same
phenomenon also applies to the $N=3m+1$ and $N=3m-1$ cases, but the avoided
crossing occurs for different $k$ labels and therefore at different energies.
Importantly, all three states involved in the avoided crossing at $\tilde{w}_c$
have strongly enhanced amplitude on the first and last site of the chain (as
well as the second and second-from last), of order unity after normalization.
All other states will be far off-resonant, and will have low amplitudes on the
endpoints proportional to the overall normalization constant for bulk
eigenvectors $\propto\sqrt{2/N}$. But the sum of outer products of eigenvectors
must resolve to the identity. This implies that as $N\to\infty$, no
off-resonant eigenvectors will contribute to the state transfer: the sum in
Eq. (<ref>) will only include resonant eigenvectors. Because only
three equally-spaced eigenvalues are involved (plus their negatives), the state
transfer must be asymptotically perfect.
Obtaining an analytical estimate of the energy splitting at the critical
impurity coupling constant is not as straightforward as it appears. The usual
method would be to start with eigenfunctions $|\psi^{(m+1)}_{\rm 1D}\rangle$
$|\psi^{(m+3)}_{\rm 1D}\rangle$ of the unperturbed Hamiltonian (<ref>)
and then calculate the mixing caused by the perturbation (<ref>),
i.e. the off-diagonal term of the mixing matrix
$\Delta\equiv\langle\psi^{(m+1)}_{\rm 1D}|H'|\psi^{(m+3)}_{\rm 1D}\rangle$.
The impurities have no support on the bare chain, however, so in principle the
energy splitting $\Delta=0$. One can nevertheless estimate $\Delta$ as
follows. The contribution to $\Delta$ from all the chain sites will be zero,
as the unperturbed eigenfunctions are orthogonal. At $\tilde{w}_c$, the
amplitude on the impurity site for the $k=m+1$ state is $-1/2\tilde{w}_c
=-\sqrt{\pi}/3^{1/4}\sqrt{N}$ after normalization (which is dominated by the
amplitudes on the first two and last two sites of the chain). Likewise, the
amplitude on the impurity site for the $k=m+3$ state is
$-\sqrt{2/N}(4\sqrt{\pi}/3^{3/4}\sqrt{N})=-4\sqrt{2\pi}/3^{3/4}N$ including the
normalization factor for $\sin(k_n)$ eigenfunctions. Because the action of the
Hamiltonian on this site returns the same amplitude (the energy is almost
unity), one obtains
\begin{equation}
\Delta\approx\frac{4\sqrt{2\pi}}{3^{3/4}N}
\frac{\sqrt{\pi}}{3^{1/4}\sqrt{N}}=\frac{4\sqrt{2}\pi}{3N^{3/2}}.
\label{eq:splitting}
\end{equation}
While the coefficient is probably not that accurate, the analytics suggest that
the energy splitting at the avoided crossing scales with the chain length as
$N^{-3/2}$. The IST time should therefore scale as $t_{\rm IST}\sim N^{3/2}$.
At this juncture the reader might well be wondering what is special about
adding impurities to the third and the third-from-last sites of the chain.
Suppose that the impurities were instead located on the second and
second-from-last sites of the chain. The left-block state, analogous to
Eq. (<ref>) is found to be
\begin{equation}
\{a_1,a_2,a_3,a_{N+1}\}
\tilde{w}\right\},
\label{eq:leftsol2}
\end{equation}
where $\gamma=2-\tilde{\lambda}^2-\tilde{w}^2$. The amplitude on the first site
can be made larger than in the bulk only if $\tilde{\lambda}\sim 1$ and
$\tilde{w}\sim 0$. The
second condition is unfortunately equivalent to the unmodified chain, and is
therefore not useful. Consider instead impurities on the fourth and
fourth-from-last sites of the chain. The left-block solution is now
\begin{eqnarray}
&&\{a_1,a_2,a_3,a_4,a_5,a_{N+1}\}\nonumber \\
\tilde{\lambda}^2-1,\tilde{\lambda}(\tilde{\lambda}^2-2),
\tilde{\lambda}^2-2+\gamma,\tilde{w}(\tilde{\lambda}^2-2)\right\},
\nonumber \\
\label{eq:leftsol4}
\end{eqnarray}
where $\gamma=3+2w^2-(4+w^2)\tilde{\lambda}^2+\tilde{\lambda}^4$. Following the
analysis above, small amplitude on the fourth and fifth sites requires
$\tilde{w}\gg 1$ and $\tilde{\lambda}\approx\sqrt{2}-1/2\sqrt{2}w^2$. To
leading order in $\tilde{w}$ one obtains
\begin{eqnarray}
&&\{a_1,a_2,a_3,a_4,a_5,a_{N+1}\}\nonumber \\
\label{eq:leftsol4b}
\end{eqnarray}
Just as for impurities on sites $i=3$ and $N-i+2$, there is a strong
enhancement of amplitude on the first and last sites. This enhancement is now
shared with four other sites ($i=2$, $3$, $N-1$, and $N-2$), however,
decreasing the total amplitude available on the target sites. This trend
continues as the impurities move further from the ends of the chain. Thus,
adding impurities to the third and third-from-last sites is optimal.
§ NUMERICAL RESULTS
The strategy pursued in this work is to determine if there exist
values of $\tilde{w}(N)$ that allow APST to occur for all $N$, but perhaps not
at the absolute minimum time allowable. Using the analytical results of the
previous section as a guide, for particular values of $\tilde{w}$ one expects
that for eigenvalues $\tilde{\lambda}^{(n)}\lesssim 1$ of the Hamiltonian the
associated eigenvectors $|\psi^{(n)}\rangle$ will have a large overlap with the
first and last sites of the chain. If the eigenvalues are reverse ordered so
that $\tilde{\lambda}^{(n)}\geq\tilde{\lambda}^{(n+1)}$ for $n=1,2,\ldots,N-1$,
then there exist values of $n\gtrsim N/3$ where
$\tilde{\lambda}^{(n)}\lesssim 1$. Define $N\equiv 3m+p$, where
$p\in\{-1,0,1\}$. Then one can
define $\tilde{\lambda}_q=\tilde{\lambda}^{(q+m)}$ which are all less than
unity for $q=1,2,\ldots$, and their associated eigenvectors $|\psi_q\rangle$.
Then IST should result for values of $\tilde{w}$ where the eigenvalues and
eigenvectors satisfy the following two criteria:
\begin{eqnarray}
\left|\frac{\Delta_2-\Delta_1}{\Delta_1}\right|&\leq&\epsilon;
\label{eq:gapdiff}\\
\label{eq:overlaps}
\end{eqnarray}
where the two successive eigenvalue gaps are
$\tilde{\Delta}_1=\tilde{\lambda}_1-\tilde{\lambda}_2$ and
$\tilde{\Delta}_2=\tilde{\lambda}_2-\tilde{\lambda}_3$, and $\epsilon\ll 1$ and
$f\lesssim 1$ are free parameters.
The first few eigenvalues
$\tilde{\lambda}_q=\lambda^{q+m}/J$ smaller
than unity ($q=1,2,\ldots,6$) are shown as a function of the impurity coupling
parameter $\tilde{w}=w/J$ for chain length $N=3m=501$ (black lines). Also shown
is the left-block eigenvalue $\tilde{\lambda}=\lambda/J=1-1/2\tilde{w}^2$
(dotted curve). The first avoided level crossing, closest to unity, is
magnified in the inset.
The Hamiltonian $H$ was diagonalized for each chain length $N$ in the range
$6\leq N\leq 501$. Figure <ref> shows the representative behavior of
the first few eigenvalues less than unity as a function of the impurity
coupling $\tilde{w}=w/J$ for $N=3m=501$. The salient features predicted by the
analytical treatment are readily observed here. The bare-chain eigenvalues
$\tilde{\lambda}_q^0=2\cos[\pi(q+m)/(N+1)]$ (for $\tilde{w}=0$) are crossed by
the eigenvalue $\tilde{\lambda}_c=1-1/2/\tilde{w}^2$ of the localized state at
finite $w$, (almost) recovering their values for large $\tilde{w}$. The $q=1$
and $q=2$ eigenvalues both follow $\tilde{\lambda}_c$ for large $\tilde{w}$,
while the $q=3$ eigenvalue approaches $2\cos[\pi(1+m)/(N+1)]$ in the same
The strong mixing between the localized and extended bulk eigenvectors gives
rise to avoided crossings, and the first such crossing (closest to unity) is
shown in the inset of Fig. <ref>. As expected, the $q=2$ eigenvalue
closely follows $\tilde{\lambda}_c$ through the first avoided crossing, as it
only weakly mixes with the $q=4$ eigenvalue here. At the value of $\tilde{w}$
where $\tilde{\lambda}_c\approx\tilde{\lambda}_2=\tilde{\lambda}_1^0$, the
$\tilde{\lambda}_1$ and $\tilde{\lambda}_3$ eigenvalues are split equally above
and below $\tilde{\lambda}_2$. Thus, $\tilde{w}_c$ both defines the point at
which $\tilde{\lambda}_c=\tilde{\lambda}_1$ and the point at which the energy
gaps coincide, $\tilde{\Delta}_1=\tilde{\Delta}_2\equiv\tilde{\Delta}$.
Figure <ref> also suggests that there are multiple values of the
impurity hopping parameter that give rise to eigenvalue resonances. The second
avoided crossing occurs when $\tilde{\lambda}_c=\tilde{\lambda}_2$. For the
$N=501$ case displayed here, the analog of result (<ref>) for the
second avoided crossing is $\tilde{w}_c'=(3^{1/4}/\sqrt{10\pi})\sqrt{N}
\approx 0.23\sqrt{N}$. While the qualitative behavior of the eigenvalue
splitting is similar, one would expect a lower IST output fidelity as the
maximum amplitude on the endpoint sites is now approximately $63\%$ lower than
for the first avoided crossing according to Eq. (<ref>).
The numerical value of the impurity coupling constant $\tilde{w}=w/J$
ensuring that the eigenvales at the first level crossing are equally spaced are
shown as a function of the chain length $N$. Top, middle, and lower curves
correspond to $N=3m+1$, $3m$, and $3m-1$ ($m$ integer), respectively. All
curves closely follow $\tilde{w}\sim\sqrt{N}$ for large $N$.
Eq. (<ref>) specifies that the spacing between successive
eigenvalues just less than unity be equal to within some tolerance $\epsilon$.
Figure <ref> shows the value of impurity parameter
$\tilde{w}=w/J$ satisfying Eq. (<ref>) at the first avoided crossing
to a tolerance $\epsilon=10^{-10}$, for $6\leq N\leq 501$. There are three
distinct curves, depending on the value of $N$. The lowest, middle, and upper
curves correspond to $N+1$ (mod 3), $N$ (mod 3), and $N-1$ (mod 3),
respectively. For the largest chain sizes considered, in the range
$400\leq N\leq 500$, each of these curves is well fit (correlation coefficient
$>0.999$) by a power law that closely resembles the analytical results,
Eqs. (<ref>), (<ref>), and (<ref>),
respectively: $\tilde{w}\approx 0.48N^{0.51}$ for $N=3m+1$,
$\tilde{w}\approx 0.33N^{0.52}$ for $N=3m$, and $\tilde{w}\approx 0.27N^{0.52}$
for $N=3m-1$. In all cases, the results are compatible with a $\sqrt{N}$
scaling for large $N$. That said, the prefactors obtained numerically are all
found to be below the analytical preductions by approximately $10\%$; it is
possible that the correspondence between analytics and numerics would tighten
up for larger $N$.
The numerical estimate of the imperfect state transfer time
$t_{\rm IST}$, based on the value of the resonant energy gaps, is shown as a
function of the chain length $N$. Top, middle, and lower curves correspond to
$N=3m+1$, $3m$, and $3m-1$ ($m$ integer), respectively. All curves closely
follow $t_{\rm IST}\sim N^{3/2}$ for large $N$.
Under the assumption that only the first three eigenvectors $|\psi_q\rangle$
with eigenvalues $\lambda_q$ just below unity (and their negative counterparts)
contribute to the transfer dynamics, the IST time can be predicted using
Eq. (<ref>). Following the discussion in Sec. <ref>, imperfect
site periodicity occurs at a time $t=2\pi\hbar/(\lambda_1-\lambda_2)
=2\pi\hbar/\Delta_1$, and parity conservation implies that the imperfect state
transfer time is half this, $t_{\rm IST}\equiv\pi\hbar/\Delta_1
=(\pi\hbar/J)/\tilde{\Delta}_1$. In rescaled time units $t=(\hbar/J)\tilde{t}$,
one obtains $\tilde{t}_{\rm IST}\equiv\pi/\tilde{\Delta}$, where
$\tilde{\Delta}=\tilde{\Delta}_1=\tilde{\Delta}_2$. The state transfer times
$t_{\rm IST}$ are shown as a function of the chain
length in the range $10\leq N\leq 501$ in Fig. <ref>. These times
are based on the energy gaps $\tilde{\Delta}_1=\tilde{\Delta}_2$ at the
critical weight plotted in Fig. <ref>. As was the case for
the critical weights, the IST time scales differently depending on the value of
$N$ (mod 3). For $N=3m$, the numerical results are best fit by the function
$\tilde{t}_{\rm IST}\approx 0.29N^{1.53}$ for large integer m; for $N=3m+1$ and
$3m-1$ one obtains
$\tilde{t}_{\rm IST}\approx 0.57N^{1.53}$ and
$\tilde{t}_{\rm IST}\approx 0.20N^{1.52}$, respectively. The power laws are all
consistent with a $N^{3/2}$ scaling of time, as predicted by the analytics
discussed in the previous section. The prefactors also appear to scale roughly
with those for the critical weights.
It is worthwhile to investigate the IST dynamics governed by
Eq. (<ref>) more closely. Keeping in mind that $|\psi_1\rangle$ and
$|\psi_3\rangle$ have even parity while $|\psi_2\rangle$ has odd parity (and
vice versa for $\lambda_q\to -\lambda_q$), so that for example
$\langle\psi_1|1\rangle=\langle\psi_1|N\rangle$ while
$\langle\psi_2|1\rangle=-\langle\psi_2|N\rangle$, one obtains
\begin{eqnarray}
\langle N|e^{-iHt/\hbar}|1\rangle&\approx&-2i\sin\left[\left(\lambda_2+\Delta
\right)t/\hbar\right]\left|\langle\psi_1|1\rangle\right|^2\nonumber \\
\nonumber \\
\langle\psi_3|1\rangle\right|^2,
\label{eq:ISTa}
\end{eqnarray}
keeping only resonant eigenvectors in the sum. Choosing $t$ such that the
coefficients of each $\left|\langle\psi_q|1\rangle\right|^2$ term are equal,
one obtains
$t_{\rm IST}=\pi\hbar/\Delta$, as expected. Then Eq. (<ref>) becomes
\begin{eqnarray}
\langle N|e^{-iHt/\hbar}|1\rangle&=&2i\sin\left(\frac{\pi\lambda_2}{\Delta}
\right)\sum_{q=1}^3\left|\langle\psi_q|1\rangle\right|^2\nonumber \\
\label{eq:ISTb}
\end{eqnarray}
where the second line is obtained by assuming that only these eigenvectors
(and their negative-eigenvalue counterparts) resolve the identity.
Equation (<ref>) appears to suggest that the maximum fidelity is
$F(t_{\rm IST})=\sin^2(\pi\lambda_2/\Delta)\approx\sin^2(\pi/\tilde{\Delta})$.
The interference arises from the fact that while the first three eigenvalues
below unity become approximately evenly spaced by $\Delta$ at resonance, the
corresponding second set of approximately evenly spaced eigenvalues above $-1$
are not necessary an integer number of $\Delta$ away from the first set.
The IST fidelity $F(t)$ is plotted as a function of time for $N=501$
at the critical impurity coupling constant $\tilde{w}\approx 8.2$. Dots
correspond to numerical data while the solid lines correspond to the envelope
of the output fidelity including only contributions from the critical
eigenvectors. The slow increase in fidelity through the IST time
$t_{\rm IST}=\pi/\tilde{\Delta}\approx 3867$ is shown in (a). A close-up of
the behavior in the vicinity of $t_{\rm IST}$ is shown in (b). Note that the
dots are so closely spaced here that they resemble an oscillating line, whereas
the solid line appears almost horizontal.
High-fidelity IST is in fact possible if the time is chosen to be slightly
above or below $\pi/\Delta$. For times in the vicinity of $t\sim t_{\rm IST}$,
Eq. (<ref>) becomes approximately
\begin{eqnarray}
\langle N|e^{-iHt/\hbar}|1\rangle&\approx&\sum_{q=1}^3\Big(
-e^{-i\lambda_2t/\hbar}\left|\langle\psi_q|1\rangle\right|^2\nonumber \\
&&\qquad + e^{i\lambda_2t/\hbar}\left|\langle\psi_q'|1\rangle\right|^2\Big).
\label{eq:ISTc}
\end{eqnarray}
Here, $|\psi_q'\rangle$ are the eigenvectors corresponding to the first
eigenvalues above $-1$ (i.e. for $\lambda_q\to -\lambda_q$); note that
$\langle 1|\psi_q'\rangle=\langle 1|\psi_q\rangle$ while
$\langle N|\psi_q'\rangle=-\langle N|\psi_q\rangle$. The right-hand side of
Eq. (<ref>) is proportional to unity if time is chosen to be
$t=(2n-1)\pi\hbar/2\lambda_2$, or $\tilde{t}=-\pi/2+r\pi$ where $r$ is an
arbitrary integer. For times near $t_{\rm IST}$, the probability on the output
site varies from zero to near unity with a period $\pi\ll t_{\rm IST}$. One can
therefore choose a time in the vicinity of $t_{\rm IST}$ at which the fidelity
should approach unity.
Figure <ref> shows the IST fidelity $F(t)$, defined in
Eq. (<ref>), as a function of time for the particular case of
$N=501$ using the optimal impurity coupling constant
$\tilde{w}_c\approx 8.197$. As expected, $F(t)$ reaches a maximum value at a
time near $\tilde{t}_{\rm IST}\approx 3867.44$, though the function oscillates
rapidly throughout this slow variation. The behaviour of the fidelity including
only resonant eigenvectors in the sum (<ref>) is shown for comparison;
only the envelope of the fidelity is plotted for clarity, as this exhibits the
same fast oscillation of the full data. Figure <ref>(a) clearly
shows that the time evolution of the output fidelity is governed almost
completely by the resonant eigenvectors. A close-up of the time sequence in the
vicinity of $t\sim t_{\rm IST}$ reveals the fast oscillation of $F(t)$; the
period is found to be $\tilde{T}_{\rm fast}\approx 3.16$ which is close to the
predicted value of $\pi$. While $F(t_{\rm IST})\approx 0.858$, the output
fidelity actually attains a maximum $F_{\rm max}\approx 0.975$ at the slightly
longer time $\tilde{t}=3867.70$. This maximum is attained for many
$\tilde{T}_{\rm fast}$ periods in the vicinity of $\tilde{t}_{\rm IST}$. In
fact, the maximum fidelity for the full dynamics slightly exceeds the value
$F\approx 0.964$ obtained from including only the resonant eigenvectors.
(color online) The maximum fidelity $F_{\rm max}$ for the state
transfer between endpoints is shown as a function of the chain length $N$. Red,
black, and blue curves correspond to $N=3m+1$, $3m$, and $3m-1$,
$3\leq m\leq 333$, respectively. Solid lines depict the exact fidelity while
dashed lines represent the result including only the six resonant eigenvectors.
The maximum fidelity $F_{\rm max}$ for quantum state transfer is shown as a
function of $N$ in Fig. <ref> for $10\leq N\leq 1000$. The data are
obtained by scanning the fidelity in the temporal region
$\tilde{t}_{\rm IST}-3\tilde{T}_{\rm fast}\leq\tilde{t}\leq
\tilde{t}_{\rm IST}+3\tilde{T}_{\rm fast}$ and recording the maximum
result for each $N$. The exact results $F_{\rm max}$, where all eigenvectors
are included in the sum (<ref>), are shown as solid lines; the
fidelities $F_{\rm max}'$ when the sum is restricted only to resonant
eigenvectors is shown as dashed lines. While $N$-dependent oscillations are
evident in the exact results, the amplitudes decrease and the wavelengths
increase with $N$; more important, though, their centers consistently follow
the restricted fidelity curves. The restricted fidelities therefore provide an
accurate representation of the exact fidelities in the thermodynamic limit
$N\to\infty$. The values of restricted fidelities for large $N$ in the range
$800\leq N\leq 1000$ are found to follow
\begin{equation}
F_{\rm max}'\approx
\begin{cases}
1-\frac{2.34}{N^{1.01}} & N=3m+1;\cr
1-\frac{11.4}{N^{1.04}} & N=3m;\cr
1-\frac{17.3}{N^{0.98}} & N=3m-1,\cr
\end{cases}
\end{equation}
with suitably chosen integer $m$. Thus, the IST fidelity approaches unity in
the thermodynamic limit, i.e. the quantum state transfer is asymptotically
perfect. According to the numerics, the state transfer error scales as
$1-F\propto N^{-1}$ for large $N$.
§ DISCUSSION AND CONCLUSIONS
In this work we have shown that asymptotically perfect quantum state transfer
is possible in uniform chains that have been modified by the addition of two
impurites, coupled to the uniform chain at the third and third-from-last sites
with strength $w$. Choosing $w\propto\sqrt{N}$, the state localized in the
vicinity of the impurity can be tuned into resonance with chain extended
states. The associated avoided level crossing gives rise to eigenstates with
large overlaps with the chain endpoints and with eigenvalues whose spacings
become approximately equal. The approximate linear spectrum together with
reflection symmetry yields approximately perfect state transfer, in a time
that scales efficiently with length, as $t_{\rm IST}\propto N^{3/2}$. Indeed,
the fidelity is found to approach unity in the thermodynamic limit
$N\to\infty$, with error scaling as $1-F\propto N^{-1}$. To our knowledge, this
is the only configuration with no external time-dependent or local control,
where a uniform chain can be made to transfer quantum information perfectly in
the limit of large system size,
While the central insights obtained from the analytical investigations are
validated by the explicit calculations, the numerical results reveal additional
information and display some important features. First and foremost, the
detailed dependence of the energy splitting on $N$ at resonance (equivalently
the amplitude of the $t_{\rm IST}\propto N^{3/2}$ scaling) was only readily
available numerically. Second, the exact time-dependence of the output
probability was found to oscillate rapidly (with period $\pi\hbar/J$) in
addition to the slow evolution toward maximum fidelity in the vicinity of
$t_{\rm IST}$, independent of $N$. This means that in a practical experiment
(with $N$ large but fixed)
the timing would have to be tested over a range of times $|t-t_{\rm IST}|\leq
(\pi/2)(\hbar/J)$ prior to using this device to transfer unknown quantum
information. Third, the exact value of the maximum fidelity is found to follow
the value obtained by including only the critical eigenvectors, but for smaller
$N$ it displays pronounced oscillations. The amplitude and frequency of these
oscillations decreases steadily with $N$, so that in the thermodynamic limit
the maximum fidelity is completely dominated by the resonant eigenvectors.
Given that the high-fidelity transfer is a direct consequence of a resonance
between the localized and extended states, one might expect the model to be
robust against random small errors in the chain coupling constants around $J$.
The errors would shift the frequencies of the extended states, so that a new
value of $w$ would need to be found to bring them back into resonance. While
this is possible in principle, in practice finding the best value of $w$ and
time could be difficult; if the errors are time-dependent the situation is even
worse. Unfortunately, numerical calculations suggest that the value of $F$
falls precipitously with noise if $w$ and $t$ are both fixed at their optimal
noise-free value. Given $J_{i,i+1}=J\pm\delta J_{i,i+1}$ with
random values $|\delta J_{i,i+1}|\leq x$, we find for $N=501$ that the average
fidelity drops to $\overline{F}\sim 0.1$ for $x\approx 0.01$.
Previous studies of perfect quantum state transfer have shown that if the
evolution of a particle on a graph exhibits PST then so will its evolution on
Cartesian powers of this graph <cit.>. Because the Cartesian square and
cube of the
uniform chain are two and three-dimensional lattices, respectively, one could
in principle extend the current model to exhibiting PST from corner to corner
of regular lattices in any dimension. That said, the two impurities in one
dimension translate to an unwieldy $4N$ impurities in two dimensions. Rather,
one could envisage arranging a sequence of impurities forming a half box of
length three centered at each corner. This would ensure the presence of a
localized state near the endpoints, which could again be tuned into resonance
through a suitable adjustment of the impurity coupling parameters. We hope to
explore this idea further in future work.
This work was supported by the Natural Sciences and Engineering Research
Council of Canada.
Fragments from Introduction:
Uniform chains: fidelity either drops off <cit.> or time grows
exponentially at fixed fidelity.
XY chains with weakly coupled first and last
qubits <cit.>. Venuti2007 was interested in generating
long-range entanglement but used the same model.
Feldman also noted XY and XXZ Hamiltonians give the same
result. Later papers also note that these are quite robust against noise and
the time scales efficiently. Yao2013 also uses a local magnetic field.
Apollaro2012 uses two weak links at either end and finds maximum fidelity of
previous results. Likewise Banchi looks only at maximum fidelity for the weak
links at the end points and finds optimal value of weak link and gives finite
APST (not approaching unity), though time scales linearly with n; interestingly
requires some channel initialization.
Application of magnetic fields near sender and
receiver <cit.>
(interestingly for Lorenzo2015 magnetic fields are on sites 3 and N-2 while
they were on sites 2 and N-1 for Lorenzo2013)
Dynamical control of (uniform) coupling (XXZ) <cit.>;
using measurements <cit.>; time-dependent control of first and
last coupling constants <cit.>
All possible coupling constants for PST <cit.>; analyzed a variety for
APST <cit.>
Show that APST is only possible for particular lengths of uniform
chains <cit.>; also shows that the APST scales exponentially in the
path length
, efficient avenues for transferring
the quantum information from one spin to In quantum computers, information is encoded into
quantum bits, also known as qubits. The ability to transfer these qubits
down some optical circuit can be very helpful in making quantum computers.
When a qubit transfers from one location on the circuit to a different location,
this is known as a state transfer. The ideal case is that these qubits will transfer
with no interference or disturbance so that they arrive at the desired location with
a wave output probability of 1; this is called Perfect State Transfer, denoted as PST. However it is
known that for linear paths of size greater than three, PST does not occur <cit.>.
When some of the information is lost in the transfer, we get Almost Perfect State Transfers,
denoted as APST.
Although transfer times can be estimated in linear path graphs, they grow exponentially
worse as path sizes increase. Therefore we want to find graphs that displayed APST but with a
better time growth rate; ideally the time will grow linearly as a function of
graph size. It is found that symmetrical tree graphs with stars attached to certain vertices
gives APST with a slower time growth rate <cit.>. From the symmetrical
star tree graphs, it can be shown that the stars can be condensed into one weighted
leaf at the vertices that the stars are attached. This design is easier to accomplish
experimentally but the weights of the edges would be restricted. Therefore the question
of how would APST change if the weights are continuous instead of integer steps will
be addressed in Section refsec:weighted-tree-graphs.
Assuming that only these three states have appreciable support on the first
(and last) sites of the chain, the criterion for APST is that the first
eigenvalue gap $\Delta_{12}\equiv\tilde{\lambda}_2-\tilde{\lambda}_1$ equals
the second
eigenvalue gap $\Delta_{23}\equiv\tilde{\lambda}_3-\tilde{\lambda}_2$ within
the level crossing.
The determination of $\tilde{\lambda}$ requires that the left-block
states (<ref>) match the bulk solutions
$|\psi_{\rm 1D}^{(n)}\rangle$ above. The equation to be satisfied is
\begin{equation}
\sin(4k_n)=\frac{\sin(5k_n)}{\gamma}\frac{\tilde{\lambda}^2-1+\gamma}
\label{eq:matchcondition}
\end{equation}
Next consider the $\tilde{w}\neq 0$ case.
It remains to be shown that by matching the left-block solution to the bulk
one may obtain the dependence of the eigenvalues with $\tilde{w}$. With the
expectation that the eigenvalues for $\tilde{w}\gg 1$ will have the form
$\tilde{\lambda}^{(n)}=2\cos(k_n)-\epsilon$ for $\epsilon\to 0^+$ and for $k_n$
of interest, then one can expand $a_4$ in Eq. (<ref>) around
$\epsilon=0$ and $\tilde{w}^{-1}=0$. To lowest order in both, one obtains
\begin{equation}
a_4\approx\frac{a_5{\rm sec}(k_n)}{4}\left[2+\epsilon{\rm sec}(k_n)\right]
\left(1-\frac{1}{\tilde{w}^2}\right).
\end{equation}
The value of $\epsilon$ is found by setting $a_4=\sin(4k_n)$ and
$a_5=\sin(5k_n)$ in the above equation. To order in $\tilde{w}^{-2}$, one then
obtains the system eigenvalues in the limit of large $\tilde{w}$:
\begin{equation}
\tilde{\lambda}^{(n)}\approx 4\cos(k_n)-4\frac{\cos^2(k_n)\sin(4k_n)}
\end{equation}
Target eigenvalues are those in the vicinity of (but just below) unity,
$\tilde{\lambda}^{(n)}=2\cos(k_n)-\epsilon\sim 1^-$, so that $k_n\gtrsim\pi/3$.
(Keep in mind that the reflection symmetry of the site array implies that all
eigenvalues have negative counterparts
$\tilde{\lambda}^{(n)}\to -\tilde{\lambda}^{(n)}$.) That said, the
$k_n=\pi n/(N+1)$ are discrete even in the limit of large $N$. There are
therefore three different cases to consider: $N=3m$, $3m+1$, and $3m+2$, where
$m=2,3,\ldots$. Consider the $N=3m$ case first. The wave vectors of interest
are indexed by $n=m+r$:
\begin{equation}
\end{equation}
where $r=0,1,2,\ldots$ for $k_r\gtrsim\pi/3$ to ensure that
$\tilde{\lambda}_r<1$. Under the assumption that the presence of $\tilde{w}$
preserves the reverse ordering of the eigenvalues with index $r$,
$\tilde{\lambda}_r\geq\tilde{\lambda}_{r+1}$. For large $m$ or $N$, one
\begin{equation}
\tilde{\lambda}_r\approx 1-\frac{1}{\tilde{w}^2}
\label{eq:eigssortof}
\end{equation}
According to Eq. (<ref>), the full spectrum must include the
eigenvalue $\tilde{\lambda}\approx 1-1/2\tilde{w}^2$, corresponding to the
$\tilde{\lambda}_r$ closest to unity, i.e. $r=0$. This can be guaranteed only
by setting the value of the weight to be
\begin{equation}
\tilde{w}\approx\frac{\sqrt{N}}{3^{1/4}\sqrt{2\pi}}\approx 0.3\sqrt{N}
\end{equation}
in the limit of large $N$.
The cases $N=3m+1$ and $N=3m+2$ cases proceed
analogously. For $N=3m+1$ for
large $N$ one obtains
\begin{equation}
\tilde{w}\approx\frac{\sqrt{N-1}}{3^{1/4}\sqrt{\pi}}\approx 0.43\sqrt{N},
\label{eq:weight1}
\end{equation}
while for $N=3m+2$ in the same limit one obtains
\begin{equation}
\tilde{w}\approx\frac{\sqrt{N-2}}{3^{3/4}\sqrt{\pi}}\approx 0.25\sqrt{N}.
\label{eq:weight2}
\end{equation}
While all weights scale as $\sqrt{N}$, the prefactor depends on the particular
choice of $N$ (mod 3), $N+1$ (mod 3), or $N+2$ (mod 3).
This yields the gap between the first and second
\begin{equation}
\Delta_1\approx\frac{3\sqrt{3}\pi}{N}.
\end{equation}
\begin{equation}
\tilde{\lambda}^{\left(N/3+r\right)}\approx 1-\frac{(3r+1)\sqrt{3}\pi}{N}.
\end{equation}
The state transfer time is found from the eigenvalue spacing,
\begin{equation}
t_{\rm APST}\approx\left(\frac{\hbar}{J}\right)\frac{N}{3\sqrt{3}}\approx\left(
\frac{\hbar}{J}\right)0.19N
\end{equation}
in the limit of large $N$. The analytics therefore suggest that almost perfect
transfer can be achieved in this system in a time linear in the chain length,
choosing the coupling to the additional spins to scale as the square root of
the length. Counterintuitively, the average transfer time per site is over
16 times lower than the PST time between only two sites $(\hbar/J)\pi$.
The cases $N=3m+1$ and $N=3m+2$ cases proceed analogously. For $N=3m+1$ for
large $N$ one obtains
\begin{eqnarray}
\tilde{\lambda}^{\left((N-1)/3+r\right)}&\approx&1-\frac{\sqrt{3}\pi(3r-1)}{N},
\end{eqnarray}
while for $N=3m+2$ in the same limit one obtains
\begin{eqnarray}
\tilde{\lambda}^{\left((N-2)/3+r\right)}&\approx&1-\frac{3\sqrt{3}\pi}{N}r.
\end{eqnarray}
In all cases the energy spacing remains the same, so that
$t\approx(\hbar/3\sqrt{3}J)N$. While all weights scale as $\sqrt{N}$, the
prefactor depends on the particular choice of $N$ (mod 3), $N+1$ (mod 3), or
$N+2$ (mod 3). It is important to underline that eigenvalue spacing is
approximately linear only for eigenvalues close to (but less than) unity.
Substituting the value of $\tilde{w}$ from Eq. (<ref>) into the
original matching condition (<ref>) yields
$\frac{4n\pi}{N}=\frac{4n\pi}{\tilde{\lambda}N}$. This is clearly only valid
for eigenvalues close to unity, consistent with the approximations made above.
Return to the left-block solution (<ref>), which for large $N$
\begin{equation}
\{a_1,a_2,a_3,a_4,a_{N+1}\}\propto\left\{\frac{1}{\tilde{\lambda}^2-1},
\frac{\tilde{\lambda}}{\tilde{\lambda}^2-1},1,\frac{1}{\tilde{\lambda}},
\frac{\tilde{w}}{\tilde{\lambda}}
\right\}
\label{eq:leftsolfin}
\end{equation}
for all $\tilde{\lambda}=\tilde{\lambda}^{(n)}$ and choices of $N$. The
enhancement of amplitude on the chain endpoints relative to those in the bulk
only exists for a small range of eigenvalues in the vicinity of unity,
applicable to only a small fraction of the eigenvectors. Consider for example
the $N=3m$ case above. After normalizing the full eigenvector (the sum of the
bulk probabilities is $N/2$), one obtains to leading order in $N\gg 1$
\begin{eqnarray}
&&\{a_1,a_2,a_3,a_4,a_{N+1}\}\nonumber \\
\frac{(3r+1)\sqrt{3}\pi}{N},\frac{3r+1}{2\tilde{w}}\right\}\nonumber \\
\label{eq:vecsclose}
\end{eqnarray}
The sum of eigenvector outer products should resolve to the identity, which
suggests that there exist at most four eigenvectors resembling
Eq. (<ref>). In fact, there must be three such vectors, and three
others for $\tilde{\lambda}=-1+\epsilon$ of the form $\{a_1,a_2,\ldots\}=
\{1/2,-1/2,\ldots\}$. The remaining eigenvectors should have asymptotically
vanishing overlap with the endpoints, so that in the large-$N$ limit the state
transfer unitary operator will be so strongly dominated by the linear-spaced
eigenvalues in the vicinity of (plus or minus) unity that the state transfer
should be asymptotically perfect.
|
1511.00071
|
[2010]Primary: 11M32, Secondary: 11F68
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
We study a double Dirichlet series of the form $\sum_{d}L(s,\chi_{d}\chi)\chi'(d)d^{-w}$,
where $\chi$ and $\chi'$ are quadratic Dirichlet characters with
prime conductors $N$ and $M$ respectively. A functional equation
group isomorphic to the dihedral group of order 6 continues the function
meromorphically to $\mathbb{C}^{2}$. A convexity bound at the central
point is established to be $(MN)^{3/8+\varepsilon}$ and a subconvexity
bound of $(MN(M+N))^{1/6+\varepsilon}$ is proven. The developed theory is used
to prove an upper bound for the smallest positive integer $d$ such
that $L(1/2,\chi_{dN})$ does not vanish, and further applications of subconvexity bounds to this problem are presented.
§ INTRODUCTION
The use of multiple Dirichlet series in number theoretic problems,
as well as the intrinsic structure they possess, have evolved the
subject to be a study in its own right. Indeed, Weyl group multiple
Dirichlet series are considered fundamental objects whose structures
are intimately linked to their analytic features <cit.>.
In this paper we are interested in the size of a certain type of double
Dirichlet series outside the region of absolute convergence, and some
of its arithmetic consequences. For ordinary $L$-functions there
is the well-known concept of convexity bound as a generic upper bound
in the critical strip, and improvements in the direction of the Lindelöf
hypothesis often have deep implications. In the situation of multiple
Dirichlet series, the notion of convexity is not so obvious to formalize.
Nevertheless there is a fairly natural candidate for a “trivial”
upper bound of a multiple Dirichlet series, and it is an important
problem to improve this estimate. This has first been carried out in <cit.> for the Dirichlet series defined by
\[
Z(s,w)=\zeta^{(2)}(2s+2w-1)\sum_{d\text{ odd}}L^{(2)}(s,\chi_{d})d^{-w}
\]
at $s=1/2+it$, $w=1/2+iu$ simultaneously in the $t,u$-aspect, where
the $^{(2)}$ superscript denotes that we are removing the Euler factor
at $2$, and $\chi_{d}$ is the Jacobi symbol $(\frac{d}{\cdot})$.
Here we study a non-archimedean analogue. We can expand the $L$-function
in the summand as a sum indexed by $n$. Instead of twisting by $n^{it}$,
$d^{iu}$, we twist by quadratic primitive Dirichlet characters $\chi(n)$,
$\chi'(d)$ of conductors $N,M$, respectively. More precisely, we
\[
\]
for sufficiently large $\Re s,\Re w$, where we write $d=d_{0}d_{1}^{2}$
with $d_{0}$ squarefree. The $P$ factors are a technical complication
necessary in order to construct functional equations, which we accomplish
thus: if we expand the $L$-function in the numerator and switch the
summation, we can relate it to a similar double Dirichlet series with
the arguments and the twisting characters interchanged, with modified
correction factors. Next, we have a functional equation obtained by
application of the functional equation for Dirichlet $L$-functions,
which maps $(s,w)$ to $(1-s,s+w-{\textstyle \frac{1}{2}})$. Applying
the switch of summation formula to this, we obtain a functional equation
mapping $(s,w)$ to $(s+w-{\textstyle \frac{1}{2}},1-w)$. These continue
$Z$ to the complex plane except for the polar lines $s=1$, $w=1$,
and $s+w=3/2$. They also generate a group isomorphic to $D_{6}$,
the dihedral group of order 6, which is isomorphic to the Weyl group
of the root system of type $A_{2}$. For this reason $Z(s,w;\chi,\chi')$
is considered a type $A_{2}$ multiple Dirichlet series <cit.>.
Interestingly, such objects are Whittaker coefficients in the Fourier
expansion of a $GL(3)$ Eisenstein series of minimal parabolic type
on a metaplectic cover of an algebraic group whose root system is
the dual root system of $A_{2}$ <cit.>. In fact, it is
conjectured that a similar relationship is true between multiple Dirichlet
series of type $A_{r}$ and $GL(r+1)$ Eisenstein series. Going down
in dimension, type $A_{2}$ multiple Dirichlet series can be constructed
from 1/2-integral weight Eisenstein series <cit.>.
As for the correction polynomials $P_{d_{0},d_{1}}^{(\chi)}(s)$,
their existence, uniqueness, and construction were studied extensively
in <cit.> and <cit.> in the more general setting of $GL(r)$
multiple Dirichlet series. They are unique in the case
of $r$ up to 3.
The notion of convexity is no longer canonical in the case of double
Dirichlet series, since our initial bounds for $Z(s,w;\chi,\chi')$
depend on what we know about bounds on their coefficients, and our
knowledge here is only partial. Nonetheless, if we use the Lindelöf
hypothesis on average (cf. Theorem <ref>),
then through careful choice of initial bounds and functional equation
applications (cf. <ref>), we obtain the bound
\[
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}};\chi,\chi')\ll(MN)^{3/8+\varepsilon},
\]
which we call the convexity bound. In this work, we present the following
subconvexity result.
For quadratic Dirichlet characters $\chi$
and $\chi'$ of conductors $N$ and $M$ which are prime or unity,
and for $\varepsilon>0$, we have the bound
\[
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it;\chi,\chi')\ll_{\varepsilon}(1+\vert t\vert)^{2/\varepsilon}(MN(M+N))^{1/6+\varepsilon}.
\]
We first point out that the $t$-aspect bound can be drastically improved,
but our focus here is on bounds in the moduli aspect.
For purposes of comparison with the convexity bound, we point out
that, via the geometric-arithmetic mean inequality, we have $(MN)^{3/8+\varepsilon}\ll(MN(M+N))^{1/4+\varepsilon}$.
This result is comparable to the subconvexity bound obtained by V.
Blomer in the archimedean case, particularly $\vert sw(s+w)\vert^{1/6+\varepsilon}$
for $\Re s=\Re w={\textstyle \frac{1}{2}}$, an improvement upon the
convexity bound with 1/4 replaced by 1/6.
A useful arithmetic application of the theory developed for $Z(s,w;\chi,\chi')$
is finding a bound for the least $d$ such that $L({\textstyle \frac{1}{2}},\chi_{dN})$
does not vanish, where $N$ is a large fixed prime. It is expected
that all ordinates of any zeros of $L(s,\chi)$ on the critical line
are linearly independent over the rationals, so that in particular
it is expected that $L({\textstyle \frac{1}{2}},\chi)$ is nonzero
for any $\chi$. Random matrix theory provides further evidence to
this conjecture: The lowest zero in families of $L$-functions (such
as $L(s,\chi_{d})$) is expected to be distributed like the “smallest”
eigenvalue (i.e., closest to unity) of a certain matrix family (depending
on the family of $L$-function). Since the corresponding measure vanishes
at zero, it suggests that the smallest zero of a Dirichlet $L$-function
is “repelled” from the real axis. In the case of quadratic twists
of $GL(2)$ automorphic forms, in particular twists of elliptic curves,
there is a connection to Waldspurger's theorem <cit.>
which states that $L({\textstyle \frac{1}{2}},f\times\chi)$ is proportional
to the squares of certain Fourier coefficients of a half-integral
weight modular form, uniformly in $\chi$. We focus here on the simpler
case of twists of Dirichlet characters. In particular, we have the
following theorem which is proven using the theory of this double Dirichlet series.
Let $N$ be an odd prime, and let $D(N)=d$
denote the smallest positive integer such that $L({\textstyle \frac{1}{2}},\chi_{dN})$
does not vanish. Then we have $D(N)\ll N^{1/2+\varepsilon}$.
It is not entirely obvious what should be regarded as the trivial
bound in this situation. The most natural approach to non-vanishing
would be to prove a lower bound for the first moment $\sum_{d\asymp X}L({\textstyle \frac{1}{2}},\chi_{dN})$
for $X$ as small as possible in terms of $N$. A straightforward
argument produces a main term of size $X\log X$ and an error term
of size $O((NX)^{1/2+\varepsilon})$ which suggests the trivial bound
$N^{1+\varepsilon}$ for the first non-vanishing twist. This is perhaps
unexpectedly weak, since the same bound holds for degree 2 $L$-functions
$L({\textstyle \frac{1}{2}},f\times\chi_{d})$, where $f$ is an automorphic
form of level $N$. Nevertheless it is not completely obvious how
to improve this in the case of Dirichlet characters $\chi_{N}$. Here we follow a modified version of a method presented in <cit.>: Let $X$ be a large positive number, and $h(y)$ be a smooth
non-negative function with support on $[1,2]$. By Mellin inversion,
we have
\[
\int_{(2)}\tilde{h}(w)Z({\textstyle \frac{1}{2}},w;\chi_{N},\psi_{1})X^{w}dw\approx\sum_{\substack{d=1}
}^{\infty}L({\textstyle \frac{1}{2}},\chi_{dN})h(d/X),
\]
where $\tilde{h}$ denotes the Mellin transform of $h$, and $\psi_{1}$
denotes the trivial character. We move the contour to $\Re w=-\varepsilon$,
picking up a double pole at $w=1$. If we apply a symmetric functional equation (<ref>) and a bound resulting from use of the Lindelöf principle on average (<ref>) to the resulting integral, then we have
\begin{equation}
\sum_{\substack{d=1}
}^{\infty}L({\textstyle \frac{1}{2}},\chi_{dN})h(d/X)\approx a_{N}X\log X+b_{N}X+O(N^{1/2+\varepsilon})\label{eq:nonvanishing-asymptotic-approx}
\end{equation}
for some coefficients $a_{N},b_{N}$. The idea now is to bound $a_{N}$
from below in terms of $N$, and choose $X$ so that the main term
is greater than the error term. Then it cannot be that $L({\textstyle \frac{1}{2}},\chi_{dN})$
vanishes for all $0<d<X$ on the left-hand side. We prove the bounds
$a_{N},b_{N}\gg N^{-\varepsilon}$ (cf. Theorem <ref>),
and thus we can choose $X=N^{1/2+\varepsilon}$. The power of the
asymptotic formula (<ref>) is that
$a_{N},b_{N}$ can be bounded below by finding an asymptotic for $\sum_{d\asymp X}L({\textstyle \frac{1}{2}},\chi_{dN})$,
but now we can take $X$ to be very large.
Of course, in the previous discussion there are significant details
suppressed for brevity, which include the aforementioned correction
factors and the error term arising from truncation of the $d$-sum.
Most nontrivial, however, are the bounds for $a_{N},b_{N}$ given
by Theorem <ref>, requiring careful treatment.
The techniques used are similar to those in <cit.>,
in which an asymptotic formula for
\sum_{0<d\leq X}L^{k}({\textstyle \frac{1}{2}},\chi_{d})
is proved, where the sum is over fundamental discriminants.
In order to prove the subconvexity bound Theorem <ref>,
we follow techniques similar to those used in <cit.>, but
we must deal with some new complications. As mentioned, we need special
correction factors in order to obtain functional equations, whereas
in the archimedean case, the correction factors are far simpler. These
correction factors appear due to the fact that the characters involved
will only be primitive when the summation variable is a fundamental
discriminant. Also, the introduction of character twists complicates
the functional equations considerably: a pair of twisting characters
$(\psi,\psi')$ will not be static under the variable transformations
of the functional equations.
We can iterate the functional equations to obtain one under the map
$(s,w)\mapsto(1-s,1-w)$. Using techniques similar to those in the
case of $L(s,\chi)$ (cf. <cit.> Theorem 5.3), we obtain the
approximate functional equation
\[
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}};\chi,\chi')\approx\sum_{d<X}\frac{L({\textstyle \frac{1}{2}},\chi_{d}\chi)\chi'(d)}{d^{1/2}}+\sum_{d\leq M^{2}N/X}\frac{L({\textstyle \frac{1}{2}},\chi_{d}\chi'\chi_{M})\chi\chi_{N}(d)}{d^{1/2}},
\]
(cf. Lemma <ref>). We can further apply the
approximate functional equation for $L$-functions,
\[
L({\textstyle \frac{1}{2}},\chi_{d}\chi)\approx\sum_{n\leq(dq)^{1/2}}\frac{\chi(n)}{n^{1/2}}.
\]
We hence see that we can roughly express $Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}};\chi,\chi')$
as a sum of double finite sums of the form
\[
S(P,Q;\chi,\chi'):=\sum_{d\leq P}\sum_{n\leq Q}\frac{\chi_{d}(n)\chi(n)\chi'(d)}{d^{1/2}n^{1/2}}
\]
for various character pairs $(\chi,\chi')$. Finally, we apply Heath-Brown's
large sieve estimate (cf. Corollary <ref>)
to obtain the subconvexity result.
We note that it is possible to extend these results beyond the case
where $M$ and $N$ are prime to arbitrary numbers, but the coefficients
of the functional equations in Theorems <ref> and
<ref> would become considerably more complicated
due to the Euler factors involved. We thus only treat the case of
prime moduli here to simplify the presentation.
Notation. The variable $\varepsilon$ will always denote
a sufficiently small positive number, not necessarily the same at
each occurrence, and the variable $A$ will denote a sufficiently
large positive number, not necessarily the same at each occurrence.
The numbers $M$ and $N$ will always denote natural numbers that
are either odd primes or unity, possibly equal. For a real function
$f$, we denote its Mellin transform by $\tilde{f}$. The trivial
character modulo unity will be denoted by $\psi_{1}$, and the primitive
character of conductor 4 shall be denoted $\psi_{-1}$. As for the
primitive characters modulo 8, we define $\psi_{2}$ as the character
that is unity at exactly 1 and 7, and we set $\psi_{-2}=\psi_{2}\psi_{-1}$.
If $\chi$ is a character, we use the notation $C_{\chi}$ to denote
its conductor.
§ PRELIMINARIES
§.§ Characters
For a positive integer $d$, we define a character on $(\mathbb{Z}/4d\mathbb{Z})^{*}$
via the Jacobi symbol by
\[
\chi_{d}(n)=\widetilde{\chi}_{n}(d)=\left(\frac{d}{n}\right).
\]
For odd positive integers $n$ and $d$, we have the following quadratic
reciprocity law for the Jacobi-Kronecker symbol (cf. <cit.>,
Theorem 4.2.1, page 197).
\[
\chi_{d}(n)=\left(\frac{d}{n}\right)
\widetilde{\chi}_{d}(n), & d\equiv1\,(\text{mod 4)};\\
\widetilde{\chi}_{d}(-n)=\widetilde{\chi}_{d}(n)\psi_{-1}(n), & d\equiv3\,(\text{mod 4)}.
\end{cases}
\]
§.§ $L$-function results
Suppose that $\chi$ is a Dirichlet character. We define
\begin{equation}
L^{(P)}(s,\chi)=L(s,\chi)\prod_{p\mid P}\left(1-\frac{\chi(p)}{p^{s}}\right).\label{eq:L-remove-primes}
\end{equation}
We define the odd sign indicator function of a Dirichlet character
$\chi$ by $\kappa=\kappa(\chi)=\frac{1}{2}(1-\chi(-1))$.
§.§ $L$-functional bounds and approximate functional equation
For a primitive character $\chi$ modulo $q$, using an absolute convergence argument
for $L(s,\chi)$ in a right half-plane and applying the functional
equation, we interpolate via the Phragmén-Lindelöf convexity principle
to obtain
\begin{equation}
[q(1+\vert\Im s\vert)]^{1/2-\Re s}, & \Re s\leq-\varepsilon;\\
{}[q(1+\vert\Im s\vert)]^{(1-\Re s)/2+\varepsilon}, & -\varepsilon<\Re s<1+\varepsilon;\\
1, & \Re s\geq1+\varepsilon,
\end{cases}\label{eq:L-bounds}
\end{equation}
away from a possible pole at $s=1$ in the case where $\chi$ is trivial.
This bound is known as the convexity bound for Dirichlet $L$-functions.
We shall also need the so-called approximate functional equation for
Dirichlet $L$-functions (cf. <cit.>, Theorem 5.3). Particularly,
if $\chi$ is a quadratic primitive character modulo odd $q$, $\psi$
is a character with conductor dividing 8, and $d_{0}$ is odd, squarefree
and coprime to $q$, then we have the weighted infinite sum
\begin{equation}
L({\textstyle \frac{1}{2}},\chi_{d_{0}}\chi\psi)=2\sum_{n=1}^{\infty}\frac{(\chi_{d_{0}}\chi\psi)(n)}{n^{1/2}}G_{\kappa}\left(\frac{n}{\sqrt{c_{0}d_{0}q}}\right),\label{eq:L-approx-functional}
\end{equation}
\begin{equation}
\begin{array}{cc}
\kappa=\kappa(\chi_{d_{0}}\chi\psi), & c_{0}=\begin{cases}
1, & d_{0}\equiv1\,(\text{mod }4),\psi=\psi_{1}\,\text{or}\, d_{0}\equiv3\,(\text{mod }4),\psi=\psi_{-1};\\
4, & d_{0}\equiv1\,(\text{mod }4),\psi=\psi_{-1}\,\text{or}\, d_{0}\equiv3\,(\text{mod }4),\psi=\psi_{1};\\
8, & \psi=\psi_{2}\text{ or }\psi_{-2},
\end{cases}\end{array}\label{eq:Lfunctional-kappa,delta0-1}
\end{equation}
and we have the weight function
\begin{equation}
G_{\kappa}(\xi)=\frac{1}{2\pi i}\int_{(2)}\frac{\Gamma(\frac{1/2+s+\kappa}{2})}{\Gamma(\frac{1/2+\kappa}{2})}\xi{}^{-s}\frac{ds}{s},\label{eq:G-def}
\end{equation}
satisfying the bound
\begin{equation}
\label{eq:G-bound}
G_{\kappa}(\xi) \ll (1+\xi)^{-A}
\end{equation}
for arbitrary $A\geq0$ (cf. <cit.>, Proposition 5.4).
We shall also make use of smooth weight functions.
We say that $w(x)$ is a smooth
weight function if it is a smooth non-negative real function supported
on $[1/4,5/4]$ and unity on $[1/2,1]$.
The following bound can be shown via sufficiently many applications
of integration by parts. We have
\begin{equation}
\tilde{w}(z)\ll_{A,\Re z}(1+\vert z\vert)^{-A}\label{eq:w-bound}
\end{equation}
for $A\geq0$, where $\tilde{w}(z)$ is the Mellin transform of $w(x)$.
§.§ Short double character sums and $L$-function moments
We shall need the following adaptation of Theorem 2 from <cit.>
which includes a character twist.
Let $\psi$ be a primitive character modulo
$j$, and for a positive integer $Q$ define $S(Q)$ to be the set
of quadratic primitive Dirichlet characters of conductor at most $Q$.
We have
\[
\sum_{\chi\in S(Q)}\vert L(\sigma+it,\chi\psi)\vert^{4}\ll_{\varepsilon}\lbrace Q+(Qj(\vert t\vert+1))^{2-2\sigma}\rbrace\lbrace Qj(\vert t\vert+1)\rbrace^{\varepsilon}
\]
for any fixed $\sigma\in[1/2,1]$ and any $\varepsilon>0$.
Removing even conductors, and using the fact that $2-2\sigma\leq1$,
we can restate this as follows.
If $\chi$ is a primitive
character with conductor $q$ and $X$ is a positive real number,
\[
\sum_{\substack{d_{0}\leq X\\
d_{0}\text{ odd, squarefree}
}\vert L(s,\chi_{d_{0}}\chi)\vert^{4}\ll_{\varepsilon}(Xq\vert s\vert)^{1+\varepsilon},\ \sigma\geq{\textstyle \frac{1}{2}}
\]
for all $\varepsilon>0$.
An important ingredient for the proof of the subconvexity bound Theorem
<ref> is a large sieve estimate for quadratic characters
due to Heath-Brown. In particular we state here Theorem 1 in <cit.>.
Let $P$ and $Q$ be positive integers,
and let $(a_{n})$ be a sequence of complex numbers. Then
\[
\left.\sum_{m\leq P}\right.^{\ast}\left|\left.\sum_{n\leq Q}\right.^{\ast}a_{n}\left(\frac{n}{m}\right)\right|^{2}\ll_{\varepsilon}(PQ)^{\varepsilon}(P+Q)\left.\sum_{n\leq Q}\right.^{\ast}\vert a_{n}\vert^{2},
\]
for any $\varepsilon>0$, where $\left.\sum\right.^{\ast}$ denotes
that the sum is over odd squarefree numbers.
Due to the nature of the double Dirichlet series we shall construct,
we shall need the following normalization of this result.
If $(a_{m}),\ (b_{n})$ are sequences
of complex numbers satisfying the bound $a_{m},\ b_{m}\ll m^{-1/2+\varepsilon}$
for some $\varepsilon>0$, and $P$ and $Q$ are positive real numbers,
\[
\sum_{\substack{m\leq P\\
m\text{ odd}
}\,\sum_{\substack{n\leq Q\\
n\text{ odd}
\]
where we have the composition $n=n_{0}n_{1}^{2}$ with $n_{0}$ squarefree,
uniformly in $P$ and $Q$, for any $\varepsilon>0$.
§.§ Gamma identities
We shall have use for the identity
\begin{equation}
\frac{\Gamma(\frac{2-z}{2})}{\Gamma(\frac{z+1}{2})}=\frac{\Gamma(\frac{1-z}{2})}{\Gamma(\frac{z}{2})}\cot\left(\frac{\pi z}{2}\right),\ z\in\mathbb{C}.\label{eq:cot-identity}
\end{equation}
By Stirling's formula, in particular (5.113) from <cit.>,
for $s\in\mathbb{C}$ with fixed real part and nonzero imaginary part,
we have
\begin{equation}
\frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})}\ll_{\Re s}(1+\vert s\vert)^{1/2-\Re s},\label{eq:stirling}
\end{equation}
away from the poles at the odd positive integers. We also have the
cotangent bound,
\begin{equation}
\cot(x+iy)=-i\,\text{sign}(y)+O(e^{-2\vert y\vert}),\ \min_{k\in\mathbb{Z}}\vert z-\pi k\vert\geq1/10.\label{eq:cot-bound}
\end{equation}
§ STRUCTURE AND ANALYTIC PROPERTIES
§.§ Switch of summation formula
The object we would like to study is
\[
\]
where $\chi$ and $\chi'$ are quadratic characters with moduli $N$
and $M$ respectively. However, in order to obtain functional equations,
we will need to augment this by some correction factors. The exact
form of these correction factors (in a more general setting) was determined
in <cit.>. Applying this theory in our case, we have the following
theorem, which we note holds for general Dirichlet characters $\chi$
and $\chi'$, though in our case we are interested in the special
case of quadratic twists.
Let $m$ and $d$ be positive integers
with $(md,2MN)=1$, and write $d=d_{0}d_{1}^{2}$ and $m=m_{0}m_{1}^{2}$,
with $d_{0}$, $m_{0}$ squarefree and $d_{1}$, $m_{1}$ positive,
and let $\chi$ and $\chi'$ be characters modulo $8\text{lcm}(M,N)$.
Then the Dirichlet polynomials
\[
P_{d_{0},d_{1}}^{(\chi)}(s)=\prod_{p^{\alpha}\Vert d_{1}}\left[\sum_{n=0}^{\alpha}\chi(p^{2n})p^{n-2ns}-\sum_{n=0}^{\alpha-1}(\chi_{d_{0}}\chi)(p^{2n+1})p^{n-(2n+1)s}\right],
\]
\[
Q_{m_{0},m_{1}}^{(\chi')}(w)=\prod_{p^{\beta}\Vert m_{1}}\left[\sum_{n=0}^{\beta}\chi'(p^{2n})p^{n-2nw}-\sum_{n=0}^{\beta}(\tilde{\chi}_{m_{0}}\chi')(p^{2n+1})p^{n-(2n+1)w}\right]
\]
satisfy the functional equations
\begin{equation}
\label{eq:Building-block-reflective}
\begin{array}{ccc}
\text{and}
\end{array}
\end{equation}
and the interchange of summation formula
\begin{equation}
\sum_{(d,2MN)=1}\frac{L^{(2MN)}(s,\chi_{d_{0}}\chi)\chi'(d)P_{d_{0},d_{1}}^{(\chi)}(s)}{d^{w}}=\sum_{(m,2MN)=1}\frac{L^{(2MN)}(w,\tilde{\chi}_{m_{0}}\chi')\chi(m)Q_{m_{0},m_{1}}^{(\chi')}(w)}{m^{s}},\label{eq:Building-block-sum-switch}
\end{equation}
for $\Re s,\Re w>1$.
This is a straightforward but lengthy computation which we supress
for brevity. We direct the reader to <cit.> for details.
In light of the above theorem, we now define our double Dirichlet
series as follows: Let $\chi$ and $\chi'$ be characters modulo $8\text{lcm}(M,N)$.
Then define
\begin{equation}
\end{equation}
We note at this point that it is easily shown that, for $d=d_{0}d_{1}^{2}$
and $m=m_{0}m_{1}^{2}$ with $\Re s,\Re w\geq{\textstyle \frac{1}{2}}$,
we have the bounds $\vert P_{d_{0},d_{1}}^{(\chi)}(s)\vert\ll d_{1}^{\varepsilon}$
and $\vert Q_{m_{0},m_{1}}^{(\chi')}(w)\vert\ll m_{1}^{\varepsilon}$.
Applying the functional equations (<ref>), we therefore have
\begin{equation}
\begin{array}{cccc}
\vert P_{d_{0},d_{1}}^{(\chi)}(s)\vert\ll\begin{cases}
d_{1}^{1-2\Re s+\varepsilon}, & \Re s<{\textstyle \frac{1}{2}};\\
d_{1}^{\varepsilon}, & \Re s\geq{\textstyle \frac{1}{2}},
\end{cases} & & & \vert Q_{m_{0},m_{1}}^{(\chi')}(w)\vert\ll\begin{cases}
m_{1}^{1-2\Re w+\varepsilon}, & \Re w<{\textstyle \frac{1}{2}};\\
m_{1}^{\varepsilon}, & \Re w\geq{\textstyle \frac{1}{2}}.
\end{cases}\end{array}\label{eq:P,Q-bounds}
\end{equation}
§.§ Functional equations
We recall that $M$ and $N$ are odd prime numbers or unity, possibly
equal. From this point on in the paper, we shall use the following
notation: Let $\chi$ and $\chi'$ be quadratic primitive characters
of squarefree conductors $k$ and $j$ respectively, where $j,k\mid\text{lcm}(M,N)$,
and let $\psi,\ \psi'$ be primitive characters with conductors dividing
8. We first derive the following expansion of the regions of absolute
convergence of the key series involved.
We have the following two series
representations of $Z(s,w;\chi\psi,\chi'\psi')$. We have
\[
\]
which is absolutely convergent on the set
\[
R_{1}^{(1)}:=\{\Re s\leq0,\Re w+\Re s>3/2\}\cup\{0<\Re s\leq1,\Re s/2+\Re w>3/2\}\cup\{\Re s,\Re w>1\},
\]
except for a possible polar line $\{s=1\}$, and
\[
\]
which is absolutely convergent on the set
\[
R_{1}^{(2)}:=\{\Re w\leq0,\Re s+\Re w>3/2\}\cup\{0<\Re w\leq1,\Re w/2+\Re s>3/2\}\cup\{\Re s,\Re w>1\},
\]
except for a possible polar line $\{w=1\}$.
This follows from applying the bounds for the Dirichlet $L$-function (<ref>) and the bounds for the correction polynomials (<ref>)
to the definition (<ref>).
We now proceed with derivation of functional equations for $Z$. Due
to the summation switch formula (<ref>),
we have
\[
\]
We can further apply the functional equation for the $Q$ factor (<ref>)
to obtain
\begin{equation}
\end{equation}
which holds for $(s,w)\in R_{1}^{(2)}$. The next step is to apply
the functional equation for Dirichlet $L$-functions in order to change
the $w$ in the $L$-function to $1-w$. This would allow us to switch
summation again to obtain $Z$ in its original form, but with a change
in variables.
We shall define the following Euler product function: For a character
$\chi^{\star}$ and a positive integer $P$, we define
\begin{equation}
K_{P}(w;\chi^{\star})=\prod_{p\mid P}\left(1-\frac{\chi^{\star}(p)}{p^{1-w}}\right)^{-1}\left(1-\frac{\chi^{\star}(p)}{p^{w}}\right).\label{eq:K-definition}
\end{equation}
Applying the Dirichlet functional equation along with (<ref>)
and the above, we now have
\begin{multline}
\pi^{w-\frac{1}{2}}\frac{\Gamma\left(\frac{1-w+\hat{\kappa}'}{2}\right)}{\Gamma\left(\frac{w+\hat{\kappa}'}{2}\right)}K_{2MN}(w;\tilde{\chi}_{m_{0}}\chi'\psi')(C_{\psi'}jm_{0})^{\frac{1}{2}-w}L^{(2MN)}(1-w,\tilde{\chi}_{m_{0}}\chi'\psi'),\label{eq:L-functional-first-functional}
\end{multline}
where $\hat{\kappa}'=\kappa(\tilde{\chi}_{m_{0}}\chi'\psi')$.
We need to break down some of these parts for further manipulation.
Recalling that $(m_{0},2MN)=1$, if $p$ is prime and does not divide
$m_{0}$, we have
\begin{equation}
\end{equation}
If for $P\in\mathbb{N}$ and a Dirichlet character $\chi^{\star}$
we set
\begin{equation}
\begin{array}{ccc}
1, & P=1;\\
\frac{\chi^{\star}(P^{2})P-P^{2}}{\chi^{\star}(P^{2})P^{2w}-P^{2}}, & \text{else},
\end{cases} & & G_{P}^{(\chi^{\star})}(w)=\begin{cases}
0, & P=1;\\
\frac{\chi^{\star}(P)(P^{2-w}-P^{1+w})}{\chi^{\star}(P^{2})P^{2w}-P^{2}}, & \text{else},
\end{cases}\end{array}\label{eq:F,G-coeff-def}
\end{equation}
then from (<ref>) and the definition (<ref>)
we have the identity
\begin{equation}
\end{equation}
which holds for prime $p$ not dividing $m_{0}$, or $p=1$. Noting
that $K_{P}(w;\chi^{\star})$ is multiplicative in $P$, and using
(<ref>) above, we now have the useful expression
\begin{multline}
K_{MN}(w;\tilde{\chi}_{m_{0}}\chi'\psi')=F_{M}^{(\chi'\psi')}\cdot F_{N}^{(\chi'\psi')}(w)+\chi_{M}(m_{0})F_{N}^{(\chi'\psi')}\cdot G_{M}^{(\chi'\psi')}(w)\\
+\chi_{N}(m_{0})F_{M}^{(\chi'\psi')}\cdot G_{N}^{(\chi'\psi')}(w)+\chi_{MN}(m_{0})G_{M}^{(\chi'\psi')}\cdot G_{N}^{(\chi'\psi')}(w).\label{eq:K-identity-3}
\end{multline}
We note that this holds true even if $M=N$. Indeed, in this case,
by the definition (<ref>), we have $K_{MN}(w;\tilde{\chi}_{m_{0}}\chi'\psi')=1$.
This is consistent with (<ref>), since according
to definition (<ref>) we have $G_{N}^{(\chi'\psi')}(w)=0$
and $F_{N}^{(\chi'\psi')}(w)=1$.
Next, we see from (<ref>) that
\begin{equation}
\frac{\Gamma\left(\frac{1-w+\hat{\kappa}'}{2}\right)}{\Gamma\left(\frac{w+\hat{\kappa}'}{2}\right)}=\frac{\Gamma\left(\frac{1-w}{2}\right)}{\Gamma\left(\frac{w}{2}\right)}\cot\left(\frac{\pi w}{2}\right)^{\hat{\kappa}'}.\label{eq:Gamma-quotient-factored}
\end{equation}
We shall find it useful to remove the dependency of $\hat{\kappa}'$
on $m_{0}$, or rather, exploit that the dependency is only on its
residue modulo 4. Hence, define $\kappa'=\kappa(\chi'\psi')$. Now
suppose that $f$ is a function of $\hat{\kappa}'$. By sieving out
by congruence classes modulo 4, we see that
\[
\]
Hence we have
\begin{equation}
\cot\left(\frac{\pi w}{2}\right)^{\hat{\kappa}'}=\frac{1}{2}(1+\psi_{-1}(m_{0}))\cot\left(\frac{\pi w}{2}\right)^{\kappa'}+\frac{1}{2}(1-\psi_{-1}(m_{0}))\cot\left(\frac{\pi w}{2}\right)^{(1-\kappa')}.\label{eq:cot-sieved}
\end{equation}
For brevity, for a character $\chi^{\star}$, we define
\[
\]
Applying the functional equation for Dirichlet $L$-functions (<ref>)
to the identity (<ref>) along with
(<ref>), and using Lemma <ref>
we now get
\begin{multline}
\times S(s,w;m,\chi\psi)\left[(1+\psi_{-1}(m_{0}))\cot\left(\frac{\pi w}{2}\right)^{\kappa'}+(1-\psi_{-1}(m_{0}))\cot\left(\frac{\pi w}{2}\right)^{(1-\kappa')}\right],\label{eq:Z-identity1}
\end{multline}
for $(s,w)\in R_{1}^{(2)}$ except for possible polar lines $\{s=1\}$
and $\{w=1\}$. There are two properties of the function $S$ which
we shall make use of, stated in the following lemma.
For two characters $\chi^{\star}$ and $\chi^{\star\star}$
and an integer $m$, the following two properties hold for $(s,w)\in R_{1}^{(2)}$,
except for possible polar lines $\{s=1\}$ and $\{w=1\}$.
\[
\begin{array}{rrcl}
\text{(i)} & S(s,w;m,\chi^{\star})\chi^{\star\star}(m) & = & S(s,w;m,\chi^{\star}\chi^{\star\star}),\\
\text{(ii)} & {\displaystyle \sum_{(m,2MN)=1}}S(s,w;m,\chi^{\star}) & = & Z(s+w-{\textstyle \frac{1}{2}},1-w;\chi^{\star},\chi'\psi').
\end{array}
\]
We can apply the identity (<ref>) and Lemma <ref>
to (<ref>), and use the bounds (<ref>)
and (<ref>) to come to the following functional equation.
Let $\chi$ and $\chi'$ be primitive Dirichlet
characters modulo squarefree $k$ and $j$ respectively, where $j,k\mid MN$,
and if $M=N$, then $j=k=M=N$, and let $\psi$ and $\psi'$ be Dirichlet
characters modulo 8. There exist functions $a_{n}^{(\chi',\psi')}(w;\psi^{\star})$
for $n\mid MN$ and $\psi^{\star}$ a Dirichlet character modulo 8
which are holomorphic except for possible poles at the positive integers,
and countably many poles on the line $\Re w=1$, bounded absolutely above by
$O((16\pi)^{\vert\Re w\vert}(1+\vert w\vert)^{1/2-\Re w})$ uniformly
in $j$ and $k$ away from the poles such that for $(s,w)\in R_{1}^{(2)}$
away from possible polar lines $\{s=1\}$ and $\{w=1\}$ we have
\begin{multline*}
Z(s,w;\chi\psi,\chi'\psi')=j{}^{1/2-w}\sum_{n\mid MN}A_{n}^{(\chi')}(w)\sum_{\psi^{\star}\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}}a_{n}^{(\chi',\psi')}(w;\psi^{\star})Z(s+w-{\textstyle \frac{1}{2}},1-w;\chi\tilde{\chi}_{n}\psi\psi^{\star},\chi'\psi'),
\end{multline*}
where $R_{1}^{(2)}$ is defined in Lemma <ref>,
we have
\[
\begin{array}{ccc}
A_{1}^{(\chi')}(w)=F_{M}^{(\chi')}(w)F_{N}^{(\chi')}(w), & & A_{M}^{(\chi')}(w)=F_{N}^{(\chi')}(w)G_{M}^{(\chi')}(w),\\
A_{N}^{(\chi')}(w)=F_{M}^{(\chi')}(w)G_{N}^{(\chi')}(w), & & A_{MN}^{(\chi')}(w)=G_{M}^{(\chi')}(w)G_{N}^{(\chi')}(w),
\end{array}
\]
and the $F$ and $G$ functions are defined in (<ref>).
By similar methods, we can obtain a second functional equation under
the transformation $(s,w)\mapsto(1-s,s+w-{\textstyle \frac{1}{2}})$
by applying the functional equation (<ref>)
to the definition (<ref>), followed by the functional
equation for $L$-functions. The result of this similarly lengthy
derivation is the following second functional equation.
Let $\chi$ and $\chi'$ be Dirichlet characters
modulo squarefree $k$ and $j$ respectively, where $j,k\mid MN$,
and if $M=N$, then $j=k=M=N$, and let $\psi$ and $\psi'$ be Dirichlet
characters modulo 8. There exist functions $b_{n}^{(\chi,\psi)}(s;\psi^{\star})$
for $n\mid MN$ and $\psi^{\star}$ a Dirichlet character modulo 8
which are holomorphic except for possible poles at the positive integers,
and countably many poles on the line $\Re s=1$, bounded absolutely above by
$O((16\pi)^{\vert\Re s\vert}(1+\vert s\vert)^{1/2-\Re s})$ uniformly
in $j$ and $k$ away from the poles such that for $(s,w)\in R_{1}^{(1)}$
away from possible polar lines $\{s=1\}$ and $\{w=1\}$ we have
\begin{multline*}
Z(s,w;\chi\psi,\chi'\psi')=k{}^{1/2-s}\sum_{n\mid MN}A_{n}^{(\chi)}(s)\sum_{\psi^{\star}\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}}b_{n}^{(\chi,\psi)}(s;\psi^{\star})Z(1-s,s+w-{\textstyle \frac{1}{2}};\chi\psi,\chi'\tilde{\chi}_n\psi'\psi^{\star}),
\end{multline*}
where $R_{1}^{(1)}$ is defined in Lemma <ref>,
and the $A$ functions are as in Theorem <ref>.
We make note of an analytic subtlety: We note that although the $a_{n}$,
$b_{n}$, and $A_{n}$ functions above have poles, they do not contribute
poles to $Z(s,w;\chi\psi,\chi'\psi')$; indeed, it will be proven
in Proposition <ref> that the only possible poles
of $Z$ are the polar lines $\{s=1\}$, $\{w=1\}$, and $\{s+w=3/2\}$.
Looking at (<ref>), though the gamma and cotangent
factors together have poles at either the even or odd positive integers,
and the $K$ factor has countably many poles along the line $\Re w=1$
(cf. (<ref>)), these poles nonetheless do not produce
poles on the right-hand side. The equation (<ref>)
essentially results from the application of the functional equation
for $L$-functions (<ref>) to the
identity (<ref>), and subsequently
sieving out by congruence classes of $m$ modulo 4. The last step
introduces coefficients with poles from the gamma and cotangent factors.
This is a manifestation of a phenomenon that is observed in the functional
equation for $L$-functions: In (<ref>),
we know that the $L$-function can only have a pole at $w=1$, yet
on the right-hand side, the gamma function produces poles which are
mitigated by the trivial zeros of the $L$-function. It is precisely
these poles which appear in the coefficients of (<ref>).
Additionally, although the $K$ function has poles, these are mitigated
by corresponding zeros of the $L$ function due to removal of the
Euler factors at primes dividing $2MN$.
It shall be useful to note the following properties of the $A$ coefficients,
which follow directly from the definitions.
Let $\chi$ be a Dirichlet character
modulo $q$. Then the following properties hold.
* For a positive integer $P$, if $(q,P)>1$ then $A_{P}^{(\chi)}(w)=0$.
* If $q=MN$ then $A_{1}^{(\chi)}(w)=1$.
* If $q=M$ then $A_{1}^{(\chi)}(w)=F_{N}^{(\chi)}(w)$ and $A_{N}^{(\chi)}(w)=G_{N}^{(\chi)}(w)$.
* If $q=N$ then $A_{1}^{(\chi)}(w)=F_{M}^{(\chi)}(w)$ and $A_{M}^{(\chi)}(w)=G_{M}^{(\chi)}(w)$.
* Moreover, the following asymptotics hold, if $P\neq1$ and $(P,q)=1$.
\[
\begin{array}{ccc}
\vert F_{P}^{(\chi)}(w)\vert\asymp\begin{cases}
1, & \Re w<1-\varepsilon;\\
P^{2-2\Re w}, & \Re w>1+\varepsilon,
\end{cases} & & \vert G_{P}^{(\chi)}(w)\vert\asymp\begin{cases}
P^{-\Re w}, & \Re w<{\textstyle \frac{1}{2}};\\
P^{\Re w-1}, & {\textstyle \frac{1}{2}}\leq\Re w<1-\varepsilon;\\
P^{1-\Re w}, & \Re w>1+\varepsilon.
\end{cases}\end{array}
\]
It shall also be useful to derive a somewhat symmetric functional equation, obtained by application of Theorem <ref>, then Theorem <ref>, followed again by Theorem <ref>. For quadratic Dirichlet characters $\rho$ and $\rho'$ of conductors $N$ and $M$ respectively which are either prime or unity, possibly equal, we have
\begin{multline}
\label{eq:functional-3}
\sum_{n,m,r\mid MN}
C_{\rho \tilde{\chi}_n}^{1-s-w}
C_{\rho' \tilde{\chi}_m}^{1/2-s}
A_m^{(\rho \tilde{\chi}_n)}(s+w-1/2)
\\
\times \sum_{\psi^{\star}, \psi^{\star\star}, \psi^{\star\star\star}\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}}
\end{multline}
where the $A$ functions are as in Theorem <ref>, the $c$ functions are holomorphic in $\mathbb{C}^2$ except for possible poles for $s$, $w$, or $s+w-1/2$ equal to positive integers, and countably many poles on the lines $\Re w = 1$, $\Re s = 1$, and $\Re s + \Re w = 3/2$, bounded absolutely above by
\begin{equation}
\label{eq:functional-3-coeffs-bound}
O((16\pi)^{2\vert \Re s \vert+2\vert \Re w \vert} (1+\vert s \vert)^{1/2-\Re s} (1+\vert w \vert)^{1/2-\Re w} (1+\vert s+w \vert)^{1-\Re s - \Re w}).
\end{equation}
§.§ Analytic continuation
We continue $Z(s,w;\chi,\chi')$ to all of $\mathbb{C}^{2}$ except
for the polar lines $s=1$, $w=1$, and $s+w=3/2$. We have
Let $\chi$ and $\chi'$ be characters modulo
$8\text{lcm}(M,N)$. The function
\[
\tilde{Z}(s,w;\chi,\chi')=(s-1)(w-1)(s+w-3/2)Z(s,w;\chi,\chi')
\]
is holomorphic in $\mathbb{C}^{2}$ and is polynomially bounded in
the sense that, given $C_{1}>0$, there exists $C_{2}>0$ such that
$\tilde{Z}(s,w;\chi,\chi')\ll[MN(1+\vert\Im s\vert)(1+\vert\Im w\vert)]^{C_{2}}$
whenever $\vert\Re s\vert,\vert\Re w\vert<C_{1}$.
We refer to the reader to <cit.> and <cit.>.
§.§ Convexity bound
The notion of convexity is not canonically defined for double Dirichlet
series as it is in the case of (single) Dirichlet series; in the latter
case, we have a single functional equation which reflects the region
of absolute convergence, and interpolating the bounds produces a convexity
bound between the two. In the case of double Dirichlet series, things
are more complicated. Firstly, our bounds in the region of absolute
convergence depend on our knowledge of the bounds on $L(s,\chi)$
on average. Secondly, we have 6 functional equations to choose from
to apply to this region. If we use the Lindelöf hypothesis on average,
namely Theorem <ref>, then we can
carefully choose a functional equation to apply in order to minimize
the resulting convexity bound from application of the Phragmén-Lindelöf
convexity principle (cf. Theorem 5.53 of <cit.>).
We shall require some initial bounds. We let $\chi$ and $\chi'$
be quadratic Dirichlet characters with conductors $k$ and $j$ respectively,
and $\psi,\psi'$ are characters modulo 8. We first assume that $\Re s=1/2$
and $\Re w=1+\varepsilon$. We apply the identity (<ref>),
the bounds (<ref>), and the Cauchy-Schwarz inequality
with the average bound of Theorem <ref>
to obtain
\begin{equation}
Z(s,w;\chi\psi,\chi'\psi')\ll(MN)^{\varepsilon}k^{1/4+\varepsilon}(1+\vert s\vert)^{1/4+\varepsilon},\Re s=1/2,\Re w=1+\varepsilon,\label{eq:convex-trivial-1}
\end{equation}
and by the switch of summation formula (<ref>)
we also obtain
\begin{equation}
Z(s,w;\chi\psi,\chi'\psi')\ll(MN)^{\varepsilon}j^{1/4+\varepsilon}(1+\vert w\vert)^{1/4+\varepsilon},\ \Re s=1+\varepsilon,\Re w=1/2.\label{eq:convex-trivial-2}
\end{equation}
We use the functional equation Theorem <ref> with
$\Re s=-\varepsilon$ and $\Re w=1+\varepsilon$ and apply (<ref>)
on the right-hand side in order to obtain a bound for $Z(s,w;\rho,\rho')$.
Looking at the coefficient bounds in Lemma <ref>,
we pick up a factor of $N^{1/2+\varepsilon}$. Further, we see that
the resulting twisting characters on the right-hand side will have
conductors $(k,j)\in\{(N,M),(N,1)\}$, so that we have
\begin{equation}
Z(s,w;\rho,\rho')\ll N^{1/2+\varepsilon}M^{1/4+\varepsilon}(1+\vert s\vert)^{1/2+\varepsilon}(1+\vert s+w\vert)^{1/4+\varepsilon},\ \Re s=-\varepsilon,\Re w=1+\varepsilon.\label{eq:pre-interpolate-1}
\end{equation}
Likewise with the functional equation Theorem <ref>
applied to (<ref>), we have the symmetric bound
\begin{equation}
Z(s,w;\rho,\rho')\ll M^{1/2+\varepsilon}N^{1/4+\varepsilon}(1+\vert w\vert)^{1/2+\varepsilon}(1+\vert s+w\vert)^{1/4+\varepsilon},\ \Re s=1+\varepsilon,\Re w=-\varepsilon.\label{eq:pre-interpolate-2}
\end{equation}
We wish to interpolate convexly between these bounds along the two
diagonal lines in $\mathbb{C}^{2}$, but we must deal with the potential
poles at $s=1$ and $w=1$. To do this we can simply multiply both
sides of the bounds by $(s-1)(w-1)$. We obtain the following bound.
For quadratic characters $\rho$ and $\rho'$ of prime moduli $N$
and $M$ respectively, we have
\[
Z(s,w;\rho,\rho')\ll[(1+\vert s\vert)(1+\vert w\vert)(1+\vert s+w\vert)]^{1/4+\varepsilon}(MN)^{3/8+\varepsilon},\ \Re s=\Re w=1/2,
\]
which we call the convexity bound for the function $Z(s,w;\rho,\rho')$.
§ APPROXIMATE FUNCTIONAL EQUATIONS
§.§ A symmetric functional equation
We introduce a succession of applications of the functional equations
in the special case of $(s,w)=(1/2,1/2-z)$ for some $z\in\mathbb{C}$
with $\Re z>0$. We recall that $\rho$ and $\rho'$ are primitive
quadratic Dirichlet characters of conductors $N$ and $M$ respectively.
We first apply Theorem <ref>, which after observing
the coefficient properties of Lemma <ref>
\begin{multline}
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}-z;\rho,\rho')=\sum_{\psi^{\star}\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}}\left[M^{z}F_{N}^{(\rho')}a_{1}^{(\rho',\psi_{1})}({\textstyle \frac{1}{2}}-z;\psi^{\star})Z({\textstyle \frac{1}{2}}-z,{\textstyle \frac{1}{2}}+z;\rho\psi^{\star},\rho')\right.\\
\left.+M^{z}G_{N}^{(\rho')}({\textstyle \frac{1}{2}}-z)a_{N}^{(\rho',\psi_{1})}({\textstyle \frac{1}{2}}-z;\psi^{\star})Z({\textstyle \frac{1}{2}}-z,{\textstyle \frac{1}{2}}+z;\psi^{\star},\rho')\right].\label{eq:special-functional-pre}
\end{multline}
We then apply the functional equation Theorem <ref>
(and again use Lemma <ref>) which further
\begin{multline}
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}-z;\rho,\rho')=\\
\sum_{\psi^{\star},\psi^{\star\star}\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}}\left[(MN)^{z}F_{N}^{(\rho')}({\textstyle \frac{1}{2}}-z)F_{M}^{(\rho)}({\textstyle \frac{1}{2}}-z)c_{1,1}^{(\rho',\rho)}(z;\psi^{\star},\psi^{\star\star})Z({\textstyle \frac{1}{2}}+z,{\textstyle \frac{1}{2}};\rho\psi^{\star},\rho'\psi^{\star\star})\right.\\
+(MN)^{z}F_{N}^{(\rho')}({\textstyle \frac{1}{2}}-z)G_{M}^{(\rho)}({\textstyle \frac{1}{2}}-z)c_{1,M}^{(\rho',\rho)}(z;\psi^{\star},\psi^{\star\star})Z({\textstyle \frac{1}{2}}+z,{\textstyle \frac{1}{2}};\rho\psi^{\star},\psi^{\star\star})\\
+M^{z}G_{N}^{(\rho')}({\textstyle \frac{1}{2}}-z)F_{M}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)F_{N}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)c_{N,1}^{(\rho',\psi_{1})}(z;\psi^{\star},\psi^{\star\star})Z({\textstyle \frac{1}{2}}+z,{\textstyle \frac{1}{2}};\psi^{\star},\rho'\psi^{\star\star})\\
+M^{z}G_{N}^{(\rho')}({\textstyle \frac{1}{2}}-z)F_{N}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)G_{M}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)c_{N,M}^{(\rho',\psi_{1})}(z;\psi^{\star},\psi^{\star\star})Z({\textstyle \frac{1}{2}}+z,{\textstyle \frac{1}{2}};\psi^{\star},\psi^{\star\star})\\
+M^{z}G_{N}^{(\rho')}({\textstyle \frac{1}{2}}-z)F_{M}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)G_{N}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)c_{N,N}^{(\rho',\psi_{1})}(z;\psi^{\star},\psi^{\star\star})Z({\textstyle \frac{1}{2}}+z,{\textstyle \frac{1}{2}};\psi^{\star},\rho'\rho\psi^{\star\star})\\
\left.+M^{z}G_{N}^{(\rho')}({\textstyle \frac{1}{2}}-z)G_{M}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)G_{N}^{(\psi_{1})}({\textstyle \frac{1}{2}}-z)c_{N,MN}^{(\rho',\psi_{1})}(z;\psi^{\star},\psi^{\star\star})Z({\textstyle \frac{1}{2}}+z,{\textstyle \frac{1}{2}};\psi^{\star},\rho\psi^{\star\star})\right],\label{eq:special-functional}
\end{multline}
\[
c_{n,m}^{(\chi,\chi')}(z;\psi^{\star},\psi^{\star\star})=a_{n}^{(\chi,\psi_{1})}({\textstyle \frac{1}{2}}-z;\psi^{\star})b_{m}^{(\chi',\psi^{\star})}({\textstyle \frac{1}{2}}-z;\psi^{\star\star}).
\]
We note that, in the case where $M=N\neq1$, we may not apply the
functional equation Theorem <ref> to $Z({\textstyle \frac{1}{2}}-z,{\textstyle \frac{1}{2}}+z;\psi^{\star},\rho')$
in (<ref>) because Theorem <ref>
is only valid if $j=k=M=N$, but here we have $k=1\neq N$. Nonetheless,
equation (<ref>) still holds: Indeed, in the
case where $M=N$, the term with $Z({\textstyle \frac{1}{2}}-z,{\textstyle \frac{1}{2}}+z;\psi^{\star},\rho')$
of (<ref>) vanishes because $G_{N}^{(\rho')}({\textstyle \frac{1}{2}-z)=0}$,
and so only the first term of (<ref>) will
remain (note also that $G_{M}^{(\rho)}({\textstyle \frac{1}{2}}-z)=0$
in this case).
Looking at (<ref>), for a quadratic character $\chi^{\star}$
whose modulus is coprime to $P$, we see that
\begin{equation}
\label{eq:G-coeff-special}
\begin{array}{ccc}
F_{P}^{(\chi^{\star})}({\textstyle \frac{1}{2}}-z) = \frac{P^{-1}-1}{P^{-2z-1}-1}
\text{and}
G_{P}^{(\chi^{\star})}({\textstyle \frac{1}{2}}-z) = P^{z-1/2}\left(\frac{\chi^{\star}(P)(1-P^{-2z})}{P^{-2z-1}-1}\right).
\end{array}
\end{equation}
Therefore, setting
\begin{equation}
\Phi:=\lbrace(\rho,\rho'),\ (\rho,\psi_{1}),\ (\psi_{1},\rho'),\ (\psi_{1},\psi_{1}),\ (\psi_{1},\rho'\rho),\ (\psi_{1},\rho)\rbrace\label{eq:Phi-def}
\end{equation}
we have
\begin{equation}
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}-z;\rho,\rho')=\sum_{\substack{(\chi,\chi')\in\Phi\\
\psi,\psi'\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}
}\beta_{\psi,\psi'}^{(\chi,\chi')}\omega_{\psi,\psi'}^{(\chi,\chi')}(z)(\gamma_{\psi,\psi'}^{(\chi,\chi')})^{z}Z({\textstyle \frac{1}{2}}+z,{\textstyle \frac{1}{2}};\chi\psi,\chi'\psi'),\label{eq:functional-with-z}
\end{equation}
where we absorb the $F_{P}^{\chi^{\star}}({\textstyle \frac{1}{2}}-z)$
factors and parenthetical expression of (<ref>)
for the $G_{P}^{\chi^{\star}}({\textstyle \frac{1}{2}}-z)$ factors,
as well as the $c_{n,m}^{(\chi,\chi')}(z;\psi^{\star},\psi^{\star\star})$
factors into the $\omega_{\psi,\psi'}^{(\chi,\chi')}(z)$ functions,
and collect the remaining factors into the $\beta_{\psi,\psi'}^{(\chi,\chi')}$
and $\gamma_{\psi,\psi'}^{(\chi,\chi')}$ coefficients. Hence we see
that for $\Re z>0$, the $\omega_{\psi,\psi'}^{(\chi,\chi')}(z)$
functions are holomorphic satisfying the bound
\begin{equation}
\omega_{\psi,\psi'}^{(\chi,\chi')}(z)\ll\left(1+\vert\Im z\vert\right)^{\Re z}\label{eq:omega-bound}
\end{equation}
uniformly in $M$ and $N$. Thus, we obtain the upper bounds in Table <ref>.
Coefficient upper bounds
$(\chi,\chi')$ $(\rho,\rho')$ $(\rho,\psi_{1})$ $(\psi_{1},\rho')$ $(\psi_{1},\psi_{1})$ $(\psi_{1},\rho'\rho)$ $(\psi_{1},\rho)$
2*$M\neq N$
$\beta_{\psi,\psi'}^{(\chi,\chi')}$ bound
1 $M^{-1/2}$ $N^{-1/2}$ $M^{-1/2} N^{-1/2}$ $N^{-1}$ $M^{-1/2} N^{-1}$
$\gamma_{\psi,\psi'}^{(\chi,\chi')}$ bound
$MN$ $M^2 N$ $MN$ $M^2 N$ $MN^2$ $M^2 N^2$
2*$M = N$
$\beta_{\psi,\psi'}^{(\chi,\chi')}$ bound
1 0 0 0 0 0
$\gamma_{\psi,\psi'}^{(\chi,\chi')}$ bound
$N^2$ 0 0 0 0 0
§.§ Approximate functional equations
The following lemma essentially takes the preceding functional equation
a step further by opening the first sum of $Z$.
There exist smooth, rapidly decaying
functions $V(\xi;t)$ and $V_{\psi,\psi'}^{(\chi,\chi')}(\xi;t)$
such that for any constant $X>0$ one has
\begin{multline*}
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it;\rho,\rho')=X^{-it}\sum_{\substack{(d,2MN)=1}
\psi,\psi'\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}
\end{multline*}
where $\beta_{\psi,\psi}^{(\chi,\chi')}$ and $\gamma_{\psi,\psi'}^{(\chi,\chi')}$
satisfy the bounds listed in Table <ref>, and we have
the bounds
\[
\begin{array}{ccc}
V_{\psi,\psi'}^{(\chi,\chi')}\left(\xi;t\right) \ll \vert\xi\vert{}^{-B}(1+\vert t\vert)^{B}
\text{and}
V(\xi;t) \ll \vert\xi\vert{}^{-B}
\end{array}
\]
uniformly in $\xi$ and $t$ for any number $B>0$.
Let $B>0$, $H$ be an even, holomorphic function with $H(0)=1$ satisfying
the growth estimate $H(z)\ll_{\Re z,A}(1+\vert z\vert)^{-A}$ for any $A>0$. We consider the integral
\begin{equation}
I(c,X,t)=\frac{1}{2\pi i}\int_{(1)}X^{cz}\left(\frac{4^{\frac{1}{2}+it+cz}-4}{4^{\frac{1}{2}+it}-4}\right)^{2}Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it+cz;\rho,\rho')H(z)\frac{dz}{z}\label{eq:I-def}
\end{equation}
for a real number $c$, a positive real number $X>0$, and a fixed
real number $t$. Examining the expression when $c=1$, the fraction
cancels the pole of the $Z$ factor at $z=1/2-it$. We apply a shift
of the contour to $\Re z=-1$, picking up the pole at $z=0$, whence
we obtain
\[
I(1,X,t)=Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it;\rho,\rho')+\frac{1}{2\pi i}\int_{(-1)}X^{z}\left(\frac{4^{\frac{1}{2}+it+z}-4}{4^{\frac{1}{2}+it}-4}\right)^{2}Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it+z;\rho,\rho')H(z)\frac{dz}{z}.
\]
We now apply a change of variables $z\mapsto-z$, arriving at
\[
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it;\rho,\rho')=I(1,X,t)+I(-1,X,t).
\]
Applying the functional equation (<ref>) and
the switch of summation formula (<ref>)
and expanding the $Z$ functions, definition (<ref>)
gives that $I(-1,X,t)$ equals
\begin{multline}
\psi,\psi'\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}
\sum_{\substack{(m,2MN)=1}
\end{multline}
\[
V_{\psi,\psi'}^{(\chi,\chi')}(\xi;t)=\frac{1}{2\pi i}\int_{(1)}\left(\frac{4^{1/2+it-z}-4}{4^{1/2+it}-4}\right)^{2}\xi^{-z+it}\omega_{\psi,\psi'}^{(\chi,\chi')}(z-it)H(z)\frac{dz}{z}.
\]
We wish to obtain an upper bound for $V_{\psi,\psi'}^{(\chi,\chi')}(\xi;t)$.
Moving the contour to $B$ (recalling that there are no poles of $\omega_{\psi,\psi'}^{(\chi,\chi')}$
in this region), and bounding by taking the absolute value of the
integrand and using the bound (<ref>), we have
\[
V_{\psi,\psi'}^{(\chi,\chi')}(\xi;t)\ll\vert\xi\vert{}^{-B}(1+\vert t\vert)^{B}
\]
uniformly in $\xi$ and $t$. Changing the summation variable from
$m$ to $d$ in (<ref>), we obtain the first term in the
statement of the lemma.
Looking at $I(1,X,t)$, we have
\[
}\frac{L^{(2MN)}({\textstyle \frac{1}{2}},\chi_{d_{0}}\rho)\rho'(d)P_{d_{0},d_{1}}^{(\rho)}({\textstyle \frac{1}{2}})}{d{}^{1/2+it}}V\left(\frac{d}{X};t\right),
\]
\[
V(\xi;t)=\frac{1}{2\pi i}\int_{(1)}\left(\frac{4^{\frac{1}{2}+it+z}-4}{4^{1/2+it}-4}\right)^{2}\xi^{-z}H(z)\frac{dz}{z}.
\]
Also, it is immediate that we have the bound $V(\xi;t)\ll\vert\xi\vert{}^{-B}$ uniformly in $\xi$ and $t$.
We now wish to truncate the sums above, accruing an error. This is
the object of the following lemma.
Let $A$ be a large positive
constant, $t$ be a real number, and $V(\xi;t)$ be a rapidly decaying
function in $\xi$ satisfying
\[
V(\xi;t)\ll\vert\xi\vert{}^{-B}(1+\vert t\vert)^{B},
\]
uniformly in $\xi$ and $t$ for any number $B>0$, let $\chi$ be
a character modulo $k$, and let $a$ be an arithmetic function satisfying $ a(d)\ll d^{\varepsilon} $ uniformly in $d$. Then we can truncate the double sum
The $L$-function is bounded asymptotically by $(d_{0}k)^{1/4+\varepsilon}$
due to the Phragmén-Lindelöf convexity bound (<ref>).
The $V$ factor is bounded by its argument to an arbitrarily large
power $-B$. Applying this gives an error that is bounded above by
\[
(1+\vert t\vert)^{2B}\sum_{d>P^{1+\varepsilon}}\frac{(d_{0}k)^{1/4+\varepsilon}}{d^{1/2-\varepsilon}}\left(\frac{d}{Y}\right)^{-B},
\]
and the result follows.
In order to bound $Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it;\rho,\rho')$,
by applying a smooth partition of unity as in <cit.>, it now
suffices to bound
\[
}\frac{L^{(2MN)}({\textstyle \frac{1}{2}},\chi_{d_{0}}\chi'\psi')\chi\psi(d)a(d)}{d{}^{1/2}}W\left(\frac{d}{Y};t\right)
\]
for $t$ a real number, $\psi,\psi'\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}$,
a smooth function $W$ with support on $[1,2]$ satisfying
\[
W(x;t)\ll_{B}x{}^{-B}(1+\vert t\vert)^{B}
\]
uniformly in $x$ and $t$ for any $B>0$, an arithmetic function
$a$ satisfying the bound $a(d)\ll d^{\varepsilon}$, and the following
conditions, according to each of the two sums in Lemma <ref>:
either $(\chi,\chi')\in\Phi$ (cf. (<ref>)) with conductors
$k$ and $j$ respectively, and
\[
1\leq Y\leq\left(\gamma_{\psi,\psi'}^{(\chi,\chi')} X^{-1}\right)^{1+\varepsilon},
\]
or $(\chi,\chi')=(\rho',\rho)$ and
\[
1\leq Y\leq X^{1+\varepsilon}.
\]
Expanding according to the Dirichlet functional equation, and further
truncating that sum expresses $D_{W,a}(Y;t,\chi',\chi)$ as a double
finite character sum, allowing us to apply Heath-Brown's large sieve
estimate Corollary <ref>. The result
of this is the following lemma.
We have the bound
\[
D_{W,a}(Y;t,\chi',\chi)\ll(1+\vert t\vert)^{2/\varepsilon}(MN)^{\varepsilon}\left(Y^{1+\varepsilon}+(Yj){}^{1/2+\varepsilon}\right)^{1/2+\varepsilon}
\]
uniformly in $t$, $Y$, $j$, and $k$.
Applying Lemma <ref> above, we have
\begin{multline*}
}\frac{L^{(2MN)}(\frac{1}{2},\chi_{d_{0}}\chi)\chi'(d)a(d)}{d{}^{1/2}}W\left(\frac{d}{Y};t\right)+O((1+\vert t\vert)^{2/\varepsilon}j^{1/4+\varepsilon}Y^{-1}).
\end{multline*}
Applying (<ref>) and (<ref>),
we have that $ D_{W,a}(Y;t,\chi',\chi) $ equals, up to an error of $ O((1+\vert t\vert)^{2/\varepsilon}j^{1/4+\varepsilon}Y^{-1}) $,
\begin{multline}
\sum_{n=1}^{\infty}\frac{(\chi\chi_{d_{0}})(n)}{n^{1/2}}G_{\kappa_{\chi'}}\left(\frac{n}{\sqrt{c_{0}d_{0}j}}\right),\label{eq:D-1}
\end{multline}
where $c_{0}$ is given in (<ref>)
and $G_{\kappa}$ is given in (<ref>). Because of the rapid
decay of $G_{\kappa}$ and $W$, we can truncate the $n$-sum at $n<\left(Yj\right){}^{1/2+\varepsilon}$.
Applying the bounds (<ref>) and $a(d)\ll d^{\varepsilon}$,
the error obtained by this is bounded by
\[
(MN)^{\varepsilon}(1+\vert t\vert)^{B}\sum_{d<Y^{1+\varepsilon}}\frac{1}{d^{1/2-\varepsilon}}\left(1+\frac{d}{Y}\right)^{-B}\sum_{n>(Yj)^{1/2+\varepsilon}}\frac{1}{n^{1/2}}\left(1+\frac{n}{\sqrt{c_{0}d_{0}j}}\right)^{-B}
\]
for any large positive number $B$. Indeed, this is bounded above
\begin{eqnarray*}
& & (MN)^{\varepsilon}(1+\vert t\vert)^{B}Y^{B}j^{B/2}\sum_{d<Y^{1+\varepsilon}}d^{-B-1/2+\varepsilon}\sum_{n>(Yj)^{1/2+\varepsilon}}n^{-B-1/2}d^{B/2}\\
& \asymp & (MN)^{\varepsilon}(1+\vert t\vert)^{B}Y^{B}j^{B/2}(Y^{1+\varepsilon})^{-B/2+1/2+\varepsilon}(Yj)^{(1/2+\varepsilon)(-B+1/2)}.
\end{eqnarray*}
We see this is bounded above by $(MN)^{\varepsilon}(1+\vert t\vert)^{B}(Yj)^{-\varepsilon B+1}$.
We choose $B$ large enough so that $\varepsilon B-1\geq1$, and so
we can choose $B=2/\varepsilon$. Thus, we have that $ D_{W,a}(Y;t,\chi',\chi) $ equals
\begin{multline*}
\sum_{\substack{n\leq(Yj)^{1/2+\varepsilon}}
+O((MN)^{\varepsilon}(1+\vert t\vert)^{2/\varepsilon}(Yj){}^{-1})+O((1+\vert t\vert)^{2/\varepsilon}j^{1/4+\varepsilon}Y^{-1}).
\end{multline*}
We wish to apply Heath-Brown's large sieve Corollary <ref>
to the double sum, but to do this, we need to separate the $n$ and
$d_{0}$ dependence on the $G$ function. Hence, we apply Mellin inversion
to render the main term above as
\begin{multline*}
}\prod_{p\mid2MN}\left(1-\frac{(\chi_{d_{0}}\chi)(p)}{p^{1/2}}\right) \\
\times \sum_{\substack{n<(Yj)^{1/2+\varepsilon}}
\end{multline*}
Looking at the summand, for $\Re s=\varepsilon$ and $\Re w=\varepsilon$,
we have
\[
\begin{array}{ccc}
(c_{0}d_{0})^{s/2}a(d)\frac{\chi'(d)}{d^{1/2+w}}\ll d^{-1/2+\varepsilon}
\text{and}
\frac{\chi(n)}{n^{1/2+s}}\ll n^{-1/2+\varepsilon},
\end{array}
\]
so that we can now apply Corollary <ref>,
and the result follows.
§ PROOF OF THEOREM <REF>
We first apply Lemma <ref> which gives
\[
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it;\rho,\rho')=X^{it}\sum_{\substack{(\chi,\chi')\in\Phi\\
\psi,\psi'\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}
\]
where the subscripts for $D$ in the first term are $V=V_{\psi,\psi'}^{(\chi,\chi')}(\xi,t)$
and $Q=Q_{d_{0},d_{1}}^{(\chi'\psi')}({\textstyle \frac{1}{2}})$,
and the subscripts for $D$ in the second term are $V=V(\xi,t)$ and
$P=P_{d_{0},d_{1}}^{(\chi\psi)}({\textstyle \frac{1}{2}})$. Applying
Lemma <ref> further gives
\begin{multline*}
Z({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}+it,\rho,\rho')\ll(1+\vert t\vert)^{2/\varepsilon}(MN)^{\varepsilon}\left[\left(X+(NX)^{1/2}\right)^{1/2+\varepsilon}\right]\\
+(1+\vert t\vert)^{2/\varepsilon}(MN)^{\varepsilon}\max_{\substack{(\chi,\chi')\in\Phi\\
\psi,\psi'\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}
}\left[\beta_{\psi,\psi'}^{(\chi,\chi')}\left(\gamma_{\psi,\psi'}^{(\chi,\chi')}X^{-1}+\left(\gamma_{\psi,\psi'}^{(\chi,\chi')}j X^{-1} \right)^{1/2}\right)^{1/2+\varepsilon}\right].
\end{multline*}
Here, $\Phi$ is given by (<ref>), and $j$ is the conductor
of $\chi'$.
We initially set $X=M^{a}N^{b}$ and eventually choose optimized values
for $a$ and $b$, depending on the bounds for the $\beta$'s and
$\gamma$'s from Table <ref>. We come to an optimal choice
of $X=M^{2/3}N^{1/3}$ and obtain the subconvexity bound of
\[
\]
(even in the case of $M=N$), compared to the convexity bound of
§ PROOF OF THEOREM <REF>
Here we present an application of the theory of double Dirichlet series as developed. Given a fixed positive prime $N$, we seek an upper bound on $d$ such that $L(1/2,\chi_{dN})$ does not vanish. We follow a modified version of a method outlined in <cit.>, and we rigorously prove an important lower bound on coefficients which arise from a residue of $Z$. Let $h(y)$ be a smooth weight function as in Definition <ref>.
Expanding as per (<ref>) and by Mellin inversion
we have
\begin{equation}
\label{eq:int-sum}
\int_{(2)}\tilde{h}(w)Z({\textstyle \frac{1}{2}},w;\chi_{N},\psi_{1})X^{w}dw
= \sum_{\substack{(d,2N)=1}
}L^{(2N)}({\textstyle \frac{1}{2}},\chi_{d_{0}N})P_{d_{0},d_{1}}^{(\chi_{N})}({\textstyle \frac{1}{2}})h({\textstyle \frac{d}{X}}).
\end{equation}
We move the contour of the integral on the left-hand side to $\Re w=-\varepsilon$,
picking up a residue at $w=1$ due to the double pole of $Z({\textstyle \frac{1}{2}},w;\chi_{N},\psi_{1})$
there. If we write its Laurent expansion as
\[
\]
then the left-hand side of (<ref>) equals
\[
[\nu_{N}\tilde{h}(1)+\mu_{N}\tilde{h}'(1)]X+\mu_{N}\tilde{h}(1)X\log X
+\int_{(-\varepsilon)}\tilde{h}(w)Z({\textstyle \frac{1}{2}},w;\chi_{N},\psi_{1})X^{w}dw.
\]
We now apply the symmetric functional equation (<ref>) with $\rho\psi=\chi_N$, $\rho'=\psi'=\psi_1$, and $(s,w)$=$(1/2,w)$ with $\Re w=-\varepsilon$ to $Z$ in the resulting integral. Using Lemma <ref> to bound the coefficients, we see that $C_{\rho\tilde{\chi}_n}=N$ when $n=1$ and is unity otherwise, and $C_{\rho'\tilde{\chi}_m}^{1/2-s}=1$ in every case because $s=1/2$. In every case the $A$ factors will always be bounded above by $O(N^\varepsilon)$, since $\Re w = -\varepsilon$. Also, $ A_m^{(\rho \tilde{\chi}_n)}(s+w-1/2) $ vanishes when $m=N$ and $n=1$. Hence, we have
\begin{multline*}
=\sum_{\psi^{\star}, \psi^{\star\star}, \psi^{\star\star\star}\in\widehat{(\mathbb{Z}/8\mathbb{Z})^{*}}}
\sum_{m,r\mid N}
\\
\times
\left(
N^{1/2-w} Z(1-w,1/2;\chi_N \tilde{\chi}_r \psi^\star \psi^{\star\star\star}, \psi^{\star\star})
+ Z(1-w,1/2;\tilde{\chi}_r \psi^\star \psi^{\star\star\star}, \tilde{\chi}_m\psi^{\star\star})
\right),
\end{multline*}
where, due also to the bound (<ref>), we have
\[
\vert
\vert
\ll
N^\varepsilon (1+\vert w \vert)^{1+\varepsilon}.
\]
We now apply the bound (<ref>), which finally gives
\[
\vert Z({\textstyle \frac{1}{2}},-\varepsilon+it;\chi_{N},\psi_{1}) \vert
\ll
(1+\vert t\vert)^{1+\varepsilon} N^{1/2+\varepsilon},
\]
which we shall apply to the integral. We thus have
\begin{equation}
}L^{(2N)}({\textstyle \frac{1}{2}},\chi_{d_{0}N})P_{d_{0},d_{1}}^{(\chi_{N})}({\textstyle \frac{1}{2}})h({\textstyle \frac{d}{X}})=a_{N}X\log X+b_{N}X+O(N^{1/2+\varepsilon})\label{eq:S-asymptotic-arbitrary-main-term}
\end{equation}
for certain coefficients $a_{N}$ and $b_{N}$. More elementary analysis
of $S(X;\chi_{N})$ via Theorem <ref> (cf.
<ref>) below gives us the lower bounds
$a_{N},b_{N}\gg N^{-\varepsilon}$. If we now assume that
\[
L^{(2N)}({\textstyle \frac{1}{2}},\chi_{dN})=L({\textstyle \frac{1}{2}},\chi_{d_{0}N})\prod_{p\mid2d_{1}N}(1-p^{-1/2})
\]
vanishes for $d\ll X$, then we get a contradiction as long as $a_{N}X\log X$
is greater than the error term. We see we can choose $X=N^{1/2+\varepsilon}$,
as required.
We essentially combine two asymptotic formulas for $S(X;\chi_{N})$:
Looking at it elementarily via Theorem <ref>
from <ref> gives us a bad error term,
but lower bounds on the coefficients. Looking at it analytically as
above allows us to take advantage of the bound (<ref>)
for $Z$ in order to obtain a smaller
error term.
§ AN ASYMPTOTIC FORMULA FOR AN $L$-FUNCTION SUM
§.§ Result
Of particular importance for the smallest nonvanishing quadratic central
value of twisted $L$-functions result is an asymptotic formula for
the weighted and twisted $L$-function sum given by $S(X;\chi)$ in
(<ref>). Indeed, we already have
an asymptotic formula, but it is important that we have a lower bound
on the main term coefficients.
Let $N$ be a natural number, $X$ a large positive real number, and
let $\chi$ be a quadratic primitive character modulo $N$. Let $h$
be a smooth weight function as defined in Definition <ref>,
and define
\[
}L^{(2N)}({\textstyle \frac{1}{2}},\chi_{d_{0}}\chi)P_{d_{0},d_{1}}^{(\chi)}({\textstyle \frac{1}{2}})h(d/X).
\]
We seek to obtain information on the main term coefficients of the
asymptotic formula (<ref>). We
shall prove the following Theorem.
There exists $\delta>0$ such that
we have the asymptotic formula
\[
S(X;\chi)=a_{N}X\log X+b_{N}X+O(N^{3/8+\varepsilon}X^{1-\delta}),
\]
\[
N^{-\varepsilon}\ll a_{N},b_{N}\ll N^{\varepsilon},
\]
uniformly in $N$.
§.§ Proof of Theorem <ref>
According to the approximate functional equation for Dirichlet $L$-functions
(<ref>), we have
\[
L({\textstyle \frac{1}{2}},\chi_{d_{0}}\chi)=2\sum_{n=1}^{\infty}\frac{(\chi_{d_{0}}\chi)(n)}{n^{1/2}}G_{\kappa}\left(\frac{n}{\sqrt{c_{0}d_{0}N}}\right),
\]
where $G_{\kappa}$ is given in (<ref>), and $c_{0}$ is
given in (<ref>). From the definition
of $P_{d_{0},d_{1}}^{(\chi)}$ in Theorem <ref>
we have
\[
P_{d_{0},d_{1}}^{(\chi)}({\textstyle \frac{1}{2}})=\sum_{f\mid d_{1}^{2}}\frac{\mu(f_{0})(\chi_{d_{0}}\chi)(f_{0})}{f_{0}^{1/2}},
\]
where we write $f=f_{0}f_{1}^{2}$ with $f_{0}$ squarefree. We shall
also use the expansion
\[
\]
Applying these expressions to $S(X;\chi)$, we have
\begin{multline}
}\sum_{f_{1}\mid d_{1}}\sum_{g\mid2f_{1}^{2}}\frac{\mu(g)(\chi_{d_{0}}\chi)(g)}{h^{1/2}}\sum_{n=1}^{\infty}\frac{(\chi_{d_{0}}\chi)(n)}{n^{1/2}}G_{\kappa}\left(\frac{n}{\sqrt{c_{0}d_{0}N}}\right)h\left(\frac{d_{0}d_{1}^{2}}{X}\right).\label{eq:S-average-L}
\end{multline}
For a subset $H\subset\mathbb{N}$, we define $S_{H}(X;\chi)$ to
be the same as the expression (<ref>) with the added
condition in the $n$-sum of $ng\in H$. We use Mellin inversion for
$G_{\kappa}$ to separate the variables, and by moving the $d_{0}$-sum
to the inside, we get that $S_{H}(X;\chi)$ equals
\begin{multline}
}\sum_{f_{1}\mid d_{1}}\sum_{\substack{g\mid2f_{1}^{2}\\
}\frac{\mu(g)\chi(g)}{g^{1/2}}\sum_{\substack{ng\in H\\
\sum_{\substack{(d_{0},2N)=1}
\end{multline}
The variables $\kappa$ and $c_{0}$ depend on the residues of $d_{0}$
and $N$ modulo 4. Thus, given $\ell\in\lbrace\pm1\rbrace$, we define
$\kappa(\ell)$ and $c_{0}(\ell)$ to be the corresponding values
for $d_{0}\equiv\ell\ (\text{mod }4)$. For convenience, for $\iota\in\lbrace\pm1\rbrace$,
we define $S_{H}(X;\chi,\iota,\ell)$ to be the same as (<ref>)
except that $\kappa$ and $c_{0}$ are replaced with $\kappa(\ell)$
and $c_{0}(\ell)$, and $\psi_{\iota}(d_{0})$ is multiplied to the
summand in the $d_{0}$-sum.
Observing that $\frac{1}{2}(1\pm\psi_{-1}(d_{0}))$ is the characteristic
function of $d_{0}\equiv\pm1\ (\text{mod }4)$, the $d_{0}$-sum in
the integrand of (<ref>) is
\[
\frac{1}{2}\sum_{\substack{(d_{0},2N)=1}
\]
whence we see that
\begin{equation}
\end{equation}
For treatment of the $d_{0}$-sum, for a positive real number $Y$
and a Dirichlet character $\psi$ we further define
\[
\]
where the sum is over squarefree $d_{0}$. With these simplifications,
we obtain that $S_{H}(X,\chi,\iota,\ell)$ equals
\begin{multline}
}\sum_{f_{1}\mid d_{1}}\sum_{\substack{g\mid2f_{1}^{2}\\
\sum_{\substack{ng\in H\\
\end{multline}
We shall choose $H=\square$ which we use to denote the set of positive
squares, and denoting the complement of $H$ in $\mathbb{N}$ by $\bar{H}$,
it is clear that
\[
\]
In order to estimate the size of $T(s;Y,\psi)$, we observe that there
are order $Y$ squarefree numbers up to $Y$. If $\psi$ is a principal
character and $\Re s=\varepsilon$, we therefore expect this sum to
be roughly of size $Y^{1+\varepsilon}$. If $\psi$ is non-principal,
the oscillations will give us a Pólya-Vinogradov type estimate.
Indeed, with this last point in mind, we step back to explain the
main idea of the proof: The sum $S_{H}(X,\chi,\psi_{\iota},\ell)$
will hence only be large when $\tilde{\chi}_{ng}\psi_{\iota}$ is
principal, that is, precisely when $H=\square$ and $\iota=1$, and
will be small otherwise. To this end, we have the following asymptotic
For $\Re s>0$, we have
\[
T(s;Y,\chi_{0}^{(m)})=\frac{\tilde{h}(1+s/2)}{\zeta(2)}\prod_{p\mid m}\left(1+\frac{1}{p}\right)^{-1}Y{}^{1+s/2}+U(s;Y,m)
\]
with $U(s;Y,m)$ holomorphic and
\[
U(s;Y,m)\ll\vert s\vert^{1/2-\Re s/2+\varepsilon}m^{\varepsilon}Y^{1/2+\Re s/2+\varepsilon},
\]
uniformly in $Y$ and $\Im s$. Further, if $\psi=\chi_{0}^{(m)}\tilde{\psi}$,
where $\tilde{\psi}$ is a nontrivial quadratic primitive character
with conductor $c$, then
\[
T(s;Y,\psi)\ll\vert cs\vert^{1/2-\Re s/2+\varepsilon}m{}^{\varepsilon}Y^{1/2+\Re s/2+\varepsilon}
\]
uniformly in $\Im s$, m, and $Y$.
Via Mellin inversion, we have
\[
T(s;Y,\chi_{0}^{(m)}\tilde{\psi})=\int_{(1+\Re s/2+\varepsilon)}\mathcal{T}_{s}(z)\tilde{h}(z)Y^{z}dz,
\]
where we have the generating function
\begin{eqnarray*}
\mathcal{T}_{s}(z) & = & \sum_{d=1}^{\infty}\frac{\mu^{2}(d)(\chi_{0}^{(m)}\tilde{\psi})(d)}{d^{z-s/2}}=\frac{L(z-s/2,\tilde{\psi})}{L(2z-s,\tilde{\psi})}\prod_{p\mid m}\left(1+\frac{1}{p^{z-s/2}}\right)^{-1}.
\end{eqnarray*}
If $\tilde{\psi}$ is the trivial character then the $L$-function
in the numerator is just the zeta function, and therefore has a pole
at $z=1+s/2$. Due to the $1/L(2z-s,\tilde{\psi})$ factor, all other
poles lie in $\Re z<\Re s/2$. We move the contour to $(1/2+\Re s/2+\varepsilon)$,
and in the case where $\tilde{\psi}$ is trivial, we pick up the residue
\begin{equation}
\frac{\tilde{h}(1+s/2)}{\zeta(2)}\prod_{p\mid m}\left(1+\frac{1}{p}\right)^{-1}Y{}^{1+s/2}.\label{eq:T-main-term}
\end{equation}
Due to (<ref>) along with the convexity bound
\[
L(z-s/2,\tilde{\psi})\ll[c(1+\vert\Im z-\Im s/2\vert)]^{1/2-\Re s/2+\varepsilon}
\]
obtained from (<ref>), we bound the resulting integral
\[
\vert cs\vert^{1/2-\Re s/2+\varepsilon}m^{\varepsilon}Y{}^{1/2+\Re s/2+\varepsilon}.
\]
§.§.§ Main term
As explained above, the main contribution will come from $S_{\square}(X;\chi,1,\ell)$,
which we will bound from below. Looking at the expansion (<ref>)
with $\iota=\ell=1$ and $H=\square$, since $ng$ is square, and
recalling that $g$ is squarefree, we have $n=gm{}^{2}$ for $m\in\mathbb{N}$,
so the inner sum is
\[
\sum_{\substack{ng\in\square\\
\]
By Lemma <ref>, the above expression is
\begin{multline*}
\frac{1}{g^{s+1/2}}\sum_{\substack{(m,N)=1}
\end{multline*}
where $U(s;X/d_{1}^{2},2gmN)$ is holomorphic for $\Re s>0$ and
\[
U(s;X/d_{1}^{2},2gmN)\ll\vert s\vert^{1/2-\Re s/2+\varepsilon}(gmN)^{\varepsilon}\left(X/d_{1}^{2}\right)^{1/2+\Re s/2}.
\]
Referring to (<ref>), and moving the contour to
$(\varepsilon)$, the error term is bounded above by
\begin{multline}
\sum_{\substack{(d_{1},2N)=1}
}\int_{(\varepsilon)}\vert Nc_{0}(\ell)\vert^{\Re s/2}\vert\tilde{G}{}_{\kappa(\ell)}(s)\vert\vert s\vert^{1/2-\Re s/2+\varepsilon}
\sum_{f_{1}\mid d_{1}}\sum_{\substack{g\mid2f_{1}^{2}\\
}\frac{(gmN)^{\varepsilon}}{m^{2s+1}}\left(\frac{X}{d_{1}^{2}}\right)^{1/2+\Re s/2}ds.\label{eq:S-square-error}
\end{multline}
Bounding the sums absolutely and using the fact that $\tilde{G}{}_{\kappa(\ell)}(s)$
decays rapidly in fixed vertical strips, we see that this is bounded
above by $N^{\varepsilon}X^{1/2+\varepsilon}$. As for the main term,
through a calculation we have the following result.
We have the identity
\[
\sum_{\substack{(m,N)=1}
\]
\[
E_{0}(s)=\frac{4}{9}(1-2^{-2s-1})\prod_{p}(1+p^{-1})(1+p^{-1}(1-p^{-2s-1})^{-1})^{-1}\prod_{p\mid N}(1+p^{-1})^{-2}(1+p^{-1}-p^{-2s-1}),
\]
\[
E_{1}(s;g)=\prod_{\substack{p\mid g\\
\]
This is a straightforward but monotonous calculation which we omit.
Applying this, we now have that $S_{\square}(X,\chi,1,\ell)$ equals, up to an error $O(N^{\varepsilon}X^{1/2+\varepsilon})$,
\begin{multline*}
\frac{2}{\zeta(2)}\int_{(\varepsilon)}\tilde{h}(1+{\textstyle \frac{s}{2}})(c_{0}(\ell)N)^{s/2}X^{1+s/2}\tilde{G}{}_{\kappa(\ell)}(s)\zeta(2s+1)E_{0}(s)
\sum_{\substack{(d_{1},2N)=1}
}d_{1}^{-2-s}\sum_{f_{1}\mid d_{1}}\sum_{\substack{g\mid2f_{1}^{2}\\
\end{multline*}
Since $E(s;g)$ is multiplicative in $g$, we can further collapse
the $g$-sum above into an Euler product. Hence we have
\begin{equation}
\end{equation}
\[
H(s) = \sum_{\substack{(d_{1},2N)=1}
}d_{1}^{-2-s}\sum_{f_{1}\mid d_{1}}\prod_{p\mid2f_{1}}\left(1-p^{-s-1}E_{1}(s;p)\right).
\]
We have the following estimates for $H$.
There exists $K>0$ such that $1/3\leq H(0)\leq K$
and $\vert H'(0)\vert\leq K$.
Let $\Re s\geq0$, and for convenience define
\[
H(s;d_{1})=\sum_{f_{1}\mid d_{1}}\prod_{p\mid2f_{1}}\left(1-p^{-s-1}E_{1}(s;p)\right).
\]
Because $0<E_{1}(0;p)\leq 4/3$, we have $1/3\leq1-p^{-1}E_{1}(0;p)<1$,
and so taking $d_{1}=1$, we have $H(0)\geq1/3$. Hence by the same
\[
\vert H(0;d_{1})\vert\leq\sum_{n\mid d_{1}}1\ll d_{1}^{\varepsilon},
\]
and so we see that $H(0)$ is absolutely bounded above. In order to
show that $H'(0)$ is absolutely bounded, because we have
\[
}d_{1}^{-2}(H'(0;d_{1})-H(0;d_{1})\log d_{1}),
\]
it suffices to show that $H'(0;d_{1})\ll d_{1}^{\varepsilon}$, and
for this, it suffices to show that $E_{1}'(0;p)$ is absolutely bounded,
which is easily seen via taking the logarithmic derivative.
We shall also need the following bounds for the $E_{0}$ function.
We have the bounds
\[
N^{-\varepsilon}\ll E_{0}(0),\ E_{0}'(0)\ll N^{\varepsilon}.
\]
We have
\[
N^{-\varepsilon} \ll d(N) \ll \frac{2}{9\zeta(2)}\prod_{p\mid N}(1+p^{-1})^{-2}
=E_{0}(0)\ll 1,
\]
and the result follows.
To treat the derivative, it is easily shown that the logarithmic derivative at $s=0$ is bounded below by a constant and above by $N^\varepsilon$, as required.
We wish to move the contour of integration of (<ref>)
from $(\varepsilon)$ to $(-\varepsilon)$. In doing so, we pick up
a double pole, since $\tilde{G}_{\kappa(\ell)}(s)$ and $\zeta(2s+1)$
have simple poles at $s=0$. The residue of this pole shall be our
main term. In order to calculate it, we shall need further analysis
of the integrand. We define
\[
\]
which is the part of the integrand which is holomorphic at $s=0$.
If the Laurent coefficients of $\zeta(2s+1)$ and $\tilde{G}_{\kappa(\ell)}$
(centred at $0$) are given by $e_{n}$ and $g_{n}$ respectively,
then the residue is
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
\begin{multline}
A'(0)=\frac{1}{2}\tilde{h}'(1)E_{0}(0)H(0)X+\frac{1}{2}\tilde{h}(1)E_{0}(0)H(0)(\log N)X\\
+\frac{1}{2}\tilde{h}(1)E_{0}(0)H(0)X\log X+\tilde{h}(1)E_{0}'(0)H(0)X+\tilde{h}(1)E_{0}(0)H'(0)X.\label{eq:A'-expanded}
\end{multline}
We now prove the following expression of the residue $R$ from the
above results.
For sufficiently large $N$, we have
\[
R=a_{N}X\log X+b_{N}X,
\]
\[
N^{-\varepsilon}\ll a_{N},b_{N}\ll N^{\varepsilon}.
\]
The first term comes from the $X\log X$ term in (<ref>).
We see that $g_{-1}=1$ due to the definition of $\tilde{G}_{\kappa}(s)$
given in (<ref>), and we also easily see that $e_{-1}=1/2$,
so that the coefficient for $A'(0)$ in (<ref>)
is positive. Now the bounds for $a_{N}$ follow from those for $E_{0}(0)$
and $H(0)$ from Lemmas <ref> and <ref>,
and the fact that $\tilde{h}(1)$ is simply a positive constant.
Next, we prove the bounds for the $X$ term coefficient. The upper
bound follows from those for $E_{0}(0)$ and $H(0)$ in Lemmas <ref>
and <ref>, and again from the fact that $\tilde{h}(1)$
and $\tilde{h}'(1)$ are constants. As for the lower bound, there
are two difficulties: First, we do not know if the coefficient for
$A(0)$ in (<ref>) is negative, which would
result in a negative $X$ term contribution from (<ref>).
Secondly, we do not know whether the $\tilde{h}'(1)$ factor in the
first term of (<ref>) is positive. Nonetheless, we
can simply compare the (positive) $X$ coefficient in the second term
of (<ref>) to that of the two terms just mentioned,
and choose $N$ large enough so that the $\log N$ factor dominates.
The lower bound then follows again from Lemmas <ref>
and <ref>.
Now we just need to bound the remaining integral (<ref>)
with contour $(-\varepsilon)$. Using the triangle inequality and
observing that $\tilde{G}{}_{\kappa(\ell)}$ decays rapidly in fixed
vertical strips, we therefore see that the integral with contour $(-\varepsilon)$
is bounded above by $N^{\varepsilon}X^{1-\varepsilon/2}.$
§.§.§ Error term
Using the same method for bounding the error term in the previous
section, we can bound $S_{\square}(X;\chi,-1,\ell)$ from above, since
its expansion according to (<ref>) will have the
factor $T(s;X/d_{1}^{2},\chi_{0}^{(2ngN)}\psi_{-1})$. By Lemma <ref>
this becomes (<ref>). It now remains to bound $S_{\bar{\square}}(X;\chi,\iota,\ell)$
from above.
According to (<ref>) with the integral contour moved
to $(3/4+\varepsilon)$, we see that $S_{\bar{\square}}(X;\chi,\iota,\ell)$ equals
\begin{multline*}
}\sum_{f_{1}\mid d_{1}}\sum_{\substack{g\mid2f_{1}^{2}\\
\sum_{\substack{ng\in\bar{\square}\\
\end{multline*}
Applying Lemma <ref>, since $ng$ is not square,
the character $\chi_{0}^{(2N)}\tilde{\chi}_{ng}\psi$ is never principal,
so we have
\[
T(s;X/d_{1}^{2},\chi_{0}^{(2N)}\tilde{\chi}_{ng}\psi_{\iota})\ll(\vert s\vert ng)^{1/4+\varepsilon}N^{\varepsilon}(X/d_{1}^{2}){}^{7/8+\varepsilon}.
\]
Now we can absolutely bound the sums and ignore the condition that
$ng$ is not square, as we did for (<ref>). We
note that the $n$-sum will absolutely converge since $\Re s=3/4+\varepsilon$,
as long as we select a small enough $\varepsilon$ in the bound for
the $T$-function above. We then arrive at a bound of
\[
S_{\bar{\square}}(X;\chi,\iota,\ell)\ll N^{3/8+\varepsilon}X^{7/8+\varepsilon},
\]
whence by (<ref>) we have sufficiently bounded
§ APPENDIX: SUBCONVEXITY BOUNDS APPLIED TO NON-VANISHING RESULTS
Though the non-vanishing result Theorem <ref> proven in <ref> does not require the subconvexity bound Theorem <ref>, one might ask whether a subconvexity bound could be applied to such a problem. We positively answer this here, outlining another method presented in <cit.>.
In <ref>, we start with (<ref>) and move the contour of the integral from $(2)$ to $(-\varepsilon)$, picking up a residue at $w=1$. Instead, we can move the contour to $(1/2)$. Now it is clear that a good subconvexity bound for $Z({\textstyle 1/2},w;\chi_N,\psi_1)$ with $\Re w = 1/2$ will produce a non-vanishing result. In our case, the subconvexity bound Theorem <ref> will only yield an upper bound of $N^{2/3+\varepsilon}$, which is worse than that already presented. However, this formulation gives motivation for developing subconvexity bounds for double Dirichlet series. In particular, an improvement to Theorem <ref> would result from an improvement in the $N$ exponent lower than $1/4+\varepsilon$ in Theorem <ref>.
§ FUNDING
This work was supported by the Ontario Graduate Scholarship Program; and the European Research Council Starting Grant [258713].
§ ACKNOWLEDGEMENTS
I sincerely thank Valentin Blomer for suggesting the problem and many useful ideas. I also sincerely thank the following individuals who provided useful suggestions and corrections: Solomon Friedberg, John Friedlander, Florian Herzig, Henry Kim, and Stephen Kudla.
|
1511.00605
|
DIMACS, Rutgers University, Piscataway NJ 08854, USA, and Department of Ecology, Evolution, & Natural Resources, Rutgers University, New Brunswick
NJ 08901, USA.
As the understanding of the importance of social contact networks in the spread of infectious diseases has increased,
so has the interest in understanding the feedback process of the disease altering the social network. While many studies
have explored the influence of individual epidemiological parameters and/or underlying network topologies on the resulting
disease dynamics, we here provide a systematic overview of the interactions between these two influences on population-level
disease outcomes. We show that the sensitivity of the population-level disease outcomes to the combination of
epidemiological parameters that describe the disease are critically dependent on the topological structure of the population's
contact network. We introduce a new metric for
assessing disease-driven structural damage to a network as a population-level outcome. Lastly, we discuss how the expected
individual-level disease burden is influenced by the complete suite of epidemiological characteristics for the circulating
disease and the ongoing process of network compromise. Our results have broad implications for prediction and mitigation of
outbreaks in both natural and human populations.
§ INTRODUCTION
Recent advances in our understanding of complex social structures have led to a re-evaluation of epidemiological processes
taking place on these structures <cit.>. Almost all infection models have been shown to behave differently on complex networks
compared to simple lattice structures or to fully mixed models <cit.>, reflecting different potential types of contact patterns
among various populations <cit.>. Typical epidemiological models <cit.> include variations of the basic SI, SIR, and SIS models,
where susceptible (S) individuals can become infected (I) upon encounter with other infected individuals and eventually either
recover with immunity (R-state where they cannot be re-infected) or without immunity (i.e. return to the susceptible state).
Each of these models is appropriate to describe varying conditions of spreading.
Thus far, however, no network-based analysis has considered cases in which the disease both generates a limited-duration
immunity that eventually lapses back into susceptibility
(due either to genetic drift, including antigenic drift <cit.> of the pathogen
or to loss of T-cell memory <cit.>, while also carrying a non-trivial risk of disease-induced mortality <cit.>.
While both of these examples focus on influenza virus <cit.>,
many pathogens exhibit this pattern of generation of immunity that later wanes.
Explicit study of such a case may, in fact, be of particular practical importance since both disease-related death and temporary
immunity will interrupt successful disease transmission over the remaining network. The dynamics between permanent removal
(i.e. death) and the temporary removal (i.e. immunity) may drive the emergence of very different global patterns in disease outcomes.
As has been well-studied <cit.>, the interplay between the network structure and the dynamic spreading process strongly influences
the outcomes of an epidemic. In lattice structures, all nodes have a similar importance to disease risk and spread because of the
spatial invariance and the homogeneous character of the system, so that these effects do not exist. In contrast, in complex network
the structure is dominated by the existence of well-connected hubs. These nodes are included in the majority of all possible paths,
so that a disease can easily reach them. Even nodes that are not well-connected can become very significant in spreading if they happen
to be in an appropriate location <cit.>. Of course, for pathogens that carries a non-trivial mortality risk, frequent infection of a node
in any network will eventually lead to its removal. This removal greatly impacts the structural features of the remaining network.
The topology becomes more hostile to spreading and areas that were easily connected through the hubs can now become protected by the
disease simply by isolation. This isolation, though, can have detrimental effects on proper communication in the network. In short,
the interplay between the exposure of nodes to infection and their asymmetrical impact of removal on both topology and dynamics,
creates a complex cycle with unusual epidemiological properties.
Disease-induced mortality itself may be especially important to consider when non-disease-related processes of network function may be
drastically diminished by disease-induced structural compromise, even though exactly such disconnection acts to diminish the probability
of transmission of future infection for the remaining nodes (by decreasing effective population density) <cit.>. In these cases, some
standard measures of structural integrity of the population may seem uncompromised (e.g. largest remaining connected component) <cit.>,
even though function can be reduced to the point of failure (e.g. increased average minimal path lengths for communication between
individuals in the population) <cit.>. To study these types of functional effects, we introduce a new measure, the stability index,
which can take into account partial structure compromise as a result of an infectious epidemic with an associated mortality risk.
This consideration may be useful in fields such as conservation biology or communication networks.
§ METHODS
§.§ The SIRDS model.
We model the spreading of a potentially fatal epidemic disease in a population. Recovery from the disease provides short-term
immunity <cit.>. The disease originates in a randomly selected node (then automatically designated in the infected class, I)
in an otherwise fully susceptible population, S. In every time-step, all the susceptible neighbors of the infected nodes become
infected with probability $\beta$ per infected neighbor. After attempting to infect their neighbors, the infected nodes leave
the I class at a recovery rate $\gamma$. These nodes either die with probability $f$, so that they fall in the D (Deceased), or recover
into the R (Recovered) class with probability $1-f$. Nodes in recovery are immune-protected, losing that protection at a rate $r$,
the rate of loss of protection.
The number of surviving nodes in this model is a function of time, $N(t)$, with the initial population being $N(0)=N$ nodes.
We describe this SIRDS process in Fig. <ref>. The equations that describe this model are:
\begin{align}\label{eq1}
\dot{S} = -\beta S I + rR \\
\dot{I} = \beta SI - \gamma I \\
\dot{D} = \gamma f I \\
\dot{R} = (1-f)\gamma I - rR \\
N(t) = S(t)+I(t)+R(t) = 1 - D(t)
\end{align}
Schematic of the SIRDS process.
The mortality probability, $f$, determines what percentage of the infected population will leave the system. The two limiting cases
of the SIRDS model correspond to a simple SIR model, when $f=1$, and to an SIRS model, when $f=0$. There are no ongoing demographic
processes, such as natural births or natural deaths in the model, so death results only as an outcome of the modeled disease.
We fix the rate $\gamma$ to $\gamma=1$, which also fixes the time-scale of the system so that our time unit throughout the
simulations is equal to the recovery time. In the following, we study the effect of the remaining parameters in the system, infection probability, $\beta$, mortality
probability, $f$, and protection loss rate, $r$.
The main quantity of interest in our current study is the mass of dead individuals, $D$, and the conditions which determine its
asymptotic, $D(\infty)$, and finite-time value, $D(t)$. We will use this quantity to further estimate the probability that the
population will retain its long-range connectivity. This is a measure of the viability of the population and its ability to
survive against spreading of fatal infections.
§.§ Network structure.
We study the SIRDS process on two typical structures for the initial population connectivity: a square two-dimensional lattice
and a scale-free network. We further split the scale-free network examples into two cases: a) random scale-free networks,
which have been used to describe large-scale human populations <cit.>, and b) self-organized networks, which may be more reflective
of emergent structures in many natural populations <cit.>. In all cases, we study small networks of size $N=200$, compromising between
an accurate order of magnitude for many natural populations of concern for ongoing persistence and sufficiency of size to enable
meaningful computational observations. We have also verified that the results are not significantly influenced when we increased the
size to $N=1000$ (see e.g. Fig. <ref>).
The random scale-free networks are created by the configuration model <cit.>, with a power-law degree distribution of exponent $\lambda=2.5$,
i.e. $P(k)\sim k^{-2.5}$. In this structure most nodes have a low degree but the hubs are quite strong, with each hub connected
to roughly 5-30% of the network.
The self-organized networks describe a different social organization, where a few nodes act as super-hubs and are connected to
almost every other node in the system <cit.>. This simulates a strongly hierarchical society, where a few `alpha' animals dominate
over the entire group (e.g. grooming behaviors in primates, <cit.>). We chose the model parameters so that we remain
consistent with previous research on such structures <cit.>. This network is built as follows: All nodes start with an initial degree $k=5$, with 5 randomly selected nodes as neighbors.
We assume that the nodes evolve their connections, with the goal of connecting to the most-connected nodes. Therefore,
at each step all nodes remove their two neighbors with the smallest degree and create new links towards two randomly selected nodes.
In this way, the nodes preferentially attach themselves to the largest hubs, which become progressively larger until at the end they
are connected to almost the entire network. We use this final form of the network as the static representation of connectivity,
and we apply the SIRDS process on this structure.
§.§ Survival probability.
We characterize the survival of nodes through the survival probability, $\Phi$. Here, $\Phi$ is defined as the probability that a node
that is alive at the present state of the spreading process will remain alive until the disease has died out. We consider this property
to be a function of either time, $\Phi(t)$, or of the percentage of removed nodes, $\Phi(p)$. These two parameters describe the state
of the network, from a different perspective. The first parameter is the number of steps, $t$, since the beginning of infection,
and this measure can be useful to determine how much time is available to intervene, independently of the current network damage.
The second parameter is the fraction $p$ of the initial network that has been removed due to the disease, independently of the time
required to reach this level of damage. In practice, we quantify the evolution of the survival probability by the point
$t_{0.9}$ or $p_{0.9}$ when $\Phi$ first becomes equal or larger to 90%, $\Phi>0.9$, i.e. after the point where more than 90% of
the remaining nodes will eventually survive.
§.§ Robustness.
The resilience of a network with regards to node removal is typically measured through the size of the largest cluster remaining,
$S_{\rm max}$, compared to the size of the largest cluster before removing any nodes <cit.>. The connectedness of a cluster does not
guarantee, though, the efficient operation of the network. For example, the fact that one node can still reach another node may not
be as important as the fact that the path length between two nodes has increased so much that it is no longer meaningful to consider
the two nodes reachable from each other. This behavior has been described by the concept of limited path percolation (LPP) <cit.>.
In LPP, two nodes which are originally at distance $\ell_{ij}$ from each other are considered to be connected after the attack only if their
new distance is smaller than $a\ell_{ij}$. The parameter $a$ indicates our tolerance of the communication distance. A value of $a=1$
requires that the original distances remain intact for the nodes to be considered connected, while $a=\infty$ implies that the distances
are no longer important and this case coincides with the typical identification of the largest connected component.
Here, we suggest a combination of two separate ideas to describe the efficiency of the remaining connected cluster.
First, instead of using the fraction of nodes in the largest cluster, we calculate the area under the dynamic calculation of this fraction
$\int_0^{p_{\rm max}} S_{\rm max}(a) dp$.
This method has been introduced to estimate the efficiency of an attack strategy, independently of the final value $S_{\rm max}$ <cit.>.
A key point in our method is that the evolution of the node removal is not described by time, for example by the number of steps.
Instead, we use the fraction of removed/dead nodes, $p$, which is a direct result of the disease-induced mortality process.
In this way we can directly compare different processes on different networks when the same number of nodes has been removed in each case.
Another quantity that characterizes the spreading process is the maximum number of nodes that have been removed due to infection, $p_{\rm max}$,
until there is no infection in the system. Notice that the largest cluster does not necessarily vanish at $p_{\rm max}$, and this is what
separates this index from the unique robustness measure, introduced in <cit.>. That definition cannot be applied if the removal process terminates
before the largest cluster vanishes. Additionally, the maximum possible value for the integral is not necessarily 0.5, and proper normalization
needs to take this fact into account. Therefore, the quantity of interest is the area under the curve of $S_{\rm max}$ from $p=0$ to $p_{\rm max}$.
We further normalize this quantity by the area under the case of minimum possible damage, $1-p$, where the only nodes that leave the spanning cluster
are those that have been physically removed. As a result, we define a stability index, $B(a)$, as
\begin{equation}
B(a) = \frac{\int_0^{p_{\rm max}} S_{\rm max}(a) dp}{\int_0^{p_{\rm max}} (1-p) dp}
\label{eq2}
\end{equation}
The limits of this expression are a) $B=1$ when the least possible damage has been done and the only nodes missing from the largest cluster
are those that have been removed by the disease, and b) $B=0$ when the largest cluster vanishes immediately after removing a few nodes.
Therefore, the stability index can characterize the extent of damage in the remaining largest cluster, independently of its final size.
The stability index coincides with the unique robustness measure <cit.> when $p_{\rm max}=1$ and $a=\infty$.
The second key idea that we use in this definition comes from the Limited Path Percolation method <cit.>, through the parameter $a$.
This parameter allows us to characterize the same cluster under varying requirements for functionality. The typical case where connectivity
between any two nodes is enough for network function, independently of how long the distance between these nodes has become,
is expressed by the value of $a=\infty$. We use this case to normalize the results. The ratio of $B(a)/B(\infty)$ then, indicates our
possible error when we decide about the survival of a group strictly from the existence of a spanning cluster. An important feature
of this error is that it includes information from the entire process and not only from the final state. For example, damage at earlier
stages leads to a smaller $B$ value.
§.§ Computational experiments
We apply the SIRDS model for three different rates of loss of protection, $r$: $r=0.05$, 0.20, and 0.50. Each of these values of $r$ indicates a different
duration of immunity for an already infected node. In the first case, a recovering node remains immune for 20 time steps, presenting a natural
obstacle for spreading over a significant amount of time. In the second case, immunity lasts for an intermediate interval of 5 steps,
while in the latter case the node becomes susceptible after only 2 steps. We independently vary the probabilities $f$ and $\beta$ from 0.05 to 1,
in steps of 0.05. For each case we average over 20 different realizations of the structure. In each realization every node serves as the infection
origin 5 times, so that each point has been averaged over a total of 100,000 simulations of the epidemic process. For each case we record the
fraction of the population, $D$, that died because of the disease and the duration of the epidemics, $T$, defined as the time from the initial
infection until when there is no infected individual. All simulations were run until the infection died out, independently of the number of steps
required to reach this stage or the number of infected/diseased nodes. The code for the simulation of the SIRDS is freely available
at <http://www.rci.rutgers.edu/ feffermn/code.php>.
Parameter ranges were chosen to explore a sufficient diversity of epidemiological characteristics to
demonstrate how different diseases may produce substantially different results, and we recommend that specific
analyses for particular diseases make use of rates tailored to the specific population/network of interest.
§.§ Notation
In Table <ref> we summarize the notation that we use throughout the paper:
Symbol Definition
$t$ Number of time steps in the simulation
$S(t)$ Number of susceptible individuals at time $t$
$I(t)$ Number of infected individuals at time $t$
$R(t)$ Number of recovering individuals at time $t$
$D(t)$ Number of dead individuals at time $t$
$N(t)$ Number of surviving individuals at time $t$
$\beta$ Infection probability
$\gamma$ Recovery rate
$f$ Probability of death for an infected individual
$r$ Loss of protection rate
$T$ Maximum epidemic duration (time until $I(t)=0$)
$p$ Fraction of dead individuals $=D(t)/N(0)$
$p_{\rm max}$ Maximum value of $p$, at $t=T$
$\lambda$ Degree exponent for the scale-free networks
$\Phi(t)$ Probability for a node to survive until $T$, given that it is alive at time $t$
$\Phi(p)$ Probability for a node to survive until $T$, given that $p$ nodes have been removed
$t_{0.9}$ Number of time steps when the survival prob. first becomes $\Phi(t)\geq0.9$
$p_{0.9}$ Fraction of removed nodes when the survival prob. first becomes $\Phi(p)\geq0.9$
$S_{\rm max}$ Fraction of surviving nodes that form the largest remaining cluster
$\ell_{ij}$ Shortest path distance between nodes i and j in the original network
$a$ Nodes are considered disconnected if their distance becomes $>a\ell_{ij}$
$B(a)$ Stability index, defined in Eq. <ref>
Definitions of the main parameters and properties used in the paper.
§ RESULTS
§.§ Fraction of dead population
For a two-dimensional lattice the picture is very similar to what we would expect from a standard SIR model (Fig. <ref>, top row).
In the SIR model (which corresponds to the SIRDS model when $f=1$), there is a sharp transition as we increase the infection probability,
$\beta$, from a `safe' population with almost no mortality to nearly complete annihilation at $\beta>0.5$. A similar pattern is observed here
in the results for the lattice.
The mortality probability, $f$, has little influence, as long as it has a value that
is not close to 0, e.g. $f>0.1$. As we increase the protection loss rate for the same infection and mortality rates,
the influence of $f$ becomes weaker and a larger part of the
population dies: in a faster recovery the nodes spend more time in the susceptible state where they can be infected rather than in
the recovering state, where they are immune.
Fraction of the population that died because of the disease, as a function of the infection probability and the death probability.
From top to bottom: two-dimensional lattices, random scale-free networks ($\lambda=2.5$), and self-organized networks.
The x-axis corresponds to the infection probability, $\beta$, and the y-axis to the death probability, $f$, of an infected node.
The columns correspond to the protection loss rate, $r$, of a node (left to right): $r=0.05$ ($N=1000$), $r=0.05$ ($N=200$), $r=0.20$, and $r=0.50$.
The threshold point for a given combination of parameters is defined as the point where the diseased fraction becomes larger than zero,
independently of the infected mass. In the plots, the threshold values for each case can be found at the point where the blue area turns into green.
The picture is quite different in random scale-free networks (Fig. <ref>, middle row). The region of total annihilation is now
restricted to high values of both $f$ and $\beta$. Here, we consider the threshold point for epidemics to be the combination of parameters
where the diseased fraction becomes larger than zero, independently of the infected mass. Even though the threshold for an epidemic outbreak remains close to $\beta=0.5$,
the inflicted damage is considerably smaller than in a lattice. The hubs are connected to a significant fraction of the network,
while the majority of the nodes have very few connections. These conditions result in efficient protection of the population.
In contrast, the hubs in self-organized networks are much stronger and are connected to almost all other nodes (Fig. <ref>, bottom row).
This makes the network more vulnerable, even for low mortality probabilities $f$. The percolation threshold for $\beta$ is considerably smaller
than in lattices and scale-free networks. The threshold value is now close to $\beta=0.1$, which indicates that it is much easier for
epidemics to occur because of the extremely centralized nature of the network. Moreover, similar results are obtained even when $f$ is very low,
in particular for large protection loss rates, $r$. Indicatively, when $\beta=0.3$ and $f=0.3$ at $r=0.50$ the infection leaves almost 55% of the
population dead, while the corresponding fraction in scale-free networks is around 6% and in lattices it is 0.3%. These differences point out
the different structural character of each system and its influence on mortality due to epidemic spreading.
The influence of the protection loss rate $r$ on the results is mainly quantitative. The behavior of the dead fraction does not change a lot
as we increase the rate of protection loss for the same structure, and the general features that we find in the plots for small $r$ also apply to those of large $r$. In the
following sections, we also find that $r$ mainly influences the numerical values of the epidemic duration and the survival probability, but has
otherwise a limited effect.
§.§ Duration of outbreak
An important feature of the spreading process is the duration of the epidemics. A longer duration leaves a much larger time window for
possible intervention, while a shorter duration may complete the maximum spreading cycle before any action can be taken.
In lattices, the duration is dictated by the value of $\beta$ and is almost independent of $f$, except for large $\beta$ and small $f$
values where we observe somewhat longer durations (Fig. <ref>, top row). This is in contrast to scale-free networks (Fig. <ref>, middle row),
where three regimes are found: a) low-$\beta$ regime: the disease lasts only for a couple of steps and dies rapidly without causing any damage,
b) intermediate-to-large $\beta$ regime and large $f$ values: the duration of the epidemics in the $N=200$ network is of the order of 10 steps.
Even though the duration is small, the damage is considerable, and c) intermediate-to-large $\beta$ regime and small $f$ values: the epidemics now
can last for more than 100 steps, even though it is not as lethal as the previous case. In self-organized networks (Fig. <ref>, bottom row) the
picture is basically the same as in scale-free networks, but now the epidemics may spread extremely fast even at large values of $\beta$. Interestingly,
even though the duration may change by an order of magnitude as we vary the mortality probability, the end result is always a large fraction of the
population dying (from 75-100%).
Duration of epidemic: time till stochastic die-out.
From top to bottom: two-dimensional lattices, random scale-free networks ($\lambda=2.5$), and self-organized networks.
The x-axis corresponds to the infection probability, $\beta$, and the y-axis to the death probability, $f$, of an infected node.
The columns correspond to the protection loss rate, $r$, of a node (left to right): $r=0.05$, $r=0.20$, and $r=0.50$.
§.§ Classification of the outbreak
Figs. <ref> and <ref> suggest the existence of roughly four regimes for the corresponding probabilities:
a) Low mortality – Low infectivity, b) Low mortality – Large infectivity, c) Large mortality – Low infectivity, and d) Large mortality – Large infectivity.
Lattices and scale-free networks are quite similar qualitatively in all these regimes, with large damage when both $\beta$ and $f$ are large.
The distinguishing feature of self-organized networks is that the dead fraction can be very large when only one of the two basic parameters, $\beta$ or $f$,
is large, even if the other one remains relatively small.
In Fig. <ref> we combine both the extent of damage and the outbreak duration. For every pair of $\beta$ and $f$ parameters we assign a color depending
on whether more than half of the population died because of the disease and whether the outbreak lasted more or less than 10 steps. In this way, there are
four possible classifications of short/long duration combined with large/small damage. Notice that the duration of the epidemic is only determined
by the number of steps until there is no infected node. A long or short duration is possible even if the end result of a spreading process
is that there are no diseased nodes.
Combination of the results on dead population and duration.
The color in the plots, as explained in the index, indicates whether the infection destroyed more or less than half the population
and whether it lasted more or less than 10 steps. Red and yellow indicate the areas of larger damage.
The case of smallest damage is when the dead fraction is low and the disease dies out quickly (blue color in the plots). A trivial result is that this behavior dominates
when $\beta$ is very small, independently of the value of $f$, because a low infection probability drastically limits the infection spread
from the node of initial disease introduction.
In lattices, the process evolves slowly, except when there is little to no node removal. This long duration can lead either to extended damage (yellow areas)
for large $\beta$ and $f$ values or to limited damage (green areas) when $f$ is smaller. Lattices never exhibit short duration/extended damage combination
(red areas) under any combination of parameters, which is the result of the extended spatial distances from the absence of hubs in the structure.
Interestingly, in scale-free networks the dominant area of short duration/little damage extends over much larger values of $\beta$,
as a result of the small damage in general, showing an alternative behavior only under large $\beta$ and $f$ values (Fig. <ref>). The case of long duration
with extended damage only emerges in a narrow range of very large infection probabilities and moderate mortality rates in scale free networks. This shows that
for scale-free networks extended damage occurs very rapidly, and only for large values of $\beta$ and $f$. Otherwise, the infection either dies out quickly or
is not able to destroy a significant part of the network.
In self-organized networks, the hubs strongly dominate the structure and create a much smaller-world. This leads to a generally short duration. For small
protection loss rates, $r=0.05$, the short duration/little damage area dominates the plot, showing extended damage only when mortality rates are extremely high.
For higher protection loss rates, a longer duration was observed for small mortality rates, $f$, which switched from small damage to higher damage as we moved
from $r=0.2$ to $r=0.5$.
§.§ The survival probability
The survival probability of a node increases, for the most part monotonically, both with time and $p$. This is the expected behavior: as time passes,
a larger number of nodes die and the network becomes increasingly sparse. Large parts of the network are effectively isolated from the disease,
so that the remaining nodes are less exposed to further infections. In Figs. <ref>a and <ref>b we compare the survival probability $\Phi$
for different topologies. We can see that in lattices, for example, the survival probability $\Phi(t)$ increases at a slower rate than
in (e.g.) self-organized networks. This indicates that if we focus on the duration of the epidemics only, a node in a lattice remains potentially
vulnerable to infections for a longer time. In self-organized networks this process is much more rapid. On the other side, when we consider $\Phi(p)$
as a function of the percentage of removed nodes, it takes a much larger number of removals for a node in self-organized networks to start feeling `safer'.
Survival probability. (a) Average survival probability for a node that has survived after $t$ steps as a function of $t$. The lines correspond to the three different topologies,
for a given set of parameter simulations. (b) Average survival probability for a node that has survived after a fraction, $p$, of nodes has died of the disease.
(c) The time $t_{0.9}$ for different parameters $r$, $f$, and $\beta$. (d) The fraction $p_{0.9}$ for different parameters $r$, $f$, and $\beta$. Combination
of the results on dead population and duration.
The results for the $t_{0.9}$ and $p_{0.9}$ points are shown in Figs. <ref>c and <ref>d for all combinations of the model parameters, $r$, $\beta$, and $f$.
It is possible that the epidemic never kills more than 10% of the population, in which case $t_{0.9}=0$, and there is no time when the probability of
survival is less than 90%. When the final percentage of the dying population is more than 90%, then the corresponding tipping point $t_{0.9}$
becomes infinite, i.e. it is certain that all nodes will die. This case appears only for $f=1$ and large values of $\beta$.
The values of $t_{0.9}$ are 0 for small values of $\beta$, practically independently of the values of $f$. Other than that, the influence of the infection
probability $\beta$ is much weaker than the mortality rate $f$. Large values of $f$ result in small values for $t_{0.9}$, i.e. there is very little time
until a node can feel safe. At small values of $f$, though, this time becomes orders of magnitude larger, and it may take more than 200 steps until the
survival probability reaches 90%. The results are rather similar among the studied topologies.
When we consider $\Phi(p)$, instead, the topology has a greater influence on the results. For the square lattice, the point $p_{0.9}$ is close to $p_{0.9}=1$
when both $\beta$ and $f$ are large. When $f$ is small this value is closer to $p_{0.9}=0.5$, and vanishes for small $\beta$. In self-organized networks, $p_{0.9}=1$
for large $\beta$ values but now $f$ is small. When $f$ is larger, then this value drops to $p_{0.9}\sim 0.8$. When $\beta$ is small, the influence of $f$ is
negligible, but the values of $p_{0.9}\sim 0.25$ are significantly higher than in the case of lattices ($p_{0.9}\sim 0$).
§.§ Functional impact to structure from disease-induced mortality
Having established how epidemics of this type function over the network structures that we study, we turn our attention into the real damage done by the infection.
The end fraction of dead individuals is an indication of what part of the social structure has survived, but the most important quantity for functional purposes
is the connectivity of the remaining network. Typically, this is described through the size of the largest remaining cluster, $S_{\rm max}$. However, as mentioned
in the Methods section, it is possible that the remaining cluster is connected but the distances are so large that communication in the network may no longer function properly.
Compared to other robustness measures, such as e.g. the largest cluster size or the unique robustness measure, the use of the stability index offers two main advantages,:
a) it can measure the structural damage even if the disease has not eliminated the largest cluster (integral calculated up to $p_{\rm max}$),
and b) the extent of damage can reflect our tolerance for the spatial expansion of communication lengths.
In scale-free networks (Fig. <ref>b), an increase of either the infection probability or the mortality rate leads to a rapid decrease of the stability
index from $B(\infty)\sim 1$ to $B(\infty)\sim 0.5$. In self-organized networks, though, the variation of the infection probability does not have the same
impact as the mortality rate. These networks are found to be more robust than the random scale-free networks, even though under the same conditions they lose
more nodes due to disease-induced deaths (Fig. <ref>). We repeated the same set of simulations for the case of $a=1.5$, i.e. two nodes are not considered
connected if their distance exceeds 1.5 times their distance in the original network. In this case (Fig. <ref>b, bottom) the dependence on $\beta$ and $f$
remained qualitatively the same, but now the values of the stability index were considerably lower.
Structural impact of the epidemic process.
(a) Calculation of the quantity $B(a)$. For a given value of $a$ we calculate the ratio of the area under the $S_{\rm max}$ curve over the area under the case
of least possible damage $1-p$. (b) The values of $B(a)$, calculated in scale-free and self-organized networks, as a function of $\beta$ and $f$. The top two
rows correspond to $a=\infty$, and the bottom two rows to $a=1.5$. (c) Comparison of the structural damage measured through the largest cluster size ($a=\infty$)
vs the damage measured through limited-path-percolation ($a=1.5$). The ratio indicates that the largest cluster may overestimate the robustness of the
social structure by a factor of up to $1.5$.
We quantified this significant variation of the stability index, $B(a)$, with decreasing $a$ (Fig. <ref>c) by comparing the ratio of the stability index
$B(\infty)/B(a=1.5)$ for pairs of ($\beta$, $f$). The differences are relatively small for small infection probabilities $\beta$, but they increase as we increase
$\beta$ and become more prominent as we increase $f$. In the most extreme case, the ratio has a value of 1.5, which means that we over-estimate the probability
of the network remaining connected by a factor of 50%. This can be crucial in border-line cases, where the existence of a connecting cluster would signal the
survival of a group, but in practice the extended damage leads to long connectivity paths that render communication difficult or impossible.
The curves reach a maximum value (i.e. maximum error) for values of $\beta$ in the range of 0.4-0.5, indicating that the error diminishes for higher
infection probabilities, where typically epidemics spread over the entire population.
§ DISCUSSION
Disease-induced mortality models can lead to a rapid extinction of the underlying population, but the conditions required for this may be far from trivial.
In particular, a scale-free network topology may accelerate spreading but it also limits the extent of the area that is susceptible to infection.
These conflicting factors can be traced to the effect of the hubs, which can easily reach different parts of the network. However, if a hub remains immune
or removed because of the disease it facilitates disease isolation and communication between different network areas becomes much more difficult.
Contrary to intuition, extensive damage in scale-free networks occurred only for very high probabilities of infection and mortality. When the hubs
become extremely dominant, such as in self-organized networks, then the dominant parameter is the infection probability rather than the mortality rate.
Even small mortality rates can lead to network destruction, as long as the infection probability remains high and preserves the infection in the system.
We see, then, that the impact of the hubs is not as straightforward as intuition may suggest from their role of bringing all network nodes closer to each other.
All the nodes in lattices are equivalent, but the fraction of removed nodes is systematically higher than in scale-free networks, under the same simulation conditions.
The epidemics duration in lattices is, of course, much longer because of their large-world character. Despite these differences, the survival probability is comparable
in both cases, and a node can feel `safe' from removal after surviving roughly 10 steps.
The fate of the infection spreading depends on the interplay between the model parameters $r$, $f$, and $\beta$ and the structure itself. In general,
small protection loss rates, $r$, protect nodes by providing temporary immunity and possibly allowing the infection to be removed from their neighborhood.
However, this change is mainly quantitative, while qualitatively the behavior remains similar as we increase $r$. Unsurprisingly, the main drivers of
the epidemics are $\beta$ and $f$. Obviously, when both the infection probability and the mortality rate are high, the infection quickly eliminates
the majority of the system. Critically, when only one of these parameters is large or if their values are relatively high, though, then the behavior
largely depends on the structure. For example, a small infection probability in lattices prevents extensive mortality even if $f$ is large.
In self-organized networks we find the opposite picture, where $\beta$ does not influence the outcome but a large mortality rate leads to extended node removal.
These outcomes reveal a much more complicated and nuanced dynamic for the spread of infectious diseases in network-structured populations than have previously been explored.
This suggests that many of the results in the literature, which have been assumed to apply generally to a diversity of diseases and a range of qualitatively similar
network structures, may actually apply only more narrowly to certain ranges of combinations of those descriptors.
We also found that if we quantify network robustness based on the size of the largest cluster only, we may over-estimate the efficiency of the network by a factor of 50%.
The natural conclusion is that survival of a connected structure does not necessarily mean that the functionality remains intact, and depending on the communication
requirements the network may have already stopped functioning as intended. To address this, we introduced the stability index suitable for describing the extent of
structural damage during a spreading process. The index can quantify the efficiency of communication in the resulting disease-affected structure by going beyond
the existence of the connected cluster, and taking into account the increase in path lengths. This index incorporates information both from the path lengths and the
dynamics of the spread of the disease. As such, it offers many distinct advantages: a) we do not assess the damage by a binary measure, i.e. the existence or not of
a spanning cluster, b) the stability index can be readily compared across different networks, since it considers the damage up to the point where the infection dies out,
c) the index takes into account the removal history, so that damage done at earlier stages leads to smaller indexes, and d) the structural damage is evaluated according
to the loss of paths, so that differences in clusters with the same number of nodes are still captured by the index.
This study demonstrates the effect of disease-induced mortality in a population, assuming it undergoes one epidemic outbreak that leaves the network weakened,
compared to its initial state. Subsequent outbreaks can accumulate additional damage on the network robustness, but now spreading starts in a different initial structure.
This results to a huge amount of possible combinations (the second disease may have different features than the first). The susceptibility of the resulting networks can be
indirectly found by using the results presented above to apply the same process to the damaged network, instead of the unperturbed structure. We plan to study the effect
of repeated epidemics in a future work.
These results imply that very specific scenarios may offer greater protection from outbreaks that could otherwise compromise populations. This conclusion may be of special
concern in the context of (re)emerging zoonotic infections where populations from multiple host species may be affected in different ways due to differences in physiological
responses to infection. Patterns in the species-to-species paths by which zoonotic diseases reach human populations are dependent on the survival of infected animal populations
at levels that permit continued circulation of disease for long enough to interact with humans (or at least other intermediate animal hosts) to enable transmission.
Such patterns are critical to the metapopulation dynamics in the ecology of infectious diseases <cit.>, and the models here presented provide greater insights into the driving
forces that may produce these patterns in ways that have gone as yet unexplored. In this way, we may provide an otherwise-missing element needed to estimate zoonotic risks based
on the interaction of epidemiology and social behavior in the reservoir species involved.
Our investigations show nontrivial interactions among the parameters of transmission and mortality risks and the network structure in a more nuanced way than is usually described
when studying disease spread on networks. Additionally, the stability metric presented extends our ability to quantify the expected practical outcomes to populations experiencing
outbreaks beyond the traditional measures to include plausibility of communication, rather than just possibility of communication. Together, these types of analyses over both
epidemiological and topological variations increase our understanding of the extent and timing of population vulnerability to outbreaks of infectious diseases.
We thank the Dept. of Homeland Security for funds in support of this research through the
CCICADA Center at Rutgers, and NSF EaSM grant No. 1049088.
Brockmann D, Helbing D.
The hidden geometry of complex, network-driven contagion phenomena.
Science. 2013;342:1337–1342.
Saramäki J, Kaski K.
Modelling development of epidemics with dynamic small-world networks.
Journal of Theoretical Biology. 2005;234:413–421.
Verdasca J, Telo da Gama MM, Nunes A, Bernardino NR, Pacheco JM, Gomes MC.
Recurrent epidemics in small world networks.
Journal of Theoretical Biology. 2005;233:553–561.
Bansal S, Grenfell BT, Meyers LA.
When individual behaviour matters: homogeneous and network models in
Journal of the Royal Society Interface. 2007;4:879–891.
Balcan D, Colizza V, Goncalves B, Hu H, Ramasco JJ, Vespignani A.
Multiscale mobility networks and the spatial spreading of infectious
Proceedings of the National Academy of Sciences of the United States
of America. 2009;106:21484–21489.
Goltsev AV, Dorogovtsev SN, Oliveira J, Mendes JF.
Localization and spreading of diseases in complex networks.
Physical review letters. 2012;109:128702.
Black AJ, House T, Keeling MJ, Ross JV.
The effect of clumped population structure on the variability of
spreading dynamics.
Journal of theoretical biology. 2014;359:45–53.
Keeling M, Eames K.
Networks and epidemic models.
J R Soc Interface. 2005;2:295–307.
Pastor-Satorras R, Vespignani A.
Epidemic spreading in scale-free networks.
Physical Review Letters. 2001;86:3200.
Eubank S, Barrett C, Beckman R, Bisset K, Durbeck L, Kuhlman C, et al.
Detail in network models of epidemiology: are we there yet?
Journal of biological dynamics. 2010;4:446–455.
Eubank S, Guclu H, Anil Kumar VS, Marathe MV, Srinivasan A, Toroczkai Z, et al.
Modelling disease outbreaks in realistic urban social networks.
Nature. 2004;429:180–184.
Sah P, Singh LO, Clauset A, Bansal S.
Exploring community structure in biological networks with random
BMC Bioinformatics. 2014;15.
Jones JH, Handcock MS.
An assessment of preferential attachment as a mechanism for human
sexual network formation.
Proceedings of the Royal Society B-Biological Sciences.
Gallos LK, Rybski D, Liljeros F, Havlin S, Makse HA.
How people interact in evolving online affiliation networks.
Physical Review X. 2012;2:031014.
Anderson R, May RM.
Infectious disease of humans.
Oxford, UK: Oxford University Press; 1991.
Smith DJ, Lapedes AS, de Jong JC, Bestebroer TM, Rimmelzwaan GF, Osterhaus AD, Fouchier RA
Mapping the antigenic and genetic evolution of influenza virus.
Science. 2004;305:371–376.
Lofgren E, Fefferman NH, Naumov YN, Gorski J, Naumova EN
Influenza seasonality: underlying causes and modeling theories.
Journal of Virology. 2007;81:5429–5436.
Franke JE, Yakubu AA.
Disease-induced mortality in density-dependent discrete-time SIS
epidemic models.
Journal of mathematical biology. 2008;57:755–790.
Shaman J, Karspeck A.
Forecasting seasonal outbreaks of influenza.
Proceedings of the National Academy of Sciences.
Bolton KJ, McCaw JM, McVernon J, Mathews JD.
Exploring alternate immune hypotheses in dynamical models of the
1918-1919 influenza pandemic.
Influenza and Other Respiratory Viruses. 2011;5:208–211.
Pastor-Satorras R, Castellano C, Van Mieghem P, Vespignani A.
Epidemic processes in complex networks.
arXiv preprint arXiv:14082701. 2014;.
Bansal S, Read J, Pourbohloul B, Meyers LA.
The dynamic nature of contact networks in infectious disease
Journal of Biological Dynamics. 2010;4:478–489.
Handcock MS, Jones JH.
Interval estimates for epidemic thresholds in two-sex network models.
Theoretical Population Biology. 2006;70:125–134.
Kitsak M, Gallos LK, Havlin S, Liljeros F, Muchnik L, Stanley HE, et al.
Identification of influential spreaders in complex networks.
Nature Physics. 2010;6:888–893.
Ferrari MJ, Bansal S, Meyers LA, Bjørnstad ON.
Network frailty and the geometry of herd immunity.
Proceedings of the Royal Society B. 2006;273:2743–2748.
Cohen R, Havlin S.
Complex Networks: Structure, Robustness and Function.
Cambridge University Press; 2010.
López E, Parshani R, Cohen R, Carmi S, Havlin S.
Limited path percolation in complex networks.
Physical Review Letters. 2007;99:188701.
Brauer F.
In: Compartmental models in epidemiology. Springer; 2008. p. 19–79.
Chakrabarti D, Wang Y, Wang C, Leskovec J, Faloutsos C.
Epidemic thresholds in real networks.
ACM Transactions on Information and System Security (TISSEC).
Fefferman NH, Ng KL.
How disease models in static networks can fail to approximate disease
in dynamic networks.
Physical Review E. 2007;76:031919.
Molloy M, Reed B.
A critical point for random graphs with a given degree sequence.
Random structures and algorithms. 1995;6:161–180.
Madden JR, Drewe JA, Pearce GP, Clutton-Brock TH.
The social network structure of a wild meerkat population: 2.
Intragroup interactions.
Behavioral Ecology and Sociobiology. 2009;64:81–95.
Cohen R, Erez K, Ben-Avraham D, Havlin S.
Resilience of the Internet to random breakdowns.
Physical Review Letters. 2000;85:4626.
Schneider CM, Moreira AA, Andrade JS, Havlin S, Herrmann HJ.
Mitigation of malicious attacks on networks.
Proceedings of the National Academy of Sciences. 2011;108:3838–3841.
Lloyd AL, Jansen VA.
Spatiotemporal dynamics of epidemics: synchrony in metapopulation
Mathematical biosciences. 2004;188:1–16.
Suzan G, Garcia-Pena GE, Castro-Arellano I, Rico O, Rubio AV, Tolsa MJ,
et al.
Metacommunity and phylogenetic structure determine wildlife and
zoonotic infectious disease patterns in time and space.
Ecology and Evolution. 2015;5:865–873.
|
1511.00090
|
$^{1}$ Department of Physics, Applied Optics Beijing Area Major Laboratory,
Beijing Normal University, Beijing 100875, China
$^{2}$ Department of Applied Physics, School of Science, Tianjin
Polytechnic University, Tianjin 300387, China
We present a one-step scheme to construct the controlled-phase gate
deterministically on remote transmon qutrits trapped in different
resonators connected by a superconducting transmission line for a
distributed quantum computing. Different from previous work on
remote superconducting qubits, the present gate is implemented with
coherent evolutions of the entire system in the all-resonance regime
assisted by the dark microwave photons to be robust against the
transmission line loss, which allows the possibility of the complex
designation of a long-length transmission line to link lots of
circuit QEDs. This gate is a fast quantum entangling operation with
a high fidelity of about $99\%$. Compare with previous works in
other quantum systems for a distributed quantum computing, under the
all-resonance regime, the present proposal does not require
classical pulses and ancillary qubits, which relaxes the difficulty
of its implementation largely.
03.67.Lx, 03.67.Bg, 85.25.Dq, 42.50.Pq
§ INTRODUCTION
Quantum computation (QC) <cit.>, as an interdisciplinary
research of computer science and quantum mechanics, has attracted
much attention in recent years. It can implement the famous Shor's
algorithm <cit.> for the factorization of an n-bit integer
exponentially faster than the classical algorithms and the Grover's
algorithm <cit.> or the optimal Long's algorithm
<cit.> for unsorted database search. Various quantum
systems have been used to implement QC, such as photons
<cit.>, nuclear magnetic
resonance <cit.>, diamond nitrogen-vacancy center
<cit.>, and cavity quantum electrodynamics
(QED) <cit.>. Among the quantum systems, circuit QED
<cit.>, composed of a superconducting qubit (SQ)
coupled to a superconducting resonator (SR), provides a good
platform for the implementation of QC because of its good ability
of the large-scale integration and the accurate manipulation on the
SQ <cit.>.
Circuit QED has been studied a lot for achieving the basic tasks of
QC on SQs or SRs, such as the construction of the single-qubit and
the universal quantum gates
<cit.>, entangled state
generation <cit.>, and the
measurement and the non-demolition detection on SQs or SRs
<cit.>. The types for integrating the SQs and
the SRs mainly contain some SQs coupled to a SR bus <cit.> or
some SRs coupled to a SR bus <cit.> or a SQ
<cit.>. At present, it is hard to integrate lots of
SQs or SRs in a quatum-bus-based processor to achieve the complex
universal QC. Further scaling up the number of SQs or SRs requires
linking the distant circuit QED systems to form a quantum network
<cit.> introduced by the
distributed quantum computing <cit.>, in which a
quantum computer can be seen as a quantum network of distant local
processors with only a few qubits and are connected by quantum
transmission lines (TL). As the key problem in the realization of
the distributed quantum computing, quantum entanglement and
universal quantum gate on remote qubits have been discussed in some
other systems <cit.>. For example, Cirac
et al. <cit.> proposed a scheme to achieve the ideal
quantum transmission between atoms trapped at spatially separated
nodes in 1997. In 2004, Xiao et al <cit.> realized the
controlled phase (c-phase) gate between two rare-earth ions embedded
in the respective microsphere cavities assisted by a single-photon
pulse in sequence. In 2011, Lü et al <cit.> proposed two
schemes to complete the entanglement generation and quantum-state
transfer between two spatially separated semiconductor quantum dot
To achieve the universal quantum gate on distant qubits trapped in
different cavities connected by the TLs, realistic flying-photon
qubit or adiabatic processes and the local operations are required.
On one hand, there are some works which studied the quantum network
by using the dark photon in the TL in other quantum systems. In
2007, Yin et al <cit.> presented some schemes to achieve
the state transfer and quantum entangling gates deterministically
between the remote multiple two-level atoms trapped in different
cavities connected by an optical fiber, in which the c-phase gate
should be completed by using the “dipole blockade" effect among
atoms in a cavity and it needs not to populate the realistic photons
in the fiber. In 2014, Clader <cit.> presented an adiabatic
scheme to transfer a microwave quantum state from one cavity to
another, assisted by an optical fiber which is robust against both
mechanical and fiber loss. On the other hand, one should transfer
the microwave photon to the optical photons to link the remote SQs.
In 2015, Yin et al. <cit.> proposed a scheme to achieve
the quantum networking of SQs based on the optomechanical interface.
To implement the distributed quantum computing on remote SQs trapped
in different SRs connected by a superconducting TL, one should
overcome the decay of the TL as the more the complicated designation
and a longer length for the TL is, the bigger the decay of the
microwave photon in it becomes. In this paper, we propose a scheme
for the construction of the c-phase gate on two remote transmon
qutrits trapped in different SRs connected by a superconducting TL
for the distributed quantum computing on SQs. Our scheme works in
the all-resonance regime by letting the frequencies of the qutrits
and the resonators equal to each other. The scheme can be achieved
with just one step assisted by the dark microwave photons in the TL,
without requiring classical pulses and ancillary qubits, which
relaxes the difficulty of its implementation in experiment largely.
Using the dark microwave photons in TL to reduce the requirement of
the quality factor of the TL allows the complex designation of a
long-length TL to link lots of remote circuit QEDs. The fidelity of
the present c-phase gate is beyond $99\%$ by using the numerical
simulation with the feasible parameters.
§ BASIC THEORIES
Let us consider a distributed quantum computing composed of two
remote superconducting qubits $q_1$ and $q_2$ trapped in two
single-mode high-quality superconducting resonators $r_a$ and $r_b$,
respectively, which are connected by a superconducting TL $r_f$,
shown in Fig. <ref>. The Hamiltonian of this device is (in the
interaction picture with $\hbar=1$)
\begin{eqnarray} %eq1
H &=& H_1^a +H_2^b + H_f^{a(b)}\nonumber\\
&=& g_{1}^{a} (a\!^{+} \sigma_{1}^{-} e^{-i\delta_{1}^{a}t}
+ a\sigma_{1}^{+}e^{i\delta_{1}^{a}t}) +\, g_{2}^{b}(b\!^{+} \sigma_{2}^{-}e^{-i\delta_{2}^{b}t}
+ b\sigma_{2}^{+}e^{i\delta_{2}^{b}t}) \nonumber \\
&& + \sum_{j=1}^{\infty} g_{f,j}^{I} \left[f_{j}\,(a^{+} +
(-1)^{j}e^{i\phi}b^{+}) + H.c. \right], \label{initial}
\end{eqnarray}
where $H_1^a$, $H_2^b$, and $H_f^{a(b)}$ are the interaction
Hamiltonians of the subsystems composed of $q_1$ and $r_a$, $q_2$
and $r_b$, and $r_f$ and $r_a$ ($r_b$), respectively. $H_f^{a(b)}$
applies to the high-finesse resonators and resonant operations over
the time scale much longer than the TL's round-trip time <cit.>.
$\delta_{J}^{I}=\omega_{I}-\omega_{J}$ ($I=a$,$b$ and
$J=1$,$2$,$f$). $\omega_a$, $\omega_b$, and $\omega_f$ are the
transition frequencies of resonators $r_a$, $r_b$, and the TL $r_f$,
respectively. $\omega_1$ and $\omega_2$ are the transition
frequencies of the qubits $q_1$ and $q_2$, respectively. $a^{+}$,
$b^{+}$, and $f^{+}$ are the creation operators of the resonators
$r_a$, $r_b$, and the TL $r_f$, respectively. $\sigma _{1}^{+}$ and
$\sigma _{2}^{+}$ are the creation operators of the transitions
$|g\rangle_{1}\leftrightarrow|e\rangle_1$ and
$|g\rangle_{2}\leftrightarrow|e\rangle_2$ of the qubits $q_{1}$ and
$q_{2}$, respectively. $|g\rangle_{1(2)}$ and $|e\rangle_{1(2)}$ are
the ground and the first excited states of the qubit $q_{1(2)}$,
respectively. $g_{1}^{a}$ and $g_{2}^{b}$ are the coupling strength
between $q_1$ and $r_a$ and that between $q_2$ and $r_b$,
respectively. $g_{f,j}^{I}$ is the coupling strength between
$r_{a(b)}$ and the mode $j$ of the TL $r_f$. $\phi$ is the phase
induced by the propagating field through the TL $r_f$ of length $l$
with the relation $\phi=2\pi\omega l/c$ in which $c$ is the speed of
(a) Setup for a distributed quantum computing composed of
two remote qubits $q_1$ and $q_2$ trapped in different resonators
$r_a$ and $r_b$ connected by a transmission line $r_f$. (b)
Illustrations of the energy splitting of the subsystem composed of
$r_a$, $r_b$, and $r_f$.
In the short TL limit $(2L\kappa_{f}^{a(b)})/(2\pi c)\leq 1$, only
one resonant mode $f$ of the TL $r_f$ interacts with the resonators'
modes ($L$ is the length of $r_f$ and $\kappa_{f}^{a(b)}$ is the
decay rate of the resonator $r_{a(b)}$ into a continuum of TL modes)
<cit.>. The Hamiltonian $H$ can be reduced to
\begin{eqnarray} %eq2
H_{int} &=& g_{1}^{a}(a^{+}\sigma_{1}^{-}e^{-i\delta_{1}^{a}t}
+ a\sigma_{1}^{+}\!e^{i\delta_{1}^{a}t}) +\, g_{2}^{b}(b^{+} \sigma_{2}^{-}e^{-i\delta_{2}^{b}t}
+ a\sigma_{2}^{+}\!e^{i\delta_{2}^{b}t}) \nonumber \\
&& +\, g_{f}^{a}(f^{+}a + fa^{+}) + g_{f}^{b}(f^{+}b + fb^{+}).
\label{cphamiltonian}
\end{eqnarray}
In the Schrödinger picture, this Hamiltonian can be rewritten as
\begin{eqnarray} %eq3
H' &=& \omega_{a} a^{+}a + \omega_{b} b^{+}b + \omega_{f} f^{+}f +
\omega_{1}\sigma _{1}^{+}\sigma _{1}^{-}
+ \omega_{2}\sigma _{2}^{+}\sigma _{2}^{-} \nonumber\\
&& +\, g_{1}^{a}(a^{+}\, \sigma _{1}^{-} + a \sigma _{1}^{+}) + g_{2}^{b}(b^{+}\, \sigma _{2}^{-} + b \sigma _{2}^{+}) \nonumber\\
&& +\, g_{f}^{a}(f^{+}\,a + f\,a^{+}) + g_{f}^{b}(f^{+}\,b +
f\,b^{+}). \label{cphamiltonian01}
\end{eqnarray}
To generate and construct the Bell state and the c-phase gate on the
remote transmon qutrits below, we consider the all-resonance
condition with
$\omega_a=\omega_b=\omega_f=\omega_{1}=\omega_{2}=\omega$ by letting
the frequencies of the qubits and the resonators and the TL equal
to each other and $g_{f}^{a}=g_{f}^{b}=g$ by letting the coupling
strength between $r_a$ and $r_f$ equal to the one of $r_b$ and
$r_f$. If one takes the canonical transformations
$C_{\pm}=\frac{1}{2}(a+b \pm \sqrt{2}f)$ and
$C=\frac{\sqrt{2}}{2}(a-b)$ <cit.>, the Hamiltonian
$H'$ can be represented as
\begin{eqnarray} %eq4
H'' &=& \omega \sigma _{1}^{+} \sigma_{1}^{-} + \omega \sigma
_{2}^{+} \sigma_{2}^{-}
+ \omega C^{+}C + \left(\omega + \sqrt{2}g\right)C_{+}C_{+}^{+}\nonumber\\
&& + \left(\omega - \sqrt{2}g\right)C_{-}C_{-}^{+} +
\frac{1}{2}\Big[g_{1}^{a}\left(C_{+} + C_{-} + \sqrt{2}c\right)\sigma_{1}^{+} \nonumber\\
&& + g_{1}^{a}\left( C_{+}^{+} + C_{-}^{+} +
\sqrt{2}C^{+}\right)\sigma_{1}^{-} + g_{2}^{b}\left( C_{+}
+ C_{-} - \sqrt{2}C\right)\!\sigma_{2}^{+} \nonumber\\
&& + g_{2}^{b}\left(C_{+}^{+} + C_{-}^{+} -
\sqrt{2}C^{+}\right)\sigma_{2}^{-}\Big]. \label{cphamiltonian02}
\end{eqnarray}
Here the modes $C$ and $C_{\pm}$ are three bosonic modes and they
are not coupled to each other. From Eq. (<ref>),
the energy level of the subsystem composed of $r_a$, $r_b$, and
$r_f$ are split into three different parts with frequencies
$\omega_{c_{+}}$, $\omega_{c_{-}}$, and $\omega_{c}$ signed by the
modes $C_{+}$, $C_{-}$, and $C$, respectively, as shown in Fig.
<ref> (b). Because of the contributions of the fields of $r_a$
and $r_b$, the three modes $C$ and $C_{\pm}$ interact with the two
qubits $q_1$ and $q_2$. When $g \gg \{g_{1}^{a},g_{2}^{b}\}$, the
excitations of modes $C_{\pm}$ are highly suppressed as $\omega \pm
\sqrt{2}g$ detune with the resonance modes ($C$, $q_1$, and $q_2$
with frequency of $\omega$) largely, which means the modes $C_{\pm}$
are the dark ones to the frequency $\omega_f$ of $r_f$, and the
Hamiltonian $H''$ can be reduced to
\begin{eqnarray} %eq5
H''' &=& \omega \sigma _{1}^{+}\sigma _{1}^{-} + \omega \sigma_{2}^{+}\sigma _{2}^{-} + \omega C^{+}C \nonumber\\
&& + \frac{1}{\sqrt{2}}\left[g_{1}^{a}\left(C \sigma_{1}^{+} + C^{+}
\sigma_{1}^{-}\right) - g_{2}^{b}\left(C \sigma_{2}^{+} +
C^{+}\sigma_{2}^{-}\right)\right]. \label{cphamiltonian03}
\end{eqnarray}
It can be written as
\begin{eqnarray} %eq6
H_{e\!f\!f} = \frac{1}{\sqrt{2}}\left[g_{1}^{a}(C \sigma_{1}^{+} +
C^{+}\sigma_{1}^{-}) - g_{2}^{b}(C \sigma_{2}^{+} +
C^{+}\sigma_{2}^{-})\right] \;\;\;\;\label{effect}
\end{eqnarray}
in the interaction picture. Here, only the mode
$C=\frac{\sqrt{2}}{2}(a-b)$ is left, which means that the TL can not
be populated in the all-resonance regime in our system. The
interaction between two remote two-energy-level qubits expressed by
$H_{e\!f\!f}$ will be used to generate and construct the
all-resonance Bell state and c-phase gate on the two remote qutrits
Illustrations of interactions between $q_1$ and $r_a$,
$r_f$ and $r_a$ ($r_b$), and $q_2$ and $r_b$, respectively, for the
construction of the c-phase gate on two remote transmon qutrits
$q_1$ and $q_2$.
§ C-PHASE GATE ON THE TWO REMOTE QUTRITS $Q_1$ AND $Q_2$
With the two transitions only discussed in the section II, one
cannot construct the one-step all-resonance c-phase gate on the two
remote transmon qutrits because of the existing of the states
$|e\rangle_{1}$ and $|e\rangle_{2}$ according to Eq.
Here, we consider the second excited energy level $|s\rangle_2$ of
$q_2$ and take
The illustrations of the interactions between $q_1$ and $r_a$, $r_f$
and $r_a$ ($r_b$), and $q_2$ and $r_b$ for constructing the c-phase
gate on $q_1$ and $q_2$ are shown in Fig. <ref>. The
Hamiltonian of the whole system can be expressed as
\begin{eqnarray} %eq7
H_{2q} &=& g_{1;ge}^{a} \left(a^{+}\sigma_{1;ge}^{-}e^{-i\delta_{1;ge}^{a}t}+a\sigma_{1;ge}^{+}e^{i\delta_{1;ge}^{a}t}\right) \nonumber\\
&& +\, g_{1;es}^{a} \left(a^{+}\sigma_{1;es}^{-}e^{-i\delta_{1;es}^{a}t}+a\sigma_{1;es}^{+}e^{i\delta_{1;es}^{a}t}\right) \nonumber\\
&& +\, g_{2,ge}^{b} \left(b^{+}\sigma_{2;ge}^{-}e^{-i\delta_{2;ge}^{b}t}+b\sigma_{2;ge}^{+}e^{i\delta_{2;ge}^{b}t}\right) \nonumber\\
&& +\, g_{2;es}^{b} \left(b^{+}\sigma_{2;es}^{-}e^{-i\delta_{2;es}^{b}t}+b\sigma_{2;es}^{+}e^{i\delta_{2;es}^{b}t}\right) \nonumber\\
&& +\, g_{f}^{a}\left(f^{+}a + fa^{+}\right) +
g_{f}^{b}\left(f^{+}b + fb^{+}\right),
\;\;\;\;\;\;\;\;\;\;\;\;\label{3energy}
\end{eqnarray}
in which $\sigma _{1(2);ge}^{+}$ and $\sigma _{1(2);es}^{+}$ are the
creation operators of the transitions
$|g\rangle_{1(2)}\leftrightarrow |e\rangle_{1(2)}$ and
$|e\rangle_{1(2)}\leftrightarrow |s\rangle_{1(2)}$ of the qutrit
$q_{1(2)}$, respectively. $g_{1(2);ge}^{a(b)}$ and
$\left(g_{1(2);es}^{a(b)}=\sqrt{2}g_{1(2);ge}^{a(b)}\right)$ are the
coupling strengths between the two transitions of $q_{1(2)}$ and
$r_{a(b)}$, respectively.
$\delta_{1(2);ge}^{a(b)}=\omega_{1(2);ge}-\omega_{a(b)}$ and
$\omega_{1(2);ge}$ ($\omega_{1(2);ef}$) is the frequency of the
transition $|g\rangle_{1(2)} \leftrightarrow |e\rangle_{1(2)}$
$\left(|e\rangle_{1(2)} \leftrightarrow |s\rangle_{1(2)}\right)$ of
the qutrit $q_{1(2)}$. $|s\rangle_{1(2)}$ is the second excited
states of $q_{1(2)}$. By taking $\omega_{1(2);ge}-\omega_{1(2);es}
\gg \{g_{1;ge}^{a},g_{2;ge}^{b}\}$, the Hamiltonian $H_{2q}$ can be
reduced to
\begin{eqnarray} %eq8
H_{2q}' &=& g_{1;ge}^{a}(a^{+}\!\!\sigma_{1;ge}^{-} +
+ g_{2;es}^{b}(b^{+} \!\sigma_{2;es}^{-} + b\sigma_{2;es}^{+}) \nonumber \\
&& +\, g_{f}^{a}(f^{+}a + fa^{+}) + g_{f}^{b}(f^{+}b + fb^{+}),
\label{2q'}
\end{eqnarray}
in which the dispersive coupling between the transition
$|e\rangle_{1} \leftrightarrow |s\rangle_{1}$ of the qutrit $q_{1}$
and $r_{a}$ and the one between the transition $|g\rangle_{2}
\leftrightarrow |e\rangle_{2}$ of the qutrit $q_{2}$ and $r_{b}$ are
Taking the same canonical transformations as the ones in Sec. II and
$g_{f}^{a}=g_{f}^{b} \gg \{g_{1;ge}^{a},g_{2;es}^{b}\}$, the
Hamiltonian $H_{2q}'$ becomes
\begin{eqnarray} %eq9
H_{e\!f\!f}' = \frac{1}{\sqrt{2}}\left[g_{1;ge}^{a} (C
\sigma_{1;ge}^{+} + C^{+} \sigma_{1;ge}^{-}) - g_{2;es}^{b}(C
\sigma_{2;es}^{+} + C^{+}\sigma_{2;es}^{-})\right]. \label{effect'}
\end{eqnarray}
Suppose that $|\psi_1\rangle=|g\rangle_1|g\rangle_2|0\rangle_c$,
$|\psi_3\rangle=|e\rangle_1|g\rangle_2|0\rangle_c$, and
$|\psi_4\rangle=|e\rangle_1|e\rangle_2|0\rangle_c$ ($|0\rangle_c
\equiv |0\rangle_a|0\rangle_b|0\rangle_f$) are the initial states of
the system undergoes the Hamiltonian $H_{e\!f\!f}'$, respectively,
one can get their evolutions as
\begin{eqnarray} %eq10
|\Psi_1(t)\rangle &=&
e^{iH_{e\!f\!f}'t}|g\rangle_1|g\rangle_2|0\rangle_c =
|g\rangle_1|g\rangle_2|0\rangle_c, \label{m1'}
\end{eqnarray}
\begin{eqnarray} %eq11
|\Psi_2(t)\rangle &=&
e^{-iH_{e\!f\!f}'t}|g\rangle_1|e\rangle_2|0\rangle_c =
|g\rangle_1|e\rangle_2|0\rangle_c, \label{m2'}
\end{eqnarray}
\begin{eqnarray} %eq12
|\Psi_3(t)\rangle &=& e^{-iH_{e\!f\!f}'t}|e\rangle_1|g\rangle_2|0\rangle_c \nonumber\\
\cos\left(\frac{g_{1;ge}^{a}}{\sqrt{2}}t\right)|e\rangle_1|g\rangle_2|0\rangle_c
\label{m3'}
\end{eqnarray}
\begin{eqnarray} %eq13
|\Psi_4(t)\rangle &=& e^{iH_{ef\!f}'t}|e\rangle_1|e\rangle_2|0\rangle_c \nonumber\\
&=& \frac{1}{G'}\left[(g_{2;es}^{b})^{2}
+(g_{1;ge}^{a})^{2}\cos\left(\sqrt{\frac{G'}{2}}t\right)\right]|e\rangle_1|e\rangle_2|0\rangle_c \nonumber\\
&&-\frac{g_{1;ge}^{a} g_{2;es}^{b}}{G'}\left[\cos\left(\sqrt{\frac{G'}{2}}t\right)-1\right]|g\rangle_1|s\rangle_2|0\rangle_c \nonumber\\
\label{m4'}
\end{eqnarray}
where $G'=(g_{1;ge}^{a})^{2}+(g_{2;es}^{b})^{2}$. From the
evolutions of the four states, one can construct the c-phase gate on
the two remote qutrits $q_1$ and $q_2$. In detail, we suppose the
initial state of the system described by $H_{e\!f\!f}'$ is
\begin{eqnarray} %eq14
|\Psi_0^{cp}\rangle &=& (\cos{\theta_1}|g\rangle_1
+\sin{\theta_1}|e\rangle_1) \otimes(\cos{\theta_2}|g\rangle_2
+\sin{\theta_2}|e\rangle_2) \otimes |0\rangle_c. \label{cp0}
\end{eqnarray}
According to Eqs. (<ref>) and (<ref>), one can keep the
states $|g\rangle_1|g\rangle_2|0\rangle_c$ and
$|g\rangle_1|e\rangle_2|0\rangle_c$ unchanged. By taking the proper
$g_{1;ge}^{a}$ and $g_{2;ge}^{b}$ to satisfy
$\frac{g_{1;ge}^{a}}{\sqrt{2}}t=(2k-1)\pi$ and
$\sqrt{\frac{G'}{2}}t=2m\pi$ ($k,m=1,2,3,\cdots$) simultaneously,
one can achieve the condition that when the state
$|e\rangle_1|g\rangle_2|0\rangle_c$ undergoes an odd number of
periods and generates a minus phase (from Eq. (<ref>)), the
state $|e\rangle_1|e\rangle_2|0\rangle_c$ undergoes an even number
of periods and keeps unchanged (from Eq. (<ref>)) meanwhile.
That is, the state of the system evolves from $|\Psi_0^{cp}\rangle$
to the final state
\begin{eqnarray} %eq15
|\Psi_f^{cp}\rangle &=& (\alpha_1|g\rangle_1|g\rangle_2 +
\alpha_2|g\rangle_1|e\rangle_2 - \alpha_3|e\rangle_1|g\rangle_2 +
\alpha_4|e\rangle_1|e\rangle_2) \otimes |0\rangle_c, \label{cpf}
\end{eqnarray}
which is just the target state after our c-phase gate operation on
$q_1$ and $q_2$ with the initial state $|\Psi_0^{cp}\rangle$. Here
$\alpha_3=\sin{\theta_1}\cos{\theta_2}$, and
$\alpha_4=\sin{\theta_1}\sin{\theta_2}$. In the basis
\}$, the matrix of the c-phase gate is
\begin{equation} %eq16
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right).
\end{equation}
The fidelity of our c-phase gate varies with $gt$ and
different $\Delta=g_{f}^{a(b)}/g_{1;ge}^{a}$.
To get the c-phase gate within a short time, we take $k=m=1$, that
Supposing that the initial state of the system is
one can get the state
after our c-phase gate operation on the two remote qutrits $q_1$ and
$q_2$ with a maximal fidelity of $99.8\%$ ($\Delta=25$), $99.6\%$
($\Delta=10$), and $98.8\%$ ($\Delta=5$) within $gt=0.705$
by using the definition
\begin{eqnarray} %eq17
F_{max}^{cp} \;=\; |\langle
\Psi_{max}^{cp}|e^{-iH_{2q}t}|\Psi_{max}\rangle|^2, \label{fidelity}
\end{eqnarray}
as shown in Fig. <ref>. Here $\Delta\equiv
§ POSSIBLE EXPERIMENTAL IMPLEMENTATION AND FIDELITY
In experiment, a high quality factor $Q\sim2\times 10^6$ of a 1D SR
has been demonstrated <cit.>. By considering the relation
between $\kappa$, $Q$, and the frequency of resonator $\omega_r$
with $\kappa=\omega_r/Q$ <cit.>, the best life time of a
microwave photon in a superconducting resonator can reach $\sim50$
$\mu$s. The coherence time of a transmon qubit <cit.> can also reach $50$ $\mu$s by using titanium
nitride <cit.>. The tunable range of the transition frequency
of a transmon qubit can reach $2.5$ GHz, which helps us to tune our
transmon qutrits to resonate (detune) with their corresponding
resonator (largely) effectively. The coupling strength between a
transmon qutrit and a SR can be realized larger than $200$ MHz
<cit.>. The anharmonicity between the two transitions of a
transmon qutrit can reach
0.72$ GHz <cit.>, which lets us ignore the detune interaction
between each qutirt and their corresponding resonator, compared with
the small coupling strength between them. As for the SRs and the TL
$r_f$, one can couple them by using the SQUID, which can reach a
coupling strength of $g_{f}^{a(b)}/(2\pi) \sim 200$ MHz
theoretically with reasonable experimental parameters
<cit.>. Moreover, one can also use the capacitance
coupling between resonators and the superconducting TL. With the
reasonable parameters
$\omega_{a}/(2\pi)=\omega_{b}/(2\pi)=\omega_{f}/(2\pi)= 6$ GHz, the
coupling capacitance $C=13.3$ fF, and the capacitance per unit
length of the transmission line and the resonators $C_r=2$ pF
<cit.>, the capacitance coupling strength can reach
$g_{f}^{a(b)}/(2\pi)=40$ MHz (which will be discussed below for the
construction of the c-phase gate with $\Delta=5$). It is worth
noticing that a coupling strength between a SR and a superconducting
TL has been realized with about $32$ MHz <cit.>.
To show the feasibility of our scheme for the construction of the
c-phase gate on two remote qutrits, we numerically simulate the
fidelity of the scheme based on the parameters realized in
experiments or predicted theoretically with reasonable experimental
The dynamics of the quantum system undergoes the Hamiltonian
$H_{2q}$ is determined by the master equation
\begin{eqnarray} %eq18
\frac{d\rho }{dt} &=& -i[H_{2q},\rho ]+\kappa_a D[a]\rho +\kappa_b D[b]\rho + \kappa_f D[f]\rho \;\;\;\;\;\; \nonumber\\
&&+\, \sum_{l=1,2}\{\gamma_{l;ge}D[\sigma_{l;ge}^{-}]\rho+\gamma_{l;es}D[\sigma_{l;es}^{-}]\rho \nonumber\\
&&+\, \gamma_{l;e}^{\phi}(\sigma_{l;ee}\rho\sigma_{l;ee}-\sigma_{l;ee}\rho/2-\rho \sigma_{l;ee}/2) \nonumber\\
\gamma_{l;s}^{\phi}(\sigma_{l;ss}\rho\sigma_{l;ss}-\sigma_{l;ss}\rho/2-\rho
\sigma_{l;ss}/2)\}. \label{masterequation}
\end{eqnarray}
Here, $\kappa_{a,b,f}$ is the decay rate of the resonator
$r_{a,b,f}$. $\gamma_{l;ge}$ ($\gamma_{l;es}$) and
$\gamma_{l;e}^{\phi}$ ($\gamma_{l;s}^{\phi}$) are the energy
relaxation and the dephase rates of the transition $|e\rangle_l
\leftrightarrow |g\rangle_l$ ($|s\rangle_l \leftrightarrow
|e\rangle_l$) of the transmon qutrits $q_l$ ($l=1,2$), respectively.
<cit.>, $\sigma_{l;ee}=|e\rangle_l\langle e|$, and
$\sigma_{l;ss}=|s\rangle_l\langle s|$. $D[L]\rho=(2L\rho
L^{+}-L^{+}L\rho-\rho L^{+}L)/2$. Because of the competition
relation between the coupling strength $g_{1;ge}^{a}$ and the decay
rates and that between the decoherence time of resonators and
qutrits, for different $\gamma_{l;ge}$ and $\kappa_{a,b,f}$, one
should choose different $g_{1;ge}^{a}$ to reach the maximal fidelity
of our scheme for constructing the c-phase gate on remote qutrits
$q_1$ and $q_2$. Besides, $\Delta$ for the simulation of our scheme
below are both larger than $15$, which indicates there are almost no
MP in the TL and the influence from the decay rate $\kappa_{f}$ of
$r_f$ is not considered here. The coupling strengths
$g_{1;ge}^{a}/(2\pi)$ chosen below for the c-phase gate construction
are the optimal ones which correspond to the highest fidelities of
the gate when we fix $g_{f}^{a(b)}/(2\pi)=200$ MHz and
$\gamma_{l;ge}^{-1}=\kappa_{a,b,f}= 50$ $\mu$s by considering the
set of discretized $g_{1;ge}^{a}$ values, varying from 1 to 100MHz
in steps of 1MHz.
To show the feasibility of our c-phase gate on remote qutrits $q_1$
and $q_2$ with decoherence time and the decay time of the qutrits
and the resonators, we numerically simulate the fidelity of
$|\Psi_{max}^{cp}\rangle$ after our c-phase gate operations on the
whole system (the initial state of the system is
$|\Psi_{max}\rangle$) by using the definition
\begin{eqnarray} % eq19
F_{cp} \;=\; \langle \Psi_{max}^{cp}|\rho(t)|\Psi_{max}^{cp}\rangle,
\label{Fcp}
\end{eqnarray}
in which the effects from the unresonant parts
\begin{eqnarray} %eq20
H_{1} &=& g_{a,1}^{e,f}(a\sigma
_{1;e,f}^{-}e^{-i\delta_{a,1}^{e,f}t})\;\;\;\;\;\;\;\; \label{cp1}
\end{eqnarray}
\begin{eqnarray} %eq21
H_{2} &=& g_{b,2}^{g,e}(b\sigma
_{2;g,e}^{-}e^{-i\delta_{b,2}^{g,e}t}) \;\;\;\;\;\;\;\; \label{cp2}
\end{eqnarray}
are considered. Parameters chosen here are
6$ GHz, $\sqrt{\frac{2}{3}}g_{2;ge}^{b}/(2\pi)=g_{1;ge}^{a}/(2\pi)=
8$ MHz. $\gamma_{l;ge}^{-1}=\kappa_{a,b,f}^{-1}= 50$ $\mu$s. As
shown in Fig. <ref> (a), the fidelity of the state
$|\Psi_{max}^{cp}\rangle$ can reach $99.28\%$ within $88.1$ ns.
(a) The fidelity of the c-phase gate on $q_1$ and $q_2$
varies with the operation time $t$. (b)-(f) The relations between
the fidelity of the gate and $g_{1;ge}^{a}$,
$\omega_{2;ge}-\omega_{2;ef}$, $\omega_{2;ef}$, $\kappa_{f}^{-1}$,
and $\gamma^{-1}$, respectively.
In the realistic experiment, parameters of the system can not match
the ones accurately chosen above. We give the influences on the
fidelity of the state $|\Psi_{max}^{cp}\rangle$ from the coupling
strength, the anharmonicity, the decoherence time, and the frequency
of the qutrits and the decay time of resonators as shown in Fig.
<ref> (b)-(f). In each figure in Fig. <ref>, parameters
are kept unchanged except for the one signed in the axis of
abscissas. The influences from $g_{1;ge}^{a}$ is shown in Fig.
<ref> (b) and the small change of $g_{1;ge}^{a}$ influences the
fidelity little. Fig. <ref> (c) indicates that the
anharmonicity of $q_2$ should be chosen to let the transition
frequency $\omega_{2;ge}$ detune with $\omega_f +
\sqrt{2}g_{f}^{a(b)}$ largely, which is required when we reduce the
Hamiltonian from $H_{2q}$ to $H_{e\!f\!f}''$. Accurate resonance
between the two remote qutrits is required as shown in Fig.
<ref> (d). In the large-scale integration of our system, the
interaction between qutrits can be turned off conveniently by tuning
the frequency of the qutrits. In Fig. <ref> (e), we give the
influences on the fidelity of the state from the decay time of
$r_f$. It can be seen that when $\kappa^{-1}_{f}>10$ ns,
$\kappa_{f}^{-1}$ influences the fidelity of the state little. To
show the possible influence from the decay rates of the resonators
and the decoherence time of the qutrits, we calculate the average
gate fidelity of the c-phase gate with different
$\Gamma^{-1}=\gamma_{l;ge}^{-1}=\kappa_{a,b,f}^{-1}$, shown in Fig.
<ref> (f) by using the average-gate-fidelity definition
\begin{eqnarray} %eq22
F = (\frac{1}{2\pi})^2 \int_0^{2\pi} \int_0^{2\pi} \langle
\Psi_{f}^{cp}|\rho(t)|\Psi_{f}^{cp}\rangle d\theta_1 d\theta_2.
\label{fidelity}
\end{eqnarray}
Here, $\rho(t)$ is the realistic density operator after our c-phase
gate operation on the initial state $\Psi_{0}^{cp}$ with the
Hamiltonian $H$. It is worth noticing that the decay time $20$
$\mu$s corresponds to the typically quality factor $Q \sim
7\times10^5$ of a 1D superconducting resonator <cit.> and the
coherence time $20$ $\mu$s of a transmon qutrit can also be readily
realized in experiment <cit.>. Although the coupling strength
$g_{f}^{a(b)}/(2\pi)$ taken here is $200$ MHz (predicted
theoretically in <cit.>) which satisfies $\Delta=25$ and
it has not been realized in experiments, if we take $\Delta=5$
($g_{f}^{a(b)}/(2\pi)=40$ MHz) and the more of the actual situation
of the life time of SRs and qutrits with $\Gamma^{-1}=20$ $\mu$s,
the fidelity of our c-phase gate can also reach a high value of
$98\%$ (compared with the fidelities between $\Delta=25$ and
$\Delta=5$ as shown in Fig. <ref>) which should be enhanced
further by taking corresponding optimal parameters. Corresponding to
the operation time of the c-phase gate construction, the length of
the superconducting TL can, in principle, reach the scale of several
§ DISCUSSION AND SUMMARY
On one hand, in order to use the dark photons in the TL to achieve
the c-phase gate on qutrits $q_1$ and $q_2$, one should take
$g_{1(2);ge}^{a(b)} \ll g_{f}^{a(b)}$. On the other hand, the small
coupling strength of $g_{1(2);ge}^{a(b)}$ does not require the large
anharmonicities of the qutrits. Moreover, to achieve our c-phase
scheme, the $\Xi$-type energy level of the qutrits is required.
Besides the transmon qutrit, the superconducting charge qutrit
<cit.> or phase qutrit <cit.> can also
been applied to our scheme. By using the transmon qutrit or the
phase qutrit with $\omega_{ge}/(2\pi)>\omega_{ef}/(2\pi)$, one
should take the proper anharmonicity of $q_2$ to let the transition
frequency $\omega_{2;ge}$ detune with $\omega_f +
\sqrt{2}g_{f}^{a(b)}$ largely. By using the charge qutrit with
$\omega_{ge}/(2\pi)<\omega_{ef}/(2\pi)$, one should take
$\omega_{2;ge}$ detune with $\omega_f - \sqrt{2}g_{f}^{a(b)}$
largely. That is, when the frequency $\omega_{2;ge} \sim \omega_f +
\sqrt{2}g_{f}^{a(b)}$, the effective Hamiltonians $H_{e\!f\!f}$ and
$H_{e\!f\!f}'$ can not be obtained as the mode $ C_{\pm}$ can not be
suppressed effectively.
In summary, we have proposed a one-step scheme to achieve the
c-phase gate on two remote transmon qutrits trapped in different
resonators connected by a superconducting TL. The scheme works in
the all-resonance regime with just one step, which leads to a fast
operation and can be demonstrated in experiment easily. Moreover,
our scheme is robust against the TL loss by using the dark microwave
photon. That is, the superconducting TL needs not to be populated,
which is convenient to be extended to a large-scale integration
condition by the complex designation of a long-length TL to link
lots of remote circuit QEDs.
§ ACKNOWLEDGES
This work is supported by the National Natural Science Foundation of
China under Grants No. 11474026 and No. 11674033, and the
Fundamental Research Funds for the Central Universities under Grant
No. 2015KJJCA01.
Nielsen Nilsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
shor Shor P W 1997 SIAM J. Sci. Statist. Comput. 26 1484
Grover Grover L K 1997 Phys. Rev. Lett. 79 325
LongGrover Long G L 2001 Phys. Rev. A 64 022307
Knill Knill E, Laflamme R and Milburn G J 2001 Nature 409 46
photon O'Brien J L, Pryde G J, White A G, Ralph T C and Branning D 2003 Nature 426 264
hyperCNOT2 Ren B C and Deng F G 2014 Sci. Rep. 4 4623
hyperCNOT3 Ren B C, Wang G Y and Deng F G 2015 Phys. Rev. A 91 032328
photon2 Wei H R and Deng F G 2013 Opt. Express 21 17671
NMR Jones J A, Mosca M and Hansen R H 1998 Nature 393 344
Long1 Long G L and Xiao L 2003 J. Chem. Phys. 119 8473
NHQCLong Feng G R, Xu G F and Long G L 2013 Phys. Rev. Lett. 110 190501
Togan Togan E, Chu Y, Trifonov A S, Jiang L, Maze J, Childress L,
Dutt M V G, Sørensen A S, Hemmer P R, Zibrov A S and Lukin M D
2010 Nature 466 730
NVgate Yang W L, Yin Z Q, Xu Z Y, Feng M and Du J F 2010 Appl. Phys. Lett. 96 241113
weiNVgate Wei H R and Deng F G 2013 Phys. Rev. A 88 042323
book Scully M O and Zubairy M S 1997 Quantum Optics (Cambridge: Cambridge University press)
Blais Blais A, Huang R S, Wallraff A, Girvin S M and
Schoelkopf R J 2004 Phys. Rev. A 69 062320
Wallraff Wallraff A, Schuster D I, Blais A, Frunzio L,
Huang R S, Majer J, Kumar S, Girvin S M and Schoelkopf R J 2004
Nature 431 162
Barends Barends R, Kelly J, Megrant A, Veitia A, Sank D, Jeffrey E, White T C, Mutus J, Fowler A G, Campbell B, Chen Y,
Chen Z, Chiaro B, Dunsworth A, Neill C, O'Malley P, Roushan P,
Vainsencher A, Wenner J, Korotkov A N, Cleland A N and Martinis J M
2014 Nature 508 500
DiCarlo DiCarlo L, Chow J M, Gambetta J M, Bishop L S, Johnson B R, Schuster D I, Majer J, Blais A,
Frunzio L, Girvin S M and Schoelkopf R J 2009 Nature
460 240
Haack Haack G, Helmer F, Mariantoni M, Marquardt F and Solano E 2010 Phys. Rev. B 82 024514
Strauch Strauch F W 2011 Phys. Rev. A 84 052313
Hua1 Hua M, Tao M J and Deng F G 2014 Phys. Rev. A 90 012328
Hua2 Hua M, Tao M J and Deng F G 2015 Sci. Rep. 5 9274
McKay McKay D C, Naik R, Reinhold P, Bishop L S and Schuster D I 2015
Phys. Rev. Lett. 114 080501
Steffen Steffen M, Ansmann M, Bialczak R C, Katz N, Lucero E, McDermott R, Neeley M, Weig E M, Cleland A N and Martinis J M 2006
Science 313 1423
Cao Cao Y, Huo W Y, Ai Q and Long G L 2011 Phys. Rev. A 84 053846
Leghtas Leghtas Z, Vool U, Shankar S, Hatridge M, Girvin S M, Devoret M H and Mirrahimi M 2013
Phys. Rev. A 88 023849
Strauch1 Strauch F W 2012 Phys. Rev. Lett. 109 210501
Strauch2 Strauch F W, Onyango D, Jacobs K and Simmonds R W 2012 Phys. Rev. A 85 022335
AWallraff Wallraff A, Schuster D I, Blais A, Frunzio L, Majer J, Devoret M H, Girvin S M and Schoelkopf R J 2005
Phys. Rev. Lett. 95 060501
Johnson Johnson B R, Reed M D, Houck A A, Schuster D I, Bishop L S, Ginossar E, Gambetta J M, DiCarlo L, Frunzio L, Girvin S M and Schoelkopf R J 2010
Nat. Phys. 6 663
Feng Feng W, Wang P Y, Ding X M, Xu L T and Li X Q 2011 Phys. Rev. A 83 042313
Majer Majer J, Chow J M, Gambetta J M, Koch J, Johnson B R, Schreier J A, Frunzio L, Schuster D I,
Houck A A, Wallraff A, Blais A, Devoret M H, Girvin S M and
Schoelkopf R J 2007 Nature 449 443
Hua3 Hua M, Tao M J and Deng F G 2015 arXiv:1507.00069.
Wu Wu C W, Gao M, Li H Y, Deng Z J, Dai H Y, Chen P X and Li C Z 2012
Phys. Rev. A 85 042301
Yang Yang C P, Su Q P and Han S Y 2012 Phys. Rev. A 86 022329
YangCP Yang C P, Su Q P, Zheng S B and Nori F 2016 Phys. Rev. A 93 042307
Pellizzari Pellizzari T 1997 Phys. Rev. Lett. 79 5242
Loo van Loo A F, Fedorov A, Lalumiére K, Sanders B C,
Blais A and Wallraff A 2013 Science 342 1494
YY Yin Y, Chen Y, Sank D, O'Malley P J J, White T C, Barends R, Kelly J, Lucero E, Mariantoni M, Megrant A, Neill C, Vainsencher A, Wenner J, Korotkov A N, Cleland A N and Martinis J M 2013
Phys. Rev. Lett. 110 107001
SJ Srinivasan S J, Sundaresan N M, Sadri D, Liu Y, Gambetta J M, Yu T, Girvin S M and Houck A A 2014
Phys. Rev. A 89 033857
MP Pechal M, Huthmacher L, Eichler C, Zeytinoǧlu S, Abdumalikov A A, Berger J S, Wallraff A and Filipp S 2014
Phys. Rev. X 4 041010
Roch Roch N, Schwartz M E, Motzoi F, Macklin C, Vijay R, Eddins A W, Korotkov A N,
Whaley K B, Sarovar M and Siddiqi I 2014 Phys. Rev. Lett.
112 170501
Mandt Mandt S, Sadri D, Houck A A and Türeci H E 2015 New J. Phys. 17 053018
Cirac Ciracf J I, Ekert A K, Huelga S F and Macchiavello C 1999
Phys. Rev. A 59 4249
Clark Clark S, Peng A, Gu M and Parkins S 2003 Phys. Rev. Lett. 91 177901
Browne Browne D E, Plenio M B and Huelga S F 2003 Phys. Rev. Lett. 91 067901
Duan1 Duan L M and Kimble H J 2003 Phys. Rev. Lett. 90, 253601
Mancini Mancini S and Bose S 2005 Phys. Rev. A 70 022307
Cirac1 Cirac J I, Zoller P, Kimble H J and Mabuchi H 1997
Phys. Rev. Lett. 78 3221
Xiao Xiao Y F, Lin X M, Gao J, Yang Y, Han Z F and Guo G C 2004
Phys. Rev. A 70 042314
Lu Lü X Y, Wu J, Zheng L L and Zhan Z M 2011 Phys. Rev. A 83 042302
Yin1 Yin Z Q and Li F L 2007 Phys. Rev. A 75 012324
Clader Clader B D 2014 Phys. Rev. A 90 012324
Yin2 Yin Z Q, Yang W L, Sun L Y and Duan L M 2015 Phys. Rev. A 91 012333
Enk van Enk S J, Kimble H J, Cirac J I and Zoller P 1999 Phys. Rev. A 59 2659
Serafini Serafini A, Mancini S and Bose S 2006 Phys. Rev. Lett. 96 010503
WLYang Yang W L, Hu Y, Yin Z Q, Deng Z J and Feng M 2011
Phys. Rev. A 83 022302
Megrant Megrant A, Neill C, Barends R, Chiaro B, Chen Y, Feigl L, Kelly J, Lucero E,
Mariantoni M, O'Malley P J J, Sank D, Vainsencher A, Wenner J, White
T C, Yin Y, Zhao J, Palmstrøm C J, Martinis J M and Cleland A N
2012 Appl. Phys. Lett. 100 113510
Koch Koch J, Yu T M, Gambetta J, Houck A A, Schuster D I, Majer J, Blais A, Devoret M H, Girvin S M and
Schoelkopf R J 2007 Phys. Rev. A 76 042319
Schreier Schreier J A, Houck A A, Koch J, Schuster D I, Johnson B R, Chow J M, Gambetta J M, Majer J, Frunzio
L, Devoret M H, Girvin S M and Schoelkopf R J 2008 Phys. Rev.
B 77 180502(R)
Reed Reed M D, DiCarlo L, Nigg S E, Sun L, Frunzio L, Girvin S M and Schoelkopf R J 2012
Nature 482 382
Chow Chow J M, Gambetta J M, Magesan E, Abraham D W, Cross A W, Johnson B R,
Masluk N A, Ryan C A, Smolin J A, Srinivasan S J and Steffen M 2014
Nat. Commun. 5 4015
Chang Chang J B, Vissers M R, Córcoles A D, Sandberg M,
Gao J S, Abraham D W, Chow J M, Gambetta J M, Rothwell M B, Keefe G
A, Steffen M and Pappas D P 2013 Appl. Phys. Lett.
103 012602
Steffen1 Steffen L, Salathe Y, Oppliger M, Kurpiers P, Baur M, Lang C, Eichler C, Hellmann G P, Fedorov A
and Wallraff A 2013 Nature 500 319
Hoi Hoi I C, Wilson C M, Johansson G, Palomaki T, Peropadre B and Delsing P 2011
Phys. Rev. Lett. 107 073601
Peropadre Peropadre B, Zueco D, Wulschner F, Deppe F, Marx A,
Gross R and Ripoll J J G 2013 Phys. Rev. B 87 134504
hu Hu Y and Tian L 2011 Phys. Rev. Lett. 106 257002
Yin Yin Y, Chen Y, Sank D, O'Malley P J J, White T C, Barends R, Kelly J, Lucero E,
Mariantoni M, Megrant A, Neill C, Vainsencher A, Wenner J, Korotkov
A N, Cleland A N and Martinis J M 2013 Phys. Rev. Lett.
110 107001
Strand Strand J D, Ware M, Beaudoin F, Ohki T A, Johnson B R, Blais A and Plourde B L T 2013
Phys. Rev. B 87 220505(R)
M Göppl M, Fragner A, Baur M, Bianchetti R, Filipp S,
Fink J M, Leek P J, Puebla G, Steffen L and Wallraff A 2008 J.
Appl. Phys. 104 113904
Nakamura Nakamura Y, Pashkin Y and Tsai J S 1999 Nature 398 786
Shnirman Shnirman A, Schön G and Hermon Z 1997 Phys. Rev. Lett. 79 2371
Martinis Martinis J M 2009 Quantum Inf. Proc. 8 81
|
1511.00529
|
§ INTRODUCTION
We all agree that complexity is everywhere. Yet, there is no agreed
definition of complexity. Perhaps complexity is so general that it
resists definition <cit.>. Still, it is useful to have formal
measures of complexity to study and compare different phenomena <cit.>.
We have proposed measures of emergence, self-organization, and complexity <cit.>
based on information theory <cit.>. Shannon information
can be seen as a measure of novelty, so we use it as a measure of
emergence, which is correlated with chaotic dynamics. Self-organization
can be seen as a measure of order <cit.>,
which can be estimated with the inverse of Shannon's information and
is correlated with regularity. Complexity can be seen as a balance
between order and chaos <cit.>, between
emergence and self-organization <cit.>.
We have studied the complexity of different phenomena for different
purposes <cit.>. Instead of searching for more
data and measure its complexity, we decided to explore different distributions
with our measures. This would allow us to study broad classes of dynamical
systems in a general way, obtaining a deeper understanding of the
nature of complexity, emergence, and self-organization. Nevertheless,
our previously proposed measures use discrete Shannon information. Even when any distribution can be discretized, this always
comes with caveats <cit.>. For this reason, we base ourselves
on differential entropy <cit.> to propose measures for continuous distributions.
The next section provides background concepts related to information and entropies. Next, discrete measures of emergence, self-organization, and complexity are reviewed <cit.>. Section 4 presents continuous versions of these measures, based on differential entropy. The probability density functions used in the experiments are described in Section 5. Section 6 presents results, which are discussed and related to information adaptation <cit.> in Section 7.
§ INFORMATION THEORY
Let us have a set of possible events whose probabilities of occurrence
are $p_{1},p_{2},\ldots,p_{n}\in P\left(X\right)$. Can we measure
the uncertainty described by the probability distribution $P\left(X\right)$?
To solve this endeavor in the context of telecommunications, Shannon proposed a measure of entropy <cit.>, which corresponds to Boltzmann-Gibbs entropy in thermodynamics. This measure as originally proposed by Shannon, possess a
dual meaning of both uncertainty and information, even when the latter
term was later discouraged by Shannon himself <cit.>. Moreover,
we encourage the concept of entropy as the average uncertainty given
the property of asymptotic equipartition (described later in this
section). From an information-theoretic perspective, entropy measures
the average number of binary questions required to determine the value
of $p_{i}$ . In cybernetics, it is related to variety <cit.>, a
measure of the number of distinct states a system can be in.
In general, entropy is discussed regarding a discrete probability distribution. Shannon extended this concept to the continuous domain with differential entropy. However, some of the properties of its discrete counterpart are not maintained. This has relevant implications for extending to the continuous domain the measures proposed in <cit.>.
Before delving into these differences, first we introduce the discrete entropy, the asymptotic equipartition property (AEP), and the properties of discrete entropy. Next, differential
entropy is described, along with its relation to discrete entropy.
§.§ Discrete Entropy
Let $X$ be a discrete random variable, with a probability mass function
$p\left(x\right)=Pr\left\{ X=x\right\} ,x\in X$ . The entropy $H\left(X\right)$
of a discrete random variable X is then defined by
\begin{equation}
H\left(X\right)=-\sum_{x\in X}p\left(x\right)\log_{2}p\left(x\right).\label{eq:DscrtEntropy}
\end{equation}
The logarithm base provides the entropy's unit. For instance, base
two measures entropy as bits, base ten as nats. If the base of
the logarithm is $ \beta $, we denote the entropy as $H_{\beta}\left(X\right)$. Unless otherwise stated, we will consider all logarithms to be of base
two. Note that entropy does not depend on the value of $X$, but on the probabilities of the possible values $X$ can take. Furthermore, Eq. <ref> can
be understood as the expected value of the information of the distribution.
§.§ Asymptotic Equipartition Property for Discrete Random Variables
In probability, the large numbers law states that, for a sequence
of n i.i.d. elements of a sample $X$, the average value of
the sample $\frac{1}{n}\sum_{i=1}^{n}X_{i}$ approximates the expected
value $\mathbb{E}\left(X\right)$. In this sense, the Asymptotic Equipartition
Property (AEP) establishes that $H\left(X\right)$ can be approximated
\[
\]
such that $n\rightarrow\infty$, and $x_{i}\in X$ are i.i.d. (independent and identically distributed).
Therefore, discrete entropy can be written also as
\begin{equation}
\end{equation}
where $\mathbb{E}$ is the expected value of $P\left(X\right).$ Consequently,
Eq. <ref> describes the expected or
average uncertainty of probability distribution $P\left(X\right).$
A final note about entropy is that, in general, any process that makes
the probability distribution more uniform increases its entropy <cit.>.
§.§ Properties of Discrete Entropy
The following are properties of the discrete entropy function. Proofs and details can be found in texbooks
* Entropy is always non-negative, $H\left(X\right)\geq0.$
* $H_{\beta}\left(X\right)=\left(\log_{\beta}a\right)H_{a}\left(X\right).$
* $H\left(X_{1},X_{2},\ldots,X_{n}\right)\leq\sum_{i=1}^{n}H\left(X_{i}\right),$
with equality iff $X_{i}$ are i.i.d.
* $H\left(X\right)\leq\log\left|X\right|,$ with equality iff $X$ is
distributed uniformly over $X$.
* $H\left(X\right)$ is concave.
§.§ Differential Entropy
Entropy was first formulated for discrete random variables, and was
then generalized to continuous random variables in which case it is
called differential entropy <cit.>. It has
been related to the shortest description length, and thus, is similar
to the entropy of a discrete random variable <cit.>. The
differential entropy $H\left(X\right)$ of a continuous random variable
$X$ with a density $f\left(x\right)$ is defined as
\begin{equation}
\end{equation}
where $S$ is the support set of the random variable. It is well-known
that this integral exists iff the density function of the random variables
is Riemann-integrable <cit.>. The Riemann integral is fundamental in modern calculus. Loosely speaking, is the approximation of the area under any continuous curve given by the summation of ever smaller sub-intervals (i.e. approximations), and implies a well-defined concept of limit <cit.>.
can also be used to denote differential entropy, and in the rest of the article,
we shall employ this notation.
§.§ Asymptotic Equipartition Property of Continuous Random Variables
Given a set of i.i.d. random variables drawn from a continuous distribution
with probability density $f\left(x\right)$, its differential entropy
$H\left(f\right)$ is given by
\begin{equation}
\end{equation}
such that $n\rightarrow\infty$. The convergence to expectation is
a direct application of the weak law of large numbers.
§.§ Properties of Differential Entropy
* $H\left(f\right)$ depends on the coordinates.
For different choices of coordinate systems for a given probability
distribution $P\left(X\right)$, the corresponding differential entropies
might be distinct.
* $H\left(f\right)$ is scale variant <cit.>.
In this sense, $H\left(af\right)=H\left(f\right)+\log_{2}\left|a\right|$,
such that $a\neq0$.
* $H\left(f\right)$is traslational invariant <cit.>.
In this sense, $H\left(f+c\right)=H\left(f\right)$.
* $-\infty\leq H\left(f\right)\leq\infty$. <cit.>.
The $H\left(f\right)$ of a Dirac delta probability distribution,
is considered the lowest $H\left(f\right)$bound, which corresponds
to $H\left(f\right)=-\infty$.
* Information measures such as relative entropy and mutual information
are consistent, either in the discrete or continuous domain <cit.>.
§.§ Differences between Discrete and Continuous Entropies
The derivation of equation <ref> comes from
the assumption that its probability distribution is Riemann-integrable.
If this is the case, then differential entropy can be defined just like discrete
entropy. However, the notion of “average uncertainty” carried
by the Eq. <ref> cannot be extended to its differential
equivalent. Differential entropy is rather a function of the parameters
of a distribution function, that describes how uncertainty changes
as the parameters are modified <cit.>.
To understand the differences between Eqs. <ref>
and <ref> we will quantize a probability density
function, and then calculate its discrete entropy <cit.>.
First, consider the continuous random variable $X$ with a probability
density function $f\left(x\right).$This function is then quantized
by dividing its range into h bins of length $\Delta$. Then,
in accordance to the Mean Value Theorem, within each $h_{i}$ bin
of size $\left[i\Delta,\left(i+1\right)\Delta\right]$, there exists
a value $x_{i}^{*}$ that satisfies
\begin{equation}
\intop_{i\Delta}^{\left(i+1\right)\Delta}f\left(x\right)dx=f\left(x_{i}^{*}\right)\Delta.
\end{equation}
Then, a quantized random variable $X_{i}^{\Delta}$is defined
\begin{eqnarray}
X_{i}^{\Delta}=x_{i}^{*} & & \text{if }i\Delta\leq X\leq\left(i+1\right)\Delta,
\end{eqnarray}
and, its probability is
\begin{equation}
\end{equation}
Consequently, the discrete entropy of the quantized variable $X^{\Delta}$,
is formulated as
\begin{eqnarray}
H\left(X^{\Delta}\right) & = & -\sum_{-\infty}^{\infty}p_{i}\log_{2}p_{i}\nonumber \\
& = & -\sum_{-\infty}^{\infty}\left(f\left(x_{i}^{*}\right)\Delta\right)\log_{2}\left(f\left(x_{i}^{*}\right)\Delta\right)\nonumber \\
& = & -\sum\Delta f\left(x_{i}^{*}\right)\log_{2}f\left(x_{i}^{*}\right)-\sum f\left(x_{i}^{*}\right)\Delta\log_{2}\Delta\nonumber \\
& = & -\log_{2}\Delta-\sum\Delta f\left(x_{i}^{*}\right)\log_{2}f\left(x_{i}^{*}\right).\label{eq:QuantizedDscrtEntrpy}
\end{eqnarray}
To understand the final form of Eq. <ref>,
notice that as the size of each bin becomes infinitesimal, $\Delta\rightarrow0$,
the left-hand term of Eq. <ref> becomes $\log_{2}\left(\Delta\right)$.
This is a consequence of
\[
\lim_{\Delta\rightarrow0}\sum_{-\infty}^{\infty}f\left(x_{i}^{*}\right)\Delta=\int_{-\infty}^{\infty}f\left(x\right)dx=1.
\]
Furthermore, as $\Delta\rightarrow0$, the right-hand side of Eq. <ref>
approximates the differential entropy of X such that
\[
\lim_{\Delta\rightarrow0}\sum_{-\infty}^{\infty}\Delta f\left(x_{i}^{*}\right)\log_{2}f\left(x_{i}^{*}\right)=\int_{-\infty}^{\infty}f\left(x\right)\log_{2}f\left(x\right)dx.
\]
Note that the left-hand side of Eq. <ref>,
explodes towards minus infinity such that
\[
\lim_{\Delta\rightarrow0}\log_{2}\left(\Delta\right)\approx-\infty,
\]
Therefore, the difference between $H\left(f\right)$ and $H\left(X^{\Delta}\right)$is
which approaches to $-\infty$ as the bin size becomes infinitesimal.
Moreover, consistently with this is the fact that the differential
entropy of a discrete value is $-\infty$ <cit.>.
Lastly, in accordance to <cit.>, the average number of
bits required to describe a continuous variable X with a n-bit
accuracy (quantization) is $H\left(X\right)+n\approx H\left(f\right)$
such that
\begin{equation}
H\left(X^{\Delta}\right)'=\lim_{\Delta\rightarrow0}H\left(X^{\Delta}\right)+\log_{2}\left(\Delta\right)\rightarrow H\left(f\right).\label{eq:nBitQuanDifferentialEntropy}
\end{equation}
§ DISCRETE COMPLEXITY MEASURES
Emergence $E$, self-organization $S$, and complexity $C$
are close relatives of Shannon's entropy. These information-based
measures, inherit most of the properties of Shannon's discrete entropy
<cit.>, being the most valuable one that, discrete
entropy quantizes the average uncertainty of a probability distribution.
In this sense, complexity $C$ and its related measures ($E$ and $S$) are based on a quantization of the average information contained by a process described by its probability distribution.
§.§ Emergence
Another form of entropy, rather related to the concept of information
as uncertainty, is called emergence $E$ <cit.>. Intuitively,
$E$ measures the ratio of uncertainty a process produces by new information that is consequence of changes in a) dynamics or b) scale
<cit.>. However, its formulation is more related to the thermodynamics entropy. Thus, it is defined as
\begin{equation}
E\equiv H\left(X\right)=-K\sum_{i=1}^{N}p_{i}\log_{2}p_{i},\label{eq:DiscreteEmergence}
\end{equation}
where $p_{i}=P\left(X=x\right)$ is the probability of the element
$i$, and $K$ is a normalizing constant.
§.§ Multiple Scales
In thermodynamics, the Boltzmann constant K, is employed to
normalize the entropy in accordance to the probability of each state. However, Shannon's entropy typical formulation
<cit.> neglects the usage of
K in Eq. <ref> (been its only constraint
that $K>0,$ <cit.>). Nonetheless, for emergence as
a measure of the average production of information for a given distribution,
K plays a fundamental role. In the cybernetic definition
of variety <cit.>, $K$ is a function
of the distinct states a system can be, i.e. the system's alphabet size.
Formally, it is defined as
\begin{equation}
\end{equation}
where $b$ corresponds to the size of the alphabet of the sample or
bins of a discrete probability distribution. Furthermore, $K$ should
guarantee that $0\leq E\leq1$, therefore, $b$ should be at least equal
to the number of bins of the discrete probability distribution.
It is also worth noting that the denominator of Eq. <ref>,
$\log_{2}\left(b\right),$ is equivalent to the maximum entropy for
a continuous distribution function, the uniform distribution. Consequently,
emergence can be understood as the ratio between the entropy for given
distribution $P\left(X\right)$, and the maximum entropy for the same
alphabet size $H\left(U\right)$ <cit.>, this is
\begin{equation}
\end{equation}
§.§ Self-Organization
Entropy can also provide a measure of system's organization, and its
predictability <cit.>. In this sense, with more uncertainty less predictability is achieved, and vice-versa. Thus, an entirely random process (e.g. uniform distribution) has the lowest organization, and a completely deterministic system one (Dirac delta distribution), has the highest. Furthermore, an extremely organized system yields no information with respect of novelty, while, on the other hand, the more chaotic a system is, the more information is yielded <cit.>.
The metric of self-organization $S$ was proposed to measure the organization
a system has regarding its average uncertainty <cit.>.
$S$ is also related to the cybernetic concept of constraint,
which measures changes in due entropy restrictions on the state space
of a system <cit.>. These constraints confine the system's behavior, increasing
its predictability, and reducing the (novel) information it provides to an
observer. Consequently, the more self-organized a system is, the less
average uncertainty it has. Formally, $S$ is defined as
\begin{equation}
\end{equation}
such that $0\leq S \leq1$. It is worth noting that, $S$ is the complement
of $E$. Moreover, the maximal $S$ (i.e. $S=1$) is only achievable
when the entropy for a given probability density function (PDF) is such that $H\left(P\left(X\right)\right)\rightarrow0$,
which corresponds to the entropy of a Dirac delta (only in the
discrete case).
§.§ Complexity
Complexity $C$ can be described as a balance between order (stability),
and chaos (scale or dynamical changes) <cit.>. More
precisely, this function describes a system's behavior in terms of
the average uncertainty produced by its probability distribution in
relation the dynamics of a system. Thus, the
complexity measure is defined as
\begin{equation}
C=4\cdot E\cdot S,\label{eq:DiscreteComplexity}
\end{equation}
such that, $0\leq C\leq1$.
§ CONTINUOUS COMPLEXITY MEASURES
As mentioned before, discrete and differential entropies do
not share the same properties. In fact, the property of discrete entropy
as the average uncertainty in terms of probability, cannot be extended
to its continuous counterpart. As consequence, the proposed continuous
information-based measures describe how the production
of information changes respect to the probability distribution parameters.
In particular, this characteristic could be employed as a feature
selection method, where the most relevant variables are those which
have a high emergence (the most informative).
The proposed measures are differential emergence ($E_{D}$), differential
self-organization ($S_{D}$), and differential complexity ($C_{D}$).
However, given that the interpretation and formulation (in terms of emergence) of discrete
and continuous $S$ (Eq. <ref>) and $C$ (Eq. <ref>) are the same, we only provide details
on $E_{D}$. The difference between $S_{D}$, $C_{D}$ and $S, C$ is that the former are defined on $E_{D}$, while the latter on $E$. Furthermore, we make emphasis in the definition of the normalizing
constant K, which play a significant role in constraining $E_{D}\in\left[0,1\right]$,
and consequently, $S_{D}$ and $C_{D}$ as well.
§.§ Differential Emergence
As for its discrete form, the emergence for continuous random variables
is defined as
\begin{equation}
\end{equation}
where, $\left[\upsilon,\zeta\right]$ is the domain, and K
stands for a normalizing constant related to the distribution's alphabet
size. It is worth noting that this formulation is highly related
to the view of emergence as the ratio of information production
of a probability distribution respect the maximum differential entropy
for the same range. However, since $E_{D}$ can be negative (i.e. entropy
of a single discrete value), we choose $E_{D}$ such that
\begin{equation}
E{}_{D} & E{}_{D}>0\\
0 & \text{otherwise}.
\end{cases}.\label{eq:DifferentialEmergencePrime}
\end{equation}
$E'{}_{D}$ is rather a more convenient function than $E{}_{D}$, as $0\leq E'{}_{D} \leq 1$.
This statement is justified in the fact that the differential entropy
of a discrete value is $-\infty$ <cit.>. In practice, differential entropy becomes negative only when the probability distribution is extremely narrow, i.e. there is a high probability for few states.
In the context of information changes due parameters manipulation, an $E{}_{D}<0$
means that the probability distribution is becoming a Dirac delta
distribution. For notation convenience, from now on we will employ
$E_{D}$ and $E'_{D}$ interchangeably.
§.§ Multiple Scales
The $K$ constant expresses the relation between uncertainty of a given $P\left(X\right)$
Defined by $H(X)$, respect to the entropy of a maximum entropy over
the same domain <cit.>. In this setup, as the uncertainty
grows, $E'_{D}$ becomes closer to unity.
To constrain the value of $H\left(X\right)=\left[0,1\right]$ in
the discrete emergence case, it was enough to establish the distribution's
alphabet size, b of Eq. <ref>, such that
$b\geq\text{\# bins}$ <cit.>. However, for any PDF,
the number of elements between a pair of points $a$ and $b$, such
that $a\neq b$, is infinite. Moreover, as the size of each bin becomes
infinitesimal, $\Delta\rightarrow0$, the entropy for each bin becomes
$-\infty$ <cit.>. Also, it has been stated
that b value should be equal to the cardinality of $X$ <cit.>,
however, this applies only to discrete emergence. Therefore,
rather than a generalization, we propose an heuristic for the selection
of a proper K in the case of differential emergence. Moreover,
we differentiate between b for $H\left(f\right)$, and b'
for $H\left(X^{\Delta}\right)'$.
As in the discrete case, K is defined as Eq. <ref>.
In order to determine the proper alphabet size b, we propose
the next algorithm:
* If we know a priori the true $P\left(X\right)$, we calculate $H\left(f\right)$,
and $b=\left|P\left(X\right)\right|$ is the cardinality within
the interval of Eq. <ref>. In this sense,
a large value will denote the cardinality of an “ghost” sample
[It is ghost, in the concrete sense that it does not exist.
Its only purpose is to provide a bound for the maximum entropy accordingly
to some large alphabet size.
* If we do not know the true $P\left(X\right)$, or we are interested
rather in $H\left(X^{\Delta}\right)'$ where a sample of finite size
is involved, we calculate b' as
b'=∑_iind(x_i) ,
such that, the non-negative function $ind\left(\cdot\right)$ is defined
1 iff P(x_i)>0
0 otherwise
For instance, in the quantized version of the standard normal distribution
($N\left(0,1\right)$), only values within $\pm3\sigma$ satisfy this
constraint despite the domain of Eq. <ref>.
In particular, if we employ $b=\left|X\right|$ rather than $b'$,
we compress the $E_D$ value as it will be shown in the next
section. On the other hand, for a uniform distribution or a power-law
(such that $0<x_{min}<x$), the whole range of points satisfies this
§ PROBABILITY DENSITY FUNCTIONS
In communication and information theory, uniform (U) and normal, a.k.a. Gaussian (G) distributions
play a significant role. Both are referent to maximum entropy:
on the one hand, U has the maximum entropy within a continuous domain;
on the other hand, G has the maximum entropy for distributions with a fixed
mean ($\mu$), and a finite support set for a fixed standard deviation
($\sigma$) <cit.>. Moreover, as mentioned
earlier, $H\left(f\right)$ is useful when comparing
the entropies of two distributions over some reference space <cit.>.
Consequently, U, but mainly G, are heavily used in the context of
telecommunications for signal processing <cit.>.
Nevertheless, many natural and man-made phenomena can be approximated with power-law (PL) distributions. These types of distributions typically
present complex patterns that are difficult to predict, making them a relevant research topic <cit.>. Furthermore, power-laws have
been related to the presence of multifractal structures in certain
types of processes <cit.>. Moreover, power-laws are tightly
related to self-organization and criticality theory, and have been
studied under information frameworks before (e.g. Tsallis', and Renyi's
maximum entropy principle) <cit.>.
Therefore, in this work we focus our attention to these three PDFs.
First, we provide a short description of each PDF, then, we summarize
its formulation, and the corresponding $H\left(f\right)$ in Table
§.§ Uniform Distribution.
The simplest PDF, as its name states, establishes that for each possible
value of X, the probability is constant over the whole support
set (defined by the range between $a$ and $b$), and 0 elsewhere.
This PDF has no parameters besides the starting and ending points
of the support set. Furthermore, this distribution appears frequently
in signal processing as white noise, and it
has the maximum entropy for continuous random variables <cit.>.
Its PDF, and its corresponding $H\left(f\right)$ are shown in first
row of Table <ref>. It is worth noting that,
as the cardinality of the domain of U grows, its differential entropy
increases as well.
§.§ Normal Distribution.
The normal or Gaussian distribution is one of the most important probability
distribution families <cit.>. It is fundamental in
the central limit theorem <cit.>, time series forecasting
models such as classical autoregressive models <cit.>,
modelling economic instruments <cit.>, encryption, modelling
electronic noise <cit.>, error analysis and statistical hypothesis testing.
Its PDF is characterized by a symmetric, bell-shaped function whose
parameters are: location (i.e. mean $\mu$), and dispersion (i.e.
standard deviation $\sigma^{2}$ ). The standard normal distribution
is the simplest and most used case of this family, its parameters
are $N\left(\mu=0,\,\sigma^{2}=1\right)$. A continuous random variable $x\in X$
is said to belong to a Gaussian distribution, $X\sim N\left(\mu,\,\sigma^{2}\right)$
, if its PDF $p\left(x\right)$ is given by the one described in the
second row of Table <ref>. As is shown in the
table, the differential entropy of G only depends on the standard
deviation. Furthermore, it is well known that its differential entropy
is monotonically increasing concave in relation to $\sigma$ <cit.>.
This is consistent with the aforementioned fact that $H\left(f\right)$
is translation-invariant. Thus, as $\sigma$ grows, so does the value
of $H\left(G\right)$, while as $\sigma\rightarrow0$ such that $0<\sigma<1$,
it becomes a Dirac delta with $H\left(f\right)\approx0$.
§.§ Power-Law Distribution.
Power-law distributions are commonly employed to describe multiple
phenomena (e.g. turbulence, DNA sequences, city populations, linguistics,
cosmic rays, moon craters, biological networks, data storage in organisms,
chaotic open systems, and so on) across numerous scientific disciplines
These type of processes are known for being scale invariant, being
the typically scales ($\alpha$, see below) in nature between one and 3.5 <cit.>.
Also, the closeness of this type of PDF to chaotic systems and
fractals is such that, some fractal dimensions are called entropy
dimensions (e.g. box-counting dimension, and Renyi entropy) <cit.>.
Power-law distributions can be described by continuous and discrete
distributions. Furthermore, Power-laws in comparison with Normal distribution,
generate events of large orders of magnitude more often, and are not
well represented by a simple mean. A Power-Law density distribution
is defined as
\begin{equation}
p\left(x\right)dx=P\left(x\leq X\leq x+dx\right)=Cx^{-\alpha}dx,
\end{equation}
such that, C is a normalization factor, $\text{\ensuremath{\alpha}}$
is the scale exponent, and $X\mid x>x_{min}>0$ is the observed
continuos random variable. This PDF diverges as $x\rightarrow0$ ,
and do not hold for all $x\ge0$ <cit.>. Thus, $x_{min}$
corresponds to lower bound of a power-law. Consequently, in Table
<ref> we provide the PDF of a Power-Law as
proposed by <cit.>, and its corresponding $H\left(f\right)$
as proposed by <cit.>.
The aforementioned PDFs, and their corresponding $H\left(f\right)$
are shown in Table <ref>. Further details about
the derivation of $H\left(f\right)$ for U, and G can be found in
<cit.>. For additional details on the differential
entropy of the power-law, we refer the reader to <cit.>.
Distribution PDF Differential EntropyUniform $p\mbox{\ensuremath{\left(x\right)}=}\begin{cases}
\frac{1}{b-a} & a\leq x\leq b\\
0 & \text{otherwise}
\end{cases}$ $H\left(p\left(x\right)\right)=\log_{2}\left(b\right)$Normal $p\left(x\right)=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-\left(x-\mu\right)^{2}}{2\sigma}}$ $H\left(p\left(x\right)\right)=\frac{1}{2}\log_{2}\left(2\pi e\sigma^{2}\right)$Power-law $p\left(x\right)=\left(\frac{\alpha-1}{x_{min}}\right)\left(\frac{x}{x_{min}}\right)^{-\alpha}$ $H\left(p\left(x\right)\right)=\log_{2}\left(x_{min}\right)-\log_{2}\left(\alpha-1\right)+\left(\frac{\alpha}{\alpha-1}\right)$
Studied PDFs (left column) with their corresponding analytical differential
entropies (right column).
§ RESULTS
In this section, comparisons of theoretical
vs quantized differential entropy for the PDFs considered are shown. Next,
we provide differential complexity results ($E'_{D}$, $S_{D}$,
and $C_{D}$) for the mentioned PDFs. Furthermore, in the case of power-laws,
we also provide and discuss the corresponding complexity measures
results for real world phenomena, already described in <cit.>.
Also, it is worth noting that, since for quantized $H\left(f\right)$
of the power-law yielded poor results, the power-law's analytical
$H\left(f\right)$ form was used.
§.§ Theoretical vs Quantized Differential Entropies
Numerical results of theoretical and quantized differential entropies are shown in Figs. <ref>
and <ref>. Analytical $H\left(f\right)$
results are displayed in blue, whereas the quantized $H\left(X^{\Delta}\right)'$
ones are shown in red. For each PDF, a sample of one million
(i.e. $1\times10^{6}\equiv\text{1M}$) points where employed for calculations.
The bin size $\Delta$ required by $H\left(X^{\Delta}\right)'$,
is obtained as the ratio $\Delta=\frac{Range}{\left|Sample\right|}$.
However, the value of $\Delta$ has considerable influence in the resulting
quantized differential entropy.
Theoretical and quantized differential entropies for the uniform and power-law
§.§.§ Uniform Distribution.
The results for U were expectable. We tested
several values of the cardinality of $P\left(X\right)$, such that
$b=2^{i}\mid i=1,\ldots,15$. Using the analytical $H\left(f\right)$
formula of Table <ref>, the quantized $H\left(X^{\Delta}\right)'$,
and $\Delta=1$ we achieved exactly the same differential entropy
values. Results for U are shown in the left side of Fig. <ref>.
As was mentioned earlier, as the cardinality of the distribution grows,
so does the differential entropy of U.
§.§.§ Normal Distribution.
Results for the Gaussian distribution were less trivial. As in the
U case, we calculate both $H\left(f\right)$ and $H\left(X^{\Delta}\right)'$,
for a fixed $\mu=0$, and modified the standard deviation parameter
such that, $\sigma=2^{i}\mid i=0,1,\ldots,14$. Notice that the first tested distribution is the
standard normal distribution.
In Fig. <ref>, results obtained
for the n-bit quantized differential entropy, and for the analytical
form of Table <ref> are shown. Moreover, we
displayed two cases of the normal distribution: the left side of Fig.
<ref> shows results for
$P\left(X\right)$ with range $\left[-50,50\right]$ and a bin size,
$\Delta=\frac{100}{1M}=1\times10^{-4}$,whereas, right side provides
results for a $P\left(X\right)$ with range $\left[-500e3,500e3\right]$
and $\Delta=1$. It is worth noting that, in the former case the quantized
differential entropy shows a discrepancy with $H\left(f\right)$ after
only $\sigma=2^{4}=16$, which quickly increases with growing $\sigma$.
On the other hand, for the latter case there is an almost perfect
match between the analytical and quantized differential entropies,
however, the same mismatch will be observed if the standard deviation
parameter is allowed to grow unboundedly $(\sigma\rightarrow\infty)$.
Nonetheless, this is a consequence of how $H\left(X^{\Delta}\right)'$
is computed. As mentioned earlier, as $\Delta\rightarrow0$ the value
of each quantized $X^{\Delta}$ grows towards $-\infty$. Therefore,
in the G case, it seems convenient employing a Probability Mass Function
(PMF) rather than a PDF. Consequently, the experimental setup of right
side image of Fig. <ref>
is employed for the calculation of the continuous complexity measures
of G.
Two comparisons of theoretical vs quantized differential entropy for
the Gaussian distribution.
§.§.§ Power-Law Distribution.
Results for the power-law distribution are shown in the right side of Fig. <ref>. In both U and G, a PMF instead of
a PDF was used to avoid cumbersome results (as depicted in the corresponding
images). However, for the power-law distribution, the use of a PDF is rather convenient.
As shown in Fig. <ref> and highlighted
by <cit.>, $x_{min}$ has a considerable impact on the value of
$H\left(f\right)$. For Fig. <ref>,
the range employed was $\left[1,50\right]$, with a bin size of $\Delta=1\times10^{-5}$,
a $x_{min}=0.99$, and modified the scale exponent parameter such
that, $\alpha=i\mid i=1,\ldots,15$. For this particular setup, we
can observe that as $\alpha$ increases, $H\left(f\right)$ and $H\left(X^{\Delta}\right)'$
decreases its value towards $-\infty$. This effect is consequence
of increasing the scale of the Power-law such that, the slope of the
function in a log-log space, approaches to zero. In this sense, with
larger $\alpha$'s, the $P\left(X\right)$ becomes closer to a Dirac delta
distribution, thus, $H\left(f\right)\rightarrow-\infty$. However,
as will be discussed later, for larger $\alpha$'s larger $x_{min}$
values are required, in order for $H\left(f\right)$ to display positive
§.§ Differential Complexity: $E{}_{D}$, $S_{D}$, and $C_{D}$
U results are trivial: $E{}_{D}=1$,
and $S_{D},C_{D}=0$. For each upper bound of U, $E'_{D}=\frac{H\left(U\right)}{H\left(U\right)}=1$,
which is exactly the same as its discrete counterpart. Thus, U results
are not considered in the following analysis.
Continuous complexity results for G and PL are
shown in Figs. <ref> and <ref>,
respectively. In the following we provide details of these measures.
§.§.§ Normal Distribution.
It was stated in Section <ref>
that, the size of the alphabet is given by the function $ind\left(P\left(X\right)\right)$.
This rule establishes a valid cardinality such that $P\left(X\right)>0$,
thus, only those states with a positive probability are considered. For
$P\left(X^{\Delta}\right)$, such operation can be performed. Nevertheless,
when the analytical $H\left(f\right)$ is used, the proper cardinality
of the set is unavailable. Therefore, in the Gaussian distribution
case, we tested two criteria for selecting the value of b:
* $\sum_{x_{i}}ind\left(\cdot\right)$ is employed for $H\left(X^{\Delta}\right)'$
* A constant with a large value ($C=1\times10^{6}$)
is used for the analytical formula of $H\left(f\right)$.
In Fig. <ref>, solid dots are used
when K is equal to the cardinality of $P\left(X\right)>0$,
whereas solid squares are used for an arbitrary large constant.
Moreover, for the quantized case of $P\left(G\right)$ , Table <ref>
shows the cardinality for each sigma, $b'_{i}$, and its corresponding
$K_{i}$. As it can be observed, for a large normalizing constant
K, a logarithmic relation is displayed for $E{}_{D}$ and $S_{D}$.
Also, the maximum $C_{D}$ is achieved for $\sigma=2^{8}=256$, which
is where $E{}_{D}=S_{D}$. However, for $H\left(X^{\Delta}\right)'$
the the maximum $C_{D}$ is found around
$\sigma=2^{1,2,3}=2,4,8$, such that $C_{D}\leq\epsilon\mid\epsilon\rightarrow0$.
A word of advise must be made here. The required cardinality to normalize
the continuous complexity measures such that $0\leq E{}_{D},S_{D},C_{D}\leq1$,
must have a lower bound. This bound should be related to the scale
of the $P\left(X\right)$ <cit.>, and the quantization
size $\Delta$. In our case, when a large cardinality $\left|U\right|=1\times10^{6}$,
and $\Delta=1$ are used, the normalizing constant flattens
$E_{D}$ results respect those obtained by $b'$; moreover, the large
constant increases $S_{D}$, and takes greater standard deviations
for achieving the maximum $C_{D}$. However, these complexity results
are rather artificial in the sense that, if we arbitrarly let $\left|U\right|\rightarrow\infty$
then trivially we will obtain $E{}_{D}=0,\, S_{D}=1,\text{ and }C_{D}=0$.
Moreover, it has been stated that the cardinality of $P\left(X\right)$
should be employed as a proper size of b <cit.>.
Therefore, when $H\left(X^{\Delta}\right)'$ is employed, the cardinality
of $P\left(X\right)>0$ must be used. On the contrary, when $H\left(f\right)$
is employed, a coarse search for increasing alphabet sizes could be
used so that the maximal $H\left(f\right)$ satisfies $\frac{H\left(f\right)}{H\left(U\right)}\leq1$.
$\sigma$ $b'=\sum ind\left(Pr\left(X\right)>0\right)$ $H\left(U\right)=\log_{2}\left(b'\right)$ $K=\frac{1}{H\left(U\right)}$$2^{0}=1$ 78 6.28 0.16$2^{1}=2$ 154 7.26 0.14$2^{2}=4$ 308 8.27 0.12$2^{3}=8$ 616 9.27 0.11$2^{4}=16$ 1232 10.27 0.10$2^{5}=32$ 2464 11.27 0.09$2^{6}=64$ 4924 12.27 0.08$2^{7}=128$ 9844 13.27 0.075$2^{8}=256$ 19680 14.26 0.0701$2^{9}=512$ 39340 15.26 0.0655$2^{10}=1024$ 78644 16.26 0.0615$2^{11}=2048$ 157212 17.26 0.058$2^{12}=4096$ 314278 18.26 0.055$2^{13}=8192$ 628258 19.26 0.0520$2^{14}=16384$ 1000000 19.93 0.050
Alphabet size $b'$, and its corresponding normalizing K constant
for the normal distribution G.
Complexity of the Gaussian distribution.
§.§.§ Power-Law Distribution
In this case, $H\left(f\right)$ rather than $H\left(X^{\Delta}\right)'$
is used for computational convenience. Although the cardinality of
$P\left(X\right)>0$ is not available, by simply substituting $p\left(x_{i}\right)>0\mid x=\left\{ 1,\ldots,1\times10^{6}\right\} $
we can see that the condition is fulfilled by the whole set. Therefore,
the large $C$ criterium, earlier detailed, is used. Still,
given that a numerical power-law distribution is given by two parameters,
a lower bound $x_{min}$ and the scale exponent $\alpha$, we depict
our results in 3D in Fig. <ref>. From left to
right, $E{}_{D},\, S_{D},\text{ and }C_{D}$ for the power-law distribution
are shown, respectively. In the three images, the same coding is used:
x-axis displays the scale exponent ($\alpha$) values, y-axis shows
$x_{min}$ values, and z-axis depicts the continuous measure
values; lower values of $\alpha$ are displayed in dark blue, turning
into reddish colors for larger exponents.
As it can be appreciated in Fig. <ref>, for small
$x_{min}$ (e.g. $x_{min}=1$) values, low emergence is produced despite
the scale exponent. Moreover, maximal self-organization ($\text{\emph{i.e.} }S_{D}=1$)
is quickly achieved (i.e. $\alpha=4$), providing a PL with at most
fair complexity values. However, if we let $x_{min}$ take larger numbers,
$E_{D}$ grows, achieving the maximal complexity (i.e. $C_{D}\approx0.8$)
of this experimental setup at $x_{min}=15,\,\alpha=1$. This behavior
is also observed for other scale exponent values, where emergence
of new information is produced as the $x_{min}$ value grows. Furthermore,
it has been stated that for $P\left(X\right)$ displays a power-law
behavior it is required that $\forall x_{i}\in P\left(X\right)\mid x_{i}>x_{min}$
<cit.>. Thus, for every $\alpha$ there should be an $x_{min}$
such that $E_{D}>0$. Moreover, for larger scale exponents, larger
$x_{min}$ values are required for the distribution shows emergence
of new information at all.
§.§ Real World Phenomena and their Complexity
Data of phenomena that follows a power law is provided in Table <ref>.
These power-laws have been studied by <cit.>, and the power-law parameters were published by <cit.>.
The phenomena in the table mentioned above compromises data from:
* Numbers of occurrences of words in the novel Moby Dick by Hermann
* Numbers of citations to scientific papers published in 1981, from
the time of publication until June 1997.
* Numbers of hits on websites by users of America Online Internet services
during a single day.
* Number of received calls to A.T.&T. U.S. long-distance telephone
services on a single day.
* Earthquake magnitudes occurred in California between 1910 and 1992.
* Distribution of the diameter of moon craters.
* Peak gamma-ray intensity of solar flares between 1980 and 1989.
* War intensity between 1816–1980, where intensity is a
formula related to the number of deaths and warring nations populations.
* Frequency of family names accordance with U.S. 1990 census.
* Population per city in the U.S. in agreement with U.S. 2000 census.
More details about these power-laws can be found in <cit.>.
For each phenomenon, the corresponding differential entropy and complexity
measures are shown in Table <ref>. Furthermore, we also provide
Table <ref> which is a color coding for complexity measures proposed in <cit.>. Five colors are employed to simplify the different value ranges of $E_D$, $S_D$, and $C_D$ results.
According to the nomenclature suggested in <cit.>,
results for these sets show that, very high complexity $0.8\leq C_{D}\leq1$
is obtained by the number of citations set (i.e. 2), and intensity
of solar flares (i.e. 7). High complexity, $0.6\leq C_{D}<0.8$ is
obtained for received telephone calls (i.e. 4), intensity of wars
(i.e. 8), and frequency of family names (i.e. 9). Fair complexity
$0.4\leq C_{D}<0.6$ is displayed by earthquakes magnitude (i.e. 5),
and population of U.S. cities (i.e. 10). Low complexity, $0.2\leq C_{D}<0.4$
is obtained for frequency of used words in Moby Dick (i.e. 1) and web
hits (i.e. 3), whereas, moon craters (i.e. 6) have very low complexity
$0\leq C_{D}<0.2$. In fact, earthquakes, and web hits, have been
found not to follow a power law <cit.>. Furthermore,
if such sets were to follow a power-law, a greater value of $x_{min}$
would be required as can be observed in Fig. <ref>.
In fact, the former case is found for the frequency of words used
in Moby Dick. In <cit.>, parameters of Table <ref>
are proposed. However, in <cit.>, another set of parameters
are estimated (i.e. $x_{min}=7,\,\alpha=1.95$). For the more recent
estimated set of parameters, a high complexity is achieved (i.e. $C_{D}=0.74$),
which is more consistent with literature about Zipf's law <cit.>.
Lastly, in the case of moon craters, the $x_{min}=0.01$ is rather
a poor choice according to Fig. <ref>. For
the chosen scale exponent, it would require at least a $x_{min}\approx1$,
for the power-law to produce any information at all. It should be noted that $x_{min}$ can be adjusted to change the values of all measures. Also, it is worth mentioning that if we were to normalize and discretize a power law distribution to calculate its discrete entropy (as in <cit.>), all power law distributions present a very high complexity, independently of $x_{min}$ and $\alpha$, precisely because these are normalized. Still, this is not useful for comparing different power law distributions.
Complexity measures for the Power-Law. Lower values of the scale exponent $\alpha$ are displayed in dark blue, colors turns into reddish for larger scale exponents.
Phenomenon $x_{min}$ $\alpha$ (Scale Exponent) $H\left(f\right)$ $E'_{D}$ $S_{D}$ $C_{D}$1 Frequency of use of words 1 2.2 1.57 red!70 0.078 blue!700.92 orange!700.292 Number of citations to papers 100 3.04 7.1 orange!70 0.36 green!700.64 blue!700.913 Number of hits on web sites 1 2.4 1.23 red!70 0.06 blue!70 0.94 orange!700.234 Telephone calls received 10 2.22 4.85 orange!70 0.24 green!70 0.76 green!70 0.745 Magnitude of earthquakes 3.8 3.04 2.38 red!70 0.12 blue!70 0.88 yellow!70 0.426 Diameter of moon craters 0.01 3.14 -6.27 red!700 blue!701 red!70 0 7 Intensity of solar flares 200 1.83 10.11 yellow!700.51 yellow!70 0.49 blue!700.998 Intensity of wars 3 1.80 4.15 orange!700.21 green!700.79 green!700.669 Frequency of family names 10000 1.94 15.44 green!700.78 orange!700.22 green!700.710 Population of U.S. cities 40000 2.30 16.67 blue!700.83 red!700.17 yellow!700.55
Power-Law parameters and information-based measures of real world
Categories for classifying $E$, $S$, and $C$.
Category Very High High Fair Low Very Low
Range $[0.8,1]$ $[0.6,0.8)$ $[0.4,0.6)$ $[0.2,0.4)$ $[0,0.2)$
Color blue!70 Blue green!70 Green yellow!70 Yellow orange!70 Orange red!70 Red
Color coding for $E_D$, $S_D$, and $C_D$ results
§ DISCUSSION
The relevance of the work presented here lies in the fact that it is now possible to calculate measures of emergence, self-organization, and complexity directly from probability distributions, without needing access to raw data. Certainly, the interpretation of the measures is not given, as this will depend on the use we make of the measures for specific purposes.
From exploring the parameter space of the uniform, normal, and scale-free distributions, we can corroborate that high complexity values require a form of balance between extreme cases. On the one hand, uniform distributions, by definition, are homogeneous and thus all states are equiprobable, yielding the highest emergence. This is also the case of normal distributions with a very large standard deviation and for power law distributions with an exponent close to zero. On the other hand, highly biased distributions (very small standard deviation in G or very large exponent in PL) yield a high self-organization, as few states accumulate most of the probability. Complexity is found between these two extremes. From the values of $\sigma$ and $\alpha$, this coincides with a broad range of phenomena. This does not tell us something new: complexity is common. The relevant aspect is that this provides a common framework to study of the processes that lead phenomena to have a high complexity <cit.>. It should be noted that this also depends on the time scales at which change occurs <cit.>.
In this context, it is interesting to relate our results with information adaptation <cit.>. In a variety of systems, adaptation takes place by inflating or deflating information, so that the “right" balance is achieved. Certainly, this precise balance can change from system to system and from context to context. Still, the capability of information adaptation has to be correlated with complexity, as the measure also reflects a balance between emergence (inflated information) and self-organization (deflated information).
As a future work, it will be interesting to study the relationship between complexity and semantic information. There seems to be a connection with complexity as well, as we have proposed a measure of autopoiesis as the ratio of the complexity of a system over the complexity of its environment <cit.>. These efforts should be valuable in the study of the relationship between information and meaning, in particular in cognitive systems.
Another future line of research lies in the relationship between the proposed measures and complex networks <cit.>, exploring questions such as: how does the topology of a network affect its dynamics? How much can we predict the dynamics of a network based on its topology? What is the relationship between topological complexity and dynamic complexity? How controllable are networks <cit.> depending on their complexity?
G.SB. was supported by the Universidad Nacional Autónoma de México (UNAM) under grant CJIC/CTIC/0706/2014.
C.G. was supported by CONACYT projects 212802, 221341, and SNI membership 47907.
Author Contributions
GSB and CG conceived and designed the experiments, GSB performed the experiments, GSB, NF, and CG wrote the paper.
Conflicts of Interest
The authors declare no conflict of interest'.
|
1511.00385
|
$^1$Vinogradov Institute of Geochemistry, Russian Academy of Sciences, Favorskii street 1a, P.O.Box 4019, 664033 Irkutsk, Russia
$^2$Irkutsk State University, Physics department, Gagarin boulevard 20, 664003 Irkutsk, Russia
Optical spectra (absorption, emission, excitation, decay) and dielectric relaxation were measured for divalent europium (and partially for ytterbium) in lanthanum fluoride crystals. Absorption of Eu$^{2+}$ contains not only asymmetric weakly structured band at 245 nm but also long-wavelength bands at 330, 380 nm. Broadband Eu$^{2+}$ emission at 600 nm appeared below 80 K, having decay time 2.2 $\mu$s at 7.5 K. Emission at 600 nm is attributed to so-called anomalous luminescence.
Bulk conductivity is directly proportional to absorption coefficient of Eu$^{2+}$ bands. Dielectric relaxation peak of LaF$_3$-EuF$_3$ is attributed to rotation of dipoles Eu$^{2+}$-anion vacancy. The long-wavelength absorption bands at 330, 380 nm are assigned to transitions from 4f$^7$ Eu$^{2+}$ ground state to states of neighbouring fluorine vacancy.
LaF$_3$ europium ytterbium absorption excitation dielectric relaxation anomalous luminescence
§ INTRODUCTION
Europium Eu$^{2+}$ ions are known as very efficient luminescence impurity in dense scintillating hosts. Europium introduced into halide crystal in divalent or trivalent states. A number of investigations are devoted to trivalent lanthanides in LaF$_3$ crystals <cit.>. At the same time authors noted the tendency of EuF$_{3}$, to reduce to EuF$_{2}$ at the high temperatures required for LaF$_{3}$ crystal growth, and the very strong broad band structure associated with Eu$^{2+}$ in the visible and ultraviolet range due to 4f - 4f5d transitions <cit.>.
Absorption bands of Eu$^{2+}$ were observed at 280 nm in LaCl$_{3}$ <cit.> and at 245 nm in LaF$_3$ <cit.>. Eu$^{2+}$ luminescence was found in LaCl$_3$ crystal at 420 nm <cit.>. Both absorption and emission are obviously due to transitions between ground 4f$^7$ and excited 4f$^6$5d$^1$ Eu$^{2+}$ states. No data on luminescence of Eu$^{2+}$ in LaF$_{3}$ were found in literature.
Besides the normal 5d-4f luminescence in most materials, the Eu$^{2+}$ (and also Yb$^{2+}$ and others) show so called "anomalous" broadband luminescence with large Stokes shift in certain crystals (see review <cit.>, <cit.>). For such crystals the excited 5d level falls into conduction band. Luminescence observed after transitions from conduction band states, which have less energy than the 5d level, to 4f level of lanthanide impurity ion <cit.>.
Divalent impurity ion has charge less than the charge of lanthanum, therefore for the electrical neutrality of the LaF$_3$ crystal the additional positive charge is needed for each divalent ion. In the absence of oxygen the charge compensation of divalent ion Ca$^{2+}$, Sr$^{2+}$ or Ba$^{2+}$ in LaF$_3$ accomplished by fluorine vacancy <cit.>. Parallel growth of ac conductivity and absorption in the visible region were observed in LaF$_3$-Sm$^{2+}$. Conductivity was attributed to Sm$^{2+}$-fluorine vacancy reorientation <cit.>. The dipoles in solids were thoroughly investigated by dielectric relaxation <cit.>.
The main topic of the present paper is to study the optical and dielectric properties of divalent Eu and Yb in LaF$_3$.
§ EXPERIMENTAL
Crystals were grown in vacuum in a graphite crucible by the Stockbarger method <cit.>. The graphite crucible contained three cylindrical cavities 10 mm in diameter and 80 mm long, which allowed growing three crystals of Ø10x50 mm dimensions with different impurity concentrations at the same time. A few percent of CdF$_2$ was added into raw materials for purification from oxygen impurity during growth. Impurity LnF$_3$ (Ln – lanthanide) was added into LaF$_3$ powder in concentration of 0.01, 0.1 and 0.3 mol.%. In LaF$_3$-YbF$_3$ crystals the Ce$^{3+}$ absorption at 245 nm and less wavelengths was found, which not influenced on Yb$^{2+}$ bands identification. The samples Ø10 mm x 2mm sawed from the grown rods and polished were typically used for measurements.
Absorption spectra in the range 190-3000 nm were taken with spectrophotometer Perkin-Elmer Lambda-950, emission spectra were measured using grating monochromator MDR2 (LOMO). Emission, excitation spectra were measured with photomodule Hamamatsu H6780-04 (185-850nm). No emission spectrum correction needs to be performed as the sensitivity only weakly changed in the region of Eu emission (400-700 nm). X-irradiation was performed using Pd-tube with 40 kV 20 mA.
As electrode contact material a silver paint (kontaktol "Kettler") was employed. Diameter of paint electrodes was around 5 mm and crystals thickness was around 2 mm. Conductivity measurements were done using immitance (RLC) meter E7-20 (“MNIPI”) in frequency range 25 Hz- 1MHz. Comparative LaF$_3$ and LaF$_3$-Eu dielectric measurements were done at room temperature. Conductivity of LaF$_3$ could be measured at mono-frequency. However, the bulk conductivity determined in this way appears to be mostly too small <cit.>. We investigate the conductivity of LaF$_3$-Sm$^{2+}$ in previous paper at frequency 1 kHz <cit.>. While the relation between conductivity of samples with different Sm$^{2+}$ concentrations remains the same, the values of measured conductivity were several time smaller. Therefore in this paper we measure true bulk LaF$_3$ conductivity from frequency dispersion <cit.>.
§ RESULTS
§.§ Optical spectra
Eu$^{3+}$ ions easily recognized in LaF$_3$ <cit.> and in many other materials by sharp red emission lines due to f-f transitions. No red luminescence due to Eu$^{3+}$ was found in all our LaF$_3$-Eu crystals at 7.5-300 K while intensive ultraviolet absorption appeared. Therefore europium impurity introduced in divalent form in our LaF$_3$ crystals.
Absorption spectrum (full curve), excitation and emission (dashed curves) of LaF$_3$-0.01 mol.% EuF$_3$ at shown temperatures. Excitation was measured for emission at 580 nm, the emission was measured for excitation at 270 nm.
Absorption spectrum of LaF$_3$-0.01 mol.% EuF$_3$ contains intensive asymmetric band at 245 nm with unresolved structure and weaker long wavelength bands at 330, 380 nm (Fig.<ref>). With increasing of EuF$_3$ doping the ultraviolet absorption becomes larger and at concentration near one percent of EuF$_3$ the crystal LaF$_3$ becomes yellow, due to absorption tail above 400 nm (see Fig.<ref>). The shape of aborption spectra does not depend on concentration of europium up to 0.3 mol.%. At higher concentration the absorption near 245 nm becomes too large. Therefore the long-wavelength bands at 300-400 nm region belong to Eu$^{2+}$ also. Absorption band at 245 nm was ascribed to Eu$^{2+}$ ions <cit.>. The authors have measured absorption spectrum up to 300 nm, which prevents observation of the long wavelength bands at 330, 380 nm, which also belong to europium Eu$^{2+}$ absorption.
Red luminescence band at 600 nm was observed at low temperature. Excitation spectrum (see Fig.<ref>) generally correlates with 245 nm absorption bands. However red luminescence was not observed with excitation into Eu$^{2+}$ long wavelength bands (see Fig.<ref>).
Temperature dependence of intensity of Eu luminescence (full curve) and decay time (dots) of LaF$_3$-0.01 % EuF$_3$ crystal.
With increasing temperature the intensity of luminescence sharply decreases above 40 K (Fig.<ref>). The decay time of red luminescence was 2.2 $\mu$s at 7.5 K. Above 50 K decay time sharply shortened similar to luminescence intensity (see Fig.<ref>).
Next most probable divalent lanthanide in LaF$_3$ is ytterbium. The Yb$^{2+}$ long-wavelength bands were observed around 360, 310 nm in alkaline-earth fluoride crystals <cit.>. The Yb$^{3+}$ in LaF$_3$ shows infrared absorption near 970 nm <cit.>. No Yb$^{2+}$ ultraviolet bands were observed in our LaF$_3$-YbF$_3$, while Yb$^{3+}$ infrared bands were grown with concentration of YbF$_3$. After x-ray irradiation of LaF$_3$-YbF$_3$ at room temperature the absorption bands at 270, 300 and 376 nm were appeared (Fig.<ref>) and increased with increasing YbF$_3$ concentration. Additionally a very intensive absorption band at 200 nm, belonging to stable at room temperature F$_3^-$ hole defects <cit.>, appeared. Evidently the bands at 270, 300 and 376 nm belong to Yb$^{2+}$ in LaF$_3$.
Absorption spectrum of LaF$_3$-0.3 wt.% YbF$_3$ at shown temperatures created by x-irradiation at 295K. LaF$_3$ crystal also contains unwanted Ce$^{3+}$, which absorbed at 245 nm and partially transformed under x-irradiation.
No luminescence in the range 400-1200 nm, which can be associated with Yb$^{2+}$, was found in x-irradiated LaF$_3$-0.3 % YbF$_3$ at temperatures down to 7.5 K.
§.§ Dielectric relaxation
Fig.<ref> presents examples of admittance plots in the complex-plane representation for cells with LaF$_3$ crystals at 295 K. The admittance plots for LaF$_3$-EuF$_3$ crystals show that the high-frequency data require an equivalent circuit composed of a frequency-independent (bulk) capacitance, Cp, in parallel with a frequency-independent (bulk) resistance, Rp. At zero point frequency is equal 25 Hz and increase till 1 MHz for last point each curve (see Fig.<ref>). The high-frequency interceptions with the real axis represents the true bulk conductances.
Complex admittance plot (Y$^*$=G$_p$+i$\omega$C$_p$) for LaF$_3$-EuF$_3$ crystals at 295 K. Frequency range 25Hz-1 MHz. Inset shows frequency dependence of tg$\delta$ for the same samples.
The increasing of europium concentration accompanied by the growth of resistive bulk conductances (see Fig.<ref>) and the increasing of the Eu$^{2+}$ absorption.
Absence of low frequency wing of tg$\delta$ (see Fig.<ref>) points on absence of steady electrical conduction of LaF$_3$-Eu. It means that all of fluorine vacancies are attached to divalent europium at room temperature.
Bulk conductance against Eu$^{2+}$ absorption at 370 nm of LaF$_3$ and LaF$_3$-EuF$_3$ crystals.
We compare the absorption and bulk conductances of LaF$_3$ and LaF$_3$-EuF$_3$ (Fig.<ref>). The measurements of absorption and conductances were done on the same samples to diminish possible errors. The linear dependence was observed up to 0.3 mol. % of EuF$_3$ dopant. Undoubtedly, the conductance of LaF$_3$-EuF$_3$ (as well as LaF$_3$-BaF$_2$ <cit.>) are due to fluorine vacancies introduced by divalent impurity.
§ DISCUSSION
The investigations of LaF$_3$ doped with divalent alkaline-earth ions Ca$^{2+}$, Sr$^{2+}$ and Ba$^{2+}$ were proved that the charge compensators are fluorine vacancies <cit.>. Introduction of divalent ions into LaF$_3$ led to increasing of ionic conductivity <cit.>, appearing the peaks of thermostimulated depolarisation <cit.> and peaks of nuclear magnetic resonance of $^{19}$F <cit.>. With increasing the Ba$^{2+}$ concentration up to 8 % the conductivity monotonically increased <cit.>. All these phenomena caused by migration of fluorine vacancies. Based on these results one could assume that charge compensator of divalent samarium is fluorine vacancy, concentration of which can be evaluated by conductivity measurements.
Both absorption and conductivity followed the logarithmic-type growth with increasing europium doping. Finally, we obtained linear increase of LaF$_3$ conductivity with increasing Eu$^{2+}$ absorption (see Fig.<ref>). The linear dependence on Fig.<ref> are plotted using absorption at 370 nm, straight lines can be obtained for any absorptions within 200-400 nm range, also. These results are proved that anion vacancy accompanied each divalent europium ion. Based on ionic thermodepolarisation <cit.> and dielectric relaxation investigations of Me$^{2+}$ doped LaF$_3$ <cit.> one could infer that anion vacancy should be in close vicinity of divalent europium.
Optical spectra of LaF$_3$-Eu$^{2+}$ has similarities with spectra of LaF$_3$-Sm$^{2+}$, investigated in our previous paper <cit.>. Indeed, the most intensive absorption bands in both cases belong to 4f$^n$-4f$^{n-1}$5d$^1$ transitions in divalent Sm or Eu. The weaker long-wave absorption bands, in which the emission was not excited, are present in both cases. In the case of Sm$^{2+}$ we attributed this absorption bands at 600 nm to transitions from 4f ground state of samarium to level of nearest anion vacancy. Preliminary unempirical calculations supported this conclusion <cit.>. Following to this the Eu$^{2+}$ bands at 330, 380 nm can be attributed to transitions 4f$^7$ - vacancy. Unempirical calculations, which are in progress now, will explain the detail of optical spectra. The absence of emissions after excitation into long-wave bands obviously related with large lattice relaxation around vacancy trapped electron.
Apart from Eu no Yb luminescence was found in LaF$_3$ crystals. According to this no Yb$^{2+}$ anomalous luminescence was found in BaF$_2$, while Yb$^{2+}$ luminescence in CaF$_2$, SrF$_2$ was observed <cit.>. It seems the absence of Yb$^{2+}$ luminescence related with larger lattice relaxation in excited state due to smaller ionic radius of Yb against that of Eu.
According to Dorenbos empirical model the 4f-5d transitions of Eu$^{2+}$ in LaF$_3$ should begin at 330 nm and 5d levels fall into the conduction band <cit.>. The absence of 4f$^{n-1}$5d$^1$ - 4f$^n$ emission of divalent europium ions and absence of fine structure of 245 nm absorption 4f$^7$ - 4f$^6$5d$^1$ band correlates with the fact that 5d level fall into conduction band.
§ CONCLUSION
Experimental results lead us to following conclusions:
- Yb$^{2+}$ has absorption bands at 270, 300 and 376 nm,
- each Eu$^{2+}$ is accompanied by fluorine vacancy at room temperature, this leads to appearance of long-wave absorption bands 330, 380 nm and dielectric relaxation peak,
- the Eu$^{2+}$ broadband luminescence in LaF$_3$ at 600 nm is an emission from relaxed conduction band states to ground Eu 4f level (anomalous emission).
§ ACKNOWLEDGMENTS
The authors gratefully acknowledge O. N. Solomein and O. V. Kozlovskii for preparation the crystals investigated in this work. The work was partially supported by RFBR grant 15-02-06666a.
In this work authors used the equipment of the Baikal Analytical Center for Collective Use, Siberian Branch, Russian Academy of Sciences.
|
1511.00140
|
The School of Mathematics
Conditional Value-at-Risk: Theory and Applications
Jakob Kisiala
Dissertation Presented for the Degree of
MSc in Operational Research
August 2015
Supervised by
Dr Peter Richtárik
This thesis presents the Conditional Value-at-Risk concept and combines an analysis that covers its application as a risk measure and as a vector norm. For both areas of application the theory is revised in detail and examples are given to show how to apply the concept in practice.
In the first part, CVaR as a risk measure is introduced and the analysis covers the mathematical definition of CVaR and different methods to calculate it. Then, CVaR optimization is analysed in the context of portfolio selection and how to apply CVaR optimization for hedging a portfolio consisting of options. The original contributions in this part are an alternative proof of Acerbi's Integral Formula in the continuous case and an explicit programme formulation for portfolio hedging.
The second part first analyses the Scaled and Non-Scaled CVaR norm as new family of norms in $\mathbb{R}^n$ and compares this new norm family to the more widely known $L_p$ norms. Then, model (or signal) recovery problems are discussed and it is described how appropriate norms can be used to recover a signal with less observations than the dimension of the signal. The last chapter of this dissertation then shows how the Non-Scaled CVaR norm can be used in this model recovery context. The original contributions in this part are an alternative proof of the equivalence of two different characterizations of the Scaled CVaR norm, a new proposition that the Scaled CVaR norm is piecewise convex, and the entire <ref>. Since the CVaR norm is a rather novel concept, its applications in a model recovery context have not been researched yet. Therefore, the final chapter of this thesis might lay the basis for further research in this area.
First of all, I would like to thank my supervisor Peter Richtárik, whose valuable feedback and ideas improved the quality of this thesis considerably. He inspired me to broaden my horizon and study topics which went beyond the syllabus. Furthermore, I would like to thank all the teaching staff who enabled me to learn a lot during my master studies.
I would also like to mention my classmates who made this year a memorable experience beyond the class room. Especially Wendy, who was always a beam of sunshine in this often cloudy and rainy city.
Own Work Declaration
I declare that this thesis was composed by myself and
that the work contained therein is my own, except where
explicitly stated otherwise in the text.
5cmEdinburgh, 21 August 2015Place, Date 4cmJakob Kisiala
CHAPTER: INTRODUCTION
This chapter presents the motivation for this thesis, gives the outline of the following chapters, and states the original contributions of the thesis.
Note that are no dedicated chapters covering a literature review or to establish notation. Rather, the literature is reviewed and notation is established in each chapter and section where it is appropriate.
§ MOTIVATION OF THE THESIS
In financial risk management, especially with practitioners, Value-at-Risk (VaR) is a widely used risk measure because its concept is easily understandable and it focusses on the down-side, i.e. tail risk. A possible definition is given by Choudhry: “VaR is a measure of market risk. It is the maximum loss which can occur with [$(\alpha \times 100)$] % confidence [...]” <cit.>.
However, despite its wide use, VaR is not a coherent risk measure. The concept of a coherent risk measure was introduced by Artzner et al. in <cit.>. They formulated that a risk measure $\rho$ is coherent if it satisfies the following axioms (see <ref> for details):
* Monotonicity
* Translation equivariance
* Subadditivity
* Positive Homogeneity
VaR is only coherent when the underlying loss distribution is normal, otherwise it lacks subadditivity. Other disadvantages of the VaR measure are that it does not give any information about potential losses in the $1 - \alpha$ worst cases and that calculating VaR optimal portfolios can be difficult, if not impossible <cit.>.
The Conditional Value-at-Risk (CVaR) is closely linked to VaR, but provides several distinct advantages. In fact, in settings where the loss is normally distributed, CVaR, VaR, and Minimum Variance (Markowitz) optimization give the same optimal portfolios <cit.>. The advantages of CVaR become apparent when the loss distribution is not normal or when the optimization problem is high-dimensional: CVaR is a coherent risk measure for any type of loss distribution.
Furthermore, in settings where an investor wants to form a portfolio of different assets, the portfolio CVaR can be optimized by a computationally efficient, linear minimization problem, which simultaneously gives the VaR at the same confidence level as a by-product. On the other hand, it is difficult to form VaR optimal portfolios, as is these settings VaR is difficult to calculate. This computationally efficient way to optimize the portfolio CVaR can also be transferred to hedging problems, in which an investment decision has been taken, but adjustments are possible so that the downside risk of the investment can be reduced. For example, <cit.>, <cit.>, <cit.>, and <cit.> used CVaR optimization to hedge risk, each one in a different setting.
What is more remarkable, is that the CVaR concept (which was developed as a financial risk measure) can be abstracted to form a new family of norms in $\mathbb{R}^n$. The Scaled and (Non-Scaled) CVaR norm can then be used as alternatives to the widely established family of $L_p$ norms. Moreover, by choosing suitable $\alpha$, the CVaR norm is equivalent to the $L_1$ and $L_{\infty}$ norm.
Having this new CVaR norm also opens up new opportunities in Big Data optimization, particularly in model or signal recovery problems. In these problems, it is the goal to reconstruct a model or signal of dimension $p$ when less than $p$ observations are available. This can be achieved by exploiting the structure of particular signals and solving a norm minimization problem using an appropriate norm. Particularly the $L_1$ and $L_{\infty}$ norm are used for two different types of models, and having the CVaR norm as another norm in $\mathbb{R}^n$ could recover further types of signals and models. To the best knowledge of the author, no research has been undertaken so far to use the CVaR norm in model recovery problems, so this might be another area of research to consider in the future.
§ OUTLINE OF THE THESIS
This thesis consists of 7 main chapters (not counting the introduction and conclusion), which concentrate on two main areas: First, the use of CVaR as a risk measure and second, the characteristics of the CVaR norm with an outlook on possible future applications. For both areas, an extensive analysis on the theory of CVaR and the CVaR norm is given, before showing how this theory can be applied in practice.
<ref> introduces the concept of CVaR as a risk measure for a univariate loss distribution. It starts by showing how VaR and CVaR are related to each other. Then, the notion of a coherent risk measure is introduced and it is shown why VaR is not coherent. <ref> then examines the mathematical definition of CVaR and shows how the CVaR can be calculated using the Convex Combination Formula. The chapter finishes by showing an alternative way to calculate CVaR, namely using Acerbi's Integral Formula.
<ref> moves from univariate to multivariate loss distributions. These loss distributions arise in portfolio optimization problems, where there are different assets, each with their own loss distribution and the investor's loss depends on his investment decision into each asset. <ref> discusses the first model that was introduced to optimize a portfolio with regards to risk (the Markowitz Model, which aims to reduce the portfolio variance). Identifying the shortcomings of the Markowitz Model gives the motivation for the next model that is considered, i.e. the Rockafellar and Uryasev Model, which optimizes the portfolio CVaR. The analysis extends the results of the CVaR analysis in the univariate case to the multivariate case and gives a linear optimization programme that minimizes the CVaR of a portfolio. This section also shows that the Markowitz Model and Rockafellar and Uryasev Model lead to the same optimal portfolio if the loss of all assets in the portfolio is normally distributed. <ref> then gives two numerical examples to demonstrate the results that were established in this chapter. First, it is shown that in certain cases CVaR and Mean-Variance optimization indeed give the same portfolio, before demonstrating that for non-normal loss distributions CVaR optimization gives a less risky portfolio that Mean-Variance optimization.
Next, <ref> shows how the CVaR optimization problem can be used to hedge tail losses from a previous investment decision. In this particular example, a scenario based on real world data is created. Simplifying assumptions are made to focus on the hedging procedure instead of the technical implementation of the hedge. For the scenario, a trader's portfolio is to be adjusted, so that the CVaR of the portfolio is minimized. Since it is an option portfolio (for which the risk manager needs a daily estimate on the portfolio variance) <ref> and <ref> give the necessary finance and risk management background. <ref> briefly describes how the portfolio is formed before <ref> explains the hedging procedure, including an explicit formulation of the hedging problem. The portfolio risk before and after hedging are compared and it is shown how the hedging procedure can improve the risk profile of the portfolio.
Moving away from the financial context, <ref> introduces two norms that are based on CVaR: the Scaled CVaR norm $C_{\alpha}^S$, and the (Non-Scaled) CVaR norm $C_{\alpha}$. For both norms, two different yet equivalent characterizations are given. <ref> then describes the properties of each norm and especially shows how their properties with regards to the parameter $\alpha$ are fundamentally different. Since these norms are fairly novel and standard algorithms to calculate them are not yet implemented in MATLAB, <ref> examines the computational efficiency of calculating the two norms, $C_{\alpha}^S$ and $C_{\alpha}$, using the two different characterizations for each.
To give a better understanding of $C_{\alpha}^S$ and $C_{\alpha}$, they are both compared to the more familiar family of $L_p$ norms in <ref>. First $C_{\alpha}^S$ is compared to $L_p^S$ norms before the $C_{\alpha}$ is analysed with regards to the parameter $\alpha$ and its proximity to $L_p$ norms.
<ref> then gives a possible application of the CVaR norm in an optimization context: model recovery using atomic norms. In model (or signal) recovery the goal is to reconstruct a $p$-dimensional model (or signal) with $n$ random measurements, such that $n < p$. For a recovery to be successful, the model must have a certain structure that can be exploited by a corresponding atomic norm. <ref> provides the background on atomic norms and convex geometry (e.g. the notions of tangent and normal cones) that is needed to explore the usefulness of the CVaR norm in this setting. <ref> states the necessary recovery conditions, more precisely the number of random measurements needed to ensure that a $p$-dimensional model can be recovered from $n$ measurements. The number of measurements $n$ is derived by using Gaussian Widths, which are quite difficult to compute directly. Therefore, <ref> states some properties of Gaussian Widths that might prove useful when establishing a bound on $n$.
The final chapter, <ref>, is completely original in the sense that it explores how the CVaR norm can be used in the context of model recovery problems. To the best knowledge of the author, no research in this particular area has been carried out before. Unfortunately, due to the limited scope of this thesis, the analysis could not be completed. Rather, this chapter should show areas of further research, with pointers towards what could be analysed in more detail. <ref> contains a conjecture about the set of atoms of the CVaR norm for a certain $\alpha$. A proposition based on the conjecture is proven, but due to the limited scope of this dissertation, the conjecture could not be proven in full. Still, a numerical experiment was carried out to identify the atoms of the CVaR norm in $\mathbb{R}^4$ and this experiment provides further evidence that the conjecture is true. <ref> is rather short, showing how a bound on the number of measurements $n$ can be derived if expressions are available for the tangent or normal cone with respect to the atoms of CVaR norm. Some numerical experiments were performed to recover simple signals using the CVaR norm in <ref>. The results are not impressive, as the experiments were limited to a certain $\alpha$ and only few special cases of signals. Analysing model recovery using the CVaR norm further could lead to different set ups, for which the results could be better.
§ ORIGINAL CONTRIBUTIONS OF THE THESIS
First of all, to the best knowledge of the author, this thesis is the first piece of work that analyses CVaR as a risk measure and the CVaR norm (including possible applications) in a unified way. There is an abundance of papers on CVaR, CVaR portfolio optimization, and further applications of CVaR as a risk measure. However, there is little research on the CVaR norm and no research on the application of the CVaR norm in the context of model recovery.
A large part of this thesis presents results of other papers. Even with established concepts, the author aims to present them in such a way that the concepts are easily understandable. Also, most plots in this paper were reproduced independently to confirm the results of other authors. But throughout the paper several original contributions are made, either by presenting new proofs to existing propositions, or by stating new propositions / conjectures. In detail, the original contributions are:
* <ref>: A new proof of Acerbi's Integral Formula (first proposed in <cit.>) to calculate CVaR is given.
* <ref>: Although this is a standard result, the author proves independently why portfolio diversification reduces risk (when measured by standard deviation). The reason to give an independent proof is that the standard introductory financial literature only shows this result for $N=2$ assets, while this thesis shows this result for $N \geq 2$ assets.
* <ref>: Although hedging using CVaR optimization was discussed by Rockafellar and Uryasev in <cit.>, they never explicitly formulated the optimization programme. This thesis clearly defines the variables and states the problem for a CVaR optimal hedge of a portfolio of options.
* <ref>: This subsection introduces a second, equivalent characterization of the Scaled CVaR norm, which was proposed by Pavlikov and Uryasev in <cit.>. The original contribution of this thesis is an alternative proof of the equivalence of the two different characterizations.
* <ref>: The piecewise convexity of the Scaled CVaR norm is a new and original proposition of this thesis, to the best knowledge of the author.
* <ref>: To the best knowledge of the author, the computational efficiency of different algorithms to calculate the Scaled and Non-Scaled CVaR norm has not been investigated before.
* <ref>: To the best knowledge of the author, the atoms (i.e. the extreme points of the unit ball) of the CVaR norm have never been explicitly stated before. This section conjectures the set of atoms of the CVaR norm for a specific $\alpha$. It shows that for different $\alpha$ the unit ball of the CVaR norm looks different, and finally a numerical experiment is performed to provide evidence for the conjecture in $\mathbb{R}^4$.
* <ref>: To the best knowledge of the author, the CVaR norm has never been analysed in the context of model recovery problems. This section performs some numerical recovery experiments to see how suitable $C_{\alpha}$ would be recover a special type of signal. Because of the close link between the CVaR norm and the $L_1$ and $L_{\infty}$ norms, it is also investigated how well the CVaR norm performs in signal recovery problems when compared to these two $L_p$ norms.
CHAPTER: CONDITIONAL VALUE-AT-RISK AS A RISK MEASURE
This chapter introduces the concept of CVaR (building on the VaR concept) in the way that it was first introduced - a financial risk measure. In <ref> the mathematical definitions of VaR and CVaR are given, followed by an intuitive description of their properties and interactions. <ref> presents the axioms that must be satisfied for a risk measure to be considered coherent. Specifically, an example is shown to prove that VaR is not subadditive - whereas for the same example, CVaR is subadditive. Finally, <ref> explores the CVaR concept in more detail, giving different algorithms and optimization programmes to calculate the CVaR of a given loss distribution in a variety of settings. <ref> states Acerbi's Integral Formula to calculate CVaR and gives an alternative proof of the formula.
§ BASIC NOTIONS IN THE VAR / CVAR FRAMEWORK
Since losses are random variables, some statistical measures need to be introduced to cover the basics for latter sections and chapters, especially the ones concerning portfolio optimization (<ref> and <ref>).
[<cit.> Expectation]
The expectation, sometimes called expected value or mean, of a random variable $X$ is defined as
\begin{align}
\E[X] & \defeq \int \limits_{- \infty}^{\infty} x f(x) dx & \text{in the continous case} \label{eqn:expectation_continuous}
\end{align}
\begin{align}
\E[X] & \defeq \sum \limits_{k = - \infty}^{\infty} k P(X = k) & \text{in the discrete case,} \label{eqn:expectation_discrete}
\end{align}
where $f(x)$ is the probability density function of $X$ and $P(X = k)$ is the probability mass function $X$.
The expectation is often denoted by the letter $\mu$, such that $\mu = \E[X]$.[Many texts apply the distinction to use $\mu$ for the population mean and $\hat{\mu}$ for the sample mean. Although the expectation of the loss variable $X$ is actually a sample mean, this dissertation will use the notation $\mu$ when talking about the expectation of losses.] $\E[X]$ provides information about the distribution of $X$; informally it can be described as the centre value around which possible values of $X$ disperse <cit.>.
[<cit.> Variance]
The variance of a random variable $X$ is defined as
\begin{align}
\text{\emph{Var}~}(X) & \defeq \E \left[ \left(X - \E [X] \right)^2 \right] . \label{eqn:variance}
\end{align}
The variance is often denoted as $\sigma^2$.[Again, many texts apply a distinction between the population variance $\sigma^2$ and the sample variance $s^2$. As in the case with the expectation, this dissertation will use the notation $\sigma^2$ when talking about the variance of losses.] Since the variance is hard to interpret as it is given in square units, the standard deviation (denoted $\sigma = \sqrt{\text{Var}(X)}$) is often used. It does not contain additional information, but is easier to interpret as $\sigma$ is given in the same units as $\mu$ <cit.>.
The standard deviation $\sigma$ (or variance $\sigma^2$) measures how strongly $X$ is dispersed around $\mu$. Small values of $\sigma$ indicate that $X$ is concentrated strongly around $\mu$, while large values of $\sigma$ mean that values of $X$ further away from $\mu$ (in either direction) are more likely.
Another important concept throughout this dissertation is Covariance.
[<cit.> Covariance]
The covariance of two random variables $X_1$ and $X_2$ is defined as
\begin{align}
\text{\emph{Cov}~}(X_1, X_2) & \defeq \E \left[ \left(X_1 - \E [X_1] \right) \left( X_2 - \E [X_2] \right) \right] . \label{eqn:covariance}
\end{align}
Covariance measures how strongly the variable $X_1$ varies together with $X_2$ (and vice versa). As a special case, $\text{Cov}(X,X) = \text{Var} (X)$. Also, if $X_1$ and $X_2$ are independent, their covariance is 0 <cit.>. As in the case with variance, the covariance is hard to interpret, as its unit is the product of the respective units of $X_1$ and $X_2$. Therefore, another measure for dependency that is derived from the covariance and variance is commonly used to express how strongly $X_1$ and $X_2$ vary together - it is called the correlation coefficient:
[<cit.> Correlation Coefficient]
The correlation coefficient of two random variables $X_1$ and $X_2$ is defined as
\begin{align}
\rho_{12} & \defeq \frac{\text{\emph{Cov}~}(X_1, X_2)}{\sqrt{\text{\emph{Var}~}(X_1)} \sqrt{\text{\emph{Var}~}(X_2)}}. \label{eqn:correlation}
\end{align}
$\rho$ always takes values between -1 and 1 and is therefore easier to interpret than covariance. If $\vert \rho_{12} \vert$ is close to 1, then there is a strong dependence between $X_1$ and $X_2$ <cit.>.
As pointed out in the introduction, Value-at-Risk (VaR) is the maximum loss that will not be exceeded at a given confidence level. This gives the following mathematical definition of VaR:
[<cit.> Value-at-Risk (VaR)]
Let $X$ be a random variable representing loss. Given a parameter $0 < \alpha < 1$, the $\alpha$-VaR of $X$ is
\begin{equation} \label{eqn:VaR}
\text{\emph{VaR}}_{\alpha} (X) \defeq \min \{ c : P (X \leq c) \geq \alpha \} \,.
\end{equation}
Given <ref>, VaR can have several equivalent interpretations <cit.>:
* $\text{VaR}_{\alpha} (X)$ is the minimum loss that will not be exceeded with probability $\alpha$.
* $\text{VaR}_{\alpha} (X)$ is the $\alpha$-quantile of the distribution of $X$.
* $\text{VaR}_{\alpha} (X)$ is the smallest loss in the $(1 - \alpha) \times 100$% worst cases.
* $\text{VaR}_{\alpha} (X)$ is the highest loss in the $\alpha \times 100$% best cases.
The general definition of CVaR is given in <ref>. At this point, only the CVaR definition for continuous random variables will be given to create a more intuitive introduction into the topic. For continuous $X$, the Conditional Value-at-Risk is the expected loss, conditional on the fact that the loss exceeds the VaR at the given confidence level:
[<cit.> Conditional Value-at-Risk (CVaR) in the continuous case]
Let $X$ be a continuous random variable representing loss. Given a parameter $0 < \alpha < 1$, the $\alpha$-CVaR of $X$ is
\begin{equation} \label{eqn:CVaR}
\text{\emph{CVaR}}_{\alpha} (X) \defeq \E [ X \mid X \geq \text{\emph{VaR}}_{\alpha} (X) ] .
\end{equation}
Alternative names for CVaR found in the literature are Average Value-at-Risk, Expected Shortfall, or Tail Conditional Expectation, although some authors make subtle distinctions between their definitions <cit.>.
<ref> shows the VaR and CVaR for a specific continuous random variable $X$. The cumulative distribution function of $X$ can be used to find $\text{VaR}_{\alpha} (X)$, and $\text{VaR}_{\alpha} (X)$ can be used in turn to calculate $\text{CVaR}_{\alpha} (X)$. [An alternative approach to find VaR and CVaR is shown in <ref>]
$\text{VaR}_{\alpha}$ and $\text{CVaR}_{\alpha}$ of a random variable $X$ representing loss.
§ COHERENT RISK MEASURES
Artzner et al. analysed risk measures in <cit.> and stated a set of properties / axioms that should be desirable for any risk measure. Any risk measure which satisfies these axioms is said to be coherent. The four axioms they stated are Monotonicity, Translation equivariance, Subadditivity, and Positive Homogeneity. For the definitions of all axioms, $X$ and $Y$ are random variables representing loss, $c \in \mathbb{R}$ is a scalar representing loss, and $\rho$ is a risk function, i.e. it maps the random variable $X$ (or $Y$) to $\mathbb{R}$, according to the risk associated with $X$ (or $Y$).
[<cit.> Monotonicity]
A risk measure $\rho$ is monotone, if for all $X$, $Y$:
\begin{equation} \label{eqn:Monotonicity}
X \leq Y \Rightarrow \rho (X) \leq \rho(Y) .
\end{equation}
[<cit.> Translation Equivariance]
A risk measure $\rho$ is translation equivariant, if for all $X$, $c$:
\begin{equation} \label{eqn:Translation_Equivariance}
\rho (X + c) = \rho(X) + c .
\end{equation}
[<cit.> Subadditivity]
A risk measure $\rho$ is subadditive, if for all $X$, $Y$:
\begin{equation} \label{eqn:Subadditivity}
\rho (X + Y) \leq \rho(X) + \rho(Y) .
\end{equation}
[<cit.> Positive Homogeneity]
A risk measure $\rho$ is positively homogeneous, if for all $X$, $\lambda \geq 0$:
\begin{equation} \label{eqn:Positive_Homogeneity}
\rho (\lambda X) = \lambda \rho(X) .
\end{equation}
Speaking in a more intuitive way, the above axioms (<ref> - <ref>) can be interpreted as follows <cit.>:
* Monotonicity: Higher losses mean higher risk.
* Translation Equivariance: Increasing (or decreasing) the loss increases (decreases) the risk by the same amount.
* Subadditivity: Diversification decreases risk.
* Monotonicity: Doubling the portfolio size doubles the risk.
VaR fails to meet the subadditivity axiom (<ref>) and is therefore criticized for not being a coherent risk measure. A simple example shows this <cit.>:
Consider two possible investments, $A$ and $B$, which have the loss profile shown in <ref>. There are three different scenarios $\xi_1, \xi_2, \xi_3$, each with associated probability $p(\xi_i)$.
$\begin{tabu}{| c | r r r |}
\hline
& \xi_1 & \xi_2 & \xi_3 \\
p(\xi_i) & 0.04 & 0.04 & 0.92 \\
\hline
A & 1000 & 0 & 0 \\
B & 0 & 1000 & 0 \\
\hline
\end{tabu}$
Losses for investments $A$ and $B$ under three scenarios.
Using <ref> to calculate the VaR at the 95 % confidence level for investments in $A$, $B$, and $A + B$ gives
\begin{align*}
\text{VaR}_{0.95} (A) = \min \{ c : P (A \leq c) \geq 0.95 \} & = 0 && ( P(A \leq 0) = 0.96 ) \,,\\
\text{VaR}_{0.95} (B) = \min \{ c : P (B \leq c) \geq 0.95 \} & = 0 & &( P(B \leq 0) = 0.96 ) \,, \text{~and} \\
\text{VaR}_{0.95} (A + B) = \min \{ c : P (A + B \leq c) \geq 0.95 \} & = 1000 \,.
\end{align*}
In this example, $\text{VaR}_{0.95} (A + B) \not \leq \text{VaR}_{0.95} (A) + \text{VaR}_{0.95} (B)$, hence VaR is not subadditive according to <ref>. Therefore, it is not a coherent risk measure in the sense of Artzner et al.
Acerbi and Tasche proved in <cit.> that CVaR in satisfies the above axioms and is therefore a coherent risk measure.[To be precise: In <cit.> Acerbi and Tasche defined Expected Shortfall (ES) and CVaR slightly differently. In the paper, they first proved that ES is a coherent risk measure and later proved that ES is identical to CVaR.] Using the previous example together with <ref> of <ref> gives
\begin{align*}
\text{CVaR}_{0.95} (A) & = 800 && (\lambda = 0.2, \text{CVaR}_{0.95}^+ (A) = 1000) \,, \\
\text{CVaR}_{0.95} (B) & = 800 && (\lambda = 0.2, \text{CVaR}_{0.95}^+ (B) = 1000) \,, \text{~and}\\
\text{CVaR}_{0.95} (A + B) & = 1000 && (\lambda = 1, \text{CVaR}_{0.95}^+ (A + B) = 0) \,.
\end{align*}
which shows that subadditivity holds for CVaR, as $\text{CVaR}_{0.95} (A + B) = 1000 \leq \text{CVaR}_{0.95} (A) + \text{CVaR}_{0.95} (B) = 1600$.
§ CLOSER ANALYSIS OF CVAR
Analysing CVaR in a wider context, one can derive CVaR from the generalized $\alpha$-tail distribution of a random variable $X$ (which represents loss). This is what Rockafellar and Uryasev did in <cit.>. While <cit.> focused on general distributions, their previous work in <cit.> concerned the CVaR of continuous loss distributions. This section will present the results of both papers in a unified way, for discrete as well as for continuous loss distributions.
Suppose that $X$ is the loss distribution, and that $F_X (z)$ is the cumulative distribution function of $X$, i.e. $F_X (z) = P (X \leq z)$. Then the generalized $\alpha$-tail distribution of is defined as <cit.>
\begin{equation} \label{eqn:generalized_alpha_tail}
F^{\alpha}_X (z) \defeq \left\{
\begin{array}{l r}
0, & \text{when~} z < \text{VaR}_{\alpha} (X) \\
\frac{F_X (z) - \alpha}{1 - \alpha}, & \text{when~} z \geq \text{VaR}_{\alpha} (X)
\end{array}
\right. .
\end{equation}
Now, if $X^{\alpha}$ is the random variable whose cumulative distribution function is $F^{\alpha}_X$ (<ref>), then the CVaR is defined as
\begin{align}
\text{CVaR}_{\alpha} (X) &\defeq \E [X^{\alpha}] , \label{def:CVaR_general_definition}
\end{align}
which leads to <ref> in the continuous case ($\text{CVaR}_{\alpha} (X) = \E [ X \mid X \geq \text{VaR}_{\alpha} (X) ]$), but is different for the discrete case <cit.>.
For discrete or non-continuous loss distributions, Rockafellar and Uryasev proposed to calculate CVaR as a weighted average, also called the Convex Combination Formula. To apply the Convex Combination Formula, one needs the $\text{VaR}_{\alpha}$ and $\text{CVaR}^+_{\alpha}$ of $X$, where $\text{CVaR}^+_{\alpha} (X)$ is the expected loss strictly greater than the $\text{VaR}_{\alpha} (X)$, i.e.,
\begin{equation} \label{eqn:CVaR_plus}
\text{CVaR}_{\alpha}^+ (X) \defeq \E [X \mid X > \text{VaR}_{\alpha} (X)] .
\end{equation}
[<cit.> CVaR as a weighted average / Convex Combination Formula]
Let $\Psi$ be cumulative probability of $\text{VaR}_{\alpha} (X)$, i.e. $ \Psi = F_X (\text{\emph{VaR}}_{\alpha} (X) )$ and define $\lambda$ as
\begin{equation*}
\lambda \defeq \frac{\Psi - \alpha}{1 - \alpha} \,,
\end{equation*}
for $0 \leq \alpha < 1$. We then have:
\begin{equation} \label{eqn:CVaR_Convex_Combination_Formula}
\text{\emph{CVaR}}_{\alpha} (X) = \lambda \text{\emph{VaR}}_{\alpha} (X) + (1 - \lambda) \text{\emph{CVaR}}_{\alpha}^+ (X) .
\end{equation}
Note that <ref> is valid for all loss distributions, including continuous ones. From <ref> it follows that $\text{CVaR}_{\alpha}$ dominates $\text{VaR}_{\alpha}$, i.e. $\text{CVaR}_{\alpha} \geq \text{VaR}_{\alpha}$. In fact, $\text{CVaR}_{\alpha} > \text{VaR}_{\alpha}$, unless $\text{VaR}_{\alpha}$ is the maximum loss possible <cit.>. Another result to emphasize is that the representation of CVaR by <ref> is rather surprising. As shown earlier, VaR is not a coherent risk measure (see <ref>) and, in fact, neither is $\text{CVaR}^+$ <cit.>. However, both these incoherent risk measures are combined in the Convex Combination Formula to yield CVaR, which is coherent and therefore has many advantageous properties <cit.>.
To provide a better understanding of the Convex Combination Formula (<ref>), an example of a discrete loss distribution will be presented. The losses $y_i$ with associated probabilities are given in <ref>.
| *7c |
i 1 2 3 4 5 6
$y_i$ 100 200 400 800 900 1000
$P ( Y = y_i )$ 0.1 0.2 0.5 0.18 0.01 0.01
Discrete loss distribution of a random variable $Y$.
Now assume the 95 % CVaR is to be determined. Since $F_Y(400) = P( Y \leq 400) = 0.8$ and $F_Y(800) = P( Y \leq 800) = 0.98$, it follows that $\text{VaR}_{0.95} (Y) = \min \{ c: P(Y \leq c) \geq 0.95\} = 800$ and $\lambda = \frac{0.98 - 0.95}{1 - 0.95} = \frac{3}{5}$. Also, $\text{CVaR}_{0.95}^+ (Y)$ can be calculated as $\frac{1}{2} \times 900 + \frac{1}{2} \times 1000 = 950$. Hence, applying <ref> gives
$$ \text{CVaR}_{0.95} (Y) = \frac{3}{5} \times 800 + \frac{2}{5} \times 950 = 860 .$$
§ ACERBI'S INTEGRAL FORMULA
Another way to express CVaR is to use Acerbi's integral formula.
[<cit.> Acerbi's Integral Formula for CVaR]
The CVaR of a random variable $X$, which represents loss, at the confidence level $\alpha$ can be expressed as
\begin{equation} \label{eqn:Acerbis_integral_formula}
\text{\emph{CVaR}}_{\alpha} (X) = \frac{1}{1 - \alpha} \int \limits_{\alpha}^{1} \text{\emph{VaR}}_{\beta}(X) \, d \beta .
\end{equation}
Hence, $\text{CVaR}_{\alpha}$ can also be interpreted as the average $\text{VaR}_{\beta}$ for $\beta \in [\alpha , 1]$ <cit.>. To demonstrate how <ref> is applied, an example with a uniform loss distribution will be given. For this example, assume that the loss is distributed continuously and uniformly between 0 and 100, i.e., $X \sim U(0,100)$. Thus, $f_X (z) = \frac{1}{100}$ for $0 \leq z \leq 100$ and 0 elsewhere. The VaR at confidence level $\beta$ is given as $\text{VaR}_{\beta} (X) = 100 \times \beta$. Then the CVaR at confidence level $\alpha$ can be calculated as
\begin{align*}
\text{CVaR}_{\alpha} (X) =& \frac{1}{1 - \alpha} \int \limits_{\alpha}^{1} \text{VaR}_{\beta} (X) \, d \beta = \frac{1}{1 - \alpha} \int \limits_{\alpha}^{1} 100 \times \beta \, d \beta \\
=& \frac{100}{1 - \alpha} \left[ \frac{1}{2} \beta^2 \right]_{\alpha}^{1} = 50 \times (1 + \alpha) .
\end{align*}
So in this example, the 90 % CVaR would be $\text{CVaR}_{0.9} (X) = 50 \times (1 + 0.9) = 95$.
§.§ A New Proof of Acerbi's Integral Formula
Although Acerbi and Tasche proved <ref> in <cit.>, another proof will be given here. Two reasons for this alternative proof are, first, that Acerbi used different definitions in his paper, and second, to show how the result can be derived in another way. To the best knowledge of the author, this alternative proof has not been published before. However, the proof given here only holds for continuous random variables and therefore lacks the generality of Acerbi's proof.
For this alternative proof, the probability density function of the generalized $\alpha$-tail distribution is needed, which can be derived from <ref> as $f^{\alpha}_X (z) = \frac{d}{dz} F^{\alpha}_X (z)$, i.e.,
\begin{equation} \label{eqn:generalized_alpha_tail_pdf}
f^{\alpha}_X (z) = \left\{
\begin{array}{l r}
0, & \text{when~} z < \text{VaR}_{\alpha} (X) \\
\frac{f_X (z)}{1 - \alpha}, & \text{when~} z \geq \text{VaR}_{\alpha} (X)
\end{array}
\right. .
\end{equation}
(Continuous case only) Starting from the very basic definition of CVaR given in <ref>, one can use integration by substitution to arrive at <ref>:
\begin{align*}
\text{CVaR}_{\alpha} (X) =& \E [X^{\alpha}] \\
=& \int \limits_{- \infty}^{\infty} z f_X^{\alpha}(z) dz \\
=& \int \limits_{- \infty}^{\text{VaR}_{\alpha}(X)} z f_X^{\alpha}(z) dz + \int \limits_{\text{VaR}_{\alpha}(X)}^{\infty} z f_X^{\alpha}(z) dz . \\
\end{align*}
Using the definition of $f_X^{\alpha}(z)$ given in <ref>, the above equality simplifies to
\begin{align*}
\text{CVaR}_{\alpha} (X) =& \int \limits_{\text{VaR}_{\alpha}(X)}^{\infty} z \frac{f_X (z)}{1 - \alpha} dz . \\
\end{align*}
Now, one can define a new variable $\beta$, such that $\beta = F_X (z)$. Differentiating $\beta$ with respect to $z$ gives
\begin{equation}
\frac{d}{dz} \beta = f_X (z) \Longleftrightarrow f_X (z) dz = d \beta . \notag
\end{equation}
Furthermore, since $X$ is continuous, there is a one-to-one relationship between $\beta$ and $z$ and by <ref>, $z$ can be expressed as $z = \text{VaR}_{\beta} (X)$. So substituting $\beta = F_X (z)$, $z = \text{VaR}_{\beta} (X)$, and adjusting the limits of the integral ($F_X ( \text{VaR}_{\alpha} (X) ) = \alpha$ and $F_X ( \infty ) = 1$) yields
\begin{align*}
\text{CVaR}_{\alpha} (X) =& \frac{1}{1 - \alpha} \int \limits_{\alpha}^{1} \text{VaR}_{\beta}(X) \, d \beta \,,
\end{align*}
which completes the proof.
CHAPTER: PORTFOLIO OPTIMIZATION USING CVAR
While <ref> introduced the CVaR concept for univariate random distributions, the concept can be extended to multivariate random distributions or random vectors as well. This will be done here with a focus on portfolio optimization, i.e. investment decisions where the investor is able to invest his funds in more than one asset. First, <ref> gives an introduction into portfolio optimization by presenting the first model that has been developed to improve decision making for portfolio investments <cit.>, namely the Markowitz or Mean Variance Model. Then, <ref> introduces the CVaR Model that has been developed by Rockafellar and Uryasev in <cit.>. It will also be explained why the CVaR Model is preferable to the Markowitz Model with regards to risk management. And finally, numerical examples will be given in <ref> to show how the two models can be applied in practice.
Before beginning with the first section, some notation will be established for the concepts that are used throughout this chapter and the rest of the dissertation.
First of all, the investor can invest in $N$ different assets. His investment decision can be represented mathematically by a decision vector $\mathbf{x} \in S \subseteq \mathbb{R}^N$. Here, $S$ represents the feasible set for investment decisions.[For example, $S$ could have the unit budget constraint $\sum_i x_i = 1$, or a concentration risk constraint $x_j \leq 0.3 \sum_i x_i \, \forall j \leq N$. In the case of the budget unit constraint, $x_3 = 0.3$ means that 30 % of available funds should be invested in asset number 3.]
To define the set of admissible portfolios $S$ for this chapter, the investor only has two constraints: He cannot short sell any assets and his decision needs to satisfy the unit budget constraint. With these considerations, the set of admissible portfolios $S$ which consists of $N$ assets can be as
\begin{equation} \label{eqn:admissible_portfolio_S}
S = \left\{ \xinR{N} : x_i \geq 0 \,\, \forall \,\, i \in \{1, 2, \dots, N\} \,, \sum \limits_{i = 1}^N x_i = 1 \right\} .
\end{equation}
Also, the returns of each asset are random. Therefore, the losses can be expressed by a random loss vector $\mathbf{r} \in \mathbb{R}^N$,[Here, the losses are the negative values of returns. Hence, a negative $r_i$ means that asset $i$ is giving the investor a profit.] so that $r_i$ is a random variable that is distributed according to the loss distribution of the $i$th asset. Note that $r_i$ and $r_j$ for $i \not = j$ do not need to have the same distribution. Furthermore, $r_i$ and $r_j$ can be correlated (and in most cases are), which is why portfolio optimization is concerned with multivariate loss distributions.
So the loss $X$ that an investor can experience is a random variable that depends on the (random) losses of each asset and also on the investment in each asset, so that $X = X(\mathbf{x}, \mathbf{r})$.
For the following considerations, the investor demands a minimum expected return. Taking $\mathbf{r}$ as the vector of random losses, $\mathbf{x}$ the vector of investment decisions, and labelling the minimum required return $R$, the minimum expected return constraint can be formulated as
\begin{align}
\mathbf{x}^T \widehat{\mathbf{r}} & \leq - R \,, \label{eqn:minimum_expected_return}
\end{align}
where $\widehat{\mathbf{r}} = \E [ \mathbf{r} ]$.
§ MEAN VARIANCE OPTIMIZATION (MARKOWITZ MODEL)
Before modern portfolio theory was introduced by Markowitz in 1952 (<cit.>), investment decisions were mostly made by an investor's belief.[Even after Markowitz's paper was published it took several decades to be adapted by the financial industry because computers did not have the necessary power to perform the calculations.] Although the expected return and variance of a single asset could be calculated, investors were not able to form optimal portfolios, i.e. assign their funds in such a way that the whole portfolio had preferable characteristics <cit.>.
The most important contribution of <cit.> is that it is favourable to diversify a portfolio because this will reduce the portfolio's standard deviation (risk) as long as the correlation between assets is less than 1. This result can be shown by a portfolio of $N$ assets <cit.>.
Assume that an investor can buy $N$ assets, with expected returns $\widehat{r}_1 \,, \dots \,, \widehat{r}_N$ and variance $\sigma^2_1 \,, \dots \,, \sigma^2_N$. Assigning $x_i$ of his funds to the $i$th asset, the investor can expect a return of
\begin{align*}
\E [\mathbf{x}^T \mathbf{r}] =& \sum \limits_{i=1}^N x_i \times \widehat{r}_i \,,
\end{align*}
which is the weighted average of expected asset returns. However, the risk for the investor can be lower than the weighted average of asset risks. To show this, the covariance matrix $\mathbf{\Sigma} \in \mathbb{R}^{N \times N}$ of the random loss vector $\mathbf{r}$ will be introduced. $\mathbf{\Sigma}$ is defined as <cit.>
\begin{equation}
\mathbf{\Sigma} \defeq \begin{bmatrix}
\text{Var}(r_1) & \text{Cov}(r_1, r_2) & \cdots & \text{Cov}(r_1, r_N) \\
\text{Cov}(r_2, r_1) & \text{Var}(r_2) & \cdots & \text{Cov}(r_2, r_N) \\
\vdots & \vdots & \ddots & \vdots \\
\text{Cov}(r_N, r_1) & \text{Cov}(r_N, r_2) & \cdots & \text{Var}(r_N) \\
\end{bmatrix} \,,
\notag
\end{equation}
where $\text{Var}(r_i) = \sigma^2_i$ was defined in <ref>. Using <ref>, $\text{Cov}(r_i, r_j)$ can be expressed as
\begin{equation}
\text{Cov}(r_i, r_j) = \rho_{ij} \sigma_i \sigma_j \,, \notag
\end{equation}
which leads to the expression below. This expression is a standard result in financial literature but has been derived independently by the author:[In the standard financial literature, e.g. <cit.>, this result is usually derived for $N = 2$ assets but not for $N > 2$.]
\begin{align*}
\sigma (\mathbf{x}^T \mathbf{r}) = \sqrt{\text{Var}(\mathbf{x}^T \mathbf{r})} =& \sqrt{\mathbf{x}^T \mathbf{\Sigma} \mathbf{x} } \\
=& \sqrt{\sum \limits_{i = 1}^N x^2_i \sigma^2_i + \sum \limits_{i = 1}^{N-1} \sum \limits_{j = i+1}^N 2 \rho_{ij}x_i x_j \sigma_i \sigma_j} \\
=& \sqrt{\sum \limits_{i = 1}^N x^2_i \sigma^2_i + \sum \limits_{i = 1}^{N-1} \sum \limits_{j = i+1}^N 2 x_i x_j \sigma_i \sigma_j - \sum \limits_{i = 1}^{N-1} \sum \limits_{j = i+1}^N 2 (1 - \rho_{ij}) x_i x_j \sigma_i \sigma_j }\\
=& \sqrt{ \left( \sum \limits_{i = 1}^N x_i \sigma_i \right)^2 - \sum \limits_{i = 1}^{N-1} \sum \limits_{j = i+1}^N 2 (1 - \rho_{ij}) x_i x_j \sigma_i \sigma_j } \\
\leq& \sqrt{ \left( \sum \limits_{i = 1}^N x_i \sigma_i \right)^2} = \sum \limits_{i = 1}^N x_i \sigma_i \,,
\end{align*}
for $\mathbf{x} \in S$. The above inequality is strict whenever $\rho_{ij} < 1 \text{~for~} i \not = j$, meaning that the portfolio risk (given by the standard deviation) is less than the weighted average of asset risks whenever the assets are not perfectly correlated (which is usually the case).
Using Markowitz's findings, a quadratic programme can be formulated to find a minimum variance portfolio. Including the constraint given by <ref>, the programme can give the investor a portfolio which offers the required minimum return at the lowest possible risk. The inputs for the model are $\widehat{\mathbf{r}}$, the expected returns of assets $1, \dots, N$ and $\mathbf{\Sigma}$, the covariance matrix. Usually these inputs have to be estimated and one possibility of estimating the entries of the covariance matrix is given in <ref> but a further discussion on parameter estimation is beyond the scope of this dissertation.
[<cit.> Minimum Variance Portfolio]
A minimum variance portfolio in the sense of <cit.> is a portfolio which can be formed by solving
min_𝐱 𝐱^T Σ 𝐱
s.t. 𝐱^T 𝐫 ≤- R
𝐱 ∈S
} ,
where $\mathbf{\Sigma}$ is the covariance matrix of the random loss vector $\mathbf{r}$, $\widehat{\mathbf{r}} = \E [ \mathbf{r} ]$, and $S$ is the set of admissible portfolios.
Since a covariance matrix $\mathbf{\Sigma}$ is always positive definite <cit.>, <ref> is a convex optimization problem. It has therefore either a unique solution or is infeasible. The only situation under which <ref> becomes infeasible is when the required expected return is higher than any single expected return of the $N$ assets under consideration.
To see how the portfolio risk changes for different expected returns, one can solve <ref> for different values of $R$ (expected minimum return) and calculate the resulting portfolio risk (standard deviation). These risk/return pairs can be used to draw the efficient frontier, which is “a graph of the lowest possible [risk] that can be attained for a given portfolio expected return” <cit.>.
For a sample portfolio of three assets with expected returns and covariance matrix
\begin{align*}
\widehat{\mathbf{r}} =& \begin{bmatrix}
-0.1073 \\
-0.0737 \\
\end{bmatrix} & \text{and} &&
\mathbf{\Sigma} =& \begin{bmatrix}
0.02778 & 0.00387 & 0.00021 \\
0.00387 & 0.01112 & -0.00020 \\
0.00021 & -0.00020 & 0.00115
\end{bmatrix} \,,
\end{align*}
the efficient frontier is shown in <ref>.
Efficient frontier for a sample portfolio.
Because of the quadratic term in the objective function of <ref>, an investor can increase his expected portfolio return with little additional risk if the portfolio has a low standard deviation to begin with. For example, increasing the expected return from 6.5 to 7 % only increases the standard deviation by 0.6 %. However, the more expected return an investor demands, the higher the increase in risk. Increasing the expected return from 9.5 to 10 % requires an additional risk of 1.7 %.
It is possible to form a portfolio with a risk/return profile that lies below the efficient frontier. However, it is not possible to form a portfolio whose risk/return profile is above or to the left of the efficient frontier in <ref> <cit.>.
§ CVAR OPTIMIZATION (ROCKAFELLAR AND URYASEV MODEL)
Despite revolutionizing risk management at its time, the Markowitz Model has some drawbacks regarding risk management. Two important disadvantages arise because it measures the risk in terms of variance of the portfolio:
* Variance is only a useful risk measure for normally (or symmetrically) distributed losses. Since variance is measured in either direction, tail losses arising from skewed loss distributions are not taken in account.
* Variance is not a coherent risk measure as it is not monotone.
The first argument is illustrated in the second scenario of <ref>, while the second argument can easily be shown by an example: Consider two random variables (both representing loss) which are normally distributed, but with different $\mu$ and $\sigma^2$: $X \sim N(\mu_X = 0, \sigma_X^2 = 2)$ and $Y \sim N(\mu_Y = 10, \sigma_Y^2 = 1)$. The probability that $X$ is bigger than $Y$ is insignificantly small. To be precise, $P(Y \leq X) = 3.9 \times 10^{-9}$. Hence, it is nearly impossible that the loss of $X$ will exceed the loss of $Y$. However, $X$ has a higher variance than $Y$, i.e. $\text{Var}(X) = 2 \geq \text{Var}(Y) = 1$, and would therefore be considered riskier if the risk were measured by the variance.
Because of this, it is preferable for a risk manager to optimize the portfolio with regards to CVaR than with regards to variance. Rockafellar and Uryasev proposed a linear programme in <cit.> to optimize the CVaR of a portfolio. They also proved that under certain conditions the CVaR optimization will give the same optimal portfolio as the minimum variance optimization. The rest of this section introduces their notation and presents their results.[Although this section follows the outline of <cit.>, the expressions are closer aligned with <cit.>.]
To derive later results, Rockafellar and Uryasev labelled the cumulative distribution function of losses $\Psi (\mathbf{x},c)$, so that for any given decision $\mathbf{x} \in S$, random asset losses $\mathbf{r} \in \mathbb{R}^n$, and loss distribution $X(\mathbf{x}, \mathbf{r})$,
\begin{align}
\Psi (\mathbf{x},c) &= F_X (c) = P ( X(\mathbf{x}, \mathbf{r}) \leq c) & \text{in the general case, and} \label{eqn:cdf_Losses_general} \\
\Psi (\mathbf{x},c) &= F_X (c) = \int \limits_{\mathbf{r}: X(\mathbf{x},\mathbf{r}) \leq c} p(\mathbf{r}) d \mathbf{r} & \text{in the continuous case,} \label{eqn:cdf_Losses_continuous}
\end{align}
where $p(\mathbf{r})$ in <ref> is the pdf for a continuous $\mathbf{r}$. The function $\Psi (\mathbf{x},c)$ can be interpreted as the probability that the losses do not exceed threshold $c$.
Continuing with the notation of $\Psi (\mathbf{x},c)$ as the threshold of losses, $\text{VaR}_{\alpha}$ and $\text{CVaR}_{\alpha}$ of an investment decision $\mathbf{x}$ can be then written as
\begin{align}
\text{VaR}_{\alpha} ( \mathbf{x} ) &= \text{VaR}_{\alpha} ( X(\mathbf{x}, \mathbf{r}) ) = \min \{ c : \Psi (\mathbf{x},c) \geq \alpha \} \label{eqn:VaR_in_terms_of_Psi} \text{, and}\\
\text{CVaR}_{\alpha} ( \mathbf{x} ) &= \text{CVaR}_{\alpha} ( X(\mathbf{x}, \mathbf{r}) ) = \E_{\mathbf{r}} [ X(\mathbf{x}, \mathbf{r}) \mid X(\mathbf{x}, \mathbf{r}) \geq \text{\emph{VaR}}_{\alpha} ( \mathbf{x} )] \label{eqn:CVaR_in_terms_of_Psi}.
\end{align}
Rockafellar and Uryasev characterized <ref> and <ref> in terms of a function
\begin{equation} \label{eqn:phi_alpha}
\phi_{\alpha} ( \mathbf{x}, c) \defeq c + \frac{1}{1 - \alpha} \E \left[ ( X( \mathbf{x}, \mathbf{r} ) - c)^+ \right],
\end{equation}
where $\E \left[ \cdot \right]$ is the expectation and $(t)^+ = \max \{ 0, t \}$. Based on <ref>, they formulated <ref>, the most important result of <cit.>.
As a function of $c$, $\phi_{\alpha} ( \mathbf{x}, c)$ is convex and continuously differentiable. The $\text{\emph{CVaR}}_{\alpha}$ of the loss associated with any $\mathbf{x} \in S$ can be determined from the formula
\begin{equation} \label{eqn:CVaR_Theorem-CVaR_from_Psi}
\text{\emph{CVaR}}_{\alpha} (\mathbf{x}) = \min_{c \in \mathbb{R}} \phi_{\alpha} ( \mathbf{x}, c) .
\end{equation}
Furthermore, let $\Phi_{\alpha}^* (\mathbf{x}) \defeq \arg \min_c \phi_{\alpha} ( \mathbf{x}, c)$, i.e. $\Phi_{\alpha}^* (\mathbf{x})$ is the set of minimizers of $\phi_{\alpha} ( \mathbf{x}, c)$. Then
\begin{equation} \label{eqn:CVaR_Theorem-VaR_from_Psi}
\text{\emph{VaR}}_{\alpha} (\mathbf{x}) = \min \{ c: c \in \Phi_{\alpha}^* (\mathbf{x}) \} .
\end{equation}
And following from <ref> and <ref>, the following equation always holds:
\begin{equation} \label{eqn:CVaR_Theorem-CVaR_Final_Expression}
\text{\emph{CVaR}}_{\alpha} (\mathbf{x}) = \phi_{\alpha} ( \mathbf{x}, \text{\emph{VaR}}_{\alpha} (\mathbf{x}) ) .
\end{equation}
The proof of <ref> is given in the appendix of <cit.>. Based on <ref>, Rockafellar and Uryasev stated another theorem, which is useful for the computational calculation to find a CVaR optimal portfolio $\mathbf{x}^* \in S$.
Let $S$ be a convex set of feasible decisions $\mathbf{x}$ and assume that $X(\mathbf{x}, \mathbf{r})$ is convex in $\mathbf{x}$. Then minimizing the $\text{\emph{CVaR}}_{\alpha}$ of the loss associated with decision $\mathbf{x} \in S$ is equivalent to minimizing $\phi_{\alpha} ( \mathbf{x}, c)$ over all $(\mathbf{x}, c) \in S \times \mathbb{R}$, in the sense that
min_𝐱 ∈S CVaR_α (𝐱) = min_(𝐱, c) ∈S ×ℝ ϕ_α ( 𝐱, c) ,
where, moreover, a pair $(\mathbf{x}^*, c^*)$ achieves the right hand side minimum if and only if $\mathbf{x}^*$ achieves the left hand side minimum and $c^* \in \Phi_{\alpha}^* (\mathbf{x})$. Therefore, in circumstances where the interval $\Phi_{\alpha}^* (\mathbf{x})$ reduces to a single point (as is typical), the minimization of $\phi_{\alpha} ( \mathbf{x}, c)$ produces a pair $(\mathbf{x}^*, c^*)$ such that $\mathbf{x}^*$ minimizes the $\text{\emph{CVaR}}_{\alpha}$ and $c^*$ gives the corresponding $\text{\emph{VaR}}_{\alpha}$.
<ref> not only gives a way to express the CVaR minimization problem in a tractable form, but also allows to calculate $\text{CVaR}_{\alpha}$ without having to calculate $\text{VaR}_{\alpha}$ first, as would have been the case with <ref>. More remarkably, finding the CVaR by using <ref>, gives the corresponding VaR as a by-product <cit.>.
Applying <ref> with <ref>, the investment decision $\mathbf{x}$ that minimizes the Conditional Value-at-Risk of a portfolio at the confidence level $\alpha$ can be expressed as <cit.>
min_𝐱 ∈S CVaR_α (𝐱) = min_𝐱 ∈S, c ∈ℝ ( c + 1/1 - α [ ( X(𝐱, 𝐫) - c )^+ ] ) .
To provide a better understanding of how to solve <ref>, a one-dimensional example will be given, i.e. there is only asset with a univariate, discrete loss distribution. Since there is only one asset to consider, $\mathbf{x} = [1]$. Because of this, it is not the goal in this example to find the optimal portfolio composition, but rather to find the VaR and CVaR using <ref>. The asset has the loss distribution of $Y$ given in <ref>. The table is reproduced below for convenience.
| *7c |
i 1 2 3 4 5 6
$y_i$ 100 200 400 800 900 1000
$P ( Y = y_i )$ 0.1 0.2 0.5 0.18 0.01 0.01
For this asset, the function $\phi_{\alpha} ( \mathbf{x}, c) = c + \frac{1}{1 - \alpha} \E \left[ ( X( \mathbf{x}, \mathbf{r} ) - c)^+ \right]$ will be drawn against $c$ to find $\text{CVaR}_{\alpha} (\mathbf{x}) = \min \limits_{c \in \mathbb{R}} \phi_{\alpha} ( \mathbf{x}, c)$ graphically. The graph of $\phi_{\alpha} ( \mathbf{x}, c)$ for $\alpha = 0.95$ is shown in <ref>.
Function value $\phi_{0.95} (c)$ of $Y$ for different values of $c$.
The graph shows that the minimum of $\phi_{\alpha} ( \mathbf{x}, c)$ occurs at $c^* = 800$. Thus, $ \min \limits_{c \in \mathbb{R}} \phi_{\alpha} ( \mathbf{x}, c) = \phi_{\alpha} ( \mathbf{x}, 800) = 860$. Hence, by <ref>, it follows that $\text{VaR}_{0.95} = 800$ and $\text{CVaR}_{0.95} = 860$, which agrees with the results of the Convex Combination Formula in <ref> as expected. Another characteristic to point out is that $\phi_{\alpha} ( \mathbf{x}, c)$ has “kinks” at points $y_i \,, i = 1, \dots, 6$ <cit.>.
<ref> is still difficult to evaluate if the loss distribution $X$ is continuous. One remedy is to use Monte Carlo Sampling to draw $K$ i.i.d. samples of the loss vector $\mathbf{r}$ ($\mathbf{r}_k \,, k \in \{1, 2, \dots, K\}$) from the distribution of $\mathbf{r}$, so that <ref> can be written in a tractable LP form <cit.>. Adding constraint <ref> to ensure a minimum expected return for the investor, the tractable LP form of the optimization problem is given as
min_c, 𝐳 c + 1/K (1 - α) ∑_k = 1^K z_k
s.t. z_k ≥𝐱^T 𝐫_k - c for k ∈{1, …, K}
z_k ≥0 for k ∈{1, …, K}
𝐱^T 𝐫 ≤- R
𝐱 ∈S
} .
Another interesting link between mean variance and CVaR optimization was established in <cit.> as well. Rockafellar and Uryasev proposed that under certain conditions, <ref> and <ref> give the same optimal portfolio.
Suppose that the loss associated with each $\mathbf{x}$ is normally distributed as holds when $\mathbf{r}$ is normally distributed. If $\alpha \geq 0.5$ and the constraint <ref> is active at solutions to <ref> and <ref>, then the solutions to those problems are the same; a common portfolio $\mathbf{x}^*$ is optimal by both criteria.
This means that under the conditions stated in the proposition, it is possible to find the minimum variance portfolio by finding the minimum CVaR portfolio. <ref> will be explored in the first scenario of <ref>.
§ NUMERICAL EXAMPLES
This section gives numerical examples for finding minimum CVaR portfolios. More precisely, the CVaR criterion will be compared to the minimum variance criterion (as formulated by Markowitz in <cit.>, see <ref>) and two scenarios will be given to show the effect of the criterion on the portfolio composition. The first scenario is adapted from <cit.> and concerns normally distributed losses. The second scenario is a theoretical construct with a positively skewed loss distribution.
§.§ First Scenario: Normally Distributed Losses
This scenario serves to display the proposition by Rockafellar and Uryasev that for certain conditions the minimum variance optimization and CVaR optimization give the same optimal portfolio $\mathbf{x}^*$:
In the example from <cit.>, three assets ($N = 3$) are available: The S&P 500 index ($x_1$), long-term US government bonds ($x_2$), and a portfolio of small cap stocks ($x_3$). The expected return of each asset and their covariance matrix is given in <ref> and <ref>, respectively.
|
1511.00013
|
1Department of Physics & Astronomy, Johns Hopkins University, Bloomberg Center, 3400 N. Charles St., Baltimore, MD 21218, USA
2Gemini Observatories, 670 N. A'Ohoku Pl., Hilo, HI 96720, USA
3Laboratoire AIM-Paris-Saclay, CEA/DSM/Irfu, Orme des Merisiers, Bat 709, F-91191 Gif sur Yvette, France
4Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
5Dartmouth College, Dept. of Physics and Astronomy, 6127 Wilder Laboratory, Hanover, NH 03755, USA
6Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China
7Department of Astronomy, School of Physics, Peking University, Beijing 100871, China
8Institut d'Astrophysique Spatiale, CNRS, Université Paris-Sud, Bat. 120-121, F-91405 Orsay, France
9Centre for Astrophysics Research, Science & Technology Research Institute, University of Hertfordshire, Hatfield AL10 9AB, UK
10Research Center for the Early Universe, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
11Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
12Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
13Gymnasium Kirschgarten, Hermann Kinkelin-Strasse 10, 4051 Basel, Switzerland
Radio emission from radio-quiet quasars may be due to star formation in the quasar host galaxy, to a jet launched by the supermassive black hole, or to relativistic particles accelerated in a wide-angle radiatively-driven outflow. In this paper we examine whether radio emission from radio-quiet quasars is a byproduct of star formation in their hosts. To this end we use infrared spectroscopy and photometry from and to estimate or place upper limits on star formation rates in hosts of $\sim 300$ obscured and unobscured quasars at $z<1$. We find that low-ionization forbidden emission lines such as [NeII] and [NeIII] are likely dominated by quasar ionization and do not provide reliable star formation diagnostics in quasar hosts, while PAH emission features may be suppressed due to the destruction of PAH molecules by the quasar radiation field. While the bolometric luminosities of our sources are dominated by the quasars, the 160 fluxes are likely dominated by star formation, but they too should be used with caution. We estimate median star formation rates to be $6-29$ $M_{\odot}$ yr$^{-1}$, with obscured quasars at the high end of this range. This star formation rate is insufficient to explain the observed radio emission from quasars by an order of magnitude, with $\log(L_{\rm radio,obs}/L_{\rm radio,SF})=0.6-1.3$ depending on quasar type and star formation estimator. Although radio-quiet quasars in our sample lie close to the 8-1000 infrared / radio correlation characteristic of the star-forming galaxies, both their infrared emission and their radio emission are dominated by the quasar activity, not by the host galaxy.
§ INTRODUCTION
The most extended, powerful and beautiful sources in the radio sky are due to synchrotron emission from relativistic jets launched by supermassive black holes in centers of galaxies <cit.>, but only a minority of active black holes produce these structures. At a given optical luminosity of the active nucleus, radio power spans many orders of magnitude, and the exact distribution of radio luminosities remains a matter of continued debate. A particularly intriguing point is whether this distribution is bimodal <cit.>: does the brighter “radio-loud” population show a well-defined luminosity separation from the fainter “radio-quiet” group, or is the distribution of radio luminosities continuous (e.g., )? This question goes to the heart of fundamental issues in black hole physics: are weak radio sources associated with supermassive black holes due to relativistic jets which are scaled down from their extended powerful analogs, or are there additional mechanisms for producing radio emission? Are all black holes actually capable of launching a relativistic jet, and do all black holes undergo such a phase?
At $\ga 1''$ resolution, the majority of quasars ($L_{\rm bol}\ga 10^{45}$ erg s$^{-1}$) are point-like radio sources with luminosities $\nu L_{\nu}[1.4{\rm GHz}]\la 10^{41}$ erg s$^{-1}$, and the origin of this emission has been the subject of recent debate <cit.>. The recent finding of a strong proportionality between the radio luminosity of radio-quiet quasars and the square of the line-of-sight velocity dispersion of the narrow-line gas <cit.> is an exciting development in this topic, offering possible clues as to the nature of the radio emission. These velocity dispersions can reach values that are much higher than those that can be confined by a typical galaxy potential, suggesting that the ionized gas is neither in static equilibrium nor in galaxy rotation. Blue-shifted asymmetries suggest that the gas is outflowing <cit.>, and interpreting the line-of-sight velocity distribution as due to the range of velocities in the outflow suggests $v_{\rm out}\sim 1000$ km s$^{-1}$.
The observed correlation between narrow line kinematics and radio luminosity suggests a physical connection between the processes that produce them. One possibility is that compact jets inject energy into the gas and launch the outflows <cit.>; another is that the winds are driven radiatively, then induce shocks in the host galaxy and the shocks in turn accelerate relativistic particles <cit.>.
A completely different approach is followed by <cit.> and <cit.> who argue that the radio emission in radio-quiet quasars is mostly or entirely due to star formation in their host galaxies. Three arguments could be put forward to support this hypothesis: (i) If the radio luminosity function is bimodal, then something other than scaled-down jets is probably responsible for the radio-quiet sources. (ii) Active galaxies with $L_{\rm bol}\la 10^{45}$ erg s$^{-1}$ tend to lie on the extension of the classical 8-1000 / radio correlation of the star-forming galaxies <cit.>. (iii) The amount of radio emission seen in high-redshift radio-quiet quasars can be explained by star formation rates $20-500 M_{\odot}$ yr$^{-1}$, which (although quite high) seem plausible for the epoch of peak galaxy formation.
Several arguments can be put forward against this hypothesis: (i) In quasars, the scatter around the radio / infrared relationship is higher than that seen in star-forming galaxies <cit.>. (ii) In quasars the infrared emission can be dominated by the quasar, rather than by the star formation <cit.>. (iii) The amount of star formation required to explain the observed radio emission in quasars may be higher than that deduced using other methods <cit.>.
<cit.> demonstrate that in radio-quiet low-luminosity active galactic nuclei (AGN) much of the observed radio luminosity is consistent with star formation in the AGN hosts. The objects in their sample have infrared luminosities $\nu L_{\nu}$[12]$\la 10^{44}$ erg s$^{-1}$. In this paper we examine AGNs with $\nu L_{\nu}$[12] from $\sim 2 \times 10^{43}$ to $\sim 10^{46}$ erg s$^{-1}$, thereby extending the analysis of <cit.> to luminosities higher by up to two orders of magnitude. Our goal is to determine whether the radio emission of quasars ($\nu L_{\nu}$[12]$\ga 10^{44}$ erg s$^{-1}$, or $L_{\rm bol}\ga 10^{45}$ erg s$^{-1}$ as per bolometric corrections by ) is due to the star formation in their host galaxies.
To this end, we estimate the rates of star formation in the hosts of quasars of different types using and data, and compare the amount of radio emission seen from these objects with that expected from star formation alone <cit.>. In Section <ref> we describe sample selection, datasets and measurements. In Section <ref>, we use far-infrared photometry to calculate star formation rates, predict the associated radio emission and compare with observations. In Section <ref> we use mid-infrared spectroscopy for a similar analysis. We discuss various difficulties in measuring star formation rates of quasar hosts in Section <ref> and summarize in Section <ref>. We use a $h$=0.7, $\Omega_m$=0.3, $\Omega_{\Lambda}$=0.7 cosmology.
Throughout the paper, we make a key distinction between far-infrared ($\ga 100\micron$) vs radio correlation and total infrared (conventionally defined over 8-1000 range) vs radio correlation. For star forming galaxies which show similar infrared spectral energy distributions, these concepts can be used interchangeably, since an accurate estimate of the total infrared luminosity can be obtained from far-infrared fluxes alone (e.g., ). However, as we add quasar contribution to both infrared and radio emission, some or all of these relationships might break down, and in particular because of the wide range of quasar spectral energy distributions their far-infrared emission and their total infrared emission are no longer strongly correlated. In Section <ref>, we investigate the fate of far-infrared vs radio and total infrared vs radio correlations in the presence of a quasar.
§ SAMPLES, OBSERVATIONS, DATA REDUCTION AND MEASUREMENTS
§.§ Type 2 and type 1 samples
Our goal is to assemble a large sample of quasars (whether optically obscured or unobscured) for which the host star formation rates can be usefully constrained with existing archival data. Furthermore, because of the sensitivity of the existing radio surveys, in order to probe the radio-quiet population we are restricted to low-redshift quasars, $z<1$. As a result, this work primarily focuses on the analysis of two quasar samples.
Our first sample consists of and follow-up of obscured (type 2) quasars from <cit.> at $z\la 0.8$. These objects are selected to have only narrow emission lines with line ratios characteristic of ionization by a hidden AGN <cit.> and are required to have $L_{\rm [OIII]}\ga 10^{41.5}$ erg s$^{-1}$. Of the 887 objects in <cit.> catalog, WISE-3 matches are available for 94% of the objects and WISE-4 matches for 87% (some of the remaining 13% are detected, but cannot be deblended from the nearby contaminants in the WISE-4 band). We calculate 12 luminosities from the WISE-3 matches, k-correcting using WISE-4 flux if available or using a median WISE-4/WISE-3 index if not <cit.>.
For this sample, we collect archival photometry and analyze new photometry as discussed in Sections <ref> and <ref> for a total of 136 objects. Furthermore, while we previously published ten spectra of type 2 quasars <cit.>, in Section <ref> we conduct an extensive archival search which allows us to significantly expand the sample and present 46 spectra here. The photometric and spectroscopic samples overlap by 28 objects. The distribution of [OIII] and mid-infrared luminosities for the parent sample and for the objects with follow-up and observations is shown in Figure <ref>.
[OIII] and 12 luminosities of type 2 quasars with 160 photometry from or (circles; Section <ref>) and spectroscopy (crosses; Section <ref>). The background grey points show the parent sample of type 2 quasars <cit.>.
Our second sample is comprised of 115 type 1 quasars at $z\la 0.5$ studied with spectroscopy by <cit.>. Of these, 90 are ultraviolet-excess Palomar-Green (PG; ) quasars, and the remaining 25 are quasars selected from the Two Micron All Sky Survey (2MASS) with red $R-K$ colors <cit.>. Thus it is a heterogeneous quasar sample that includes objects with a range of extinction, from $A_V\simeq 0$ to $\la 5$ mag <cit.>, but overall it is dominated by type 1 (broad-line) sources. In the few cases of narrow-line (type 2) classification in the optical, broad emission lines and strong quasar continuum are seen in the near-infrared <cit.>. Hereafter, we refer to these objects collectively as type 1 sources, sometimes making a distinction between `blue' and `red' as necessary, according to whether they are drawn from the PG sample or the 2MASS sample.
Out of 115 type 1 quasars, all but one have complete 4-band photometry from Wide-field Infrared Survey Explorer (WISE; ). We calculate rest-frame 12 mid-infrared luminosities by power-law-interpolating between WISE-3 and WISE-4 bands, and then estimate bolometric luminosities by applying bolometric correction of 8.6 from <cit.>. spectroscopy is available for all 115 objects <cit.>. For red type 1 quasars, we collect archival photometry (Section <ref>) and for blue type 1 quasars we use recently published photometry (Section <ref>), so that 114 out of 115 objects have far-infrared photometric data.
The redshifts and mid-infrared luminosity distributions of type 1 and type 2 samples are similar, as shown in Figure <ref>. For 39 of the 115 type 1 quasars, [OIII] luminosity measurements are available in the catalog of <cit.>. For this subsample, we find $L_{\rm [OIII]}=10^{42.3\pm 0.6}$ erg s$^{-1}$ (average and standard deviation), similar to the range probed by the type 2 sample with follow-up infrared data ($10^{42.5\pm 0.5}$ erg s$^{-1}$).
Redshift and mid-infrared luminosity distributions of both quasar samples discussed in this paper. Left: PG (filled blue circles) and 2MASS (open red circles) type 1 quasars with spectroscopy presented by <cit.> and and photometry. Right: type 2 quasars with 160 photometry from or (circles) and spectroscopy (crosses) and with parent sample <cit.> in grey. We estimate the bolometric luminosities of type 1 quasars from their 12 monochromatic luminosities (right axis of the left panel) using <cit.> bolometric correction of 8.6 (which ranges between 7.8 and 9.3 depending on the assumed spectral energy distribution). Type 2 quasars likely have higher bolometric corrections.
Type 2 quasars are redder in the mid-infrared than type 1 quasars <cit.>. Specifically, the infrared power-law index between rest-frame 5 and 12 luminosities $\beta$ (defined as $\nu L_{\nu}\propto \lambda^{\beta}$) is $-0.05\pm 0.30$ (mean and standard deviation) for type 1 quasars from <cit.>, whereas for the type 2 quasars from <cit.> catalog it is $\beta=0.78\pm 0.45$. Furthermore, the ratio of 12 luminosities to $L_{\rm [OIII]}$ is 0.5 dex higher in type 1 quasars than in type 2s at the same emission line luminosity (Zakamska et al. in prep.). Both these factors suggest that 12 luminosity is not an isotropic measure of quasar luminosity and that type 2 quasars are obscured even at mid-infrared wavelengths. The bolometric corrections of type 2 quasars are therefore likely to be higher than those of type 1 quasars, perhaps by as much as a factor of $\sim 3$ (which would be necessary to reconcile the infrared-to-[OIII] ratios of type 1s and type 2s), but as they remain uncertain we do not show them in Figure <ref>.
§.§ Far-infrared photometry with
Composite spectral energy distribution models (e.g., ) attempt to decompose emission from active galactic nuclei into a component powered by the black hole and a component powered by star formation in the host galaxy, and to use these measurements to determine the power of both processes. At the heart of these methods is the empirical notion that dust heated by the black hole accretion is warmer than that heated by starlight, and thus a quasar-dominated spectral energy distribution peaks at shorter wavelengths than that of a star-forming galaxy <cit.>. Therefore, observing longward of thermal peaks maximizes sensitivity to star formation and minimizes contamination by the quasar.
We cross-correlate the 887 type 2 quasars from <cit.> against the Heritage Archive and we find 62 distinct sources with nominal coverage at 160 (the longest available wavelength) by the Multiband Imaging Photometer for (MIPS; ), and we download the corresponding 100 distinct Astronomical Observation Requests (AOR). We use filtered (mfilt, mfunc) post basic calibrated data (PBCD) products to perform point-spread function (PSF) photometry. To this end, we generate PSF models using STinyTim (MIPS Instrument Handbook, 2011) and develop an analytic approximation to them using piece-wise Airy functions; a detailed description of the PSF is available in <cit.>.
With this in hand, we perform PSF photometry of the First Look Survey, which allows us to calibrate our PSF fitting procedure against the catalog of 160 sources by <cit.>. We find that our measurements are systematically fainter than theirs by 24%, which we attribute to color corrections which they applied and we did not. We take their fluxes to be `true' values and correct the systematic offset using a constant multiplicative factor (analogous to their use of color corrections), after which we find excellent agreement between their fluxes and ours within their stated absolute uncertainty of 25%. In the absence of color information, we cannot tailor our color corrections to a specific target. Having thus calibrated our PSF photometry procedure, we apply it to the MIPS-160 data of type 2 quasars.
Of the 62 sources, 11 have poor enough data quality (covered on the edges of big scans, gaps in coverage overlapping with the source location) that we do not consider them. The 51 sources with acceptable data quality are listed in Table <ref>; of these, 12 are detected, both as evaluated by the improvement in reduced $\chi^2$ over a continuum-only fit and by visual inspection. Following <cit.>, we adopt 25% as the photometric uncertainty. The median value of detected flux is 101 mJy. For the remaining sources we derive upper limits by fitting PSFs at multiple random locations within the field of the object and deriving the standard deviation of the fitted fluxes, which is taken as a 1$\sigma$ limit for point-source detection. In Table <ref>, we give 5$\sigma$ upper limits derived using this procedure. The median upper limit is 84 mJy.
We then select all good observations of the 39 non-detected sources by choosing only those with the reported uncertainty in the vicinity of the object of $<0.4$ MJy/sr, which roughly corresponds to a 5$\sigma$ sensitivity for point source detection of 280 mJy. We then make cutouts from these data centered on the known positions of our sources and we stack them using error-weighted averaging. We find a strong ($\sim 10\sigma$) detection in the stacked image, with a PSF flux of 23 mJy, which we take to be an estimate of the mean flux of non-detected sources. We also conduct a null test, in which all images to be stacked are randomly offset by several pixels from the source position. There is no source detection in the null test stack.
The sample of type 2 quasars with archival MIPS-160 data is heterogeneous, as described in Table <ref>. 25 objects constitute the full content of our targeted program (GO-3163, PI Strauss); they were selected based on [OIII]$\lambda$5007Å luminosity ($L_{\rm [OIII]}\ge 10^{42.5}$ erg s$^{-1}$), tend to be at relatively high redshifts $(z\ga 0.30)$ and show low rates ($<20\%$) of MIPS-160 detection. Three were observed by other groups because they are powerful radio galaxies with strong enough line emission to make it into the [OIII]-selected sample of <cit.>. Five objects at low redshifts were observed by other groups as candidate Ultraluminous Infrared Galaxies (ULIRGs) or type 2 quasars, and these are strongly detected with high fluxes. 18 objects are covered serendipitously by observations of other targets or calibration observations. Thus is it not surprising to have a few bright detections (in particular, nearby objects selected by other observers as ULIRG candidates) supplemented with many objects that are much fainter.
For the 25 objects covered by our program GO-3163, we also have MIPS-70 measurements performed in 2006 (previously unpublished). For MIPS-70, we computed fluxes by aperture photometry using MOPEX with aperture radius of $16''$ and applying aperture corrections derived from mosaicked images. The statistical errors were estimated from rms fluctuations of backgrounds. The color correction was applied assuming power-law flux density with the slope of $-1$. Twelve of the objects are detected at $>3\sigma$ level (whereas only four in the same program GO-3163 are detected in MIPS-160). In this paper we use these 12 detected sources to estimate infrared colors of type 2 quasars in Section <ref>, leaving a detailed analysis of the spectral energy distributions for future. The MIPS-24 observations in this program have been superseded by WISE-4 data.
As for the type 1 sample, the majority of blue quasars were observed by as described in the next section and in <cit.>. Since data supersedes MIPS-160 data, we do not rematch the blue quasars to the archive. All 25 red type 1 quasars are covered by archival MIPS-160 observations from two programs: 11 objects were observed by PI F. Low as a follow-up of 2MASS-selected quasars, and the remaining 14 objects were observed by PI G. Rieke as a follow-up of the most luminous quasars known at $z<0.3$. We analyze the photometry of these 25 sources in the same way as we do the type 2 sample and include them in Table <ref>. 17 objects are detected with a median flux of 139 mJy and for 8 objects we give upper limits with a median value of 114 mJy.
§.§ Far-infrared photometry from
We proceed to photometry of type 2 quasars from <cit.>. Our sample is assembled from two programs of pointed observations. In the first one (PI Zakamska), we obtained pointed observations of seven [OIII]-luminous sources ($L_{\rm [OIII]}\ge 10^{43.0}$ erg s$^{-1}$, median $L_{\rm [OIII]}=10^{43.2}$ erg s$^{-1}$) whose optical line emission was studied in detail by <cit.>. In the second (PI Ho), we obtained pointed observations of 90 sources roughly matched in redshift, infrared luminosity and [OIII] luminosity to the PG sample (Figure <ref>) and sampling the full range of [OIII] luminosities ($L_{\rm [OIII]}=10^{41.7-43.4}$ erg s$^{-1}$) of the parent sample of <cit.>. Similarly deep photometry was obtained in both programs using the Photodetector Array Camera and Spectrometer (PACS) in the mini-scan map mode at 70 and 160 and Spectral and Photometric Imaging Receiver (SPIRE) at 250. All our targets are assumed to be point sources at resolution (the full width at half maximum of the point spread function is 12).
For the smaller program (PI Zakamska), we use Level 2 PACS and SPIRE observations produced by standard pipeline reduction procedures (described in Chapter 7 of the PACS observing manual and in Chapter 5 of SPIRE data handbook). Source confusion is not an issue in PACS bands: at 0.7 mJy <cit.>, confusion at 160 is well below our $1\sigma$ sensitivity of 2.5 mJy. We perform aperture photometry in the Interactive Processing Environment (HIPE) version 10.0 around the optical positions (known to better than 0.1, with absolute pointing error of 0.81, ). We use the AnnularSkyAperturePhotometry task within HIPE and apply aperture corrections using the PhotApertureCorrectionPointSource task. We detect all seven sources at 70 and six of them at 160 at above 3$\sigma$, with median fluxes of 22 mJy and 16 mJy, respectively. This photometry is presented in Table <ref>.
For the SPIRE images, we use an extraction and photometry task in HIPE that implemented the SUSSExtractor algorithm described by <cit.>. We do not detect any sources in the SPIRE bands, where our nominal $1\sigma$ point-source sensitivity is slightly below the confusion limit, 6 mJy at 250 <cit.>, and our images are indeed confusion-limited. Extrapolating our measured PACS-160 fluxes to the SPIRE-250 band using $F_{\nu}\propto \nu^{4.5}$ typical of the long-wavelength spectrum of star-forming galaxies <cit.>, we find that the median flux in SPIRE-250 is expected at the $\sim 3$ mJy level, below the confusion limit, thus the lack of detections is not surprising.
data for type 2 quasars from the larger program (PI Ho) will be presented in their entirety by Petric et al. (in prep). Here we use exclusively 160 PACS fluxes from this program obtained using aperture photometry tools in HIPE in a manner similar to that described in <cit.>. In this program, 90 objects were observed, 76 of them were detected and for the remaining 14 we set 4$\sigma$ upper limits.
For blue type 1 quasars photometric data are published in <cit.> and we use their 160 fluxes here. 85 objects were observed, 69 of them were detected and for the remaining 16 we use upper limits from <cit.>. Out of the remaining five blue type 1s from the sample of <cit.>, four have 160 photometry from or ISO in the literature <cit.>, and we include them in our analysis.
§.§ Spectroscopic observations
Mid-infrared spectra of galaxies contain a wealth of information on star formation processes and on the nuclear activity, including the emission features of polycyclic aromatic hydrocarbons (PAHs; ) and the low-ionization and high-ionization ionic emission lines ([NeVI]$\lambda$7.65, [SIV]$\lambda$10.51, [NeII]$\lambda$12.81, [NeV]$\lambda$14.32, [NeIII]$\lambda$15.56, ). Our analysis is based on the Space Telescope Infrared Spectrograph (IRS; ) low-resolution spectra of quasars of different types. For type 1 blue PG quasars and red 2MASS quasars, we use published spectra and analysis by <cit.>. As for type 2 quasars, we cross-correlate the type 2 quasar sample <cit.> against IRS data using Heritage Archive.
We find 46 type 2 quasars from <cit.> with IRS spectra of varying quality within 3 of the optical position. In Table <ref> we list type 2 quasars with mid-infrared spectroscopic measurements as well as comments on how these objects were selected for follow-up spectroscopy. The majority were targeted by various groups as type 2 quasar candidates. Ten of them were from our own program <cit.> and were selected based on [OIII] luminosity and infrared flux ($L_{\rm [OIII]}>10^{42.6}$ erg s$^{-1}$, $F_{\nu}$[8]$>1.5$ mJy, $F_{\nu}$[24]$>6$ mJy). Other programs selected targets based on X-ray properties and optical or infrared luminosity diagnostics. Thus the sample is a fairly representative subsample of the <cit.> sample of type 2 quasars (Figure <ref>). Depending on the redshifts of the targets and on which IRS gratings were used for the observations, the wavelength coverage ranges from $\sim$5-13 in the rest-frame (12 objects) to $\sim$5-25 (the rest of the sample). Example spectra are shown in Figure <ref>. There are 28 objects in common between the type 2 sample with IRS spectra and the type 2 sample with 160 photometric data; these sources are discussed in Section <ref>.
Example spectra of three type 2 quasars from our sample: from top to bottom, a spectrum with relatively strong PAHs, a power-law-dominated spectrum and a silicate-absorption-dominated spectrum. On the right, we show PAH[11.3] fits with fixed shape of the PAH feature taken from <cit.> and third order polynomial continuum. Pink shading shows the expected locations of PAH complexes, vertical blue lines show positions of some of the brightest emission lines. The broad absorption feature extending from $\sim 8$ to $\sim 13$ in SDSS J1154+6138 is due to silicates.
As the majority of the sources are point-like at resolution, we use the enhanced data products described in Chapter 9 of the IRS Instrument Handbook. In a handful of cases where several spectra of the same target are returned by the search engine (perhaps because the IRS coordinates are slightly offset from one another) we combine the spectra into one using error-weighting. We inspect all spectra to make sure that the short- and the long-wavelength (SL and LL) spectra stitch together well in the region of the overlap. Because the LL grating has a larger aperture than the SL one, for extended sources some of the flux may be missed by the SL grating <cit.>; furthermore, even for point sources slight relative mis-alignment of the gratings would result in a greater loss of flux from the SL slit. In 10 cases we apply a multiplicative factor $>$1 to the SL spectrum to bring it into the agreement with the LL spectrum; only in 5 of these cases is the adjustment greater than 10%.
With these spectra in hand, we double-check their absolute flux calibrations. Because we primarily use PAH[11.3] fluxes in the analysis which follows, absolute flux calibration around this wavelength is particularly important. We convolve the spectra with the Wide-Field Infrared Survey Explorer (WISE) filter curves from <cit.>, obtain synthetic fluxes in the WISE-3 band (effective wavelength 11.6) and compare those with observed WISE-3 fluxes. They show excellent agreement, with the average ratio between synthetic fluxes and observed fluxes of 0.03 dex and the standard deviation among the 46 objects of 0.04 dex. We therefore take 0.05 dex (12%) to be the absolute flux calibration uncertainty for these sources.
To calculate PAH fluxes, we cut out $\la$3-wide chunks of the spectrum and model them using a polynomial continuum and Drude profiles with profile shapes and widths taken from <cit.>. Drude (or damped harmonic oscillator) profiles arise in the Drude theory of conductivity <cit.> and are found to be very well matched to ultra-violet and infrared opacity curves of small dust particles <cit.>. For PAH complexes, such as those at 11.3 and at 7.7, the relative amplitudes of the components within the complex are fixed to their ratios in the template spectrum of normal star-forming galaxies <cit.>. For example, within the 11.3 complex the amplitude ratio of the 11.23 and 11.33 components is fixed to 1.25:1. Depending on the model for the local continuum, from a constant to a cubic polynomial, the number of fit parameters varies from two to five, with the amplitude being the only parameter that describes the intensity of the PAH feature (since the functional shape of the feature remains fixed). Example fits are shown in Figure <ref>. Overall the 11.3 and the 6.2 features are reproduced well, but the quality of fits of the 7.7 feature is poor. Contributing factors are a strong [NeVI]7.652 emission line blended with the PAH complex and a poorly anchored continuum, whose shape is complicated by silicate absorption. We therefore do not use the results from the PAH[7.7] fits.
The mid-infrared continua of obscured quasars show a wide range of behavior of the silicate feature centered at 9.7, from deep absorption to occasional emission <cit.>. We measure the apparent strength of silicate absorption defined as $S[9.7\micron]=-\ln(f_{\rm obs}[9.7\micron]/f_{\rm cont}[9.7\micron])$, where $f_{\rm obs}$ is the observed flux density at 9.7 and $f_{\rm cont}$ is the estimate of silicate-free continuum obtained by power-law interpolation between $5.3-5.6\micron$ and $13.85-14.15\micron$. Negative values of $S[9.7\micron]$ indicate silicate emission, while positive values indicate absorption, with $S[9.7\micron]\ga 1$ for the 10% most absorbed sources. This method is similar to that used by <cit.>, except we do not use a continuum point at 7.7 even in the cases of weak PAH emission. The apparent strength of Si absorption is closely related to, but not identical to the optical depth of Si dust absorption; depending on the poorly known continuum opacity of the dust at these wavelengths, the actual optical depth is $\simeq (1.2-1.5)\times$ the apparent strength of the Si feature <cit.>.
Forbidden lines [NeII]$\lambda$12.81, [NeIII]$\lambda$15.56, [NeV]$\lambda$14.322, [NeVI]$\lambda$7.652 and [SIV]10.51 are measured by fitting Gaussian profiles plus an underlying linear continuum. As these lines are not spectrally resolved, their widths $\sigma$ in the observer's frame are fixed to the order-dependent instrumental resolution tabulated by <cit.>. We cut out a $3\sigma-$wide piece of spectrum centered on the emission line in question and perform a three-parameter fit, with two parameters describing the continuum and one parameter for the line amplitude. We allow for an 0.03 variation in the line centroid to account for the wavelength calibration uncertainty <cit.>. Because our fits for PAH emission features and forbidden emission lines are linear in all parameters, we use the standard error as the estimate of the standard deviation of the parameter estimate.
In addition to the 46 type 2 quasars with spectra, we use 115 IRS spectra for all type 1 quasars from <cit.>. As was described in the beginning of Section <ref>, 90 of these are optically selected blue PG quasars and 25 are near-infrared-selected quasars of varying optical types. The IRS sample was assembled from several dedicated programs and archival search as described by <cit.>, and their spectra were analyzed in detail using methods similar to ours. In each of the three subsamples (blue type 1 quasars, red type 1 quasars, type 2 quasars) the detection rate of the 11.3 PAH feature is $\sim$ 50%, and we use upper limits on PAH fluxes in the remaining objects.
§.§ Radio data
We cross-match all objects with spectroscopic or photometric infrared data with the Faint Images of Radio Sky at Twenty cm survey (FIRST; ) within 3 of the optical position. FIRST used the Very Large Array to produce a catalog of the radio sky at 1.4 GHz with a resolution of 5, subarcsec positional accuracy, rms sensitivity of 0.15 mJy and catalog threshold of $\sim 1.0$ mJy for point sources. When a source is covered by the FIRST data but there is no catalog detection, we estimate the flux density upper limit as 5$\times$rms flux density at source position$+0.25$ mJy, with the last term included to correct for the CLEAN bias <cit.>.
In cases of no FIRST coverage (7% of type 2 quasars and 25% of type 1 quasars), we use the NRAO VLA Sky Survey (NVSS; ), which is a 1.4 GHz survey covering the entire sky north of $-40\deg$ with a resolution of 45, positional accuracy of better than 7, rms sensitivity of $\sim 0.4$ mJy and catalog threshold of $\sim 2.3$ mJy. We match within 15 of the optical position and in case of non-detections, calculate the upper limit as 5$\times$rms flux density at source position$+0.4$ mJy <cit.>.
Our matching procedure implies that for extended radio sources – a minority of our sample – we are sensitive only to the core fluxes, not to the extended lobes. Inclusion of lobe emission would increase the observed radio luminosities quoted in this paper, but only for a minority of sources. Most objects are point-like at the resolution of FIRST and NVSS <cit.>, with only 10%-20% sources (both in the type 2 and the type 1 samples) showing integrated fluxes significantly higher than peak fluxes. The radio detection rates are 73% for the type 2 sample with far-infrared photometry, 85% for the type 2 sample with IRS spectroscopy, and 59% for the type 1 sample.
All radio luminosities quoted in this paper are K-corrected to rest-frame 1.4 GHz using equation
\begin{equation}
L_{\rm radio,obs}\equiv\nu L_{\nu}[1.4{\rm GHz}] =4\pi D_L^2 \nu F_{\nu}(1+z)^{-1-\alpha},
\end{equation}
where $\nu=1.4$ GHz, $D_L$ is the luminosity distance, $F_{\nu}$ is the observed flux density at 1.4GHz, and $\alpha$ is the power-law spectral index defined as $F_{\nu}\propto \nu^{\alpha}$. Radio-quiet quasars at $z\sim 0.5$ are too faint to be detectable by any large radio surveys other than FIRST and NVSS, which have data only at 1.4 GHz, so we cannot measure $\alpha$ from archival data. Values between $-0.5$ and $-1.0$ for the radio-quiet population were suggested in the literature <cit.>; unless specified otherwise, we assume $\alpha=-0.7$. For a source at $z=0.5$ with a fixed observed flux density $F_{\nu}$, varying $\alpha$ from -0.7 in its typical range between -0.5 and -1 results in a 10% uncertainty in $L_{\rm radio,obs}$.
§ STAR FORMATION RATES OF QUASAR HOSTS FROM PHOTOMETRY
Dust that produces infrared emission of quasars and star forming galaxies is heated by the radiation from the accretion disk or from young stars. Because of the very high optical depths involved, all incoming radiation at optical and ultraviolet wavelengths is absorbed in a thin layer close to the source of the emission and then thermally reprocessed thereafter. Therefore, it is unlikely that any differences between radiation fields in active and star forming galaxies may be responsible for the noticeable differences in the infrared spectral energy distributions.
Instead, the biggest difference between quasar-heated and star-formation-heated dust is that the latter is distributed over the entire galaxy on $D_{\rm gal}\ga 1$ kpc scales, whereas dust heated by an AGN is concentrated on scales $D_{\rm qso}\la 10$ pc even in luminous objects <cit.>. A star-forming galaxy and an active nucleus of similar bolometric luminosities ($L\propto D^2 T^4$) would have different characteristic dust temperatures, $T_{\rm gal}/T_{\rm qso}\sim \sqrt{D_{\rm qso}/D_{\rm gal}}\sim 0.1$. This crude scaling is borne out by far-infrared observations of star-forming galaxies, whose characteristic temperature is $T_{\rm gal} \simeq 25$ K, and of AGN, where the bulk of the thermal emission is produced with $T_{\rm qso}\gg 100$ K <cit.>. Beyond this basic temperature distinction, a variety of shapes of the spectral energy distributions can be produced due to the differences in the geometric distribution of dust (compact vs diffuse, spherical vs non-spherical, clumpy vs non-clumpy, etc.), its amount, and its orientation relative to the observer <cit.>.
Because of the steep decline of the modified black body function at wavelengths greater than those that correspond to the thermal peak, in composite sources with similar contributions from the active nucleus and the star forming host galaxy the mid-infrared emission tends to be dominated by the active nucleus and the far-infrared emission ($\lambda\ga 100\micron$) is dominated by star formation <cit.>. But in quasars even the longest wavelength emission probed by observations can be dominated by emission from hot (presumably quasar-heated) dust <cit.>.
Our approach is therefore to use far-infrared observations to obtain strict upper limits on the quasar hosts' star formation rates. To minimize the contribution from the quasar – insofar as it is possible – we use the longest wavelength observations available to us. In practice, we use 160 data either from or from . We then assume that all of the observed far-infrared emission is due to star formation, and calculate the corresponding star formation rates and the expected radio luminosities <cit.>. This predicted radio luminosity is an upper limit on the amount of radio emission that can be generated by star formation. We then compare these predictions with the observed radio emission. By using a variety of templates to estimate star formation rates, we ensure that our results are robust to varying the assumed spectral energy distribution of a star-forming galaxy.
Finally, our measurements are predicated on the assumption that the far-infrared fluxes in star-forming galaxies are measuring the instantaneous rates of star formation. This is not always true <cit.>, in that previously formed stars can continue to illuminate left-over dust even after star formation rates have declined. But because this effect results in an over-estimate of star formation rate when using far-infrared fluxes, it is consistent with our upper-limit approach.
The main result of this section is presented in Figure <ref> which demonstrates that the radio emission due to star formation in the quasar hosts is inadequate – by almost an order of magnitude – to explain the observed radio emission. Below we describe in detail the steps involved in this comparison and in the cross-checking of this result we performed using a variety of methods.
Left: Results from the far-infrared photometric data from and 160 observations of type 2 quasars from <cit.>. We plot radio luminosity expected due to star formation in the quasar hosts vs the observed radio luminosity. Four points corresponding to radio-loud sources with $L_{\rm radio,obs}>3\times 10^{41}$ erg s$^{-1}$ and $L_{\rm radio,SF}<10^{41}$ erg s$^{-1}$ are off the scale of the plot to the right. Vertical bars show that all points represent upper limits on host star formation rates (and thus upper limits on the associated expected radio luminosity). Grey points with horizontal error bars denote points that are not detected by FIRST / NVSS, whereas black symbols correspond to radio detections. In the cases where radio emission is detected (74% of the objects), star formation is insufficient to account for the observed radio emission, with the median $\log(L_{\rm radio,obs}/L_{\rm radio,SF})=1.0$. Right: Same calculation for type 1 quasars. Blue points are for PG quasars (predominantly data from ) and red for 2MASS quasars (MIPS-160 data) detected in the radio, and grey points are for radio non-detections. For these objects, the median $\log(L_{\rm radio,obs}/L_{\rm radio,SF})=1.1$. Uncertainties in radio fluxes as less than 15%.
§.§ The infrared-radio correlation of star-forming galaxies
The key to making an accurate comparison between the observed radio luminosity and that predicted from star formation in the host galaxy is a careful calibration between the far-infrared luminosities, star formation rates and radio luminosities due to star formation in star-forming galaxies without an active black hole. The strong correlation between these values is due to massive young stars which dominate the ultraviolet continuum most easily absorbed by interstellar dust, resulting in a `calorimetric measure' of star formation rates <cit.>. The same young stars explode as supernovae, resulting in acceleration of cosmic rays which produce the observed radio emission <cit.>.
The tightest correlation is between the total infrared luminosity of star formation (by convention, often integrated between 8 and 1000, $L_{\rm 8-1000\micron, SF}\equiv L_{\rm IR,SF}$) and radio luminosity, which we take from Figure 3 of <cit.>:
\begin{equation}
\log L_{\rm radio,SF}{\rm [erg\, s^{-1}]}=26.4687+1.1054 \log (L_{\rm IR, SF}/L_{\odot}).\label{eq_ir_radio}
\end{equation}
In this equation and hereafter, the radio luminosity is the monochromatic luminosity at rest-frame 1.4 GHz, $L_{\rm radio}\equiv \nu L_{\nu}$[1.4GHz].
To reassure ourselves that this relationship applies to galaxies with a wide range of star formation rates – including the high star formation rates we confront in infrared-luminous sources discussed in this paper – we double-check this conversion against data for two samples of star-forming galaxies analyzed completely independently from <cit.> by several different groups. At low luminosities, we use nearby galaxies from <cit.>. At high luminosities, we use the Great Observatories All-Sky Luminous Infrared Galaxy Survey (GOALS; ). In both cases, we take advantage of the 8-1000 infrared luminosities tabulated by <cit.> and <cit.> and obtain radio luminosities from the NVSS.
Because the GOALS sample may contain active galactic nuclei, we restrict our comparison to those objects that have mid-infrared classifications consistent with pure star formation, by requiring the rest-frame equivalent width (EW) of the polycyclic aromatic hydrocarbon emission at 6.2 to be above 0.3 <cit.>. Although not a perfect diagnostic, this measure is reasonably well correlated with ionization-line diagnostics of AGN activity <cit.>. Furthermore, many GOALS galaxies are found in mergers, and the total infrared luminosities of these sources include all components found within the $\sim 5\arcmin$ beam of the Infrared Astronomical Satellite <cit.> which is significantly larger than the NVSS beam ($\sim 45\arcsec$). Thus, in widely separated mergers the infrared emission could include multiple interacting components, while the radio emission would be coming from only one of them. To make sure we compare fluxes from similar apertures, we further restrict our comparison to objects that are either in single non-interacting hosts or in late-stage mergers (`N' and `d' classifications of ), excluding pairs and triples at all interaction stages. For these objects, using NVSS fluxes ensures that the total radio emission is taken into account.
Overall we find good agreement between the infrared-radio correlation reported by <cit.> and that displayed by these two samples of star-forming galaxies which sample three orders of magnitude in infrared luminosity. Given a measurement of the total infrared luminosity, the standard deviation of the radio luminosity of two samples around the best-fit correlation is 0.16 dex (for galaxies) and 0.24 dex (for GOALS galaxies), which we take to be the practical measure of the uncertainty in the $L_{8-1000\micron}$-radio correlation.
In quasars, the total infrared luminosity may be dominated by the activity in the nucleus, and without many additional assumptions we cannot obtain a measurement of total infrared luminosity due to star formation alone. Thus our challenge is to make the best guess of the upper limit on the star formation rate and on the associated radio emission from just one photometric datapoint at 160. To this end we use the calibrations presented by <cit.> for star forming galaxies derived from deep MIPS data:
\begin{eqnarray}
\log (L_{\rm IR, SF}/L_{\odot}) = 1.16 + 0.92 \log (\nu L_{\nu}[70\micron]/L_{\odot});\label{eq_sym1}\\
\log (L_{\rm IR, SF}/L_{\odot}) = 1.49 + 0.90 \log (\nu L_{\nu}[160\micron]/L_{\odot}).\label{eq_sym2}
\end{eqnarray}
These relationships are well calibrated even in the highly star-forming regime.
Thus our calculation of the expected radio emission due to star formation involves the following steps. We use equations (<ref>)-(<ref>) to convert a far-infrared photometric detection to the total luminosity of star formation, and then we use equation (<ref>) to derive the expected radio luminosity. To verify that the scaling relationships give the correct answer for star forming galaxies, we apply this method to the GOALS sample in Figure <ref>. We use 160 photometry from <cit.>. Since these authors concentrate on the nearby ($z<0.083$) subsample, we can use equation (<ref>) directly without any need for k-corrections.
Comparison between predicted radio emission due to star formation obtained from 160 fluxes from scaling relationships (<ref>) and (<ref>) and the observed radio emission for GOALS galaxies <cit.>, most of which are dominated by star formation. Black points show 160 detections, with 0.24 dex error bars which reflect the conversion of total infrared luminosity to predicted radio due to star formation for GOALS galaxies and grey points show 160 upper limits. We find excellent agreement between the observed and predicted radio luminosities in galaxies with 160 detections, with a standard deviation around the 1:1 relationship (dotted line) of 0.18 dex and a mean difference of 0.03 dex. The three most significant outliers below the dotted line all contain active nuclei, which presumably contribute excess radio emission over that associated with star formation alone.
We find an excellent agreement between the observed radio luminosity and that predicted from the 160 flux via the scaling relationships. Only three points show a significant ($>0.4$ dex) excess of radio emission over that predicted from the 160 fluxes, and all three turn out to contain luminous active nuclei (MCG-03-34-064 is a Seyfert 1 galaxy, and NGC 5256 and NGC 7674 are Seyfert 2s, ) which likely contribute radio emission in excess of that due to star formation in the host. Excluding these three sources, we find that the median / average ratio of the observed-to-predicted flux is 0.01 / 0.03 dex, and the standard deviation around the 1:1 relationship is 0.18 dex. Therefore, we assume that the typical uncertainty in our method of predicting radio emission due to star formation from 160 band fluxes is about 0.2 dex, which is the combination of the standard deviation around the correlation and the typical photometric error of 160 observations (20-25%, or $<0.1$ dex). Encouraged by such excellent agreement between predicted and observed radio fluxes in luminous star-forming galaxies, we apply the same method to quasars in the next section.
§.§ Radio emission in quasars is not due to star formation
We use the observed 160 fluxes (or upper limits) of the quasars in our samples to derive the upper limit on their total infrared luminosity due to star formation using equations (<ref>)-(<ref>). Because the relations are given at rest-frame 70 and 160, and the spectral slopes of our targets are unknown, instead of performing k-corrections on the data we linearly interpolate the slopes and the normalizations of equations (<ref>)-(<ref>) between rest-frame 70 and 160 depending on the redshift of each target, thereby establishing a relation between monochromatic luminosity and the total luminosity of star formation at rest-frame wavelength of 160/$(1+z)$. We then use equation (<ref>) to derive an upper limit on the radio emission due to star formation and compare with the observed amount.
The results for both type 1 and type 2 quasar samples are shown in Figure <ref>. Unlike GOALS galaxies in Figure <ref>, almost all quasars in our sample show significantly higher radio luminosities than those expected from star formation (and furthermore the predicted radio emission is a strict upper limit on the star formation contribution, as reflected in the one-sided error bars, as not all of the 160 continuum is due to star formation). Among the quasars which have strong detections in the radio, the median ratio between observed radio luminosity and that expected from star formation is an order of magnitude: $\log (L_{\rm radio,obs}/L_{\rm radio,SF})=1.1$ for type 1 quasars and $1.0$ for type 2s. The standard deviation of this ratio is 0.7 dex for type 2 quasars and 1.1 dex for type 1s, though in the latter case the ratio is not log-normally distributed and the standard deviation is artificially inflated by eight radio-loud sources with $L_{\rm radio}\ge 10^{41}$ erg s$^{-1}$. Removing these, we find a standard deviation of 0.7 dex for type 1s as well. In any case, the standard deviation is much greater than the $\sim$ 0.2 dex standard deviation in the calibrations of star formation rates in star-forming galaxies and the $\sim 0.1$ dex 160 flux uncertainty for detections.
Our main conclusion is that the star formation in quasar hosts falls short of explaining the observed radio emission in quasars by about an order of magnitude. This is consistent with the study by <cit.> who found, using the spectral energy decomposition methods, that the observed radio luminosities were in all cases well above the calculated star formation component in their sample. Quasars in their sample are somewhat less luminous than ours, so the effect is likely more pronounced in our case: at the same star formation rate <cit.>, an increase in the quasar contribution would make the radio/160 ratio more discrepant from that measured in star forming galaxies.
It is more difficult to draw conclusions from the radio non-detections in the FIRST and / or NVSS surveys (shown as grey points in Figure <ref>), because in this case both the predicted radio luminosity due to star formation and the observed luminosity are only available as upper limits. Nonetheless, these objects do not alter our main conclusion. We have stacked the FIRST images of the non-detected sources in Figure <ref>, left, and obtained a strong point source detection with a mean peak flux of 0.4 mJy/beam. This estimate is in excellent agreement with our previous finding in Stripe 82 <cit.>, where we were able to detect all FIRST-undetected sources in a more sensitive survey <cit.> with fluxes about a factor of 2 below the limit of the FIRST survey ($\sim 1$ mJy). If in Figure <ref> all sources without radio detections have typical fluxes of 0.4 mJy, then we can again calculate the excess of observed radio emission over the upper limit on radio emission from star formation, which we find to be $\log (L_{\rm radio,obs}/L_{\rm radio,SF})=0.59$ for type 1 quasars and $=0.74$ for type 2s.
To make sure that our results are robust to changes in the spectral energy distribution of star formation, we use several star formation templates available in the literature to recalculate the total infrared luminosity of star formation. We use seven templates, five from nearby star forming galaxies by <cit.> and two from $z=1-2$ star forming galaxies by <cit.>. We scale the templates (properly adjusted for redshift) to reproduce the observed 160 fluxes of our sources, with one fitting parameter – the overall luminosity of the template. For each template, we obtain the total infrared luminosity $L_{\rm IR, SF}$ due to star formation by integrating the scaled template between 8 and 1000, and of the seven results we pick the highest one, in keeping with the strict upper limits approach, which we convert to the expected radio luminosity <cit.>. The results are qualitatively similar to those obtained via scaling relations from <cit.> and shown in Figure <ref>, though the star formation rates obtained using the template method are systematically higher by about 0.2 dex. The reason for this is that we pick the most conservative template – the one that gives us the highest star formation rate at a given 160 flux. Even with this method, the observed radio luminosities of quasars are in excess of those predicted from star formation, with $\log (L_{\rm radio,obs}/L_{\rm radio,SF})=0.6$.
§.§ Contribution of the active nucleus to the far-infrared flux
Figure <ref> shows the comparison between the spectral energy distribution of one of our obscured quasars and the star-forming galaxy templates. The spectral energy distribution is assembled from photometric data from SDSS, WISE and , and the seven star formation templates <cit.> are scaled to match the 160 observation. The spectral energy distribution of this object peaks at significantly shorter wavelengths (between 10 and 20) than that of any of the star formation templates (between 50 and 150); this is typical of our targets.
The spectral energy distribution of one of the obscured quasars from <cit.> and <cit.>. Seven star-formation templates from <cit.> and <cit.> are scaled to match the longest wavelength point. While it is clear that the overall spectral energy distribution of the object is inconsistent with any of the star formation templates (with excess emission at mid-infrared wavelengths likely due to quasar-heated dust), we can use the longest wavelength detection to place strict upper limits on the star formation rate in the quasar host galaxy.
In Figure <ref>, left, we show infrared colors of our sources (black) compared with those of template star-forming galaxies (red) which are placed at the same redshift range as our targets ($z=0.24-0.73$). Quasars from our sample have noticeably warmer / bluer colors in the infrared than do star-forming galaxies. In principle, if we knew the spectral energy distribution of a `pure quasar' (i.e., a component that included the circumnuclear obscuring material where heating is dominated by the nucleus but excluded the larger host galaxy where heating is dominated by the stars) then from the observed colors of our objects we could determine the fractional contribution of the AGN and the host galaxy to each spectral energy distribution.
Left: Infrared colors of type 2 quasars: black points and upper limits for and observations of type 2 quasars detected at 70 in our targeted deep observations. Dotted red lines show infrared colors of seven star formation templates <cit.> placed at $z=0.24-0.73$. Solid blue lines show the same for six obscured quasar templates: A – `Hot DOGs' <cit.>, B – `featureless AGN' <cit.> and `obscuring torus' <cit.>, C – `type 2 quasar' <cit.>, D – `silicate AGN' <cit.> and E – Mrk231 <cit.>. Magenta curves with dots show the color locus of the linear combinations, for three different redshifts, of a star formation template with the HotDOG template. Dots mark 0, 10%, 20%, etc. contribution to the total 8-1000 infrared luminosity. All model colors include convolution of the templates with filter curves. Right: For three different redshifts (from top to bottom, $z=0.24$, 0.49 and 0.73 – the bracketing redshifts of our sample, plus one value in the middle of the range), the relationship between the SF contribution to the apparent 160 flux as a function of its contribution to the bolometric luminosity. Even when star formation contributes only 20% to the bolometric luminosity, over half of the apparent 160 flux is due to star formation for the redshifts of our sample.
To this end, we collect obscured AGN templates from the literature, including three from SWIRE <cit.>: Mrk231 which has a power-law-like spectral energy distribution in the mid-infrared, `obscuring torus' which has a steep cutoff both at short and long wavelengths, and `type 2 quasar', which is obtained by heavy reddening of a type 1 quasar spectral energy distribution. These are supplemented by two more templates from <cit.>: `featureless AGN' that do not show silicate absorption, and `silicate AGN' which do. Finally, we also include the median spectral energy distribution of hot dust-obscured galaxies (HotDOGs) from <cit.>. The observations of these extremely luminous high-redshift obscured quasar candidates cover rest-frame wavelengths $\la 100$, so in order to compute the 160/70 colors we extrapolate the HotDOG template using a modified Rayleigh-Jeans spectrum with $\beta=1.5$ <cit.> beyond 100. The model colors of AGN templates at $z=0.24-0.73$ are shown with blue lines in Figure <ref>.
Intriguingly, half of the AGN templates (C – `type 2 quasar', D – `silicate AGN', and E – Mrk231) are redder / colder than the observed colors of type 2 quasars. It is likely that a significant fraction of the total luminosity of these templates is due to the host galaxy instead of the active nucleus, so their colors are between those of the type 2 quasars in our sample and those of the star forming galaxies. `Featureless AGN' and `torus' templates (they have similar colors; marked B) and `HotDOGs' (marked A) have colors that are much closer to those observed in our sample.
The magenta curves mark a linear combination of one of the star formation templates with the HotDOG template, going from 100% of the 8–1000 luminosity dominated by star formation to 100% dominated by the HotDOG template. The colors of our type 2 quasars are roughly consistent with such linear combinations if the quasar contributes at least half of $L_{\rm IR, 8-1000\mu m}$. Thus the observed infrared colors of type 2 quasars suggest that the bolometric luminosities of our objects are likely dominated by the quasar, not star formation in the host galaxy.
However, because the spectral energy distribution of the quasar template declines so steeply beyond the peak, even a small fractional contribution of star formation is sufficient to dominate the observed 160 flux, as shown in Figure <ref>, right. As little as 20% contribution of star formation to the total infrared luminosity $L_{\rm IR, 8-1000\mu m}$ is sufficient for it to contribute more than 50% of the 160 flux at the redshifts of our sample. The lower the redshift, the longer is the rest wavelength probed by the 160 observations, and the smaller the contribution of star formation required to dominate at that wavelength.
Unfortunately, these calculations do not allow us to unambiguously decompose the infrared spectral energy distribution of our sources into a quasar and star formation component, and to turn our upper limits on star formation rates into actual measurements of star formation rates. The primary reason is that the decomposition is sensitive to the assumed template for the quasar contribution, which is clear from the diversity of colors of AGN templates in Figure <ref>, left. A slight shift of the peak of the AGN template to longer wavelengths results in a larger contribution of the AGN to the 160 flux and to a smaller required contribution of star formation. The AGN templates are in turn sensitive to the geometry of the obscuring material (smooth vs clumpy, geometrically thin vs geometrically thick) and the relative orientation of the observer to the obscuring structure <cit.>.
<cit.> conduct detailed spectral energy distribution decomposition of a subsample of 20 type 2 quasars from <cit.> with HST observations most of which are also presented in this paper. The average and the standard deviation of the luminosities of the 20 objects are $L_{\rm [OIII]}=10^{43.1\pm 0.4}$ erg s$^{-1}$ and $\nu L_{\nu}$[12]$=10^{45.0\pm 0.5}$ erg s$^{-1}$, so they represent the luminous end of the objects probed in this paper. Spectral energy distribution fits with CIGALE <cit.> and DecompIR <cit.> suggest that in this subsample, the bolometric luminosities are dominated by the AGN (derived AGN fractions are $0.7\pm 0.2$ and $0.8\pm 0.15$ with the two methods, respectively). Nonetheless the median contribution of star formation to the observed 160 flux is 91%, and the star formation rates derived by <cit.> are therefore similar to those we present here as upper limits. These conclusions are in agreement with our analysis based on far-infrared colors. While we continue treating our 160-derived star formation rates as upper limits, we keep in mind that they are likely close to the actual star formation rates, even though the bolometric luminosities of our sources are dominated by the quasar.
§.§ Star formation rates of quasar hosts
We compare the upper limits on star formation rates among the different subsamples of quasars discussed in this paper in Figure <ref>. To convert from infrared luminosities of star formation to star formation rates, we use the calibration from <cit.> which is slight modification of that of <cit.> and assumes Salpeter initial mass function. In this section we make a distinction between radio-quiet and radio-loud AGN by applying a simple luminosity cut $\nu L_{\nu}[1.4{\rm GHz}]=10^{41}$ erg s$^{-1}$ <cit.>. All sources (except one) with radio luminosities above this cutoff are detected in FIRST / NVSS, so we do not have to worry about upper limits on radio detections in potentially radio-loud sources. Of the 186 type 1 and type 2 quasars with data, twelve are radio-loud by this criterion.
Distributions of upper limits on infrared luminosities of star formation as derived from 160 fluxes. For radio-quiet type 1 quasars (magenta, sparsely shaded), the median (average) and the standard deviation are $\log (L_{\rm IR,SF,upper}/L_{\odot})=10.90(10.86)\pm 0.58$, corresponding to the median upper limit on star formation rate of 11.3$M_{\odot}$ yr$^{-1}$. Just for the blue type 1 quasars with deep observations <cit.>, we find $\log (L_{\rm IR,SF,upper}/L_{\odot})=10.67(10.72)\pm 0.57$ and 6.3$M_{\odot}$ yr$^{-1}$. For radio-quiet type 2 quasars (solid black), $\log (L_{\rm IR,SF,upper}/L_{\odot})=11.24(11.22)\pm 0.54$. Excluding shallow observations which include a lot of 160 non-detections and using only observations (grey, densely shaded), we find $\log (L_{\rm IR,SF,upper}/L_{\odot})=11.03(11.04)\pm 0.47$, corresponding to median upper limit on star formation rate of 18.1$M_{\odot}$ yr$^{-1}$. Type 1 and type 2 radio-loud sources (green, solid fill) with observations nominally show $\log (L_{\rm IR,SF,upper}/L_{\odot})=11.42(11.52)\pm 0.52$, but their 160 fluxes can be boosted by synchrotron emission and thus are unreliable measures of star formation.
The first striking result is that the nominal star formation rates are higher in radio-quiet type 2 quasars than in radio-quiet type 1s. Part of this is due to the heterogeneity of our sample: a third of the type 2 quasar sample was observed with -160, and these data have shallower observations and higher confusion limits than -160. As a result, 39 out of 51 type 2 quasars observed with -160 are not detected. To make a better-matched comparison between the two samples, we consider only type 2 quasars observed with -160, the majority of which are detected. These objects still show appreciably higher star formation rates (or rather, upper limits on star formation rates) than type 1s with similarly deep observations and similar intrinsic luminosities and redshifts. Specifically, the median upper limits on star formation rate derived for hosts of blue type 1 quasars is 6$M_{\odot}$ yr$^{-1}$, whereas that of type 2 quasar hosts is 18$M_{\odot}$ yr$^{-1}$. (The nominal median upper limit on star formation in red type 1 quasars is even higher, 32$M_{\odot}$ yr$^{-1}$, but it is based on shallower MIPS-160 observations, with a third of the sample undetected.) As we discuss in Section <ref>, even though our method technically only allows us to place an upper limit on the star formation rate, the actual values are likely close to the derived upper limits ($\sim 90\%$), so in the following discussion we make the assumption that these star formation upper limits are representative of the actual star formation rates.
It is now well-established that star formation rates in obscured quasars are higher than those in unobscured ones <cit.>. This result appears to hold whether star formation is calculated from photometric or spectroscopic indicators. This conclusion is not well-explained by the classical orientation-based unification model, in which type 1 and type 2 quasars should occupy similar host galaxies. The difference in host star formation rates may be due to evolutionary effects: this observation could represent direct evidence that type 2 quasars are more likely to be found in dust-enshrouded environments characteristic of an ongoing starburst <cit.>, as suggested by many models of galaxy formation <cit.>.
An alternative explanation for a higher star formation rates in hosts of type 2 quasars is that the selection of these objects is biased toward gas-rich galaxies. It is usually assumed that AGN obscuration happens on circumnuclear scales ($\ll 1$ kpc), and it is not clear whether AGN obscuration is directly connected with the geometry of the host galaxy. In type 2 quasars, which occupy predominantly elliptical hosts, no clear relationship emerges between the presence or absence of the galactic disk and obscuration and their relative orientation <cit.>. However, in less luminous type 2 AGN there are indications that at least some of the obscuration is occurring on the galactic scales by the gas and dust in the galaxy disk <cit.>. If this is a typical situation in type 2 quasars, then they would be preferentially found in more gas-rich and by extension more star-forming galaxies than type 1s.
Finally, it is possible that type 1 and type 2 quasars discussed in this paper have different bolometric luminosities. While they have similar values of $\nu L_{\nu}$[12], this measure could underestimate the luminosities of type 2 quasars which show evidence for obscuration even at mid-infrared wavelengths <cit.>. Therefore, if the type 2s are significantly more luminous than the type 1s, and if the host star formation rate increases with quasar luminosity, then the observed difference in the host star formation rates may be due to the intrinsic differences between the two samples, but this scenario is unlikely given the very flat relationship between the AGN luminosity and the host star formation rates <cit.>. Another possibility is that intrinsically more luminous type 2s (with apparent mid-infrared luminosities similar to those of type 1s) dominate their 160 fluxes, resulting in higher measured nominal star formation rates, but that is also unlikely in light of the spectral energy decomposition results by <cit.> who find that 91% of 160 emission is due to star formation even in the luminous quasars as discussed in the previous section.
Far-infrared star formation indicators are dominated by obscured star formation and may therefore miss unobscured star formation, which is normally estimated from the ultra-violet luminosities. Ultra-violet measures of star formation are extremely difficult to obtain in type 1 quasars, where the quasar in the nucleus of the galaxy makes identification of circumnuclear star formation all but impossible. Off-nuclear stellar populations, when measured with proper accounting for possible quasar light scattered off the interstellar medium into the observer's line of sight, tend to show post-starburst features <cit.> rather than active star formation. In type 2 quasars, despite nuclear obscuration, ultra-violet emission is dominated by quasar scattered light <cit.>. When this contribution is accounted for, the median ultra-violet rates of star formation in type 2 quasar hosts are $\la 3M_{\odot}$ yr$^{-1}$ <cit.>, about an order of magnitude lower than those derived from far-infrared luminosities. This is in line with the typical balance between obscured and unobscured star formation found at $z<1$ <cit.>. Therefore, we conclude that unobscured star formation is unlikely to affect our results.
Another result apparent from Figure <ref> is that the nominal upper limits on star formation rates of radio-loud quasars appear to be higher than those of radio-quiet quasars, which is contrary to many previous studies demonstrating that of all types of AGN, these objects are associated with the lowest rates of star formation <cit.>. The reason for this discrepancy is that the 160 fluxes of these objects can be boosted by the synchrotron emission associated with the jet, not by the star formation in the host, particularly in type 1 quasars (four out of seven radio-loud objects) where jet emission might be expected to be beamed. As an example, we take a fiducial radio-loud quasar at $z=0.33$ with $\nu L_{\nu}$[1.4GHz]$=10^{43}$ erg s$^{-1}$ and spectral index of synchrotron emission of $\alpha=-0.5$ ($F_{\nu}\propto \nu^{\alpha}$). Such object would appear as a 2.3 Jy radio source at 1.4 GHz, and if its synchrotron spectrum continues all the way to the wavelengths of our far-infrared observations then its 160 flux due to the jet would be 61 mJy. With our one-band observations, we would calculate a star formation rate of 41 $M_{\odot}$ yr$^{-1}$. The exact contribution of synchrotron emission to the 160 flux depends sensitively on the spectral index and on whether the synchrotron spectrum continues all the way to these high frequencies. We conclude that upper limits on star formation rates derived from 160 data are not very meaningful for radio-loud objects; their star formation rates are likely lower than those reported in Figure <ref>.
§ STAR FORMATION RATES OF QUASAR HOSTS FROM SPECTROSCOPY
In this section we continue investigating star formation rates in quasar host galaxies (and the resulting radio emission), but now using mid-infrared spectroscopic data. We use polycyclic aromatic hydrocarbon features and low-ionization emission lines as potential star formation diagnostics.
§.§ Calibration of PAHs as star formation diagnostics
Polycyclic aromatic hydrocarbon (PAH) emission dominates mid-infrared spectra of star-forming and starburst galaxies and is typically seen as a star-formation indicator <cit.>. Various studies have presented calibrations between PAH luminosities and infrared luminosities of star formation or star formation rates <cit.>, but often either the PAH luminosities or the infrared luminosities or the star formation rates are not measured exactly the same way as what we do here, resulting in significant systematic uncertainties. Most notably, some authors use broad-band 8 fluxes <cit.> which likely overestimate the amount of PAH[7.7] emission. Others (e.g., ) measure PAH luminosities spectroscopically by subtracting an interpolated continuum, which likely results in an underestimate of the PAH fluxes since Drude-like PAH profiles <cit.> have significant flux in the wings. For example, for the 6.2 feature in ultraluminous infrared galaxies <cit.> we calculate the PAH flux using the Drude-fitting method, as well as by subtracting a linear continuum interpolated between 5.95 and 6.55, and we find that the former is a factor of 2 higher than the latter.
Of the PAH / star formation calibrations presented in the literature, the one that utilizes the measures closest to ours is presented by <cit.>. These authors calculate PAH luminosities by fitting Drude profiles to the PAH features, similarly to our procedure, and they find an appropriate star formation template with the same PAH luminosity to calculate the star formation luminosity between 8 and 1000 . Their final conversion between PAH[11.3] and $L_{\rm IR, SF}$ is well described by
\begin{equation}
\log({\rm PAH[11.3\micron]}/L_{\odot})=-0.7842+0.8759\log(L_{\rm IR,SF}/L_{\odot}).\label{eq_shi}
\end{equation}
This calibration can then be supplemented by the IR-to-radio correlation from <cit.> listed in equation (<ref>) to obtain a direct relationship between PAH luminosities and radio emission due to star formation:
\begin{equation}
\log L_{\rm radio,SF}{\rm [erg\, s^{-1}]}=27.4584+1.2620 \log ({\rm PAH[11.3\micron]}/L_{\odot}).\label{eq_pah}
\end{equation}
The infrared luminosities due to star formation can be converted to star formation rates <cit.>.
To double-check calibrations (<ref>)-(<ref>), we use nearby star-forming galaxies from <cit.> who tabulated $L_{\rm IR, SF}$ values measured from the Infrared Astronomical Satellite (IRAS; ) data. Of the 22 objects presented in their paper, we exclude four with the optical signature of an active nucleus. For the remaining 18, we download the reduced spectra from the Spitzer-ATLAS database <cit.>, convolve them with IRAS-12 and IRAS-25 filter curves from <cit.> to calculate synthetic IRAS fluxes and compare them with those observed. We then augment the spectra by a factor necessary to match the synthetic fluxes to the observed fluxes and use the resulting flux-calibrated spectra to measure PAH luminosities using our Drude-fitting methods. We also obtain the highest listed 1.4 GHz radio fluxes for these galaxies from the NASA/IPAC Extragalactic Database. We assume that any major flux discrepancies between the listed 1.4 GHz measurements are due to varying resolutions of the radio data; for these nearby sources, we prefer the lowest resolution, highest flux observations which are more likely to capture extended radio flux.
On the high-luminosity end, we use a subsample of GOALS galaxies from <cit.> dominated by star formation (EW of PAH[6.2] $>0.3$) and hosted by single or late-stage merger hosts <cit.>, as was described in Section <ref>. We further restrict our analysis to objects with silicate absorption strength $<1.1$, to avoid any biases in PAH measurements due to our approach of not correcting PAH fluxes for extinction. Although <cit.> present PAH flux measurements, we have found that even minor differences between their and our fitting procedures result in a noticeable systematic offset in PAH/$L_{\rm IR,SF}$ and PAH ratios. Therefore, we re-measure PAH features in these objects using exactly the same tools as those used for type 1 and type 2 quasars. We download the IRS enhanced data products from the Heritage Archive, stitch together the SL and LL orders, flux-calibrate using IRAS-12 and IRAS-25 fluxes and measure PAHs using our Drude-fitting methods. The radio fluxes of these sources are available in <cit.> and in NVSS. Flux calibration of <cit.> and <cit.> spectra against IRAS data is necessary only because some of these objects are so nearby that they are extended beyond the IRS slits. For type 2 quasar spectra, most of which appear as point sources to , the default flux calibration provided for the enhanced data products of IRS is in excellent agreement with and WISE photometry, and the flux calibration step is unnecessary.
As seen in Figure <ref>, we find good agreement between the scaling relationships (<ref>) and (<ref>) and the actual measurements in these two samples of star-forming galaxies. Given a measurement of PAH[11.3], our adopted calibrations (<ref>) and (<ref>) predict the total infrared and radio luminosities for star-forming galaxies with a standard deviations of $\la 0.3$ dex. Thus the PAH vs star formation conversion appears to have somewhat higher intrinsic spread than the conversion between 160 flux and $L_{\rm IR,SF}$ and $L_{\rm radio, SF}$, which has a standard deviation of $\la 0.2$ dex. Much of this spread likely reflects the intrinsic dispersion of galaxy properties, as the PAH features are strongly detected in all sources, the quality of the data are high, and PAH measurement uncertainties are only a few per cent for this sample.
Double-checking the PAH vs infrared and radio correlations for star-forming galaxies. The solid lines show our adopted calibrations (<ref>) and (<ref>). Crosses show star-forming galaxies from <cit.>, in black for results obtained using IRAS-12 for bolometric flux calibration and in grey using IRAS-25. Circles show the star-forming subsample of the GOALS galaxies from <cit.>. Star-forming galaxies of low and high luminosities show a good agreement with our adopted calibrations, with standard deviations $\la 0.3$ dex.
§.§ PAH measurements of star formation in quasar hosts
As the sources in our sample are quasars with $L_{\rm bol}\ga 10^{45}$ erg s$^{-1}$, their continuum emission in the mid-infrared is dominated by the thermal emission of quasar-heated dust, with PAH features sometimes visible on top. In this Section, we start by using the 11.3 PAH feature exclusively and in Section <ref> we discuss the reliability of this measure. In Figure <ref>, left, we show the predicted radio luminosity due to star formation for type 1 quasars analyzed by <cit.>, and in the right panel we show the same for type 2 quasars.
Results from PAH-based measures of star formation. Left: Radio luminosities expected from star formation and the observed radio luminosities for PG (solid blue circles) and 2MASS (open red circles) quasars from the sample of <cit.>. Objects not detected in the radio are shown in grey. Right: same for the type 2 quasars from <cit.>. The majority of all quasars lie below the 1:1 dashed line, indicating that star formation in quasar host galaxies is insufficient to account for the observed radio emission from quasars, by 1.1-1.3 dex on average.
With the exception of a handful of problematic objects, our task for the <cit.> sample is straightforward: we take their measured PAH-derived total luminosities of star formation, convert them to the expected radio luminosity of star formation (eq. <ref>) and compare with the observed values. We use most of their $L_{\rm IR, SF}$ values, with the following exceptions. We add an upper limit to star formation for PG 0003+158, where we use their upper limit on PAH[11.3]. Furthermore, for 2MASS J130700.66+233805.0 and 2MASS J145331.51+135358.7 we do not use their PAH[7.7] measurements which are overestimated due to unmodeled ice absorption and instead use their PAH[11.3] measurements; this results in a decrease of the calculated $L_{\rm IR,SF}$ for the former object. For type 2 quasars, we use our own PAH[11.3] measurements and convert them to the expected radio emission using equation (<ref>).
Figure <ref> makes it clear that if PAH[11.3] is a good measure of the host galaxies' star formation rates, then in all quasar samples (blue type 1s, red 2MASS-selected quasars and obscured type 2s) the observed radio emission is well in excess of that expected from the star formation, which is consistent with the results we obtained from photometric measures of star formation in Section <ref>. The median / mean / sample standard deviation ratio of the observed radio emission to that predicted from star formation is 1.1 dex / 1.2 dex $\pm$ 0.9 dex for quasars in the left panel and 1.1 dex / 1.3 dex $\pm$ 1.1 dex for quasars in the right panel (only objects with radio detections were taken into account).
PAHs are detected in half of each subsample – blue type 1s, red type 1s and type 2s. Among the detections, the median star formation rates follow the trend seen in photometric data (lower star formation rates in the blue type 1 subsample than in the other two): 6.7, 26 and 29 $M_{\odot}$ yr$^{-1}$, correspondingly. Based on the same dataset, it was pointed out by <cit.> that red type 1 quasars occupy more actively star-forming hosts than blue type 1s. We convert non-detections to upper limits on star formation rates, but in some cases the quality of the data make them not very constraining, and the median upper limits are 35, 64 and 17 $M_{\odot}$ yr$^{-1}$.
§.§ Issues with PAH measures of star formation
§.§.§ PAHs and dust obscuration
It is not clear to what extent PAH emission is affected by intervening dust absorption. <cit.> showed that in star-forming ultraluminous infrared galaxies, ratios of PAH complexes at different wavelengths are correlated with the strength of the silicate absorption feature, in a manner consistent with PAH-emitting regions being affected by an amount of obscuration similar to (though somewhat smaller than) that affecting the mid-infrared continuum; this possibility was also pointed out by <cit.> and <cit.>. But this trend is not borne out in a large sample of lower-luminosity GOALS galaxies <cit.>. Perhaps only the most luminous, most compact starbursts follow a relatively simple geometry in which both the PAH-emitting regions and the thermal-continuum-emitting regions are enshrouded in similar amounts of cold absorbing gas, whereas in more modest star-forming environments the different components are mixed with one another. Therefore, it remains unclear whether PAH emission is affected by absorption, and any absorption correction would be rather uncertain.
Because type 2 quasars show weak silicate absorption <cit.>, any putative absorption corrections to PAH fluxes are relatively minor: in 90% of our sample, the peak absorption strength is $S[9.7\micron]<1$, so the optical depth at 11.3 is less than 0.6 and the correction to PAH flux would be less than 30%, with a median of 8%. But more importantly, in type 2 quasars most of the mid-infrared continuum is due to the quasar-heated dust, and therefore the strength of silicate absorption reflects the geometry of this circumnuclear component. Correcting PAH emission which is likely produced in different spatial regions using this value of silicate absorption appears meaningless, so we do not attempt it.
If star formation in quasar hosts is extremely obscured and if PAH-emitting regions are buried inside optically thick layers of dust, then our methods would underestimate PAH emission, and thus the star formation rate and the expected radio emission. In order for such absorption to completely account for a 1.2 dex offset between the observed and the predicted radio emission, we would need an optical depth of 2.8 at the wavelength of the 11.3 feature, which corresponds to the strength of the silicate absorption of about $S[9.7\micron]\simeq 4$. Only 6% of ultraluminous infrared galaxies have an apparent absorption strength above 3 <cit.>, so such high average values of extinction toward PAH-emitting regions are implausible. Therefore, it is unlikely that using the PAH method we underestimate the star formation rates in quasar hosts by the amount necessary to explain the difference between the observed and predicted radio emission.
§.§.§ Effect of the quasar radiation field on PAHs
The harsh radiation field of the quasar and shocks due to quasar-driven outflows may destroy PAH-emitting molecules and therefore suppress PAH emission <cit.>. As a result, by using PAH luminosities we may be underestimating the star formation rates in quasar hosts. To evaluate this possibility, in Figure <ref> we investigate the ratios of PAH[6.2] to PAH[11.3] in star-forming galaxies and in quasars. Because these ratios are related to the size distribution of the aromatic molecules, differences between PAH ratios found in star-forming galaxies and those found in quasars may indicate that the quasar has an impact on the PAH-emitting particles and make PAH-based star formation rates suspect.
Left: PAH[6.2]/PAH[11.3] ratios in star-forming galaxies from our comparison samples (crosses: , circles: , squares: ) and the median and 68% range derived from these data (solid horizontal lines) and those presented by <cit.>, shown in dotted horizontal lines. Right: PAH ratios in the comparison samples (, grey), whether or not they have any signs of an AGN, and those of type 2 quasars in our spectroscopic sample (black points with error bars). The horizontal lines denote the mean and the standard deviation for type 2 quasars (solid lines) and the same for Seyfert galaxies (dotted lines) as presented by <cit.>. PAH ratios of type 2 quasars are more in line with those in Seyferts than those in star-forming galaxies, although there is not a strong correlation between the PAH ratios and the relative contribution of the active nucleus.
In the left panel of Figure <ref>, we demonstrate that our measurements of the median and the standard deviation of the PAH[6.2]/PAH[11.3] ratio among star forming galaxies are in excellent agreement with those of <cit.>, whose PAH-fitting procedures are very similar to ours. For this comparison, we pre-selected objects without any optical or infrared signatures of an AGN, so that we are evaluating purely star-forming galaxies. In the right panel, we now include PAH measurements of sources with varying degree of AGN contribution, and demonstrate the dependence of the PAH ratios as a function of the EW of PAH[6.2], a common measure of the fractional AGN contribution to the bolometric budget <cit.>. Because AGN tend to produce power-law-like emission in the mid-infrared, while star-forming galaxies (which have lower dust temperatures) rarely have strong continuum at these wavelengths, the EW of PAH[6.2] is high in star-forming galaxies and low in AGN, with 0.3 being the typical dividing line <cit.>.
We see that type 2 quasars display relatively low EW of PAH[6.2], in agreement with our previous conclusion that their bolometric luminosities are dominated by AGN activity. Furthermore, despite large measurement errors, we find that the PAH[6.2]/PAH[11.3] ratio is suppressed in type 2 quasars, and again we see excellent quantitative agreement between our measurements of the median ratio and its standard deviation and those of <cit.>. The relatively low PAH[6.2]/PAH[11.3] in type 2 quasars cannot be explained by extinction which would increase this ratio: due to the silicate feature centered at 9.7 which extends over a wide wavelength range, dust opacity is higher at 11.3 than at 6.2 <cit.>. Therefore, we confirm that the PAH ratios appear to be affected by the quasar radiation field. However, it is not clear how much of an effect quasar radiation field may have specifically on the PAH[11.3]-derived star formation rates. <cit.> argue that PAH[11.3] feature may be less affected than PAH[6.2] which traces smaller easily destroyed grains, but <cit.> have argued that PAH[11.3], too, can be suppressed by the quasar emission resulting in an underestimate of the star formation rate.
This is suggested by <cit.> who show that the star formation rates of type 1 quasar hosts derived from far-infrared luminosities are a factor of two-three higher than those derived from PAH[11.3] luminosities. Thus either the far-infrared fluxes are contaminated by the quasar resulting in an overestimate by this factor, or the PAH-emitting particles are destroyed, resulting in an underestimate by this factor, or perhaps both are true to a lesser extent. In our sample of type 2 quasars, there is unfortunately limited overlap between objects with 160 photometry and IRS spectroscopy (28 objects, most of them with upper limits in at least one of these measurements). In the few that do have both measures of star formation (Figure <ref>), we do not detect any noticeable offset from the locus of star-forming galaxies, suggesting that both 160 fluxes and PAH[11.3] luminosities provide consistent measures of star formation rates in these objects.
Following <cit.>, we directly compare infrared luminosities of star formation in type 2 quasar hosts as derived from PAH[11.3] and from the 160 flux. Solid circles show type 2 quasars with both a 160 detection and a PAH[11.3] detection, whereas open circles denote upper limits in one of these measures (those objects with upper limits in both PAH[11.3] and 160 flux are excluded). The solid line shows the locus expected for star forming galaxies. In type 1 quasars, <cit.> find that 160 photometry systematically overestimates star formation rates, or PAH[11.3] measurements systematically underestimate them, or both. This is not borne out by the data for type 2 quasars, although the sample is too small for a conclusive investigation.
§.§ Star formation rates from [NeII] and [NeIII] lines
Low-ionization forbidden emission lines are frequently used as a star-formation diagnostic. In Figure <ref> we examine the utility of [NeII]$\lambda$12.81 and [NeIII]$\lambda$15.56 for use as star formation diagnostics in our sample. The line ratios of neon are clearly affected by the presence of the quasar: [NeIII]/[NeII] ratios are significantly higher in our sample of type 2 quasars than in the comparison star-forming galaxies (left). Furthermore, [NeV]$\lambda$14.32 and [NeVI]$\lambda$7.65 are frequently strongly detected in our sources, with [NeV]/[NeII]$\sim 1$, whereas these lines are rarely seen at all in star-forming galaxies <cit.>.
Left: Neon ratios in type 2 quasars (solid black points) and in GOALS galaxies (open grey points, ), as a function of the equivalent width of the PAH[6.2] feature. Objects with EW of PAH[6.2] of $<0.3\micron$ (which includes the majority of type 2 quasars) are thought to be bolometrically dominated by the AGN. We find that these sources tend to show significantly higher [NeIII]/[NeII] ratios than those seen in pure star-forming galaxies, suggesting that AGN contributes to neon ionization. Right: Two mid-infrared star-formation diagnostics, PAH[11.3] luminosity and [NeII] luminosity, plotted against each other for star-forming galaxies from the GOALS sample (open grey points) and for type 2 quasars (solid black points and arrows), along with two star formation calibrations provided in the literature: the solid line is from <cit.> and the dotted line is from <cit.>. GOALS star forming galaxies show excellent agreement with both calibrations, but type 2 quasars show either an excess of [NeII] or a deficit of PAH[11.3] by about 0.35 dex, or a factor of 2.2. We choose to compare [NeII] (which is an over-estimate of star formation) with PAH[11.3] (which may be an underestimate of star formation) rather than 160 fluxes (which provide upper limits on star formation) because these two values bracket actual star formation rates.
Thus [NeIII] and higher ionization lines are clearly affected by the presence of the quasar. In Figure <ref>, right, we explore whether [NeII] can still be safely used as a star formation diagnostic or whether it, too, has an appreciable contribution from the quasar. Using GOALS star-forming galaxies, we reproduce with high accuracy the PAH vs [NeII] calibrations from the literature <cit.>. The median offset between GOALS galaxies and these relationships is only $\sim 0.04$ dex, with a standard deviation around the relationships of $\sim 0.17$ dex, i.e., the quality of the PAH-[NeII] correlation is comparable to, or better, than the correlations between other star formation indicators we have discussed here.
In contrast, type 2 quasars lie systematically above the star formation relations, with a median offset of about 0.3 dex, or a factor of 2. Therefore, either the quasar contributes appreciably to the [NeII] ionization, or PAH[11.3] features are suppressed, or both. Because we have not found any offset between PAH-derived and 160-derived star formation rates (Figure <ref>), we suggest that the destruction of PAH[11.3] may not be the dominant effect and that instead the quasar dominates the photoionization of [NeII]. This finding is similar to that of <cit.> who found that low-ionization optical emission lines, e.g., [OII]$\lambda$3727Å, may also be dominated by quasar photoionization and be poor star formation indicators.
However, the possible contribution of the quasar photoionization to the [NeII] emission does not modify our main conclusions regarding the excess of radio emission over that predicted due to star formation. Even if we were to use [NeII] as a star formation diagnostic, we would still underpredict the amount of radio emission in type 2 quasars by 0.42 / 0.78 dex (median / mean), or a factor of 2.6 / 6.0.
§ DISCUSSION: EFFECTS OF QUASAR ON STAR FORMATION AND ITS DIAGNOSTICS
Although spectral energy distribution decomposition methods assume that an AGN component and a star-forming component are simply added together to produce the overall galaxy spectrum, the two components could affect each other in a much less linear way. For example, a powerful but compact circumnuclear starburst could be further heated by the AGN, raising the apparent dust temperature beyond those encountered in star-formation templates and leading the observer to classify the source as quasar-dominated. A dusty galaxy with modest star formation rates could host a quasar; with a fortuitous geometric distribution of dust, the quasar can be hidden from view and lead to strong far-infrared emission from dust on $\ga$ kpc scales, which can be mistaken for high rates of star formation. PAH-emitting particles could be destroyed by the harsh radiation from the AGN, while low-ionization emission line regions associated with star formation can get photo-ionized by the quasar, suppressing these star-formation diagnostics.
We find clear evidence for some of these effects: quasars likely suppress some PAH emission, enhance the low-ionization emission-line diagnostics, and contribute to the far-infrared emission, thereby likely affecting all mid- and far-infrared diagnostics of star formation. To minimize these effects, we choose the longest wavelength observations available to us (160) which have the smallest fractional contamination by the AGN, PAH[11.3] which is less affected by the AGN than PAH[6.2] and less contaminated by high-ionization line emission than PAH[7.7], and [NeII]$\lambda$12.81. These star formation measures agree with one another to within a factor of two even in quasar hosts (with [NeII] likely the most strongly affected by the quasar), giving some credence to our measured rates of star formation. Therefore, we find it highly unlikely that we have underestimated star formation rates in quasar hosts by a factor of 10, the value required to bring the observed radio fluxes in agreement with those expected from star formation alone.
The impact of quasars on the evolution of their host galaxies has emerged as a key question in modern galaxy formation models. Active black holes are now suspected in limiting galaxy masses via quasar feedback <cit.>, and determining the progression of star formation activity during and after an episode of black hole activity remains an interesting and unsolved problem in observational astronomy. Some studies suggest that long-term average AGN luminosity and star formation are strongly correlated, potentially due to a common supply of cold gas <cit.>. In individual AGN, the correlations between AGN luminosity and their hosts' star formation rates are quite weak <cit.>, which can largely be explained by short-term fluctuations in AGN luminosity <cit.>. Intriguingly, some studies have reported an apparent suppression of star formation in quasar hosts <cit.>, although in some cases this can be attributed to limited survey volumes and sample sizes <cit.>. In any case, a potential detection of a suppression in star formation for quasar hosts must be distinguished from over-ionization or destruction of the star-formation diagnostics by the quasar. This is necessary in order to be able to measure the impact of the quasar on its host galaxy.
Quasars with luminosities $\ga 10^{46}$ erg s$^{-1}$ are luminous enough to easily ionize the gas over galaxy-wide scales <cit.>, and quasar-driven outflows <cit.> can lead to galaxy-wide shocks <cit.>. While circumnuclear obscuration could shield parts of the galaxy from direct quasar emission, outflows and shocks might find ways around this obstacle <cit.>. Physical removal of the interstellar medium suppresses subsequent star formation activity, but it would also be interesting to detect the effect of the quasar on on-going star formation in the regions affected by quasar radiation and / or quasar-driven winds. One possibility is that star formation proceeds as usual, but emission line diagnostics of star formation are strongly affected by quasar photoionization or shock excitation. This could be the origin of the decreasing [OII]$\lambda$3728Å/[OIII]$\lambda$5007Å ratio both as a function of quasar luminosity <cit.> and narrow-line kinematics <cit.>. If this hypothesis is correct, then one can hide significant amounts of star formation by exposing these regions to the quasar radiation field which would bias star formation diagnostics. Another possibility is that quasar radiation and / or quasar-driven outflows are in fact suppressing on-going star formation. Developing star formation diagnostics that can distinguish between these scenarios in hosts of luminous AGN remains an interesting challenge.
§ CONCLUSIONS
The correlation between radio luminosity and narrow line gas kinematics in radio-quiet quasar host galaxies <cit.> suggests that there may be a physical connection between the two. This correlation has renewed the debate over the origin of the radio emission in radio-quiet quasars. One hypothesis, proposed by <cit.> and others, cites star formation in the host galaxy as the driving mechanism behind the observed radio emission from radio quiet quasars. The median radio luminosity of radio-quiet obscured quasars is $\nu L_{\nu}$[1.4 GHz]$=1\times 10^{40}$ erg s$^{-1}$ <cit.>, which would require 300 $M_{\odot}$ yr$^{-1}$ worth of star formation <cit.>. While this is not impossible, it seems fairly high, so in this paper we ask whether the actual amount of star formation in the host galaxies is sufficient to produce the observed radio emission.
We use archival samples of star-forming galaxies to revisit calibrations of different star formation indicators: radio emission (eq. <ref>), single-band infrared fluxes (eq. <ref>-<ref>), and PAH[11.3] luminosities (eq. <ref>-<ref>). As long as uniform measurement methods are used – which is particularly important for PAH emission – we find excellent agreement between published calibrations and archival data, with a spread in different correlations of $\sim 0.2$ dex. With these star-formation calibrations in hand, we measure star formation rates in the hosts of quasars of different types at $z<1$ using photometric and spectroscopic data from the and telescopes.
Using infrared colors of type 2 quasars, we demonstrate that the bolometric luminosities of the objects in our samples – estimated to range from $10^{45}$ to $10^{47}$ erg s$^{-1}$, Figure <ref> – are dominated ($>50$%) by quasar emission; however, even in objects with 80% quasar contribution to the bolometric budget, more than half of the 160 emission is due to star formation because the spectral energy distribution of starlight-heated dust peaks at much longer wavelengths than the quasar-heated dust which is concentrated on much smaller spatial scales. Thus we conclude that while our 160 measurements strictly speaking yield upper limits on star formation, the measured rates are close to the actual values. This is confirmed by spectral energy distribution modeling of a subsample of type 2 quasars drawn from the upper decade of our luminosity distribution <cit.>.
Using 160 fluxes in $\sim 245$ obscured and unobscured quasars, we find a broad distribution of star formation rates in quasar hosts, from median values of 6$M_{\odot}$ yr$^{-1}$ (in hosts of blue type 1 quasars) to 18$M_{\odot}$ yr$^{-1}$ (in type 2 hosts). The difference in star formation rates in hosts of type 1 and type 2 quasars has been seen in other observations <cit.> and remains a challenge to the standard geometric unification model. Although statistics of molecular gas observations are still limited, existing data suggest similar availability of cold gas in hosts of type 1 and type 2 quasars <cit.>, and the relatively low star formation rates of type 1 quasar hosts is in tension with the availability of cold gas in them, suggesting low star formation efficiency <cit.>.
We then use mid-infrared spectra of $\sim 160$ quasars of different types to estimate the star formation rates of their hosts using spectroscopic diagnostics, especially PAH emission. We find a broad distribution of star formation rates, with a median of $\la 30$ $M_{\odot}$ yr$^{-1}$. We find that PAH ratios in quasars differ from those in star forming galaxies, in excellent quantitative agreement with the findings of <cit.>, and we use PAH[11.3] as our primary spectroscopic star formation indicator. Further evaluating [NeII]$\lambda$12.8 and [NeIII]$\lambda$14.3 lines as star formation indicator, we find both to be strongly affected and likely dominated by quasar photo-ionization.
Regardless of the method used to estimate star formation in quasar hosts, we find that even the most generously computed star formation rates are insufficient to explain the observed radio emission, by about an order of magnitude. Depending on the measurement method, we measure the ratio of observed radio luminosity to that predicted due to star formation alone $L_{\rm radio,obs}/L_{\rm radio,SF}$ of 0.6$-$1.3 dex. Thus radio emission in radio-quiet quasars is unlikely to be dominated by star formation in quasar hosts and is likely associated with the quasars. Our results are in agreement with other evaluations of the star formation contribution to the radio emission of radio quiet quasars <cit.>.
<cit.> find that in AGN with $\nu L_{\nu}$[12]$<10^{44}$ erg s$^{-1}$ both the radio emission and the 12 emission appear to closely follow the star-formation locus and they conclude that both 12 and radio emission are associated with star formation (see also ). In Figure <ref> we show the discrepancy between the observed and the predicted radio emission as a function of the infrared luminosity. There is a slight trend of an increasing discrepancy toward higher 12 luminosity. Thus it is possible that for low-luminosity AGN ($\nu L_{\nu}$[12]$\ll 10^{44}$ erg s$^{-1}$) the radio emission is a good measure of star formation rates in the host galaxy as <cit.> suggest, though some contribution to both radio and 12 emission from the AGN at $10^{43}<\nu L_{\nu}$[12]$<10^{44}$ erg s$^{-1}$ appears likely in light of our results.
The discrepancy between the observed radio emission and that expected from star formation as a function of the 12 luminosity. Grey points are for radio non-detections, black points for radio-detected type 2 quasars, blue for PG type 1s and red for 2MASS type 1s. The discrepancy between the predicted and the observed radio emission increases with infrared luminosity (Spearman rank probability of the null hypothesis of no correlation is P[NH]$=10^{-3}$).
Regardless of the mechanism responsible for producing radio emission in the radio-quiet majority of quasars, this emission would be in addition to the radio emission produced by the host galaxy. This is well supported by Figures <ref>, <ref> and <ref>, which show that rarely does the observed radio emission fall below the levels predicted to be due to star formation (and when it does, not by much). The objects whose $L_{\rm radio, obs}$ are closest to the $L_{\rm radio, SF}$ are those where the radio emission associated with the quasar is weaker than that due to star formation in the host galaxy (there are several such examples in this study and in ). Figure <ref> suggests that as the quasar luminosity increases, there are fewer such `radio-silent' objects.
There are clear cases where quasar emission dominates all bands, including far-infrared <cit.>, and more ambiguous objects in which the lowest temperature of the dust (as measured from the spectral energy distribution) is higher than typical values seen in star-forming galaxies <cit.> which could also be the case of a quasar dominating far-infrared emission. As we demonstrate in this paper, the radio emission in quasars is not a good estimate of star formation. Thus both $8-1000$ fluxes and radio emission – used together or separately – may strongly overestimate star formation rates of quasar hosts <cit.>, and additional cross-checks or full spectral energy distribution decomposition are required <cit.>. Calculating star formation rates in quasar hosts is further complicated by a possibly reduced star formation efficiency which would bias diagnostics based on cold gas detection <cit.> and, for the unobscured star formation, by the strong contribution of direct or scattered light to the observed ultra-violet emission <cit.>.
While this work rules our star formation as the origin of most radio emission from quasars, we cannot distinguish between radio emission due to compact weak jets and radio emission due to wide-angle winds. Radiatively-driven winds can produce sufficient amount of radio emission only in quasars with $L_{\rm bol}\ga 3\times 10^{45}$ erg s$^{-1}$, which appears to be the threshold luminosity for driving galaxy-wide winds <cit.>. Precise measurement of the radio luminosity function of these objects at faint radio luminosities is difficult because they are rare, so fairly high-redshift sources need to be observed with high sensitivity. Recent such work indicating a break in the radio luminosity function of quasars at fixed optical luminosity <cit.> suggests that different mechanisms are likely responsible for the emission of radio-quiet and radio-loud sources.
This observation, and the radio vs gas kinematics correlation <cit.>, lead us to favor radiatively-driven winds as the ultimate origin of the radio emission <cit.>. In this scenario, relativistic particles are accelerated on the shocks driven into the interstellar medium by the expanding wind. It may be testable by multi-wavelength observations <cit.>, radio spectral measurements <cit.> and high-resolution radio imaging. The problem of distinguishing radio emission from compact jets from radio emission as a bi-product of radiatively driven has proven especially difficult because the two mechanisms are similar in terms of energetics <cit.> and because a radiatively driven wind can inflate bubbles which mimic double-lobed radio morphologies, making morphology an unreliable jet / wind diagnostic <cit.>.
We have demonstrated that quasars and star-forming galaxies lie on different far-infrared / radio correlations: quasars show an order of magnitude more radio emission than do star-forming galaxies with the same 160 luminosities. However, multiple groups have demonstrated that quasars, star-forming galaxies and composite sources may lie on the same total (8-1000) infrared / radio correlations <cit.>, though possibly the spread <cit.> or the normalization <cit.> of the correlation may change slightly in the quasar-dominated regime. If radio emission of radio-quiet quasars is dominated by jets, then the infrared/radio correlation can only be explained by a strong coupling between accretion processes and jet production <cit.>, but finding jets on the same infrared / radio correlation as star-forming galaxies is quite surprising. If radio emission of radio-quiet quasars is dominated by radiatively driven winds, then in both quasars and star-forming galaxies the radio emission is produced by relativistic particles accelerated on shocks. What is surprising in this case is to find quasars and star-forming galaxies to be converting the same fraction of their bolometric power into shocks via completely different mechanisms <cit.>.
NLZ is grateful to E.Quataert, C.-A. Faucher-Giguère, J.Nims and the referee J. Mullaney for useful discussions. NLZ and RCH acknowledge support from the Herschel Science Center under JPL contracts No. 1475252 and No. 1471850, respectively. KL is supported by the Johns Hopkins University Dean's Undergraduate Research Award. RCH acknowledges support from an Alfred P. Sloan Research Fellowship and the Dartmouth Class of 1962 Faculty Fellowship. LCH acknowledges support by the Chinese Academy of Science through grant No. XDB09030102 (Emergence of Cosmological Structures) from the Strategic Priority Research Program and by the National Natural Science Foundation of China through grant No. 11473002. MO is supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and Grant-in-Aid for Scientific Research from the JSPS (26800093).
This research made use of Tiny Tim/Spitzer, developed by John Krist for the Spitzer Science Center. The Center is managed by the California Institute of Technology under a contract with the National Aeronautics and Space Administration (NASA). This research has made use of the NASA / IPAC Infrared Science Archive and NASA/IPAC Extragalactic Database (NED), which are operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.
160 photometry and photometric constraints on star formation
ID subsample $z$ $L$[12] $F_{\nu}$[160] SFR $L_{\rm radio,SF}$ $L_{\rm radio,obs}$ comment
SDSS J004340.12-005150.2 T2 0.5849 45.04 -360.3 567 40.34 -40.23 Spitzer, calibration
SDSS J005009.81-003900.6 T2 0.7276 46.05 112.7 300 40.03 41.04 Spitzer, ULIRG
SDSS J005621.72+003235.8 T2 0.4840 45.24 -72.9 95 39.47 40.97 Spitzer, type 2 quasar (PI Strauss)
SDSS J010523.62+011321.4 T2 0.2049 43.89 -230.8 55 39.20 -39.20 Spitzer, calibration
SDSS J012341.47+004435.9 T2 0.3990 44.80 -75.8 69 39.32 40.91 Spitzer, type 2 quasar (PI Strauss)
Table presented in full in the electronic edition; a small portion given here for guidance regarding format and content.
`Subsample' describes whether the object belongs to the type 2 sample (T2) or red type 1 sample (T1red). data for blue type 1 quasars are published by <cit.>, and data for type 2 quasars will be available in Petric et al. (in prep.)
$L$[12] is given in units of $\log$($\nu L_{\nu}$[12], erg s$^{-1}$) and is set to -100 for objects undetected in both WISE-12 and WISE-22.
$F_{\nu}$[160] measured in mJy are positive for detections (we use 25% as an estimate of the absolute uncertainty) and negative for 5$\sigma$ upper limits.
SFR gives the upper limit on the star formation rate in $M_{\odot}$ yr$^{-1}$ calculated in Section <ref>.
$L_{\rm radio,SF}$ is the upper limit on the radio emission due to star formation, given in units of $\log$($\nu L_{\nu}$[1.4GHz], erg s$^{-1}$).
$L_{\rm radio,obs}$ is the observed radio luminosity (or in the case of negative values, 5$\sigma$ upper limits) in units of $\log$($\nu L_{\nu}$[1.4GHz], erg s$^{-1}$).
Comment column describes whether the object was in the or in the sample, whether it was targeted for pointed observations, and if so, why, or if it was covered serendipitously in observations of other targets (with the survey name given in parentheses if applicable).
IRS spectroscopy of type 2 quasars
ID $z$ $S$[9.7] SL $L_{\rm radio,obs}$ $L$[NeII] $L$[NeIII] PAH PAH comment
factor [6.2] [11.3]
SDSS J004252.56+153246.8 0.1175 0.64 1.000 39.56 1.19 1.84 6.09 7.20 line-selected type 2 AGN
SDSS J005009.81-003900.6 0.7276 0.49 1.000 41.04 -10.48 21.14 -11.89 -32.60 ULIRG
SDSS J005621.72+003235.8 0.4840 2.52 1.000 40.97 5.01 NaN -7.82 15.94 type 2 quasar (PI Zakamska)
SDSS J012341.47+004435.9 0.3990 -0.02 1.000 40.91 1.85 NaN -8.84 -5.15 type 2 quasar (PI Zakamska)
SDSS J080224.35+464300.6 0.1206 0.36 1.300 39.69 0.80 1.55 -1.19 1.88 line-selected type 2 AGN
Table presented in full in the electronic edition; a small portion given here for guidance regarding format and content.
$S$[9.7] is the dimensionless strength of the silicate feature (positive for absorption, negative for emission), similar to optical depth and defined in Section <ref>; typical systematic uncertainty in this measurement is 0.2 <cit.>.
`SL factor' is the multiplicative factor applied to the short-low IRS orders to bring them in agreement with the long-low orders.
$L$ of Ne and PAH emission features is given in units of $10^{42}$ erg s$^{-1}$, positive for detections, negative for 3$\sigma$ upper limits, `NaN' for lacking spectral coverage.
$L_{\rm radio,obs}$ is the observed radio luminosity (or in the case of negative values, 5$\sigma$ upper limits) in units of $\log$($\nu L_{\nu}$[1.4GHz], erg s$^{-1}$).
Comment column describes why the object was targeted for IRS observations.
|
1511.00245
|
1Department of Astronomy, Oskar Klein Centre, Stockholm University, AlbaNova,
SE–106 91 Stockholm, Sweden
2Astrophysics Research Centre, School of Mathematics and Physics, Queen's University Belfast, Belfast, BT7 1NN, UK .
The observational effects of the 'Infrared Catastrophe' are discussed in view of the very late observations of the Type Ia SN 2011fe.
Our model spectra at 1000d take non-local radiative transfer into account, and find that this has a crucial impact on the spectral formation.
Although rapid cooling of the ejecta to a few 100 K occurs also in these models, the late-time optical/NIR flux is brighter by 1-2 magnitudes due to redistribution of UV emissivity, resulting from non-thermal excitation and ionization. This effect brings models into better agreement with late-time observations of SN 2011fe and other Type Ia supernovae, and offers a solution to the long standing discrepancy between models and observations.
The models show that spectral formation shifts from Fe2 and Fe3 at 300d to Fe1 at 1000d, which explains the apparent wavelength shifts seen in SN2011fe. We discuss effects of time dependence and energy input from ${}^{57}$Co, finding both to be important at 1000d.
§ INTRODUCTION
The infrared catastrophe (IRC) of Type Ia supernovae (SNe) was predicted already in 1980 by <cit.>. It occurs at $\sim$500 days as a result of the change in cooling of the ejecta from transitions in the optical to fine-structure transitions in the mid- and far-infrared. Up to $\sim$500
days cooling is dominated by optical/NIR [Fe2-III] lines with excitation temperatures $\sim (1-3)\times10^4$ K. As radioactive energy input and temperature decrease, at $\sim 2000$ K the cooling is replaced by mid-IR fine-structure lines with excitation temperatures $\sim 500$ K.
Because the cooling is insensitive to temperature between $\sim$500-2000 K, the temperature drops dramatically until the Boltzmann factors for the fine structure lines become effective and the temperature stabilises at a few 100 K. At this point the thermal emission has moved completely to the mid/far-infrared.
<cit.> renewed this calculation,
but found a strong discrepancy with the late observations of SN 1972E.
Later studies <cit.> confirmed this and did not find much evidence for the IRC from either photometry or spectra.
The modeling in these papers was based on the <cit.> code, here referred to as the KF98 code.
The model light curves predicted large drops in most photometric bands caused by the IRC, in disagreement with observations.
The redistribution of radiation caused by the multiple scattering and fluorescence could, however, not be treated.
The nearby Type Ia SNe 2011fe and 2014J offer unique possibilities to study Ia SNe at late times with photometry
and spectra up to $\sim 1000$ days.
From photometry up to $\sim 950$ days <cit.> found a decline in the optical broadly consistent with that expected for ${}^{56}$Co decay, with no signs of an IRC.
In <cit.> a spectrum taken at 1034 days was discussed with tentative line identifications.
From a comparison with a $\sim 300$ day spectrum it was clear that the spectrum at this epoch was still dominated by Fe lines, but it was not clear if these were from Fe1 or Fe2. This is important as a direct probe of the physical conditions of the ejecta.
<cit.> present a similar discussion based on a spectrum 981 days after explosion. These late epochs also offer a possibility to test explosion models from their nucleosynthesis, in particular the formation of isotopes
such as ${}^{57}$Ni, $^{55}$Fe and ${}^{44}$Ti
<cit.>. From photometry of SN 2012cg at $\sim 3$ years <cit.> claim evidence for power input by ${}^{57}$Co, although the absence of spectral modeling makes the bolometric correction uncertain.
In this Letter we discuss a solution of the apparent contradiction between models of the IRC and observations, based on detailed spectral synthesis of realistic explosion models.
§ MODELS
To calculate spectra and light curves
we use two different codes. One is based on KF98, updated with extended atomic data. This code includes time dependence and calculates the ionization and temperature of the ejecta, including non-thermal excitations, and local scatterings treated with the Sobolev approximation. This gives an accurate calculation of the temperature and ionization, but does not take into account the multiple scatterings which redistribute UV radiation into optical radiation. As discussed in <cit.>, this is important in the phase when non-thermal processes dominate the ionization and excitation.
To treat this we use the code by <cit.> in its latest version <cit.>, here referred to as the JFK11+ code. This treats the combined NLTE and radiative transfer problem for all relevant ions from hydrogen to nickel, but does not include time dependence. These two approaches are therefore complementary.
As input explosion models we study the 1D W7 model by <cit.> and the N100 3D delayed detonation model by <cit.>, which we make a 1D version of. For W7 we use the composition and density structure computed by <cit.>. The ejecta are zoned into 30 (KF98 calculation) and 180 (JFK11+) shells, with the larger number needed to resolve
the radiative transfer in the JFK11+ models.
For N100 we
map the 3D model to a 1D version with the following recipe, which minimises the microscopic mixing between the major burning zones, in contrast to a straight spherical average at the different radii: For each cell in the 3D model the two most abundant elements are identified. This produces a set of 8 discrete composition classes. For each of these classes the composition is averaged by mass over all cells belonging to
that class. The ejecta are then divided into shells in velocity up to $25,000 \kms$, with 40 shells for KF98 and 240 shells for JFK11+. For each radial shell the two composition classes with the highest mass within that velocity range in the 3D model are identified, and the shell is split into two equal-density subshells with corresponding compositions and relative masses (more massive innermost). Compared to the original N100 model it conserves the total elemental masses at $\la 10\%$, with exception of ${}^{58}$Ni, with a total mass $0.099 \Msun$, compared to $0.069 \Msun$ in the original model. This is a consequence of using only two composition zones per radial shell.
Radioactive input (gamma-rays, leptons and X-rays) from ${}^{56}$Ni, ${}^{57}$Ni and ${}^{44}$Ti and their daughter isotopes are included <cit.>.
At 1000 days ${}^{56}$Ni and ${}^{57}$Ni contribute roughly equal to the deposition, depending on the exact ${}^{57}$Ni mass. The ${}^{56}$Ni mass is $ 0.59 \ \msun$ in W7 and $ 0.60 \ \msun$ in N100, within the observed range for SN 2011fe, $ 0.5-0.6 \ \msun$ <cit.>. There is no independent observational constraint on the ${}^{57}$Ni mass. The delayed detonation models by
<cit.> have $(1.1-3.4)\times 10^{-2} \Msun$, with N100 at $1.8\times 10^{-2} \Msun$,
while W7 has $2.4 \times 10^{-2} \Msun$ <cit.>.
In comparisons with the observations we have used a distance of 6.4 Mpc to M 101 <cit.>, and $E_{\rm B-V} = 0$ <cit.>.
§ RESULTS
In the upper panel of Fig. <ref> we show the temperature of typical composition zones of the W7 ejecta as function of time for the KF98 model. The temperature in all cases shows a drop from $(5 -10)\times 10^3$ K to $\la 100$ K.
The epoch when this
occurs differs from 400 to 800 days, depending on the composition. The C/O zone is cooling more slowly than the others, but reaches the lowest temperatures after 1000 days. This is a combination of low density, higher ionization and low cooling efficiency of the abundant ions. The cooling is here dominated by [C2] 157.7$\mu$m and [O3] 88.34$\mu$m (excitation temperatures 91 K and 164 K, respectively), while the Si and Fe rich zones cool by fine-structure lines of higher energy, 410 K and 553 K for [Si2] 34.81$\mu$m and [Fe2] 25.99$\mu$m, respectively. The cooling of these zones therefore drops at higher temperature compared to the C/O zone.
Upper panel: Temperature evolution of selected zones in the ejecta of the W7 model calculated with the KF98 code. Note the dramatic drop at $400-800$ days indicating the transition from cooling by optical lines to cooling by mid-IR lines. The mean velocities of the zones are 3000 $\kms$ (${}^{54}$Fe/${}^{58}$Ni), 6800 $\kms$ (${}^{56}$Ni/He), 10,900 $\kms$ (Si/S), 13,000 $\kms$ (O/Si) and 15,000 $\kms$ (C/O).
Lower panel: Synthetic light curves in the V (green) and R (red) bands, for the same model together with observations in B and V at 0-652d <cit.>, g and r at 930d <cit.>, and g and r at 1034d (triangles, T2015). The open stars at 1034d are synthetic g and r magnitudes for the JFK11+ W7 model. The dashed lines show the V and R band light curves for a KF98 model in steady state, showing time-dependence to become important after 800-900d.
In the lower panel of Fig. <ref> we show the V and R band light curves of the W7 model with the KF98 code. The IRC is here seen as the drop at $\sim 500$ days in both bands. A comparison with the magnitudes of <cit.> and <cit.> of SN 2011fe clearly shows the discrepancy
after the IRC.
To demonstrate the importance of the radiative transfer treatment and the UV to optical conversion we show in Fig. <ref> the W7 spectrum from UV to NIR, with and without non-local radiative transfer using the JFK11+ code.
Lower panel: Comparison of the energy distribution, $\lambda F_\lambda$, of W7 with (blue) and without (red) multiple scattering, both calculated with the JFK11+ code.
Upper panel: Line optical depth of the ejecta.
Without non-local radiative transfer we see extremely strong UV lines below $\sim 4000$ Å, mainly resulting from non-thermal ionizations to Fe2 and Fe3, followed by recombinations in the UV, as well as excitations of Fe1-II. In particular, $\sim 1/3$ of all Fe1 excitations occur in a single transition, at $2348.3$ Å, followed mainly by the emission of lines at 2999.5 Å and $1.443 \mu$m. The latter dominates the NIR range (Fig. <ref>) and the non-thermal scenario may be tested by searching for this signature in SN 2014J. The optical/NIR luminosity is low, as most non-thermal excitation and ionizations lead to UV emission.
Including non-local scattering and fluorescence
the forest of optically thick lines (upper panel of Fig. <ref> ) decreases the UV flux below 4000 Å by a factor 5.7, while the flux between 4000 - 10000 Å and between $1 -2.5 \mu$m increases by factors 3.4 and 3.2, respectively. Both multiple scatterings and fluorescence contribute to transferring the UV emissivity to optical and NIR flux. For example, [Ca2] 7300 Å and the Ca2 triplet are both powered by fluorescence following absorptions in the H and K lines.
In Fig. <ref> we show the observed spectrum of SN 2011fe together with the W7 and N100 spectra at 1034d calculated with the JFK11+ code. The models have been multiplied by a factor of 2,
a correction still needed to match the optical luminosity. As we discuss later, this can plausibly
be attributed to some combination of too low ${}^{57}$Ni mass
in the W7 model and lack of freeze-out in the JFK+11 code.
The spectrum is dominated by Fe1 lines as well as
[Ca2] $\wll 7291, 7324$ and the Ca IR triplet.
In T2015 the feature at $\sim 6300$ Å was interpreted as either [O1] $\wll 6300, 6364$ or [Fe1] $\wll 6359, 6231, 6394$. In our synthetic spectra of both W7 and N100 it is dominated by Fe1, with some contribution of Si1-II in N100. There is therefore no need for any low velocity oxygen. The feature at $\sim 5900$ Å is a mix of Na D and Fe I in N100. Unfortunately, Na is missing in the unburned high velocity region in W7, although already Fe I alone tends to overproduce this.
Optical spectrum of SN 2011fe at 1034 days from T2015 (red) together with the JFK11+ model spectrum (black) from the W7 (upper panel) and the N100 delayed detonation mode (lower panel). The contributions of Fe1 (green), Fe2 (blue) and Ca2 (magenta) are also shown.
Both spectra are multiplied by a factor 2.0.
Calculating synthetic photometry for these spectra we find that the g and r band discrepancies between the photometry from the T2015 spectrum and the W7 model decrease from 3.0 mag and 2.2 mag in the KF98 models to 0.36 mag and 0.89 mag, respectively, in the JFK11+ model (not including the factor 2 scaling in Fig. <ref>)
We note that the g-band contains most of the optical flux. Given the uncertainties discussed below, we thus find a resolution to the long-standing problem of the IRC not being observed in Type Ia supernovae:
The omission of non-local radiative transfer effects in previous models seriously underestimates the optical/NIR flux at $t \ga 500$ days.
We also note that the features at $\sim 4300$ Å and $\sim 5200$ Å are well reproduced in wavelength, and there is no need for any ejecta asymmetry, as was suggested in T2015 from a comparison with the 300 day spectrum. This is a consequence of the shift from a Fe2-III dominated spectrum to one dominated by Fe1.
There are many similarities of the 1000 days spectrum here to the conditions at $\sim 8$ years in the Fe-core of SN 1987A, discussed by <cit.>. The densities are similar,
$\sim 10^4 \ccm$, and the dominant excitation and ionization is by leptons from the radioactive decays. Most of this energy is deposited in the iron rich regions in the core, resulting in Fe1 dominated spectra.
From <cit.> one can see that the emissions following non-thermal excitations and ionizations below $\sim 3000$ Å escapes in line gaps longwards of $\sim 4000$ Å.
Because of the low temperature, thermal, collisional processes are not important at this epoch, with the exception of the mid/far-IR fine-structure lines, which account for $\sim 50 - 80 \%$ of the energy output (see below). This energy emerges mainly in [Fe2] $25.99 \mu$m, and weaker [Fe3] $ 22.93 \mu$m, [Fe1] $ 24.04 \mu$m, [Si2] $ 34.81\mu$m and [Fe2] $35.35 \mu$m. This part of the spectrum is unaffected by scattering.
The main discrepancy in our modeling are the general level of the optical flux, which is a factor of $\sim 2$ fainter than observed.
In N100 there are also significant differences in the line ratios, in particular the ratio of the Ca2 lines to the Fe1 4300, 5200 Å multiplets. In the W7 model there are also too strong line features at $\sim$ 2600, $\sim 3700$ Å and at $\sim 4000$ Å. There are, however, several factors which may contribute to these shortcomings.
An important factor is the quality of atomic data, especially for Fe1, where most excitation cross sections by non-thermal electrons are missing. Instead, we use the Bethe approximation, which is reasonable at high energy for permitted lines, but less accurate at low energies.
Fortunately, most excitations go to permitted transitions.
Recombinations to both Fe1 and Fe2 are
responsible for a comparable flux to the excitations. Rates have been calculated for a large number of specific levels <cit.>, but half of the total rate is to higher levels, and is in our model atom distributed among the highest levels according to statistical weights, introducing an uncertainty in individual lines.
Uncertainty in the explosive burning conditions is associated with at least a factor 2 uncertainty in
the ${}^{57}$Ni mass, which
accounts for$\ga 50\%$ of the energy input at this epoch. The density distribution at high velocity also shows a large variation between different models. An underestimate of the 'Fe' abundances and/or the density at high velocities results in an underestimate of the UV-scattering, which may
be the reason for the model overproduction of lines below $\sim 4000$ Å.
The most important factor is probably the steady state assumption in the JFK11+ models. Because of the low density the balance between heating and cooling, as well as between ionization and recombination, may break down, usually known as 'freeze-out' <cit.>. From a comparison of a steady-state and a fully time-dependent calculation of the W7 model with the KF98 code, we find that later than $\sim 700$ days time dependent effects become increasingly important (Fig. <ref>).
This leads to an underestimate of the flux by 0.35 and 0.86 mags in the V and R bands respectively at 1000 days in the steady-state compared to the time dependent case. We note the larger difference in the R-band, as is also seen for the JFK11+ model when compared to observations.
Taking these uncertainties
and especially the freeze-out,
into account, the model provides a satisfactory reproduction of the main properties of the 1034 day spectrum and photometry.
As a further test we show in Fig. <ref> the JFK11+ model for W7 at 331d, compared with observations from T2015.
In general, there is agreement between the model and the strongest features. The optical spectrum at this epoch is mainly dominated by blends of [Fe2-III] and weaker features of Ca2 and [Ni2]. The overproduction of the [Ni2-III] lines between 6000-8200 Å indicates that W7 contains too much stable Ni. The analysis by <cit.> also favoured relatively small amounts of stable nickel in the ejecta.
The too strong Fe3 multiplet at $\sim 4700$ Å suggests a too high ionization of the ejecta. Despite these shortcomings the model can qualitatively reproduce the spectrum showing that one and the same model works at both 300d and 1000d.
Optical spectrum of SN 2011fe at 331 days from T2015 (red) together with the JFK11+ model spectrum from the W7 model (black). Individual contributions are the same as in Fig. <ref> with the addition of Fe3 (cyan) and Ni2-III (yellow).
§ DISCUSSION
After the IRC, the temperature is far too low for thermal excitations of anything except for the mid/far-IR fine structure lines. Because of the concomitant decrease in the ionization
an increasing fraction of the energy goes into non-thermal excitations and ionizations rather than heating <cit.>.
Recombinations, following the non-thermal ionizations to Fe2, are mainly to high-excitation levels of Fe1. This is also the case for the excitations, where the high-lying permitted transitions have the largest cross sections for the non-thermal electrons and positrons. Therefore both these processes result in populations of primarily high levels, which de-excite by permitted transitions in the UV.
In Fig. <ref> we show the change of ionization with density at 1000 days for a model where we calculate the state of ionization and energy deposition for a pure iron plasma, but ignore all radiative transfer effects and only include cooling by the [Fe1-II] fine-structure lines. Because photoionizations are not included the degree of ionization may be somewhat underestimated, but the qualitative features should be correct. We assume M(${}^{56}$Ni) $= 0.6 \Msun$ and M(${}^{57}$Ni)$= 2 \times 10^{-2} \Msun$ and constant energy input per unit mass. Gamma-rays are neglected, but make only a minor contribution.
Electron fraction $X_e$ (black, dashed) and energy fractions going into excitations (blue), ionizations (red) and heating (black, solid) by non-thermal electrons and positrons for a pure iron plasma at 1000d. The vertical dash-dotted line is the density at 1000 days for a uniform sphere of $1.4 \Msun$ expanding at $10^4 \kms$ .
From this we see that in the range $1 \times 10^3 - 4 \times 10^5 \ccm$ the electron fraction, $X_{\rm e}$, varies from $\sim 1.0$ to $\sim 0.07$.
Because the densities below $10^4 \kms$ vary between $\sim 5\times 10^3 \ccm$ to $\sim 5\times 10^4 \ccm$
in the explosion models (for no clumping),
the dominance of Fe1 features should be a robust result at these epochs. There is simply not enough radioactive power to maintain a higher ionization state. Between $1 \times 10^3 - 4 \times 10^5 \ccm$ the heating efficiency varies between $\epsilon_{\rm h} \approx 0.4-0.8$. Because the local ionization is roughly $\propto (X_{56+57} / \rho)^{1/2}$, where $X_{56+57}$ is the initial abundance of the radioactive ${}^{56}$Ni and ${}^{57}$Ni isotopes, $\epsilon_{\rm h} $ depends on the density where the radioactive isotopes are abundant, which differs between different explosion models. With the JFK11+ code the deposition weighted heating efficiencies are $\epsilon_{\rm h} \approx 0.76$ for W7 and $\epsilon_{\rm h} \approx 0.73$ for N100.
Because the fraction going into heating is emitted as mid/far-IR radiation, while the rest is converted to optical/NIR emission,
an increase in the ionization and excitation efficiency with density can have a strong effect on the optical flux.
Clumping could increase the optical flux of our models and
help alleviate the discrepancy with the observations <cit.>.
§ CONCLUSIONS
In this Letter we have studied the importance of radiative transfer in the nebular phase of Type Ia SNe.
Without non-local radiative transfer, only $\sim 5\%$ of the deposited energy emerges in the optical/NIR
at 1000d, the rest being in the MIR ($\sim 80\%$) and UV ($\sim 15 \%$). Non-local scattering and fluoresence, however, converts most of the UV flux into the optical/NIR, raising this by a factor $\sim 4$ to $\sim 20 \%$.
Because of the dominance of non-thermal processes and the UV to optical conversion there is therefore no contradiction between the drop in temperature
caused by the IRC and a sustained optical flux roughly consistent with the radioactive decay. Between $50-80 \%$ of the non-thermal deposition still goes to heating leading to mid-IR line cooling, in particular by [Fe2] $\wl 25.99 \mu$m. With this model improvement, we find a good general reproduction of the spectrum of SN 2011fe at 1000 days with a spectrum dominated by Fe1,
which explains the 'shift' of some lines at this epoch compared to the thermally dominated Fe2-III spectrum at 300 days, discussed by T2015.
We also find a strong need of radioactive input from ${}^{57}$Co at $\sim 1000$ days. Without this the optical spectrum would be underproduced by a factor $\sim 4$, compared to the factor $\sim 2$ in our models, which include $\sim 2 \times 10^{-2} \msun$ of ${}^{57}$Ni. A ${}^{57}$Ni input at this level together with the freeze-out effects appears to be necessary to give the observed optical flux level at this late epoch.
We are grateful to Markus Kromer, Stefan Taubenberger, Melissa Graham and Ken Nomoto for discussions and for models and to the referee for very useful comments. This research was supported by the Swedish Research Council and
National Space Board.
Axelrod, T. S. 1980, PhD thesis, California Univ., Santa Cruz.
[Fransson et al.(1996)Fransson, Houck, &
Fransson, C., Houck, J., & Kozma, C. 1996, in IAU Colloq. 145:
Supernovae and Supernova Remnants, ed. T. S. Kuhn, 211
[Fransson & Kozma(1993)]Fransson1993
Fransson, C., & Kozma, C. 1993, , 408, L25
[Graham et al.(2015)]Graham2015 Graham, M. L., Nugent,
P. E., Sullivan, M., et al. 2015, arXiv:1502.00646
[Graur et al.(2015)Graur, Zurek, Shara, &
Graur, O., Zurek, D., Shara, M. M., & Riess, A. G. 2015, ArXiv
e-prints 1505.00777
[Iwamoto et al.(1999)Iwamoto, Brachwitz, Nomoto,
Kishimoto, Umeda, Hix, & Thielemann]Iwamoto1999
Iwamoto, K., Brachwitz, F., Nomoto, K., et al. 1999, , 125, 439
[Jerkstrand et al.(2015)Jerkstrand, Ergon, Smartt,
Fransson, Sollerman, Taubenberger, Bersten, &
Jerkstrand, A., Ergon, M., Smartt, S. J., et al. 2015, , 573, A12
[Jerkstrand et al.(2011)Jerkstrand, Fransson, &
Jerkstrand, A., Fransson, C., & Kozma, C. 2011, , 530, A45
[Kerzendorf et al.(2014)Kerzendorf, Taubenberger,
Seitenzahl, & Ruiter]Kerzendorf2014
Kerzendorf, W. E., Taubenberger, S., Seitenzahl, I. R., & Ruiter,
A. J. 2014, , 796, L26
[Kozma & Fransson(1992)]Kozma1992
Kozma, C., & Fransson, C. 1992, , 390, 602
& Fransson(1998a)]Kozma1998a Kozma, C., & Fransson, C. 1998, , 496, 946
[Kozma & Fransson(1998b)]Kozma1998b
—. 1998, , 497, 431
[Kozma et al.(2005)Kozma, Fransson, Hillebrandt,
Travaglio, Sollerman, Reinecke, Röpke, &
Kozma, C., Fransson, C., Hillebrandt, W., et al. 2005, , 437, 983
[Leloudas et al.(2009)Leloudas, Stritzinger, Sollerman,
Burns, Kozma, Krisciunas, Maund, Milne, Filippenko, Fransson,
Ganeshalingam, Hamuy, Li, Phillips, Schmidt, Skottfelt,
Taubenberger, Boldt, Fynbo, Gonzalez, Salvo, &
Leloudas, G., Stritzinger, M. D., Sollerman, J., et al. 2009, ,
505, 265
[Mazzali et al.(2015)]Mazzali2015 Mazzali, P. A.,
Sullivan, M., Filippenko, A. V., et al. 2015, , 450, 2631
[Nahar et al.(1997)Nahar, Bautista, & Pradhan]Nahar1997
Nahar, S. N., Bautista, M. A., & Pradhan, A. K. 1997, , 479, 497
[Nomoto et al.(1984)Nomoto, Thielemann, &
Nomoto, K., Thielemann, F.-K., & Yokoi, K. 1984, , 286, 644
[Nugent et al.(2011)]Nugent2011 Nugent, P. E., Sullivan,
M., Cenko, S. B., et al. 2011, , 480, 344
[Pereira et al.(2013)Pereira, Thomas, Aldering, Antilogus,
Baltay, Benitez-Herrera, Bongard, Buton, Canto, Cellier-Holzem,
Chen, Childress, Chotard, Copin, Fakhouri, Fink, Fouchez,
Gangler, Guy, Hillebrandt, Hsiao, Kerschhaggl, Kowalski,
Kromer, Nordin, Nugent, Paech, Pain, Pécontal, Perlmutter,
Rabinowitz, Rigault, Runge, Saunders, Smadja, Tao,
Taubenberger, Tilquin, & Wu]Pereira2013
Pereira, R., Thomas, R. C., Aldering, G., et al. 2013, , 554, A27
[Röpke et al.(2012)]Ropke2012 Röpke, F. K.,
Kromer, M., Seitenzahl, I. R., et al. 2012, , 750, L19
[Seitenzahl et al.(2009)Seitenzahl, Taubenberger, &
Seitenzahl, I. R., Taubenberger, S., & Sim, S. A. 2009, , 400, 531
[Seitenzahl et al.(2013)Seitenzahl, Ciaraldi-Schoolmann,
Röpke, Fink, Hillebrandt, Kromer, Pakmor, Ruiter, Sim, &
Seitenzahl, I. R., Ciaraldi-Schoolmann, F., Röpke, F. K., et al.
2013, , 429, 1156
[Shappee & Stanek(2011)]Shappee2011
Shappee, B. J., & Stanek, K. Z. 2011, , 733, 124
[Sollerman et al.(2004)Sollerman, Lindahl, Kozma, Challis,
Filippenko, Fransson, Garnavich, Leibundgut, Li, Lundqvist,
Milne, Spyromilio, & Kirshner]Sollerman2004
Sollerman, J., Lindahl, J., Kozma, C., et al. 2004, , 428, 555
[Taubenberger et al.(2015)Taubenberger, Elias-Rosa,
Kerzendorf, Hachinger, Spyromilio, Fransson, Kromer, Ruiter,
Seitenzahl, Benetti, Cappellaro, Pastorello, Turatto, &
Taubenberger, S., Elias-Rosa, N., Kerzendorf, W. E., et al. 2015,
, 448, L48 (T2015)
[Tsvetkov et al.(2013)Tsvetkov, Shugarov, Volkov,
Goranskij, Pavlyuk, Katysheva, Barsukova, & Valeev]Tsvetkov2013
Tsvetkov, D. Y., Shugarov, S. Y., Volkov, I. M., et al. 2013,
Contributions of the Astronomical Observatory Skalnate Pleso, 43, 94
|
1511.00395
|
Status Update of the Majorana Demonstrator Neutrinoless Double Beta Decay Experiment
J. Gruszko, M. Buuck, C. Cuesta, J.A. Detwiler, I.S. Guinn, J. Leon, and R.G.H. Robertson
N. Abgrall, A.W. Bradley, Y-D. Chan, S. Mertens, and A.W.P. Poon
I.J. Arnquist, E.W. Hoppe, R.T. Kouzes, B.D. LaFerriere, and J.L. Orrell
F.T. Avignone III
A.S. Barabash, S.I. Konovalov, and V. Yumatov
F.E. Bertrand, A. Galindo-Uribarri, D.C. Radford, R.L. Varner, B.R. White, and C.-H. Yu
V. Brudanin, M. Shirchenko, S. Vasilyev, E. Yakushev, and I. Zhitnikov
M. Busch
D. Byram, B.R. Jasinski, and N. Snyder
A.S. Caldwell, C.D. Christofferson, C. Dunagan, S. Howard, A.M. Suriano
P.-H. Chu, S.R. Elliott, J. Goett, R. Massarczyk, K. Rielage, and W. Xu
Yu. Efremenko
H. Ejiri
T. Gilliss, G.K. Giovanetti, R. Henning, M.A. Howe, J. MacMullin, S.J. Meijer, C. O'Shaughnessy, J. Rager, B. Shanks, J.E. Trimble, and K. Vorren
M.P. Green
V.E. Guiseppe, D. Tedeschi, and C. Wiseman
K.J. Keeter
M.F. Kidd
R.D. Martin
E. Romero-Romero
K. Vetter
J.F. Wilkerson
Neutrinoless double beta decay searches play a major role in determining neutrino properties, in particular the Majorana or Dirac nature of the neutrino and the absolute scale of the neutrino mass. The consequences of these searches go beyond neutrino physics, with implications for Grand Unification and leptogenesis. The Majorana Collaboration is assembling a low-background array of high purity Germanium (HPGe) detectors to search for neutrinoless double-beta decay in $^{76}$Ge. The Majorana Demonstrator, which is currently being constructed and commissioned at the Sanford Underground Research Facility in Lead, South Dakota, will contain 44 kg (30 kg enriched in $^{76}$Ge) of HPGe detectors. Its primary goal is to demonstrate the scalability and background required for a tonne-scale Ge experiment. This is accomplished via a modular design and projected background of less than 3 cnts/tonne-yr in the region of interest. The experiment is currently taking data with the first of its enriched detectors.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
In a number of even-even nuclei, $\beta$ decay is energetically forbidden, but the second-order weak process of 2$\nu$ double-$\beta$ decay is allowed. In this rare decay, first proposed by Goeppert-Mayer in 1935 <cit.>, $(A, Z) \rightarrow (A, Z+2) + 2 e^{-} +2 \nu$. If the neutrino is a Majorana particle, neutrinoless double-$\beta$ decay could also occur via the exchange of a light Majorana neutrino, or by other mechanisms <cit.>. This decay, $(A, Z) \rightarrow (A, Z+2) + 2\beta$, violates lepton number and provides a model-independent test of the nature of the neutrino.
The rate of neutrinoless double-$\beta$ ($0\nu\beta\beta$) decay via light Majorana neutrino exchange is given by
$$ (T_{1/2}^{0\nu})^{-1} = G^{0\nu}|M_{0\nu}|^{2}\left(\frac{\langle m_{\beta\beta} \rangle}{m_e}\right)^2 $$
where $G^{0\nu}$ is a phase space factor, $M_{0\nu}$ is the nuclear matrix element, and $m_e$ is the electron mass. $\langle m_{\beta\beta} \rangle$ is the effective Majorana mass of the exchanged neutrino, $\langle m_{\beta\beta} \rangle = |\sum\limits_{i=1}^3 U^2_{ei}m_i|,$ where $U_{ei}$ specifies the admixture of neutrino mass eigenstate $i$ in the electron neutrino. Because $\langle m_{\beta\beta} \rangle$ depends on the oscillation parameters, both the overall neutrino mass and the mass hierarchy can contribute to the observed rate (see Fig. <ref>). Provided the nuclear matrix elements are understood, $0\nu\beta\beta$ decay experiments could establish an absolute scale for the neutrino mass.
Left: $\langle m_{\beta\beta} \rangle$, and therefore the $0\nu\beta\beta$ rate, depend on the mass hierarchy and overall mass scale. <cit.> Right: The experimental signature of $0\nu\beta\beta$ decay as it would appear in $^{76}$Ge.
The experimental signature of such a decay would be a peak in energy at the endpoint of the two-neutrino mode spectrum with width defined by the experiment's energy resolution, as shown in Fig. <ref>. Given the extremely low rates predicted (current limits indicate $T_{1/2} > 10^{25}$ years) and irreducible background due to the 2$\nu$ mode, high source mass, high efficiency, excellent resolution and extremely low backgrounds in the signal region are key.
In the previous generation of experiments, the most sensitive limits on $0\nu\beta\beta$ decay came from the IGEX <cit.> and Heidelberg-Moscow <cit.> experiments, both using $^{76}$Ge. An observation of $0\nu\beta\beta$ decay was claimed by a subgroup of the Heidelberg-Moscow collaboration <cit.>. Recent searches carried out in $^{76}$Ge (GERDA <cit.>), in $^{136}$Xe (EXO-200 <cit.>, KamLAND-ZEN <cit.>), and in $^{130}$Te (CUORE-0 <cit.>) have set limits that do not support such a claim.
§ THE MAJORANA DEMONSTRATOR
§.§ Overview
The Majorana Demonstrator (MJD) <cit.> is an array of enriched and natural germanium detectors that will search for the $0\nu\beta\beta$ decay of $^{76}$Ge. Its main goal is to achieve backgrounds of 3 counts/tonne-year in the 4 keV region of interest (ROI) around the 2039 keV Q$_{\beta\beta}$ of $^{76}$Ge $0\nu\beta\beta$ decay after analysis cuts. This background level, which scales to 1 count/ROI-t-y in a tonne-scale experiment, is the most aggressive background goal of any current experiment. MJD's additional goals are to establish the feasibility of constructing and fielding modular arrays of germanium detectors and to conduct searches for other physics beyond the standard model, such as WIMP dark matter and axions.
Left: A model of the Majorana DEMONSTRATOR, showing the shielding and veto system. Right: The cryostat of Module 1, being sealed while inside the glovebox.
Photo from http://pics.sanfordlab.org/
MJD consists of 44 kg of P-type point-contact (PPC) germanium detectors. 29.7 kg of this material is enriched to 87% $^{76}$Ge, contained in 35 detectors provided by ORTEC, and the remaining 15 kg is natural-abundance germanium, contained in 24 BEGe detectors from Canberra. The PPC geometry was chosen for two reasons: its small ($\sim$1 pF) capacitance, which leads to good energy resolution ($<$ 2.3 keV FWHM at 1333 keV) and low energy thresholds ($\mathcal{O}$(a few keV)), and its localized weighting potential, which allows for multi-site event rejection.
MJD uses a modular design, making it naturally scalable to tonne-scale and allowing for staged deployment. Two ultra-clean electroformed copper cryostats each contain seven “strings” of three to five detectors each. The two cryostats have independent vacuum and cryogenic systems, and share a compact passive shield and 4$\pi$ active muon veto system. See Fig. <ref>. Implementation is divided into three stages: (1) The Prototype Cryostat, used for R&D, which was made from OFHC copper and contained ten natural Ge detectors, (2) Module 1, which contains 16.8 kg of enriched Ge (20 detectors) and 5.7 kg of natural Ge (9 detectors), and (3) Module 2, which contains 12.6 kg of enriched Ge (14 detectors) and 9.4 kg of natural Ge (15 detectors).
§.§ Background Reduction
MJD is being built and housed at the 4850' level (4260 m.w.e. overburden) of the Sanford Underground Research Facility (SURF) <cit.> in Lead, South Dakota, to reduce cosmogenic activation of materials and muon flux in the experiment. Construction is being done in a class 1000 cleanroom to limit contamination from natural radioactivity, with ultra-low background components (i.e. all parts internal to the cryostats) assembled in a class 10 nitrogen-purged glovebox, to reduce radon contamination. Most low-background parts are machined in an underground shop to control contamination and cosmogenic activation, and tracked extensively <cit.>.
A breakdown of expected background contributions in the Majorana Demonstrator.
To reduce background contributions from natural radioactivity, the Majorana Collaboration has conducted an extensive assay campaign using gamma counting, NAA, ICP-MS, and GDMS. Most components of the experiment are made from ultra-clean copper that has been electroformed underground at SURF and Pacific Northwest National Laboratories. Current assays indicate that this copper has less than 0.1 $\mu$Bq/kg of activity in each of the uranium and thorium decay chains. All other components are low-mass and use low-background materials. Based on assay values and upper limits, the background rate is projected to be $<$ 3.5 counts/ROI-t-y. See Fig. <ref> for details.
Left: The energy spectrum for a natural BEGe detector in the Prototype Cryostat shows that multi-site events are cut effectively, while retaining high efficiency for a nearby single-site peak. Right: In prototype cryostat calibration runs, pulse shape discrimination methods are tuned to retain 90% of known single-site events, found in the $^{208}$Th double escape peak. They reject > 90% of known multi-site events, found in the $^{208}$Th single escape peak. At Q$_{\beta\beta}$, the Compton continuum is reduced by 50%.
Powerful background-rejection analysis techniques further enable the Demonstrator's physics reach. The goal of such techniques is to reject multiple interaction site events due to gamma rays, since they are not plausible double-beta decay candidate events. Using pulse shape discrimination methods <cit.>, it is possible reject 90% of multi-site events while retaining 90% of single-site events and reducing the Compton continuum at $Q_{\beta\beta}$ by approximately 50%, as seen in Fig. <ref>.
§ STATUS OF THE MAJORANA DEMONSTRATOR
The Prototype Module, which included a commercial copper cryostat, was used for research and development of mechanical systems, fabrication and cleaning processes, and assembly procedures. It took data in the shield from July 2014 to June 2015, and has now been decommissioned.
Module 1, which included the first of two ultra-clean cryostats, was moved into the shield at the end of May 2015. It is now in commissioning with 23 of 29 detectors operating, giving 14 kg of enriched mass and 3.7 kg of natural germanium. A calibration spectrum for one of the enriched detectors in Module 1 can be seen in Fig. <ref>. The module is currently taking data without the inner electroformed copper shield in place. It will be removed from the shield and return to the glovebox once more, to allow for the installation of low-background gaskets and repair of the inoperable detectors. Blinded data-taking with Module 1 will begin shortly.
A $^{228}$Th calibration with an enriched detector in Module 1, with incomplete shielding in place.
Module 2, the final stage of the Demonstrator, is in construction; the first detector strings have been built, and the vacuum system is nearing completion. Module 2 is planned to be in commissioning by the end of 2015.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics. We acknowledge support from the Particle Astrophysics Program of the National Science Foundation. This research uses these US DOE Office of Science User Facilities: the National Energy Research Scientific Computing Center and the Oak Ridge Leadership Computing Facility. We acknowledge support from the Russian Foundation for Basic Research. We thank our hosts and colleagues at the Sanford Underground Research Facility for their support. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1256082.
M. Goeppert-Mayer,
Phys. Rev. 48, 512 (1935).
F. T. Avignone, III, S. R. Elliott and J. Engel,
Rev. Mod. Phys. 80, 481 (2008)
[arXiv:0708.1033 [nucl-ex]].
A. Schubert and J. F. Wilkerson,
PhD thesis, The University of Washington (2012)
C. E. Aalseth et al. [IGEX Collaboration],
Phys. Rev. D 65, 092007 (2002)
C. E. Aalseth et al.,
Phys. Rev. D 70, 078302 (2004)
L. Baudis et al.,
Phys. Rev. Lett. 83, 41 (1999)
H. V. Klapdor-Kleingrothaus et al.,
Eur. Phys. J. A 12, 147 (2001)
M. Agostini et al. [GERDA Collaboration],
Phys. Rev. Lett. 111, no. 12, 122503 (2013)
[arXiv:1307.4720 [nucl-ex]].
J. B. Albert et al. [EXO-200 Collaboration],
Nature 510, 229 (2014)
[arXiv:1402.6956 [nucl-ex]].
A. Gando et al. [KamLAND-Zen Collaboration],
Phys. Rev. C 86, 021601 (2012)
[arXiv:1205.6372 [hep-ex]].
K. Alfonso et al. [CUORE Collaboration],
Phys. Rev. Lett. 115, no. 10, 102502 (2015)
[arXiv:1504.02454 [nucl-ex]].
N. Abgrall et al. [Majorana Collaboration],
Adv. High Energy Phys. 2014, 365432 (2014)
[arXiv:1308.1633 [physics.ins-det]].
J. Heise,
J. Phys. Conf. Ser. 606, no. 1, 012015 (2015)
[arXiv:1503.01112 [physics.ins-det]].
N. Abgrall et al. [Majorana Collaboration],
Nucl. Instrum. Meth. A 779, 52 (2015).
D. Budjas, M. Barnabe Heider, O. Chkvorets, N. Khanbekov and S. Schonert,
JINST 4, P10007 (2009)
[arXiv:0909.4044 [nucl-ex]].
|
1511.00350
|
Let $G$ be a connected graph with maximum degree $\Delta$. Brooks' theorem
states that $G$ has a $\Delta$-coloring unless $G$ is a complete graph or an
odd cycle. A graph $G$ is degree-choosable if $G$ can be properly
colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors.
In the context of list coloring, Brooks' theorem can be strengthened to the
following. Every connected graph $G$ is degree-choosable unless each block of
$G$ is a complete graph or an odd cycle; such a graph $G$ is a Gallai
This degree-choosability result was further strengthened to Alon–Tarsi orientations;
these are orientations of $G$ in which the number of spanning Eulerian
subgraphs with an even number of edges differs from the number with an odd
number of edges. A graph $G$ is degree-AT if $G$ has an Alon–Tarsi
orientation in which each vertex has indegree at least 1.
Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also
, , and
Schauz showed that a connected graph is degree-AT if and only if it is not a
Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected
graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such
that $G$ has no Alon–Tarsi orientation in which each vertex has indegree at
least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the
characterization is simple to state.
§ INTRODUCTION
Brooks' theorem is one of the fundamental results in graph coloring.
For every connected graph $G$, it says that $G$ has a $\Delta$-coloring
unless $G$ is a complete graph $K_{\Delta+1}$ or an odd cycle. When we seek to
prove coloring results by induction, we often want to color a subgraph $H$
where different vertices have different lists of allowable colors (those not
already used on their neighbors in the coloring of $G-H$). This gives rise to
list coloring. Vizing <cit.> and, independently, ,
Rubin, and Taylor <cit.> extended Brooks' theorem to list coloring. They
proved an analogue of Brooks' theorem when each vertex $v$ has $\Delta$
allowable colors (possibly different colors for different vertices).
, Rubin, and Taylor <cit.> and Borodin <cit.>
strengthened this Brooks' analogue to the following result, where a
Gallai treeGallai tree is a connected graph in which each
block is a complete graph
or an odd cycle.
If $G$ is connected and not a Gallai tree, then for any list assignment $L$
with $|L(v)|=d(v)$ for all $v\in V(G)$, graph $G$ has a proper coloring
$\varphi$ with $\varphi(v)\in L(v)$ for all $v$.
The graphs in Theorem A are
It is easy to check that every Gallai tree is not degree-choosable. So the set
of all connected graphs that are not degree-choosable are precisely the Gallai trees.
, , and Schauz <cit.> extended this characterization to the setting of
Alon–Tarsi orientations.
For any digraph $D$, a spanning Eulerian subgraph is one in which each vertex
has indegree equal to outdegree. The parity of a spanning Eulerian
subgraph is the parity of its number of edges. For an orientation of a graph
$G$, let EE (resp. EO) denote the number of even (resp. odd) spanning Eulerian
subgraphs. An orientation is Alon–TarsiAlon–Tarsi orientation-.4cm
(or AT) if EE and EO differ.
A graph $G$ is $f$-AT$f$-AT, $k$-AT if it has an Alon–Tarsi
orientation $D$ such that $d^+(v)\le f(v)-1$ for each vertex $v$. In
particular, $G$ is degree-ATdegree-AT (resp. $k$-AT) if
it is $f$-AT, where $f(v)=d(v)$ (resp. $f(v)=k$) for all $v$. Similarly, a
graph $G$ is $f$-choosable$f$-choosable if $G$ has a proper
coloring $\varphi$ from any list assignment $L$ such that $|L(v)|=f(v)$ for all
$v\in V(G)$. Alon and Tarsi <cit.> used algebraic methods to prove the
following theorem for choosability. Later, Schauz <cit.>
strengthened the result to paintability, which we discuss briefly in
Section <ref>.
For a graph $G$ and $\func{f}{V(G)}{\IN}$,
if $G$ is $f$-AT, then $G$ is also $f$-choosable.
In this paper we characterize those graphs $G$ with a specified vertex $x$
that are not $f$-AT, where $f(x)=d(x)-1$ and $f(v)=d(v)$ for all other $v\in
V(G)$. All such graphs are formed from a few 2-connected building blocks, by
repeatedly applying a small number of operations.
Most of the work in the proof is spent on the case when $G$ is 2-connected.
This result is easy to state, so we include it a bit later in the introduction,
as our target:mainLemmaMain Lemma.
Near the end of Section <ref>, with a little more work we extend our
target:mainLemmaMain Lemma, by removing the hypothesis of
2-connectedness, to characterize
all pairs $(G,h_x)$ that are not AT. This result is
Theorem <ref>.
This line of research began with Gallai, who studied the minimum number of edges
in an $n$-vertex $k$-critical graph $G$. Since $G$ has minimum degree at least
$k-1$, clearly $|E(G)|\ge\frac{k-1}2n$. Gallai <cit.> improved this
bound by classifying all connected subgraphs that can be induced by vertices of
degree $k-1$ in a $k$-critical graph. By Theorem A, all such graphs are Gallai
trees. Here, we consider graphs $G$ that are critical with respect to Alon–Tarsi
orientation. Specifically, $G$ is not $(k-1)$-AT, but every proper subgraph is;
such graphs are $k$-AT-critical. The characterization of degree-AT
graphs shows that, much like $k$-critical graphs, in a $k$-AT-critical graph
$G$, every connected subgraph induced by vertices of degree $k-1$ must be a
Gallai tree. Our main result characterizes the subgraphs that can be induced by
vertices of degree $k-1$, together with a single vertex of degree $k$. Thus, it
is natural to expect that this result will lead to improved lower bounds on
the number of edges in $n$-vertex $k$-AT-critical graphs.
Similar to that for
degree-AT, our characterization remains unchanged in the contexts of
list-coloring and paintability, as we show in Section <ref>. We
see a sharp contrast when we consider
graphs $G$ with two specified vertices $x_1$ and $x_2$ that are not $f$-AT,
where $f(x_i)=d(x_i)-1$ for each $i\in \{1,2\}$ and $f(v)=d(v)$ for all other
$v\in V(G)$. For Alon–Tarsi orientations, we have more than 50 exceptional
graphs on seven vertices or fewer. Furthermore, the characterizations for
list-coloring, paintability, and Alon–Tarsi orientations all differ.
We consider graphs with vertices labeled by natural numbers; that is, pairs
$(G,h)$ where $G$ is a graph and $\func{h}{V(G)}{\IN}$. We focus on the case
when $h(x)=1$ for some $x$ and $h(v)=0$ for all other $v$; we denote this
labeling as $h_x$. $h_x$-.45cm We say that $(G, h)$ is
is AT if $G$ is $(d_G - h)$-AT.
When $H$ is an induced subgraph of $G$, we simplify notation by referring to
the pair $(H, h)$ when we really mean $\parens{H, h\restriction_{V(H)}}$.
Given a pair $(G,h)$ and a specified edge $e\in E(G)$, when we stretch
$e$stretch $e$, we form $(G',h')$ from $(G,h)$ by subdividing $e$ twice and setting
$h'(v_i)=0$ for each of the two new vertices, $v_1$ and $v_2$ (and $h'(v)=h(v)$
for all other vertices $v$). In Section <ref>, we prove a
target:SubdivideTwiceStretching Lemma, which shows that if $(G,h)$ is not
AT and $e\in E(G)$, then stretching $e$ often yields another pair $(G',h')$
that is also not AT. Thus, stretching plays a key role in our main result.
It is easy to check that the three pairs $(G,h)$ shown in
Figure <ref> are not AT
(and we do this below, in Proposition <ref>). Let $\D$ $\D$ be the
collection of all pairs formed from the graphs in Figure <ref> by stretching each
bold edge 0 or more times. The target:SubdivideTwiceStretching Lemma
implies that each pair in $\D$ is not AT. Our
target:mainLemmaMain Lemma is that these are
the only pairs $(G,h_x)$, where $G$ is 2-connected and neither complete
nor an odd cycle, such that $(G,h_x)$ is not AT, for some vertex $x\in V(G)$.
Let $G$ be 2-connected and let $x \in V(G)$.
Now $(G,h_x)$ is AT if and only if
(1) $d(x)=2$ and $G-x$ is not a Gallai tree; or
(2) $d(x)\ge 3$, $G$ is not complete, and $(G,h_x) \not \in \D$.
The characterization of degree-choosable graphs has been applied
to prove a variety of graph coloring
results <cit.>.
Likewise, we think our main results in this paper may be helpful in proving
other results for Alon–Tarsi orientations, such as giving better lower bounds
on the number of edges in $k$-AT-critical graphs.
_BoldEdgeStyle = [line width=3]
[scale = 11]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 0.650, y = 0.550, L = $0$]v0
[style = labeledStyle, x = 0.850, y = 0.700, L = $0$]v1
[style = labeledStyle, x = 1.050, y = 0.550, L = $0$]v2
[style = labeledStyle, x = 0.850, y = 0.950, L = $1$]v3
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v1)(v2)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v1)(v3)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v3)(v0)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v3)(v2)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[scale = 11]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 0.650, y = 0.550, L = $0$]v0
[style = labeledStyle, x = 0.850, y = 0.700, L = $0$]v1
[style = labeledStyle, x = 1.050, y = 0.550, L = $0$]v2
[style = labeledStyle, x = 0.850, y = 0.950, L = $1$]v3
[style = labeledStyle, x = 0.750, y = 0.750, L = $0$]v4
[style = labeledStyle, x = 0.850, y = 0.800, L = $0$]v5
[style = labeledStyle, x = 0.950, y = 0.750, L = $0$]v6
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v1)(v2)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v5)(v3)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v5)(v1)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v4)(v3)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v4)(v0)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v6)(v3)
[style = _BoldEdgeStyle, label = , labelstyle=auto=right, fill=none](v6)(v2)
[scale = 11]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 1.000, y = 0.250, L = $0$]v0
[style = labeledStyle, x = 1.200, y = 0.250, L = $0$]v1
[style = labeledStyle, x = 0.850, y = 0.400, L = $0$]v2
[style = labeledStyle, x = 1.000, y = 0.400, L = $0$]v3
[style = labeledStyle, x = 1.350, y = 0.400, L = $0$]v4
[style = labeledStyle, x = 1.200, y = 0.400, L = $0$]v5
[style = labeledStyle, x = 1.100, y = 0.600, L = $1$]v6
[label = , labelstyle=auto=right, fill=none](v0)(v2)
[label = , labelstyle=auto=right, fill=none](v0)(v3)
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v2)(v3)
[label = , labelstyle=auto=right, fill=none](v4)(v1)
[label = , labelstyle=auto=right, fill=none](v5)(v1)
[label = , labelstyle=auto=right, fill=none](v5)(v4)
[label = , labelstyle=auto=right, fill=none](v6)(v2)
[label = , labelstyle=auto=right, fill=none](v6)(v3)
[label = , labelstyle=auto=right, fill=none](v6)(v4)
[label = , labelstyle=auto=right, fill=none](v6)(v5)
Three pairs $(G,h_x)$ that are in $\D$. In each case $x$ is labeled 1
and all other vertices are labeled 0. Each other pair in $\D$ can be formed
from one of these pairs by repeatedly stretching one or more bold edges.
To conclude this section, we show that each pair in $\D$ is not AT.
If $(G,h_x)\in \D$, then $(G,h_x)$ is not AT.
For each pair $(G,h_x)\in \D$, we construct a list assignment $L$ such that
$|L(x)|=d(x)-1$ and $|L(v)|=d(v)$ for all other $v\in V(G)$, but $G$ has no
proper coloring from $L$. Now $(G,h_x)$ is not AT, by the contrapositive of
Theorem B.
Let $(G,h_x)$ be some stretching of the leftmost pair in
Figure <ref>. Assign the
list $\{1,2,3\}$ to each of the vertices on the unbolded triangle and assign the
list $\{1,2\}$ to each other vertex. If $G$ has some coloring from these lists,
then vertex $x$, labeled 1 in the figure, must get color 1 or 2; by symmetry,
assume it is 1. Along each path from $x$ to the triangle, colors must
alternate $2, 1, \ldots$. Each of the paths from $x$ to the triangle has odd
length; thus, color 1 is forbidden from appearing on the triangle. So $G$ has
no coloring from $L$. Now let $(G,h_x)$ be some stretching of the center pair
in Figure <ref>. The proof is identical to the first case, except that each path
has even length, so if $x$ gets color 1, then color 2 is forbidden on the triangle.
Finally, consider the rightmost pair in Figure <ref>. Here $d(x)=4$ and $d(v)=3$
for all other $v\in V(G)$. Thus, it suffices to show that $G$ is not
3-colorable. Assume that $G$ has a 3-coloring and, by symmetry, assume that $x$
is colored 1. Now colors 2 and 3 must each appear on two neighbors of $x$.
Thus, the two remaining vertices must be colored 1. Since they are adjacent,
this is a contradiction, which proves that $G$ is not 3-colorable.
§ SUBGRAPHS, SUBDIVISIONS, AND CUTS
When , , and Schauz characterized degree-AT graphs, their proof relied
heavily on the observation that a connected graph $G$ is degree-AT if and only
if $G$ has some induced subgraph $H$ such that $H$ is degree-AT. Below, we
reprove this easy lemma, and also extend it to our setting of pairs $(G,h_x)$.
Let $G$ be a connected graph and let $H$ be an induced subgraph of $G$. If $H$
is degree-AT, then also $G$ is degree-AT. Similarly, if $x\in V(H)$ and
$(H,h_x)$ is AT, then also $(G,h_x)$ is AT. Further, if $x\notin V(H)$,
$d_G(x)\ge 2$, and $(H,h_x)$ is AT, then $(G,h_x)$ is AT.
Suppose that $H$ is degree-AT, and let $D'$ be an orientation of $H$ showing
this. Extend $D'$ to an orientation $D$ of $G$ by orienting all edges away from
$H$, breaking ties arbitrarily, but consistently. Now every directed cycle in
$D$ is also a directed cycle in $D'$ (and vice versa), so $G$ is degree-AT.
The proof of the second statement is identical. The proof of the third
statement is similar, but now if some edge $xy$ has endpoints equidistant from
$H$, then $xy$ should be oriented into $x$.
Recall that, given a pair $(G,h)$ and a specified edge $e\in E(G)$, when we
stretch $e$, we form $(G',h')$ from $(G,h)$ by subdividing $e$ twice
and setting $h'(v_i)=0$ for each of the two new vertices, $v_1$ and $v_2$
(and $h'(v)=h(v)$ for all other vertices $v$). By repeatedly stretching edges,
starting from the three pairs in Figure <ref>, we form all pairs
$(G,h_x)$, where $G$ is 2-connected and $(G,h_x)$ is not AT.
The following lemma will be useful for proving this.
Form $(G',h')$ from $(G,h)$ by stretching some edge $e\in E(G)$.
(1) if $(G,h)$ is AT, then $(G', h')$ is AT; and
(2) if $(G', h')$ is AT, then either $(G,h)$ is AT or $(G-e,h)$ is AT.
Suppose $e = u_1u_2$ and call the new vertices $v_1$ and $v_2$ so that $G'$ contains
the induced path $u_1v_1v_2u_2$. For (1), let $D$ be an orientation of $G$ showing
that $(G,h)$ is AT. By symmetry we may assume $u_1u_2 \in E(D)$. Form an
orientation $D'$ of $G'$ from $D$ by replacing $u_1u_2$ with the directed path
$u_1v_1v_2u_2$. We have a natural parity preserving bijection between the spanning
Eulerian subgraphs of $D$ and $D'$, so we conclude that $(G', h')$ is AT.
For (2), let $D'$ be an orientation of $G'$ showing that $(G',h')$ is AT.
Suppose $G'$ contains the directed path $u_1v_1v_2u_2$ or the directed path
$u_2v_2v_1u_1$. By symmetry, we can assume it is $u_1v_1v_2u_2$. Now form an
orientation $D$ of $G$ by replacing $u_1v_1v_2u_2$ with the directed edge
$u_1u_2$. As above, we have a parity preserving bijection between the spanning
Eulerian subgraphs of $D$ and $D'$, so we conclude that $(G, h)$ is AT. So
suppose instead that $G'$ contains neither of the directed paths $u_1v_1v_2u_2$
and $u_2v_2v_1u_1$. Now no spanning Eulerian subgraph of $D'$ contains a cycle
passing through $v_1$ and $v_2$. So, the spanning Eulerian subgraph counts of
$D'$ are the same as those of $D' - v_1 - v_2$. However, this gives an
orientation of $G-e$ showing that $(G-e, h)$ is AT.
Given a pair $(G,h)$ that is not AT, the target:SubdivideTwiceStretching Lemma suggests a way to construct a larger graph $G'$ such that $(G',h')$ is
not AT. In some cases, we can also use the
target:SubdivideTwiceStretching Lemma to construct a smaller graph
$\widehat{G}$ such that $(\widehat{G},h)$ is not AT.
Specifically, we have the following.
If $e$ is an edge in $G$ such that $(G,h)$ is not AT and $(G-e, h)$ is not AT,
then stretching $e$ gives a pair $(G',h')$ that is not AT. Further,
let $G$ be a graph with an induced path $u_1v_1v_2u_2$ such that $d_G(v_1) =
d_G(v_2) = 2$. If $(G,h)$ is AT, where $h(v_1) = h(v_2) = 0$, and
$(G-v_1-v_2,h)$ is not AT, then
\[\parens{(G - v_1 - v_2) + u_1u_2,
h\restriction_{V(G) \setminus \set{v_1, v_2}}} \text{ is AT.}\]
The first statement is immediate from the target:SubdivideTwiceStretching Lemma. Now we prove the second. Suppose $(G,h)$ satisfies the
hypotheses. Applying part (2) of the target:SubdivideTwiceStretching Lemma
shows that either $\parens{G - v_1 - v_2, h\restriction_{V(G) \setminus
\set{v_1, v_2}}}$ is AT or $\parens{(G - v_1 - v_2) + u_1u_2,
h\restriction_{V(G) \setminus \set{v_1, v_2}}}$ is AT.
By hypothesis, the former is false. Thus, the latter is true.
With standard vertex coloring, we can easily reduce to the case where $G$ is
2-connected. If $G$ is a connected graph with two blocks, $B_1$ and $B_2$,
meeting at a cutvertex $x$, then we can color each of $B_1$ and $B_2$
independently, and afterward we can permute colorings to match at $x$.
For Alon–Tarsi orientations, the situation is not quite as simple. However,
the following lemma plays a similar role for us.
Let $A_1, A_2 \subseteq V(G)$, and $x\in V(G)$ be such that $A_1\cup A_2=V(G)$ and $A_1 \cap
A_2 = \set{x}$. If $G[A_i]$ is $f_i$-AT for each $i \in \{1,2\}$, then $G$ is
$f$-AT, where $f(v) = f_i(v)$ for each $v \in V(A_i-x)$ and $f(x) = f_1(x) + f_2(x)
- 1$. Going the other direction, if $G$ is $f$-AT, then $G[A_i]$ is $f_i$-AT
for each $i \in \{1,2\}$, where $f_i(v) = f(v)$ for each $v \in V(A_i-x)$ and
$f_1(x) + f_2(x) \le f(x) + 1$.
We begin with the first statement.
For each $i \in \{1,2\}$, choose an orientation $D_i$ of $A_i$ showing that $A_i$
is $f_i$-AT. Together these $D_i$ give an orientation $D$ of $G$. Since no cycle
has vertices in both $A_1-x$ and $A_2-x$, we have
\begin{align*}
EE(D) - EO(D) &= EE(D_1)EE(D_2) + EO(D_1)EO(D_2) - EE(D_1)EO(D_2) - EO(D_1)EE(D_2) \\
&= (EE(D_1) - EO(D_1))(EE(D_2) - EO(D_2)) \\
&\ne 0.
\end{align*}
Hence $G$ is $f$-AT.
Now we prove the second statement. Suppose that $G$ is $f$-AT and choose an
orientation $D$ of $G$ showing this.
Let $D_i = D[A_i]$ for each $i \in \{1,2\}$. As above, we have $0 \ne
EE(D) - EO(D) = (EE(D_1) - EO(D_1))(EE(D_2) - EO(D_2))$. Hence, $EE(D_1) -
EO(D_1) \ne 0$ and $EE(D_2) - EO(D_2) \ne 0$. Since the indegree of $x$ in
$D$ is the sum of the indegree of $x$ in $D_1$ and the indegree of $x$ in
$D_2$, the lemma follows.
§ DEGREE-AT GRAPHS AND AN EXTENSION LEMMA
Recall that our target:mainLemmaMain Lemma relies on a
characterization of degree-AT graphs.
As we mentioned in the introduction, a description of degree-choosable graphs
was first given by Borodin <cit.> and , Rubin, and
Taylor <cit.>. , , and Schauz <cit.> later extended the
proof from <cit.> to Alon–Tarsi orientations. This proof relies on
Rubin's Block lemma, which states that every 2-connected graph $G$ contains an
induced even cycle with at most one chord, unless $G$ is complete or an
odd cycle. For variety, and completeness, we include a new proof; it extends
ideas of Kostochka, Stiebitz, and Wirth <cit.> from list-coloring to
Alon–Tarsi orientations. For this proof we need the following very special
case of a key lemma in <cit.>. When vertices $x$ and $y$ are
adjacent, we write $x\adj y$; otherwise $x\nonadj y$.
Let $G$ be a graph and $x \in V(G)$ such that $H$ is connected, where $H \DefinedAs G-x$.
If there exist $z_1, z_2 \in V(H)$ with $N_H[z_1] = N_H[z_2]$ such that $x \adj
z_1$ and $x \nonadj z_2$, then $G$ is $f$-AT where $f(x) = 2$ and $f(v) =
d_G(v)$ for all $v \in V(H)$.
Order the vertices of $H$ with $z_1$ first and $z_2$ second so that every
vertex, other than $z_1$, has at least one neighbor preceding it.
Orient each edge of $H$ from its earlier endpoint toward its later endpoint.
Orient $xz_1$ into $z_1$ and orient all other
edges incident to $x$ into $x$. Let $D$ be the resulting orientation.
Clearly, $d_{D}^+(v) \le f(v) - 1$ for all $v \in V(D)$. So, we just need to
check that $EE(D) \ne EO(D)$.
Since $xz_1$ is the only edge of $D$ leaving
$x$, and $D-x$ is acyclic, every spanning Eulerian subgraph of $D$ that has
edges must have edge $xz_1$.
Consider an Eulerian subgraph $A$ of $D$ containing $xz_1$. Since $z_1$
has indegree $1$ in $A$, it must also have outdegree $1$ in $A$. We show
that $A$ has a mate $A'$ of opposite parity.
If $z_2 \in A$ then $z_1z_2w \in A$, for some $w$, so we form
$A'$ from $A$ by removing $z_1z_2w$ and adding $z_1w$.
If instead $z_1z_2\notin A$, then $z_2 \not \in A$ and $z_1w \in
A$ for some $w \in N_H[z_1]-z_2$, so we form $A'$ from $A$ by removing $z_1w$ and
adding $z_1z_2w$.
Hence exactly half of the Eulerian subgraphs of $D$ that contain edges are
even. Since the edgeless spanning subgraph of $D$ is an even Eulerian
subgraph, we conclude that $EE(D) = EO(D) + 1$. Hence $G$ is $f$-AT.
We use the previous lemma to give a new proof of the characterization of
degree-AT graphs.
A connected graph $G$ is degree-AT if and only if it is not a Gallai tree.
We begin with the “only if” direction. Neither odd cycles nor complete graphs
are degree-choosable. Thus, by target:thmBTheorem B, they are not
degree-AT. By induction on the number of blocks, Lemma <ref>
implies that no Gallai tree is degree-AT.
Now, the “if” direction.
Suppose there exists a connected graph that is not a Gallai tree, but is also not
degree-AT. Let $G$ be such a graph with as few vertices as possible.
Since $G$ is not degree-AT, no induced subgraph $H$ of $G$ is
degree-AT by the target:InducedSubgraphSubgraph Lemma.
Hence, for any $v \in V(G)$ that is not a cutvertex, $G-v$ must be a Gallai
tree by minimality of $|G|$.
If $G$ has more than one block, then for endblocks $B_1$ and $B_2$, choose
noncutvertices $w\in B_1$ and $x\in B_2$. By the minimality of $|G|$, both
$G-w$ and $G-x$ are Gallai trees. Since every block of $G$ appears either as a
block of $G-w$ or as a block of $G-x$, every block of $G$ is either complete or
an odd cycle. Hence, $G$ is a Gallai tree, a contradiction. So instead $G$
has only one block, that is, $G$ is $2$-connected. Further, $G-v$ is a Gallai
tree for all $v \in V(G)$.
Let $v$ be a vertex of minimum degree in $G$. Since $G$ is $2$-connected,
$d_G(v) \ge 2$ and $v$ is adjacent to a noncutvertex in every endblock of $G-v$.
If $G-v$ has a complete block $B$ with noncutvertices $x_1,x_2$ where $v \adj
x_1$ and $v \nonadj x_2$, then we can apply Lemma <ref>
to conclude that $G$ is degree-AT, a
contradiction. So, $v$ must be adjacent to every noncutvertex in every
complete endblock of $G-v$.
Suppose $d_G(v) \ge 3$. Now no endblock of $G-v$ can be an odd cycle of
length at least $5$ ($G$ would have vertices of degree $3$ and also $d_G(v) \ge
4$, contradicting the minimality of $d_G(v)$). Let $B$ be a smallest complete
endblock of $G-v$. Now for a noncutvertex $x \in V(B)$, we have $d_G(x) =
|B|$ and hence $d_G(v) \le |B|$.
If $G-v$ has at least two endblocks, then $2(|B|-1) \le |B|$, so $d_G(v)
\le |B| = 2$, a contradiction. Hence, $G-v = B$ and $v$ is joined to $B$, so
$G$ is complete, which is a contradiction.
Thus, we have $d_G(v) = 2$. Suppose $G-v$ has at least two endblocks.
Now it has exactly two and $v$ is adjacent to one noncutvertex in each.
Neither of the endblocks can be odd cycles of length at least five, since then
we can get a smaller counterexample by the target:SubdivideTwiceStretching Lemma. Since $v$ is adjacent to every noncutvertex in every
complete endblock of $G-v$, both endblocks must be $K_2$. But now either
$G=C_4$ (which is degree-AT, by orienting the cycle consistently) or we can get
a smaller counterexample by the target:SubdivideTwiceStretching Lemma.
So, $G-v$ must be $2$-connected. Since $G-v$ is a Gallai tree, it is either
complete or an odd cycle. If $G-v$ is not complete, then we can get a smaller
counterexample by the target:SubdivideTwiceStretching Lemma. So, $G-v$
is complete and $v$ is adjacent to every noncutvertex of $G-v$; that is, $G$ is
complete, a contradiction.
§ WHEN H IS 1 FOR ONE VERTEX
In this section, we prove our target:mainLemmaMain Lemma.
For a graph $G$ and $x \in V(G)$ recall that
$\func{h_x}{V(G)}{\IN}$ is defined as $h_x(x) = 1$ and $h_x(v) = 0$ for all $v
\in V(G-x)$. We classify the connected graphs $G$ such that $(G,h_x)$ is AT for
some $x \in V(G)$. We begin with the case when $G$ is 2-connected, which takes
most of the work. At the end of the section, we extend our characterization to
all connected graphs.
We will show that for most 2-connected graphs $G$ and vertices $x\in V(G)$, the
pair $(G,h_x)$ is AT. Specifically, this is true for all pairs except those in
$\D$, defined in the introduction. In view of the
target:InducedSubgraphSubgraph Lemma, for a 2-connected graph $G$ and
$x\in V(G)$, to show that $(G,h_x)$ is AT it suffices to
find some induced subgraph $H$ such that $(H,h_x)$ is AT.
The subgraphs $H$ that we consider all have $d_H(x)=0$ or $d_H(x)\ge 3$. This
motivates the next lemma, which allows us to reduce to the case $d_G(x)\ge 3$.
If $G$ is a connected graph and $x \in V(G)$ with $d_G(x) = 2$, then $(G,h_x)$
is AT if and only if $G-x$ is degree-AT.
Let $D$ be an orientation of $G$ showing that $(G,h_x)$ is AT. Now
$d_{D}^-(x) = 2$, so no spanning Eulerian subgraph contains a cycle
passing through $x$. Therefore, the Eulerian subgraph counts in $G-x$ are
different and $G-x$ is degree-AT. The other direction is immediate from the
target:InducedSubgraphSubgraph Lemma.
Lemma <ref>, together with Lemma <ref>,
proves Case (1) of our target:mainLemmaMain Lemma. Before we can prove Case
(2), we need a few more definitions and lemmas.
A $\theta$-graph$\theta$-graph consists of two vertices joined by
three internally disjoint paths, $P_1$, $P_2$, and $P_3$. When we write $h_x$
for a $\theta$-graph, we always assume that $d(x)=3$. We will see
shortly that if $H$ is a $\theta$-graph with $d_H(x)=3$, then $(H,h_x)$ is AT.
Thus, the target:InducedSubgraphSubgraph Lemma implies that
if $(G,h_x)$ is not AT, then $G$ has no induced $\theta$-graph $H$ with $d_H(x)=3$.
A $T$-graph$T$-graph is formed from vertices $x, z_1, z_2, z_3$,
by making the $z_i$
pairwise adjacent, and joining each vertex $z_i$ to $x$ by a path $P_i$ (where
the $P_i$ are disjoint). Equivalently, a $T$-graph is formed from $K_4$ by
subdividing each of the edges incident to $x$ zero or more times.
Similar to the proof characterizing degree-AT graphs in <cit.>,
our approach in proving our target:mainLemmaMain Lemma
is to find an induced subgraph $H$ such that $(H,h_x)$ is AT, and apply the
target:InducedSubgraphSubgraph Lemma.
Thus, we need the following lemma about pairs $(H,h_x)$ that are AT.
The pair $(G,h_x)$ is AT whenever (i) $G$ is a $\theta$-graph, (ii) $G$ is a
$T$-graph and two paths $P_i$ have lengths of opposite parities, or (iii) $G$ is
formed from a $T$-graph by adding an extra vertex with neighborhood
In each case, we give an AT orientation $D$ of $G$ such that $d_D^-(v)\ge
h_x(v)+1$ for each $v\in V(G)$.
Case (i). Orient the edges of each path $P_i$ consistently, with $P_1$ and
$P_2$ into $x$ and $P_3$ out of $x$; this orientation satisfies the degree
requirements. Further, it has exactly three spanning Eulerian subgraphs,
including the empty subgraph. Thus, $EE+EO$ is odd, so $EE\ne EO$.
Case (ii). Let $P_1$ and $P_2$ be two paths with opposite parities. As before,
orient the edges of each path consistently, with $P_1$ and $P_2$ into $x$ and
$P_3$ out of $x$. Orient the three additional edges as $\vec{z_1z_2},
\vec{z_2z_3}$, and $\vec{z_3z_1}$. The resulting digraph $D$ has four spanning
Eulerian subgraphs, 3 of one parity and 1 of the other. Note that the empty
subgraph and the subgraph $\{\vec{z_1z_2}, \vec{z_2z_3}, \vec{z_3z_1}\}$ have
opposite parities. Further, the parities are the same for the two subgraphs
consisting of the directed cycles $xP_3z_3z_1P_1$ and $xP_3z_3z_1z_2P_2$. So,
$EE\ne EO$.
Case (iii). The simplest instance of this case is when $G=K_5-e$. Now
$(G,h_x)$ is AT by Lemma <ref>. In fact, that proof gives the
stronger statement that there exists an orientation $D$ satisfying the degree
requirements such that $EE(D)=EO(D)+1$. In particular, $EE+EO$ is odd.
To handle larger instances of this case, we repeatedly subdivide edges incident
to $x$ and orient each of the resulting paths consistently, and in the direction
of the corresponding edge in $D$. The resulting orientation satisfies the
degree requirements. Further, the sum $EE+EO$ remains unchanged, and thus odd.
Hence, still $EE\ne EO$.
Let $G$ be a $T$-graph. Let $P$ be a path of $G$ where all internal
vertices of $P$ have degree 2 in $G$ and one endvertex of $P$ has degree 2 in
$G$. Form $G'$ from $G$ by adding a path $P'$ (of length at least 2) joining
the endvertices of $P$. Now $(G', h_x)$ is AT.
We can assume that $G$ is not AT; otherwise, we are done by the
target:InducedSubgraphSubgraph Lemma. By symmetry, assume $P$ is a
subpath of $P_3$. First, we get an orientation of $G$ with indegree at least 1
for all vertices and $d^-(x) = 2$. Orient $P_1$ from $z_1$ to $x$, $P_2$ from
$z_2$ to $x$, $P_3$ from $x$ to $z_3$, and the triangle as $\vec{z_1z_2},
\vec{z_2z_3}$, and $\vec{z_3z_1}$. To get an orientation of $G'$, orient the
new path $P'$ consistently, and opposite of $P$. Now the only directed cycle
containing edges of $P'$ is $P'P$. Since the Eulerian subgraph counts are
equal for $G$, they differ by 1 for $G'$.
Now we can prove Case (2) of our target:mainLemmaMain Lemma. For
reference, we restate it.
[scale = 8]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 0.650, y = 0.550, L = $0$]v0
[style = labeledStyle, x = 0.850, y = 0.700, L = $0$]v1
[style = labeledStyle, x = 1.050, y = 0.550, L = $0$]v2
[style = labeledStyle, x = 0.850, y = 0.950, L = $1$]v3
[style = labeledStyle, x = 0.750, y = 0.750, L = $0$]v4
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v1)(v2)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[label = , labelstyle=auto=right, fill=none](v4)(v0)
[label = , labelstyle=auto=right, fill=none](v4)(v3)
[label = , labelstyle=auto=right, fill=none](v3)(v1)
[label = , labelstyle=auto=right, fill=none](v3)(v2)
[scale = 8]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 0.650, y = 0.550, L = $0$]v0
[style = labeledStyle, x = 0.850, y = 0.700, L = $0$]v1
[style = labeledStyle, x = 1.050, y = 0.550, L = $0$]v2
[style = labeledStyle, x = 0.850, y = 0.950, L = $1$]v3
[style = labeledStyle, x = 0.750, y = 0.750, L = $0$]v4
[style = labeledStyle, x = 0.950, y = 0.750, L = $0$]v5
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v1)(v2)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[label = , labelstyle=auto=right, fill=none](v4)(v0)
[label = , labelstyle=auto=right, fill=none](v4)(v3)
[label = , labelstyle=auto=right, fill=none](v5)(v2)
[label = , labelstyle=auto=right, fill=none](v5)(v3)
[label = , labelstyle=auto=right, fill=none](v3)(v1)
The pair $(G,h_x)$ is AT, when $G$ is formed from $K_4$ by subdividing
one or two edges incident to $x$.
[scale = 8]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 0.650, y = 0.550, L = $0$]v0
[style = labeledStyle, x = 0.850, y = 0.700, L = $0$]v1
[style = labeledStyle, x = 1.050, y = 0.550, L = $0$]v2
[style = labeledStyle, x = 0.850, y = 0.950, L = $1$]v3
[style = labeledStyle, x = 0.850, y = 0.600, L = $0$]v4
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v1)(v2)
[label = , labelstyle=auto=right, fill=none](v1)(v3)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[label = , labelstyle=auto=right, fill=none](v3)(v0)
[label = , labelstyle=auto=right, fill=none](v3)(v2)
[label = , labelstyle=auto=right, fill=none](v4)(v1)
[label = , labelstyle=auto=right, fill=none](v4)(v0)
[label = , labelstyle=auto=right, fill=none](v4)(v2)
[scale = 8]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 0.650, y = 0.550, L = $0$]v0
[style = labeledStyle, x = 0.850, y = 0.700, L = $0$]v1
[style = labeledStyle, x = 1.050, y = 0.550, L = $0$]v2
[style = labeledStyle, x = 0.850, y = 0.950, L = $1$]v3
[style = labeledStyle, x = 0.750, y = 0.750, L = $0$]v4
[style = labeledStyle, x = 0.850, y = 0.800, L = $0$]v5
[style = labeledStyle, x = 0.950, y = 0.750, L = $0$]v6
[style = labeledStyle, x = 0.850, y = 0.600, L = $0$]v7
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v1)(v2)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[label = , labelstyle=auto=right, fill=none](v4)(v0)
[label = , labelstyle=auto=right, fill=none](v4)(v3)
[label = , labelstyle=auto=right, fill=none](v5)(v1)
[label = , labelstyle=auto=right, fill=none](v5)(v3)
[label = , labelstyle=auto=right, fill=none](v6)(v2)
[label = , labelstyle=auto=right, fill=none](v6)(v3)
[label = , labelstyle=auto=right, fill=none](v7)(v1)
[label = , labelstyle=auto=right, fill=none](v7)(v0)
[label = , labelstyle=auto=right, fill=none](v7)(v2)
(a) The pair $(G,h_x)$ is AT, where $G=K_5-xy$.
(b) The pair $(G,h_x)$ is AT, where $G$ is formed from $K_5-e$ by
subdividing each edge incident to $x$.
Let $G$ be 2-connected, and choose $x\in V(G)$ with $d(x)\ge 3$.
Now $(G,h_x)$ is AT if and only if $G$ is not complete and $(G,h_x) \not \in \D$.
When $(G,h_x)\in \D$ the lemma holds by Proposition <ref>.
Now let $G$ be 2-connected, choose $x\in V(G)$ with $d(x)\ge 3$, and suppose that
$(G,h_x)\notin \D$. Since $G-x$ is connected, let $H'$ be a smallest connected
subgraph of $G-x$ containing three neighbors of $x$; call these neighbors $w_1$,
$w_2$, and $w_3$. Consider a spanning tree $T$ of $H'$. Since $H'$ is minimum,
each leaf of $T$ is among $\{w_1, w_2, w_3\}$. If $T$ is a path, then $H'$ is also a
path. Otherwise, $T$ is a subdivision of $K_{1,3}$. Let $s$ be the vertex with
$d_T(s)=3$. If $E(G)-E(T)$ has any edge with both ends outside of $N(s)$, then
we can delete some vertex in $N(s)$ and remain connected, contradicting the
minimality of $H'$. Similarly, if $N(s)$ contains at least two edges, then
$H'-s$ still connects, $w$, $y$, and $z$. Now let $H$ be the subgraph of $G$
induced by $V(H')\cup\{x\}$. Note that $H$ is either a $\theta$-graph (if $H'$
is a tree) or a $T$-graph (if $H'$ has one extra edge in $N(s)$).
If $H$ is a $\theta$-graph, then $(G,h_x)$ is AT, by
Lemma <ref>.i and
the target:InducedSubgraphSubgraph Lemma.
So assume $H$ is a $T$-graph. Let $z_1$, $z_2$, $z_3$ be the vertices
of degree 3 (other than $x$), and let $P_1$, $P_2$, and $P_3$ denote the paths
from $x$ to $z_1$, $z_2$, and $z_3$; when we write $V(P_i)$, we exclude $x$ and
$z_i$, so possibly $V(P_i)$ is empty for one or more $i\in\{1,2,3\}$.
If any two of $P_1$, $P_2$, and $P_3$ have lengths with opposite parities, then
we are done by Lemma <ref>.ii; so assume not.
Now $(H,h_x)\in \D$, so we can assume that $V(G-H) \ne \emptyset$. Choose $u
\in V(G-H)$, and let $H_u$ be a minimal $2$-connected induced subgraph of $G$
that contains $V(H) \cup \set{u}$. By the target:InducedSubgraphSubgraph Lemma and Lemma <ref>, $G-x$ is a Gallai tree.
Thus, so is $H_u-x$; in particular, the block $B_u$ of $H_u-x$ containing $u$
is complete or an odd cycle. Therefore, we either have (i) $V(B_u) \cap V(H) =
\set{z_1, z_2,z_3}$ or (ii) $V(B_u) \cap V(H) \subseteq P_i \cup \set{z_i}$
for some $i \in \{1,2,3\}$.
Suppose (i) happens. Now $N_G(u) \cap V(H_u - x) = \set{z_1,z_2,z_3}$. If $x
\nonadj u$, then $(G,h_x)$ is AT by the target:InducedSubgraphSubgraph Lemma and Lemma <ref>.iii. If $x \adj u$, then $x$ must have odd
length paths to each $z_i$, by Lemma <ref>.ii, with $u$ in the role
of some $z_i$. Further, $x \adj z_i$ for all $i \in \{1,2,3\}$, since
otherwise $(G,h_x)$ is AT by the target:InducedSubgraphSubgraph Lemma,
Lemma <ref>.iii, and the target:SubdivideTwiceStretching Lemma. So, $H=K_4$ and $H_u=K_5$. This implies that (ii) cannot happen for
any vertex in $V(G-H)$, since if $V(B_u)\cap V(H)=\{z_i\}$ for some $i$, then
$(G,h_x)$ is AT by Lemma <ref>.i
and the target:InducedSubgraphSubgraph Lemma).
So (i) happens for every vertex in $V(G-H)$; in particular, $V(G-H)$ is joined
to $\set{x,z_1,z_2,z_3}$. Since $G$ is not complete, $G-x$ must contain an induced
copy of Figure <ref>(a); hence, $(G,h_x)$ is AT by
Lemma <ref>.iii and the target:InducedSubgraphSubgraph Lemma.
Assume instead that (ii) happens for every vertex in $V(G-H)$, including $u$.
By symmetry, assume that $V(B_u) \cap V(H) \subseteq P_1$. Let $z_1P_1
= v_1v_2\cdots v_{\ell}$, where $v_{\ell}\adj x$. First, assume that $B_u$ is
an odd cycle of length at least $5$. If there is $u' \in V(B_u)\setminus V(H)$
with $u' \adj x$, then $G$ contains a $\theta$-graph and $(G,h_x)$ is AT, by
Lemma <ref>.i and the target:InducedSubgraphSubgraph Lemma. So, we may assume that $u' \nonadj x$ for all $u' \in V(B_u)\setminus
V(H)$. Now we are done by Lemma <ref> and the
target:InducedSubgraphSubgraph Lemma.
So instead we assume that $B_u$ is complete. If $V(B_u) \cap V(H)=\set{v_\ell}$,
then $G$ has an induced $\theta$-graph $J$, where $d_J(x)=d_J(v_\ell)=3$, so we are
done by Lemma <ref>.i and the target:InducedSubgraphSubgraph Lemma. Thus, we must have $V(B_u) \cap V(H) = \set{v_{j}, v_{j+1}}$ for some
$j \in \irange{\ell-1}$. In particular, $B_u$ is a triangle. If $u\nonadj x$,
then $(G,h_x)$ is AT by the target:InducedSubgraphSubgraph Lemma and
Lemma <ref>. So we conclude that $u\adj x$, which requires
$j=\ell-1$, by the minimality of $H$. Hence, $H_u$ is formed from a $T$-graph
by adding a vertex $u$ that is adjacent to $x$ and also to the vertices of a
$K_2$ endblock $D_u$ of $H-x$.
Suppose there are distinct vertices $u_1, u_2\in V(G-H)$
adjacent to vertices of the same $K_2$ endblock.
Now $G$ contains an induced copy of Figure <ref>(a), so
$(G,h_x)$ is AT by Lemma <ref>.iii and the
target:InducedSubgraphSubgraph Lemma. Thus, each $K_2$ endblock has at
most one such $u$.
Let $t$ be the number of $K_2$ endblocks in $H-x$.
By construction, $t\le 3$; this implies that $|V(G - H)| \le t\le 3$.
If $t = 0$, then $G = H = K_4$, which contradicts that $G$ is not complete.
If $t=1$, then $G = H_u$, for the unique $u \in V(G-H)$; this is the Moser
spindle, shown in Figure <ref>(c). So, assume that $t \in
\set{2,3}$. By symmetry, assume that for each $i\in\{1,2\}$ there exists $u_i$
such that $V(B_{u_i})\subseteq P_i\cup\{z_i\}$. Now the subgraph induced by
$\set{u_2}\cup V(H-P_1)$ is reducible by Lemma <ref>.
So, again we are done by the target:InducedSubgraphSubgraph Lemma.
Taken together, Lemmas <ref>
and <ref>, with
Lemma <ref>, prove our target:mainLemmaMain Lemma.
However, this characterizaton requires that $G$ be 2-connected.
Now we extend our result to the more general case, when $G$ need only be
connected. We use the following two definitions. Let $G$ be a graph, $x$
a vertex of $G$, and $B$ a block of $G$. An $x$-lobe of
$G$$x$-lobe is a maximal subgraph $A$ such that $A-x$ is connected. A
$B$-lobe of $G$$B$-lobe is a maximal subgraph $A$ such that
$A-B$ is connected, and $A$ includes a single vertex of $B$.
If $G$ is connected and $x \in V(G)$, then $(G, h_x)$ is not AT if and only if
(1) $G$ is a Gallai tree; or
(2) $d(x) = 1$; or
(3) $d(x) = 2$ and $G-x$ has a component that is a Gallai tree; or
(4) $x$ is not a cutvertex, for the block $B$ of $G$ containing $x$,
we have $(B,h_x) \in \D$, and every other block of $G$ is complete or
an odd cycle; or
$x$ is a cutvertex, all but at most one $x$-lobe of $G$, say $A$, is a Gallai
tree, and either:
(i) $d_A(x) = 1$; or
(ii) $d_A(x)=2$ and $A-x$ is a Gallai tree; or
(iii) for the block $B$ of $A$ containing $x$,
we have $(B,h_x)\in \D$ and all $B$-lobes of $A$ are Gallai trees.
First, we check that if any of Cases (1)–(5) hold, then $(G, h_x)$ is not AT.
Cases (1) and (2) are immediate. Case (3) follows from
Lemma <ref>.
Consider Case (4).
By Proposition <ref>, we know $(B,h_x)$ is not AT.
Now $(G,h_x)$ is not AT by repeated application of Lemma <ref>.
Finally, Case (5) follows from Cases (2), (3), and (4), by
Lemma <ref>.
Now, for the other direction, suppose $(G, h_x)$ is not AT and none of Cases
(1)–(5) hold. By Lemma <ref>, and not (2) and not (3), we
must have $d(x) \ge 3$.
Suppose $x$ is a cutvertex. Now, by not (5), either (a) at least two $x$-lobes
of $G$ are not Gallai trees or (b) $(H, h_x)$ is AT for some $x$-lobe $H$ of
$G$. In each case, $(G,h_x)$ is AT by Lemma <ref>, which is a
So assume instead that $x$ is not a cutvertex. Suppose the block $B$ of $G$
containing $x$ is complete or $(B,h_x) \in \D$. By not (1) and not (4),
some $B$-lobe $H$ of $G$ is not a Gallai tree. Since $H$ is a subgraph of
$G-x$, and $G-x$ is connected, Lemma <ref> and the
target:InducedSubgraphSubgraph Lemma imply that $G-x$ is degree-AT;
hence, $(G,h_x)$ is also AT. So, we conclude that $B$ is not complete and
$(B,h_x)\notin \D$.
First suppose that $d(x)=2$. By not (3), we know that $G-x$ is not a Gallai
tree. Lemma <ref> implies that $G-x$ is degree-AT.
So, again, the target:InducedSubgraphSubgraph Lemma shows that $(G,h_x)$
is AT. Now assume instead that $d(x)\ge 3$. Since $(B,h_x)\notin \D$, now
Lemma <ref> implies that $(B,h_x)$ is AT;
once more, the target:InducedSubgraphSubgraph Lemma implies that $(G,h_x)$
is AT.
§ CHOOSABILITY AND PAINTABILITY
As we mentioned in the introduction, Alon and Tarsi showed that if a graph $G$
is $f$-AT, then $G$ is also $f$-choosable. Online list coloring, also
called painting is similar to list coloring, but now the list for each
vertex is progressively revealed, as the graph is colored.
Schauz <cit.> extended the Alon–Tarsi theorem, to show that
if $G$ is $f$-AT, then $G$ is also $f$-paintable (which we define formally
below). In this section, we use our
characterization of pairs $(G,h_x)$ that are not AT to prove characterizations
of pairs $(G,h_x)$ that are not paintable and that are not choosable. More
precisely, a pair $(G,h_x)$ is choosablechoosable pair-.3cm
if $G$ has a proper coloring from its lists $L$ whenever $L$ is such that
$|L(x)|=d(x)-1$ and $|L(v)|=d(v)$ for all other $v$; otherwise $(G,h_x)$ is
not choosable. A pair being paintablepaintable
pair-.3cm is defined analogously. We characterize all pairs $(G,h_x)$,
where $G$ is connected and $(G,h_x)$ is not choosable (resp. not paintable).
In fact, we will see that these characterizations, for both choosability and
paintability, are identical to that for pairs that are not AT.
For completeness, we include the following definition of $f$-paintable.
Schauz <cit.> gave a more intuitive (yet equivalent) definition,
in terms of a two player game. We say that $G$ is $f$-paintable
$f$-paintable if either (i) $G$ is empty or (ii) $f(v) \ge 1$ for all
$v \in V(G)$ and for every $S \subseteq V(G)$ there is an independent set $I
\subseteq S$ such that $G-I$ is $f'$-paintable where $f'(v) \DefinedAs f(v)$
for all $v \in V(G) - S$ and $f'(v) \DefinedAs f(v) - 1$ for all $v \in S - I$.
Since all pairs $(G,h_x)$ that are AT are also both paintable and choosable, it
suffices to show that every pair $(G,h_x)$ that is not AT is also not choosable
(here we use that if a pair is paintable, then it is also choosable).
For every connected graph $G$, the pair $(G,h_x)$ is not choosable if and only
if $(G,h_x)$ is not AT. Thus, the same characterization holds for pairs that
are not paintable.
As noted above, every pair that is AT is also choosable and paintable. Thus, it
suffices to show that each pair $(G,h_x)$ in Theorem <ref> is not
To show that Gallai trees are not degree-choosable, assign to each block $B$ a
list of colors $L_B$ such that $|L_B|=d_B(x)$ for each $x\in V(B)$; further, for
all distinct blocks $B_1$ and $B_2$, we require that $L_{B_1}$ and $L_{B_2}$ are
disjoint. For each $v\in V(G)$, let $L(v)=\cup_{B_i\ni v}L_{B_i}$. To show
that $G$ is not colorable from these lists, we use induction on the number of
blocks. Let $B$ be an endblock and $x$ a cutvertex in $B$. Let
$G'=G\setminus(V(G)-x)$. Since $B$ is complete or an odd cycle, $B$ has no
coloring from $L_B$. Thus any coloring $\varphi$ of $G$ from $L$ does not use
$L_B$ on $x$. Hence, $\varphi$ gives a coloring $\varphi'$ of $G'$ from its
lists $L'$, where $L'(x)=L(x)\setminus L_B$ and $L'(v)=L(v)$ for all $v\in
V(G)\setminus V(B)$. This coloring $\varphi'$ of $G'$ contradicts the induction
hypothesis. Thus, $G$ has no coloring from $L$.
Here we use a similar approach. Consider a pair $(G,h_x)$ that satisfies one of
Cases (1)–(5) in Theorem <ref>. We show that $(G,h_x)$ is not
choosable. Case (1) is immediate by the previous paragraph.
Case (2) is immediate, since $|L(x)|=0$. For Case (3), give lists to the Gallai
tree of $G-x$ as above; now let $L(x)=\{c\}$ for some new color $c$, and add $c$
to the list of each neighbor of $x$. Again $G$ cannot be colored from $L$. For
Case (4), assign lists to $V(B)$ as in Proposition <ref> and to the
other blocks as above. Again, $G$ has no coloring from
these lists. Finally, consider Case (5). Assign lists for all blocks outside
of $A$ as above, and assign lists for $A$ as above in Case (2), (3), or (4).
To conclude this section, we consider labelings $h_{x,y}$, where $h_{x,y}(x)=
h_{x,y}(y)=1$ and $h_{x,y}(v)=0$ for all other $v\in V(G)$.
We show that the set of pairs $(G,h_{x,y})$ that are not AT differs from the set
of that are not paintable. Further, both sets differ from
the set of pairs that are not choosable.
It suffices to give a pair $(G_1,h_{x,y})$ that is choosable but not paintable
and a second pair $(G_2,h_{x,y})$ that is paintable but not AT.
[scale = 11]
VertexStyle = []
EdgeStyle = []
labeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw]
unlabeledStyle=[shape = circle, minimum size = 6pt, inner sep = 1.2pt, draw, fill]
[style = labeledStyle, x = 0.850, y = 0.800, L = $0$]v0
[style = labeledStyle, x = 0.750, y = 0.700, L = $1$]v1
[style = labeledStyle, x = 0.950, y = 0.700, L = $1$]v2
[style = labeledStyle, x = 0.800, y = 0.550, L = $0$]v3
[style = labeledStyle, x = 0.900, y = 0.550, L = $0$]v4
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[label = , labelstyle=auto=right, fill=none](v3)(v1)
[label = , labelstyle=auto=right, fill=none](v3)(v2)
[label = , labelstyle=auto=right, fill=none](v3)(v4)
[label = , labelstyle=auto=right, fill=none](v4)(v1)
[label = , labelstyle=auto=right, fill=none](v4)(v2)
[style = labeledStyle, x = 0.850, y = 0.800, L = $0$]v0
[style = labeledStyle, x = 0.750, y = 0.700, L = $1$]v1
[style = labeledStyle, x = 0.950, y = 0.700, L = $1$]v2
[style = labeledStyle, x = 0.800, y = 0.550, L = $0$]v3
[style = labeledStyle, x = 0.900, y = 0.550, L = $0$]v4
[label = , labelstyle=auto=right, fill=none](v1)(v0)
[label = , labelstyle=auto=right, fill=none](v2)(v0)
[label = , labelstyle=auto=right, fill=none](v3)(v1)
[label = , labelstyle=auto=right, fill=none](v3)(v2)
[label = , labelstyle=auto=right, fill=none](v4)(v1)
[label = , labelstyle=auto=right, fill=none](v4)(v2)
The pair on the left is choosable, but not paintable.
The pair on the right is paintable, but not AT.
The pair $(G_1,h_{x,y})$ on the left in Figure <ref> is choosable,
but not paintable. The pair $(G_2,h_{x,y})$ on the right in
Figure <ref> is paintable, but not AT.
Let $(G_1,h_{x,y})$ denote the pair on the left, where $x$ and $y$ are the
vertices labeled 1. Let $(G_2,h_{x,y})$ denote the pair on the right, where
$x$ and $y$ are the vertices labeled 1.
We first show that $(G_1,h_{x,y})$ is choosable. Let $L$ denote the list
assignment. If there exists $c\in L(x)\cap L(y)$, then use $c$ to color $x$ and
$y$, and color the remaining vertices greedily. So suppose there does not
exist such a color $c$. Let $z$ be a vertex in both triangles and note that
there exist $c\in (L(x)\cup L(y))\setminus L(z)$. By symmetry, assume that
$c\in L(x)$. Color $x$ with $c$, and color $G_1-x$ greedily, starting with the
vertex of degree 2 and ending with $z$.
We now show that $(G_1,h_{x,y})$ is not paintable. Let $S$ be the vertices of one
triangle. By definition, there must be $I \subseteq S$ such that $G_1-I$ is
$f'$-paintable, where $f'(v) \DefinedAs f(v)$ for $v \in V(G_1) - S$ and $f'(v)
\DefinedAs f(v) - 1$ for $v \in S - I$. $I$ must have one vertex, $w$. There
are two choices for $w$; either $w$ is in two triangles or not. If $w$ is not
in two triangles, then $G_1 - w$ is a triangle with a pendant edge, where
the vertices on the triangle all have list size 2, so $G_1-w$ is not paintable.
If $w$ is one of the vertices in two triangles, then $G_1-w$ is a 4-cycle with
list sizes alternating $1, 2, 1, 2$. Again $G_1-I$ is not paintable (nor choosable).
To see that $(G_2,h_{x,y})$ is not AT, note that any good orientation would need
indegrees summing to at least 7, but $G_2$ has only 6 edges. Now we show that
$(G_2,h_{x,y})$ is paintable. Note that $G_2$ is isomorphic to $K_{2,3}$, the complete
biparite graph. Call the parts $X$ and $Y$, with $|X|=2$ and $|Y|=3$. If $S$
includes at least two vertices of $X$ or at least two vertices of $Y$, take $I$
to be an independent set of size at least 2. It is easy to check that $G-I$ is
paintable, since it induces either an independent set or a path, where each
endvertex has more colors than neighbors. So assume that $S$ contains at most
one vertex from each of $X$ and $Y$. If $S$ contains a vertex of $X$, then
color it. The resulting graph is paintable, since it is a claw, $K_{1,3}$,
with at most one leaf having a single color and all other vertices having two
colors. Finally, suppose $S$ contains only a single vertex of $Y$. Let $I=S$.
The resulting graph is $C_4$, which is degree-paintable (since it is degree-AT).
A graph is unstretched unstretched if it has no induced path
$u_1v_1v_2u_2$ where $d(v_1)=d(v_2)=2$ (as in Corollary <ref>).
We finish with the following question.
Are there only finitely many unstretched, 2-connected graphs $G$ such that
$(G,h_{x,y})$ is not choosable (resp. paintable, AT)? More generally, let
$h_{x_1,\ldots,x_k}$ be a labeling that assigns 1 to vertices $x_1,\ldots,x_k$
and 0 to all others. Are there only finitely many unstretched, 2-connected
graphs $G$ such that $(G, h_{x_1,\ldots,x_k})$ is not choosable
(resp. paintable, AT)?
|
1511.00316
|
We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of $C^*$-algebras. We demonstrate explicitly how duality is implemented as a natural isomorphism between quantum field theories. We apply this formalism to develop a fully covariant quantum theory of self-dual fields.
*Report no.: EMPG–15–13
*Keywords: Abelian gauge theory, differential cohomology, Dirac charge quantization,
Abelian duality, self-dual Abelian gauge fields, algebraic quantum field theory
*MSC 2010: 81T13, 81T05, 53C08, 55N20
§ INTRODUCTION AND SUMMARY
Dualities in string theory have served as a rich source of
conjectural relations between seemingly disparate situations in
mathematics and physics, particularly in some approaches to quantum
field theory. Heuristically, a `duality' is an equivalence between two
descriptions of the same quantum theory in different classical terms,
and it typically involves an interchange of classical and quantum
data. The prototypical example is electric/magnetic duality of Maxwell
theory on a four-manifold $M$: Magnetic flux is discretized at the
classical level by virtue of the fact that it originates as the
curvature of a line bundle on $M$, whereas electric flux
discretization is a quantum effect arising via Dirac charge
quantization. The example of electric/magnetic duality in Maxwell
theory has a generalization to any spacetime dimensionality, of relevance to
the study of fluxes in string theory, which we may collectively refer
to as `Abelian duality'. The configuration spaces of these (generalized) Abelian gauge theories
are mathematically modeled by suitable (generalized) differential cohomology groups,
see e.g. <cit.> for reviews.
In this paper we will describe a new perspective on Abelian duality by
combining methods from Cheeger-Simons differential cohomology and
locally covariant quantum field theory; this connection between
Abelian gauge theory and differential cohomology was originally pursued
by <cit.>. The quantization of Abelian gauge theories was described
from a Hamiltonian perspective by <cit.>, where the
representation theory of Heisenberg groups was used to define the quantum Hilbert
space of an Abelian gauge theory in a manifestly duality invariant way.
In the present work we shall instead build the semi-classical configuration space for dual gauge field
configurations in a fully covariant fashion, which agrees with that proposed
by <cit.> upon fixing a Cauchy surface $\Sigma$ in
a globally hyperbolic spacetime $M$, but which is manifestly independent of the
choice of $\Sigma$. Following the usual ideas of algebraic quantum field theory,
we construct not a quantum Hilbert space of states but rather a $C^*$-algebra of
quantum observables; the requisite natural presymplectic structure also agrees with
that of <cit.> upon fixing a Cauchy surface $\Sigma$,
but is again independent of the choice of $\Sigma$. Our approach thereby lends a
new perspective on the phenomenon of Abelian duality, and it enables a rigorous
(functorial) definition of quantum duality as a natural isomorphism between quantum
field theory functors. An alternative rigorous perspective on Abelian duality has been recently proposed
by <cit.> using the factorization algebra approach to (Euclidean) quantum field theory.
We do not yet understand how to describe the full duality groups,
i.e. the analogues of the $SL(2,\bbZ)$ S-duality group of Maxwell theory, as this
in principle requires a detailed understanding of the automorphism groups of our
quantum field theory functors <cit.>, which is beyond the scope of the present paper.
Our approach also gives a novel and elegant formulation of the quantum theory
of self-dual fields, which is an important ingredient in the
formulation of string theory and supergravity: In two dimensions the self-dual gauge
field is a worldsheet periodic chiral scalar field in heterotic string theory whose quantum Hilbert space
carries representations of the usual (affine) Heisenberg algebra; in six
dimensions the self-dual field is an Abelian gerbe connection which lives on the worldvolume
of M5-branes and NS5-branes, and in the evasive superconformal
$(2,0)$ theory whose quantum Hilbert space should similarly carry
irreducible representations of the corresponding Heisenberg group; in ten dimensions
the self-dual field is the Ramond-Ramond four-form potential of Type IIB supergravity.
The two generic issues
associated with the formulation of the self-dual field theory are: (a) The lack of covariant local
Lagrangian formulation of the theory (without certain choices,
cf. <cit.>); and: (b) The reconciliation of the
self-duality equation with Dirac quantization requires the
simultaneous discretization of both electric and magnetic fluxes in
the same semi-classical theory. Our quantization of Abelian gauge theories
at the level of algebras of quantum observables eludes both of these
problems. In particular, the noncommutativity of torsion fluxes
observed by <cit.> is also straightforwardly
evident in our approach. As in <cit.>, our quantization
of the self-dual field does not follow from the approach developed in the rest of this paper.
Other Abelian self-dual gauge theories can be analyzed starting
from generalized differential cohomology theories fulfilling a suitable self-duality property,
e.g. differential K-theory, see <cit.> for the Hamiltonian point of view.
An approach closer to the one pursued in the present paper is possible also in these cases
provided one has suitable control on the properties of the relevant generalized differential cohomology theory.
In addition to being cast in a manifestly covariant framework, another improvement on
the development of <cit.> is that our approach does not
require the spacetime to admit compact Cauchy surfaces. Our main technical achievement
is the development of a suitable theory of Cheeger-Simons differential characters with compact support
and Pontryagin duality, in a manner which does not destroy the Abelian
duality. As the mathematical details of this theory are somewhat involved and of independent
interest, they have been delegated to a companion paper <cit.> to which we frequently
refer. The present paper focuses instead on the aspects of interest in physics.
The outline of the remainder of this paper is as follows. In Section <ref> we
introduce and analyze the semi-classical configuration spaces of dual gauge fields in the
language of differential cohomology; our main result is the identification of this space
with the space of solutions of a well-posed Cauchy problem which agrees with the description
of <cit.>, but in a manifestly covariant fashion and without
the assumption of compactness of Cauchy surfaces. In Section <ref> we
analogously study a suitable space of dual gauge field configurations of spacelike compact
support, and show in Section <ref> that it is isomorphic to a suitable Abelian group
of observables defined in the spirit of smooth Pontryagin duality as in <cit.>.
In Section <ref> we consider the quantization of the semi-classical gauge theories
and the extent to which they satisfy the axioms of locally covariant quantum field
theory <cit.>; we show that, just as in <cit.>, our quantum
field theory functors satisfy the causality and time-slice axioms but violate the locality
axiom.[The violation of locality is due to topological properties of the spacetime $M$ and
owes to the fact that differential cohomology constructs the pertinent configuration spaces
as gauge orbit spaces. As a matter of fact, all approaches to gauge theory in the context of
general local covariance <cit.> exhibit at least some remnant of the failure of locality,
see <cit.>.
There are indications that the tension between locality and gauge theory can be solved
by means of homotopical techniques (in the context of model categories),
see <cit.> for the first steps towards this goal.]
In Section <ref> we show that dualities extend to the
quantum field theories thus defined. In Section <ref> we apply our formalism to give a
proper covariant formulation of the quantum field theory of a self-dual field. An appendix at the
end of the paper provides some technical details of constructions which are used in the main text.
§ DUAL GAUGE FIELDS
In this section we describe and analyze the configuration spaces of
the (higher) gauge theories that will be of interest in this paper.
Their main physical feature is a discretization of both electric and magnetic
fluxes, which is motivated by Dirac charge quantization.
To simplify notation, we normalize both
electric and magnetic fluxes so that they are quantized in the same
integer lattice $\bbZ\subset \bbR$.
Because Dirac charge quantization arises as a quantum effect
(i.e. it depends on Planck's constant $\hbar$, which in our
conventions is equal to $1$),
we shall use the attribution “semi-classical” for the gauge
field configurations introduced below. In this paper all manifolds are implicitly assumed
to be smooth, connected, oriented and of finite type, i.e. they admit a finite good cover.
§.§ Semi-classical configuration space
Let $M$ be a manifold. The integer cohomology group $\H^k(M;\bbZ)$ of degree $k$
is an Abelian group which has a (non-canonical) splitting
$\H^k(M;\bbZ)\simeq \Hf^k(M;\bbZ)\oplus \Ht^k(M;\bbZ)$ into free and torsion
subgroups, respectively. Let $\Omega_\bbZ^k(M)\subset \Omega^k(M)$
denote the closed differential $k$-forms on $M$ with integer periods.
Below we recall the definition of Cheeger-Simons differential characters <cit.>.
A degree $k$ Cheeger-Simons differential character on a manifold $M$
is a group homomorphism $h : Z_{k-1}(M)\to \bbT$
from the group $Z_{k-1}(M)$ of $k-1$-cycles on $M$ to the circle group $\bbT := \bbR/\bbZ$
for which there exists a differential form $\omega_h\in\Omega^k(M)$ such that
h(γ) = ∫_γ ω_h , ∀γ∈C_k(M) ,
where $\del \gamma$ denotes the boundary of the $k$-chain $\gamma$.
The Abelian group of Cheeger-Simons differential characters is denoted by $\dH^k(M;\bbZ)$.
For a modern perspective on differential cohomology which includes the Cheeger-Simons model
see <cit.>. We use the degree conventions of <cit.> in which the curvature
of a differential character in $\dH^k(M;\bbZ)$ is a $k$-form.
The assignment of $\dH^k(M;\bbZ)$ to each manifold $M$
is a contravariant functor
^k( - ;) : 𝖬𝖺𝗇^⟶
from the category $\Man$ of manifolds to the category $\Ab$ of Abelian groups. For
notational convenience, we simply denote by $f^\ast$ the group homomorphism
$\dH^k(f;\bbZ) : \dH^k( M^\prime ;\bbZ) \to \dH^k( M ;\bbZ) $ for any
smooth map $f: M\to M^\prime$.
The functor (<ref>) comes together with four natural transformations which are given by
the curvature map $\cu : \dH^k( - ;\bbZ) \Rightarrow \Omega^k_{\bbZ}(-)$,
the characteristic class map $\ch : \dH^k( - ;\bbZ) \Rightarrow \H^k(-;\bbZ)$,
the inclusion of topologically trivial fields $\iota : \Omega^{k-1}(-)/\OmegaZ^{k-1}(-) \Rightarrow \dH^k( - ;\bbZ) $
and the inclusion of flat fields $\kappa: \H^{k-1}(-;\bbT)\Rightarrow
\dH^k( - ;\bbZ) $, where $\bbT=\bbR / \bbZ$ is the circle group.
The (functorial) diagram of Abelian groups
0 [d] 0 [d] 0[d]
0 [r] ^̋k-1(M;)/^k-1(M;) [r]^-κ̃ [d]
Ω^k-1(M)/^k-1(M) [r]^-[d]_-ι Ω^k-1(M) [r] [d]^-⊆ 0
0 [r] ^̋k-1(M;) [r]^-κ[d]
^k(M;) [r]^-[d]_- ^k(M) [r] [d]^-[ · ] 0
0 [r] ^k(M;) [r] [d]
^̋k(M;) [r] [d]
^k(M;) [r] [d] 0
0 0 0
is a commutative diagram whose rows and columns are short exact sequences.
In the remainder of this paper we shall take $M$ to be a time-oriented $m$-dimensional globally hyperbolic
Lorentzian manifold, which we regard as `spacetime'; for a thorough
discussion of Lorentzian geometry including global hyperbolicity see
e.g. <cit.>, while a brief overview can be found in e.g. <cit.>.
The semi-classical configuration space $\Conf^k(M;\bbZ)$ of interest to us is obtained as the pullback
^k(M;) @–>[d] @–>[r] ^m-k(M;) [d]^-
^k(M;) [r]_- Ω^k(M)
By definition, any element $(h,\tilde h) \in \Conf^k(M;\bbZ)\subseteq \dH^k(M;\bbZ) \times \dH^{m-k}(M;\bbZ) $
has the property that the curvature of $h$ is the Hodge dual of the
curvature of $\tilde h$, i.e. $\cu\, h = \dcu\, \tilde h$.
We may interpret this condition as being responsible for the
quantization of electric fluxes: the de Rham cohomology class
of the Hodge dual curvature $\dcu \,h$ is also an element in $\Hf^{m-k}(M;\bbZ)$
and hence electric fluxes are quantized in the same lattice
$\bbZ\subset \bbR$ as magnetic fluxes.
In a similar fashion, we introduce the semi-classical topologically trivial fields $\TT^k(M;\bbZ)$ as the pullback
^k(M;) @–>[d] @–>[r] Ω^m-k-1(M)/^m-k-1(M)
[r]_- Ω^k(M)
To simplify notation we will adopt the following useful convention:
For any graded Abelian group $A^\sharp = \bigoplus_{k\in\bbZ} \, A^k$, we introduce
A^p,q := A^p ×A^q .
Using (<ref>) we introduce
a new commutative diagram of Abelian groups with exact rows and columns,
whose central object is the semi-classical configuration space $\Conf^k(M;\bbZ)$.
Consider the two group homomorphisms
_1: ^k(M;) ⟶^k ∩∗^m-k(M) , (h,h̃) ⟼ h = h̃
_1: ^k(M;) ⟶Ω^k-1 ∩∗Ω^m-k-1(M) , ([A],[Ã]) ⟼A = ∗Ã .
Then the diagram of Abelian groups
0 [d] 0 [d] 0[d]
0 [r] ^̋k-1,m-k-1(M;)/^k-1,m-k-1(M;)
[r]^-κ̃×κ̃ [d]
^k(M;) [r]^-_1 [d]_-ι×ι
Ω^k-1 ∩∗ Ω^m-k-1(M)
[r] [d]^-⊆ 0
0 [r] ^̋k-1,m-k-1(M;) [r]^-κ×κ [d]
^k(M;) [r]^-_1 [d]_-×
^k ∩∗ ^m-k(M)
[r] [d]^-([ · ],[∗^-1 · ]) 0
0 [r] ^k,m-k(M;) [r] [d]
^̋k,m-k(M;) [r] [d]
^k,m-k(M;) [r] [d] 0
0 0 0
is a commutative diagram whose rows and columns are short exact sequences.
Commutativity of this diagram follows by construction. Hence we focus on proving that the rows and columns are exact.
The bottom row and the left column are exact because they are Cartesian products of exact sequences.
Injectivity of $\iota \times \iota$, $\kappa \times \kappa$
and $\tilde \kappa \times \tilde \kappa$ is immediate by (<ref>).
Let us now show that $\dd_1$ and $\cu_1$ are surjective.
Given $\dd A = \ast\, \dd \tilde A$ for $A \in \Omega^{k-1}(M)$
and $\tilde A \in \Omega^{m-k-1}(M)$, we note that $([A], [\tilde A])$ is an element of $\TT^k(M; \bbZ)$
and $\dd_1 ([A], [\tilde A]) = \dd A =\ast\, \dd \tilde A$, thus showing that $\dd_1$ is surjective.
A similar argument applies to $\cu_1$ using surjectivity of $\cu: \dH^p(M;\bbZ) \to \OmegaZ^p(M)$
for $p = k$ and for $p = m-k$.
To show that $([\,\cdot\,],[\ast^{-1}\,\cdot\,])$ is also surjective,
let us take any $(z,\tilde z) \in \Hf^{k,m-k}(M;\bbZ) \subseteq \H^{k,m-k}(M;\bbR)$
and recall that by de Rham's theorem it can be presented as $(z,\tilde z) = ([\omega],[\tilde \omega])$,
for some $\omega \in \OmegaZ^k(M)$ and $\tilde \omega \in \OmegaZ^{m-k}(M)$.
Let $\de = (-1)^{m\, (k-1)}\, \ast\dd\,\ast: \Omega^k(M) \to \Omega^{k-1}(M)$ denote the codifferential.
We solve the equations $[\de \theta] = [\omega] \in \H^k_\dR(M)$
and $[\de \tilde \theta\, ]=[\tilde \omega] \in \H^{m-k}_\dR(M)$
for $\theta \in \Omega^{k+1}(M)$ and $\tilde \theta \in \Omega^{m-k+1}(M)$.[To show that
a solution exists, let us introduce the d'Alembert operator
$\Box = \de \, \dd + \dd \, \de$ and consider its retarded/advanced
Green's operators $G^\pm$, cf. <cit.>.
Let us also consider a partition of unity $\{\chi_+,\chi_-\}$ on $M$ such that
$\chi_\pm$ has past/future compact support, see <cit.> for a definition of these support systems.
Then $\theta = G (\dd \chi_+ \wedge \omega)$ is a solution, where $G = G^+ - G^-$ is the causal propagator.
In fact $\de \theta = \de \, \dd (G^+ (\chi_+ \, \omega) + G^- (\chi_-
\, \omega))
= \omega - \dd \, \de (G^+ (\chi_+ \, \omega) + G^- (\chi_- \, \omega))$.
A similar argument applies to $\tilde \theta$.]
Introducing $F = \de \theta + \ast\, \de \tilde \theta$, we find $[F] = [\de \theta] = [\omega] \in \H^k_\dR(M)$
and $[\ast^{-1} F] = [\de \tilde \theta\, ] = [\tilde \omega] \in \H^{m-k}_\dR(M)$;
in particular, both $F$ and $\ast^{-1} F$ have integral periods since so do $\omega$ and $\tilde \omega$.
We conclude that $F \in \OmegaZ^k \cap \ast\, \OmegaZ^{m-k}(M)$.
Surjectivity of $\ch \times \ch$ follows from what we have already shown above
and by using a diagram chasing argument.
Take any $(x,\tilde x) \in \H^{k,m-k}(M;\bbZ)$. Mapping to the corresponding free group
and recalling that both $([\,\cdot\,],[\ast^{-1}\,\cdot\,])$ and $\cu_1$ are epimorphisms,
we find $(h,\tilde h) \in \Conf^k(M;\bbZ)$ whose image along $([\,\cdot\,],[\ast^{-1}\,\cdot\,]) \circ \cu_1$
matches the image of $(x,\tilde x)$ in $\Hf^{k,m-k}(M;\bbZ)$.
By exactness of the bottom row,
$(\ch\, h,\ch\, \tilde h)$ differs from $(x, \tilde x)$ by an element
$(t,\tilde t\, )$
of the torsion subgroup $\Ht^{k,m-k}(M;\bbZ)$, i.e. $(x, \tilde x) =
(\ch\, h + t,\ch\, \tilde h + \tilde t\, )$.
Exactness of the left column allows us to find a preimage $(u,\tilde
u) \in \H^{k-1,m-k-1}(M;\bbT)$ for $(t,\tilde t\, )$.
Commutativity of the diagram then implies that $(h + \kappa\, u,\tilde h + \kappa\, \tilde u)$
is a preimage of $(x,\tilde x)$ via $\ch \times \ch$.
We still have to check that the first two rows and the last two columns are exact at their middle objects.
This is a straightforward consequence of the exactness of the corresponding rows and columns in (<ref>).
To better motivate the semi-classical configuration space $\Conf^k(M;\bbZ)$
we establish below its relation with Maxwell theory. For this purpose we consider the case $m=4$ and $k=2$.
The usual Maxwell equations (without external sources) for the Faraday tensor $F \in \Omega^2(M)$
are $\dd F = 0$ and $\dd \ast F = 0$. These equations are invariant under electric-magnetic duality,
i.e. under the exchange of $F$ and $\ast F$. The standard approach to gauge theory
consists in the replacement of $F$ with the curvature of (the isomorphism class of) a circle bundle with connection
(equivalently, a differential cohomology class in degree 2).
In this framework, however, $\ast F$ does not have any geometric interpretation,
hence the original electric-magnetic duality of Maxwell theory is lost passing to gauge theory.
Nevertheless, one can present Maxwell equations in an equivalent way,
which is however better suited for a gauge theoretic extension preserving electric-magnetic duality:
F = ∗F̃ , F = 0 , F̃ = 0 .
Interpreting both $F$ and $\tilde F$ as the curvatures of circle bundles with connections,
the semi-classical configuration space $\Conf^2(M;\bbZ)$ is obtained
and the original electric-magnetic duality of Maxwell theory is lifted to $\Conf^2(M;\bbZ)$,
see Section <ref> for the situation in arbitrary spacetime dimension and degree.
Notice that the semi-classical configuration space has the same local “degrees of freedom” as Maxwell theory.
In fact, on a contractible spacetime $\Conf^2(M;\bbZ)$ reduces to
the top-right corner in diagram (<ref>).
Since exact and closed forms are the same on a contractible manifold, Maxwell theory is recovered.
In conclusion, the semi-classical configuration space $\Conf^2(M;\bbZ)$
is a gauge theoretic extension of Maxwell theory that carries the same local information,
however preserving electric-magnetic duality by matching
the relevant topological (as opposed to local) data in a suitable way.
As a by-product, any configuration $(h,\tilde h) \in \Conf^2(M;\bbZ)$
realizes the discretization of magnetic and electric fluxes,
which arise as the characteristic classes $\ch \,h, \ch\, \tilde h \in \H^2(M;\bbZ)$.
This argument can be made general for higher gauge theories in arbitrary spacetime dimension.
The semi-classical configuration space is a contravariant functor
^k(-;): _m^⟶
from the category $\Loc_m$ of time-oriented $m$-dimensional globally hyperbolic Lorentzian manifolds
with causal embeddings[A causal embedding $f: M \to M^\prime$
between time-oriented $m$-dimensional globally hyperbolic Lorentzian manifolds
is an orientation and time-orientation preserving isometric embedding,
whose image is open and causally compatible,
i.e. $J^\pm_{M^\prime}(f(p)) \cap f(M) = f(J^\pm_M(p))$ for all $p
\in M$; here $J^\pm_M(p)$ denotes the causal future/past of $p\in M$
consisting of all points of $M$ which can be reached by a
future/past-directed smooth causal curve stemming from $p$, see <cit.>.] as morphisms
to the category $\Ab$ of Abelian groups. For notational convenience, we simply denote by $f^\ast$
the group homomorphism $\Conf^k(f;\bbZ): \Conf^k(M^\prime;\bbZ) \to
\Conf^k(M;\bbZ)$ associated with a morphism
$f: M \to M^\prime$ in $\Loc_m$.
§.§ Cauchy problem
We will now show that the semi-classical configuration space
$\Conf^k(M;\bbZ)$ is the space of solutions of a well-posed Cauchy problem.
Let us start by recalling a well-known result for the Cauchy problem
of the Faraday tensor, see e.g. <cit.>
and also <cit.> for details on
how to treat initial data of not necessarily compact
support. For the related Cauchy problem of the gauge potential see <cit.>.
Throughout this paper $\Sigma$ will denote a smooth
spacelike Cauchy surface of $M$ with embedding $\iota_\Sigma: \Sigma \to M$ into $M$.
For each $(B,\tilde B) \in \Omega^{k,m-k}_\dd(\Sigma)$ (where the subscript ${}_\dd$ denotes closed forms),
there exists a unique solution $F \in \Omega^k(M)$ to the initial value problem
\begin{align}
\dd F & = 0~, \qquad \iota_\Sigma^\ast F = B~, \\[4pt]
\dd \ast^{-1} F & = 0~, \qquad \iota_\Sigma^\ast \ast^{-1} F = \tilde B~,
\end{align}
whose support is contained
in the causal future and past of the support of the initial data, i.e.$\supp\, F \subseteq J(\supp\, B \cup \supp\, \tilde B)$.
We consider also the similar well-posed initial value problem for $\tilde F \in \Omega^{m-k}(M)$ given by
\begin{align}
\dd \tilde F & = 0~, \qquad \iota_\Sigma^\ast \tilde F = \tilde B~, \\[4pt]
\dd \ast \tilde F & = 0~, \qquad \iota_\Sigma^\ast \ast \tilde F = B~,
\end{align}
where the initial data are also specified by $(B,\tilde B) \in \Omega^{k,m-k}_\dd(\Sigma)$.
Given now any initial data $(B,\tilde B) \in \Omega^{k,m-k}_\dd(\Sigma)$,
let us consider the corresponding unique solutions $F$ and $\tilde F$ of
the Cauchy problems (<ref>) and (<ref>).
This implies that $F - \ast \tilde F$ solves the Cauchy problem (<ref>) with vanishing initial data,
and therefore $F = \ast \tilde F$.
We further show that, given initial data $(B,\tilde B) \in \Omega^{k,m-k}_\bbZ(\Sigma)$ with integral periods,
the corresponding solution $F$ of the Cauchy problem (<ref>)
is such that both $F$ and $\ast^{-1} F$ have integral periods.
For this, using the results of Lemma <ref> (i)
we can express each $k$-cycle $\gamma \in Z_k(M)$ as
$\gamma = \iota_{\Sigma\,\ast} \, \pi_{\Sigma\,\ast} \gamma + \del
h_\Sigma \gamma$, and hence
∫_γ F = ∫_π_Σ ∗ γ
ι_Σ^∗F + ∫_h_Σγ F
= ∫_π_Σ ∗ γ B ∈ .
Similarly, for each $m{-}k$-cycle $\tilde \gamma \in Z_{m-k}(M)$ we have
∫_γ̃ ∗^-1 F
= ∫_π_Σ ∗ γ̃ ι_Σ^∗ ∗^-1 F
+ ∫_h_Σγ̃ ∗^-1 F
= ∫_π_Σ ∗ γ̃ B̃ ∈ .
Conversely, given $F \in \Omega^k_\bbZ \cap \ast \Omega^{m-k}_\bbZ(M)$
we have $\iota_\Sigma^\ast F \in \Omega^k_\bbZ(\Sigma)$
and $\iota_\Sigma^\ast \ast^{-1} F \in \Omega^{m-k}_\bbZ(\Sigma)$.
Summing up, we obtain
The embedding $\iota_\Sigma: \Sigma \to M$ of $\Sigma$ into
$M$ induces an isomorphism of Abelian groups
@<3pt>[rr]^-(ι_Σ^∗,ι_Σ^∗ ∗^-1)
Ω^k,m-k_(Σ) @<1pt>[ll]^-_Σ ,
whose inverse $\solve_\Sigma$ is the map assigning to initial data
$(B,\tilde B) \in \Omega^{k,m-k}_\bbZ(\Sigma)$ the corresponding unique solution
$F \in \Omega^k_\bbZ \cap \ast \Omega^{m-k}_\bbZ(M)$
of the Cauchy problem (<ref>).
Let us consider the central row of the diagram (<ref>).
Taking into account also naturality of $\kappa$ and $\cu$, one finds that the diagram of Abelian groups
0 [r] ^̋k-1,m-k-1(M;)
[r]^-κ×κ [d]_-ι_Σ^∗×ι_Σ^∗
^k(M;) [r]^-_1 [d]_-ι_Σ^∗×ι_Σ^∗
[r] [d]^-(ι_Σ^∗,ι_Σ^∗ ∗^-1) 0
0 [r] ^̋k-1,m-k-1(Σ;) [r]_-κ×κ
^k,m-k(Σ;) [r]_-×
Ω^k,m-k_(Σ) [r] 0
commutes and its rows are short exact sequences.
Using also Lemma <ref> (ii), Corollary <ref>
and the five lemma, we obtain
The embedding $\iota_\Sigma: \Sigma \to M$ induces an isomorphism of Abelian groups
^k(M;) [rr]^-ι_Σ^∗×ι_Σ^∗
^k,m-k(Σ;) .
We can interpret the result of Theorem <ref>
as establishing the well-posedness of the initial value problem for $(h,\tilde h) \in \dH^{k,m-k}(M;\bbZ)$ given by
\begin{align}\label{eqCauchyPbl}
\cu\, h = \ast \, \cu\, \tilde h~, \qquad \iota_\Sigma^\ast h = h_\Sigma~, \quad \iota_\Sigma^\ast \tilde h = \tilde h_\Sigma~,
\end{align}
for initial data $(h_\Sigma,\tilde h_\Sigma) \in \dH^{k,m-k}(\Sigma;\bbZ)$.
It follows that the semi-classical configuration space $\Conf^k(M;\bbZ)$ arises as
the space of solutions of this Cauchy problem.
If $M$ has compact Cauchy surfaces $\Sigma$, we can easily endow $\Conf^k(M;\bbZ)$
with the structure of a presymplectic Abelian group induced by the ring structure $\cdot$ on differential characters,
see <cit.>. For this, we define the circle-valued presymplectic structure
σ: ^k(M;) ×^k(M;)
⟶ , ((h,h̃) , (h', h̃' )) ⟼(ι_Σ^∗(h̃ ·h'
- h̃' ·h))[Σ] ,
where $[\Sigma] \in \H_{m-1}(\Sigma)$ denotes the fundamental class of $\Sigma$.
Using compatibility between the ring structure on differential characters
and the natural transformations $\iota$, $\kappa$, $\cu$ and $\ch$,
one can show that $\sigma$ is in fact independent of the choice of $\Sigma$.
Fixing any Cauchy surface $\Sigma$ and using the isomorphism given in Theorem <ref>,
the presymplectic structure (<ref>) can be induced to initial data and thereby agrees with the
one constructed by <cit.> from a Hamiltonian perspective.
However, in contrast to <cit.>
our construction does not depend on the choice of a Cauchy surface, i.e. it is generally covariant.
As we show in Section <ref>, the assumption of compactness of the Cauchy surfaces
can be dropped, provided that one introduces a suitable support restriction
on the semi-classical gauge fields.
§ DUAL GAUGE FIELDS WITH SPACELIKE COMPACT SUPPORT
In this section we introduce and analyze a suitable Abelian group
$\Conf^k_{\rm sc}(M;\bbZ)$
of semi-classical gauge fields of spacelike compact support.
Similarly to the case of the usual quantum field theories on curved spacetimes,
such as Klein-Gordon theory, the role played by $\Conf^k_{\rm sc}(M;\bbZ)$ will be dual
to that of the semi-classical configuration space $\Conf^k(M;\bbZ)$; in fact,
we shall show in Section <ref> that elements in $\Conf^k_{\rm sc}(M;\bbZ)$
define functionals (i.e. classical observables) on $\Conf^k(M;\bbZ)$ which are
group characters $\Conf^k(M;\bbZ) \to \bbT$. This dual role of the semi-classical
gauge fields of spacelike compact support will be reflected mathematically
in the fact that $\Conf^k_{\rm sc}(-;\bbZ): \Loc_m \to \Ab$ is a covariant functor, while
$\Conf^k(-;\bbZ): \Loc_m^\op \to \Ab$ is contravariant.
The correct definition of $\Conf^k_{\rm sc}(M;\bbZ)$ is a
very subtle point because, in contrast to the standard examples
like Klein-Gordon theory, the Abelian group $\Conf^k_{\rm sc}(M;\bbZ)$
cannot be presented as a subgroup of $\Conf^k(M;\bbZ)$, see
Remark <ref> below.
We give a definition of $\Conf^k_{\rm sc}(M;\bbZ)$ in terms of relative differential cohomology
and frequently refer to the companion paper <cit.>
for further technical details.
§.§ Semi-classical configuration space
Let $K\subseteq M$ be a compact subset.
In analogy to (<ref>), we define the Abelian group $\Conf^k(M,M \setminus J(K);\bbZ)$
of semi-classical gauge fields on $M$ relative to $M \setminus J(K)$ as the pullback
^k(M,M ∖J(K);) @–>[d] @–>[rr]
^m-k(M,M ∖J(K);) [d]^-
^k(M,M ∖J(K);) [rr]_- Ω^k(M,M ∖J(K))
where $\dH^{p}(M,M \setminus J(K);\bbZ)$ denote the relative differential cohomology groups
and $\Omega^k(M,M \setminus J(K))$ denotes the group of relative differential forms,
see <cit.> for the definitions and our conventions.
We shall make frequent use of the short exact sequence
0 [r] ^̋k-1,m-k-1(M,M ∖J(K);)
^k(M,M ∖J(K);)
Ω^k_∩∗Ω^m-k_(M,M ∖J(K))
[r] 0
for relative semi-classical gauge fields, which immediately follows
from <cit.> and <cit.>
by imitating the proof of Theorem <ref>.
One may heuristically think of semi-classical gauge fields on $M$
relative to $M\setminus J(K)$ as fields
on $M$ which “vanish” outside of the closed light-cone $J(K)$ of $K$.
However, strictly speaking this interpretation is not correct: There is
a group homomorphism $I : \Conf^k(M,M \setminus J(K);\bbZ) \to \Conf^k(M;\bbZ)$
which is induced by the group homomorphisms (denoted with abuse of notation by the same symbols)
$I : \dH^p(M,M\setminus J(K);\bbZ) \to \dH^p(M,\bbZ)$ that restrict
relative differential characters from relative cycles to cycles by precomposing
them with the homomorphism $Z_{p-1}(M) \to Z_{p-1}(M,M\setminus J(K))$, cf. <cit.>.
By <cit.> and Theorem <ref> below,
we observe that the homomorphism $I : \Conf^k(M,M \setminus J(K);\bbZ) \to \Conf^k(M;\bbZ)$
is not necessarily injective, which implies that $\Conf^k(M,M \setminus J(K);\bbZ)$ is in general not a subgroup of
We define the Abelian group $\Conf^k_{\rm sc}(M;\bbZ)$
of semi-classical gauge fields of spacelike compact support
by formalizing the intuition that for any element $(h,\tilde h)\in \Conf^k_{\rm sc}(M;\bbZ)$
there should exist a sufficiently large compact subset $K \subseteq M$
such that $(h,\tilde h)$ can be represented as an element in $\Conf^k(M,M \setminus J(K);\bbZ)$.
Let us denote by $\mathcal{K}_M$ the directed set of compact subsets of $M$
and notice that the assignment $\Conf^k(M,M \setminus J(-);\bbZ): \mathcal{K}_M \to \Ab$
is a diagram of shape $\mathcal{K}_M$.[For this, we use the group homomorphisms
$Z_{p-1}(M,M\setminus J(K^\prime\, )) \to Z_{p-1}(M,M\setminus J(K))$ of relative cycles
which exist for any $K\subseteq K^\prime$.]
Then the intuition is formalized by taking the colimit of this diagram, i.e. we define
the semi-classical gauge fields of spacelike compact support by
^k_(M;) := (^k(M,M ∖J(-);): 𝒦_M →) .
The colimit in (<ref>) can be equally well computed by
restricting to the directed set $\mathcal{K}_\Sigma$ of compact subsets
of any smooth spacelike Cauchy surface $\Sigma$ of $M$.
In fact, denoting by $\mathcal{C}_M$ the directed set of closed subsets of $M$,
one notices that the map $\mathcal{K}_M \to \mathcal{C}_M$, $K \mapsto J(K)$,
preserves the preorder relation induced by inclusion.
In particular, we may interpret the functor $\Conf^k(M,M \setminus J(-);\bbZ): \mathcal{K}_M \to \Ab$
as the composition of the functors $\Conf^k(M,M \setminus -;\bbZ): \mathcal{C}_M \to \Ab$
and $J: \mathcal{K}_M \to \mathcal{C}_M$; then $\mathcal{K}_\Sigma \subseteq \mathcal{K}_M$ is cofinal
with respect to $J: \mathcal{K}_M \to \mathcal{C}_M$.
In fact, for each $K \subseteq M$, we have $J(K) \subseteq J(K_\Sigma)$ for $K_\Sigma = J(K) \cap \Sigma$,
which is by construction a compact subset of $\Sigma$. This observation provides the isomorphism
\begin{equation}
\Conf^k_\sc(M;\bbZ) \simeq \colim \big(\Conf^k(M,M \setminus J(-);\bbZ): \mathcal{K}_\Sigma \to \Ab\big)~.
\end{equation}
Similarly to Remark <ref>, there is a group homomorphism (denoted with abuse of notation
by the same symbol)
I : ^k_sc(M;) ⟶^k(M;) ,
which is however in general not injective, see <cit.>
and Corollary <ref> below. Hence semi-classical gauge fields
of spacelike compact support cannot in general be faithfully represented as elements in
the semi-classical configuration space $\Conf^k(M;\bbZ)$.
§.§ Cauchy problem
Consider any compact subset $K \subseteq \Sigma$.
Taking into account the support property of the Cauchy problem considered in Theorem <ref>
and applying arguments similar to those in Section <ref> to the relative case,
in particular (<ref>) and (<ref>) (see also Lemma <ref> (i)),
one concludes that, given initial data
$(B,\tilde B) \in \OmegaZ^{k,m-k}(\Sigma,\Sigma \setminus K)$, the Cauchy problem (<ref>)
has a unique solution $F \in \OmegaZ^k \cap \ast\OmegaZ^{m-k}(M,M \setminus J(K))$.
This observation leads us to the relative version of Corollary <ref>.
The embedding $\iota_\Sigma: \Sigma \to M$ induces an isomorphism of Abelian groups
Ω^k_∩∗Ω^m-k_(M,M ∖J(K))
@<3pt>[rr]^-(ι_Σ^∗,ι_Σ^∗ ∗^-1)
@<1pt>[ll]^-_Σ ,
whose inverse $\solve_\Sigma$ is the map assigning
to initial data $(B,\tilde B) \in \Omega^{k,m-k}_\bbZ(\Sigma,\Sigma \setminus K)$ the corresponding
unique solution $F \in \Omega^k_\bbZ \cap \ast \Omega^{m-k}_\bbZ(M,M \setminus J(K))$
of the Cauchy problem (<ref>).
Using (<ref>) and <cit.>,
and the fact that relative differential cohomology is a functor (in a suitable sense, see <cit.>),
we conclude that the diagram of Abelian groups
0 [r] ^̋k-1,m-k-1(M,M ∖J(K);)
[r]^-κ×κ [d]_-ι_Σ^∗×ι_Σ^∗
^k(M,M ∖J(K);)
[r]^-_1 [d]_-ι_Σ^∗×ι_Σ^∗
Ω^k_∩∗Ω^m-k_(M,M ∖J(K))
[r] [d]^-(ι_Σ^∗,ι_Σ^∗ ∗^-1) 0
0 [r] ^̋k-1,m-k-1(Σ,Σ∖K;) [r]_-κ×κ
^k,m-k(Σ,Σ∖K;) [r]_-×
Ω^k,m-k_(Σ,Σ∖K) [r] 0
commutes and its rows are short exact sequences.
Using also Lemma <ref> (ii), Corollary <ref>
and the five lemma, we obtain the relative version of Theorem <ref>.
The embedding $\iota_\Sigma: \Sigma \to M$ induces an isomorphism of Abelian groups
^k(M,M ∖J(K);) [rr]^-ι_Σ^∗×ι_Σ^∗
^k,m-k(Σ,Σ∖K;) .
Taking the colimit of (<ref>) over the directed set $\mathcal K_\Sigma$ of compact subsets of $\Sigma$
and recalling Remark <ref> we find that the diagram of Abelian groups
0 [r] ^̋k-1,m-k-1_sc(M;)
[r]^-κ×κ [d]_-ι_Σ^∗×ι_Σ^∗
^k_sc(M;) [r]^-_1 [d]_-ι_Σ^∗×ι_Σ^∗
Ω^k_sc, ∩∗Ω^m-k_sc,(M)
[d]^-(ι_Σ^∗,ι_Σ^∗ ∗^-1) 0
0 [r] ^̋k-1,m-k-1_c(Σ;) [r]_-κ×κ
^k,m-k_c(Σ;) [r]_-×
Ω^k,m-k_c,(Σ) [r] 0
commutes, its rows are short exact sequences and its vertical arrows are isomorphisms.
The subscript $_{\rm c}$ denotes compact support
and the various groups of this diagram are defined by these colimits.[For a
detailed presentation of differential characters with compact support, see <cit.>.]
This shows that $\Conf^k_{\rm sc}(M;\bbZ)$ is the space of solutions
of the Cauchy problem (<ref>) for $(h,\tilde h)\in \dH^{k,m-k}_{\rm sc}(M;\bbZ)$
with initial data in $\dH^{k,m-k}_{\rm c}(\Sigma;\bbZ)$.
The embedding $\iota_\Sigma: \Sigma \to M$ induces an isomorphism of Abelian groups
^k_sc(M;) [rr]^-ι_Σ^∗×ι_Σ^∗
^k,m-k_c(Σ;) .
The assignment of the
Abelian groups $\Conf_{\rm sc}^k(M;\bbZ)$ to objects $M$ in $\Loc_m$
is a covariant functor
^k_sc(-;): _m ⟶ .
The group homomorphism $f_\ast := \Conf^k_{\rm sc}(f;\bbZ) : \Conf^k_{\rm sc}(M;\bbZ)
\to \Conf^k_{\rm sc}(M^\prime;\bbZ)$ associated with a morphism
$f : M\to M^\prime$ in $\Loc_m$ is constructed in Lemma <ref>.
With a similar construction as in Lemma <ref>, we obtain two more functors
^̋k-1,m-k-1_sc(-;) : _m ⟶ ,
Ω^k_sc, ∩∗Ω^m-k_sc,(-) : _m ⟶ .
Using these constructions one can further show that the
short exact sequence in the first row of the diagram (<ref>)
is natural, i.e. for any morphism $f : M\to M^\prime$ in $\Loc_m$, the diagram
of Abelian groups
0 [r] ^̋k-1,m-k-1_sc(M;)
[r]^-κ×κ [d]_-f_∗
^k_sc(M;) [r]^-_1 [d]_-f_∗
Ω^k_sc, ∩∗Ω^m-k_sc,(M)
[r] [d]^-f_∗ 0
0 [r] ^̋k-1,m-k-1_sc(M^';)
^k_sc(M^';) [r]_-_1
Ω^k_sc, ∩∗Ω^m-k_sc,(M^' )
[r] 0
commutes. Here we also use the notation $f_\ast$ for the group homomorphisms
$\H^{k-1,m-k-1}_{\rm sc}(f;\bbT) $ and $\Omega^k_{{\rm sc},\bbZ} \cap
\ast \Omega^{m-k}_{{\rm sc},\bbZ}(f) $.
§ OBSERVABLES FOR DUAL GAUGE FIELDS
In this section we introduce and analyze a suitable Abelian group
$\Obs^k(M;\bbZ)$ of basic semi-classical observables.
In general, observables are given by functionals on the configuration
space of the field theory.
Recalling that the semi-classical configuration space $\Conf^k(M;\bbZ)$ is an Abelian group,
there is a distinguished class of observables given by the
group characters $\Conf^k(M;\bbZ)^\star := \Hom_{\Ab}(\Conf^k(M;\bbZ),\bbT)$.
However, generic group characters define observables that are too singular
for quantization, hence it is reasonable to impose a suitable regularity condition
in the spirit of smooth Pontryagin duality <cit.>.
After defining and analyzing the smooth Pontryagin dual $\Obs^k(M;\bbZ)$ of $\Conf^k(M;\bbZ)$,
we shall show that it is isomorphic to the Abelian group $\Conf^k_{\rm sc}(M;\bbZ)$
of semi-classical gauge fields of spacelike compact support.
Generalizing the constructions of Remark <ref>
to the case of not necessarily compact Cauchy surfaces, we obtain a natural presymplectic structure
on the Abelian group of semi-classical observables $\Obs^k(M;\bbZ)$.
We shall analyze properties of these presymplectic Abelian groups
in view of the axioms of locally covariant quantum field theory <cit.>.
§.§ Semi-classical observables
We shall begin by imposing a suitable regularity condition on the
Abelian group of group characters $\Conf^k(M;\bbZ)^\star $
of the semi-classical configuration space.
The Abelian group $\Obs^k(M;\bbZ)$ of semi-classical observables
is the following subgroup of $\Conf^k(M;\bbZ)^\star$:
A group character $\varphi \in \Conf^k(M;\bbZ)^\star$ is an element in
$\Obs^k(M;\bbZ)$ if and only if there exists
$\omega = \ast \tilde \omega \in \Omega^k_{{\rm sc},\bbZ} \cap \ast\Omega^{m-k}_{{\rm sc},\bbZ}(M)$
and a smooth spacelike Cauchy surface $\Sigma$ of $M$ such that
= ∫_Σ ( Ã ∧ω- (-1)^k (m-k) A ∧ω̃) ,
for all semi-classical topologically trivial fields
$([A],[\tilde A]) \in \TT^k(M;\bbZ)$.
We now prove that Definition <ref> does not depend on the choice
of Cauchy surface $\Sigma$ used to
evaluate the integral (<ref>).
For this, notice that
$\omega = \ast \tilde \omega \in \Omega^k_{{\rm sc},\bbZ} \cap \ast\Omega^{m-k}_{{\rm sc},\bbZ}(M)$
implies $\Box \, \omega =0$ and $\Box \, \tilde\omega =0$, where $\Box
= \de \, \dd + \dd\, \de$
is the d'Alembert operator. By <cit.> there exists
$\tilde\beta \in\Omega^{m-k}_{\rm c}(M)$ such
that $\tilde\omega = G\tilde\beta$, where
$G = G^+ - G^-$ is the causal propagator and $G^\pm$ are the retarded/advanced Green's operators
of $\Box$.
We further have $\omega = \ast\tilde \omega = G{\ast}\tilde\beta$.
Because of $\dd \omega =0$ and $\dd\tilde\omega=0$,
there exist $\alpha\in\Omega^{k+1}_{\rm c}(M)$ and $\tilde\alpha\in\Omega^{m-k+1}_{\rm c}(M)$
such that $\dd {\ast}\tilde \beta = \Box \alpha$ and $\dd\tilde\beta = \Box\tilde\alpha$.
Using these observations, and realizing $\Sigma$ as the boundary of $J^-(\Sigma)\subseteq M$
and also as the boundary of $J^+(\Sigma)\subseteq M$ (with opposite orientation),
we can rewrite (<ref>) as
φ((ι×ι)([A],[Ã])) =
∫_Σ (
à ∧G∗β̃- (-1)^k (m-k) A ∧Gβ̃ )
=∫_J^-(Σ) ( Ã ∧G^+∗β̃- (-1)^k (m-k) A ∧G^+β̃
+ ∫_J^+(Σ) ( Ã ∧G^-∗β̃- (-1)^k (m-k) A ∧G^-β̃
= ∫_M ( (-1)^m-k à ∧α- (-1)^k (m-k) (-1)^k A∧α̃) ,
where we have also used $\dd A = \ast \dd \tilde A$. It then follows that (<ref>) is independent
of the choice of Cauchy surface because (<ref>) shows that it can be written
as an integral over spacetime $M$.
There is an alternative but equivalent definition of the
Abelian group $\Obs^k(M;\bbZ)$ of semi-classical observables
which employs the notion of smooth Pontryagin duality developed in <cit.>.
Taking the smooth Pontryagin dual of the pullback diagram (<ref>) which defines
the semi-classical configuration space $\Conf^k(M;\bbZ)$, we define
an Abelian group $\Conf^k(M;\bbZ)^\star_\infty$ (called the smooth Pontryagin dual of $\Conf^k(M;\bbZ)$)
via the pushout
Ω_c^k(M) [d]_-^⋆ [r]^-()^⋆ ^m-k(M;)^⋆_∞ @–>[d]
^k(M;)^⋆_∞ @–>[r] ^k(M;)^⋆_∞
where $\dH^p(M;\bbZ)^{\star}_{\infty}$ denotes the smooth Pontryagin dual of $\dH^p(M;\bbZ)$.
This pushout may be realized explicitly as the quotient
^k(M;)^⋆_∞= ^k(M;)^⋆_∞⊕^m-k(M;)^⋆_∞/{ ^⋆ω⊕- ()^⋆ω : ω∈Ω_c^k(M)} .
One can show that the Abelian group $\Conf^k(M;\bbZ)^\star_\infty $ is isomorphic
to the Abelian group $\Obs^k(M;\bbZ)$ of semi-classical observables given in
Definition <ref>. As we do not need this isomorphism in
this paper, we refrain from writing it out explicitly. Let us just point out that
the elements of the smooth Pontryagin dual are in particular continuous group characters.
In fact, on account of <cit.>,
all differential cohomology groups on a manifold of finite type are Fréchet-Lie groups
that are (non-canonically) isomorphic to the Cartesian product of a torus, a torsion group,
a discrete lattice in a Euclidean space (all finite dimensional) and a Fréchet vector space of differential forms.
This observation allows one to conclude that the elements of the smooth Pontryagin dual
are continuous group characters with respect to the Fréchet topology mentioned above.
We now show that the assignment of the Abelian groups $\Obs^k(M;\bbZ)$ of semi-classical observables
to objects $M$ in $\Loc_m$ is a covariant functor
^k(-;): _m ⟶ .
For this, note that
the assignment of character groups $\Conf^k(M;\bbZ)^\star = \Hom_{\Ab}(\Conf^k(M;\bbZ),\bbT)$
(without the regularity condition of Definition <ref>)
to objects $M$ in $\Loc_m$ is a covariant functor
$\Conf^k(-;\bbZ)^\star : \Loc_m\to\Ab$: Given any morphism
$f : M\to M^\prime$ in $\Loc_m$, functoriality of the semi-classical configuration spaces
provides us with a group homomorphism $f^\ast = \Conf^k(f;\bbZ): \Conf^k(M^\prime;\bbZ)\to \Conf^k(M;\bbZ)$,
which we can dualize to a group homomorphism (called pushforward)
$f_\ast :=\Conf^k(f;\bbZ)^\star = (f^\ast)^\star: \Conf^k(M;\bbZ)^\star\to \Conf^k(M^\prime;\bbZ)^\star$
between the character groups. It remains to show that these group homomorphisms
induce group homomorphisms $f_\ast : \Obs^k(M;\bbZ)\to \Obs^k(M^\prime;\bbZ)$, i.e.
that pushforwards preserve the regularity condition of Definition <ref>.
Let $\varphi \in \Obs^k(M;\bbZ)$ and
$\omega = \ast \tilde \omega \in \Omega^k_{{\rm sc},\bbZ} \cap \ast \Omega^{m-k}_{{\rm sc},\bbZ}(M)$
be as in Definition <ref>.
Exploiting the Cauchy problem described by Corollary <ref>,
we can easily push forward $\omega$ and $\tilde\omega$ to
$f_\ast \omega = \ast f_\ast \tilde\omega \in \Omega^k_{{\rm sc},\bbZ} \cap \ast \Omega^{m-k}_{{\rm sc},\bbZ}(M^\prime\, )$
by pushing forward the initial data from a Cauchy surface $\Sigma\subseteq M$
to a suitable Cauchy surface $\Sigma^\prime\subseteq M^\prime$.[
A suitable Cauchy surface $\Sigma^\prime\subseteq M^\prime$ can be constructed follows:
Let $U\subseteq \Sigma$ be an open neighborhood
of $\supp\,\omega \cap \Sigma$ with compact closure.
Then the image $f(\, \overline{U}\, )\subseteq M^\prime$ of the closure $\overline{U}$ of $U$
is a spacelike and acausal compact submanifold with boundary, and we can take
$\Sigma^\prime$ to be any Cauchy surface extending $f(\,
\overline{U}\, )$; see <cit.>
for the existence of such a Cauchy surface.
By construction, we have
= φ((ι×ι)([f^∗A],[f^∗Ã]))
= ∫_Σ ( f^∗Ã ∧ω-
(-1)^k (m-k) f^∗A ∧ω̃)
= ∫_Σ^' (Ã ∧f_∗ω- (-1)^k (m-k) A ∧f_∗ω̃) ,
which shows that $f_\ast \varphi\in \Obs^k(M^\prime;\bbZ)$ as required.
§.§ Observables from spacelike compact gauge fields
We shall now show that the Abelian group $\Conf^k_{\rm sc}(M;\bbZ)$ of
semi-classical gauge fields of spacelike compact support is isomorphic to the Abelian group
$\Obs^k(M;\bbZ)$ of semi-classical observables introduced in Definition <ref>.
Using the techniques which allow us to establish this isomorphism,
we shall also prove that $\Obs^k(M;\bbZ)$ is large enough to separate points of the semi-classical configuration space
$\Conf^k(M;\bbZ)$, i.e. for $(h,\tilde h), (h',\tilde h'\, )\in \Conf^k(M;\bbZ)$ we have
$\varphi\big((h,\tilde h)\big) = \varphi\big((h',\tilde h'\, )\big)$ for all
$\varphi \in \Obs^k(M;\bbZ)$ if and only if $(h,\tilde h) = (h',\tilde h'\,)$.
By <cit.>, for any smooth
spacelike Cauchy surface $\Sigma$ of $M$ there is a $\bbT$-valued pairing
$\ips{\cdot}{\cdot}_{\rm c} : \dH^{m-p}(\Sigma;\bbZ) \times \dH^p_{\rm c}(\Sigma;\bbZ)\to\bbT$
between differential cohomology and compactly supported differential cohomology. Using the isomorphisms
given in Theorem <ref> and Corollary <ref>, we define
a $\bbT$-valued pairing between $\Conf^k(M;\bbZ)$ and $\Conf^k_{\rm sc}(M;\bbZ)$ by
··: ^k(M;) ×^k_sc(M;) ⟶ ,
((h,h̃) , (h',h̃' )) ⟼ι_Σ^∗h̃ι_Σ^∗h' _c-
(-1)^k (m-k) ι_Σ^∗hι_Σ^∗h̃' _c .
In Lemma <ref> we show that this pairing does not depend on
the choice of Cauchy surface $\Sigma$ and we prove its naturality in the sense that
for any morphism $f: M \to M^\prime$ in $\Loc_m$ the diagram of Abelian groups
[d]_-𝕀×f_∗^k(M^';) ×^k_sc(M;) [rr]^-f^∗×𝕀 ^k(M;) ×^k_sc(M;)[d]^-··
^k(M^';) ×^k_sc(M^';)[rr]_-··
By partial evaluation, the pairing (<ref>) allows us to define group characters on
$\Conf^k(M;\bbZ)$: For any $(h',\tilde h'\, )\in \Conf^k_{\rm sc}(M;\bbZ) $
there is a group character
· (h',h̃' ) : ^k(M;)⟶ , (h,h̃)⟼(h,h̃) (h' ,h̃' ) .
The next result in particular allows us to separate points of the semi-classical configuration space
$\Conf^k(M;\bbZ)$ by using only such group characters.
The pairing $\ips{\cdot}{\cdot}$ introduced in (<ref>) is weakly non-degenerate.
Recalling Theorem <ref> and Corollary <ref>,
the pullback along the embedding $\iota_\Sigma:\Sigma\to M$ provides isomorphisms
$\Conf^k(M;\bbZ) \simeq \dH^{k,m-k}(\Sigma;\bbZ)$
and $\Conf^k_{\rm sc}(M;\bbZ) \simeq \dH^{k,m-k}_{\rm c}(\Sigma;\bbZ)$.
Using these isomorphisms, the pairing (<ref>) corresponds precisely
to the pairing
··_Σ^ : ^k,m-k(Σ;) ×^k,m-k_c(Σ;) ⟶ ,
((h,h̃) , (h',h̃' )) ⟼h̃h' _c - (-1)^k (m-k) hh̃' _c
between initial data on $\Sigma$. The proof then follows from weak non-degeneracy
of the pairing $\ips{\cdot}{\cdot}_{\rm c} : \dH^{m-p}(\Sigma;\bbZ) \times \dH^p_{\rm c}(\Sigma;\bbZ)\to\bbT$,
cf. <cit.>.
Finally, we show that the partial evaluation (<ref>)
establishes an isomorphism between $\Conf^k_{\rm sc}(M;\bbZ)$
and the Abelian group $\Obs^k(M;\bbZ)$ of semi-classical
observables introduced in Definition <ref>.
The group homomorphism
: ^k_sc(M;) ⟶^k(M;) , (h',h̃' ) ⟼· (h',h̃' ) ,
is an isomorphism which provides a natural isomorphism
between the functors $\Conf^k_{\rm sc}(-;\bbZ): \Loc_m \to \Ab$
and $\Obs^k(-;\bbZ): \Loc_m \to \Ab$.
We first have to show that the group character
$\ips{\cdot\, }{\, (h',\tilde h'\, )}$ satisfies the regularity condition of Definition <ref>,
for any $(h',\tilde h'\, ) \in \Conf^k_{\rm sc}(M;\bbZ)$. This follows
from <cit.>: For any $([A],[\tilde A]) \in \TT^k(M)$, we find
(ι×ι)([A],[Ã]) (h',h̃' )
= ∫_Σ (Ã ∧ h' - (-1)^k (m-k) A
∧ h̃' ) .
Moreover, (<ref>) is injective due to Proposition <ref>.
To show that (<ref>) is surjective,
let us take any $\varphi \in \Obs^k(M;\bbZ)$ and choose a smooth spacelike Cauchy surface $\Sigma$ of $M$.
Using the isomorphism established in Theorem <ref> and recalling Definition <ref>,
there exists a unique smooth character $\varphi_\Sigma \in \dH^{k,m-k}(\Sigma;\bbZ)^\star_\infty$
such that $\varphi_\Sigma \circ (\iota_\Sigma^\ast \times \iota_\Sigma^\ast) = \varphi$.
Using further the character duality proven in <cit.>,
there exists a unique $(h'_\Sigma,\tilde h_\Sigma') \in \dH^{k,m-k}_{\rm c}(\Sigma;\bbZ)$
such that $\ips{\cdot\, }{\, (h'_\Sigma,\tilde h'_\Sigma)}_{\Sigma}^{}
= \varphi_\Sigma$ by (<ref>).
Using Corollary <ref>, we introduce $(h',\tilde h'\, ) \in \Conf^k_{\rm sc}(M;\bbZ)$ as the unique
element with $(\iota_\Sigma^\ast \times \iota_\Sigma^\ast)(h',\tilde
h'\, ) = (h'_\Sigma,\tilde h'_\Sigma)$.
Then the definition of $\ips{\cdot}{\cdot}$ given in
(<ref>) implies that $\varphi = \ips{\cdot\, }{\,
(h',\tilde h'\, )}$.
It remains to prove that the established isomorphism is natural.
Let $f: M \to M^\prime$ be a morphism in $\Loc_m$.
Recall that the pushforward $f_\ast: \Obs^k(M;\bbZ) \to \Obs^k(M^\prime;\bbZ)$ is given by the Pontryagin dual
of the pullback $f^\ast: \Conf^k(M^\prime;\bbZ) \to \Conf^k(M;\bbZ)$
and that Lemma <ref> establishes naturality
of the pairing $\ips{\cdot}{\cdot}$. Therefore, for all $(h',\tilde
h'\, ) \in \Conf^k_{\rm sc}(M;\bbZ)$, one has
$f_\ast\ips{\cdot\, }{\, (h',\tilde h'\, )} = \ips{f^\ast \cdot\, }{\,
(h',\tilde h'\, )} = \ips{\cdot\, }{\, f_\ast(h',\tilde h'\, )}$,
i.e. the diagram
^k_sc(M;) [r]^- [d]_-f_∗ ^k(M;) [d]^-f_∗
^k_sc(M^';) [r]_- ^k(M^';)
commutes and hence (<ref>) is a natural isomorphism.
§.§ Presymplectic structure
We will introduce a natural $\bbT$-valued presymplectic structure $\tau$
on the Abelian group $\Obs^k(M;\bbZ)$ of semi-classical observables.
In this way we obtain a functor $(\Obs^k(-;\bbZ),\tau): \Loc_m \to
\PSAb$ valued in the category $\PSAb$
of presymplectic Abelian groups (with group homomorphisms preserving the presymplectic structures as morphisms).
This will be the main input for Section <ref>,
where the quantization of the semi-classical model described by $(\Obs^k(-;\bbZ),\tau)$ will be addressed.
Let $\obs: \Conf^k_{\rm sc}(M;\bbZ) \to \Obs^k(M;\bbZ)$ be the isomorphism
introduced in Proposition <ref> and $I : \Conf^k_{\rm
sc}(M;\bbZ) \to \Conf^k(M;\bbZ) $ the group homomorphism
given in (<ref>). Then
τ: ^k(M;) ×^k(M;)
⟶ , (φ,φ' )
⟼I(^-1 φ)^-1
defines a presymplectic structure on $\Obs^k(M;\bbZ)$ whose radical is $\obs(\ker I)$.
Up to the isomorphism $\obs^{-1}: \Obs^k(M;\bbZ) \to \Conf^k_{\rm sc}(M;\bbZ)$,
the mapping $\tau$ is given by
σ: ^k_sc(M;) ×^k_sc(M;)
⟶ , ((h,h̃) , (h',h̃' )) ⟼I(h,h̃) ( h',h̃' ) .
Recalling (<ref>), we observe that $\sigma$ is
$\bbZ$-bilinear and hence so is $\tau$.
Antisymmetry follows from <cit.>.
Since the pairing $\ips{\cdot}{\cdot}: \Conf^k(M;\bbZ) \times \Conf^k_{\rm sc}(M;\bbZ) \to \bbT$
is weakly non-degenerate by Proposition <ref>, the radical of
$\sigma$ coincides with the kernel of $I$ which implies that the radical of $\tau$ is $\ker(I \circ \obs^{-1}) = \obs(\ker I)$.
If $M$ has compact Cauchy surfaces $\Sigma \subseteq M$, the group homomorphism
$I : \Conf^k_{\rm sc}(M;\bbZ) \to \Conf^k(M;\bbZ) $
is the identity. In fact, with $\Sigma$ compact, $J(\Sigma) = M$ entails
that the diagram whose colimit defines $\Conf^k_{\rm sc}(M;\bbZ)$
has $\Conf^k(M,M \setminus J(\Sigma);\bbZ) = \Conf^k(M;\bbZ)$ as its terminal object.
In particular, $\tau$ is actually weakly symplectic for globally hyperbolic Lorentzian manifolds with compact Cauchy surfaces.
In this case (<ref>) coincides with the (pre)symplectic structure
described in Remark <ref>.
Our next task is to prove that the presymplectic structure $\tau$ introduced in Proposition <ref> is natural,
so that we can interpret $(\Obs^k(-;\bbZ),\tau)$ as a functor from $\Loc_m$ to $\PSAb$.
Let $f: M \to M^\prime$ be a morphism in $\Loc_m$.
Then the diagram of Abelian groups
^k(M;) ×^k(M;) [dr]^-τ[dd]_-f_∗×f_∗
^k(M^';) ×^k(M^';) [ur]_-τ
Recalling from Proposition <ref> that $\obs: \Conf^k_{\rm sc}(-;\bbZ) \Rightarrow \Obs^k(-;\bbZ)$
is a natural isomorphism, it is enough to prove commutativity of the diagram
^k_sc(M;) ×^k_sc(M;)
^k_sc(M^';) ×^k_sc(M^';) [ur]_-σ
for $\sigma$ given in (<ref>). Making use of naturality of the pairing
$\ips{\cdot}{\cdot}$ (cf. Lemma <ref>), we obtain
σ(f_∗(h,h̃) , f_∗(h',h̃' )) = I f_∗(h,h̃) f_∗(h',h̃' )
= f^∗ I f_∗(h,h̃) (h',h̃' )
= I (h,h̃) (h',h̃' ) = σ((h,h̃) , (h',h̃' )) ,
for all $(h,\tilde h), (h,\tilde h'\, )\in \Conf^k_{\rm sc}(M;\bbZ)$.
In the third equality we used $f^\ast\circ I\circ f_\ast = I$ which is proven in Lemma <ref>.
§.§ Locally covariant field theory
We analyze properties of the functor $(\Obs^k(-;\bbZ),\tau): \Loc_m \to \PSAb$ from the point of view
of the axioms of locally covariant field theory <cit.>.
[Causality axiom]
Let $M_1 \stackrel{f_1}{\longrightarrow} M \stackrel{f_2}{\longleftarrow} M_2$ be
a diagram in $\Loc_m$ such that the images of $f_1$ and $f_2$ are causally disjoint, i.e.$J(f_1(M_1)) \cap f_2(M_2) = \emptyset$.
Then the presymplectic structure $\tau: \Obs^k(M;\bbZ) \times \Obs^k(M;\bbZ) \to \bbT$
vanishes on ${f_1}_\ast(\Obs^k(M_1;\bbZ)) \times {f_2}_\ast(\Obs^k(M_2;\bbZ))$.
Using again the natural isomorphism
$\obs: \Conf^k_{\rm sc}(-;\bbZ) \Rightarrow \Obs^k(-;\bbZ)$,
it is equivalent to prove the analogous statement for $\sigma$ given in (<ref>).
Given $(h,\tilde h)\in \Conf^k_{\rm sc}(M_1;\bbZ)$ and $(h',\tilde h'\, )\in \Conf^k_{\rm sc}(M_2;\bbZ)$,
naturality of the pairing $\ips{\cdot}{\cdot}$ implies
σ(f_1_∗(h,h̃) , f_2_∗(h',h̃' ))=
f_2^∗ I f_1_∗(h,h̃) (h',h̃' )
and the proof follows from $f_{2}^\ast\circ I \circ {f_{1}}_\ast =0$, see Lemma <ref>.
[Time-slice axiom]
Let $f: M \to M^\prime$ be a Cauchy morphism, i.e. a $\Loc_m$-morphism
whose image $f(M)$ contains a smooth spacelike Cauchy surface of $M^\prime$.
Then $f_\ast: \Obs^k(M;\bbZ) \to \Obs^k(M^\prime;\bbZ)$ is an isomorphism.
Take any smooth spacelike Cauchy surface $\Sigma^\prime \subseteq f(M)$ of $M^\prime$
and note that its preimage $\Sigma = f^{-1}(\Sigma^\prime\, )$ is a smooth spacelike Cauchy surface of $M$.
The diagram of Abelian groups
^k_sc(M;) [r]^-f_∗ [d]_-ι_Σ^∗×ι_Σ^∗
^k,m-k_c(Σ;) [r]_-f_Σ∗ ×f_Σ∗
commutes, its vertical arrows are isomorphisms (cf. Corollary <ref>)
and its bottom horizontal arrow is an isomorphism since, by restriction,
$f$ induces an orientation preserving isometry $f_\Sigma: \Sigma \to \Sigma^\prime$.
Hence $f_\ast$ is an isomorphism and, by using again the natural isomorphism
$\obs: \Conf_{\rm sc}^k(-;\bbZ) \Rightarrow \Obs^k(-;\bbZ)$, we find
that $f_\ast: \Obs^k(M;\bbZ) \to \Obs^k(M^\prime;\bbZ)$ is an isomorphism.
[Violation of the locality axiom]
Let $f: M \to M^\prime$ be a morphism in $\Loc_m$.
Then $f_\ast:\Obs^k(M;\bbZ) \to \Obs^k(M^\prime;\bbZ)$ is injective
if and only if $f_\ast: \H^{k-1,m-k-1}_{\rm sc}(M;\bbT)\to \H^{k-1,m-k-1}_{\rm sc}(M^\prime;\bbT)$ is injective.
For $m=2$ and $k=1$ the latter is always the case, while for $m \geq 3$ and $k \in \{1, \ldots, m-1\}$
there is at least one morphism in $\Loc_m$ violating injectivity.
Using again the natural isomorphism
$\obs: \Conf^k_{\rm sc}(-;\bbZ) \Rightarrow \Obs^k(-;\bbZ)$,
we can replace $f_\ast:\Obs^k(M;\bbZ) \to \Obs^k(M^\prime;\bbZ)$ in the statement by
$f_\ast:\Conf_{\rm sc}^k(M;\bbZ) \to \Conf_{\rm sc}^k(M^\prime;\bbZ)$.
The proof of the first part follows easily from
the commutative diagram of short exact sequences
given in Remark <ref> and the fact that
$f_\ast: \Omega^k_{{\rm sc},\bbZ} \cap \ast \Omega^{m-k}_{{\rm sc},\bbZ}(M)\to
\Omega^k_{{\rm sc},\bbZ} \cap \ast \Omega^{m-k}_{{\rm sc},\bbZ}(M^\prime\, )$
is always injective.
To prove the second part, we notice that there is a chain of isomorphisms
^̋k-1,m-k-1_sc(M;)≃^̋k-1,m-k-1_c(Σ;) ≃^̋m-k,k(Σ;)^⋆≃^̋m-k,k(M;)^⋆ ,
where $^\star$ denotes Pontryagin duality. The first isomorphism
is from (<ref>), the second is
presented in <cit.> and
the third simply follows from homotopy invariance of cohomology and $M\simeq \bbR\times \Sigma$.
Hence the counterexamples to injectivity provided
in <cit.> can be used to prove the present claims.
For the case $m=2$ and $k=1$, see the argument preceding <cit.>.
The next theorem summarizes the results obtained in this section
in view of the standard axioms of locally covariant field theory <cit.>.
In particular, we stress that the locality axiom,
which requires $f_\ast: \Obs^k(M;\bbZ) \to \Obs^k(M^\prime;\bbZ)$
to be injective for all $\Loc_m$-morphisms $f: M \to M^\prime$, does not hold in general.
The functor $(\Obs^k(-;\bbZ),\tau): \Loc_m \to \PSAb$
satisfies the causality and time-slice axioms of locally covariant field theory,
however the locality axiom is satisfied only in the case $m=2$ and $k=1$,
while it is violated for $m \geq 3$ and $k \in \{1, \ldots, m-1\}$.
§ QUANTIZATION
The quantization of the semi-classical gauge theory $(\Obs^k(-;\bbZ),\tau): \Loc_m \to \PSAb$
can be easily performed by using the well-established techniques
of CCR-algebras, see <cit.> and also <cit.>
for details. Loosely speaking, given any $m$-dimensional spacetime $M$,
we assign to the presymplectic Abelian group $(\Obs^k(M;\bbZ),\tau)$ the
$C^\ast$-algebra $\mathfrak{CCR}(\Obs^k(M;\bbZ),\tau)$
that is generated by Weyl symbols $\mathrm{W}(\varphi)$, for all $\varphi\in \Obs^k(M;\bbZ)$,
which satisfy the Weyl relations
W(φ) W(φ' ) = exp( 2π i τ(φ,φ' )) W(φ+φ' ) , W(φ)^∗= W(-φ) ,
for all $\varphi,\varphi' \in \Obs^k(M;\bbZ)$; by $\exp( 2\pi \, {\rm i} \,(\, \cdot\,)) : \bbT=\bbR/\bbZ \to \bbC$
we denote the embedding of the circle group into the complex numbers.
More precisely, the CCR-functor $\mathfrak{CCR} : \PSAb \to \CAlg$
from the category of presymplectic Abelian groups to the category of $C^\ast$-algebras
is constructed in detail in <cit.>. Composing the
functor $(\Obs^k(-;\bbZ),\tau): \Loc_m \to \PSAb$ with the CCR-functor, we obtain
a functor from $\Loc_m$ to $\CAlg$ which, according to <cit.>,
should be interpreted as a quantum field theory. We can thereby define a family
of quantum field theories by setting
𝔄^k := ℭℭℜ∘(^k(-;),τ) : 𝖫𝗈𝖼_m⟶ ,
which depend on the degree $k\in \{1,\dots, m-1\}$ of the gauge theory.
The properties of the semi-classical gauge theory from Theorem <ref>
are preserved by quantization (see e.g. the arguments in <cit.>),
which leads us to
The functor $\mathfrak{A}^k : \Loc_m \to \CAlg$ enjoys the following properties:
* Quantum causality axiom: Let $M_1 \stackrel{f_1}{\longrightarrow} M \stackrel{f_2}{\longleftarrow} M_2$ be
a diagram in $\Loc_m$ such that the images of $f_1$ and $f_2$ are causally disjoint.
Then the subalgebras ${f_1}_\ast (\mathfrak{A}^k(M_1)) $ and ${f_2}_\ast (\mathfrak{A}^k(M_2)) $
of $\mathfrak{A}^k(M) $ commute.
* Quantum time-slice axiom: Let $f: M \to M^\prime$ be a Cauchy morphism.
Then $f_\ast: \mathfrak{A}^k(M) \to \mathfrak{A}^k(M^\prime\, )$ is an isomorphism.
* Violation of the quantum locality axiom: Let $f: M \to M^\prime$ be a morphism in $\Loc_m$.
Then $f_\ast:\mathfrak{A}^k(M) \to \mathfrak{A}^k(M^\prime\, )$ is injective
if and only if $f_\ast: \H^{k-1,m-k-1}_{\rm sc}(M;\bbT)\to \H^{k-1,m-k-1}_{\rm sc}(M^\prime;\bbT)$ is injective.
For $m=2$ and $k=1$ the latter is always the case,
while for $m \geq 3$ and $k \in \{1, \ldots, m-1\}$
there is at least one morphism in $\Loc_m$ violating injectivity.
§ QUANTUM DUALITY
In this section we show that there exist dualities between the quantum field theories
defined in (<ref>). These dualities will be described at the functorial level
and therefore hold true for all spacetimes $M$ in a coherent (natural) way.
In order to motivate our definition of duality given below,
let us recall that a quantum field theory is a functor $\mathfrak{A} : \Loc_m\to \CAlg$
from the category of $m$-dimensional spacetimes to the category of $C^\ast$-algebras.
The collection of all $m$-dimensional quantum field theories
is therefore described by the functor category $[\Loc_m,\CAlg]$;
objects in this category are functors $\mathfrak{A} : \Loc_m\to \CAlg$
and morphisms are natural transformations $\eta : \mathfrak{A}\Rightarrow \mathfrak{A}'$.
In physics one calls the functor category $[\Loc_m,\CAlg]$ the “theory space”
of $m$-dimensional quantum field theories which, being a category,
comes with a natural
notion of equivalence of theories.
A duality between two quantum field theories $\mathfrak{A}, \mathfrak{A}' : \Loc_m\to \CAlg$
is a natural isomorphism $\eta : \mathfrak{A}\Rightarrow \mathfrak{A}'$.
We shall now construct explicit dualities between the quantum field theories
$\mathfrak{A}^k$ and $\mathfrak{A}^{m-k}$ given in (<ref>), for all
$m\geq 2$ and $k \in\{1,\dots,m-1\}$. Our strategy is to define first the dualities
at the level of the semi-classical configuration spaces (<ref>),
and then lift them to the presymplectic Abelian groups and ultimately to the corresponding
quantum field theories. For any object $M$ in $\Loc_m$ we define a group homomorphism
ζ: ^m-k(M;)⟶^k(M;) , (h,h̃)
⟼(h̃, -(-1)^k (m-k) h) ,
which interchanges (up to a sign) the roles of $h\in \dH^{m-k}(M;\bbZ)$ and $\tilde h\in \dH^{k}(M;\bbZ)$.
We interpret the mapping (<ref>) physically as exchanging the `electric' and `magnetic' sectors
of the Abelian gauge theory. The map (<ref>) defines a natural isomorphism
$\zeta: \Conf^{m-k}(-;\bbZ) \Rightarrow \Conf^{k}(-;\bbZ)$ because its components are clearly isomorphisms
and for any morphism $f : M\to M^\prime$ in $\Loc_m$ the diagram of Abelian groups
[d]_-f^∗ ^m-k(M^';) [r]^-ζ ^k(M^';) [d]^-f^∗
^m-k(M;) [r]_-ζ ^k(M;)
We now dualize (<ref>) with respect to the weakly non-degenerate pairing
(<ref>): Define a group homomorphism $\zeta^\star : \Conf^{k}_{\rm sc}(M;\bbZ)\to \Conf^{m-k}_{\rm sc}(M;\bbZ)$
by the condition
(h,h̃) ζ^⋆(h',h̃' ) := ζ(h,h̃) (h',h̃' ) ,
for all $(h',\tilde h'\, )\in \Conf^{k}_{\rm sc}(M;\bbZ)$ and $(h,\tilde h)\in \Conf^{m-k}(M;\bbZ)$.
A quick calculation shows that
ζ^⋆: ^k_sc(M;)⟶^m-k_sc(M;) , (h',h̃' )⟼( -(-1)^k (m-k) h̃',h' ) .
The mapping (<ref>) defines a natural isomorphism
$\zeta^\star : \Conf^{k}_{\rm sc}(-;\bbZ)\Rightarrow \Conf^{m-k}_{\rm sc}(-;\bbZ)$, i.e.
for any morphism $f: M\to M^\prime$ in $\Loc_m$
the diagram of Abelian groups
[d]_-f_∗^k_sc(M;)[r]^-ζ^⋆ ^m-k_sc(M;)[d]^-f_∗
^k_sc(M^';) [r]_-ζ^⋆ ^m-k_sc(M^';)
commutes. We next observe that (<ref>) preserves the presymplectic structure
(<ref>): A quick calculation shows that
σ(ζ^⋆(h,h̃) , ζ^⋆(h',h̃' )) = σ((h,h̃) , (h', h̃' )) ,
for all $(h,\tilde h), (h',\tilde h'\, )\in \Conf^{k}_{\rm sc}(M;\bbZ)$.
Using also the natural isomorphisms $\obs : \Conf^p_{\rm sc}(-;\bbZ) \Rightarrow \Obs^p(-;\bbZ)$
given in Proposition <ref>,
for $p=k$ and $p=m{-}k$,
we find that $\zeta^\star$ defines a natural isomorphism (denoted by the same symbol)
ζ^⋆: (^k(-;),τ) ⟹(^m-k(-;),τ)
between functors from $\Loc_m$ to $\PSAb$.
We can now state the main result of this section.
The $C^\ast$-algebra homomorphism
η:= ℭℭℜ(ζ^⋆) : 𝔄^k(M) ⟶𝔄^m-k(M)
a duality between the two quantum field theories $\mathfrak{A}^k,\mathfrak{A}^{m-k} : \Loc_m\to \CAlg$.
We need to show that $\eta$ defines a natural isomorphism
$\eta : \mathfrak{A}^k\Rightarrow \mathfrak{A}^{m-k}$. Naturality of
$\eta$ is a direct consequence of naturality of $\zeta^\star$ and the fact that $\mathfrak{CCR}$ is a functor,
in particular it preserves compositions. As functors preserve isomorphisms it then follows
that $\eta$ is a natural isomorphism.
For $m=2k$ the duality of Theorem <ref> becomes a self-duality, i.e. a natural automorphism
$\eta : \mathfrak{A}^k \Rightarrow \mathfrak{A}^{k}$.
§ SELF-DUAL ABELIAN GAUGE THEORY
In dimension $m=2k$ it makes sense to demand the self-duality condition
h = h
for a differential character $h\in\dH^k(M;\bbZ)$. Applying the Hodge operator $\ast$
to both sides of (<ref>) we obtain
h = -(-1)^k^2 h = -(-1)^k^2 h ,
which implies that for $k$ even the only solutions to (<ref>) are flat fields $h = \kappa(t)$, for
$t\in \H^{k-1}(M;\bbT)$. In the following we shall focus on the physically much richer and interesting
case where $k\in 2\bbZ_{\geq0}+1$ is odd.
The Abelian group of solutions to the self-duality equation (<ref>) is denoted
^k(M;) := {h∈^k(M;) : h = h} .
There is a monomorphism
diag : ^k(M;) ⟶^k(M;) , h⟼(h,h)
to the semi-classical configuration space introduced in (<ref>).
Given any smooth spacelike Cauchy surface $\Sigma$ of $M$ with embedding $\iota_{\Sigma} : \Sigma \to M$,
we compose (<ref>) with the isomorphism of Theorem <ref>
and obtain a monomorphism
(ι^∗_Σ ×ι^∗_Σ)∘diag : ^k(M;)
⟶^k,k(Σ;) , h⟼(ι^∗_Σ h , ι^∗_Σ h) ,
whose image is given by the diagonal in $\dH^{k,k}(\Sigma;\bbZ)$: Given any
$(h_{\Sigma},h_{\Sigma})\in \dH^{k,k}(\Sigma;\bbZ)$ in the diagonal,
consider the unique solution $(h,\tilde h)\in \Conf^k(M;\bbZ)$
of $\cu\,h =\dcu\,\tilde h$ with initial data $\iota_{\Sigma}^\ast h= h_{\Sigma}$ and
$\iota^\ast_{\Sigma}\tilde h = h_{\Sigma}$, cf. Theorem <ref>.
Then $(h-\tilde h, \tilde h-h)\in \Conf^k(M;\bbZ)$ satisfy $\iota_\Sigma^\ast(h-\tilde h)=0$
and $\iota_\Sigma^\ast(\tilde h-h)=0$, hence $\tilde h=h$ by using again Theorem <ref>.
We have thereby obtained an isomorphism
of Abelian groups
ι^∗_Σ : ^k(M;)⟶^k(Σ;) ,
which we may interpret as in (<ref>) as establishing the well-posedness of the initial value problem
for $h\in\dH^k(M;\bbZ)$ given by
h = h , ι^∗_Σh = h_Σ ,
with initial datum $h_{\Sigma}\in\dH^k(\Sigma;\bbZ)$.
Similar statements hold true for the Abelian group of solutions of spacelike compact support
to the self-duality equation (<ref>), denoted by
^k_sc(M;) := {h ∈^k_sc(M;) : h = h} .
In particular, there is a monomorphism
diag : ^k_sc(M;) ⟶_sc^k(M;) , h⟼(h,h)
to the Abelian group of semi-classical gauge fields of spacelike compact support introduced in (<ref>).
Using Corollary <ref>, one easily shows that
ι^∗_Σ : ^k_sc(M;)⟶^k_c(Σ;)
is an isomorphism, which we may interpret as establishing
the well-posedness of the initial value problem (<ref>)
for $h\in\dH_{\rm sc}^k(M;\bbZ)$ of spacelike compact support
and initial datum $h_{\Sigma}\in\dH_{\rm c}^k(\Sigma;\bbZ)$ of compact support.
Similarly to (<ref>), there is a weakly non-degenerate $\bbT$-valued pairing
··_𝔰𝔡 : ^k(M;)×^k_sc(M;)
⟶ , (h,h' )⟼ι^∗_Σ hι^∗_Σh' _c^ ,
which is independent of the choice of Cauchy surface $\Sigma$ of $M$.[This is
demonstrated by a proof similar to that of Lemma <ref>.]
Thus there is a monomorphism
_𝔰𝔡 : ^k_sc(M;)⟶^k(M;)^⋆ , h'⟼·h' _𝔰𝔡
to the character group of $\sdConf^k(M;\bbZ)$,
whose image is denoted $\sdObs^k(M;\bbZ)$ and called
the Abelian group of semi-classical observables on $\sdConf^k(M;\bbZ)$.
Analogously to Proposition <ref> we define a $\bbT$-valued presymplectic structure
τ_𝔰𝔡 : ^k(M;)×^k(M;)⟶ , (φ,φ' )⟼I(_𝔰𝔡^-1φ)_𝔰𝔡^-1φ' _𝔰𝔡 .
Up to the isomorphism $\obs_{\mathfrak{sd}}^{-1} : \sdObs^k(M;\bbZ)\to \sdConf^k_{\rm sc}(M;\bbZ)$
induced by (<ref>), the presymplectic structure reads as
σ_𝔰𝔡 : ^k_sc(M;)×^k_sc(M;) ⟶ , (h,h' ) ⟼I(h)h' _𝔰𝔡 .
The radical of $\sigma_{\mathfrak{sd}}$ coincides with the kernel
of $I : \sdConf^k_{\rm sc}(M;\bbZ) \to \sdConf^k(M;\bbZ) $, hence the radical of $\tau_{\mathfrak{sd}} $
is $\obs_{\mathfrak{sd}}(\mathrm{ker} \,I)$.
Using arguments similar to those of Section <ref>, one can show that the presymplectic Abelian groups
$(\sdObs^k(M;\bbZ),\tau_{\mathfrak{sd}})$ for the self-dual field theory are functorial, i.e.we have constructed a functor
(^k(-;),τ_𝔰𝔡) : _2k ⟶ .
Composing with the CCR-functor from Section <ref> we obtain quantum field theories
𝔰𝔡𝔄^k := ℭℭℜ∘(^k(-;),τ_𝔰𝔡) : _2k⟶ ,
for all $k\in 2\bbZ_{\geq0} +1$, which quantize the self-duality equation (<ref>).
Using similar arguments as those of Section <ref>,
one can show that these quantum field theories satisfy the same
properties as those listed in
Theorem <ref>.
The functor $\mathfrak{sdA}^k : \Loc_{2k} \to \CAlg$ enjoys the following properties:
* Quantum causality axiom: Let $M_1 \stackrel{f_1}{\longrightarrow} M \stackrel{f_2}{\longleftarrow} M_2$ be
a diagram in $\Loc_{2k}$ such that the images of $f_1$ and $f_2$ are causally disjoint.
Then the subalgebras ${f_1}_\ast (\mathfrak{sdA}^k(M_1)) $ and ${f_2}_\ast (\mathfrak{sdA}^k(M_2)) $
of $\mathfrak{sdA}^k(M) $ commute.
* Quantum time-slice axiom: Let $f: M \to M^\prime$ be a Cauchy morphism.
Then $f_\ast: \mathfrak{sdA}^k(M) \to \mathfrak{sdA}^k(M^\prime\, )$ is an isomorphism.
* Violation of the quantum locality axiom: Let $f: M \to M^\prime$ be a morphism in $\Loc_{2k}$.
Then $f_\ast:\mathfrak{sdA}^k(M) \to \mathfrak{sdA}^k(M^\prime\, )$ is injective
if and only if $f_\ast: \H^{k-1}_{\rm sc}(M;\bbT)\to \H^{k-1}_{\rm sc}(M^\prime;\bbT)$ is injective.
For $k=1$ the latter is always the case, while for $k \in 2\bbZ_{\geq0}+3 $
there is at least one morphism in $\Loc_{2k}$ violating injectivity.
We address the question how the self-dual quantum field theories $\mathfrak{sdA}^k$, which quantize
the self-duality equation (<ref>), are related to the self-dualities of
the quantum field theories $\mathfrak{A}^k$ established in Corollary <ref>.
Let $k\in 2\bbZ_{\geq0}+1$ and consider any object $M$ in $\Loc_{2k}$.
The self-duality (<ref>) on $\Conf^k_{\rm sc}(M;\bbZ)$ then reduces
to $\zeta^\star(h',\tilde h'\, ) = (\tilde h',h'\, )$, i.e. it simply interchanges $h'$ and $\tilde h'$.
The Abelian group of invariants under this self-duality is given by the diagonal
^k_sc(M;)^inv :=
{(h',h̃' )∈^k_sc(M;) : ζ^⋆(h',h̃' ) = (h',h̃' ) }
={(h',h' )∈^k_sc(M;)} ,
which by (<ref>) is isomorphic to $\sdConf^k_{\rm sc}(M;\bbZ)$.
Restricting the presymplectic structure (<ref>) to
the invariants $\Conf^k_{\rm sc}(M;\bbZ)^{\mathrm{inv}}$ then yields
σ: ^k_sc(M;)^inv×^k_sc(M;)^inv⟶ , ((h,h) , (h',h' ))⟼2 σ_𝔰𝔡(h,h' ) ,
where $\sigma_{\mathfrak{sd}}$ is the presymplectic
structure on $\sdConf^k_{\rm sc}(M;\bbZ)$ given in (<ref>).
Due to the prefactor $2$, it follows that $(\sdConf^k_{\rm sc}(M;\bbZ),\sigma_{\mathfrak{sd}})$
and $(\Conf^k_{\rm sc}(M;\bbZ)^{\mathrm{inv}},\sigma)$ are not
isomorphic as presymplectic Abelian groups, but only as Abelian groups.
Moreover, the $C^\ast$-algebras $\mathfrak{sdA}^k(M)$ and $\mathfrak{A}^k(M)^{\mathrm{inv}}$
(i.e. the $C^\ast$-subalgebra of $\mathfrak{A}^k(M)$ which is generated by the invariant Weyl symbols
$\mathrm{W}(\obs(h',h'\, ))$, for all $(h',h'\, )\in\Conf^k_{\rm sc}(M;\bbZ)^{\mathrm{inv}} $)
are in general not isomorphic.
Thus even though the quantum field theories $\mathfrak{sdA}^k : \Loc_{2k}\to \CAlg$
and $\mathfrak{A}^k(-)^{\mathrm{inv}} : \Loc_{2k}\to \CAlg$ are similar,
they are strictly speaking not isomorphic. In particular, due to effects which are
caused by $\bbZ_2$-torsion elements in the cohomology groups $\H^k(M;\bbZ)$, the latter theory typically
has a bigger center than the former theory. An explicit example of this fact is illustrated below.
Fix any $k \in 2\bbZ_{\geq 0}+3$ and consider the lens space $L = \mathbb{S}^{2k-3}/\bbZ_2$
obtained as the quotient of the $2k{-}3$-sphere $\mathbb{S}^{2k-3}$ by the antipodal $\bbZ_2$-action.
Take any object $M$ in $\Loc_{2k}$ which admits a smooth spacelike Cauchy surface $\Sigma$ diffeomorphic
to $\bbT^2 \times L$, where $\bbT^2$ is the $2$-torus. Since the Cauchy surface
$\Sigma$ is compact, the notion of spacelike compact support becomes irrelevant for this spacetime
$M$ and in particular the homomorphism $I : \sdConf^k_{\rm sc}(M;\bbZ) \to \sdConf^k(M;\bbZ) $
reduces to the identity. Using standard results on the homology groups of lens spaces,
see e.g. <cit.>, and the universal coefficient theorem
for cohomology, one shows that $\H^{k-1}(L;\bbZ) \simeq \bbZ_2$. Using the Künneth theorem
we find that $\H^k(\Sigma;\bbZ)$ has a direct summand $(\bbZ_2)^2$.
In particular, there exists $t \in \H^k(\Sigma;\bbZ)$ such that $t \neq 0$, but $2 t = 0$.
Recalling that $\ch$ is surjective (cf. (<ref>)),
we find $f \in \dH^k(\Sigma;\bbZ)$ such that $\ch\, f = t$.
It follows that there exists $A \in \Omega^{k-1}(\Sigma)$ such that $\iota[A] = 2 f$.
Introducing $h_\Sigma = f - \iota[A/2] \in \dH^k(\Sigma;\bbZ)$,
by construction we obtain $h_\Sigma \neq 0$ (otherwise $t$ would be trivial) and $2 h_\Sigma = 0$.
Solving the initial value problem (<ref>) provides $h \in \mathfrak{sdC}^k(M;\bbZ)$
with $h \neq 0$, but $2 h = 0$.
In fact, $2 h \in \mathfrak{sdC}^k(M;\bbZ)$ solves (<ref>)
with initial datum $2 h_\Sigma = 0$.
Since $\Sigma$ is compact, the presymplectic structure (<ref>) is weakly non-degenerate.
In particular, being non-zero, $h \in \mathfrak{sdC}^k(M)$ is not in the radical.
Conversely, taking into account also $(h,h) \in \Conf^k(M;\bbZ)^\mathrm{inv}$,
we find $\sigma((h,h),(h',h'\, )) = 2 \sigma_\mathfrak{sd}(h,h'\, ) =
\sigma_\mathfrak{sd}(2 h,h'\, ) = 0$
for all $(h',h'\, ) \in \Conf^k(M)^{\mathrm{inv}}$. This shows that the center of $\mathfrak{A}^k(M)^{\mathrm{inv}}$
is bigger than that of $\mathfrak{sdA}^k (M)$ for this particular spacetime $M$.
§ ACKNOWLEDGMENTS
We thank Ulrich Bunke for helpful discussions and the anonymous referees for their comments and remarks.
This work was supported in part by the Action MP1405 QSPACE from the
European Cooperation in Science and Technology (COST).
The work of C.B. is partially supported by the Collaborative Research Center (SFB)
“Raum Zeit Materie”, funded by the Deutsche Forschungsgemeinschaft (DFG, Germany).
The work of M.B. is supported partly by a Research Fellowship of the Della Riccia Foundation (Italy)
and partly by a Postdoctoral Fellowship of the Alexander von Humboldt Foundation (Germany).
The work of A.S. is supported by a Research Fellowship of the Deutsche Forschungsgemeinschaft (DFG, Germany).
The work of R.J.S. is partially supported by the Consolidated Grant ST/L000334/1
from the UK Science and Technology Facilities Council.
§ TECHNICAL LEMMAS
In this appendix we prove five lemmas which are used in the main text.
Some of these proofs are rather technical and also make use of results in the companion
paper <cit.>, in which case we give precise references.
Let $M$ be a time-oriented $m$-dimensional globally hyperbolic Lorentzian manifold
and $\Sigma$ a smooth spacelike Cauchy surface of $M$.
Consider the embedding $\iota_{\Sigma} : \Sigma \to M$ of $\Sigma$ into $M$,
and the projection $\pi_{\Sigma} : M\to \Sigma$ of $M$ onto $\Sigma$ which is induced by a
choice of diffeomorphism $M\simeq \bbR\times \Sigma$ such that $\iota_{\Sigma}(\Sigma) \simeq \{0\}\times \Sigma$,
cf. <cit.>. Then:
(i) Inducing $\iota_{\Sigma}$ and $\pi_{\Sigma}$ to smooth singular chains, i.e.
$\iota_{\Sigma\,\ast} : C_{\sharp}(\Sigma) \to C_{\sharp}(M)$ and $\pi_{\Sigma\,\ast} : C_{\sharp}(M)\to C_{\sharp}(\Sigma)$,
we have $\pi_{\Sigma\,\ast}\circ \iota_{\Sigma\,\ast} = \id$
and $\iota_{\Sigma\,\ast}\circ \pi_{\Sigma\,\ast} - \id = \del\circ h_{\Sigma} + h_{\Sigma}\circ\del$,
for a chain homotopy $h_{\Sigma} : C_{\sharp} (M)\to C_{\sharp+1}(M)$. In particular, $\iota_{\Sigma}$ and $\pi_{\Sigma}$
induce isomorphisms on smooth singular homology:
_̋♯(M) @<3pt>[r]^-π_Σ ∗ _̋♯(Σ) @<1pt>[l]^-ι_Σ ∗ .
(ii) Let $G$ be an Abelian group. Inducing $\iota_{\Sigma}$ and $\pi_{\Sigma}$ to smooth singular $G$-valued cochains,
i.e. $\iota_{\Sigma}^\ast : C^{\sharp}(M;G) \to C^{\sharp}(\Sigma;G)$ and
$\pi_{\Sigma}^{\ast} : C^{\sharp}(\Sigma;G)\to C^{\sharp}(M;G)$,
we have $\iota_{\Sigma}^\ast \circ \pi_{\Sigma}^\ast = \id$
and $\pi_{\Sigma}^\ast\circ \iota_{\Sigma}^\ast - \id = \cdel\circ h_{\Sigma}^\star + h_{\Sigma}^\star\circ\cdel$,
for a cochain homotopy $h_{\Sigma}^\star : C^{\sharp} (M;G)\to C^{\sharp-1}(M;G)$.
In particular, $\iota_{\Sigma}$ and $\pi_{\Sigma}$
induce isomorphisms on smooth singular cohomology with coefficients in $G$:
^̋♯(M;G) @<-1pt>[r]_-ι_Σ^∗ ^̋♯(Σ;G) @<-3pt>[l]_-π_Σ^∗ .
We shall denote points by $x\in\Sigma$ and $(t,x)\in M\simeq \bbR\times \Sigma$.
By construction we have $\pi_{\Sigma}\circ \iota_\Sigma = \id_{\Sigma}$.
Notice further that $\iota_{\Sigma}\circ \pi_{\Sigma}$ and the identity $\id_M$
are homotopic via
H_Σ : [0,1]×M⟶M , (s , (t,x) )⟼(s t,x) .
As usual, see for example the proof of <cit.>, this homotopy induces
the desired chain homotopy $h_{\Sigma} : C_{\sharp} (M)\to C_{\sharp+1}(M)$, which proves item (i).
Item (ii) then follows by defining $h_\Sigma^\star = \Hom(h_\Sigma,G): C^{\sharp} (M;G)\to C^{\sharp-1}(M;G)$.
Under the same hypotheses as in Lemma <ref>, let $K\subseteq \Sigma$
be a compact subset. Then:
(i) Inducing $\iota_{\Sigma}$ and $\pi_{\Sigma}$ to relative smooth singular chains, i.e.
$\iota_{\Sigma\,\ast} : C_{\sharp}(\Sigma,\Sigma\setminus K) \to C_{\sharp}(M,M\setminus J(K))$
and $\pi_{\Sigma\,\ast} : C_{\sharp}(M,M\setminus J(K))\to C_{\sharp}(\Sigma,\Sigma\setminus K)$,
we have $\pi_{\Sigma\,\ast}\circ \iota_{\Sigma\,\ast} = \id$
and $\iota_{\Sigma\,\ast}\circ \pi_{\Sigma\,\ast} - \id =
\del\circ h_{\Sigma} + h_{\Sigma}\circ\del$, for a chain homotopy $h_{\Sigma} : C_{\sharp} (M,M\setminus J(K))
\to C_{\sharp+1}(M,M\setminus J(K))$. In particular, $\iota_{\Sigma}$ and $\pi_{\Sigma}$
induce an isomorphism on relative smooth singular homology:
_̋♯(M,M∖J(K)) @<3pt>[r]^-π_Σ ∗
_̋♯(Σ,Σ∖K) @<1pt>[l]^-ι_Σ ∗ .
(ii) Let $G$ be an Abelian group. Inducing $\iota_{\Sigma}$ and $\pi_{\Sigma}$ to relative
smooth singular $G$-valued cochains,
i.e. $\iota_{\Sigma}^\ast : C^{\sharp}(M,M\setminus J(K);G) \to C^{\sharp}(\Sigma,\Sigma\setminus K;G)$ and
$\pi_{\Sigma}^{\ast} : C^{\sharp}(\Sigma,\Sigma\setminus K ;G)\to C^{\sharp}(M,M\setminus J(K);G)$,
we have $\iota_{\Sigma}^\ast \circ \pi_{\Sigma}^\ast = \id$
and $\pi_{\Sigma}^\ast\circ \iota_{\Sigma}^\ast - \id
= \cdel\circ h_{\Sigma}^\star + h_{\Sigma}^\star\circ\cdel$,
for a cochain homotopy $h_{\Sigma}^\star : C^{\sharp} (M,M\setminus J(K);G)\to C^{\sharp-1}(M,M\setminus J(K);G)$.
In particular, $\iota_{\Sigma}$ and $\pi_{\Sigma}$
induce an isomorphism on relative smooth singular cohomology with coefficients in $G$:
^̋♯(M,M∖J(K);G) @<-1pt>[r]_-ι_Σ^∗
^̋♯(Σ,Σ∖K;G) @<-3pt>[l]_-π_Σ^∗ .
Notice that $\iota_{\Sigma} : \Sigma \to M$ maps $\Sigma\setminus K$ to $M\setminus J(K)$ because
$\Sigma$ is by assumption spacelike.
Moreover, $\pi_{\Sigma} : M \to \Sigma$ maps $M\setminus J(K)$ to $\Sigma\setminus K$:
Assume there exists $(t,x)\in M\setminus J(K)$ such that $\pi_{\Sigma}(t,x) = x\in K$;
then the curve $\gamma : [0,1] \to M\,,~s\mapsto (s\,t,x)$ connecting $(0,x)$ with $(t,x)$
is timelike, hence $(t,x)\in J(K)$ which is a contradiction.
Similarly, the homotopy (<ref>) restricts
to $H_{\Sigma} : [0,1]\times \big(M\setminus J(K)\big) \to M\setminus J(K)$.
The rest of the proof then follows that of Lemma <ref>.
Let $f: M \to M^\prime$ be a morphism in $\Loc_m$
and denote by $\mathcal{K}_M$ the directed set of compact subsets of $M$.
Consider the natural transformation
f^∗: ^k(M^',M^'∖J(f(-));) ⟹^k(M,M ∖J(-);)
between functors from $\mathcal{K}_M$ to $\Ab$ induced by $f$.
Then, for each smooth spacelike Cauchy surface $\Sigma$ of $M$,
the restriction of $f^\ast$ to the directed set $\mathcal{K}_\Sigma \subseteq \mathcal{K}_M$
of compact subsets of $\Sigma$ is a natural isomorphism.
In particular, we can consider the natural transformation
(f^∗)^-1: ^k(M,M ∖J(-);) ⟹^k(M^',M^'∖J(f(-));)
between functors from $\mathcal{K}_\Sigma$ to $\Ab$.
Then the pushforward for semi-classical gauge fields of spacelike compact support
\begin{equation}\label{eqPushforwardSCConf}
f_\ast: \Conf^k_{\rm sc}(M;\bbZ) \longrightarrow \Conf^k_{\rm sc}(M^\prime;\bbZ)
\end{equation}
is canonically induced by the colimit prescription in (<ref>) restricted to $\mathcal{K}_\Sigma$,
see also Remark <ref>, and by the universal property of colimits.
For each $K \subseteq M$ compact, we note that $f: M \to M^\prime$ in $\Loc_m$ induces
an open embedding $f:(M,M \setminus J(K)) \to (M^\prime,M^\prime \setminus J(f(K)))$ of
pairs which is compatible with the inclusions of the given submanifolds.
Looking at an inclusion $K \subseteq K^\prime$ of compact subsets of $M$,
one realizes that both $(M,M \setminus J(-))$ and $(M^\prime,M^\prime \setminus J(f(-)))$
are functors from $\mathcal{K}_M$ to $\Pair^\op$, the opposite category
of the category $\Pair$ of pairs of manifolds
with submanifold preserving smooth maps as morphisms;
moreover, $f:(M,M \setminus J(-)) \Rightarrow (M^\prime,M^\prime \setminus J(f(-)))$
is a natural transformation between these functors.
Therefore, applying the functor $\dH^{k,m-k}(-;\bbZ): \Pair^\op \to \Ab$,
cf. <cit.>,
we obtain the pullback along $f$ as a natural transformation
\begin{equation}
f^\ast: \dH^{k,m-k}(M^\prime,M^\prime \setminus J(f(-));\bbZ) \Longrightarrow \dH^{k,m-k}(M,M \setminus J(-);\bbZ)~
\end{equation}
between functors from $\mathcal{K}_M$ to $\Ab$.
Since $f$ is a morphism in $\Loc_m$, hence in particular an isometry,
and $\cu$ is a natural transformation for relative differential characters,
see <cit.>,
$f^\ast$ maps relative semi-classical gauge fields on $M^\prime$ to relative
semi-classical gauge fields on $M$,
so that we obtain the natural transformation displayed in (<ref>).
We will now show that the restriction to $\mathcal{K}_\Sigma$
of the natural transformation (<ref>) is a natural isomorphism.
For each $K \subseteq \Sigma$ compact, we choose an open neighborhood $U \subseteq \Sigma$ of $K$
with compact closure $\overline U$. We denote by $j: U \to \Sigma$ the open embedding
induced by the inclusion. Observing that $f(\, \overline U\, ) \subseteq M^\prime$
is a spacelike and acausal compact submanifold with boundary of $M^\prime$,
by <cit.> there is a smooth spacelike Cauchy surface
$\Sigma^\prime$ of $M^\prime$ extending $f(\, \overline U\, )$
whose embedding in $M^\prime$ is denoted by $\iota_{\Sigma^\prime}: \Sigma^\prime \to M^\prime$.
We also denote by $f_U: U \to \Sigma^\prime$ the open embedding induced by the restriction of $f$.
By construction, the diagram
(U,U ∖K) [rr]^-f_U [d]_-j (Σ^',Σ^'∖f(K))
(Σ,Σ∖K) [d]_-ι_Σ
(M,M ∖J(K)) [rr]_-f (M^',M^'∖J(f(K)))
in the category $\Pair$ commutes.
Therefore, applying the functor $\dH^{k,m-k}(-;\bbZ): \Pair^\op \to \Ab$
and recalling that the pullback along $f$ maps relative semi-classical gauge fields to relative semi-classical gauge fields,
we obtain a new commutative diagram
^k(M^',M^'∖J(f(K));) [rr]^-f^∗ [dd]_-ι_Σ^'^∗
^k(M,M ∖J(K);) [d]^-ι_Σ^∗
^k,m-k(Σ,Σ∖K;) [d]^-j^∗
^k,m-k(Σ^',Σ^'∖f(K);) [rr] _f_U^∗
^k,m-k(U,U ∖K;)
in the category of Abelian groups $\Ab$.
Using Theorem <ref> and the excision theorem <cit.>
we find that the vertical and bottom horizontal arrows are isomorphisms.
In particular, note that $f_U$ can be factored as the composition of a diffeomorphism onto its image
followed by the inclusion of its image into the target, hence by excision
$f_U^\ast: \dH^k(\Sigma^\prime,\Sigma^\prime \setminus f(K);\bbZ) \to \dH^k(U,U \setminus K;\bbZ)$
is an isomorphism. It follows that the top horizontal arrow is an isomorphism too.
The proofs of our final two lemmas will rely extensively on Lemma <ref>.
In particular, we will adopt the following approach.
Starting from a semi-classical gauge field of spacelike compact support
and unraveling the directed colimit in (<ref>), we will represent it by a gauge field
relative to the complement of $J(K)$ for a suitable compact subset $K$ of a smooth spacelike Cauchy surface.
Then we use Lemma <ref> to represent the pushforward
of the given spacelike compact gauge field by the image under the inverse of the pullback
of the corresponding relative gauge field.
The pairing (<ref>) does not depend on the choice of Cauchy surface $\Sigma$.
Moreover, for any morphism $f: M \to M^\prime$ in $\Loc_m$ the diagram of Abelian groups
[d]_-𝕀×f_∗^k(M^';) ×^k_sc(M;) [rr]^-f^∗×𝕀
^k(M;) ×^k_sc(M;)[d]^-··
^k(M^';) ×^k_sc(M^';)[rr]_-··
We first prove independence of the pairing (<ref>) on the choice of Cauchy surface $\Sigma$
used to evaluate it. For this, we choose any two smooth spacelike Cauchy surfaces $\Sigma$
and $\Sigma^\prime$ of $M$. Let $(h,\tilde h) \in \Conf^k(M;\bbZ)$, $K \subseteq \Sigma$ compact
and $(h',\tilde h'\, ) \in \Conf^k(M,M \setminus J(K);\bbZ)$.
Note that $K^\prime = \Sigma^\prime \cap J(K)$ is compact and that
$J(K)\subseteq J(K^\prime\, )$.
Let $\mu$ denote the unique element of $\H_{m-1}(\Sigma,\Sigma \setminus K)$
which agrees with the orientation of $\Sigma$ for each point of $K$.
Similarly, let $\mu^\prime$ denote the unique element of
$\H_{m-1}(\Sigma^\prime,\Sigma^\prime \setminus K^\prime\, )$
which agrees with the orientation of $\Sigma^\prime$ at each point of $K^\prime$.
By means of the isomorphisms in Lemma <ref>, we can compare $\mu$
with $\tilde \mu = \pi_{\Sigma\,\ast} \, \iota_{\Sigma^\prime\,\ast} \mu^\prime \in \H_{m-1}(\Sigma,\Sigma \setminus K)$.
Since the orientations of both $\Sigma^\prime$ and $\Sigma$
are chosen consistently with the orientation and time-orientation of $M$,
for each point of $K$, $\tilde \mu$ agrees with the orientation of $\Sigma$.
In particular <cit.> entails that
$\tilde \mu = \mu \in \H_{m-1}(\Sigma,\Sigma \setminus K)$, therefore
$\iota_{\Sigma^\prime\,\ast} \mu^\prime = \iota_{\Sigma\,\ast} \mu \in \H_{m-1}(M,M \setminus J(K))$
by Lemma <ref>. Let $\nu \in Z_{m-1}(\Sigma,\Sigma \setminus K)$
and $\nu^\prime \in Z_{m-1}(\Sigma^\prime,\Sigma^\prime \setminus
K^\prime\, )$ be cycles
representing $\mu$ and $\mu^\prime$ respectively. Hence we obtain $\gamma \in C_m(M,M \setminus J(K))$
such that $\iota_{\Sigma \ast} \nu - \iota_{\Sigma^\prime \ast} \nu^\prime = \del \gamma$.
Taking into account also <cit.>, we get
\begin{multline}
\ip{\iota_{\Sigma }^\ast (h,\tilde h)\, }{\, \iota_{\Sigma }^\ast( h',\tilde h'\, )}_\Sigma
- \ip{\iota_{\Sigma' }^\ast( h,\tilde h)\, }{\, \iota_{\Sigma' }^\ast( h',\tilde h'\, )}_{\Sigma^\prime}
= \big(\, \tilde h \cdot h' - (-1)^{k\, (m-k)} \, h \cdot \tilde h'\,\big)(\del \gamma) \\[4pt]
= \int_\gamma \, \cu\big(\tilde h \cdot h' - (-1)^{k\, (m-k)}\, h \cdot \tilde h'\, \big) \mod \bbZ~,
\end{multline}
where the subscripts $_\Sigma$ and $_{\Sigma^\prime}$ denote the Cauchy surfaces
which have been used to evaluate the pairing (<ref>).
Furthermore, using <cit.>
together with the identities $\cu\, h = \ast \, \cu\, \tilde h$ and $\cu\, h' = \ast \, \cu\, \tilde h'$, one has
(h̃ ·h' - (-1)^k (m-k) h ·h̃' )
= h̃ ∧∗ h̃' - h̃'
∧∗ h̃ = 0 .
To complete the proof, we show that the diagram (<ref>) commutes.
Let $(h,\tilde h) \in \Conf^k(M^\prime;\bbZ)$ and $(h',\tilde h'\, ) \in \Conf^k_{\rm sc}(M;\bbZ)$.
Consider a smooth spacelike Cauchy surface $\Sigma$ of $M$ and $K \subseteq \Sigma$ compact
such that $(h',\tilde h'\, ) \in \Conf^k(M,M \setminus J(K);\bbZ)$.
Since $f$ is a morphism in the category $\Loc_m$, the induced map
$f: (M,M \setminus J(K)) \to (M^\prime,M^\prime \setminus J(f(K)))$ is a morphism in $\Pair$,
in particular an open embedding between the $m$-manifolds $M$ and $M^\prime$
which maps the open subset $M \setminus J(K)$ to $M' \setminus f(J(K))$. We
represent $f_\ast (h',\tilde h'\, ) \in \Conf^k_{\rm sc}(M^\prime;\bbZ)$
by $(f^\ast)^{-1} (h',\tilde h'\, ) \in \Conf^k(M^\prime,M^\prime \setminus J(f(K));\bbZ)$,
and interpret $(h',\tilde h'\, )$ as its representative in $\Conf^k(M,M \setminus J(K);\bbZ)$.
Recalling the proof of Lemma <ref>,
we consider an open neighborhood $U \subseteq \Sigma$ of $K$ with compact closure $\overline{U}$
and we extend $f(\, \overline U\, )$ to a smooth spacelike Cauchy surface $\Sigma^\prime$ of $M^\prime$.
With the same notation, we then find
(h,h̃) f_∗(h',h̃' )
= ι_Σ^'^∗(h,h̃)
ι_Σ^'^∗ (f^∗)^-1 (h',h̃' )_Σ'
= ι_Σ^'^∗(h,h̃) (f_U^∗)^-1
j^∗ ι_Σ^∗(h',h̃' )_Σ' .
This follows from the definition of the pairing (<ref>) and the diagram in (<ref>).
Note that $f_{U \ast}: \dH^{k,m-k}_{\rm c}(U;\bbZ) \to \dH^{k,m-k}_{\rm c}(\Sigma^\prime;\bbZ)$ is defined as the colimit
over the directed set $\mathcal K_U$ of compact subsets of $U$ of the inverse of the natural isomorphism
$f_U^\ast: \dH^{k,m-k}(\Sigma^\prime,\Sigma^\prime \setminus f_U(-)) \Rightarrow \dH^{k,m-k}(U,U \setminus -)$,
see <cit.>.
We further use commutativity of the diagram
^k(Σ^';) ×^m-k_c(U;)
[rr]^-f_U^∗×𝕀 [d]_-𝕀×f_U ∗
^k(U;) ×^m-k_c(U;) [d]^-··_c
^k(Σ^';) ×^m-k_c(Σ^';) [rr]_-··_c
which is shown in <cit.>.
It then follows that
(h,h̃) f_∗(h',h̃' )
= ι_Σ^'^∗(h,h̃) f_U ∗ j^∗ ι_Σ^∗(h',h̃' )_Σ'
= f_U^∗ ι_Σ^'^∗(h,h̃) j^∗ ι_Σ^∗(h',h̃' )_U .
From diagram (<ref>) we have $\iota_{\Sigma^\prime} \circ f_U = f \circ \iota_\Sigma \circ j$,
and by using the analogue of the diagram (<ref>) for the open embedding $j: U \to \Sigma$ we find
(h,h̃) f_∗(h',h̃' )
= j^∗ ι_Σ^∗ f^∗(h,h̃) j^∗ ι_Σ^∗(h',h̃' )_U
= ι_Σ^∗ f^∗(h,h̃) j_∗ j^∗ ι_Σ^∗(h',h̃' )_Σ .
Similarly to $f_{U \ast}$, the group homomorphism $j_\ast: \dH^{k,m-k}_{\rm c}(U;\bbZ) \to \dH^{k,m-k}_{\rm c}(\Sigma;\bbZ)$
is obtained as the colimit over $\mathcal K_U$ of the inverse of the natural isomorphism
$j^\ast: \dH^{k,m-k}(\Sigma,\Sigma \setminus -;\bbZ) \Rightarrow \dH^{k,m-k}(U,U \setminus -;\bbZ)$. It follows that
(h,h̃) f_∗(h',h̃' )
= ι_Σ^∗ f^∗(h,h̃) (j^∗)^-1 j^∗ ι_Σ^∗(h',h̃' )_Σ
=f^∗(h,h̃) (h',h̃' ) ,
where for the last equality we used the definition of the pairing (<ref>).
Recalling (<ref>),
the group homomorphisms $I : \Conf^k_{\rm sc}(M;\bbZ)\to \Conf^k(M;\bbZ)$, for objects $M$ in $\Loc_m$,
enjoy the following properties:
(i) $f^\ast\circ I \circ f_\ast = I$, for all morphisms $f : M\to M^\prime$ in $\Loc_m$.
(ii) $f^\ast_2\circ I \circ {f_{1}}_\ast =0$, for all diagrams $M_1\stackrel{f_1}{\longrightarrow} M
\stackrel{f_2}{\longleftarrow} M_2$ in $\Loc_m$ such that the images of $f_1$ and $f_2$ are causally disjoint.
Let us start with statement (i). For $(h',\tilde h'\, ) \in \Conf^k_{\rm sc}(M;\bbZ)$,
we show that $f^\ast\, I\, f_\ast (h',\tilde h'\, ) = I (h',\tilde h'\, )$.
For a fixed smooth spacelike Cauchy surface $\Sigma$, let $K \subseteq \Sigma$ be compact
with $(h',\tilde h'\, ) \in \Conf^k(M,M \setminus J(K);\bbZ)$.
By <cit.> one has
$f^\ast\circ I = I\circ f^\ast: \Conf^k(M^\prime,M^\prime \setminus J(f(K));\bbZ) \to \Conf^k(M;\bbZ)$, and so we find
f^∗ I (f^∗)^-1 (h',h̃' ) = I f^∗ (f^∗)^-1 (h',h̃' ) = I (h',h̃' ) .
This equation corresponds to $f^\ast\, I \, f_\ast (h',\tilde h'\, ) = I (h',\tilde h'\, )$ when $(h',\tilde h'\, )$
is regarded as an element of $\Conf^k_{\rm sc}(M;\bbZ)$.
For statement (ii), let $(h',\tilde h'\, ) \in \Conf^k_{\rm sc}(M_1;\bbZ)$.
Choosing a smooth spacelike Cauchy surface $\Sigma$ of $M_1$, we find a compact subset $K$ of $\Sigma$
such that $(h',\tilde h'\, ) \in \Conf^k(M_1,M_1 \setminus J(K);\bbZ)$.
As above, we represent $f_{1 \ast} (h',\tilde h'\, ) \in \Conf^k_{\rm sc}(M;\bbZ)$
by $(f_1^\ast)^{-1} (h',\tilde h'\, ) \in \Conf^k(M,M \setminus J(f_1(K));\bbZ)$.
For each pair of cycles $(z,\tilde z) \in Z_{k-1,m-k-1}(M_2)$,
the pushforward $f_{2 \ast} (z,\tilde z) \in Z_{k-1,m-k-1}(M)$ is supported inside $f_2(M_2)$.
By assumption $f_2(M_2) \subseteq M \setminus J(f_1(M_1)) \subseteq M \setminus J(f_1(K))$,
hence $f_{2 \ast} (z,\tilde z) = 0 $ in $ Z_{k-1,m-k-1}(M,M \setminus J(f_1(K)))$.
In particular, $I \, (f_1^\ast)^{-1} (h',\tilde h'\, )$ vanishes when evaluated on $f_{2 \ast} (z,\tilde z)$.
Since this is the case for any $(z,\tilde z) \in Z_{k-1,m-k-1}(M_2)$,
we conclude that $f_2^\ast\, I \, (f_1^\ast)^{-1} \, (h',\tilde h'\, ) = 0$ and
hence also $f_2^\ast\, I \, f_{1 \ast} (h',\tilde h'\, ) = 0$.
C. Bär,
“Green-hyperbolic operators on globally hyperbolic spacetimes,"
Commun. Math. Phys. 333 (2015) 1585
[arXiv:1310.0738 [math-ph]].
C. Bär and C. Becker,
“Differential characters,”
Lect. Notes Math. 2112, Springer (2014).
C. Bär and K. Fredenhagen (eds.),
“Quantum field theory on curved spacetimes,”
Lect. Notes Phys. 786 (2009) 1.
C. Bär, N. Ginoux and F. Pfäffle,
“Wave equations on Lorentzian manifolds and quantization,”
Zürich, Switzerland: Eur. Math. Soc. (2007)
[arXiv:0806.1036 [math.DG]].
C. Becker, M. Benini, A. Schenkel and R. J. Szabo,
“Cheeger-Simons differential characters with compact support and Pontryagin duality,”
arXiv:1511.00324 [math.DG].
C. Becker, A. Schenkel and R. J. Szabo,
“Differential cohomology and locally covariant quantum field theory,”
arXiv:1406.1514 [hep-th].
J. K. Beem, P. Ehrlich and K. Easley,
“Global Lorentzian geometry,”
CRC Press (1996).
D. Belov and G. W. Moore,
“Holographic action for the self-dual field,”
M. Benini, C. Dappiaggi, T.-P. Hack and A. Schenkel,
“A $C^*$-algebra for quantized principal $U(1)$-connections on globally hyperbolic Lorentzian manifolds,”
Commun. Math. Phys. 332 (2014) 477
[arXiv:1307.3052 [math-ph]].
M. Benini, C. Dappiaggi and A. Schenkel,
“Quantized Abelian principal connections on Lorentzian manifolds,”
Commun. Math. Phys. 330 (2014) 123
[arXiv:1303.2515 [math-ph]].
M. Benini, C. Dappiaggi and A. Schenkel,
“Quantum field theory on affine bundles,”
Ann. Henri Poincaré 15 (2014) 171
[arXiv:1210.3457 [math-ph]].
M. Benini, A. Schenkel and R. J. Szabo,
“Homotopy colimits and global observables in Abelian gauge theory,”
Lett. Math. Phys. 105 (2015) 1193
[arXiv:1503.08839 [math-ph]].
A. N. Bernal and M. Sanchez,
“Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes,”
Commun. Math. Phys. 257 (2005) 43
A. N. Bernal and M. Sanchez,
“Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions,”
Lett. Math. Phys. 77 (2006) 183
R. Brunetti, K. Fredenhagen and R. Verch,
“The generally covariant locality principle: A new paradigm for local quantum field theory,”
Commun. Math. Phys. 237 (2003) 31
J. Cheeger and J. Simons,
“Differential characters and geometric invariants,”
Lect. Notes Math. 1167 (1985) 50.
C. Dappiaggi and B. Lang,
“Quantization of Maxwell's equations on curved backgrounds and general local covariance,”
Lett. Math. Phys. 101 (2012) 265
[arXiv:1104.1374 [gr-qc]].
C. Dappiaggi and D. Siemssen,
“Hadamard States for the Vector Potential on Asymptotically Flat Spacetimes,”
Rev. Math. Phys. 25 (2013) 1350002
[arXiv:1106.5575 [gr-qc]].
C. Elliott,
“Abelian duality for generalised Maxwell theories,”
arXiv:1402.0890 [math.QA].
C. J. Fewster,
“Endomorphisms and automorphisms of locally covariant quantum field theories,”
Rev. Math. Phys. 25 (2013) 1350008
[arXiv:1201.3295 [math-ph]].
C. J. Fewster and B. Lang,
“Dynamical locality of the free Maxwell field,”
Ann. Henri Poincaré 17 (2016) 401
arXiv:1403.7083 [math-ph].
D. S. Freed,
“Dirac charge quantization and generalized differential cohomology,”
Surv. Diff. Geom. VII (2000) 129
D. S. Freed, G. W. Moore and G. Segal,
“The uncertainty of fluxes,”
Commun. Math. Phys. 271 (2007) 247
D. S. Freed, G. W. Moore and G. Segal,
“Heisenberg groups and noncommutative fluxes,”
Ann. Phys. 322 (2007) 236
A. Hatcher,
“Algebraic topology,”
Cambridge University Press (2002).
F. R. Harvey, H. B. Lawson, Jr. and J. Zweck,
“The de Rham-Federer theory of differential characters and character duality,"
Amer. J. Math. 125 (2003) 791
J. Manuceau, M. Sirugue, D. Testard and A. Verbeure,
“The smallest $C^\ast$-algebra for canonical commutation relations,”
Commun. Math. Phys. 32 (1973) 231.
B. O'Neill,
“Semi-Riemannian geometry with applications to relativity,”
Academic Press (1983).
K. Sanders, C. Dappiaggi and T. P. Hack,
“Electromagnetism, Local Covariance, the Aharonov-Bohm Effect and Gauss' Law,”
Commun. Math. Phys. 328 (2014) 625
[arXiv:1211.6420 [math-ph]].
J. Simons and D. Sullivan,
“Axiomatic characterization of ordinary differential cohomology,"
J. Topol. 1 (2008) 45
R. J. Szabo,
“Quantization of higher Abelian gauge theory in generalized differential cohomology,”
PoS ICMP 2012 (2012) 009
[arXiv:1209.2530 [hep-th]].
|
1511.00200
|
1,2]Galen D. Reed
Corresponding author email: [email protected]
1,2]Cornelius von Morze
3]Alan S. Verkman
1,2]Bertram L. Koelsch
1]Myriam M. Chaumeil
4]Michael Lustig
1,2]Sabrina M. Ronen
5]Jeff M. Sands
1,2]Peder E. Z. Larson
1]Zhen J. Wang
6,7]Jan Henrik Ardenkjær Larsen
1,2]John Kurhanewicz
1,2]Daniel B. Vigneron
[1]Department of Radiology and Biomedical Imaging, University of California San Francisco, San Francisco, California, USA
[2]Graduate Group in Bioengineering, University of California San Francisco, San Francisco, California, USA, and
University of California Berkeley, Berkeley, California, USA
[3]Departments of Medicine and Physiology, University of California San Francisco, San Francisco, California, USA
[4]Department of Electrical Engineering and Computer Sciences, University of California Berkeley, Berkeley, California, USA
[5]Department of Medicine, Renal Division, Emory University, Atlanta, Georgia, USA
[6]GE Healthcare, Brøndby, Denmark
[7]Department of Electrical Engineering, Technical University of Denmark, Kongens Lyngby, Denmark
In vivo spin spin relaxation time ($T_2$) heterogeneity of hyperpolarized 13C urea in the rat kidney was investigated. Selective quenching of the vascular hyperpolarized 13C signal with a macromolecular relaxation agent revealed that a long-$T_2$ component of the 13C urea signal originated from the renal extravascular space, thus allowing the vascular and renal filtrate contrast agent pools of the 13C urea to be distinguished via multi-exponential analysis. The $T_2$ response to induced diuresis and antidiuresis was performed with two imaging agents: hyperpolarized 13C urea and a control agent hyperpolarized bis-1,1-(hydroxymethyl)-1-13C-cyclopropane-$^2\textrm{H}_8$. Large $T_2$ increases in the inner-medullar and papilla were observed with the former agent and not the latter during antidiuresis suggesting that $T_2$ relaxometry may be used to monitor the inner-medullary urea transporter (UT)-A1 and UT-A3 mediated urea concentrating process. Two high resolution imaging techniques - multiple echo time averaging and ultra-long echo time sub-2 mm$^3$ resolution 3D imaging - were developed to exploit the particularly long relaxation times observed.
§ INTRODUCTION
Urea is primary end product of nitrogen metabolism in mammals, and humans typically produce more than 20 g of urea per day <cit.>. Concentrating urea to more than 100 times plasma levels for efficient excretion of this large osmotic load while minimizing water loss is one of the primary functions of the mammalian kidney <cit.>. Renal urea handling is a multistep process beginning with filtration of the blood at the glomerulus followed by a countercurrent multiplication in the medulla. The countercurrent exchange is assisted by urea transporters (UT) expressed in the descending thin limb of the loops of Henle (UT-A2 isoform), the erythrocytes (UT-B isoform), and the inner two thirds of the inner medullary collecting ducts (UT-A1 and UT-A3 isoforms) <cit.>. The inner medullary collecting duct (IMDC) transporters UT-A1 and UT-A3 increase the effective permeability of the tubular wall in the presence of vasopressin. This allows for urea to freely pass into the inner medullary interstitial fluid (IF) where it can accumulate to greater than 1 M concentration when water conservation is important.
Imaging renal solute handling could be a potentially valuable tool for the study of renal function. Several magnetic resonance imaging (MRI) studies have demonstrated non-invasive sodium detection with 23Na-detected MRI <cit.>. Indirect urea detection via changes in the 1H water resonance after radio frequency (RF) saturation of the urea amide frequency has been demonstrated <cit.>. Direct magnetic resonance imaging (MRI) of intravenously-injected 13C-labeled urea <cit.> and numerous other small molecules <cit.> has recently been enabled using dynamic nuclear polarization (DNP) <cit.>. In this process, the isotopically-enriched molecule is doped with an organic radical, cooled to liquid helium temperatures, and irradiated at microwave frequencies to achieve polarizations many orders of magnitude above thermal equilibrium values. After dissolution, the polarization decays exponentially with a time constant $T_1$ which is typically on the order of 10-90 s for 13C-labeled carbonyl sites <cit.>. This method has been utilized to generate background-free angiograms in preclinical models using highly biocompatible contrast agents <cit.>.
Recently, a method for in vivo imaging of hyperpolarized 13C urea with improved spatial resolution was presented which used a $T_2$-weighted steady state free precession (SSFP) sequence in combination with supplementary 15N labeling of the urea amide groups <cit.>. $T_2$ mapping experiments performed in Sprague Dawley rats gave uniform urea $T_2$ values of approximately 1 s throughout the animal with the exception of the kidneys, where values between 4 s and 15 s were noted <cit.>. Given the importance if urea in renal solute handling process as well as the paucity of literature discussing the physiological significance of in vivo 13C $T_2$ contrast, this current study presents measurements for further investigation of the 13C urea $T_2$ heterogeneity. In the first set of experiments, a chase infusion of an intravascular macromolecular relaxation agent was performed after hyperpolarized 13C urea injection but prior to $T_2$ measurement. This experiment was designed to selectively quench the vascular 13C urea signal while monitoring its effects on the 13C $T_2$ signal decay. A multi-exponential fitting algorithm tracked the changes in signal intensity of the short- and long-$T_2$ components both with and without the macromolecular relaxation agent. In a second set of experiments, hyperpolarized 13C $T_2$ mapping experiments were performed on rats on induced antidiuresis and osmotic diuresis with two imaging agents: hyperpolarized 13C urea and hyperpolarized bis-1,1-(hydroxymethyl)-1-13C-cyclopropane-$^2\textrm{H}_8$ (commonly abbreviated as HMCP or HP001). In the antidiuresis state, the kidneys are in water conservation mode, and the inner medullary transporters UT-A1 and UT-A3 aid to maximally concentrate urea in the inner medulla.
§ MATERIALS AND METHODS
§.§ Animal Handling
Animal studies were performed under a protocol approved by the University of California San Francisco Institutional Animal Care and Utilization Committee (IACUC). Sprague Dawley rats (mean mass 400 g) were anesthetized with a 1.7% isofluorane / oxygen mixture under a constant flow rate of 1 liter per minute. Animals were imaged in the supine position inside the birdcage coil and thermally insulated via heat pad. Contrast agents were injected via lateral tail vain catheters. Rats were housed three per cage at the UCSF Laboratory Animal Resource Center (LARC). Experiments were conducted between the hours of 5 p.m. and 1 a.m.
The hyperpolarized 13C $T_2$ mapping experiment. Hyperpolarized 13C-labeled substrates were injected via lateral tail vain catheter inside the MRI scanner. The $T_2$ mapping sequence acquired coronal projection images at 0.9 s echo time intervals while playing 180$^\circ$ pulses for 18 s. Each image was then corrected for respiratory motion via rigid translation in the superior / inferior direction. The dynamic images were then denoised using an SVD-based thresholding in the space / time dimensions. The signal component at each $T_2$ was estimated using a regularized version of the $T_2$ non-negative least squares method, and the long and short $T_2$ signal components were isolated by integrating the $T_2$ distribution.
§.§ Hardware
Imaging experiments were conducted in a GE 3T clinical MRI (GE Medical Systems, Waukesha, WI) equipped with a rat-sized dual-tuned 1H / 13C transceiver birdcage RF coil (8 cm inner diameter) placed on the patient table. An Oxford Instruments Hypersense polarizer (Oxford Instruments, Oxford, UK) was used for dissolution DNP experiments.
§.§ Sample Preparation
Isotopically enriched [13C,15N$_2$]urea and bis-1,1-(hydroxy- methyl)-1-13C-cyclopropane-$^2\textrm{H}_8$ were each doped with the trityl radical OX063 (Oxford Instruments, Abingdon, UK) and Dotarem (Guerbet, Roissy, France) as described previously <cit.>. Supplementary urea 15N labeling was necessary for the $T_2$ increase afforded by the elimination of the scalar coupling of the second kind relaxation pathway <cit.>. Bovine serum albumin (BSA) conjugated with an average of 23 gadolinium / diethylene triamine pentaacetic acid (GdDTPA) chelates per BSA molecule (abbreviated BSA-GdDTPA, molecular weight $\sim$ 85 kDa) was synthesized using methods previously described <cit.>.
§.§ 13C MRI Acquisition
*$T_2$ mapping
13C $T_2$ mapping was performed using sequences previously described <cit.>. Dynamic projection images were acquired in the coronal plane with 1 mm in-plane resolution, $14\times7$ cm $FOV$, 13 ms $TR$, and 70 phase encodes per image giving a temporal resolution of 910 ms and 18.2 s total acquisition time for all 20 echoes. Images were reconstructed with a simple 2D Fourier transform without spatial filtering.
§.§ Relaxometric Data Analysis
*Motion correction
Periodic respiratory motion caused a 1-2 mm offset which was largely resolved along the superior / inferior (SI) axis of the animal. This motion can be seen in the supplementary videos S1 and S2. To correct for this observed shift, a simple search algorithm was developed in which each image was aligned with its previous time point. Given that the motion was primarily 1D, a brute-force search was implemented which translated each image in 1 mm increments over $\pm 1$ cm from the initial location along the SI axis (for a total of 20 sampling points). At each position, the normalized mutual information (MI) was calculated between the floating image and the previous time point thus generating an MI versus translation curve as shown in Fig 1. The shift which maximized this curve was then applied.
*Subspace Denoising
Outside of the kidneys, the majority of the 13C signal disappeared after the first few echoes (Fig 1). This led to low rank data matrices in the spatiotemporal dimensions. A singular value decomposition (SVD)-based denoising <cit.> could therefore be utilized to better condition the ill-posed problem of multi-exponential estimation <cit.>. The 3-dimensional images ($x,y,t$) were concatenated along the $x$ and $y$ dimensions forming the matrix $I(r,t)$. Singular values less than the largest 7 (out of 20) were set to zero. This SVD-thresholded matrix was then used for the regularized $T_2$ estimation (Fig 1).
*Multi-Exponential Analysis
The $T_2$ non-negative least squares ($T_2$NNLS) algorithm <cit.> was used for quantitative analysis of the time-decay data. This method uses a least-squares inversion with non-negativity constraints to estimate $s(T_2)$, the signal's $T_2$ distribution at each pixel. $T_2$NNLS is appealing since it requires no a priori assumptions on the number of decay-modes. However, regularization, necessary for management of noise amplification, imposes assumptions of the shape of the distribution. $L_2$ norm regularization produces smoothly varying distributions, whereas $L_1$ norm regularization generates a sparse $T_2$ spectra <cit.>. We observed similar image appearance when using $L_1$- and $L_2$-regularized inversion of separation of short- and long-$T_2$ signal components, but the sparse $T_2$ spectra was easier to interpret visually (for example, see Fig. 3d). The inverse problem with $L_1$ constraints minimized
\begin{equation}
||As - y||^2 + \lambda ||s||_1,
\end{equation}
where the matrix $A$ has the elements
\begin{equation}
A_{i,j} = e^{-t_j/T_{2,i}}.
\end{equation}
$T_{2,i}$ is an array of 128 logarithmically-spaced $T_2$ values ranging from .3 to 20 s, $t_j$ are the echo times (20 regularly spaced from 0.5 to 19 s), and $y_j$ is the detected pixel $SNR$ values of the $j$th echo. The regularization parameter $\lambda$ was chosen to be 0.01% of $\lambda_{\textrm{max}}$, where $\lambda_{\textrm{max}}$ the maximum possible value for which the solution is non-zero: $\lambda_{\textrm{max}} = 2\textrm{ max} \left(A^T y\right)$ <cit.>.
From the distribution $s(T_2)$, the signal $s_{\alpha,\beta}$ in the range $T_2\in [\alpha,\beta]$ was calculated by integrating the $s(T_2)$ from $\alpha$ to $\beta$. A mean $\langle T_2\rangle$ measure in the range $T_2\in [\alpha,\beta]$ was estimated by the first moment of the distribution:
\begin{equation}\label{eq_meant2}
\langle T_2\rangle = \exp\left\{\frac{\int_\alpha^\beta\! s(T_2)\log T_2\,dT_2}{\int_\alpha^\beta\! s(T_2)\,dT_2}\right\}.
\end{equation}
The integration limits $\alpha$ and $\beta$ were used to isolate short- or long-$T_2$ species.
§.§ 13C Urea / BSA-GdDTPA Relaxometry
$^{13}$C Imaging Experiments
These experiments were designed to selectively quench the vascular 13C urea polarization while measuring the differential attenuation of the short and long $T_2$ signal components. Analogous to previously reported experiments <cit.>, a gadolinium (Gd) carrier molecule was infused after the hyperpolarized 13C injection but prior to 13C detection. The paramagnetic Gd greatly reduces the $T_1$ of the 13C molecule thus causing rapid 13C polarization loss in regions where the 13C-labeled molecule and the Gd carrier are in close proximity. This polarization loss manifests as lowered signal during 13C imaging. This study used bovine serum albumin as the carrier which was conjugated with and average of 23 Gd / diethylene triamine pentaacetic acid chelates (abbreviated BSA-GdDTPA). This a well-characterized blood pool agent has a high molecular weight (85 kDa) which prohibits extravasation and glomerular filtration on the sub minute timescale <cit.>.
To estimate the efficacy of hyperpolarized 13C signal quenching, the 13C $T_1$ relaxivity of [13C,15N$_2$]urea with respect to BSA-GdDTPA was measured. BSA-GdDTPA was titrated in approximately 0.2 mM increments into a into a 1 mL vial containing 1 M [13C,15N$_2$]urea. At each titration point, the 13C urea $T_1$ was measured via saturation recovery experiments at $B_0=3$ T, $T=27^\circ$C. A linear fit was performed on the [BSA-GdDTPA] versus $1/T_1$ data points. Using this measured curve in conjunction with model-based estimates of the rats' blood volumes, the in vivo $T_1$ of 13C urea within the blood pool was roughly estimated.
In this study, 4 rats were imaged after 2 separate 13C urea injections: one with the BSA-GdDTPA chaser and one without to act as the control experiment. Due to the persistence of the BSA-GdDTPA in the blood pool, the chaser experiment was always performed after the control. Fig 3a shows a schematic of the experimental timeline. Rats were injected with 3 mL, $150$ mM hyperpolarized [13C,15N$_2$]urea solution over 12 s. The hyperpolarized urea was then allowed to diffuse for 28 s, and the 13C $T_2$ mapping sequence was initiated 40 s after the beginning of injection. 2 hours later, a second [13C,15N$_2$]urea infusion was performed over 12 s. The hyperpolarized urea was then allowed to diffuse for 20 s, then 1 mL, $0.59$ mM BSA-GdDTPA (15 mM GdDTPA) was injected over 1 s. The chaser was allowed to diffuse for 7 s, and then the 13C $T_2$ mapping sequence was initiated again 40 s after the beginning of the urea injection.
Statistical Analysis
Image noise was estimated by the standard deviation of a signal-free region of the 13C images (prior to image denoising), and all image pixels were normalized by this measurement. All pixels of the 13C urea images with first-time-point-SNR greater than 10 were included in analysis. Multi exponential calculations were performed on each pixel after image alignment and denoising. The short- and long-$T_2$ signal components were isolated from the $T_2$ distributions calculated at each pixel using the integration limits ($\alpha = 0.3$ s, $\beta = 2.5$ s) for the short-$T_2$ component and ($\alpha =2.5$ s, $\beta= 20$ s) for the long-$T_2$ component. In this way, 4 maps were calculated for each animal: short- and long-$T_2$ signal maps for each 13C urea and 13C urea + BSA-GdDTPA experiment. In each animal, the mean pixel signal of each map was computed, and two paired t-tests were performed: one comparing the mean short-$T_2$ SNR of the 13C urea images with and without BSA-GdDTPA chaser, and the other comparing the mean long-$T_2$ SNR of the 13C urea images with and without BSA-GdDTPA chaser. $p<0.05$ was used as the significance criteria. Additionally, all pixels from the short-$T_2$ and long-$T_2$ maps from all 4 animals were binned by signal intensity and plotted on top of each other as a aid for visualization of the BSA-GdDTPA chaser effects.
$^1$H MRI
For a qualitative visualization of the Gd carrier distribution, 1H-detected, $T_1$-weighted images were acquired 5 minutes post infusion of BSA-GdDTPA. This was followed by an additional injection of 1 mL, $0.5$ mM low molecular weight (940 Da) Gd-DTPA without attached albumin (Magnevist, Bayer Schering, Berlin). Both Gd agents were imaged the same 1H spoiled gradient echo (SPGR) sequence (flip angle = 35, $TE/TR=1.4/7$ ms, 3 averages, .8 mm isotropic resolution).
§.§ Diuresis / Antidiuresis Relaxometry
$^{13}$C Imaging Experiments
These experiments were designed to measure the $T_2$ of two agents - hyperpolarized 13C urea and hyperpolarized 13C HMCP - in response to induced antidiuresis and osmotic diuresis in rats. In the antidiuresis state, the kidney is in maximally concentrating mode, and high levels of circulating vasopressin activate the inner medullary transporters UT-A1 and UT-A3 in order to maximally concentrate urea while minimizing water loss. 3 rats were imaged with both agents; HMCP served as a control since this molecule is not expected to be affected by urea transporters. The methods for inducing diuresis and antidiuresis were identical to those described previously <cit.>. To induce antidiuresis, the rats were deprived of food and water for an overnight period of $16$ hours. To induce osmotic diuresis, the rats were first deprived of food and water for $16$ hours and then allowed free access to aqueous glucose (10% by mass) solution for 9 hours. In each experiment, the urea injection (3 mL, $150$ mM hyperpolarized [13C,15N$_2$]urea) was performed at least 2 hours prior to the HMCP injection (3 mL, $125$ mM hyperpolarized HMCP). The $T_2$ mapping sequence was initiated 40 s after the beginning of injection with identical acquisition parameters aside from a 4.5 kHz resonance frequency offset between urea and HMCP.
Statistical Analysis
Both kidneys were manually delineated on all 13C images after alignment and denoising, and only pixels within the kidney were included in analysis. Following pixel selection, a $T_2$ distribution was computed for each pixel. A single mean $\langle T_2 \rangle$ value was calculated at each pixel from this distribution by calculating the first moment over the full distribution ($\alpha = 0.3$ s, $\beta = 20$ s). The use of the full distribution was motivated by the difficulty in comparing two molecules with differing relaxation properties. 4 maps were computed for each of the 3 animals: $\langle T_2 \rangle$ maps for urea and HMCP in diuresis and antidiuresis states. Histograms including all pixels binned by $\langle T_2 \rangle$ were plotted for each agent to visualize the effect of diuresis and antidiuresis states. For quantitative comparison, the mean of the upper-90th percentile was computed for each animal. Due to the inward $\langle T_2 \rangle$ gradient observed with both agents, this operation selected pixels within the center of the kidney. Two paired t-tests were then performed: the first compared the mean upper-90th percentile of $\langle T_2 \rangle$ of urea between diuresis and antidiuresis, and the second compared the mean upper-90th percentile of $\langle T_2 \rangle$ of HMCP between diuresis and antidiuresis. $p<0.05$ was used as the significance criteria.
$^1$H MRI
Animals were imaged in the coronal and axial planes using a $T_2$-weighted $^1$H fast spin echo (FSE) sequence with $TR=1$s, $TE=$100 ms, 32 echoes, 0.8 mm in-plane resolution.
§.§ High Resolution 13C Imaging Techniques
*Multiple Echo Time Averaging
The persistence of signal across multiple echo times allowed for image averaging to increase the effective $SNR$. This summation was performed after motion correction and $SVD$ denoising.
*3D Imaging
As an alternative multi-exponential fitting, the MRI acquisition may be designed with a long echo time to filter out the short-$T_2$ signal components. One method of achieving this is using 2 spatial dimensions of phase encoding (for the acquisition of a full 3D image) with a rasterized Cartesian ordering in which the echo time is half the total acquisition time. A 3D SSFP acquisition was initiated at 20 s, 25 s, and 30 s after the beginning of urea injection. This sequence used a large flip angle ($\theta=120^\circ$) and extremely long echo time (3.5 s) allowing for editing out the vascular signal with enhancement of the long-$T_2$ filtrate. The un-aliased $FOV$ was chosen to cover only the kidneys in the 2 phase encoded dimensions. Since the images were sampled below the Nyquist cutoff frequency in these dimensions, the signal filtering of the ultra long echo time acquisition was utilized to reduce aliasing from the vascular signal. A $(42,42,14)$ acquisition matrix was acquired over a $(5,5,1.7)$ cm $FOV$ in (L-R,S-I,A-P) coordinates yielding a 1.2 mm isotropic pixel length. 588 phase encodes were acquired with $TR=12$ ms for a 7 s total scan time.
§ RESULTS
§.§ 13C Urea / BSA-GdDTPA Relaxometry
The $T_1$ relaxivity of the BSA-GdDTPA complex on 13C urea was estimated to be $77\pm10$ mM$^{-1}$s$^{-1}$ with respect to the BSA carrier, or $3.1\pm.4$ mM$^{-1}$s$^{-1}$ per GdDTPA chelate ($R^2=.97$) from saturation recovery experiments. Fig. 3e shows the measured $T_1$ relaxivity curve with error bars indicating $T_1$ measurement uncertainty from the intrinsic signal to noise ratio. The estimated rat blood volume was $27\pm3$ mL <cit.>. Therefore, at the expected in vivo [BSA-GdDTPA] of $\sim0.024$ mM, the urea $T_1$ should be approximately 0.5 s. Although this calculation is extremely rough since it ignores circulation or potentially differing relaxivity in vivo and in vitro, the 7 s delay should have been adequate for any hyperpolarized 13C urea to undergo several $T_1$ time constants of decay leading to large polarization loss when in contact with BSA-GdDTPA.
Differential attenuation of the 13C urea signal after a chase injection of the intravascular agent BSA-GdDTPA. Large field-of-view images are shown on the top, and the bottom panels are zoomed to the kidney. a) First time-point 13C urea MRI image. b) 13C urea MRI acquired 8 s after BSA-GdDTPA infusion shows strong suppression of the vascular signal and interlobular arteries (red arrow). c) 1H MRI acquired 5 minutes after BSA-GdDTPA infusion. The interlobular arteries show positive contrast in this image (green arrow). d) 1H MRI acquired 5 minutes after GdDTPA infusion. In contrast to BSA-GdDTPA (mass $\sim85$ kDa), GdDTA (mass $\sim938$ Da) is freely filtered at the glomerulus (yellow arrow).
Figure 2 shows a first echo 13C urea image with (Fig. 2a) and without (Fig. 2b) the chaser injection of the macromolecular BSA-GdDTPA relaxation agent. This image shows a large suppression of the vascular 13C urea signal throughout as well as darkening of the interlobular branches of the renal artery (Fig 2b). The darkening of the renal arterial branches indicates that the 7 s delay was adequate not only for both bolus arrival of the BSA-GdDTPA to the kidneys, but also for the BSA-GdDTPA to cause substantial 13C urea polarization loss. These same arterial branches show up bright in a $T_1$-weighted $^1$H image (Fig 2c). No renal perfusion was detected with the BSA-GdDTPA agent up to 5 minutes after initial infusion. In contrast, the post-GdDTPA image (Fig 2d) shows renal perfusion and bladder accumulation since this agent is freely filtered at the glomerulus <cit.>.
13C urea $T_2$ relaxometry after quenching the vascular signal.
a) Timeline of the substrate injections and imaging. The left column shows images from the control experiment, and the center column shows the post BSA-GdDTPA images. b) 13C urea signal outside of the kidneys has $T_2$ less than 2.5 s which was strongly attenuated by BSA-GdDTPA. c) The long-$T_2$ urea signal component was confined to the kidneys and was unaffected by the BSA-GdDTPA chaser. d) $T_2$ distributions of single pixels selected from the center of the kidneys showing this short-$T_2$ signal attenuation (red arrow). e) The 13C urea relaxation rate $1/T_1$ decreases linearly with BSA-GdDTPA concentration with a slope of $77\pm10$ mM$^{-1}$s$^{-1}$. f) $T_1$ weighted 1H imaging 5 minutes post BSA-GdDTPA infusion. g) $T_1$ weighted 1H imaging post GdDTPA infusion. h) Single pixel $T_2$ decay curves (corresponding to the distributions in d).
The dynamic 13C urea images acquired under $T_2$ decay conditions initially showed greatly reduced signal at early echo times when accompanied by the BSA-GdDTPA chaser. At later echo times, however, the images converge and look nearly identical. This effect is most easily visualized in the supplemental video S1: by the 7th time point (corresponding to a 6.8 s echo time) only urea within the kidneys is visible in both images, and the signal's spatial variation is nearly identical in the images with and without the BSA-GdDTPA chaser. The similarity persisted in all experiments until the end of imaging acquisition. At this echo time, both images show urea signal throughout the cortex and medulla, and a dark-rim is present at the outer stripe of the outer medulla. The later echo images were acquired up to 25 s after the infusion of the BSA-GdDTPA chaser and provide strong evidence that the slowly decaying 13C urea signal component emanated from regions inaccessible to the BSA-GdDTPA.
13C urea $T_2$ relaxometry after quenching the vascular signal.
Pixel distributions of the short-$T_2$ (a) and long-$T_2$ (b) of the 13C urea signal with and without BSA-GdDTPA chaser. All pixels with first time point SNR $>10$ from 4 animals are displayed in these histograms. c) shows the range of the mean of these signal components for the 4 animals scanned. A paired t-test show significant ($p<0.05$) attenuation of the short-$T_2$ by the BSA-GdDTPA chaser (shown in the left 2 boxes).
Multi exponential relaxometry quantified this dissimilarity at early echoes and similarity at late echoes. Fig 3b and 3c show the short-$T_2$ and long-$T_2$ signal components, respectively, with the latter appearing identical in the 13C urea images with and without BSA-GdDTPA. Fig 3h shows a single decay curve from a pixel selected from the center of the kidney with semi-log axes. The two decay modes show up as peaks in the $T_2$ distributions in Fig 3d, with the short-$T_2$ component being significantly reduced by the BSA-GdDTPA chaser. Binning the all the pixels from all animals scanned also clearly showed this differential attenuation of the short- and long-$T_2$ signal components (Fig. 4a, 4b). The mean of the short- and long-$T_2$ components for the 4 animals is shown in the box plot in Fig. 4c. The whiskers represent the minimum and maximum of the mean of the short- and long-$T_2$ maps. The paired t-test showed significant ($p<.05$) diminishing of the SNR of the short-$T_2$ 13C urea signal component over all animals.
§.§ Diuresis / Antidiuresis Relaxometry
Fig. 5 shows the long-$T_2$, extravascular 13C urea. During osmotic diuresis, the urea remains largely in the outer medulla and cortex at the imaging start time used (40 s after the beginning of a 12 s injection). During antidiuresus, a larger fraction of the urea is collected in the inner medulla and papilla consistent with the enhanced inner-medullary urea pool during antidiuresis indicative if the UT-A1 and UT-A3 transporters (Fig 5, left). When the acquisition delay allowed for inner-medullary urea accumulation, large $T_2$ increases were observed due to a strong inward $T_2$ gradient were observed with the long-$T_2$ component of 13C urea (for example, see Fig 10 for a $T_2$ distribution of the extravascular 13C urea). Given this $T_2$ gradient, relaxometry is a sensitive detection method for observing urea transporter facilitated concentrating ability. Supplementary video S2 shows the dynamic 13C urea at multiple echo times. During antidiuresis, not only is a greater inner medullary accumulation of the 13C urea observed, but the signal persists to very late echo times.
Extravascular 13C urea during antidiuresis and diuresis.
A larger fraction of the urea is collected in the inner medulla and papilla consistent with the enhanced inner-medullary urea pool during antidiuresis indicative if the UT-A1 and UT-A3 transporters (left). During diuresis, the urea is primarily in the cortex and outer medulla (right) at the time of imaging.
Hyperpolarized 13C urea and HMCP relaxometry during antidiuresis and diuresis.
Hyperpolarized 13C urea and HMCP images are shown the left and right columns, respectively. HMCP showed increased medullary $T_2$ but the effect did not vary between diuresis (b) and antidiuresis (a) states. c) decay curves selected from a single pixel in the inner medulla. The red arrow shows the persistence of signal to late echo times.
Fig. 6 shows $T_2$ exponential relaxometry performed with both imaging agents in rats on induced diuresis and antidiuresis. Rather than selecting the short- or long-$T_2$ components, the mean $\langle T_2\rangle$ (calculated by equation <ref> with $\alpha = 0.3$ s, $\beta = 20$ s) was used to simplify analysis. A large $\langle T_2\rangle$ increase was observed in the inner medulla and renal papilla with 13C urea during antidiuresis (Fig. 6a). This $\langle T_2\rangle$ increase in the kidneys' central region was not observed in diuresis (Fig. 6b). The signal distribution and $\langle T_2\rangle$ of HMCP did not change significantly between antidiuresis and diuresis (Fig. 6a,b, right). This effect can also be seen in the pixel histogram $\langle T_2\rangle$ distributions (selected from pixels within the kidney) in Figure 7.
Both agents showed $\langle T_2\rangle$ lengthening in the inner medulla and papilla. Unlike 13C urea, HMCP showed a small $\langle T_2\rangle$ reduction in the cortex and outer medulla compared to the blood. HMCP had a much higher $\langle T_2\rangle$ in the vascular pool (4 s) compared to urea (1 s). This effect is most easily visualized in the outer margins of the images in Fig. 6a,b (right). With 13C urea, $\langle T_2\rangle$ values greater than 2 s were only observed within the kidney.
Pixel $\langle T_2\rangle$ distributions of 13C urea and HMCP relaxometry during diuresis and antidiuresis.
a) 13C urea and b) HMCP pixels from the kidneys of 3 animals. The distribution extends to high $\langle T_2\rangle$ values for 13C urea in antidiuresis indicating 13C urea concentration in the inner medullary collecting ducts and medullary interstitium. This effect was not observed with HMCP. c) the mean upper 90th percentile plotted for all animals.
§.§ High Resolution 13C Imaging Techniques
After image alignment, denoising, and echo time averaging, signal in the ureters could be observed in osmotic diuresis conditions with both 13C urea and HMCP (see Fig 8). The inner lumen of the rat ureter has a diameter between 50 and 150 $\mu$m <cit.>, so a 1 mm width pixel (acquired perpendicular plane to the ureter axis) contains on the order of 10 nL intra-ureter fluid. Although the signal was extremely faint, the persistence of signal allowed for detectable ureter structure. HMCP had a much stronger signal than 13C urea in the ureters likely due to its longer $T_1$. Ureters were only visible in 2 of the 3 diuresis scans for HMCP and 1 of 3 scans for 13C urea. The long $T_1$ of HMCP enabled urinary bladder collection to be imaged at over 3 minutes post injection (image not shown).
13C ureter imaging during diuresis.
10 echo times (from 5 s to 14 s) were averaged after alignment and denoising to improve signal. Ureters could be observed in diuresis with both urea and HMCP (arrows). High urea reabsorption and reduced UT-A1 and UT-A3 activity leads to substantial cortical and outer medullary signal compared to HMCP.
Fig. 9 shows 3D images acquired at 1.2 mm isotropic resolution (1.73 mm$^3$ pixel volume). These images are zoomed to a single kidney. The efficacy of the blood pool suppression is evidenced by the low background signal as well as the dark interlobular arteries (magenta arrows on the image panels). The outer stripe of the outer medulla (OSOM) enhanced later than the cortex and inner stripe of the outer medulla (ISOM, see the 20 s and 25 s time points). Once inner medullary accumulation occurred, the inner medulla (IM) and papilla were bright due to the sequence weighting (see simulations in Fig 9a, right). These images highlight the sensitivity in rats of imaging timing for desired contrast agent distribution. These images also represent, to the authors' best knowledge, the first in vivo hyperpolarized 13C images acquired at sub-2 mm$^3$ isotropic resolution.
13C urea imaging at sub-2 mm$^3$ isotropic resolution
a) Sequence design (left) and signal simulations (right 3 panels) of a 1.7 mm$^3$ isotropic resolution image. Simulations show the signal response expected from regions with $T_2=1.3$ s (vascular pool), $T_2=4$ s (cortex / outer medulla), and $T_2=10$ s (inner medulla / papilla). Blue arrows indicate sequence parameters used, and thus the signal response expected in the images. b) The long effective echo time (4 s) of a 3D acquisition was utilized for blood pool signal suppression and for encoding the 13C urea at 1.2 mm isotropic resolution. b) 13C urea images acquired at three different delay times after three different injections. Blood pool suppression is evidenced by the dark interlobular arteries visible on both the 13C urea and 1H fast spin echo images (magenta arrows).
§ DISCUSSION
Comparison of 1H and 13C MRI of the Kidney
Hyperpolarized 13C MRI is a background-free imaging method that allows direct detection the labeled contrast agent in the kidney, whereas Gd-enhanced 1H MRI utilizes the enhanced relaxation rate of water for indirect detection. The latter method, therefore, always has signal contribution from regions without contrast agent, but this contribution is typically minimized with strong $T_1$ weighting of the acquisition. The experiments in this study highlight that relaxation is also an extremely important factor for the analysis of hyperpolarized 13C images. Regions of hypo-intense signal may stem not only from low contrast agent concentration, but may also arise from $T_1$ or $T_2$ shortening, and this effective contrast can be exacerbated by the acquisition.
In both hyperpolarized 13C and Gd-enhanced 1H MRI, the molecular label carrier influences the renal contrast agent distribution. Since GdDTPA is not significantly reabsorbed along the nephron <cit.>, the signal likely arises primarily from the lumen of the tubules and collecting ducts as well as the glomerular blood supply. A significant fraction of urea in the filtrate, however, is reabsorbed along the nephron. Fig. 8 shows qualitatively the high reabsorption of urea compared to HMCP in rats on induced diuresis. Urea (mass $\sim63$ Da) and HMCP (mass $\sim110$ Da) are both freely filtered at the glomerulus, but the late echo time urea image (Fig 8, bottom left) still shows substantial cortical signal. It is worth noting that other commonly used endogenous molecular probes for hyperpolarized 13C MRI such as pyruvate and lactate are also freely filtered but are nearly completely reabsorbed under normal conditions <cit.>, so significant collection in the inner medulla and renal pelvis would not be expected. Urea is an endogenous metabolic waste product, and HMCP is a non endogenous molecule, so the kidney attempts to excrete both of these agents.
$T_2$ of extravascular 13C urea
$T_2$-weighted 1H localizer is shown on the left. The right image is a $T_2$ map isolated to the renal filtrate ($T_2>2.5$ s). The cortex (C) and inner stripe of the outer medulla (ISOM) gave lower $T_2$ values than the outer stripe of the outer medulla (OSOM) and inner medulla (IM). These $T_2$ variations are potentially indicative of slow 13C urea exchange between the medullary interstitial fluid and the vasa recta.
The spatial variation of 13C $\langle T_2 \rangle$ is compared with a $T_2$-weighted spin echo 1H MRI in Fig. 10. In this figure, the mean $\langle T_2 \rangle$ was calculated to represent that of the extravascular urea pool ($T_2>2.5$ s). The 1H $T_2$ correlates well with water content <cit.> and increases with edema <cit.>, while 13C urea undergoes a strong relaxation enhancement from erythrocytes <cit.>. Unlike contrast-enhanced imaging with GdDTPA which is not significantly reabsorbed, the extravascular urea signal likely arises from the tubular lumen as well as the interstitial fluid (IF) due to high permeability along the proximal tubule, the thin limbs of the loop of Henle, and the inner-medullary collecting ducts <cit.>. The relative contribution of tubular and interstitial pools on the short timescale of the 13C imaging experiment is unknown, but the spatial $\langle T_2 \rangle$ variation (Fig. 10, right) could suggest a large contribution from the IF. The cortex (C) and inner stripe of the inner medulla (ISOM) showed $\langle T_2 \rangle$ values less than the outer stripe of the outer medulla (OSOM) and inner medulla (IM). A potential explanation is the urea exchange between the medullary interstitium and the peritubular microvascular supply. Urea within the IF is in exchange with the vasa recta, a process which is critical for efficiency of the countercurrent exchange mechanism. The density of the peritubular vascular network differs within the various renal segments. The peritubular vascular tissue fraction in the cortex and ISOM is nearly double that of the OSOM and IM <cit.>. The medullary microvascular blood supply is sparse, receives less than 1% of the total renal blood flow <cit.>, and occupies less than 0.4% of the total tissue volume <cit.>. However, the vasa recta are highly permeable to urea, and the apparent $\langle T_2 \rangle$ lengthening of the 13C urea isolated to the filtrate in the sparse miscrovascular regions (OSOM, IM) and shortening in the dense microvascular regions (C, ISOM) potentially indicate that the $T_2$ variations observed indicate a slow exchange of urea between the interstitium and the microvascular supply.
Comparison with prior hyperpolarized 13C NMR measurements
The 13C urea $T_1$ relaxivity of the macromolecular relaxation BSA-GdDTPA measured here ($3.1\pm.4$ mM$^{-1}$s$^{-1}$ per GdDTPA, $B_0 = 3$ T) is more than an order of magnitude higher than that reported with Gadodiamide and [1-13C]pyruvate ($0.19\pm.01$ mM$^{-1}$s$^{-1}$, $B_0 = 4.7$ T) <cit.>. This is likely attributable to the increased relaxivity from the slow correlation time of the macromolecule Gd carrier <cit.> but may also signify some preferential binding of urea to albumin.
Based on measurements in this and prior studies, 13C urea (with 15N labeling) experiences strong relaxation enhancement in plasma ($T_2 = 11$ s) and whole blood ($T_2 = 5$ s) compared to aqueous solution ($T_2 = 24$ s) <cit.>. Although the strongest in vitro relaxation enhancement was observed in whole blood, this value is more than double that measured in vivo ($T_2 = 1.2$ s) potentially indicating a large decrease in the apparent $T_2$ due to flow. Relaxation enhancement in erythrocytes is likely due to a combination of high blood viscosity and high erythrocyte membrane permeability <cit.>; the latter is mediated by the erythrocyte isoform of the urea transporter UT-B. A 20% reduction in the diffusion coefficient of urea has been reported in UT-B expressed tissue xenografts compared to controls likely indicating an increased intracellular pool size <cit.>. Controlled experiments in erythrocytes reported a changes in the resonant frequency <cit.> and reduction of the longitudinal $T_1$ relaxation time <cit.> of the intracellular 13C urea. These results are concordant with a shortening of the vascular $T_2$ due to erythrocyte permeability.
The vascular $T_2$ of HMCP measured here ($T_2=4$ s at $B_0=3$ T) is in agreement with a "worst case" value ($T_2=1.3$ s at $B_0=2.35$ T) reported previously using SSFP <cit.> but is over an order of magnitude higher than in vivo measurements ($T_2=0.4$ s at $B_0=9.4$ T) reported using adiabatic refocusing pulses with surface coils <cit.> suggesting further study in the dependence of apparent $T_2$ measurements on $B_0$, contrast agent uptake in tissue, and acquisition type.
Experimental Limitations
Multicomponent $T_2$ relaxometry has well-known difficulties in accurately resolving closely-spaced $T_2$ values at the SNR, scan times, and echo spacing permissible on clinical MRI scanners <cit.>. In this study, the SNR limitation was somewhat exacerbated by the polarization loss from $T_1$ decay during the long delay periods between injection and imaging necessary for renal contrast agent accumulation. The observed $T_2$ decay times were significantly longer than those typical of in vivo $^1$H MRI which permitted the coarse temporal sampling required by the single-shot, 1 mm resolution planar readout. Furthermore, $T_2$ differences between intra- and extravascular 13C urea differed by a factor of at least 4. Decreasing the resolution would allow finer temporal sampling and increased SNR for the stabilization of $T_2$ distribution estimation <cit.>. $T_2$ values less than the sampling time of 910 ms are not expected to be resolved accurately. In non-selective in vivo CPMG experiments performed after infusion of hyperpolarized 13C urea with finer temporal sampling (10 ms), we observed approximately 30% of the total signal had a $T_2$ of 300 ms or less.
In each experiment, the rat was given two 3 mL injections of hyperpolarized contrast agents spaced at least 2 hours apart. For the macromolecular relaxation experiments, the rat was given 2 additional 1 mL injections of Gd contrast. The total injected volume equals approximately 1/4 the animals' total blood volume and had an unknown affect on cardiac output, glomerular filtration rate (GFR), and renal concentrating capacity. Additionally, the urea mixture contained glycerol which, in large doses, is known to cause GFR reduction in rats <cit.>. Although it is difficult to conclude that this large aggregate injection volume had no transient effect on renal function, we did not observe imaging evidence of differing renal concentrating capacity with repeat injections when the imaging was started at a constant duration after the injection.
Polarization variability will lead to random errors in comparing absolute SNR between experiments. Although prior measurements showed less than a 15% variability in 13C urea polarizations
when polarization and transport time are kept consistent <cit.>, this potential random error will almost certainly be minimized with the use of automated transport injectors <cit.> and magnetically shielded transport pathways <cit.>.
Systematic errors may arise in quantitative $T_2$ mapping due to transmitter strength mis-calibrations when using the transient phase of the SSFP signal. As derived by Scheffler, the exponential decay envelope of the signal is described by the positive eigenvalue <cit.>
\begin{equation}
\lambda_1 = \frac{1}{2}\left( \left(E_1-E_2\right)\cos\theta +\sqrt{4E_1E_2 + \left(E_1-E_2\right)^2\cos^2\theta} \right),
\end{equation}
with $E_1 = e^{-TR/T1}$, $E_2 = e^{TR/T_2}$. The transmitter offset may be modeled as $\theta = 180^\circ + \delta \theta$, and a Taylor expansion of $\lambda_1$ gives
\begin{equation}
\lambda_1\approx a_0 + a_1 \delta\theta + a_2 \delta\theta^2 + ...
\end{equation}
\begin{align}
a_0 &= E_2 \\
a_1 &= 0 \\
a_2 &= \frac{E_2\left(E_1-E_2\right)}{E_1+E_2}
\end{align}
Non-ideal $\pi$ pulses will cause some apparent lengthening of the measured $T_2$ by introducing some $T_1$ weighting. However, these errors show up only as second or higher even order terms of $\delta \theta$. The second order term is minimized when $E_1\approx E_2$, and this condition is expected to be better approximated in the longer $T_2$ regions. Assuming $T_2 =1.5$ s, $T_1 = 20$ s, flip angle errors $\delta\theta/\theta$ up to $20\%$ cause the apparent decay time to differ from $T_2$ by less than $10\%$.
Potential Clinical Utility
Although $T_2$-weighted imaging has been a standard clinical MRI evaluation for three decades <cit.>, only a few studies have investigated $T_2$ contrast for in vivo hyperpolarized 13C agents imaging <cit.>. In addition to chemical shift <cit.> and diffusion <cit.> sensitive imaging techniques, $T_2$ relaxometry could be a very useful tool for probing the microenvironment of hyperpolarized 13C molecules in vivo since it yields high signal thus allowing for high resolution encoding.
Given the importance of urea in the urine concentrating mechanism, high resolution imaging of renal urea handling could be a powerful tool for the investigation of renal physiology. This imaging technique could be applicable to the monitoring diuretic drugs which act on urea transporters <cit.> or to study the effects of antineoplastic drugs whose side effects include reduced urea concentrating ability <cit.>. Radiologically, this method could address the inherent difficulty of renal perfusion evaluation on patients with impaired renal function and chronic kidney disease since virtually all commonly used iodinated CT contrast agents and gadolinium-based MRI contrast agents pose some hazard of acute renal failure in these patients <cit.>. Although urea clearance is well known to be an inaccurate marker for GFR estimation due to its significant reabsorption <cit.>, 13C urea MRI could provide a qualitative assessment of renal perfusion such as is regularly performed on transplantation candidates, before and after ablation for renal cell carcinoma, and for the assessment of congenital urological abnormalities <cit.>. All experiments in this study were performed on a clinical MRI scanner using infused urea doses which have been shown to be safe for humans with far advanced renal failure <cit.>.
§ CONCLUSION
High resolution imaging of two key steps of the renal urea handling process was enabled by a hyperpolarized 13C relaxometry. Selective quenching of the vascular hyperpolarized 13C signal with a macromolecular relaxation agent revealed that a long-$T_2$ component of the 13C urea signal originated from the renal extravascular space, thus allowing the vascular and filtrate pools of the 13C urea to be distinguished via multi-exponential analysis. The $T_2$ response to induced diuresis and antidiuresis was performed with two imaging agents: hyperpolarized 13C urea and a control agent hyperpolarized bis-1,1-(hydroxymethyl)-1-13C-cyclopropane-$^2\textrm{H}_8$. Large $T_2$ increases in the inner-medullar and papilla were observed with the former agent and not the latter during antidiuresis suggesting that $T_2$ relaxometry may be used to monitor the inner-medullary urea transporter (UT)-A1 and UT-A3 mediated urea concentrating process. Two high resolution imaging techniques - multiple echo time averaging and ultra-long echo time sub-2 mm$^3$ resolution 3D imaging - were developed to exploit the particularly long relaxation times observed.
|
1511.00043
|
Recent applications of Stackelberg Security Games (SSG),
from wildlife crime to urban crime, have employed machine
learning tools to learn and predict adversary behavior
using available data about defender-adversary
interactions. Given these recent developments, this paper commits to an approach of directly
learning the response function of the adversary. Using the PAC model,
this paper lays a firm theoretical foundation for learning in SSGs (e.g., theoretically
answer questions about the numbers of samples required to learn adversary behavior) and
provides utility guarantees when the learned adversary model is used to plan the defender's strategy.
The paper also aims to answer practical questions such as how much more
data is needed to improve an adversary model's accuracy. Additionally, we explain a recently
observed phenomenon that prediction accuracy
of learned adversary behavior is not enough to discover
the utility maximizing defender strategy. We
provide four main contributions: (1) a PAC model of learning adversary response functions in SSGs;
(2) PAC-model analysis of the learning of key, existing bounded rationality
models in SSGs; (3) an entirely new approach to adversary modeling based on
a non-parametric class of response functions with PAC-model analysis and
(4) identification of conditions under which computing the best
defender strategy against the learned adversary behavior
is indeed the optimal strategy. Finally, we conduct
experiments with real-world data from a national park
in Uganda, showing the benefit of our new adversary modeling approach
and verification of our PAC model predictions.
§ INTRODUCTION
Stackelberg Security Games (SSGs) are arguably the best example of the application of the Stackelberg game model in the real world. Indeed, numerous successful deployed applications <cit.> (LAX airport, US air marshal)
and extensive research on related topics <cit.> provide evidence about the generality of the SSG framework. More recently, new application domains of SSGs, from wildlife crime to urban crime,
are accompanied by significant amounts of past data of recorded defender strategies and adversary reactions.
This has enabled the learning of adversary behavior from such data <cit.>. Also, analysis of these datasets and human subject experiment studies <cit.> have revealed that modeling bounded rationality of the adversary enables the defender to further optimize her allocation of limited security resources. Thus, learning the adversary's bounded rational behavior and computing defender strategy based on the learned model has become an important area of research in SSGs.
However, without a theoretical foundation for this learning and strategic planning problem, many issues that arise in practice cannot be explained or addressed. For example, it has been recently observed that in spite of good prediction accuracy of the learned models of adversary behavior, the performance of the defender strategy that is computed against this learned adversary model is poor <cit.>. A formal study could also answer several other important questions that arise in practice, for example, (1) How many samples would be required
to learn a “reasonable” model of adversary behavior in a given SSG?
(2) What utility bound can
be provided when deploying the best defender strategy that is computed
against the learned adversary model?
Motivated by the learning of adversary behavior from data in recent applications <cit.>, we adopt the framework in which the defender first learns the response function of the adversary (adversary behavior) and then optimizes against the learned response. This paper is the first theoretical study of the adversary bounded rational behavior learning problem and the optimality guarantees (utility bounds) when computing the best defender strategies against such learned behaviors.
Indeed, unlike past theoretical work on learning in SSGs (see related work) where reasoning about adversary response happens through payoff and rationality, we treat the response of the bounded rational adversary as the object to be learned.
Our first contribution is using the Probably Approximately Correct (PAC) model <cit.> to analyze the learning problem at hand. A PAC analysis yields sample complexity, i.e., the number of samples required to achieve a given level of learning guarantee. Hence, the PAC analysis allows us to address the question of required quantity of samples raised earlier. While PAC analysis is fairly standard for classifiers and real valued functions (i.e., regression) it is not an out-of-the-box approach. In particular, PAC-model analysis of SSGs brings to the table significant new challenges. To begin with, given that we are learning adversary response functions, we
must deal with the output being a probability
distribution over the adversary's actions, i.e., these response functions are vector-valued. We appeal to the framework of Haussler <cit.> to study the PAC learnability of vector-valued response functions. For SSGs, we first pose the learning problem in terms of maximizing the likelihood of seeing the attack data, but without restricting the formulation to any particular class of response functions. This general PAC framework for learning adversary behavior in SSGs enables the rest of the analysis in this paper.
Our second contribution is an analysis of the SUQR model of bounded rationality adversary behavior used in SSGs, which posits a class of parametrized response functions with a given number of parameters (and corresponding features). SUQR is the best known model of bounded rationality in SSGs, resulting in multiple deployed applications <cit.>. In analyzing SUQR, we advance the state-of-the-art in the mathematical techniques involved in PAC analysis of vector-valued function spaces. In particular, we provide a technique to obtain sharper sample complexity for SUQR than simply directly applying Haussler's original techniques. We decompose the given SUQR function space into two (or more) parts, performing PAC analysis of each part and finally combining the results to obtain the sample complexity result (which scales as $T \log T$ with $T$ targets) for SUQR (see details in Section <ref>).
Our third contribution includes an entirely new behavioral model specified by the non-parametric Lipschitz (NPL) class of response functions for SSGs, where the only restriction on NPL functions is Lipschitzness. The NPL approach makes very few assumptions about the response function, enabling the learning of a multitude of behaviors albeit at the cost of higher sample complexity. As NPL has never been explored in learning bounded rationality models in SSGs, we provide a novel learning technique for NPL. We also compute the sample complexity for NPL.
Further, we observe in our experiments that the power to capture a large variety of behaviors enables NPL to perform better than SUQR with the real-world data from Queen Elizabeth National Park (QENP) in Uganda.
Our fourth contribution is to convert the PAC learning guarantee into a bound on the utility derived by the defender when planning her strategy based on the learned adversary behavior model. In the process, we make explicit the assumptions required from the dataset of adversary's attacks in response to deployed defender mixed strategies in order to discover the optimal (w.r.t. utility) defender strategy.
These assumptions help explain a puzzling phenomenon observed in recent literature on learning in SSGs <cit.>—in particular that learned
adversary behaviors provide good prediction accuracy, but the best defender
strategy computed against such learned behavior may not perform well in practice.
The key is that the dataset for learning must not simply record a large number of attacks against few defender strategies
but rather contain the attacker's response against a variety of defender
mixed strategies. We discuss the details of our assumptions and its implications for the strategic choice of defender's actions in Section <ref>.
We also conduct experiments with real-world poaching data from the QENP in Uganda (obtained from <cit.>) and data collected from human subject experiments. The experimental results support our theoretical conclusions about the number of samples required for different learning techniques. Showing the value of our new NPL approach, the NPL approach outperforms all existing approaches in predicting the poaching activity in QENP. Finally, our work opens up a number of exciting research directions, such as studying learning of behavioral models in active learning setting and real-world application of non-parametric models.
[Due to lack of space, some proofs in this paper are in the online Appendix: <http://bit.ly/1l4n3s1>]
§ RELATED WORK
Learning and planning in SSGs with rational adversaries has been studied in two recent papers <cit.>, and in Stackelberg games by Letchford et al. letchford2009learning and Marecki et al. marecki2012playing. All these papers study the learning problem under an active learning framework, where the defender can choose the strategy to deploy within the learning process. Also, all these papers study the setting with perfectly rational adversaries. Our work differs as we study bounded rational adversaries in a passive learning scenario (i.e., with given data) and once the model is learned we analyze the guarantees of planning against the learned model.
Also, our focus on SSGs differentiates us from recent work on PAC learnability in co-operative games <cit.>, in which the authors study PAC learnability of the value function of coalitions with perfectly rational players.
Also, our work is orthogonal to adversarial learning <cit.>, which studies game theoretic models of an adversary attacking a learning algorithm.
PAC learning model has a very rich and extensive body of work <cit.>. The PAC model provides a theoretical underpinning for most standard machine learning techniques. We use the PAC framework of Haussler <cit.>. For the parametric case, we derive sharp sample complexity bounds based on covering numbers using our techniques rather than bounding it using the standard technique of pseudo-dimension <cit.> or fat shattering dimension <cit.>. For the NPL case we use results from <cit.> along with our technique of bounding the mixed strategy space of the defender; these results <cit.> have also been used in the study of Lipschitz classifiers <cit.> but we differ as our hypothesis functions are real vector-valued.
§ SSG PRELIMINARIES
This section introduces the background and preliminary notations for SSGs. A summary of notations used in this paper is presented in Table <ref>.
An SSG
is a two player Stackelberg game between a defender (leader) and an adversary (follower) <cit.>.
The defender wishes to protect $T$ targets with a limited number of security resources $K$ ($K <\!< T$).
For ease of presentation, we restrict ourselves to the scenario with no scheduling constraints (see Korzhyk et al. Korzhyk10). The defender's pure strategy is to allocate each resource to a target.
A defender's mixed-strategy $\tilde x$ ($\forall j \in \mathcal{P}.~ \tilde x_{j}\in [0,1], \sum_{j=1}^{\mathcal{P}}\tilde x_{j}=1)$ is then defined as a probability distribution over the set of all possible pure strategies $\mathcal{P}$. An equivalent description (see Korzhyk et al. Korzhyk10) of these mixed strategies are coverage probabilities over the set of targets: $x$ ($ \forall i \in T.~x_{i}\in [0,1], \sum_{i=1}^{T}x_{i}\leq K)$. We refer to this latter description as the mixed strategy of the defender.
A pure strategy of the adversary is defined as attacking a single target. The adversary's mixed strategy is then a categorical distribution over the set of targets. Thus, it can be expressed as parameters $q_i$ $(i\in T)$ of a categorical distribution such that $0 \leq q_i \leq 1$ and $\sum_i q_i = 1$. The adversary's response to the defender's
mixed strategy is given by a function $q: X \rightarrow Q$, where $Q$ is the space of all mixed strategies of the adversary. The matrix $U$ specifies the payoffs of the defender, and her expected utility is $x^T U q(x)$ when she plays a
mixed strategy $x \in X$.
Bounded Rationality Models:
We discuss the SUQR model and its representation for the analysis in this paper below.
Building on prior work on quantal response <cit.>, SUQR <cit.> states that given $n$ actions, a human player plays action $i$ with probability $q_i \propto e^{w \cdot v}$, where $v$ denotes a vector of feature values for choice $i$ and $w$ denotes the weight parameters for these features.
The model is equivalent to conditional logistic regression <cit.>. The features are specific to the domain, e.g., in case of SSG applications, the set of features include the coverage probability $x_i$, the reward $R_i$ and penalty $P_i$ of target $i$. Since, other than the coverage $x$, remaining features are fixed for each target in real world data, we assume a target-specific feature $c_i$ (which may be a linear combination of rewards and penalties) and analyze the following generalized[This general form is harder to analyze than the standard SUQR form in which the exponent function (function of $x_i, R_i, P_i$) for all $q_i$ is same: $w_1 x_i + w_2 R_i + w_3 P_i$. For completeness, we derive the results for the standard SUQR form in the Appendix.] form of SUQR with parameters $w_1$ and $c_i$'s: $q_i (x) \propto e^{w_1 x_i + c_i}$. As $\sum_{i=1}^T q_i(x) = 1$, we have:
$$q_i(x) = \frac{e^{w_1 x_i + c_i}}{ \sum_{j=1}^{T} e^{w_1 x_j + c_j}}.$$
Equivalent Alternate Representation:
For ease of mathematical proofs, using standard techniques in logistic regression, we take $q_T \propto e^0$, and hence, $q_i \propto e^{w_1 (x_i - x_T) + (c_i - c_T)}$. To shorten notation, let $c_{iT} = c_i - c_T$, $x_{iT} = x_i - x_T$. By multiplying the numerator and denominator by $e^{w_1 x_T + c_T}$, it can be verified that $q_i(x) = \frac{e^{w_1 x_{iT} + c_{iT}}}{ e^0 + \sum_{j=1}^{T-1} e^{w_1 x_{jT} + c_{jT}}} = \frac{e^{w_1 x_i + c_i}}{ \sum_{j=1}^{T} e^{w_1 x_j + c_j}}$.
§ LEARNING FRAMEWORK FOR SSG
Notation Meaning
$T, K$ Number of targets, defender resources
$d_{l_p}(o, o')$ $l_p$ distance between points $o,o'$
$\bar{d_{l_p}}(o, o')$ Average $l_p$ distance: $=d_{l_p}(o, o')/n$
$X$ Instance space (defender mixed strategies)
$Y$ Outcome space (attacked target)
$A$ Decision space
$h \in \mathcal{H}$ $h:X \rightarrow A$ is the hypothesis function
$\mathcal{N}(\epsilon, \mathcal{H}, d)$ $\epsilon$-cover of set $\mathcal{H}$ using distance $d$
$\mathcal{C}(\epsilon, \mathcal{H}, d)$ capacity of $\mathcal{H}$ using distance $d$
$r_h(p), \hat{r}_h(\vec{z})$ true risk, empirical risk of hypothesis $h$
$d_{L^1(P, d)} (f, g)$ $L_1$ distance between functions $f,g$
$q^p(x)$ parameters of true attack distribution
$q^h(x)$ parameters of attack distr. predicted by $h$
First, we introduce some notations: given two $n$-dimensional points $o$ and $o'$, the $l_p$ distance $d_{l_p}$ between the two points is: $d_{l_p}(o,o') =||o - o'||_p = (\sum_{i=1}^n |o_i - o'_i|^p)^{1/p}$. In particular, $d_{l_\infty}(o,o') = ||o - o'||_\infty = \max_i |o_i - o'_i|$. Also, $\bar{d_{l_p}} = d_{l_p}/n$.
$KL$ denotes the Kullback-Leibler divergence <cit.>.
We use the learning framework of Haussler Haussler1992, which includes an instance space $X$ and outcome space $Y$. In our context, $X$ is same as the space of defender mixed strategies $x\in X$. Outcome space $Y$ is defined as the space of all possible categorical choices over a set of $T$ targets (i.e., choice of target to attack) for the adversary.
Let $\attackletter_i$ denote the attacker's choice to attack the $i^{th}$ target. More formally $\attackletter_i = \langle \attackletter^1_i, \ldots, \attackletter^T_i \rangle$, where $\attackletter^j_i =1$ for $j=i$ and otherwise $0$. Thus, $Y = \{\attackletter_1, \ldots, \attackletter_T\}$. We will use $y$ to denote any general element of $Y$. To give an example, given three targets $T_1$, $T_2$ and $T_3$, $Y=\{\attackletter_1, \attackletter_2, \attackletter_3\} = \{\langle 1,0,0 \rangle, \langle 0,1,0 \rangle, \langle 0,0,1 \rangle\}$, where $\attackletter_1$ denotes $\langle 1,0,0 \rangle$, i.e., it denotes that $T_1$ was attacked while $T_2$ and $T_3$ were not attacked, and so on. The training data are samples drawn from $Z = X\times Y$ using an unknown probability distribution, say given by density $p(x,y)$. Each training data point $(x, y)$ denotes the adversary's response $y \in Y$ (e.g., t1 or attack on target 1) to a particular defender mixed strategy $x \in X$. The density $p$ also determines the true attacker behavior $q^p(x)$ which stands for the conditional probabilities of the attacker attacking a target given $x$ so that $q^p(x) = \langle q^p_1(x), \ldots, q^p_T(x) \rangle$, where $q^p_i(x) = p(\attackletter_i | x)$.
Haussler Haussler1992 also defines a decision space $A$,
a space of hypothesis (functions) $\mathcal{H}$ with elements $h:X \rightarrow A$ and a loss function $l:Y \times A \rightarrow \mathbb{R}$. The hypothesis $h$ outputs values in $A$ that enables computing (probabilistic) predictions of the actual outcome. The loss function $l$ captures the loss when the real outcome is $y \in Y$ and the prediction of possible outcomes happens using $a \in A$.
Example 1: Generalized SUQR For the parametric representation of generalized SUQR in the previous section and considering our 3-target example above, $\mathcal{H}$ contains vector valued functions with $(T-1)=2$ components that form the exponents of the numerator of prediction probabilities $q_1$ and $q_2$. $\mathcal{H}$ contains two components, since the third component $q_3$ is proportional to $e^0$ as discussed above. That is, $\mathcal{H}$ contains functions of the form: $\langle w_1 (x_1 - x_3) + c_{13}, w_1 (x_2 - x_3) + c_{23} \rangle $; $\forall x\in X$. Also, $A$ is the range of the functions in $\mathcal{H}$,
i.e., $A \subset \mathbb{R}^2$. Then, given $h(x) = \langle a_1, a_2 \rangle$, the prediction probabilities $q^h_1(x), q^h_2(x), q^h_3(x)$ are given by $q^h_i(x) = \frac{e^{a_i}}{1 + e^{a_1} + e^{a_2}}$ (assume $a_3 = 0$).
PAC learnability: The learning algorithm aims to learn a $h \in \mathcal{H}$ that minimizes the true risk of using the hypothesis $h$.
The true risk $\textstyle r_h (p)$ of a particular hypothesis (predictor) $h$, given density function $p(x,y)$ over $Z = X\times Y$,
is the expected loss of predicting $h(x)$ when the true outcome is $y$:
$$ r_h (p) = \int p(x,y) l(y,h(x)) \, dx \, dy $$
Of course, as $p$ is unknown the true risk cannot be computed. However, given (enough) samples from $p$, the true risk can be estimated by the empirical risk.
The empirical risk $\hat{r}_h(\vec{z})$, where $\vec{z}$ is a sequence of $m$ training samples from $Z$, is defined as: $\hat{r}_h(\vec{z}) = 1/m \sum_{i=1}^m l(y_i, h(x_i))$. Let $h^*$ be the hypothesis that minimizes the true risk, i.e., $r_{h^*}(p) = \inf \{r_h(p) ~|~ h \in \mathcal{H}\}$ and let $\hat{h}^*$ be the hypothesis that minimizes the empirical risk, i.e., $\hat{r}_{\hat{h}^*}(\vec{z}) = \inf \{\hat{r}_h(\vec{z}) ~|~ h \in \mathcal{H}\}$. The following is the well-known PAC learning result <cit.> for any empirical risk minimizing (ERM) algorithm $\mathcal{A}$ yielding hypothesis $\mathcal{A}(\vec{z})$:
\begin{array}{c}
\mbox{If }Pr(\forall h \in \mathcal{H}. |\hat{r}_h(\vec{z}) - r_h(p)| < \alpha/3 ) > 1 - \delta/2\\
\mbox{and } Pr(|\hat{r}_{\mathcal{A}(\vec{z})}(\vec{z}) - \hat{r}_{\hat{h}^*}(\vec{z})| < \alpha/3 ) > 1 - \delta/2\\
\mbox{then } Pr( |r_{\mathcal{A}(\vec{z})}(p) - r_{h^*}(p)| < \alpha ) > 1 - \delta
\end{array}
The final result states that output hypothesis $\mathcal{A}(\vec{z})$ has true risk $\alpha$-close to the lowest true risk in $\mathcal{H}$ attained by $h^*$ with high probability $1 - \delta$ over the choice of training samples.
The first pre-condition states that it must be the case that for all $h \in \mathcal{H}$ the difference between empirical risk and true risk is $\frac{\alpha}{3}$-close with high probability $1 - \frac{\delta}{2}$. The second pre-condition states that the output $\mathcal{A}(\vec{z})$ of the ERM algorithm $\mathcal{A}$ should have empirical risk $\frac{\alpha}{3}$-close to the lowest empirical risk of $\hat{h}^*$ with high probability $1 - \frac{\delta}{2}$. A hypothesis class $\mathcal{H}$ is called $(\alpha, \delta)$-PAC learnable if there exists an ERM algorithm $\mathcal{A}$ such that $\mathcal{H}$ and $\mathcal{A}$ satisfy the two pre-conditions. In this work, our empirical risk minimizing algorithms find $\hat{h}^*$ exactly (upto precision of convex solvers, see Section <ref>), thus, satisfying the second pre-condition; hence, we will focus more on the first pre-condition. As the empirical risk estimate gets better with increasing samples, a minimum number of samples are required to ensure that the first pre-condition holds (see Theorem <ref>). Hence we can relate $(\alpha, \delta)$-PAC learnability to the number of samples.
Modeling security games: Having given an example for generalized SUQR, we systematically model learning of adversary behavior in SSGs using the PAC framework for any hypothesis class $\mathcal{H}$. We assume certain properties of functions $h \in \mathcal{H}$ that we present below. First, the vector valued function $h \in \mathcal{H}$ takes the form
$$h(x) = \langle h_1(x) , \ldots, h_{T-1}(x) \rangle.$$
Thus, $A$ is the product space $A_1 \times \ldots, A_{T-1}$.
Each $h_i(x)$ is assumed to take values between $[-\frac{M}{2}, \frac{M}{2}]$, where $M >\!>1$, which implies $A_i = [-\frac{M}{2}, \frac{M}{2}]$. The prediction probabilities induced by any $h$ is
q^h(x) = \langle q^h_1(x), \ldots, q^h_T(x) \rangle
$, where $q^h_i(x) = \frac{e^{h_i(x)}}{1 + \sum_i e^{h_i(x)}}$ (assume $h_T(x) = 0$). Next, we specify two classes of functions that we analyze in later sections. We choose these two functions classes because (1) the first function class represents the widely used SUQR model in literature <cit.> and (2) the second function class is very flexible as it capture a wide range of functions and only imposes minimal Lipschitzness constraints to ensure that the functions are well behaved (e.g., continuous).
Parametric $\mathcal{H}$: In this approach we model generalized SUQR. Generalizing from Example 1, the functions $h \in \mathcal{H}$ take a parametric form where each component function is $h_i(x) = w_{1} x_{iT} %(x_{i} - x_T)
+ c_{iT}$.
Non-parametric Lipschitz (NPL) $\mathcal{H}$: Here, the only restriction we impose on functions $h \in \mathcal{H}$ is that each component function $h_i$ is $L$-Lipschitz where $L \leq \hat{K}$, for given and fixed constant $\hat{K}$. We show later (Lemma <ref>) that this implies that $q^h$ is Lipschitz also.
Next, given the stochastic nature of the adversary's attacks, we use a loss function (same for parametric and NPL) such that minimizing the empirical risk is equivalent to maximizing the likelihood of seeing the attack data. The loss function $l:Y \times A \rightarrow \mathbb{R}$ for actual outcome $\attackletter_i$ is defined as:
\begin{equation}\label{losseq}
l(\attackletter_i, a) = -\log \big( e^{a_i}/1 + \sum_{j=1}^{T-1} e^{a_j} \big ).
\end{equation}
It can be readily inferred that minimizing the empirical risk (recall $\hat{r}_h(\vec{z}) = 1/m \sum_{i=1}^m l(y_i, h(x_i))$) is equivalent to maximizing the log likelihood of the training data.
§ SAMPLE COMPLEXITY
In this section we derive the sample complexity for the parametric and NPL case, which provides an indication about the amount of data required to learn the adversary behavior. First, we present a general result about sample complexity bounds for any $\mathcal{H}$, given our loss $l$. This result relies on sample complexity results in <cit.>. The bound depends on the capacity $\mathcal{C}$ of $\mathcal{H}$, which we define after the theorem. The bound also assumes an ERM algorithm which we present for our models in Section <ref>.
Assume that the hypothesis space $\mathcal{H}$ is permissible[As noted in Haussler: “This is a measurability condition defined in Pollard (1984) which need not concern us in
practice.”]. Let the data be generated by $m$
independent draws from $X \times Y$ according to $p$. Then, assuming existence of an ERM algorithm and given our loss $l$ defined in Eq. <ref>, the least $m$ required to ensure $(\alpha, \delta)$-PAC learnability is (recall $\bar{d_{l_1}}$ is average $l_1$ distance)
\frac{576M^2}{\alpha^2} \Big( \log \frac{1}{\delta} + \log \big(8\mathcal{C}(\frac{\alpha}{96T}, \mathcal{H}, \bar{d_{l_1}}) \big) \Big)
Haussler <cit.> present a result of the above form using a general distance metric defined on the space $A$ for any loss function $l$:
\rho(a, b) = \max_{y \in Y} |l(y, a) - l(y, b)|
The main effort in this proof is relating $\rho$ to $\bar{d_{l_1}}$ for our choice of the loss function $l$ given by Equation <ref>. We are able to show that $\rho(a,b) \leq 2T \bar{d_{l_1}} (a, b)$ for our loss function. Our result then follows from this relation (details in Appendix).
The above sample complexity result is stated in terms of the capacity $\mathcal{C}(\alpha/96T, \mathcal{H}, \bar{d_{l_1}})$. Thus, in order to obtain the sample complexity of the generalized SUQR and NPL function spaces we need to compute the capacity of these function spaces. Therefore, in the rest of this section we will
focus on computing capacity $\mathcal{C}(\alpha/96T, \mathcal{H}, \bar{d_{l_1}})$ for both the generalized SUQR and NPL hypothesis
space. First, we need to define capacity $\mathcal{C}$ of functions spaces, for which we start by defining the covering number $\mathcal{N}$ of function spaces.
Let $d$ be a pseudo metric for the set $\mathcal{H}$. For any $\epsilon > 0$, an $\epsilon$-cover for $\mathcal{H}$
is a finite set $\mathcal{F} \subseteq \mathcal{H}$ such that for any $h \in \mathcal{H}$
there is a $f \in \mathcal{F}$ with $d(f, h) \leq \epsilon$, i.e., any element in $\mathcal{H}$ is at least $\epsilon$-close to some element of the cover $\mathcal{F}$.
The covering number $\mathcal{N}(\epsilon, \mathcal{H}, d)$ denotes the size of the
smallest $\epsilon$-cover for set $\mathcal{H}$ (for the pseudo metric $d$).
We now proceed to define a pseudo metric $d_{L^1(P, d)}$ on $\mathcal{H}$ with respect to any probability measure $P$ on $X$ and any given pseudo-metric $d$ on $A$. This pseudo-metric is the expected (over $P$) distance (with $d$) between the output of $f$ and $g$.
d_{L^1(P, d)} (f, g) = \int_X d(f(x), g(x)) \; dP(x) \quad \forall f, g \in \mathcal{H}
Then, $\mathcal{N}(\epsilon, \mathcal{H}, d_{L^1(P, d)})$ is the covering number for $\mathcal{H}$ for the pseudo metric $d_{L^1(P, d)}$.
However, to be more general, the capacity of function spaces provides a “distribution-free” notion of covering number. The capacity $\mathcal{C}(\epsilon, \mathcal{H}, d)$ is:
\mathcal{C}(\epsilon, \mathcal{H}, d) = \sup_P \; \{\mathcal{N}(\epsilon, \mathcal{H}, d_{L^1(P, d)})\}
Capacity of vector valued function:
The function spaces (both parametric and NPL) we consider are vector valued.
Haussler Haussler1992 provides an useful technique to bound the capacity for vector valued function space $\mathcal{H}$ in terms of the capacity of each of the component real valued function space. Given $k$ functions spaces $\mathcal{H}_1, \ldots, \mathcal{H}_k$ with functions from $X$ to $A_i$, he define the free product function space $\times_i \mathcal{H}_i$ with functions from $X$ to $A= A_1 \times \ldots A_k$ as $\times_i \mathcal{H}_i = \{\langle h_1, \ldots, h_k \rangle ~|~ h_i \in \mathcal{H}_i \}$, where $\langle h_1, \ldots, h_k \rangle (x) = \langle h_1(x), \ldots, h_k(x) \rangle$. He shows that:
\begin{equation} \label{freebound}
\mathcal{C}(\epsilon, \times_i \mathcal{H}_i, \bar{d_{l_1}}) < \prod_{i=1}^{k} \mathcal{C}(\epsilon, \mathcal{H}_i, d_{l_1})
\end{equation}
Unfortunately, a straightforward application of the above result does not give as tight bounds for capacity in the parametric case as the novel direct sum decomposition of function spaces approach we use in the sub-section. Even for the NPL case where the above result is used we still need to compute $\mathcal{C}(\epsilon, \mathcal{H}_i, d_{l_1})$ for each component function space $\mathcal{H}_i$.
§.§ Parametric case: Generalized SUQR
Recall that
the hypothesis function $h$ has $T-1$ component functions $w_1 x_{iT} + c_{iT}$. However, the same weight $w_1$ in all component functions implies that $\mathcal{H}$ is not a free product of component function spaces, hence we cannot use Equation <ref> directly. However, if we consider the space of functions, say $\mathcal{H}'$, in which the $i^{th}$ component function space $\mathcal{H}'_i$ is given by $w_i x_{iT} + c_{iT}$ (note $w_i$ can be different for each $i$) then we can use Equation <ref> to bound $\mathcal{C}(\epsilon, \mathcal{H}', \bar{d_{l_1}})$. Also, the fact that $\mathcal{H} \subset \mathcal{H}'$ allows upper bounding $\mathcal{C}(\epsilon, \mathcal{H}, \bar{d_{l_1}})$ by $\mathcal{C}(\epsilon, \mathcal{H}', \bar{d_{l_1}})$. But, this approach results in a weaker $T \log (\frac{T}{\alpha} \log \frac{T}{\alpha})$ bound (detailed derivation using this approach is in Appendix) than the technique we use below. We obtain a $T \log (\frac{T}{\alpha})$ result below in Theorem <ref>.
We propose a novel approach that decomposes $\mathcal{H}$ into a direct sum of two functions spaces (defined below), each of which capture the simpler functions $w_1x_{iT}$ and $c_{iT}$ respectively. We provide a general result about such decomposition which allows us to bound $\mathcal{C}(\epsilon, \mathcal{H}, \bar{d_{l_1}})$. We start with the following definition.
Direct-sum semi-free product of function spaces $\mathcal{G} \subset \times_i \mathcal{G}_i$ and $\times_i \mathcal{F}_i$ is defined as $\mathcal{G} \oplus \times_i \mathcal{F}_i = \{\langle g_1+f_1, \ldots, g_{T-1}+f_{T-1} \rangle ~|~ \langle g_1, \ldots, g_{T-1} \rangle \in \mathcal{G} , \langle f_1, \ldots, f_{T-1} \rangle \in \times_i \mathcal{F}_i\}$.
Applying the above definition for our case, $\mathcal{G}_i$ contains functions of the form $w x_{iT}$ ($w$ taking different values for different $g_i \in \mathcal{G}_i$). A function $\langle g_1, \ldots, g_{T-1} \rangle \in \times_i \mathcal{G}_i$ can have different weights for each component $g_i$, and thus we consider the subset $\mathcal{G} = \{\langle g_1, \ldots, g_{T-1} \rangle ~|~ \langle g_1, \ldots, g_{T-1} \rangle \in \times_i \mathcal{G}_i, \mbox{same coefficient $w$ for all $g_i$} \}$. $\mathcal{F}_i$ contains constant valued functions of the form $c_{i T}$ ($c_{i T}$ different for different functions $f_i \in \mathcal{F}_i$). Then, $\mathcal{H} = \mathcal{G} \oplus \times_i \mathcal{F}_i$.
Next, we prove a general result about direct-sum semi-free products:
If $\mathcal{H}$ is the direct-sum semi-free product $\mathcal{G} \oplus \times_i \mathcal{F}_i$
\mathcal{C}(\epsilon,\mathcal{H}, \bar{d_{l_1}}) < \mathcal{C}(\epsilon/2, \mathcal{G}, \bar{d_{l_1}}) \prod_{i=1}^{T-1} \mathcal{C}(\epsilon/2, \mathcal{F}_i, d_{l_1})
Fix any probability distribution over $X$, say $P$. For brevity, we write $k$ instead of $T-1$.
Consider an $\epsilon/2$-cover $U_i$ for each $\mathcal{F}_i$; also let $V$ be an $\epsilon/2$-cover for $\mathcal{G}$. We claim that $V \oplus \times_i U_i $ is an $\epsilon$-cover for $\mathcal{G} \oplus \times_i \mathcal{F}_i$. Take any function $h = \langle g_1+f_1, \ldots g_k+f_k \rangle$. Find functions $f'_i \in U_i$ such that $d_{L^1(P, d_{l_1})}(f_i, f'_i) < \epsilon/2$. Similarly, find function $g' = \langle g'_1, \ldots g'_k\rangle \in V$ such that $d_{L^1(P, \bar{d_{l_1}})}(g, g') < \epsilon/2$ where $g = \langle g_1, \ldots g_k\rangle$. Let $h' = \langle g'_1+f'_1, \ldots g'_k+f'_k \rangle$. Then,
\begin{array}{l}
\displaystyle d_{L^1(P, \bar{d_{l_1}})}(h, h') \\
\quad = \displaystyle\int_X \frac{1}{k} \sum_{i=1}^k d_{l_1}(g_i(x) + f_i(x), g'_i(x) + f'_i(x)) \; dP(x)\\
\quad \leq \displaystyle\int_X \frac{1}{k} \sum_{i=1}^k d_{l_1}(g_i(x) , g'_i(x)) + d_{l_1}(f_i(x) , f'_i(x)) \; dP(x)\\
\quad = \displaystyle d_{L^1(P, \bar{d_{l_1}})}(g , g') + \frac{1}{k} \sum_{i=1}^k d_{L^1(P, d_{l_1})}(f_i , f'_i)\\
\quad < \epsilon/2 + \epsilon/2 = \epsilon
\end{array}
Thus, the size of $\epsilon$-cover for $\mathcal{G} \oplus \times_i \mathcal{F}_i$ is bounded by $|V|\prod_i |U_i|$.
\begin{array}{l}
\mathcal{N}(\epsilon, \mathcal{G} \oplus \times_i \mathcal{F}_i, d_{L^1(P, \bar{d_{l_1}})}) < |V|\prod_i |U_i| \\
\qquad = \mathcal{N}(\epsilon/2, \mathcal{G}, d_{L^1(P, \bar{d_{l_1}})}) \prod_{i=1}^{k} \mathcal{N}(\epsilon/2, \mathcal{F}_i, d_{L^1(P, d_{l_1})})
\end{array}
Taking $\sup$ over probability distribution $P$ on both sides of the above inequality we get our desired result about capacity.
Next, we need to bound the capacity of $\mathcal{G}$ and $\mathcal{F}_i$ for our case.
We assume the range of all these functions ($g_i, f_i$) to be $[-\frac{M}{4}, \frac{M}{4}]$ (so that their sum $h_i$ lies in $[-\frac{M}{2}, \frac{M}{2}]$). We can obtain sharp bounds on the capacities of $\mathcal{G}$ and $\mathcal{F}_i$ decomposed from $\mathcal{H}$ in order to obtain sharp bounds on the overall capacity.
$\mathcal{C}(\epsilon, \mathcal{G}, \bar{d_{l_1}}) \leq M/4\epsilon$ and $\mathcal{C}(\epsilon, \mathcal{F}_i, d_{l_1}) \leq M/4\epsilon$.
First, note that $x_{iT} = x_i - x_T$ lies between $[-1, 1]$ due to the constraints on $x_i, x_T$.
Then, for any two functions $g, g' \in \mathcal{G}$ we can prove following result: $d_{L^1(P, \bar{d_{l_1}})}(g, g') \leq | (w - w')|$ (details in Appendix).
Also, note that since the range of any $g = w(x_i - x_T)$ is $[-\frac{M}{4}, \frac{M}{4}]$ and given $x_i - x_T$ lies between $[-1, 1]$, we can claim that $w$ lies between $[-\frac{M}{4}, \frac{M}{4}]$. Thus, given the distance between functions is bounded by the difference in weights, it enough to divide the $M/2$ range of the weights into intervals of size $2\epsilon$ and consider functions at the boundaries. Hence the $\epsilon$-cover has at most $M/4\epsilon$ functions.
The proof for constant valued functions $\mathcal{F}_i$ is similar, since its straightforward to see the distance between two functions in this space is the difference in the constant output. Also, the constants lie in $[-\frac{M}{4}, \frac{M}{4}]$, Then, the argument is same as the $\mathcal{G}$ case.
Then, plugging the result of Lemma <ref> (substituting $\epsilon/2$ for $\epsilon$) into Lemma <ref> we obtain
\textstyle \mathcal{C}(\epsilon, \mathcal{H}, \bar{d_{l_1}}) < (M/2\epsilon)^T
Having bounded $\mathcal{C}(\epsilon, \mathcal{H}, \bar{d_{l_1}})$, we use Theorem <ref> to obtain:
The generalized SUQR parametric hypothesis class $\mathcal{H}$ is $(\alpha, \delta)$-PAC learnable with sample complexity[In the Appendix, we show that for standard SUQR (simpler than our generalized SUQR) the sample size is $O\big(\frac{1}{\alpha^2} ( \log\frac{1}{\delta} + \log \frac{T}{\alpha} )\big)$]
O\Big(\big(\frac{1}{\alpha^2}\big) \big ( \log (\frac{1}{\delta}) + T\log (\frac{T}{\alpha}) \big)\Big)
The above result shows a modest $T \log T$ growth of sample complexity with increasing targets, suggesting the parametric approach can avoid overfitting limited data with increasing number of targets; however, the simplicity of the functions captured by this approach (compared to NPL) results in lower accuracy with increasing data, as shown later in our experiments on real-world data.
§.§ Non-Parametric Lipschitz case
Recall that $\mathcal{H}$ for the NPL case is defined such that each component function $h_i$ is $L$-Lipschitz where $L \leq \hat{K}$.
Consider the functions spaces $\mathcal{H}_i$ consisting of real valued $L$-Lipschitz functions where $L \leq \hat{K}$. Then, $\mathcal{H} = \times_i \mathcal{H}_i$. Then, using Equation <ref>:
\textstyle \mathcal{C}(\epsilon, \mathcal{H}, \bar{d_{l_1}}) \leq \prod_{i=1}^{T-1} \mathcal{C}(\epsilon, \mathcal{H}_i, d_{l_1})
Next, our task is to bound $\mathcal{C}(\epsilon, \mathcal{H}_i, d_{l_1})$. Consider the sup-distance metric between real valued functions: $d_{l_\infty}(h_i, h_i') = \sup_X |h_i(x) - h_i'(x)|$ for $h_i, h_i' \in \mathcal{H}_i$.
Note that $d_{l_\infty}$ is independent of any probability distribution $P$, and for all functions $h_i, h_i'$ and any $P$, $d_{L^1(P, d_{l_1})}(h_i, h_i') \leq d_{l_\infty}(h_i, h_i')$. Thus, we can infer <cit.> that for all $P$, $\mathcal{N}(\epsilon, \mathcal{H}_i, d_{L^1(P, d_{l_1})}) \leq \mathcal{N}(\epsilon, \mathcal{H}_i, d_{l_\infty})$ and then taking sup over $P$ (recall $\mathcal{C}(\epsilon, \mathcal{H}_i, d_{l_1}) = \sup_P \; \{\mathcal{N}(\epsilon, \mathcal{H}_i, d_{L^1(P, d_{l_1})})\}$) we get
\begin{equation} \label{ontheway}
\mathcal{C}(\epsilon, \mathcal{H}_i, d_{l_1}) \leq \mathcal{N}(\epsilon, \mathcal{H}_i, d_{l_\infty})
\end{equation}
We bound $\mathcal{N}(\epsilon, \mathcal{H}_i, d_{l_\infty})$ in terms of covering number for $X$ (recall $X = \{x ~|~ x \in [0,1]^T, \sum_i x_i \leq K\}$) using results from <cit.>.
\mathcal{N}(\epsilon, \mathcal{H}_i, d_{l_\infty}) \leq \Big(2 \Big\lceil \frac{M}{\epsilon} \Big\rceil + 1 \Big) \cdot 2^{\mathcal{N}(\frac{\epsilon}{2\hat{K}}, X, d_{l_\infty})}
To use the above result, we still need to bound $ \mathcal{N}(\epsilon, X, d_{l_\infty})$. We do so by combining two remarkable results about Eulerian number $\eulerian{T}{k}$ <cit.> ($k$ has to be integral).
* Laplace laplace <cit.> discovered that the volume of $X_k = \{x | x \in [0,1]^T, k-1 \leq \sum_i x_i \leq k \}$ is $\eulerian{T}{k} / T!$. Thus, if $X_K = \cup_{k=1}^K X_k$, then $vol(X_K) = \sum^{K}_{k = 1} vol(X_k) = \sum^{K}_{k = 1} \eulerian{T}{k} / T!$.
* Also, it is known <cit.> that $\eulerian{T}{k} / T! = F_T(k) - F_T(k-1)$, where $F_T(x)$ is the CDF of the probability distribution of $S_T = U _1 + \ldots + U_T$ and each $U_i$ is a uniform random variable on $[0, 1)$.
Combining these results, $vol(X_{K+1}) = F_T(K+1)$. The volume of a $l_{\infty}$ ball of radius $\epsilon$ ($l_{\infty}$ ball is a hypercube) is $(2\epsilon)^T$ <cit.>. Then, the number of balls that fit tightly (aligned with the axes) and completely inside $X_{K+1}$ is bounded by $F_T(K+1)/(2\epsilon)^T$. Since $\epsilon <\!< 1$, these balls cover $X_K = X$ completely and the tight packing ensures that the center of the balls forms an $\epsilon$-cover for $X$.
Then, bounding $F_T(K+1)$ using Bernstein's inequality about concentration of random variables we get:
For $K+1 \leq 0.5T$ (recall $K <\!< T$)
\mathcal{N}(\epsilon, X, d_{l_\infty}) \leq e^{\frac{-3T(0.5-(K+1)/T)^2}{1-(K+1)/T}} / (2\epsilon)^T
Plugging the above result into Lemma <ref> and then using that in Equation <ref>, we bound $\mathcal{C}(\epsilon, \mathcal{H}_i, d_{l_1})$. Finally, Equation <ref> gives a bound on $\mathcal{C}(\epsilon, \mathcal{H}, \bar{d_{l_1}})$ that we use in Theorem <ref> to obtain
The non-parametric hypothesis class $\mathcal{H}$ is a $(\alpha, \delta)$-PAC learnable with sample complexity
O\Big(\big(\frac{1}{\alpha^2} \big) \big( \log (\frac{1}{\delta}) + (\frac{T^{T+1}}{\alpha^T}) \big) \Big)
The above result shows that the sample complexity for NPL grows fast with $T$ suggesting that NPL may not be the right approach to use when the number of targets is large.
§ LEARNING ALGORITHM
As stated earlier, our loss function was designed so that the learning algorithm (empirical risk minimizer in PAC framework) was same as maximizing log likelihood of data. Indeed, for
, the standard MLE approach can be used
to learn the parameters (weights) and has been used in literature <cit.>.
However, for
NPL, which has no parameters, maximizing likelihood only
$h(x)$ for those
mixed strategies $x$ that are
in the training data.
Hence we present a novel two step learning algorithm for the NPL case. In the first step, we estimate the most likely value for $h_i(x)$ (for each $i$) for each $x$ in the training data, ensuring that for any pair $x, x'$ in the training data, $|h_i(x) - h_i(x')| \leq \hat{K} ||x - x'||_1$. In the second step, we construct the function
$h_i$ with the least Lipschitz constant subject to the constraint that $h_i$ takes the values for the training data output by the first step.
More formally, assume the training data has $s$ unique values for $x$
in the training set and let these values be $x^1, \ldots, x^{s}$. Further, let there be $n_j$ distinct data points against
$x^j$, i.e., $n_j$ attacks against mixed strategy $x^j$. Denote by $ n_{j,i}$ the number of attacks at each target $i$ when
$x^j$ was used. Let $h_{ij}$ be the variable that stands for the estimate of value $h_i(x^j)$; $i \in \{1, \ldots, T\}$, $j \in \{1, \ldots, s\}$. Fix $h_{Tj} = 0$ for all $j$. Then, probability of attack on target $i$ against mixed strategy $x^j$ is given by $q_{ij} = \frac{e^{h_{ij}}}{\sum_i e^{h_{ij}}}$. Thus, the log likelihood of the training data is $\sum_{j=1}^{s} \sum_{i=1}^T n_{j, i} \log q_{ij}$. Let $Lip(\hat{K})$ denote the set of $L$-Lipschitz functions with $L \leq \hat{K}$. Using our assumption that $h_i \in Lip(\hat{K})$, the following optimization problem provides the most likely $h_{ij}$ :
\begin{array}{r l}
\displaystyle\max_{h_{ij}} & \displaystyle\sum_{j=1}^{s} \sum_{i=1}^T n_{j,i} \log \frac{e^{h_{ij}}}{\sum_i e^{h_{ij}}} \\
\mbox{subject to} & \displaystyle\forall i, j, j', ~|h_{ij} - h_{ij'}| \leq \hat{K}||x^j - x^{j'}||_1 \\
& \displaystyle\forall i, j, ~ -M/2 \leq h_{ij} \leq M/2
\end{array}
Given solution $h_{ij}^*$ to the above problem, we wish to construct the solution $h_i$ such that its Lipschitz constant (given by $K_{h_i}$) is the lowest possible subject to $h_i$ taking the value $h_{ij}^*$ for $x^j$. Such a construction provides the most smoothly varying solution given the training data, i.e., we do not assume any more sharp changes in the adversary response than what the training data provides.
\min_{h_i \in Lip(\hat{K})} K_{h_i} \; \mbox{ subject to } \; \forall i,j.~h_i(x^j) = h_{ij}^* \quad (\sf{MinLip})
The above optimization is impractical to solve computationally as uncountably many constraints are required to relate $K_{h_i}$ to $h_i$, Fortunately, we obtain an analytical solution:
The following is a solution for problem $\sf{MinLip}$
h_i(x) = \textstyle \min_j \{h_{ij}^* + K^*_i||x - x^j||_1\}
where $K^*_i = \max_{j, j': j\neq j'} \frac{| h_{ij}^* - h_{ij'}^*|}{||x^j - x^{j'}||_1}$
Observe that due to the definition of $K^*$ any solution to $\sf{MinLip}$ will have Lipschitz constant $\geq K^*$. Thus, it suffices to show that the Lipschitz constant of $h_i$ is $K^*$, to prove that $h_i$ is a solution of $\sf{MinLip}$, which we show in the Appendix.
Note that for any point $x^j$ is the training data we have $h_i(x^j) = h_{ij}^*$. Then the value of $h_i(x)$ for a $x$ not in the training set and close to $x^j$ is quite likely be the $h_i(x^j)$ plus the scaled distance $K_i^*||x - x^j||_1$ showing the value of $x$ is influenced by nearby training points.
§ UTILITY BOUNDS
Next, we bound the difference between the optimal utility and the utility derived from planning using the learned $h$. The utility bound is same for the parametric and NPL case. Recall that the defender receives the utility $x U q^p(x)$ when playing strategy $x$. We need to bound the difference between the true distribution $q^p(x)$ and the predicted distribution $q^h(x)$ of attacks in order to start analyzing bounds on utility.
Thus, we transform the PAC learning guarantee about the risk of output $h$ to a bound on $||q^p(x) - q^h(x)||_1$. As the PAC guarantee only bounds the risk between predicted $h$ and the best hypothesis $h^*$ in $\mathcal{H}$, in order to relate the true distribution $q^p$ and predicted distribution $q^h$, the lemma below assumes a bounded KL divergence between the distribution of the best hypothesis $q^{h^*}$ and the true distribution $q^p$.
Assume $E[\mbox{KL}( q^p(x) ~||~ q^{h^*}(x))] \leq \epsilon^*$. Given an ERM $\mathcal{A}$ with output $h =\mathcal{A}(\vec{z}) $ and guarantee $Pr( |r_{h}(p) - r_{h^*}(p)| < \alpha ) > 1 - \delta$, with prob. $\geq 1 - \delta$ over training samples $\vec{z}$ we have
Pr(||q^p(x) - q^h(x)||_1 \leq \sqrt{2} \Delta) \geq 1 - \Delta
where $\Delta = (\alpha + \epsilon^*)^{1/3}$ and $x$ is sampled using density $p$.
Utility bound:
Next, we provide an utility bound, given the above guarantee about learned $h$.
Let the optimal strategy computed using the learned adversary model $h$ be $\tilde{x}$, i.e., $\tilde{x}^T U q^h(\tilde{x}) \geq x'^T U q^h (x')$ for all $x'$. Let the true optimal defender mixed strategy be $x^*$ (optimal
w.r.t. true attack distribution $q^p(x)$), so that the maximum defender utility is $x^{*T} U q^p(x^*)$. Let $B(x, \epsilon)$ denote the $l_1$ ball of radius $\epsilon$ around $x$.
We make the following assumptions:
* $h_i$ is $\hat{K}$-Lipschitz $\forall i$ and $q^p$ is $K$-Lipschitz in $l_1$ norm.
* $\exists \mbox{ small }\epsilon$ such that $Pr(x \in B(x^*, \epsilon)) > \Delta$ over choice of $x$ using $p$.
* $\exists \mbox{ small } \epsilon$ such that $Pr(x \in B(\tilde{x}, \epsilon)) > \Delta $ over choice of $x$ using $p$.
While the first assumption is mild [Lipschitzness is a mild restriction on function classes.], the last two assumptions for small $\epsilon$ mean that the points $x^*$ and $\tilde{x}$ must not lie in low density regions of the distribution $p$ used to sample the data points. In other words, there should be many defender mixed strategies in the data of defender-adversary interaction that lie near $x^*$ and $\tilde{x}$. We discuss the assumptions in details after the technical results below.
Given these assumptions, we need Lemma 7 that relates assumption (1) to Lipschitzness of $q^h$ in order to obtain the utility bound.
If $h_i$ is $\hat{K}$-Lipschitz then
$\forall x, x' \in X.~||q^h(x) - q^h(x')||_1 \leq 3 \hat{K} ||x - x'||_1$, i.e., $q^h(x)$ is $3\hat{K}$-Lipschitz.
Then, we can prove the following:
Given above assumptions and the results of Lemma <ref> and <ref>, with prob. $\geq 1- \delta$ over the training samples the expected utility $\tilde{x}^T U q^h(\tilde{x})$ for the learned $h$ is at least
x^{*T} U q^p(x^*) - (K+1)\epsilon - 2\sqrt{2}\Delta - 6\hat{K} \epsilon
Discussion of assumptions: A puzzling phenomenon observed in recent work on learning in SSGs is that good prediction accuracy of the learned adversary behavior is not a reliable indicator of the defender's performance in practice <cit.>.
The additional assumptions, over and above the PAC learning guarantee, are made to bound the utility deviation from the optimal utility point towards the possibility of such occurrences. Recall that the second assumption requires the existence of many defender mixed strategies in the dataset near the utility optimal
strategy $x^*$. Of course $x^*$ is not known apriori, hence in order to guarantee utility close to the highest possible utility the dataset must contain defender mixed strategies from all regions of the
mixed strategy space; or at-least if it is known that some regions of the mixed strategies dominate other parts in terms of utility then it is enough to have mixed strategies from these regions. Thus, following our assumption, better utility can be achieved by collecting attack data against a variety of mixed strategies rather than many attacks against few mixed strategies.
Going further, we illustrate with a somewhat extreme example where violating our assumptions can lead to this undesirable phenomenon.
For the purpose of illustration, consider the extreme example where probability distribution $p$ (recall data points are sampled using $p$) puts all probability mass on $x_0$, where the the utility for $x_0$ is much lower than $x^*$. Hence, the dataset will contain only one defender mixed strategy $x_0$ (with many attacks against it). Due to Lipschitzness (assumption 1), the large utility difference between $x_0$ and $x^*$ implies that $x_0$ is not close to $x^*$ which in turn violates assumption 2. This example provides a very good PAC guarantee since there is no requirement for the learning algorithm to predict accurately for any other mixed strategies (which occur with zero probability) in order to have good prediction accuracy. The learning technique needs to predict well only for $x_0$ to achieve a low $\alpha, \delta$. As a result the defender strategy computed against the learned adversary model may not be utility maximizing because of the poor predictions for all defender mixed strategies other than the low utility yielding $x_0$. More generally, good prediction accuracy can be achieved by good predictions only for the mixed strategies that occur with high probability.
Indeed, in general, the prediction accuracy in the PAC model (and any applied machine learning approach) is not a reliable indicator of good prediction over the entire space of defender mixed strategies unless, following our assumption 2, the dataset has attacks against strategies from all parts of the mixed strategy space. However, in past work <cit.> researchers have focused on gathering a lot of attack data but on limited number of defender strategies. We believe that our analysis, in addition to providing a principled explanation of prior observations, provides guidance towards methods of discovering the defender's utility maximizing strategy.
[SUQR, Uganda data, coarse-grained prediction]
[NPL, Uganda data, coarse-grained prediction]
[SUQR, Uganda data, fine-grained prediction]
[NPL, Uganda data, fine-grained prediction]
[Generalized SUQR, Uganda Data, fine-grained prediction]
[Generalized SUQR, Uganda Data, coarse-grained prediction]
[Parametric, AMT data]
[NPL, AMT data]
Results on Uganda and AMT datasets for both the parametric and NPL learning settings.
§ EXPERIMENTAL RESULTS
We show experimental results on two datasets: (i) real-world poaching data from QENP (obtained from <cit.>); (ii) data from human subjects experiments on AMT (obtained from <cit.>), to estimate prediction errors and the amount of data required to reduce the error for both the parametric and NPL learning settings. Also, we compare the NPL approach with both the standard as well as the generalized SUQR approach and show that: (i) the NPL approach, while computationally slow, outperforms the standard SUQR model
for Uganda data; and (ii) the performance of generalized SUQR is in between NPL and standard SUQR.
For each dataset, we conduct four experiments with 25%, 50%, 75% and 100% of the original data. We create 100 train-test splits in each of the four experiments per dataset. For each train-test split we compute the average prediction error $\alpha$ (average difference between the log-likelihoods of the attacks in the test data using predicted and actual attack probabilities). We report the $\alpha$ value at the $1-\delta$ percentile of the 100 $\alpha$ values, e.g., reported $\alpha = 2.09$ for $\delta=0.1$ means that 90 of the 100 test splits have $\alpha <2.09$.
§.§ Real-World Poaching data
We first present results of our experiments with real-world poaching data.
The dataset obtained contained information about features such as ranger patrols and animal densities, which are used as features in our SUQR model, and the poachers' attacks, with 40,611 total observations recorded by rangers at various locations in the park. The park area was discretized into 2423 grid cells, with each grid cell corresponding to a 1km$^2$ area within the park. After discretization, each observation fell within one of 2423 target cells and the animal densities and the number of poaching attacks within each target cell were then aggregated. The dataset contained 655 poachers' attacks in response to the defender's strategy for 2012 at QENP. Although the data is reliable because the rangers recorded the latitudes and longitudes of the location for each observation using a GPS device, it is important to note that this data set is extremely noisy because of: (i) Missing observations: all the poaching events are not recorded because the limited number of rangers cannot patrol all the areas in this park all the time; (ii) Uncertain feature values: the animal density feature is also based on incomplete observations of animals; (iii) Uncertain defender strategy: the actual defender mixed strategy is unknown, and hence, we estimate the mixed strategies based on the provided patrol data.
In this paper, we provide two types of prediction in our experiments: (i) fine-grained and (ii) coarse-grained. First, to provide a baseline for our error measures, we use the same coarse-grained prediction approach as reported by Nguyen et al. <cit.>, in which the authors only predict whether a target will be attacked or not. The results for coarse-grained predictions with our performance metric ($\alpha$ values for different $\delta$) are shown in Figs. <ref>, <ref> and <ref>. Next, in the fine-grained prediction approach we predict the actual number of attacks on each target in our test set; these results are shown in Figs. <ref>, <ref> and <ref>. In <cit.>, the authors used a particular metric for prediction performance called the area under the ROC curve (AUC), which we will discuss later in the section.
From our fine-grained and coarse-grained prediction approaches, we make several important observations. First, we observe that
$\alpha$ decreases with increasing sample size at a rate proportional to
$\frac{1}{\sqrt m}$, where $m$ is the number of samples.
For example, in Fig. <ref>, the $\alpha$ values corresponding to $\delta=0.01$ (black bar) for the 25%, 50%, 75% and 100% data are 2.81, 2.38, 2.18 and 1.8 respectively, which fits $\frac{1}{\sqrt m}$with a goodness-of-fit, i.e., $r^2$=0.97. This observation supports the relationship between $\alpha$ and $m$ shown in Theorem 3 and can be used to approximately infer the number of samples required to reduce the prediction error to a certain value. For example, assuming we collect same number of samples (=655) per year, to reduce $\alpha$ from 1.8 to 1.64, we would require two more years of data. Note here that $\alpha$ is in log-scale and hence the decrease is significant. It is also worth noting here that, for a random classifier,
we observe a value of $\alpha=6.3$ for $\delta=0.01$ while performing coarse grained predictions with 25% data. This is more than $\alpha=2.81$ for $\delta=0.01$ obtained for our standardized SUQR model while performing coarse grained predictions with 25% data. As $\alpha$ is in the log-scale, the increase in error is actually more than two-fold.
Our second observation is that $\alpha$ values for fine-grained predictions (e.g., 2.9 for $\delta=0.01$ and 100% data for the standardized SUQR model in Fig. <ref>) are understandably higher than the corresponding values for coarse-grained predictions (1.8 for $\delta=0.01$ and 100% data for SUQR in Fig. <ref>) because in the fine-grained case we predict the exact number of attacks.
Third, we observe that the performance of the generalized SUQR model (e.g., 2.47 for $\delta=0.01$ and 100% data in Fig. <ref>) is better in most cases than that of the standardized SUQR approach (2.9 for $\delta=0.01$ and 100% data in Fig. <ref>), but worse than the NPL approach (2.15 for $\delta=0.01$ and 100% data in Fig. <ref>).
Finally, we observe that our NPL model performs better than its parametric counterparts in predicting future poaching attacks for the fine-grained case (see example in previous paragraph), indicating that the true adversary behavior model may indeed be more complicated than what can be captured by SUQR.
Relation to previous work: In earlier work, Nguyen et al. <cit.> uses the area under curve (of a ROC curve) metric to demonstrate the performance of their approaches.
The AUC value of 0.73 reported in their paper is an alternate view of our $\alpha, \delta$ metric for the coarse grained prediction approach.
While there has been alternate analysis in terms of measuring prediction performances with the AUC metric in earlier papers, in our paper we have shown new trends and insights with the $\alpha, \delta$ metric through analysis from the PAC model perspective, which is missing in earlier work. We show: (i) sample complexity results and the relationship between increasing number of samples and the reduction in prediction error for each of our models; (ii) the differences in errors while learning a vector valued response function (fine-grained prediction) as opposed to classifying targets as attacked or not (coarse-grained prediction); and (iii) comparison of the performance of our new NPL model with other parametric approaches in terms of both fine-grained and coarse-grained predictions and its effectiveness on real-world poaching data which was not shown in previous work.
Uganda Parametric Uganda NPL AMT Parametric AMT NPL0.7188 121.24 0.91 123.4
Runtime results (in secs.) for one train-test split
§.§ AMT data
Here we show fine-grained prediction results on real-world AMT data obtained from <cit.>
to demonstrate the performance of both our approaches on somewhat cleaner data. This dataset is cleaner than the Uganda data because: (i) all attacks are observed, and (ii) animal densities and deployed defender strategies are known.
The dataset consisted of 16 unique mixed strategies.
There were an average of 40 attack data points per mixed strategy. Each attack had been conducted by an unique individual recruited on AMT.
We used attack data corresponding to 11 randomly chosen mixed strategies for training and data for the remaining mixed strategies for testing.
are shown in Figs. <ref>
and <ref>.
We observe that: (i) $\alpha$ values in this case are lower as compared to the Uganda data as the AMT data is cleaner; and (ii) the NPL model's performance on this dataset is poor as compared to SUQR due to, (a) low number of samples in AMT data, and (b) real-world poacher behavior may be more complicated than that of AMT participants and hence SUQR in this case was able to better capture AMT participants' behavior with limited number of samples.[In the Appendix we show that, on simulated data, $\alpha$ does indeed approach zero and NPL outperforms SUQR with enough samples.]
Runtime: While running on Matlab R2015a on an Intel Core i7-5500 [email protected], 8GB RAM machine with a 64-bit Windows 10, on average, the NPL computation takes longer than the parametric setting, as shown in Table <ref>.
§ CONCLUSION
Over the last couple of years, a lot of work has used learning methods to learn bounded rationality adversary behavioral model, but there has been no formal study of the learning process and its implication on the defender's performance. The lack of formal analysis also means that many practical questions go unanswered. We have advanced the state of the art in learning of adversary behaviors in SSGs, in terms of their analysis and implications of such learned behaviors on defender's performance.
While we used the PAC framework, it is not an out of the box approach. We needed
to come up with innovative techniques to obtain sharp bounds for our case. Furthermore, we also provide a new non-parametric learning approach which showed promising results with real-world data. Finally, we provided a principled explanation of observed phenomenon that prediction accuracy is not enough to guarantee good defender performance. We explained why datasets with attacks against a variety of defender mixed strategies helps in achieving good defender performance. Finally, we hope this work leads to more theoretical work in learning of adversary models and the use of non-parametric models in the real world.
The Appendix is structured as follows: Section <ref> contains the missing proofs, Section <ref> contains the result of the applicability of our techniques for Stackelberg games, Section <ref> constains results about the sample complexity of standard SUQR, Section <ref> contains the weaker sample complexity bound result for the generalized SUQR model derived using the approach of Haussler and Section <ref> contains additional experiments.
§ PROOFS
Proof of Theorem <ref>
First, Haussler uses the following pseudo metric $\rho$ on $A$ that is defined using the loss function $l$:
$$\begin{array}{c}\rho(a, b) = \max_{y \in Y} |l(y, a) - l(y, b)|. \end{array}$$
To start with, relying on Haussler's result, we show
$$Pr(\forall h \in \mathcal{H}. |\hat{r}_h(\vec{z}) - r_h(p)| < \frac{\alpha}{3}) \geq 1- 4\mathcal{C}\Big(\frac{\alpha}{48}, \mathcal{H}, \rho \Big)e^{-\frac{\alpha^2m}{576M^2}}$$
Choose $\alpha = \alpha'/4M$ and $\nu = 2M$ in Theorem 9 of <cit.>. Using property (3) (Section 2.2, <cit.>) of $d_v$ we obtain $|r - s| \leq \epsilon$ whenever $d_v(r, s) \leq \alpha'$. Using this directly in Theorem 9 of Haussler Haussler1992 we obtain the desired result above.
Note the dependence of the above probability on $m$ (the number of samples), and compare it to the first pre-condition in the PAC learning result. By equating $\delta/2$ to $4\mathcal{C}(\alpha/48, \mathcal{H}, \rho )e^{-\frac{\alpha^2m}{576M^2}}$, we derive the sample complexity as
m \geq \frac{576M^2}{\alpha^2} \log \frac{8\mathcal{C}(\alpha/48, \mathcal{H}, \rho )}{\delta}
We wish to compute a bound on $\mathcal{C}(\epsilon, \mathcal{H}, \rho)$ in order to use the above result to obtain sample complexity. First, we prove that
\rho \leq 2T d_{\bar{l_1}}$ for the loss function we use. This result is used to bound $\mathcal{C}(\epsilon, \mathcal{H}, \rho)$, since, it is readily verified from definition that $\mathcal{C}(\epsilon, \mathcal{H}, \rho) \leq \mathcal{C}(\epsilon/2T, \mathcal{H}, d_{\bar{l_1}})$. Such a bounding directly gives
m \geq \frac{576M^2}{\alpha^2} \log \frac{8\mathcal{C}(\alpha/96T, \mathcal{H}, \rho )}{\delta}
Below we prove that $
\rho \leq 2T d_{\bar{l_1}}$.
Given the loss function defined above, we have
$\rho(a,b) \leq 2 \max_i |a_i - b_i| \leq 2 \sum_i |a_i - b_i| \leq 2T d_{\bar{l_1}} (a, b) $
By definition, $\rho(a,b) = \max_i \Big|-a_i + b_i + \log\frac{1 + \sum_{i=1}^{T-1} e^{a_i}}{1 + \sum_{i=1}^{T-1} e^{b_i}} \Big | \leq \max_i|a_i - b_i | + \big| \log\frac{1 + \sum_{i=1}^{T-1} e^{a_i}}{1 + \sum_{i=1}^{T-1} e^{b_i}} \Big |$.
There is $j$ and $k$ such that $max_r = \frac{e^{a_j}}{e^{b_j}} \geq \frac{e^{a_i}}{e^{b_i}}$ for all $i$ and $min_r = \frac{e^{a_k}}{e^{b_k}} \leq \frac{e^{a_i}}{e^{b_i}}$ for all $i$. Thus,
$$\log \frac{1 + min_r t}{ 1 + t} \leq \log\frac{1 + \sum_{i=1}^{T-1} e^{a_i}}{1 + \sum_{i=1}^{T-1} e^{b_i}} \leq \log \frac{1 + max_r t}{1 +t }$$
where $t = \sum_{i=1}^{T-1} e^{b_i}$. The greatest positive value of the RHS is $\log max_r \leq |a_j - b_j|$ and least negative value possible for LHS is $\log min_r \geq -|a_k - b_k|$. Thus,
$$ \big | \log\frac{1 + \sum_{i=1}^{T-1} e^{a_i}}{1 + \sum_{i=1}^{T-1} e^{b_i}} \big| \leq \max_i \big| a_i - b_i \big|$$
Hence, we obtain $\rho(a,b) = \max_i | l(y_i, a) - l(y_i, b)| \leq 2 \max_i |a_i - b_i|$, and the last inequality is trivial.
Thus, using the above result we get
m \geq \frac{576M^2}{\alpha^2} \log \frac{8\mathcal{C}(\alpha/96T, \mathcal{H}, d_{\bar{l_1}} )}{\delta}
Proof of Lemma <ref>
First, note that $x_{iT} = x_i - x_T$ lies between $[-1, 1]$ due to the constraints on $x_i, x_T$.
Then, for any two functions $g, g' \in \mathcal{G}$ we have the following result:
\begin{array}{l}
d_{L^1(P, \bar{d_{l_1}})}(g, g') = \\
\quad = \displaystyle\int_X \frac{1}{T-1} \sum_{i=1}^{T-1} d_{l_1}( w(x_i - x_T), w'(x_i - x_T) ) \; dP(x)\\
\quad = \displaystyle\int_X \frac{1}{T-1} \sum_{i=1}^{T-1} | (w - w')(x_i - x_T)| \; dP(x)\\
\quad \leq \displaystyle\int_X \frac{1}{T-1} \sum_{i=1}^{T-1} | (w - w')| \; dP(x) \, = \, | (w - w')|\\
%\quad \\
\end{array}
Also, note that since the range of any $g = w(x_i - x_T)$ is $[-\frac{M}{4}, \frac{M}{4}]$ and given $x_i - x_T$ lies between $[-1, 1]$, we can claim that $w$ lies between $[-\frac{M}{4}, \frac{M}{4}]$. Thus, given the distance between functions is bounded by the difference in weights, it enough to divide the $M/2$ range of the weights into intervals of size $2\epsilon$ and consider functions at the boundaries. Hence the $\epsilon$-cover has at most $M/4\epsilon$ functions.
The proof for constant valued functions $\mathcal{F}_i$ is similar, since its straightforward to see the distance between two functions in this space is the difference in the constant output. Also, the constants lie in $[-\frac{M}{4}, \frac{M}{4}]$, Then, the argument is same as the $\mathcal{G}$ case.
Proof of Lemma <ref>
First, the space of functions $\hat{\mathcal{H}} = \{h/\hat{K} ~|~ h \in \mathcal{H}_i \}$ is Lipschitz with Lipschitz constant $\leq 1$ and $|h_i(x)| \leq M/2\hat{K}$. Clearly $\mathcal{N}(\epsilon, \mathcal{H}_i, d_{l_\infty}) \leq \mathcal{N}(\epsilon/\hat{K}, \hat{\mathcal{H}}, d_{l_\infty})$. Using the following result from <cit.>: for any Lipschitz real valued function space $\mathcal{H}$ with constant $1$, any positive integer $s$ and any distance $d$
\mathcal{N}(\epsilon, \mathcal{H}, d_{l_\infty}) \leq \Big(2\Big\lceil \frac{M(s+1)}{2\hat{K}\epsilon} \Big\rceil + 1\Big) \cdot
(s+1)^{\mathcal{N}(\frac{s\epsilon }{s+1}, X, d)}$$
we get the bound on $\mathcal{N}(\epsilon/\hat{K}, \hat{\mathcal{H}}, d_{l_\infty})$ by choosing $s=1$ and $d = d_{l_\infty}$, and hence obtain the desired bound on $\mathcal{N}(\epsilon, \mathcal{H}_i, d_{l_\infty})$.
Proof of Lemma <ref>
For ease of notation, we do the proof with $k$ standing for $K+1$.
Let $Y_i = U_i - 0.5$, then
$|Y| \leq 1/2$
$S_T - 0.5T = \sum_i Y_i$. Using Bernstein's inequality with the fact that $E[Y_i^2] = 1/12$
P(\sum_i Y_i = S_T - 0.5T \leq - t) \leq e^{\frac{-0.5t^2}{T/12 + t/6}}
$P(S_T \leq 0.5T - t) \leq e^{\frac{0.5t^2}{T/12 + t/6}}$.
Take $k = 0.5T - t$, and hence $t = 0.5T - k = T(0.5 - k/T)$. Hence,
$$P(S_T \leq k) \leq e^{\frac{-3T(0.5-k/T)^2}{1-k/T}}$$
Proof of Theorem <ref>
Given the results of Lemma <ref>, we get the sample complexity is of order
\frac{1}{\alpha^2}\Big ( \log\frac{1}{\delta} + T \big(\mathcal{N}(\frac{\alpha}{T}, X, d_{l_1}) \big) \Big )
Now, suing result of Lemma <ref>, we get the required order in the Theorem. We wish to note that if $K/T$ is a constant then the $O(e^{-T})$ in Lemma <ref> gets swamped by the $T^T$ term. However, in practice for fixed $T$, this term does provide lower actual complexity bound than what is indicated by the order.
Proof of Lemma <ref>
Observe that due to the definition of $K^*$ any solution to $\sf{MinLip}$ will have Lipschitz constant $\geq K^*$. Thus, it suffices to show that the Lipschitz constant of $h_i$ is $K^*$, to prove that $h_i$ is a solution of $\sf{MinLip}$. Take any two $x, x'$. If the min in the expression for $h_i$ occurs for the same $j$ for both $x, x'$ then $|h_i(x) - h_i(x')|$ is given by $K^*|||x - x^{j}||_1 - ||x' - x^{j}||_1|$. By application of triangle inequality
-||x - x'||_1 \leq ||x - x^{j}||_1 - ||x' - x^{j}||_1 \leq ||x - x'||_1
Thus, $|h_i(x) - h_i(x')| \leq K^* ||x - x'||_1$.
For the other case when the min for $x$ occurs at some $j$ and min for $x'$ at some $j'$ we have the following: $h_i(x') = h_{ij'} + K^*||x' - x^{j'}||_1$ and $h_i(x) = h_{ij} + K^*||x - x^{j'}||_1$ . Also, due to the min, $h_i(x') \leq h_{ij} + K^*||x' - x^{j}||_1 = h_i(x) + K^*||x' - x^{j}||_1 - K^*||x - x^{j'}||_1$. Thus, we get
h_i(x') - h_i(x) \leq K^*(||x' - x^{j}||_1 - ||x - x^{j}||_1) \leq K^* ||x' - x||_1
Using the symmetric case inequality for $x$ we get
h_i(x) - h_i(x') \leq K^*(||x - x^{j}||_1 - ||x' - x^{j}||_1) \leq K^* ||x - x'||_1
Combining both these we can claim that $|h_i(x) - h_i(x')| \leq K^* ||x' - x||_1$.
Thus, we have proved that $h_i$ is $K^*$ Lipschitz, and hence a solution of $\sf{MinLip}$.
Proof of Lemma <ref>
Let $p_X$ be the marginal of $p(x,y)$ for space $X$. Define the expected entropy $E[H(x)] = \int p_X(x) \sum_{i=1}^T I_{y = \attackletter_i} q^p_i (x) \log q^p_i (x) \, dx$. Given the loss function, we know that $ r_h(p) = -\int p(x, y) \sum_{i=1}^T I_{y = \attackletter_i} \log q^h_i (x) \, dx \, dy$. This is same as
$-\int p_X(x) \sum_{i=1}^T I_{y = \attackletter_i} q^p_i (x) \sum_{i=1}^T I_{y = \attackletter_i} \log q^h_i (x) \, dx \, dy$. This reduces to
$-\int p_X(x) \sum_{i=1}^T I_{y = \attackletter_i} q^p_i (x) \log q^h_i (x) \, dx \, dy$.
Thus, we have
E[H(x)] + r_h(p) = \int p_X(x) \sum_{i=1}^T I_{y = \attackletter_i} q^p_i (x) \log \frac{q^p_i (x)}{q^h_i (x)} \, dx \, dy
Hence, we obtain
E[H(x)] + r_h(p) = E[\mbox{KL}(q^p(x) ~||~ q^h(x))]
Hence, $|r_h(p) - r_{h^*}(p)|$ is equal to
|E[\mbox{KL}( q^p(x) ~||~ q^h(x))] - E[\mbox{KL}( q^p(x) ~||~ q^*(x))]|
Thus, from the assumptions, we get
$E[\mbox{KL}( q^p(x) ~||~ q^h(x))] \leq \alpha + \epsilon^*$ with probability $ \geq 1 - \delta$. Next, using Markov inequality, with probability $\geq 1 - \delta$
$$Pr(\mbox{KL}( q^p(x) ~||~ q^h(x)) \geq (\alpha + \epsilon^*)^{2/3}) \leq (\alpha + \epsilon^*)^{1/3}$$
that is using the notation $\Delta = (\alpha + \epsilon^*)^{1/3}$, with probability $\geq 1 - \delta$
$$Pr(\mbox{KL}( q^p(x) ~||~ q^h(x)) \leq \Delta^{2/3}) \geq 1 - \Delta^{1/3}$$
Using Pinkser's inequality we get $(1/2) ||q^p(x) - q^h(x)||_1^2 \leq \mbox{KL}( q^p(x) || q^h(x))$. That is, the event $\mbox{KL}( q^p(x) ~||~ q^h(x)) \leq \Delta^{2/3}$ implies the event $||q^p(x) - q^h(x)||_1 \leq \sqrt{2} \Delta$. Thus, $Pr(||q^p(x) - q^h(x)||_1 \leq \sqrt{2} \Delta) \geq Pr(\mbox{KL}( q^p(x) ~||~ q^h(x)) \leq \Delta^{2/3})$. Thus, we obtain: with probability $\geq 1 - \delta$,
Pr(||q^p(x) - q^h(x)||_1 \leq \sqrt{2} \Delta) \geq 1 - \Delta
Proof of Lemma <ref>
We know that $q^h_i(x) = \frac{e^{h_i(x)}}{\sum_ j e^{h_j(x)}}$ (assume $h_T(x) = 0$). Thus,
|q^h_i(x) - q^h_i(x')| = q^h_i(x')|e^{h_i(x) - h_i(x')} \frac{\sum_j e^{h_j(x')}}{\sum_j e^{h_j(x)}} - 1|
Let $r$ denote $ \frac{\sum_j e^{h_j(x')}}{\sum_j e^{h_j(x)}}$. There is $l$ and $k$ such that $max_r = \frac{e^{h_l(x')}}{e^{h_l(x)}} \geq \frac{e^{h_j(x')}}{e^{h_j(x)}}$ for all $j$ and $min_r = \frac{e^{h_k(x')}}{e^{h_k(x)}} \leq \frac{e^{h_j(x')}}{e^{h_j(x)}}$ for all $j$. Then, $min_r \leq r \leq max_r$ First, note that due to our assumption that for each $i$ $|h_i(x') - h_i(x)| \leq \hat{K} ||x' - x||_1$, we have
e^{- \hat{K} ||x' - x||_1} \leq min_r \leq r \leq max_r \leq e^{\hat{K} ||x' - x||_1}
Using the Lipschitzness we can also claim that $e^{- \hat{K} ||x' - x||_1} \leq e^{h_i(x) - h_i(x')} \leq e^{\hat{K} ||x' - x||_1}$. Thus,
e^{- 2\hat{K} ||x' - x||_1} \leq e^{h_i(x) - h_i(x')} \cdot r \leq e^{2\hat{K} ||x' - x||_1}
Since, $e^{- 2\hat{K} ||x' - x||_1} < 1$ and $e^{2\hat{K} ||x' - x||_1} > 1$ we have
|e^{h_i(x) - h_i(x')} r - 1| \leq \max(|e^{- 2\hat{K} ||x' - x||_1} - 1|, |e^{2\hat{K} ||x' - x||_1} - 1|)
Also, it is a fact that $|e^y - 1| \leq 1.5|y|$ for $|y| \leq 3/4$. Thus, we obtain
|e^{h_i(x) - h_i(x')} r - 1| \leq 3 \hat{K} ||x' - x||_1 \mbox{ for } 2\hat{K} ||x' - x||_1 \leq 3/4
Thus, $||q^h(x') - q^h(x)||_1 = \sum_i |q^h_i(x) - q^h_i(x')| = \sum_i q^h_i(x')|e^{h_i(x) - h_i(x')} \frac{\sum_j e^{h_j(x')}}{\sum_j e^{h_j(x)}} - 1| \leq (\sum_i q^h_i(x')) 3 \hat{K} ||x' - x||_1$ for $ \hat{K} ||x' - x||_1 \leq 3/8$. Since $\sum_i q^h_i(x') = 1$, we have
||q^h(x') - q^h(x)||_1 \leq 3 \hat{K} ||x' - x||_1 \mbox{ for } ||x' - x||_1 \leq 3/8\hat{K}
In other words $q^h$ is locally $3\hat{K}$-Lipschitz for every $l_1$ norm ball of size $3/8\hat{K}$. The following allows us to prove global Lipschitzness.
Any locally $L$-Lipschitz function $f$ for every $l_p$ ball of size $\delta_0$ on a compact convex set $X \subset \mathbb{R}^n$ is Lipschitz on the set $X$. The Lipschitz constant is also $L$.
Take any two points $x, y \in X$, the straight line joining $x,y$ lies in $X$ (as $X$ is convex). Also, a finite number of balls of size $\delta_0$ cover $X$ (due to compactness). Thus, there are finitely many points $x= z_1, \ldots, z_\mu = y$ on the line from $x, y$ such that $d_{l_p}(z_i, z_{i+1}) \leq \delta_0$. Further, since these points lie on a straight line we have
\begin{array}{l}
d_{l_p}(x, y) = \sum_1^{\mu-1} d_{l_p}(z_i, z_{i+1})\end{array}
Then, let any metric $d$ be used to measure distance in the range space of $f$, thus, we get
\begin{array}{l l}
d(f(x), f(y)) & \leq \sum_1^{\mu-1} d(f(z_i) , f(z_{i+1})) \\
&\leq \sum_1^{\mu-1} L d_{l_p}(z_i, z_{i+1}) \\
& = L d_{l_p}(x, y)
\end{array}
Since in our case the defender mixed strategy space is compact and convex and $q^h(x)$ satisfies the above lemma with $L = 3\hat{K}$ and $\delta_0 = 3/8\hat{K}$, $q^h(x)$ is $3\hat{K}$-Lipschitz.
Proof of Theorem <ref>
Coupled with the guarantee that with prob. $\geq 1- \delta$, $Pr(||q^p(x) - q^{h}(x)||_1 \leq \sqrt{2} \Delta) \geq 1 - \Delta$, the assumptions guarantee that with prob. $\geq 1- \delta$ for the learned hypothesis $h$ there must exist a $x' \in B(x^*, \epsilon)$ such that
$||q^p(x') - q^{h}(x')||_1 \leq \sqrt{2} \Delta$ and there must exist $x'' \in B(\tilde{x}, \epsilon)$ such that
$||q^p(x'') - q^{h}(x'')||_1 \leq \sqrt{2} \Delta$.
First, for notational ease let $\gamma$ denote $\sqrt{2} \Delta$.
The following are immediate using triangle inequality, with the results $||q^p(x') - q^{h}(x')||_1 \leq \gamma$ and $||q^p(x'') - q^{h}(x'')||_1 \leq \gamma$ and the Lipschitzness assumptions
\begin{array}{c}
||q^p(x^*) - q^{h}(x')||_1 \leq K \epsilon + \gamma \quad (\mbox{opt}x^*) \\
||q^p(\tilde{x}) - q^{h}(x'')||_1 \leq 3\hat{K} \epsilon + \gamma \quad (\mbox{opt}\tilde{x})
\end{array}
We call $\tilde{x}^T U q^h(\tilde{x}) \geq x'^T U q^h (x') $ as equation opt$h$.
Thus, we bound the utility loss as following
\begin{array}{l}
x^{*T} U q^p(x^*) - \tilde{x}^T U q^p(\tilde{x}) \\
= x^{*T} U q^p(x^*) - \tilde{x}^T U q^{h}(\tilde{x}) + \tilde{x}^T U q^{h}(\tilde{x}) - \tilde{x}^T U p(y/\tilde{x}) \\
\leq x^{*T} U q^p(x^*) - x'^T U q^{h}(x') + \tilde{x}^T U q^{h}(\tilde{x}) - \tilde{x}^T U p(y/\tilde{x}) \\
\qquad \mbox { using }\mbox{opt}h\\
= (x^* -x')^T U q^p(x^*) + x'^T U (q^p(x^*) - q^{h}(x')) + \\
\quad \tilde{x}^T U q^{h}(\tilde{x}) - \tilde{x}^T U q^p(\tilde{x}) \\
\leq \epsilon + (K\epsilon + \gamma) + \tilde{x}^T U q^{h}(\tilde{x}) - \tilde{x}^T U q^p(\tilde{x}) \\
\qquad \mbox{ using } x' \in B(x^*, \epsilon), \mbox{opt}x^*\\
= ((K+1)\epsilon + \gamma) + \tilde{x}^T U (q^{h}(\tilde{x}) - q^{h}(x'')) + \\
\quad \tilde{x}^T U (q^{h}(x'') - q^p(\tilde{x}))\\
\leq (K+1)\epsilon + \gamma + 6\hat{K} \epsilon + \gamma \\
\qquad \mbox{ using } x'' \in B(\tilde{x}, \epsilon) \mbox{ with Lipschitz }q^{h}, \mbox{opt}\tilde{x}
\end{array}
§ EXTENSION TO STACKELBERG GAMES
Our technique extends to Stackelberg games by noting that the single resource case $K=1$ with $T-1$ targets gives
$\sum_{i=1}^{T-1} x_i \leq 1$. This directly maps to a probability distribution over $T$ actions.
The $x_i$'s with $x_{T} = 1 - \sum_{i=1}^{T-1} x_i$ is the probability of playing an action. With this set-up now the security game is a standard Stackelberg game, but where the leader has $T$ actions and follower has $T-1$ actions.
Thus, in order to capture the general Stakelberg game, for the adversary, we assume $N$ actions for the adversary (instead of $T-1$ above). Then, similar to security games $q_1, \ldots, q_N$ denotes the adversary's probability of playing an action. Thus, the function $h$ now outputs vectors of size $N-1$ (instead of $O(T)$), i.e., $A$ is a subset of $N-1$ dimensional Euclidean space. The model of security game in the PAC framework extends as is to this Stackelberg setup, just with $h(x)$ and $A$ being $N-1$ dimensional. The rest of the analysis proceeds exactly as for security games for both parametric and non-parametric case, by replacing the $T$ corresponding to the adversary's action space by $N$. Since, the proof technique is exactly same, we just state the final results. Thus, for a Stackelberg game with $T$ leader actions and $N$ follower actions, the bound for Theorem <ref> becomes
\frac{576M^2}{\alpha^2} \log \frac{8\mathcal{C}(\alpha/96N, \mathcal{H}, d_{\bar{l_1}} )}{\delta}
It can be seen from the proof for the parametric part that the sample complexity does not depend on the dimensionality of $X$, but only on the dimensionality of $A$.
Hence, the sample complexity results from generalized SUQR parametric case is
O\big(\frac{1}{\alpha^2} ( \log\frac{1}{\delta} + N\log \frac{N}{\alpha} )\big)
and for the non-parametric case, which depends on both dimensionality of $X$ and $T$, the sample complexity is
O\big(\frac{1}{\alpha^2} ( \log\frac{1}{\delta} + \frac{N^{T+1}}{\alpha^T} )\big)
§ ANALYSIS OF STANDARD SUQR FORM
For SUQR the rewards and penalties are given and fixed.
Let the rewards be given and fixed $r = \langle r_1, \ldots, r_T \rangle$ (each $r_i \in [0, r_{\max}], r_{\max} > 0$), and the penalty values are $p = \langle p_1, \ldots, p_T \rangle$ (each $p_i \in [0, p_{\min}], p_{\min} < 0$). Thus, the output of $h$ is
\begin{array}{l}
h(x) = \\
\langle w_{1} x_{1T} + w_2 r_{1T} + w_3 p_{1T}, \ldots, \\w_{1}x_{T-1 T} + w_2 r_{T-1T} + w_3 p_{T-1T} \rangle
\end{array}
where $r_{iT} = r_i - r_T$ and same for $p_{iT}$. Note that in the above formulation all the component functions $h_i(x)$ have same weights. We can consider the function space $\mathcal{H}$ as the following direct-sum semi-free product $\mathcal{G} \oplus \mathcal{F} \oplus \mathcal{E} = \{\langle g_1+f_1+e_1, \ldots, g_{T-1}+f_{T-1}+e_{T-1} \rangle ~|~ \langle g_1, \ldots, g_{T-1} \rangle \in \mathcal{G}, \langle f_1, \ldots, f_{T-1} \rangle \in \mathcal{F}, \langle e_1, \ldots, e_{T-1} \rangle \in \mathcal{E}\}$, where each of $\mathcal{G}, \mathcal{F}, \mathcal{E}$ is defined below. $\mathcal{G} = \{ \langle g_1, \ldots, g_{T-1} \rangle ~|~ \langle g_1, \ldots, g_{T-1} \rangle \in \times_i \mathcal{G}_i, \mbox{ all $g_i$ have same weight} \}$ where $\mathcal{G}_i$ has functions of the form $w x_{iT}$. $\mathcal{F} = \{ \langle f_1, \ldots, f_{T-1} \rangle ~|~ \langle f_1, \ldots, f_{T-1} \rangle \in \times_i \mathcal{F}_i, \mbox{ all $f_i$ have same weight} \}$ where $\mathcal{F}_i$ has constant valued functions of the form $w r_{iT}$. $\mathcal{E} = \{ \langle e_1, \ldots, e_{T-1} \rangle ~|~ \langle e_1, \ldots, e_{T-1} \rangle \in \times_i \mathcal{E}_i, \mbox{ all $e_i$ have same weight} \}$ where $\mathcal{E}_i$ has constant valued functions of the form $w p_{iT}$.
Consider an $\epsilon/3$-cover $U_e$ for $\mathcal{E}$, an $\epsilon/3$-cover $U_f$ for $\mathcal{F}$ and $\epsilon/3$-cover $U_g$ for $\mathcal{G}$. We claim that $U_e \times U_f \times U_g$ is an $\epsilon$-cover for $\mathcal{E} \oplus \mathcal{F} \oplus \mathcal{G}$. Thus, the size of the $\epsilon$-cover for $\mathcal{E} \oplus \mathcal{F} \oplus \mathcal{G}$ is bounded by $|U_e| |U_f| |U_g|$. Thus,
\mathcal{N}(\epsilon, \mathcal{H}, d_{\bar{l_1}}) \leq \mathcal{N}(\epsilon/3, \mathcal{G}, d_{\bar{l_1}}) \mathcal{N}(\epsilon/3, \mathcal{F}, d_{\bar{l_1}}) \mathcal{N}(\epsilon/3, \mathcal{E}, d_{\bar{l_1}})
Taking sup over $P$ we get
\mathcal{C}(\epsilon, \mathcal{H}, d_{\bar{l_1}}) \leq \mathcal{C}(\epsilon/3, \mathcal{G}, d_{\bar{l_1}}) \mathcal{C}(\epsilon/3, \mathcal{F}, d_{\bar{l_1}}) \mathcal{C}(\epsilon/3, \mathcal{E}, d_{\bar{l_1}})
Now, we show that $U_e \times U_f \times U_g$ is an $\epsilon$-cover for $\mathcal{H} = \mathcal{E} \oplus \mathcal{F} \oplus \mathcal{G}$
Fix any $h \in \mathcal{H} = \mathcal{E} \oplus \mathcal{F} \oplus \mathcal{G}$. Then, $h = e + f + g$ for some $e \in \mathcal{E}, f \in \mathcal{F}, g \in \mathcal{G}$. Let $e' \in U_e$ be $\epsilon/3$ close to $e$, $f' \in U_f$ be $\epsilon/3$ close to $f$ and $g' \in U_g$ be $\epsilon/3$ close to $g$.
\begin{array}{l}
\displaystyle d_{L^1(P, d_{\bar{l_1}})}(h, h') \\
\quad = \displaystyle\int_{X} \frac{1}{k} \sum_{i=1}^k d_{l_1}(h_i(x), h'_i(x)) \; dP(x) \quad\\
\quad \leq \displaystyle\int_{X} \frac{1}{k} \sum_{i=1}^k d_{l_1}(g_i(x), g'_i(x)) \\
\qquad \displaystyle + d_{l_1}(f_i(x), f'_i(x)) + d_{l_1}(e_i(x), e'_i(x)) \; dP(x) \\
\quad = d_{L^1(P, d_{\bar{l_1}})}(g, g') + d_{L^1(P, d_{\bar{l_1}})}(f, f') + d_{L^1(P, d_{\bar{l_1}})}(e, e')\\
\quad \leq \epsilon
\end{array}
Similar to Lemma <ref>, it is possible to show that for any probability distribution $P$, for any function $g, g'$ $d_{\bar{l_1}}(g, g') \leq |w - w'|$ and $f, f'$ $d_{\bar{l_1}}(f, f') \leq |w - w'|r_{max}$ and $e, e'$ $d_{\bar{l_1}}(e, e') \leq |w - w'||p_{min}|$. Assume each of the functions have a range $[-M/6, M/6]$ (this does not affect the order in terms of $M$). Given, these ranges $w$ for $g$ can take values in $[-M/6, M/6]$, $w$ for $g$ can take values in $[-M/6r_{max}, M/6r_{max}]$ and $w$ for $g$ can take values in $[-M/6|p_{min}|, M/6|p_{min}|]$. To get a capacity of $\epsilon/3$ it is enough to divide the respective $w$ range into intervals of $2\epsilon/3$, and consider the boundaries. This yields an $\epsilon/3$-capacity of $M/2\epsilon$, $M/2\epsilon r_{max}$ and $M/2\epsilon |p_{min}|$ for $\mathcal{G}$, $\mathcal{F}$ and $\mathcal{E}$ respectively.
\mathcal{C}(\epsilon, \mathcal{H}, d_{\bar{l_1}}) \leq (M/2\epsilon)^3 \frac{1}{r_{max} |p_{min}|}
Plugging this in sample complexity from Theorem <ref> we get the results that the sample complexity is
O\big(\frac{1}{\alpha^2} ( \log\frac{1}{\delta} + \log \frac{T}{\alpha} )\big)
§ ALTERNATE PROOF FOR GENERALIZED SUQR SAMPLE COMPLEXITY
As discussed in the main paper we use the function space $\mathcal{H}'$ with each component function space $\mathcal{H}'_i$ given by $w_i x_{iT} + c_{iT}$. Then, we can directly use
Equation <ref>. We still need to bound $\mathcal{C}(\epsilon, \mathcal{H}'_i, d_{l_1})$. For this, we note the set of functions $w_i x_{iT} + c_{iT}$ has two free parameters $w_i$ and $c_i$, thus, this function space is a subset of the vector space of functions of dimension two (two values needs to represent each function). Using the pseudo-dimension technique <cit.> we know that for psuedo-dimension $d$ of function space $\mathcal{H}_i$ we get
$$\mathcal{C}(\epsilon, \mathcal{H}'_i, d_{l_1}) \leq 2 (\frac{eM}{\epsilon} \log \frac{eM}{\epsilon})^d$$
Also, we know <cit.> that pseudo-dimension is equal to the vector space dimension if the function class is a subset of a vector space. Therefore, for our case $d = 2$. Therefore, using Equation <ref> we get
\mathcal{C}(\epsilon, \mathcal{H}', d_{l_1}) \leq 2^T (\frac{eM}{\epsilon} \log \frac{eM}{\epsilon})^{2T}
Plugging this result in Theorem <ref> we get the sample complexity of
$$O\Big(\big(\frac{1}{\alpha^2}\big) \big ( \log (\frac{1}{\delta}) + T\log (\frac{T}{\alpha} \log \frac{T}{\alpha}) \big)\Big)$$
§ EXPERIMENTAL RESULTS
Here we provide additional experimental results on the Uganda, AMT and simulated datasets.
The AMT dataset consisted of 32 unique mixed strategies, 16 of which were deployed for one payoff structure and the remaining 16 for another. In the main paper, we provided results on AMT data for payoff structure 1. Here, in Figs. <ref> and <ref>, we show results on the AMT data for both the parametric (SUQR) and NPL learning settings on payoff structure 2.
For running experiments on simulated data, we used the same mixed strategies and features as for the AMT data, but simulated attacks, first using the actual SUQR model and then using a modified form of the SUQR model. Figs. <ref> and <ref> show results on simulated data on payoff structures 1 and 2 for the parametric cases, when the data is generated by an adversary with an SUQR model with true weight vector reported in Nguyen et. al Nguyen13analyzingthe ($(w_1, w_2, w_3)=(-9.85, 0.37, 0.15)$ ($c_i = w_2 R_i + w_3 P_i$)). Similar results for the NPL model are shown in Figs. <ref> and <ref> respectively. We can see that the NPL approach performs poorly with only one or five samples as expectied but improves significantly as more samples are added. To further show its potential, we modified the true adversary model of generating attacks from SUQR to the following: $q_i \propto e^{w_1 x_i^2 + c_i}$, i.e., instead of $x_i$, the adversary reasons based on $x_i^2$. We considered the same true weight vector to simulate attacks. Then, we observe in Figs. <ref> (for payoff structure 1) and <ref> (for payoff structure 2 data), that $\alpha$ approaches a value closer to zero for 500 or more sample. Also, the NPL model performs better than the parametric model with 500 or more samples. This shows that the NPL approach is more accurate when the true adversary does not satisfy the simple parametric logistic form, indicating that when we don't know the true function of the adversary's decision making process, adopting a non-parametric method to learn the adversary's behavior is more effective.
[AMT Parametric Results Payoff 2]
[AMT Nonparametric Results Payoff 2]
[Simulated Data Payoff 1 - Parametric results]
[Simulated Data Payoff 2 - Parametric results]
[Simulated Data Payoff 1 - Nonparametric results]
[Simulated Data Payoff 2 - Nonparametric results]
[Parametric vs Non-parametric results on Simulated (for various sample sizes) data from payoff 1 when the true adversary model is different from the parametric learned function]
[Parametric vs Non-parametric results on Simulated (for various sample sizes) data from payoff 2 when the true adversary model is different from the parametric learned function]
Results on Uganda, AMT and simulated datasets for the parametric and non-parametric cases respectively.
|
1511.00545
|
*maintheoremMain Theorem
In honor of Marty Golubitsky on the occasion of his seventieth birthday.
In this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by lauterbach2014equivariant and lauterbach2010do we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem.
Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples.
§ INTRODUCTION
lauterbach2010do have looked at the Ize conjecture:
Let $V$ be a real, linear and absolutely irreducible representation of a finite group or a compact Lie group $G$. Then there exists an isotropy subgroup $H \le G$ with odd-dimensional fixed point space.
They proved that this conjecture is not true by presenting three infinite families of finite groups acting on $\RR^4$, such that any of these groups has only nontrivial isotropy subgroups whose corresponding fixed point spaces are two-dimensional. They also show that for equivariant bifurcations with any group in the first two families at least one of the nontrivial isotropy types is generically symmetry breaking (in the sense of field1990symmetry). In their construction each of these families relates to a compact Lie group, which contains all the groups in the family, however these Lie groups do not a play a substantial role in the analysis.
Concerning dimensions of representation spaces which are small multiples of 4 they provide tables presenting computational results on counterexamples to the Ize conjecture including the three mentioned families. It turns out that there are, besides the three families, many more potential counterexamples to the Ize conjecture (however there are no proofs yet). The bifurcation question for all of these groups is completely open. In lauterbach2014equivariant the third family is analysed including the question concerning the generic bifurcations. Based on this information a new family of infinitely many finite groups acting on $\RR^8$ is constructed. For both cases in dimension 4 and in dimension 8 it is shown that generically the (only) nontrivial isotropy type is symmetry breaking in the sense of Field and Richardson. Again there is a compact Lie group which plays no visible role in this context. In part 4 of Theorem B in <cit.>, stated that this Lie group is a counter example to the Ize conjecture. However no proof for this statement is provided and in fact it is not correct as one can easily see.
In this paper we investigate infinite families of finite groups whose orders do not form an arithmetic progression as in the previous examples. Moreover we construct a new family acting on $\RR^4$ and based on this family a second family acting on $\RR^8$ which has a single nontrivial isotropy type and the dimension of its fixed point space is four dimensional. We prove that this isotropy type is generically symmetry breaking. The proofs are substantially different from the previous ones, here we make essential use of the Lie groups containing the groups in the family. The general question whether counterexamples to Ize's conjecture possess isotropy types which are generically symmetry breaking is open, but our technique might provide a tool to either construct counterexamples or to provide proofs.
§ MAIN RESULTS
lauterbach2010do have constructed three families of groups of orders $16\ell$ with $\ell\in 2\NN+1$, acting absolutely irreducibly on $\RR^4$ and leading to counterexamples to the Ize conjecture. In <cit.>, continues this work and constructs a family of groups of order $64\ell$ with $\ell \in 2\NN+1$ acting absolutely irreducibly on $\RR^8$ with only even-dimensional fixed point subspaces. In this paper we construct groups of order $8\m$ where $\m$ is odd and of the form
\begin{equation}
\label{decm}
\m=a\cdot b \quad \text{with } a, b \in 2\NN +1 \text{ and } \gcd (a,b) = 1
\end{equation}
(This sequence is listed in the On-Line Encyclopedia of Integer Sequences as sequence A061346 (<http://www.oeis.org>)). These groups act absolutely irreducibly on $\RR^4$ and we use them to construct groups twice their size acting absolutely irreducibly on $\RR^8$. For this step $a$ needs to be of a special form guaranteeing the existence of square roots of $-1$ modulo $a$:
Let $a_i = 1 \mod 4$ be prime and $s_i \in \NN$ for $i=1, \ldots , r$. Furthermore let
\[a = \prod_{i=1}^r a_i^{s_i}. \]
Then there exists $\rho \in \NN$ such that
\[ \rho^2 = -1 \mod a.\]
For further use we denote the set of such $a$ by $\A$:
\begin{align*}
\label{eqA}
\A &= \left\lbrace \prod_{i=1}^r a_i^{s_i} \mid r \in \NN; a_i \text{ prime}, a_i = 1 \mod 4, s_i \in \NN \text{ for } i=1, \ldots , r \right\rbrace \\
&= \lbrace 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, \ldots \rbrace \notag
\end{align*}
(The sequence of these $\m$ is a subsequence of the one listed as sequence A257591 in the On-Line Encyclopedia of Integer Sequences (<http://www.oeis.org>)). The groups acting on $\RR^8$ have precisely one nontrivial isotropy type which has a $4$-dimensional fixed point space and therefore lead to counterexamples to the Ize conjecture in dimension $8$.
The construction in lauterbach2010do relies heavily on the biquaternionic characterization of elements in $\SO{4}$ presented in conway2003on. We briefly recall the necessary notations. Denote the space of quaternions with the standard basis $\lbrace1,\ii,\jj,\kk \rbrace$ where $\ii^2 = \jj^2 = \kk^2 = -1$ by $\HH$ and let $\Q \subset \HH$ be the set of unitary quaternions. The group of ordered pairs of such quaternions $\Q\times \Q$ is isomorphic to the $4$-dimensional spin group . Identifying $\HH$ with $\RR^4$ in the obvious manner,
\[ x= \left( x_1, x_2, x_3, x_4 \right)^T \leftrightarrow x_1 + \ii x_2 + \jj x_3 + \kk x_4 , \]
these pairs correspond to elements in 4 via
\[ \widetilde{[l,r]}\colon x \mapsto \bar{l}xr.\]
conway2003on show that this is a two-to-one map on 4 where the needed identification is $\widetilde{[1,1]}=\widetilde{[-1,-1]}$ yielding $\widetilde{[l,r]}=\widetilde{[-l,-r]}$, which obviously both map a point $x$ to the same image point. The authors have used this characterization to classify the closed subgroups of 4 in terms of the biquaternionic notation.
It is a subtle yet very important observation that this map is – when taking the identification into account – a bijection but it is not a group homomorphism. Following chillingworth2015molien we define a similar map via
\begin{equation}
\label{smallrep}
[l,r]\colon x \mapsto lx \bar{r}.
\end{equation}
This provides an isomorphism $\Q \times \Q \to \SO{4}$ using the same identification $[1,1] = [-1,-1]$. Therefore we may view it as a group representation. Note that this isomorphism corresponds to the application of $\widetilde{[l,r]}^{-1}$ with the map given in conway2003on as $\bar{l} = l^{-1}$ for unitary quaternions. The application of $\widetilde{[l,r]}$ therefore yields an anti-representation. The tilde notation is obsolete from now on.
In a similar manner we can construct a map $\Q \times \Q \to \OO{4} \setminus \SO{4}$ using
\[ *[l,r] \colon x \mapsto l \bar{x} \bar{r}. \]
We do not need the explicit definition of this map but the fact that it is two-to-one as well turns out to be helpful.
We may now define the groups, we want to study in more detail. Let
\[ e_p = e^{\frac{\pi \ii}{p}} \]
be one of the primitive $p$-th root of $-1$ in $\CC$ and denote a group that is generated by the elements $g_1, g_2, \ldots$ by $\langle g_1, g_2, \ldots \rangle.$ Choose $a, b\in 2\NN+1$ such that they are relatively prime and define
\begin{equation}
\label{H_ab}
\Hab = \left\langle [e_a,1], [1,e_b], [1,\jj], [\jj,1] \right\rangle.
\end{equation}
We summarize results on the structure and the $4$-dimensional representation of these groups in the following theorems.
* For each odd $\m \in \NN$ and each decomposition $\m=a \cdot b$ as in (<ref>) forms a subgroup of 4 of order $8\m$.
* Let $b<b'$ where $b'$ is odd and relatively prime to $a$. If $b$ divides $b'$, is a subgroup of $H_{a,b'}$, i.e.
\[\Hab \le H_{a,b'}. \]
* The action of on $\HH$ as defined in (<ref>) is absolutely irreducible. It has precisely two nontrivial isotropy types. The corresponding fixed point spaces are $2$-dimensional.
We use this construction to define the family $\Hh_a$ for each $a \in 2\NN+1$:
\begin{equation*}
\label{smallfamily}
\Hh_a = \left\lbrace \Hab \mid b \in 2\NN+1 \text{ and } \gcd(a,b)=1 \right\rbrace.
\end{equation*}
The last result of <ref> allows us to generate a one-dimensional Lie group for each suitable $a$ as follows:
* Let $a \in 2\NN+1$. Then the set
\begin{equation*}
\Ha = \overline{\bigcup_{H \in \Hh_a} H} = \left\langle [e_a,1], [1,\jj], [\jj,1], [1,e^{\psi\ii}] \mid \psi \in S^1 \right\rangle
\end{equation*}
forms a compact Lie group of dimension $1$. Its action on $\HH$ is absolutely irreducible and it possesses isotropy subgroups with one-dimensional fixed point space.
* Let $a, a' \in 2\NN+1$ odd with $a < a'$. If $a$ divides $a'$, is a subgroup of $\mathbf{H}_{a'}$, i.e.
\[ \Ha \le \mathbf{H}_{a'}. \]
In the same manner this gives rise to a Lie group of dimension $2$:
The set
\begin{equation*}
\label{smalllie2}
\mathbf{H} = \overline{\bigcup_{a \in 2\NN+1} \Ha} = \left\langle [1,\jj], [\jj,1], [e^{\ii \phi},1], [1,e^{\ii \psi}] \mid \phi, \psi \in S^1 \right\rangle
\end{equation*}
forms a compact Lie group of dimension $2$.
To perform the final step in the construction of the groups acting on $\RR^8$ we need the matrix representatives of the generating elements of the groups and denote them as follows:
\[ \left[e_a,1\right] \leftrightarrow c, \quad \left[1,e_b\right] \leftrightarrow d, \quad [1,\jj] \leftrightarrow q, \quad [\jj,1] \leftrightarrow s. \]
We then look at $8$-dimensional representations of the groups constructed so far and extend them so that the representation becomes absolutely irreducible. Let $a \in \A$ and $b \in 2\NN+1$ such that $a$ and $b$ are relatively prime as before. Choose $\rho$ as in Proposition <ref>. Without loss of generality we may assume $\rho$ to be odd. If $\rho^2 = -1 \mod a$ then the same holds for $-\rho$. Since $a$ is odd, either $\rho$ or $-\rho$ is odd. We define a group as follows:
\begin{equation}
\label{smallmatrep}
\tilde{H} = \left\langle [e_a,1]^\rho, [1,e_b], [1,\jj]^3, [\jj,1] \right\rangle.
\end{equation}
This is obviously a subgroup of . Furthermore
\begin{equation*}
\left(\left([e_a,1]^\rho \right)^{-1}\right)^\rho = [e_a,1] \quad \text{and} \quad
[\jj,1]^2 [1,\jj]^3 = [1,\jj]
\end{equation*}
so the other inclusion holds as well and therefore the groups are equal: $\tilde{H} = \Hab.$ Hence we can define a representation of on $\RR^4$ where $[e_a,1]$ and $[1,\jj]$ act as $[e_a,1]^\rho$ and $[1,\jj]^3$ respectively.
We are interested in the direct sum of these two representations. It defines a group action of on $\RR^8$ which is obviously reducible. To guarantee absolute irreducibility we need to supplement the set of generators of with an element $v$ which exchanges the blocks of the two representations. We define its action on $\RR^8$ as follows: let $x, y \in \RR^4$ then
\[ v \begin{pmatrix}
x \\
\end{pmatrix} = \begin{pmatrix}
\mathbbm{1} _4 y \\
[ \jj , 1 ] x
\end{pmatrix}. \]
Since $v$ cannot be displayed properly in terms of pairs of unitary quaternions we focus on the matrix representation from now on. To meet the assumptions we have made on the $8$-dimensional representation we define the matrix generators as block matrices as follows (see (<ref>) and (<ref>)):
\begin{equation*}
\label{bigMatrixRep}
\begin{alignedat}{5}
C(a) = C &= \begin{pmatrix}
c & 0 \\
0 & c^\rho
\end{pmatrix}& &\quad
D(b)& & =D& &= \begin{pmatrix}
d & 0 \\
0 & d
\end{pmatrix} \\
Q &= \begin{pmatrix}
q & 0 \\
0 & -q
\end{pmatrix}& &\quad
& &\mathrel{\phantom{=}} S & &= \begin{pmatrix}
s & 0 \\
0 & s
\end{pmatrix} \\
V & = \begin{pmatrix}
0 & \mathbbm{1} _4 \\
s & 0
\end{pmatrix}.
\end{alignedat}
\end{equation*}
With these we define the groups acting on $\RR^8$ in terms of matrix generators
\[ \Gab = \left\langle C, D, Q, S, V \right\rangle \]
where the dependence on $a$ and $b$ lies in the matrices $C$ and $D$.
We obtain similar results on the structure and the $8$-dimensional representations as in the $4$-dimensional case.
* For each $a \in \A$ and $b \in 2\NN+1$ with $\gcd(a,b)=1$ as above the group forms a subgroup of 8 of order $16\m$ where $\m = a \cdot b$.
* Let $b<b'$ where $b'$ is odd and relatively prime to $a$. If $b$ divides $b'$, is a subgroup of $G_{a,b'}$, i.e.
\[ \Gab \le G_{a,b'}. \]
* The natural $8$-dimensional representation of is absolutely irreducible. It has precisely one nontrivial isotropy type. The corresponding fixed point space is $4$-dimensional.
We define families of these groups for every suitable $a$ as well
\begin{equation*}
\label{family}
\GG_a = \left\lbrace \Gab \mid b \ne 1 \text{ odd and } \gcd(a,b)=1 \right\rbrace.
\end{equation*}
In a similar manner as in the $4$-dimensional case we can generate compact Lie groups of dimension $1$ from the groups for every $a \in \A$. To do so we adapt notation of the generating matrices to characterize arbitrary rotations. Denote the $2$-dimensional rotation matrix by an angle $\psi$ by $\rr(\psi)$ and write
\[
d(\psi) =
\begin{pmatrix}
\rr(- \psi) & 0 \\
0 & \rr(\psi)
\end{pmatrix},\\
\]
where $\psi \in S^1$ (compare this to (<ref>)) and
\[
D(\psi) =
\begin{pmatrix}
d(\psi) & 0 \\
0 & d(\psi)
\end{pmatrix}.
\]
Let $a \in \A$. Then the set
\begin{equation*}
\label{liegroup}
\Ga = \overline{\bigcup_{G\in \GG_a} H } = \left\langle C, Q, S, V, D(\psi) \mid \psi \in S^1 \right\rangle
\end{equation*}
forms a compact Lie group of dimension 1. Its natural $8$-dimensional representation is absolutely irreducible and it possesses isotropy subgroups with one-dimensional fixed point spaces.
Furthermore we investigate the equivariant structure and the bifurcation behaviour of -symmetric systems and obtain the final result.
Let $a \in \A$ with $a>5$ and $G \in \GG_a$. The $8$-dimensional representation of $G$ has no quadratic equivariants. The space of cubic equivariants $P_{G}^3\left(\RR^8, \RR^8 \right)$ is $5$-dimensional. A basis is given by the maps $E_1, \ldots , E_5$ (see <ref>). Furthermore these are equivariant with respect to the Lie groups .
For the natural $8$-dimensional representation of $G \in \Ga$ with $5<a$ the only nontrivial isotropy type is generically symmetry breaking. Systems that are symmetric with respect to this representation generically have nontrivial symmetry breaking branches of steady states that are hyperbolic within the fixed point spaces.
The bifurcation result holds true for the groups as well. The two nontrivial isotropy types of the $4$-dimensional representation are generically symmetry breaking. The proof uses exactly the same techniques as the proof for the main theorem. But as this is no new result on counterexamples to the Ize conjecture in $4$-dimensional representations, we omit the details and only present the proof for the $8$-dimensional case.
§ PRIME NUMBERS OF THE FORM $\BOLDSYMBOL{1 \MOD 4}$
To construct the groups acting absolutely irreducibly on $\RR^8$ from the ones acting on $\RR^4$ it is crucial that we restrict ourselves to numbers $a$ which are products of prime numbers of the form $1 \mod 4$. We quote some number theoretic results first that eventually deliver square roots of $-1$ in suitable congruences. The first and easiest result, which is proved using Wilson's theorem, can be found for example in hardy1968an. For a more thorough historical discussion of this question see gauss1870disquisitiones.
Let $a$ be a prime number of the form $a= 1 \mod 4$. Then there exists $\rho \in \NN$ such that
\[ \rho^2 = -1 \mod a.\]
The next step is to apply a method based on Hensel's lemma (see eisenbud1995commutative or milne2006elliptic for the formulation that is used here) that provides the same result for prime powers.
Let $\tilde{a}$ be a prime number of the form $\tilde{a}= 1 \mod 4$ and $a= \tilde{a}^s$ for some $s\in \NN$. Then there exists $\rho \in \NN$ such that
\[ \rho^2 = -1 \mod a.\]
We use Hensel's lemma in the formulation given in milne2006elliptic with the polynomial
\[ f(X) = X^2+1 .\]
Performing an induction we obtain zeros in congruences of arbitrary powers of $a$, since we have at least the zero modulo $a$ from Proposition <ref>.
Now we can apply the Chinese remainder theorem (see for example eisenbud1995commutative) to obtain the result for arbitrary products of prime powers. Hence this completes the proof for <ref>.
§ FAMILIES OF GROUPS
§.§ Representation on $\boldsymbol{\RR^4}$
In a first step towards the proof of the results on the $4$-dimensional representation we investigate the structure of the groups as defined before (see (<ref>)). Note that the generators are subject to several relations which we summarize in the following lemma.
The trivial relations are
\begin{align*}
[e_a,1][1,e_b] &= [1,e_b][e_a,1], &\quad [e_a,1][1,\jj] &= [1,\jj][e_a,1], &\quad [1,e_b][\jj,1] &= [\jj,1][1,e_b], \\
[1,\jj][\jj,1] &= [\jj,1][1,\jj], &\quad [1,\jj]^4 &= [1,1], &\quad [\jj,1]^4 &= [1,1].
\end{align*}
The fact that $e_p \cdot \jj = \jj \cdot \bar{e_p} = \jj \cdot e_p^{-1}$ yields
\[ [e_a,1][\jj,1] = [\jj,1][e_a,1]^{-1} = [\jj,1][e_a,1]^{2a-1}, \quad [1,e_b][1,\jj] = [1,\jj][1,e_b]^{-1} = [1,\jj][1,e_b]^{2b-1}. \]
From the identification $[-1,-1] = [1,1]$ we obtain
\[ [e_a,1]^a = [1,\jj]^2, \quad [1,e_b]^b = [1,\jj]^2, \quad [\jj,1]^2 = [1,\jj]^2. \]
These relations allow us to write every group element $h \in \Hab$ in the form
\begin{equation}
\label{smallgroupel}
h = [e_a,1]^{k_1} [1,e_b]^{k_2} [1,\jj]^{l_1} [\jj,1]^{l_2}
\end{equation}
where $k_1 \in \ZZ/a\ZZ, k_2 \in \ZZ/b\ZZ, l_1 \in \ZZ/4\ZZ$ and $l_2 \in \ZZ/2\ZZ$. We present the proof for <ref> in the following lemmas.
For each odd $\m \in \NN$ and each decomposition $\m=a \cdot b$ as in (<ref>) the group forms a subgroup of 4 of order $8\m$.
Comparing with Table 4.2 in conway2003on and using their notation we find
\[ \Hab = \pm \left[ D_{2a}, D_{2b} \right] \]
where $D_{2n}$ is the dihedral group of order $2n$. This group is of order $2 \cdot 2a \cdot 2b = 8\m$. In the notation of conway2003on, the $\pm [\ldots]$ is reflected in a factor $2$ in the group orders.
The order of can also be derived directly from the representation of group elements in terms of the generators (<ref>).
The definition of $\Hab$ is symmetric in $a$ and $b$:
\[ \Hab \cong H_{b,a}. \]
However choosing different decompositions for a value of $\m$ leads to groups of the same order which are not necessarily isomorphic. At the end of this section we have listed some concrete examples (see Table <ref>).
Let $b<b'$ where $b'$ is odd and relatively prime to $a$. If $b$ divides $b'$, is a subgroup of $H_{a,b'}$, i.e.
\[\Hab \le H_{a,b'}. \]
Let $b$ and $b'$ be as assumed above. There exists $q \in \NN$ with $b' = bq$. Then
\[ e_{b'}^q = e^{\frac{\pi \ii q}{b'}} = e^{\frac{\pi \ii}{b}} = e_b. \]
So we obtain $[1, e_b] \in H_{a,b'}$ and therefore
\[\Hab \le H_{a,b'}. \]
The same result holds for the parameter $a$. The proof is completely analog to the one of the previous lemma.
In the next step we consider the action of on $\HH$ (see (<ref>)).
To prove absolute irreducibility of the representation we follow the strategy of lauterbach2010do from where we use Lemma 6.2 (for the necessary background on representation theory see Chapter 4 in chossat2000methods).
The action of on $\HH$ is absolutely irreducible.
We want to use the two two-to-one maps from the ordered pairs of unitary quaternions $\Q \times \Q$ to $\SO{4}$ and $\OO{4} \setminus \SO{4}$ respectively to find linear maps that commute with the group action. Let $[l,r] \in \Q \times \Q$ commute with . Consider the group element $[1,\jj]$. If $[l,r]$ commutes with $[1,\jj]$, then $r$ commutes with $\jj$. This yields $r=r_1 + r_2 \jj$ with $r_1,r_2 \in \RR$. Furthermore $[1,e_b] = \left[1, \cos \left(\pi/b \right) + \sin \left(\pi/b \right) \ii \right] \in \Hab$ and the relation $r \cdot e_b = e_b \cdot r$ yields $r_2 = 0$, since $\sin \left(\pi/b \right) \ne 0$. Therefore $r \in \RR$ and as $r$ is a unitary quaternion this gives $r= \pm 1$. Performing the same calculations for $l$ using the elements $[\jj,1]$ and $[e_a,1]$, we obtain $l = \pm 1$ as well. All pairs of unitary quaternions that commute with are $[\pm 1, \pm 1]$. Application of the two-to-one maps from $\Q \times \Q$ to $\SO{4}$ and $\OO{4} \setminus \SO{4}$ respectively yields that the only elements of $\OO{4}$ commuting with the group action are $\pm \mathbbm{1}$. Lemma 6.2 from lauterbach2010do implies absolute irreducibility.
In the following lemmas we investigate the isotropy of the action of on $\HH$. Using Lemmas 6.3 and 6.4 from lauterbach2010do as well as the relations of generating elements (<ref>) we may prove:
Let $h = [e_a,1]^{k_1} [1,e_b]^{k_2} [1,\jj]^{l_1} [\jj,1]^{l_2} \in \Hab$ as in (<ref>) with $k_1 \in \rkr{a}, k_2 \in \rkr{b}$ as well as $l_1 \in \lbrace 1,3 \rbrace$ and $l_2=1$. Then
* $h$ fixes a $2$-dimensional subspace of $\HH$;
* $h$ is of order $2$;
* For $l_1=1$ the fixed point space of $h$ is
\[ \left\langle \begin{pmatrix}
\cos \left(\frac{1}{2} \left(\frac{k_1}{a} - \frac{k_2}{b}\right)\pi \right) \\
\sin \left(\frac{1}{2} \left(\frac{k_1}{a} - \frac{k_2}{b}\right)\pi \right) \\
0 \\
\end{pmatrix}, \begin{pmatrix}
0 \\
0 \\
\cos \left( \frac{1}{2} \left(\frac{k_1}{a} + \frac{k_2}{b}\right)\pi \right) \\
\sin \left( \frac{1}{2} \left(\frac{k_1}{a} + \frac{k_2}{b}\right)\pi \right)
\end{pmatrix} \right\rangle. \]
For $l_1=3$ the fixed point space of $h$ is
\[ \left\langle \begin{pmatrix}
\cos \left(\frac{1}{2} \left(\frac{k_1}{a} - \frac{k_2}{b} +1 \right)\pi \right) \\
\sin \left(\frac{1}{2} \left(\frac{k_1}{a} - \frac{k_2}{b} +1 \right)\pi \right) \\
0 \\
\end{pmatrix}, \begin{pmatrix}
0 \\
0 \\
\cos \left( \frac{1}{2} \left(\frac{k_1}{a} + \frac{k_2}{b} +1 \right)\pi \right) \\
\sin \left( \frac{1}{2} \left(\frac{k_1}{a} + \frac{k_2}{b} +1 \right)\pi \right)
\end{pmatrix} \right\rangle. \]
The previous lemma describes restrictions on the exponents in the representation (<ref>) that guarantee nontrivial isotropy. These are in fact all the elements with nontrivial fixed point spaces. Using Lemmas 6.5 and 6.6 in lauterbach2010do, we see that elements which do not meet these restrictions fix the origin only.
Let $h = [e_a,1]^{k_1} [1,e_b]^{k_2} [1,\jj]^{l_1} [\jj,1]^{l_2} \in \Hab \setminus \lbrace[1,1]\rbrace$ as in (<ref>) with $k_1 \in \rkr{a}, k_2 \in \rkr{b}$ as well as $l_1 \not\in \lbrace 1,3 \rbrace$ or $l_2\ne1$. Then $h$ fixes only $0 \in \HH$.
The form of group elements that have nontrivial fixed points from <ref> guarantees that nontrivial isotropy subgroups can only contain one such element. The product of two different elements with fixed point space can not fix a point besides $0$.
The nontrivial isotropy subgroups of are generated by precisely one group element.
To shorten notation we want to name the two types of isotropy subgroups as follows for the rest of this subsection
\begin{align*}
K &= \langle h \rangle = \langle [e_a,1]^{k_1} [1,e_b]^{k_2} [1,\jj] [\jj, 1] \rangle \\
K' &= \langle h' \rangle = \langle [e_a,1]^{k_1} [1,e_b]^{k_2} [1,\jj]^3 [\jj, 1] \rangle.
\end{align*}
The isotropy groups $K$ and $K'$ are conjugate either to $\langle [1, \jj][\jj, 1] \rangle = \langle [\jj, \jj] \rangle$ or to $\langle [1, \jj]^3 [\jj,1] \rangle = \langle -[\jj, \jj] \rangle$.
This can be calculated directly using the relations on the generating elements of $K = \langle h \rangle$ and $H' = \langle h' \rangle$ and the fact that $a$ and $b$ are odd. For $K$ we obtain
\begin{align*}
h &= \left( [1, \jj]^2 \left( [e_a,1]^{\frac{a+1}{2}} \right)^2 \right)^{k_1} \left( [1,\jj]^2 \left( [1,e_b]^{\frac{b+1}{2}} \right)^2 \right)^{k_2} [1,\jj] [\jj, 1] \\
% &= [e_a,1]^{k_1 \frac{a+1}{2}} [1,e_b]^{k_2 \frac{b+1}{2}} \left( [1,\jj]^2 \right)^{k_1 + k_2} [1,\jj] [\jj, 1] [1,e_b]^{-k_2 \frac{b+1}{2}} [e_a,1]^{-k_1 \frac{a+1}{2}} \\
&= \tilde{h} \left( [1,\jj]^2 \right)^{k_1 + k_2} [1,\jj] [\jj, 1] \tilde{h}^{-1}
\end{align*}
\[ \tilde{h} = [e_a,1]^{k_1 \frac{a+1}{2}} [1,e_b]^{k_2 \frac{b+1}{2}} .\]
\[ \left( [1,\jj]^2 \right)^{k_1 + k_2} = \begin{cases}
[1,1] \quad &\text{for } k_1+k_2 \text{ even}, \\
[1, \jj]^2 \quad &\text{for } k_1+k_2 \text{ odd},
\end{cases} \]
this yields the claim for $K$. The proof for $K'$ is completely alike.
The previous lemma completes the proof for <ref> on the groups and their $4$-dimensional representations. The proofs for the Lie group structure are straightforward from the corresponding properties of the finite groups, where the $2$-dimensional Lie group (<ref>) arises in the same manner as the one-dimensional Lie group (<ref>).
* The closure of the union of the groups over all suitable $b$ is the smallest group that contains the elements $\left\lbrace [e_a,1], [1,\jj], [\jj,1], [1,e^{\psi\ii}] \mid \phi \in S^1 \right\rbrace$ (note that $[1,e^{(\pi + \psi)\ii}] = [-1,e^{\psi\ii}]$ for $\psi \in [0, \pi)$). Write it as follows
\[ \Ha = \left\langle [e_a,1], [1,\jj], [\jj,1], [1,e^{\psi\ii}] \mid \psi \in S^1 \right\rangle. \]
This is a compact $1$-dimensional Lie group. It contains an element $[e_a,e_a]$ which fixes $\langle 1, \ii \rangle$ as a real subspace of $\HH$. Furthermore $[\jj, \jj] \in \Hab$ fixes the real subspace $\langle 1, \jj \rangle$. Thus the subgroup generated by these two elements $\langle [e_a,e_a], [\jj, \jj] \rangle$ fixes the one-dimensional real subspace $\langle 1 \rangle \subset \HH$.
* Write and $\mathbf{H}_{a'}$ in terms of generators. Then for $a,a' \in 2\NN+1$ with $a < a'$ and $a$ divides $a'$ we obtain $[e_a,1] \in \mathbf{H}_{a'}$. The claim follows as in the proof of Theorem <ref>.
For the construction of the groups acting on $\RR^8$ we need the matrix representation of the groups with respect to the standard basis of $\RR^4$. It can be calculated directly via applying the generators to the basis elements. One obtains
\begin{equation}
\label{MatrixRep}
\begin{alignedat}{3}
c &=
\begin{pmatrix}
\rr\left( \frac{\pi}{a} \right) & 0 \\
0 & \rr\left( \frac{\pi}{a} \right)
\end{pmatrix}& &\quad
d& &=
\begin{pmatrix}
\rr\left( -\frac{\pi}{b} \right) & 0 \\
0 & \rr\left( \frac{\pi}{b} \right)
\end{pmatrix} \\
q &= \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{pmatrix}& &\quad
s& &= \begin{pmatrix}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{pmatrix}
\end{alignedat}
\end{equation}
where $r(\psi)$ is again the $2$-dimensional rotation matrix by an angle $\psi$. One can see that the first two elements correspond to blockwise rotations in two coordinates respectively.
The matrix generators are subject to the same relations as the corresponding pairs of quaternions (see Lemma <ref>): $cd = dc, cq = qc, ds = sd, qs = sq, q^4 = \mathbbm{1}_4, s^4 = \mathbbm{1}_4, cs = sc^{-1} = sc^{2a-1}, dq = qd^{-1} = qd^{2b-1}, c^a = q^2, d^b = q^2, s^2 = q^2$.
The group algebra software GAP <cit.> allows to check some of the stated results for low group orders. Among other things, GAP provides a classification scheme for small groups. The GAP-identifiers are composed of two integers. The first one is the group order and the second one enumerates the isomorphism classes of groups of the given order. Here we present the identifiers of the first few groups within our classification which are relevant for our subsequent analysis.
$\m$ 15 35 39 45 51 55 65 65
$(a,b)$ (5,3) (5,7) (13,3) (5,9) (17,3) (5,11) (5,13) (13,5)
GAP-id. [120, 10] [280, 9] [312, 17] [360, 9] [408, 9] [440, 19] [520,13] [520,13]
GAP-identifiers of for small values of $\m$. GAP identifies groups by their order in the first position and an enumeration of the isomorphism classes in the second position. Note the symmetry in $a$ and $b$ in the case $\m = 65$ where both factors are in $\A$. In cases like this we have multiple choices for $a$. Bear in mind that there are more complicated cases in which we have more than two choices for $a$ and $b$ for the same value of $\m$. In these cases, a change of the parameters does not necessarily lead to the same groups. However, the smallest groups for which this occurs are already far beyond the reach of the SmallGroup library from GAP.
§.§ Representation on $\boldsymbol{\RR^8}$
We want to investigate the groups in a similar manner as the groups before. First of all we can calculate relations on the generators. The relations on $C,D,Q$ and $S$ are the same as for $c,d,q$ and $s$ because of the blockdiagonal structure. The relations containing $V$ can be calculated using the ones for the small matrices (see Lemma <ref>).
The blockdiagonal generators of are subject to the following relations: $CD = DC, CQ = QC, DS = SD, QS = SQ, Q^4 = \mathbbm{1}_8, S^4 = \mathbbm{1}_8, CS = SC^{-1} = SC^{2a-1}, DQ = QD^{-1} = QD^{2b-1}, C^a = Q^2, D^b = Q^2, S^2 = Q^2$.
Adding $V$ yields
\begin{align*}
VC &= C^\rho V &\quad
CV &= VC^{-\rho} = VC^{2a - \rho} &\quad
VD &= DV \\
VQ &= Q^3 V &\quad
VS &= SV &\quad
V^8 &= \mathbbm{1}_8 &\quad
V^2 &= S.
\end{align*}
This is the point where the fact that $\rho$ is odd becomes important. If it were even $C^a$ would not be equal to $Q^2=-\mathbbm{1}_8$ but
\[ C^a = \begin{pmatrix}
- \mathbbm{1}_4 & 0 \\
0 & \mathbbm{1}_4
\end{pmatrix}. \]
Just as before this allows us to write every element $g \in \Gab$ in the form
\begin{equation}
\label{groupel}
g = C^{k_1} D^{k_2} Q^{l_1} S^{l_2} V^m
\end{equation}
with $k_1 \in \rkr{a}, k_2 \in \rkr{b}, l_1 \in \rkr{4}, l_2 \in \rkr{2}$ and $m \in \rkr{2}$. Using the calculations for the groups we may state similar results on the structure and isotropy of the groups .
For each $a \in \A$ and $b \in 2\NN+1$ with $\gcd(a,b)=1$ the group forms a subgroup of 8 of order $16\m$ where $\m = a \cdot b$.
The elements $C,D,Q$ and $S$ generate a group that is isomorphic to . Addition of $V$ to the set of generators gives two copies of this group. Therefore $ \left| \Gab \right| = 2 \left| \Hab \right| = 16\m$. Once again this can be calculated directly from the form of the group elements (<ref>).
Let $b<b'$ where $b'$ is odd and relatively prime to $a$. If $b$ divides $b'$, is a subgroup of $G_{a,b'}$, i.e.
\[ \Gab \le G_{a,b'}. \]
This follows directly from the second statement in Theorem <ref>.
In contrast to the $4$-dimensional case, we do not obtain the same result for the parameter $a$, which is due to the power $\rho$ in the definition of the generating element $C$. This power is not necessarily the same for $a$ and $a'$ when $a$ divides $a'$. However in this case is isomorphic to a subgroup of $G_{a',b}$.
The natural $8$-dimensional representation of is absolutely irreducible.
Let $L \colon \RR^8 \to \RR^8$ be a linear map in matrix representation that commutes with the group action of . We write $L$ in form of a block matrix
\[ L = \begin{pmatrix}
L_{1,1} & L_{1,2} \\
L_{2,1} & L_{2,2}
\end{pmatrix} \]
where $L_{i,j} \colon \RR^4 \to \RR^4$ for $i,j \in \lbrace 1,2 \rbrace$. In a first step we want to show that $L_{1,2}=L_{2,1}=0$. Then we can use absolute irreducibility of the $4$-dimensional representation of to prove the claim. Using the commutativity assumption and the structure of the generating matrices we obtain
\begin{equation*}
\begin{alignedat}{6}
L_{1,2}qs &=& &-qsL_{1,2} & &\quad \text{from} \quad& &LQ& &=& &QL \text{ and } LS = SL, \\
L_{1,2}c^\rho &=& &cL_{1,2} & &\quad \text{from} \quad& &LC& &=& &CL, \\
L_{1,2}s &=& &L_{2,1}& &\quad \text{from} \quad& &LV& &=& &VL.
\end{alignedat}
\end{equation*}
The first relation yields that $L_{1,2}$ is of the form
\[ L_{1,2} = \begin{pmatrix}
0 & * & 0 & * \\
* & 0 & * & 0 \\
0 & * & 0 & * \\
* & 0 & * & 0
\end{pmatrix}. \]
We want to apply the second relation and remember that $c$ is the representation matrix of $[e_a,1]$ on $\HH$. Thus we calculate the power of $c$ to be
\[ c^\rho =
\begin{pmatrix}
\rr\left( \frac{\rho \pi}{a} \right) & 0 \\
0 & \rr\left( \frac{\rho \pi}{a} \right)
\end{pmatrix}
. \]
Note that the entries of this matrix contain the real and imaginary part of $e_a^\rho$:
\begin{equation*}
\cos \left(\frac{\rho \pi}{a}\right) = \Re \left(e_a^\rho\right), \quad
\sin \left(\frac{\rho \pi}{a}\right) = \Im \left(e_a^\rho\right).
\end{equation*}
Now we make use of the special choice of the power $\rho$ to prove that these can not match the real and imaginary part of $e_a$. Since $\rho \in \lbrace 0, \ldots, a-1 \rbrace$, we obtain that the only chance for $\Re \left(e_a^\rho\right) = \Re \left(e_a\right)$ is for $\rho = 1$ or $\rho = 2a-1$ which both contradict the fact that $\rho^2 = -1 \mod a$. Considering the imaginary parts, we obtain that the only possibility for $\Im \left(e_a^\rho\right) = \Im \left(e_a\right)$ is if $\rho = 1$ or $\rho = a-1$. Once again this contradicts the choice of $\rho$. Therefore
\begin{equation*}
\cos \left(\frac{\rho \pi}{a}\right) \ne \cos \left(\frac{\pi}{a}\right) \quad \text{and} \quad
\sin \left(\frac{\rho \pi}{a}\right) \ne \sin \left(\frac{\pi}{a}\right).
\end{equation*}
Omitting the details, this allows us to compute that the remaining entries of $L_{1,2}$ are zero as well. Together with the last relation this yields $L_{1,2}=L_{2,1}=0$.
Therefore we obtain two linear maps $L_{i,i} \colon \RR^4 \to \RR^4$ for $i=1,2$ that commute with the action of . From the absolute irreducibility of this action we know that $L$ is of the form
\[ L = \begin{pmatrix}
\gamma \mathbbm{1}_4 & 0 \\
0 & \delta \mathbbm{1}_4
\end{pmatrix} \]
with $\gamma, \delta \in \RR$. Commutation with $V$ yields $\gamma = \delta$.
In the next step we investigate the isotropy of the $8$-dimensional representation of . Note that the corresponding results on mostly rely on the relations of the generating elements. Hence they can be adapted almost directly.
Let $g \in \Gab \setminus \lbrace \mathbbm{1}_8 \rbrace$ be written in the form (<ref>). Then $g$ fixes a point $x \in \RR^8 \setminus \lbrace 0 \rbrace$ if and only if $l_1 \in \lbrace 1,3 \rbrace, l_2 = 1$ and $m=0$.
For $m=0$ the claim follows directly from Lemmas <ref> and <ref> since the other elements keep the two -blocks intact. Therefore we consider elements of the form (<ref>) with $m=1$:
\[ g = C^{k_1} D^{k_2} Q^{l_1} S^{l_2} V. \]
Suppose $x = (\zeta, \eta)$ with $\zeta,\eta \in \RR^4$ such that $gx = x$. Using the structure of the generating matrices, this yields
\[ gx = C^{k_1} D^{k_2} Q^{l_1} S^{l_2} \left(\eta, s\zeta \right) = (\zeta, \eta). \]
Since $C,D,S$ and $Q$ keep the block structure intact, we may split this into two equations:
\begin{align*}
c^{k_1} d^{k_2} q^{l_1} s^{l_2} \eta &= \zeta \\
c^{\rho k_1} d^{k_2} q^{3 l_1} s^{l_2 + 1} \zeta &= \eta.
\end{align*}
Inserting the second equation in the first one, we obtain
\[ \left(c^{k_1} d^{k_2} q^{l_1} s^{l_2} \right) \left( c^{\rho k_1} d^{k_2} q^{3 l_1} s^{l_2 + 1} \right) \zeta = \zeta. \]
Using the relations on the matrix representation of the generators of (Lemma <ref>), we may then calculate
\[ c^{k_1} d^{k_2} q^{l_1} s^{l_2} c^{\rho k_1} d^{k_2} q^{3 l_1} s^{l_2 + 1} = c^{k_1+ (-1)^{l_2}\rho k_1} d^{k_2 + (-1)^{l_1} k_2} q^{2 l_2} s. \]
Since the power of $q$ is even, Lemma <ref> yields $\zeta = 0$. Inserting this in the second equation gives $\eta = 0$ which completes the proof.
Note that we can use the formulas to compute basis elements of the fixed point spaces (<ref>) in the case $m=0$ as well. We only have to take the powers of $c$ and $q$ in the second block of the matrices $C$ and $Q$ into account. The fixed point spaces are obviously $4$-dimensional.
Concerning the isotropy subgroups of we obtain the same result as in <ref> from the fact that $\langle C,D,Q,S \rangle$ is isomorphic to :
The nontrivial isotropy subgroups of are generated by precisely one group element.
Once more we want to make use of shorter notations. We therefore write
\begin{align*}
K &= \langle g \rangle = \left\langle C^{k_1} D^{k_2} Q S \right\rangle \\
K' &= \langle g' \rangle = \left\langle C^{k_1} D^{k_2} Q^3 S \right\rangle
\end{align*}
for the two types of nontrivial isotropy subgroups for the rest of this subsection. Using the element $V$, we may now show, that in the $8$-dimensional case we obtain only one isotropy type:
All nontrivial isotropy subgroups of are conjugate to $\langle QS \rangle$.
All nontrivial isotropy subgroups are generated by either $g$ or $g'$ which both do not contain a factor $V$. We may therefore use Lemma <ref> and the fact that $C,D,Q$ and $S$ are subject to the same relations as $[e_a,1], [1,e_b], [1, \jj]$ and $[\jj, 1]$. This yields that every nontrivial isotropy subgroup is conjugate to either $\langle QS \rangle$ or $\langle Q^3S \rangle$. These two subgroups are conjugate by $V$:
\[ VQSV^{-1} = Q^3SVV^{-1} = Q^3S. \]
Thus all nontrivial isotropy subgroups are conjugate to $\langle QS \rangle$.
This completes the proof for <ref>. Similarly to the $4$-dimensional case these considerations leave the results on the Lie group structure straightforward and we may state the proof of <ref>:
The claim follows in the same way as in the proof for the groups . Let $\lbrace \xi_1, \ldots, \xi_8 \rbrace$ be the standard basis of $\RR^8$. The Lie group contains the elements $C$ and $D\left(\pi/a\right)$ and their product $C D\left(\pi/a\right)$ fixes the subspace $\left\langle \xi_1, \xi_2 \right\rangle$. Furthermore $QS$ fixes the subspace $\left\langle \xi_1, \xi_3, \xi_6, \xi_8 \right\rangle$. Thus the subgroup $\left\langle C D\left(\pi/a\right), QS \right\rangle$ generated by these two elements fixes the subspace $\left\langle \xi_1 \right\rangle$.
As mentioned before the subgroup relation for the one-dimensional Lie groups as in <ref> does not hold because of the exponent $\rho$ in the construction of the matrix $C$. Furthermore we do not obtain Lie groups of dimension $2$ when considering the closure of the union of the one-dimensional Lie groups over all $a \in \A$. The reason for this structural difference lies in the power $\rho$ as well. It is a nonconstant natural number depending on the angle which is a rational multiple of $\pi$. As such it has no smooth – more precisely, not even a continuous – continuation on all angles $\phi \in S^1$ and therefore prevents a smooth structure on the matrices $C$ for all angles. It is unknown whether there exist $2$-dimensional Lie groups containing all the groups $\Ga$.
We provide the GAP-identifiers for the first groups (compare to Table <ref>):
$\m$ 15 35 39 45 51 55 65 65
$(a,b)$ (5,3) (5,7) (13,3) (5,9) (17,3) (5,11) (5,13) (13,5)
GAP-id. [240, 101] [560, 94] [624, 130] [720, 98] [816, 97] [880, 130] [1040,105] [1040,112]
GAP-identifiers of for small values of $\m$. Note that for the groups the symmetry in $a,b$ is broken in the case of both factors being in $\A$. This is due to the different construction of the matrix $C$ from $c$.
§ EQUIVARIANT STRUCTURE FOR GA
Since we are interested in bifurcation problems on $\RR^8$ with -symmetry for suitable $a$ and $b$, we have to investigate smooth -equivariant maps on $\RR^8$. Using methods from character and invariant theory, we are able to compute dimensions of spaces of homogeneous equivariant polynomial maps for up to third degree. Then we determine the generating functions for the corresponding spaces. This allows us to gain insight in the general bifurcation behaviour of equations with -symmetry which we will investigate further in the next section.
For a group $\Gamma$ acting on the real space $W$ define its character as follows
\[ \chi \colon \Gamma \to \RR, \quad g \mapsto \tr{g} \]
and denote the space of smooth $\Gamma$-equivariant maps by $C_\Gamma^\infty \left( W, W \right)$. It is well known that the symmetric functions form a module which contains the equivariant polynomials as a dense subset (see for example chossat2000methods or field2007dynamics). The space of homogeneous equivariant polynomial maps of degree $d$ shall be denoted by $P_\Gamma^d (W,W)$. To gather information about the equivariant structure of a given representation one often looks at the so called Molien series, a formal power series that carries information about dimensions of these spaces. It is defined as follows
\[ \sum_{d = 0}^{\infty} R_d z^d\]
where $R_d = \dim P_\Gamma^d (W,W)$ is the number of linearly independent equivariant polynomial maps of degree $d$ to which we refer as Molien coefficients. In a similar way we consider invariant polynomials from the represention space into the real numbers. These are in a close relationship to the equivariant polynomial maps. We denote the space of invariant homogeneous polynomials of degree $d$ by $\Pi_\Gamma^d (W)$. Then we obtain for example that for every $p\in \Pi_\Gamma^d(W)$ the gradient $\nabla p$ is an equivariant polynomial map: $\nabla p \in P_{\Gamma}^{d-1}\left(W,W\right)$. For more details on this matter and the connection between invariant and equivariant polynomials see chossat2000methods. The corresponding formal power series
\[ \sum_{d = 0}^{\infty} r_d z^d\]
with $r_d = \dim \Pi_\Gamma^d(W)$ is called Molien series as well. Molien's theorem states a way to calculate these formal power series but it is often difficult to do so. That is why we use a slightly different approach.
§.§ Computation of Molien coefficients
We are especially interested in the equivariant structure for low degree polynomial maps. sattinger1979group proves a formula by which we can calculate the $R_d$ for a single $d$ without having to deal with the Molien series. This formula also follows from the results in zhilinskii1989tensor. Although it is impractical for large values of $d$ it is very helpful in the cases which we consider. For $g\in \Gamma$ defines the quantity
\[ \chi_{(d)}(g) = \sum_{\sum_{k=1}^{d} k\cdot i_k = d} \frac{\chi^{i_1}(g) \cdots \chi^{i_d} \left( g^d \right)}{1^{i_1}i_1 ! 2^{i_2}i_2 ! \cdots d^{i_d} i_d !} \]
and obtains
\begin{equation}
\label{eqsat2}
R_d = \int_{\Gamma} \chi_{(d)}(g) \chi(g) dg.
\end{equation}
Note that in the case of a finite group the integral becomes a normed sum. To compute single Molien coefficients $r_d$ for the invariant polynomials there exists a similar formula that can easily be derived from the calculations in zhilinskii1989tensor:
\[ r_d = \int_{\Gamma} \chi_{(d)}(g) dg .\]
For the bifurcation analysis we are only interested in the equivariant structure and we will see later that we only need the data for degrees up to $d=3$ (see Section <ref>). For these cases we can apply formula (<ref>) with reasonable effort. In the case of an absolutely irreducible representation, the only linear maps commuting with the group action are multiples of the identity, therefore we immediately obtain $R_1 = 1$. Furthermore $\chi_{(2)}$ reads
\[ \chi_{(2)} (g) = \frac{1}{2} \left( \chi(g^2) + \chi^2 (g)\right). \]
To calculate $\chi_{(3)}$ using $ i_1 + 2i_2 + 3i_3 = 3$ we have the choices $(3,0,0), (1,1,0)$ and $(0,0,1)$ for $(i_1, i_2, i_3)$. Therefore we get
\[ \chi_{(3)} (g) = \frac{1}{3!}\chi^3(g) + \frac{1}{2} \chi(g)\chi(g^2) + \frac{1}{3} \chi(g^3). \]
We want to use formula (<ref>) to calculate $R_2$ and $R_3$ for the groups . Let $a \in \A$ and $b \in 2\NN+1$ with $\gcd(a,b)=1$ be natural numbers as before. In a first step we investigate the character $\chi: \Gab \to \RR$ for an arbitrary element $g \in \Gab$. It is very useful to notice that the character is a class function, i.e. it is invariant under conjugation. We have seen that $g$ can be written in the form
\[ g = C^{k_1} D^{k_2} Q^{l_1} S^{l_2} V^m \]
with $k_1 \in \rkr{a}, k_2 \in \rkr{b}, l_1 \in \rkr{4}, l_2 \in \rkr{2}$ and $m \in \rkr{2}$ (see (<ref>)). Recall how is constructed from and note that $C, D, Q$ and $S$ keep the two -blocks intact. This yields that every element $g$ with $m=1$ is of the form
\[ g = \begin{pmatrix}
0 & * \\
* & 0
\end{pmatrix} \]
and therefore $\chi (g) = 0$. Hence we may restrict to the case $m=0$:
\[ g = C^{k_1} D^{k_2} Q^{l_1} S^{l_2}. \]
These elements are of the form
\begin{equation}
\label{bigblock}
g = \begin{pmatrix}
h & 0 \\
0 & h'
\end{pmatrix}
\end{equation}
with $h, h' \in \Hab$ (in matrix representation) and for their character we obtain
\[ \chi (g) = \chi_4 (h) + \chi_4 (h') \]
where $\chi_4 \colon \Hab \to \RR$ denotes the character of the $4$-dimensional representation of . Investigating this character provides us with the needed result. To do so we make use of both the biquaternionic and the matrix representation of .
Similar to the form of the group element $g \in \Gab$ we may characterize
\[ h = c^{k_1} d^{k_2} q^{l_1} s^{l_2} \]
for $h \in \Hab$ (see (<ref>)). In a similar manner as before we obtain
\[ h = \begin{pmatrix}
0 & * \\
* & 0
\end{pmatrix} \]
if $l_1 + l_2$ is odd and therefore $\chi_4 (h) = 0$ in this case. Consider $l_1 = 3$ and $l_2 = 1$. In Lemma <ref> we have seen that elements of this form are conjugate to either $qs = \diag{(1,-1,1,-1)}$ or $-qs$ and therefore $\chi_4 (h) = 0$. For $l_1 = 1$ and $l_2=1$ we have
\[ h = c^{k_1} d^{k_2} q s = - c^{k_1} d^{k_2} q^{3} s \]
and by linearity of the character $\chi_4 (h) = 0$ as well. The remaining two cases are $l_1 \in \lbrace 0,2 \rbrace$ and $l_2 = 0$. Once more note that
\[ c^{k_1} d^{k_2} q^2 = - c^{k_1} d^{k_2} = -h \]
for $h = c^{k_1} d^{k_2}$ and we may make use of the linearity again. The matrix $h$ corresponds to the group element $\left[ e_a^{k_1}, e_b^{k_2} \right]$ and we compute it to be
\[ h =
\begin{pmatrix}
\rr\left(\left(\frac{k_1}{a} - \frac{k_2}{b} \right) \pi\right) & 0 \\
0 & \rr\left(\left(\frac{k_1}{a} + \frac{k_2}{b} \right) \pi\right)
\end{pmatrix}
. \]
From now on let
\[ \eta = \frac{k_1}{a} \pi \quad \text{and} \quad \nu = \frac{k_2}{b} \pi. \]
Then we obtain
\[ \chi_4 (h) = 4 \cos \left(\eta \right)\cos \left(\nu \right). \]
Summarizing this yields the only nonzero cases for $l_1 \in \lbrace 0,2 \rbrace$ and $l_2 = 0$ giving
\[ \chi_4 (h) = (-1)^{\frac{l_1}{2}} 4 \cos \left(\eta \right)\cos \left(\nu \right). \]
Returning back to $g \in \Gab$ with $m=0$ and using the block structure (<ref>) we obtain
\begin{align*}
h &= c^{k_1} d^{k_2} q^{l_1} s^{l_2} \\
h' &= c^{\rho k_1} d^{k_2} q^{3 l_1} s^{l_2}
\end{align*}
for the -blocks. We investigate the same cases for the powers as before. If $l_1 + l_2$ is odd, then so is $3l_1 + l_2$ and therefore $\chi_4 (h') = 0$ giving $\chi (g) = 0$. If $l_2=1$ and $l_1$ is odd then so is $3l_1$ and in the same manner we obtain $\chi (g) = 0$. For $l_2 = 0$ and $l_1$ even we obtain
\[ 3l_1 = \begin{cases}
0 &\quad \text{for} \quad l_1 = 0, \\
6 = 2 \mod 4 &\quad \text{for} \quad l_1 = 2.
\end{cases} \]
This yields
\[ \chi (g) = (-1)^{\frac{l_1}{2}} 4 \left( \cos \left(\eta \right) + \cos \left(\rho \eta \right) \right) \cos \left(\nu \right) \]
if $l_1 \in \lbrace 0,2 \rbrace$ and $l_2=0$ and $\chi (g) = 0$ in all other cases.
Knowing the character for every element $g \in \Gab$ allows us to calculate the quantities $\chi_{(d)}$. Note that we only need them for group elements with $\chi (g) \ne 0$ because of the corresponding factor in the dimension formula (<ref>). To perform these calculations for $d=2, 3$ we still need to consider $\chi\left(g^d\right)$. Note that for $l_1 \in \lbrace 0,2 \rbrace$ and $l_2=m=0$ we obtain
\[ g^2 = C^{2 k_1} D^{2 k_2} Q^{2 l_1} = C^{2 k_1} D^{2 k_2}, \]
using the relations on the generating elements, and therefore
\[ \chi (g^2) = 4 \left( \cos \left(2 \eta \right) + \cos \left(2 \rho \eta \right) \right) \cos \left(2 \nu \right). \]
In an analogue way we obtain
\[ g^3 = C^{3 k_1} D^{3 k_2} Q^{l_1} \]
and therefore
\[ \chi (g^3) = (-1)^{\frac{l_1}{2}} 4 \left( \cos \left(3 \eta \right) + \cos \left(3 \rho \eta \right) \right) \cos \left(3 \nu \right). \]
We can then put the parts together to obtain $\chi_{(d)}$ for $d=2,3$ which we use to calculate $R_2$ and $R_3$. The remaining steps are a subtle computation using calculation rules for cosine and the geometric sum formula. The details shall be omitted at this point but can be found in the Appendix (<ref>). Performing the calculations we obtain:
The dimensions $R_d=\dim P_{\Gab}^d\left(\RR^8, \RR^8 \right)$ for $d=1,2,3$ are
\begin{align*}
R_1 &= 1, \\
R_2 &= 0, \\
R_3 &= \begin{cases}
8 &\quad \text{for} \quad a=5, \\
5 &\quad \text{else}.
\end{cases}
\end{align*}
§.§ Equivariant maps in the case $\boldsymbol{a=5}$ and $\boldsymbol{b=3}$
We want to determine the equivariant structure up to third degree for the smallest group we can construct with the method presented in Sections <ref> and <ref> which is $G_{5,3}$. The groups in the family $\GG_5$ form a special case in our considerations as we have seen from the calculations of the Molien coefficients. We point out when this is important in a remark at the end of the section. By irreducibility we already know that the only linear equivariants are scalar multiples of the identity. Furthermore we have no quadratic $G_{5,3}$-symmetric maps on $\RR^8$, since $R_2=0$ and the space of cubic equivariants is $8$-dimensional.
There are several ways to find equivariant maps of a given degree. sattinger1979group investigates some simple examples. lauterbach2010do describe methods for groups that are constructed in a similar way as the ones we consider using complex polynomials. More general results and computer algebra systems can be found in gatermann1999computer, gatermann1991software and gatermann1996grobner. We have chosen an elementary method to calculate a basis using general homogeneous polynomials and having Maple <cit.> solve for the coefficients under the assumption of equivariance with respect to the generating matrices. We obtain eight linearly independent polynomial maps $E_1, \ldots ,E_8$ that prove to meet the symmetry condition. They can be found in the Appendix in <ref>.
§.§ The general case
We want to use the results for the case $a=5, b=3$ to obtain the full picture for all groups. By construction of the groups it follows that the only dependence on the parameters $a$ and $b$ is in the matrices $C(a)$ and $D(b)$. A short calculation shows that the vector fields $E_1, \ldots , E_5$ remain equivariant with respect to the matrices $C(a)$ and $D(b)$ with arbitrary $a \in \A$ and $b \in 2\NN+1$ such that $\gcd(a,b)=1$. We may even prove that $E_1, \ldots, E_5$ are equivariant with respect to a matrix $D(\psi)$ that describes an arbitrary angle of rotation $\psi \in S^1$. This gives us the final result on the equivariant structure and hence completes the proof of <ref>.
* The dimension of $P_{\Ga}^d\left(\RR^8, \RR^8 \right)$ is at most the dimension of $P_{\Gab}^d\left(\RR^8, \RR^8 \right)$ with $\Gab \le \Ga$. As a consequence we obtain $P_{\Ga}^3\left(\RR^8, \RR^8 \right) = P_{\Gab}^3\left(\RR^8, \RR^8 \right)$.
* A similar statement holds true for the matrices $C$. We define the matrix $c(\phi)$ to be the representing matrix of $[e^{\phi \ii},1]$ and $\tilde{C}(\phi, \phi')$ as the diagonal blockmatrix of $c(\phi), c(\phi')$ for arbitrary distinct angles $\phi, \phi' \in S^1$. This leads to a compact $3$-dimensional Lie group
\[ \tilde{\mathbf{G}} = \left\langle Q, S, V, \tilde{C}(\phi, \phi'), D(\psi) \mid \phi, \phi', \psi \in S^1 \right\rangle. \]
It is easy to see that $E_1, \ldots, E_5$ are equivariant with respect to $\tilde{\mathbf{G}}$. Therefore the space $P_{\tilde{\mathbf{G}}}^3\left(\RR^8, \RR^8 \right)$ is also generated by these vectorfields. But as mentioned before $\tilde{\mathbf{G}}$ is not obtained from the closure of the union of all .
* The Lie group $\tilde{\mathbf{G}}$ contains the matrices $C(a)$ and $D(b)$ for arbitrary values of $a$ and $b$. This could have served as proof for the -equivariance of $E_1 ,\ldots, E_5$.
* At first it may appear odd that the number of linearly independent cubic equivariant polynomials is different in the case $a=5$. But from equivariance with respect to the Lie group $\tilde{\mathbf{G}}$ it follows that the dimension of cubic equivariants has to become stationary for some value of $a$. This occurs at the first step from $a=5$ to $a=13$. Therefore, when investigating -symmetric dynamical systems, we need to take care of the case $a=5$ separately.
§ GENERIC SYMMETRY BREAKING BIFURCATIONS
In this section we want to investigate bifurcation problems on $\RR^8$ which are symmetric with respect to the groups that we have constructed before. In order to do so, we use methods proposed by field1996symmetry, field2007dynamics and field1990symmetry. The authors use techniques from equivariant transversality to develop a complete geometric theory on equivariant dynamics. It allows us, similar to the equivariant branching lemma, to obtain results on bifurcations in generic equations that are symmetric with respect to a given representation. The basic principle is that it suffices to investigate Taylor expansions up to some critical degree to gather information on the dynamical behaviour. Since these polynomials are equivariant as well, we can apply methods from invariant theory to calculate possible terms in the expansion, which is what we have done in the previous section using formula (<ref>). The authors even prove that we can always find such a critical degree $d$ in which the branching of solutions is fully determined. We say that the equivariant bifurcation problems are $d$-determined. However we will not go that far here, as we see that the cubic truncation suffices to prove the bifurcation result. For this reason we do not try to establish determinacy statements. In our case we can apply a polar blowing-up technique from the texts mentioned above to find a nontrivial branch of solutions bifurcating off the trivial one. All the methods used in this section are formulated in Chapter 4 of field2007dynamics, where we can also find the technical details that we partly omit here. Furthermore we use a slight modification of this approach which respects the restriction on fixed point spaces of isotropy subgroups. This will be pointed out explicitly when we make use of it. As the equivariant structure forms a special case for $a=5$ (compare to the previous section), we restrict ourselves to $a \in \A$ with $a > 5$ for the rest of this section.
§.§ Normalized families of equivariant vector fields
Following the notation of field2007dynamics we let ${\mathcal{V}\left(\RR^8,G\right) = \CG}$ be the set of smooth $G$-equivariant vector fields, for $G\in \GG_a$ and $5<a\in \A$, depending on a real parameter. The action of $G$ on the product space is defined to be only on the first component. We equip the function space with the $C^\infty$-topology and subsets with the induced topology. For $f\in \mathcal{V}\left(\RR^8, G \right)$ we define the $1$-parameter family $\lbrace f_\lambda \rbrace_\lambda$ of smooth $G$-symmetric vector fields on $\RR^8$ by $f_\lambda = f( \cdot, \lambda)$. By equivariance we get
\begin{align*}
f(0,\lambda) &= 0 \quad \text{for every } \lambda\in\RR,\\
D_1f(0,\lambda) & = \sigma_f(\lambda) \mathbbm{1} _8
\end{align*}
with $\sigma_f \in C^\infty\left(\RR\right)$. This set of zeros will be called the branch of trivial zeros and we are looking for solution branches bifurcating off this branch as we vary $\lambda$. As long as $\sigma_f(\lambda) \ne 0$ we can use the implicit function theorem to obtain a neighbourhood $U$ of $(0,\lambda)$ such that the only zeros in $U$ are trivial. We are therefore interested in points $\lambda_0 \in \RR$ with $\sigma_f(\lambda_0)=0$ to find nontrivial solutions. Generically in such a point $f$ will satisfy $\sigma_f'(\lambda_0) \ne 0$ which we will assume from now on. Furthermore we can assume $\lambda_0 = 0$ without loss of generality and use the inverse function theorem to reparametrize $\lambda$ so that $\sigma_f(\lambda) = \lambda$ for $\lambda$ near $0$. The extension of $\sigma_f$ to all the real numbers in the same manner does not impose a loss of generality, since we are only interested in branching close to the trivial solution. These considerations motivate the restriction to the closed affine linear subspace
\[ \VV = \VV \left(\RR^8, G\right) = \left\lbrace f \in \mathcal{V} \left( \RR^8, G \right) \mid \sigma_f
(\lambda) = \lambda, \lambda \in \RR \right\rbrace \]
of normalized families of smooth $G$-equivariant vector fields on $\RR^8$. For $f\in \VV$ we may write
\[ f_\lambda (x) = f(x,\lambda) = \lambda x + F_\lambda(x)\]
using Taylor's theorem, to which we refer as normalized bifurcation problem.
§.§ Nonradial equivariant polynomial maps
To follow the methods of Field the next step is to find $G$-equivariant polynomial maps on $\RR^8$ which are nonradial. A polynomial map $P$ is called radial if it is of the form
\[ P = p \mathbbm{1}_8 \]
where $p \colon \RR^8 \to \RR$ is an invariant polynomial. We call $d=d(G,\RR^8)$ the smallest degree in which nonradial equivariant polynomial maps exist. We have seen before from the Molien coefficients (Section <ref>) and <ref> in the Appendix that $d = 3$. For $f\in \VV$ let $R$ be the Taylor polynomial of $f_0$ of order $d$ at the origin. Using Taylor's theorem we obtain
\begin{equation*}
\label{normfam}
f(x,\lambda) = \lambda x + R(x) + F_1 (x) + \lambda F_2 (x,\lambda)
\end{equation*}
with $F_1(x) = f(x,0)-R(x) = \mathcal{O}(\|x\|^{d+1})$ and $F_2(x,\lambda) = f(x,\lambda) - R(x) - F_1(x) = \mathcal{O}(\|x\|^d)$ (see field2007dynamics).
It is a well known fact (that can be recalled from the results of Chapter 5 in chossat2000methods) that the homogeneous terms in the Taylor expansion of an equivariant vector field are equivariant as well. The linear part $\lambda \mathbbm{1}_8$ of the Taylor polynomial of $f_\lambda$ vanishes for $\lambda = 0$. If we look again at the Molien coefficients, we find that there are no quadratic $G$-equivariant polynomial maps. Therefore $R \in P_{G}^3\left(\RR^8, \RR^8 \right)$ which is generated by $E_1, \ldots , E_5$ and hence $R$ must be of the form
\[ R= \alpha E_1 + \beta E_2 + \gamma E_3 + \delta E_4 + \epsilon E_5 \quad \text{with} \quad \alpha, \beta, \gamma, \delta, \epsilon \in \RR. \]
§.§ Phase vector fields and fixed point subspaces
As mentioned in the introduction to this section, the major technical tool to find nontrivial solution branches is a polar blowing-up technique. Using general polar coordinates we can decompose the normalized function $f\in \VV$ into a spherical part – a smooth vector field on the unit sphere – and a radial part perpendicular to the sphere. A suitable solution for the spherical part with the radial coordinate $0$ can then be generalized to other radial values, using the implicit function theorem, leading to nontrivial solutions. But first, we adapt Field's method in such a way, that it applies in fixed point spaces of isotropy subgroups. In our case this reduces the dimension by four, which is convenient for the following computations.
As we have seen before (Theorem <ref>) each group $G \in \GG_a$ has precisely one nontrivial isotropy type $[K]$ which contains the conjugate subgroups of
\[ K = \left\langle QS \right\rangle = \left\langle \diag{1,-1,1,-1,-1,1,-1,1} \right\rangle.\]
The subgroup $K$ obviously fixes elements of the subspace
\[ \fix{K} = \lbrace x \in \RR ^8 \mid x_2=x_4=x_5=x_7=0 \rbrace \cong \RR^4.\]
Utilizing the symmetry property of $f$, it therefore suffices to consider $\fix{K}$. Denote the coordinates by
\[ y = \left(y_1, y_2, y_3, y_4 \right) \in \fix{K}. \]
By equivariance $f_\lambda$ fixes $\fix{K}$ for each $f\in \VV$:
\[ kf_\lambda(x) =f_\lambda(kx) = f_\lambda(x) \]
for $k \in K$ and $x\in \fix{K}$. Therefore $f_\lambda(x) \in \fix{K}$ for all $x\in \fix{K}$. With this we may now restrict $f_\lambda$ and $R$ to $\fix{K}$, which is the part that does not appear in the texts by Field and Field & Richardson. Since the blow-up method does not interfere with this reduction, we may perform it in the fixed point space just as well.
To investigate bifurcation behaviour in $\fix{K}$ we calculate the so called phase vector field of the cubic equivariant polynomial maps. For $R$ as before restricted to $\fix{K}$ it is defined as the vector field
\[ \pvf{R}(y) = R(y) - \left\langle R(y), y \right\rangle y \quad \text{with} \quad y \in S^3 \subset \fix{K}\]
which is the tangential component of the restriction of $R$ to the unit sphere $S^3$ in $\RR^4$. Up to a factor depending on the radial coordinate, it is equal to the spherical part of $R$. Furthermore the phase vector field $\pvf{R}$ coincides with the spherical part of $f$ if the radial coordinate is $0$, which is the starting point for the blow-up technique. Note that the phase vector field of a radial polynomial vanishes and therefore cannot provide any information on solutions of the original equation. The projection on the phase vector field is a linear map from $P_{\Gab}^3\left(\RR^4, \RR^4 \right)$ to $P_{\Gab}^5\left(S^3, S^3 \right)$ so
\[\pvf{R} = \alpha \pvf{E_1} + \beta \pvf{E_2} + \gamma \pvf{E_3} + \delta \pvf{E_4} + \epsilon \pvf{E_5}. \]
The phase vector fields of $E_1, \ldots E_5$ can be found in the appendix (see <ref>).
As we have seen in <ref> the cubic equivariants $E_1, \ldots, E_5$ are -symmetric as well and the same holds for $\pvf{E_1}, \ldots, \pvf{E_5}$. Hence they leave the fixed point spaces of isotropy subgroups of invariant. In <ref> we have proved that has subgroups with one-dimensional fixed point spaces. These intersect the sphere in two points and therefore directly lead to zeros of the phase vector fields $\pvf{E_1}, \ldots, \pvf{E_5}$. For example the group
\[ \left\langle CD\left( - \frac{\pi}{a} \right)^{\rho}, QS \right\rangle < \Ga \]
fixes the one-dimensional subspace $\langle (0, \ldots, 0, 1)^T \rangle \subset \RR^8$. This is a subspace of $\fix{K}$ as well and reads $\langle (0,0,0,1)^T \rangle$ in the corresponding coordinates. Thus $y_0 = (0,0,0,1)^T$ and $-y_0$ are common zeros of $\pvf{E_1}, \ldots, \pvf{E_5}$ and therefore $\pvf{R}(\pm y_0) = 0$ for any linear combination. The Jacobian of $\pvf{R}(y_0)$ has the eigenvalues $-\alpha + \delta, -\alpha+\gamma, -\alpha+\beta$ and $-2\alpha$.
So we see that $y_0$ is a hyperbolic zero of $\pvf{R}$ if $\alpha \ne 0$, $\alpha \ne \beta$, $\alpha \ne \gamma$ and $\alpha \ne \delta$. These conditions are met for an open and dense subset of $\RR^5$ and therefore $y_0$ is generically a hyperbolic zero for $\pvf{R}$. This allows us to start the blow-up technique which, using the implicit function theorem, provides us with nontrivial hyperbolic solutions to $f = 0$ depending on the value of the radial coordinate. These can be reformulated into a solution curve bifurcating off the trivial solution where the direction of branching is $y_0$. By construction this new branch of solutions lies in the fixed point space $\fix{K}$ meaning that the isotropy type $[K]$ is symmetry breaking. For the technical details of the blow-up method see Lemma 4.8.1. and its proof from field2007dynamics with the fact that $\langle R(y_0), y_0 \rangle = \alpha$ which is generically not zero. As we have seen, the branching of steady states occurs for a generic bifurcation problem. This completes the proof for the main theorem on bifurcations with -symmetry.
§ THE SPECIAL CASE $A=5$
To conclude the above considerations we want to briefly discuss the special case $a=5$ and point out that it is not so special after all. As we have seen in <ref> the main difference between the groups $G_{5,b}$ and (for admissible values of $b$) lies in the structure of equivariant polynomial maps. This in turn influences the argumentation to prove the bifurcation result. In <ref> we have computed the space of equivariant cubic polynomial maps $P_{G_{5,b}}^3 \left(\RR^8, \RR^8 \right)$ to be generated by the maps $E_1, \ldots, E_8$ (see <ref>). Furthermore, in <ref>, we have seen that the corresponding spaces are generated by $E_1,\ldots,E_5$ whenever $a>5$. A major aspect for the proof of the bifurcation result is the fact that these maps are equivariant not only with respect to the groups but also with respect to the Lie groups . A short calculation shows that this holds true for the maps $E_6, E_7$ and $E_8$ as well. But, on the contrary to the case $a>5$, the additional maps are not equivariant with respect to the largest Lie group $\tilde{\mathbf{G}}$ that we have considered.
Nevertheless we may use the same technique to investigate the bifurcation behavior in the presence of $G_{5,b}$ symmetry as before. We only sketch the proof here since it is completely analog to the one before. We consider the cubic truncation of a normalized bifurcation problem
\[
R = \sum_{i=1}^{8} \alpha_i E_i \quad \text{with} \quad a_i \in \RR
\]
and restrict to the fixed point subspace
\[ \fix{K} = \lbrace x \in \RR ^8 \mid x_2=x_4=x_5=x_7=0 \rbrace \cong \RR^4.\]
Then we consider the corresponding phase vector field
\[ \pvf{R}(y) = R(y) - \left\langle R(y), y \right\rangle y \quad \text{with} \quad y \in S^3 \subset \fix{K}\]
which by the same argumentation as before – one-dimensional fixed point space of an isotropy subgroup of $\mathbf{G}_5$ – has the zero $y_0 = (0,0,0,1)^T$. This is once again generically hyperbolic and thus we can apply the polar blowing up method to obtain a branch of zeros for the bifurcation equation bifurcating off the trivial solution.
Summing up we see that the case $a=5$, even though it has to be treated separately, does not imply significant differences. It provides the same bifurcation result which can be proved using the same techniques. The main point of interest lies in the structure of the equivariant maps as $P_{\Gab}^3\left(\RR^8, \RR^8 \right) \subset P_{G_{5,b}}^3\left(\RR^8, \RR^8 \right)$ as a proper subspace.
§ FURTHER GROUPS AND EVEN DIMENSIONAL REPRESENTATIONS
In this paper we have constructed groups of order $16\m$ – here $\m=a\cdot b$ with $a,b \ne 1$, relatively prime, odd and $a>5$ being a product of prime powers of the form $1 \mod 4$ – with an $8$-dimensional absolutely irreducible representation that provide counterexamples to the Ize conjecture. Furthermore lauterbach2014equivariant describes groups of order $64+128\ell$ with $\ell\in \NN$ with the same purpose. In both cases there are generically symmetry breaking isotropy types. But comparing these results to the GAP-calculations provided by in lauterbach2010do we still expect a vast number of counterexamples to the Ize conjecture in $8$ dimensions that have not yet been investigated systematically. It is an open task to find a reasonable ordering for the groups in terms of their orders and to obtain information on the dynamics in their $8$-dimensional representations.
In order to do so, a first step could be to slightly adapt the construction of in such a way that we define the $8\times8$ generating matrix $Q$ to be
\[ Q = \begin{pmatrix}
q & 0 \\
0 & q
\end{pmatrix}\]
instead of the second block being $-q$. We obtain two isotropy types in this case for which it would be interesting to determine whether both of them are generically symmetry breaking. Another task is to determine the role of the $3$-dimensional Lie group containing all the . This has not yet been sufficiently investigated.
Furthermore we see from Tables 5-10 in lauterbach2010do that there are further groups acting absolutely irreducibly in dimensions $4,8,12,16$ and $20$ that appear to lead to counterexamples to the Ize conjecture but none with the same property in dimensions $2,6,10,14$ and $18$ (at least for small group orders). The authors formulate the conjecture: “For dimensions $N = 0 \mod 4$, there are infinitely many groups acting absolutely irreducibly on $\RR^N$ that have no isotropy subgroups with odd-dimensional fixed point spaces. But for dimensions $N = 2 \mod 4$, there are no such groups”. There is some evidence for this conjecture to be true as the GAP calculations do not provide counterexamples in the second case for groups of order up to $1000$. Furthermore ruan2011fixed proves the claim for dimension $6$ under the mild additional assumption that the groups are solvable.
These intermediate steps and the conjecture of lauterbach2010do would provide some major insight in the question if absolute irreducible group actions lead to generically symmetry breaking isotropy types. This interpretation of the Ize conjecture from the dynamical systems point of view would be a significant contribution to the understanding of bifurcations in the presence of symmetry. However such a general statement is still far from being proved.
R.L. would like to thank U. Kühn for some helpful discussions on modular
§ APPENDIX
§.§ Calculations of Molien coefficients
In this section we want to fill the gaps that were left in Section <ref> in the calculations of
\[ R_d = \frac{1}{\left| \Gab \right|} \sum_{g \in \Gab} \chi_{(d)} (g) \chi (g) \]
for $d=2,3$. First of all let
\[ \eta = \frac{k_1}{a} \pi \quad \text{and} \quad \nu = \frac{k_2}{b} \pi \]
and note that
\begin{equation*}
\chi_{(2)} (g) = 2 \left( \cos \left( 2 \eta \right) + \cos \left(2 \rho \eta \right) \right) \cos \left(2 \nu \right) %\\
+ 8 \left( \cos \left(\eta \right) + \cos \left(\rho \eta \right) \right)^2 \cos \left(\nu \right)^2
\end{equation*}
which only depends on $k_1$ and $k_2$. Remember that the only nonzero terms occur for $l_1 \in \lbrace 0,2 \rbrace$ and $l_2 = m = 0$. This allows us to calculate
\[
R_2 = \frac{1}{16ab} \sum_{k_1 = 0}^{a-1} \sum_{k_2 = 0}^{b-1} \sum_{l_1 \in \lbrace 0,2 \rbrace } (-1)^{\frac{l_1}{2}} 4 \chi_{(2)}(g) \left( \cos \left(\eta \right) + \cos \left(\rho \eta \right) \right) \cos \left(\nu \right).
\]
Because of the factor $(-1)^{\frac{l_1}{2}}$ whereas the rest of each summand is independent of $l_1$ these two summands cancel each other. This directly yields
There are no quadratic equivariant maps for the $8$-dimensional representation of :
\[R_2 = 0. \]
The case $d=3$ is much more complicated. We compute
\begin{align*}
\chi_{(3)} (g)
&= (-1)^{\frac{l_1}{2}} \frac{32}{3} \left( \cos \left(\eta \right) + \cos \left(\rho \eta \right) \right)^3
\cos \left(\nu \right)^3 \\
&\phantom{=} \mathrel{+} (-1)^{\frac{l_1}{2}} 8 \left( \cos \left(2 \eta \right) + \cos \left(2 \rho \eta \right) \right)
\left( \cos \left(\eta \right) + \cos \left(\rho \eta \right) \right) \cdot \cos \left(2\nu \right) \cos \left(\nu \right) \\
&\phantom{=} \mathrel{+} (-1)^{\frac{l_1}{2}} \frac{4}{3} \left( \cos \left(3 \eta \right) + \cos \left(3 \rho \eta \right) \right)
\cos \left(3\nu \right).
\end{align*}
This depends on $l_1$ therefore the terms do not cancel out as easily as in the case $d=2$. To be able to compute $R_3$, we state two technical lemmas first.
Let $w \in \NN$ and $l \in \ZZ$. Then
\[ \sum_{k=0}^{w-1} \cos 2l\frac{k}{w} \pi =
\begin{cases}
0 &\quad \text{for} \quad w \nmid l , \\
w &\quad \text{else}.
\end{cases} \]
We want to use this lemma to calculate sums of such cosine terms that contain an even factor in front of $(k/w) \pi$. We have to distinguish whether this factor is an integer multiple of $2w$. The following lemma performs this distinction in the occurring cases.
Let $a \in \A$ and $\rho$ be chosen as in Proposition <ref> and odd (compare to the construction of ). Then
* $\rho-1 \ne 0 \mod 2a$;
* $2(\rho-1) \ne 0 \mod 2a$;
* $\rho+1 \ne 0 \mod 2a$;
* $2(\rho+1) \ne 0 \mod 2a$;
* $2\rho \ne 0 \mod 2a$;
* $4\rho \ne 0 \mod 2a$;
* $\rho-3 = 0 \mod 2a$ if and only if $a = 5$ and $\rho=3$;
* $3\rho-1 \ne 0 \mod 2a$;
* $\rho+3 \ne 0 \mod 2a$;
* $3\rho+1 = 0 \mod 2a$ if and only if $a = 5$ and $\rho=3$.
* Suppose $\rho -1 = 0 \mod 2a$. Then $\rho = 1 \mod a$ and $\rho^2=1 \mod a$ which is a contradiction to the choice of $\rho$.
* Suppose $2(\rho -1) = 0 \mod 2a$. Then $\rho -1 = 0 \mod a$ and $\rho = 1 \mod a$. The contradiction follows as before.
* Suppose $\rho + 1 = 0 \mod 2a$. Then $\rho = -1 \mod a$ and $\rho^2 = 1 \mod a$ which is again a contradiction.
* Suppose $2(\rho +1) = 0 \mod 2a$. Then $\rho +1 = 0 \mod a$ and $\rho = -1 \mod a$. This is a contradiction as before.
* Suppose $2 \rho = 0 \mod 2a$. Then $\rho = 0 \mod a$ which contradicts the choice of $\rho$.
* Suppose $4 \rho = 0 \mod 2a$. Then $2\rho = 0 \mod a$ and therefore $4 \rho^2 = 0 \mod a$. But $\rho$ was chosen so that $4 \rho^2 = -4 \mod a$. This is impossible, since $a$ is odd.
* Suppose $\rho - 3 = 0 \mod 2a$. Then $\rho = 3 \mod a$ and $\rho^2 = 9 \mod a$. But $\rho^2 = -1 \mod a$ so $9 = -1 \mod a$. The only choice is $a = 5$, since $a$ is odd. The corresponding odd $\rho$ is $3$.
* Suppose $3 \rho -1 = 0 \mod 2a$. Then $3\rho = 1 \mod a$ and $\rho = 3\rho^2 = -3 \mod a$. This yields $\rho^2 = 9 \mod a$ and as before $a=5$. Then $\rho = -3 = 2 \mod a$ which is not odd and therefore a contradiction.
* Suppose $\rho + 3 = 0 \mod 2a$. Then $\rho = -3 \mod a$ and the contradiction follows as before.
* Suppose $3 \rho + 1 = 0 \mod 2a$. Then $3 \rho = -1 \mod a$ and therefore $\rho = -3 \rho^2 = 3 \mod a$. As before we obtain $a = 5$ and $\rho =3$.
These two technical lemmas together with multiple applications of the calculation rules for cosine allow us to compute $R_3$ explicitly and we obtain
There are eight linearly independent cubic equivariant maps for the $8$-dimensional representation of if $a=5$ and five linearly independent cubic equivariant polynomial maps for all other $a \in \A$:
\[R_3 = \begin{cases}
8 &\quad \text{for} \quad a=5, \\
5 &\quad \text{else}. \\
\end{cases} \]
§.§ Cubic equivariant maps
$ E_1(x) = \begin{pmatrix}
\left(x_1^2 + x_2^2 \right) x_1 \\
\left(x_1^2 + x_2^2 \right) x_2 \\
\left(x_3^2 + x_4^2 \right) x_3 \\
\left(x_3^2 + x_4^2 \right) x_4 \\
\left(x_5^2 + x_6^2 \right) x_5 \\
\left(x_5^2 + x_6^2 \right) x_6 \\
\left(x_7^2 + x_8^2 \right) x_7 \\
\left(x_7^2 + x_8^2 \right) x_8
\end{pmatrix} $
$ E_5(x) = \begin{pmatrix}
-x_3x_5x_7 - x_3x_6x_8 - x_4x_5x_8 + x_4x_6x_7 \\
x_3x_5x_8 - x_3x_6x_7 - x_4x_5x_7 - x_4x_6x_8 \\
-x_1x_5x_7 - x_1x_6x_8 + x_2x_5x_8 - x_2x_6x_7 \\
-x_1x_5x_8 + x_1x_6x_7 - x_2x_5x_7 - x_2x_6x_8 \\
x_1x_3x_7 + x_1x_4x_8 - x_2x_3x_8 + x_2x_4x_7 \\
x_1x_3x_8 - x_1x_4x_7 + x_2x_3x_7 + x_2x_4x_8 \\
x_1x_3x_5 - x_1x_4x_6 + x_2x_3x_6 + x_2x_4x_5 \\
x_1x_3x_6 + x_1x_4x_5 - x_2x_3x_5 + x_2x_4x_6
\end{pmatrix} $
$ E_2(x) = \begin{pmatrix}
\left(x_3^2 + x_4^2 \right) x_1 \\
\left(x_3^2 + x_4^2 \right) x_2 \\
\left(x_1^2 + x_2^2 \right) x_3 \\
\left(x_1^2 + x_2^2 \right) x_4 \\
\left(x_7^2 + x_8^2 \right) x_5 \\
\left(x_7^2 + x_8^2 \right) x_6 \\
\left(x_5^2 + x_6^2 \right) x_7 \\
\left(x_5^2 + x_6^2 \right) x_8
\end{pmatrix} $
$ E_6(x) = \begin{pmatrix}
-x_1x_3x_6 + x_1x_4x_5 + x_2x_3x_5 + x_2x_4x_6 \\
x_1x_3x_5 + x_1x_4x_6 + x_2x_3x_6 - x_2x_4x_5 \\
-x_1x_3x_8 + x_1x_4x_7 + x_2x_3x_7 + x_2x_4x_8 \\
x_1x_3x_7 + x_1x_4x_8 + x_2x_3x_8 - x_2x_4x_7 \\
x_3x_5x_8 + x_3x_6x_7 + x_4x_5x_7 - x_4x_6x_8 \\
x_3x_5x_7 - x_3x_6x_8 - x_4x_5x_8 - x_4x_6x_7 \\
-x_1x_5x_8 - x_1x_6x_7 - x_2x_5x_7 + x_2x_6x_8 \\
-x_1x_5x_7 + x_1x_6x_8 + x_2x_5x_8 + x_2x_6x_7
\end{pmatrix} $
$ E_3(x) = \begin{pmatrix}
\left(x_5^2 + x_6^2 \right) x_1 \\
\left(x_5^2 + x_6^2 \right) x_2 \\
\left(x_7^2 + x_8^2 \right) x_3 \\
\left(x_7^2 + x_8^2 \right) x_4 \\
\left(x_3^2 + x_4^2 \right) x_5 \\
\left(x_3^2 + x_4^2 \right) x_6 \\
\left(x_1^2 + x_2^2 \right) x_7 \\
\left(x_1^2 + x_2^2 \right) x_8
\end{pmatrix} $
$ E_7(x) = \begin{pmatrix}
-2x_5x_7x_8 - x_6x_7^2 + x_6x_8^2 \\
-x_5x_7^2 + x_5x_8^2 + 2x_6x_7x_8 \\
x_5^2x_8 + 2x_5x_6x_7 - x_6^2x_8 \\
x_5^2x_7 - 2x_5x_6x_8 - x_6^2x_7 \\
x_1^2x_4 + 2x_1x_2x_3 - x_2^2x_4 \\
-x_1^2x_3 + 2x_1x_2x_4 + x_2^2x_3 \\
2x_1x_3x_4 + x_2x_3^2 - x_2x_4^2 \\
-x_1x_3^2 + x_1x_4^2 + 2x_2x_3x_4
\end{pmatrix} $
$ E_4(x) = \begin{pmatrix}
\left(x_7^2 + x_8^2 \right) x_1 \\
\left(x_7^2 + x_8^2 \right) x_2 \\
\left(x_5^2 + x_6^2 \right) x_3 \\
\left(x_5^2 + x_6^2 \right) x_4 \\
\left(x_1^2 + x_2^2 \right) x_5 \\
\left(x_1^2 + x_2^2 \right) x_6 \\
\left(x_3^2 + x_4^2 \right) x_7 \\
\left(x_3^2 + x_4^2 \right) x_8
\end{pmatrix} $
$ E_8(x) = \begin{pmatrix}
x_3^2x_8 - 2x_3x_4x_7 - x_4^2x_8 \\
-x_3^2x_7 -2x_3x_4x_8 + x_4^2x_7 \\
x_1^2x_6 - 2x_1x_2x_5 - x_2^2x_6 \\
-x_1^2x_5 - 2x_1x_2x_6 + x_2^2x_5 \\
2x_1x_7x_8 + x_2x_7^2 - x_2x_8^2 \\
x_1x_7^2 - x_1x_8^2 - 2x_2x_7x_8 \\
-2x_3x_5x_6 - x_4x_5^2 + x_4x_6^2 \\
-x_3x_5^2 + x_3x_6^2 + 2x_4x_5x_6
\end{pmatrix} $
Cubic quivariant maps $E_1, \ldots, E_8$ for $G_{5,3}$.
§.§ Phase vector fields
$ \pvf{E_1} (y) = \begin{pmatrix} -y_{1}\, \left( {y_{1}}^{4}+{y_{2}}^{4}+{y_{
3}}^{4}+{y_{4}}^{4}-{y_{1}}^{2} \right) \\ \noalign{\medskip}-y_{2}\, \left( {y_{1}}^{4}+{y_{2}}^{4}+{y_{3}}^{
4}+{y_{4}}^{4}-{y_{2}}^{2} \right) \\ \noalign{\medskip}-y_{3}\, \left( {y_{1}}^{4}
+{y_{2}}^{4}+{y_{3}}^{4}+{y_{4}}^{4}-{y_{3}}^{2} \right)
\\ \noalign{\medskip}-y_{4}\, \left( {y_{1}}^{4}
+{y_{2}}^{4}+{y_{3}}^{4}+{y_{4}}^{4}-{y_{4}}^{2} \right) \end{pmatrix} $
$ \pvf{E_2} (y) = \begin{pmatrix} -y_{1}\, \left( 2\,{y_{1}}^{2}{y_{2}}^{2}+2
\,{y_{3}}^{2}{y_{4}}^{2}-{y_{2}}^{2} \right)
\\ \noalign{\medskip}-y_{2}\, \left( 2\,{y_{1}}^{2}{y_{2}}^{2}+2\,{y_{
3}}^{2}{y_{4}}^{2}-{y_{1}}^{2} \right) \\ \noalign{\medskip}-y_{3}\, \left( 2\,{y_{1}}^
{2}{y_{2}}^{2}+2\,{y_{3}}^{2}{y_{4}}^{2}-{y_{4}}^{2} \right)
\\ \noalign{\medskip}-y_{4}\, \left( 2\,{y_{1}}^
{2}{y_{2}}^{2}+2\,{y_{3}}^{2}{y_{4}}^{2}-{y_{3}}^{2} \right)
\end{pmatrix} $
$ \pvf{E_3} (y) = \begin{pmatrix} -y_{1}\, \left( {y_{1}}^{2}{y_{3}}^{2}+{y_{4
}^{2} \right) \\ \noalign{\medskip}-y_{2}\,
\left( {y_{1}}^{2}{y_{3}}^{2}+{y_{4}}^{2}{y_{1}}^{2}+{y_{3}}^{2}{y_{2
}}^{2}+{y_{2}}^{2}{y_{4}}^{2}-{y_{4}}^{2} \right)
\\ \noalign{\medskip}-y_{3
}\, \left( {y_{1}}^{2}{y_{3}}^{2}+{y_{4}}^{2}{y_{1}}^{2}+{y_{3}}^{2}{y
_{2}}^{2}+{y_{2}}^{2}{y_{4}}^{2}-{y_{2}}^{2} \right)
\\ \noalign{\medskip}-y_{4}\, \left( {y_{1}}^{2}
y_{4}}^{2}-{y_{1}}^{2} \right) \end{pmatrix} $
$ \pvf{E_4} (y) = \begin{pmatrix} -y_{1}\, \left( {y_{1}}^{2}{y_{3}}^{2}+{y_{1
}^{2} \right) \\ \noalign{\medskip}-y_{2}\,
\left( {y_{1}}^{2}{y_{3}}^{2}+{y_{1}}^{2}{y_{4}}^{2}+{y_{3}}^{2}{y_{2
}}^{2}+{y_{2}}^{2}{y_{4}}^{2}-{y_{3}}^{2} \right)
\\ \noalign{\medskip}-y_{3
}\, \left( {y_{1}}^{2}{y_{3}}^{2}+{y_{1}}^{2}{y_{4}}^{2}+{y_{3}}^{2}{y
_{2}}^{2}+{y_{2}}^{2}{y_{4}}^{2}-{y_{1}}^{2} \right)
\\ \noalign{\medskip}-y_{4}\, \left( {y_{1}}^{2}
y_{4}}^{2}-{y_{2}}^{2} \right) \end{pmatrix} $
$ \pvf{E_5} (y) = \begin{pmatrix} -y_{2}\,y_{3}\,y_{4}
\\ \noalign{\medskip}-y_{1}\,y_{3}\,y_{4}\\ \noalign{\medskip}y_{1}\,y_{2}\,y_{4}
\\ \noalign{\medskip}y_{1}\,y_{2}\,y_{3}
\end{pmatrix} $
Phase vector fields of $E_1, \ldots, E_5$ restricted to $S^3 \subset \fix{K}$.
|
1511.00563
|
We exhibit a family of $3$-uniform hypergraphs with the property that their $2$-colour Ramsey numbers grow polynomially in the number of vertices, while their $4$-colour Ramsey numbers grow exponentially. This is the first example of a class of hypergraphs whose Ramsey numbers show a strong dependence on the number of colours.
§ INTRODUCTION
The Ramsey number $r_k(H)$ of a $k$-uniform hypergraph $H$ is the smallest $n$ such that any $2$-colouring of the edges of the complete $k$-uniform hypergraph $K_n^{(k)}$ contains a monochromatic copy of $H$. Similarly, for any $q \geq 2$, we may define a $q$-colour Ramsey number $r_k(H; q)$.
One of the main outstanding problems in Ramsey theory is to decide whether the Ramsey number for complete $3$-uniform hypergraphs is double exponential. The best known bounds, due to Erdős, Hajnal and Rado <cit.>, state that there are positive constants $c$ and $c'$ such that
\[2^{c t^2} \leq r_3(K_t^{(3)}) \leq 2^{2^{c' t}}.\]
Paul Erdős has offered $500 for a proof that the upper bound is correct, that is, that there exists a positive constant $c$ such that $r_3(K_t^{(3)}) \geq 2^{2^{c t}}$. Some evidence that this may be true was given by Erdős and Hajnal (see, for example, <cit.>), who showed that the analogous bound holds for $4$ colours, that is, that there exists a positive constant $c$ such that $r_3(K_t^{(3)}; 4) \geq 2^{2^{c t}}$. In this paper, we show that this evidence may not be so compelling by finding a natural class of hypergraphs, which we call hedgehogs, whose Ramsey numbers show a strong dependence on the number of colours.
The hedgehog $H_t$ is the $3$-uniform hypergraph with vertex set $[t + \binom{t}{2}]$ such that for every pair $(i, j)$ with $1 \leq i < j \leq t$ there is a unique vertex $k > t$ such that $ijk$ is an edge. We will sometimes refer to the set $\{1, 2, \dots, t\}$ as the body of the hedgehog. Our main result is that the $2$-colour Ramsey number $r_3(H_t)$ grows as a polynomial in $t$, while the $4$-colour Ramsey number $r_3(H_t; 4)$ grows as an exponential in $t$.
If $H_t$ is the $3$-uniform hedgehog with body of order $t$, then
$r_3(H_t) \leq 4 t^3$,
there exists a positive constant $c$ such that $r_3(H_t; 4) \geq 2^{ct}$.
For the intermediate $3$-colour case, we show that the answer is intimately connected with a special case of the multicolour Erdős–Hajnal conjecture <cit.>. This conjecture states that for any complete graph $K$ with a fixed $q$-colouring of its edges, there exists a positive constant $c(K)$ such that any $q$-colouring (with the same $q$ colours) of the edges of the complete graph on $n$ vertices with no copy of $K$ contains a clique of order $n^{c(K)}$ which receives only $q-1$ colours. Though this conjecture is known to hold in a number of special cases (see, for example, Section 3.3 of <cit.>), the best known general result, due to Erdős and Hajnal themselves, says that there exists a positive constant $c(K)$ such that any $q$-colouring of the edges of the complete graph on $n$ vertices with no copy of $K$ contains a clique of order $e^{c(K) \sqrt{\log n}}$ which receives only $q-1$ colours.
We will be concerned with the particular case where $q = 4$ and the banned configuration $K$ is a rainbow triangle with one edge in each of the first three colours.
Definition. Let $F(t)$ be the smallest $n$ such that every $4$-colouring of the edges of $K_n$, in red, blue, green and yellow, contains either a rainbow triangle $K$, with one edge in each of red, blue and green, or a clique of order $t$ with at most $3$ colours.
We will show that $r_3(H_t;3)$ is bounded above and below by polynomials in $F(t)$ (strictly speaking, the upper bound is a polynomial in $F(t^3)$, but, provided $F(t)$ does not jump pathologically, this will be at most polynomial in $F(t)$). Since the result of Erdős and Hajnal mentioned in the previous paragraph implies that $F(t) \leq t^{c \log t}$ for some constant $c$, this in turn shows that $r_3(H_t; 3) \leq t^{c \log t}$ for some constant $c$. Moreover, the Erdős–Hajnal conjecture holds in this case if and only if there is a polynomial upper bound for $r_3(H_t;3)$.
If $H_t$ is the $3$-uniform hedgehog with body of order $t$, then
$r_3(H_t;3) =O(t^4 F(t^3)^2)$,
$r_3(H_t; 3) \geq F(t)$.
In particular, there exists a constant $c$ such that $r_3(H_t;3) \leq t^{c \log t}$.
We will prove Theorem <ref> in the next section and Theorem <ref> in Section <ref>. We conclude by discussing a number of interesting questions that arose from our work.
§ THE BASIC DICHOTOMY
In this section, we prove Theorem <ref>. We begin by proving that the $2$-colour Ramsey number of $H_t$ is at most $4 t^3$.
Proof of Theorem <ref>(i):
Let $n = 4t^3$. We will show that every red/blue-colouring of the complete $3$-uniform hypergraph on $n$ vertices contains a monochromatic copy of $H_t$. To begin, we define a partial colouring of the edges of the complete graph on the same vertex set. We will colour an edge $uv$ red if there are fewer than $\binom{t}{2} + t$ red triples containing $u$ and $v$. Similarly, we colour $uv$ blue if there are fewer than $\binom{t}{2} + t$ blue triples containing $u$ and $v$. To find a monochromatic $H_t$, it will clearly suffice to find a subset of order $t$ containing no red edge or no blue edge, since we can consider this set as the body of the hedgehog and embed the spines greedily.
We claim that no vertex is contained in $2t^2$ red edges and $2t^2$ blue edges. Suppose, on the contrary, that $u$ is such a vertex and let $V_R$ and $V_B$ be the vertices which are connected to $u$ in red and blue, respectively. Since it is easy to see that no edge can be coloured both red and blue, $V_R$ and $V_B$ are disjoint. Moreover, for each vertex $v$ in $V_R$, since $uv$ is contained in fewer than $\binom{t}{2} + t$ red triples, there are at least
\[|V_B| - \binom{t}{2} - t > \frac{|V_B|}{2}\]
vertices $w$ in $V_B$ such that $uvw$ is blue. This implies that more than half of the triples $uvw$ with $v \in V_R$ and $w \in V_B$ are blue. However, by first considering vertices $w$ in $V_B$, the same argument also shows that more than half of these triples are red, a contradiction.
We now assign a colour to each vertex in the graph, colouring it red if it is contained in fewer than $2t^2$ red edges and blue otherwise. In the latter case, the claim of the last paragraph shows that it will be contained in fewer than $2t^2$ blue edges. By the pigeonhole principle, at least half the vertices in the graph have the same colour, say red. That is, we have a subset of order at least $n/2$ such that every vertex is contained in fewer than $2t^2$ red edges. By Brooks' theorem, we conclude that this set contains a subset of order $n/4t^2$ containing no red edge. Since $n/4t^2 \geq t$, this is the required set.
We will now show that the $4$-colour Ramsey number of $H_t$ is at least $2^{ct}$ for some positive constant $c$. This is clearly sharp up to the constant in the exponent.
Proof of Theorem <ref>(ii):
A standard application of the first moment method gives a positive constant $c$ such that, for every integer $t \geq 4$, there is a $4$-colouring $\chi$ of the edges of the complete graph on $2^{ct}$ vertices with the property that every clique of order $t$ contains all $4$ colours.
We now $4$-colour the edges of the complete $3$-uniform hypergraph on the same vertex set by colouring the triple $uvw$ with any colour which is not contained within the set $\{\chi(u, v), \chi(v,w), \chi(w,u)\}$. Suppose now that there is a monochromatic copy of $H_t$ with colour $1$, say, and let $u_1, u_2, \dots, u_t$ be the body of this copy. Then, in the original graph colouring $\chi$, none of the edges $u_i u_j$ with $1 \leq i < j \leq t$ received the colour $1$. However, this contradicts the property that every set of order $t$ contains all $4$ colours.
§ THREE COLOURS AND THE ERDŐS–HAJNAL CONJECTURE
To prove Theorem <ref>(i), we require two lemmas. The first is a result of Spencer <cit.> which says that any $3$-uniform hypergraph with few edges contains a large independent set.
If $H$ is a $3$-uniform hypergraph with $n$ vertices and $e$ edges, then $\alpha(H) = \Omega(n^{3/2}/e^{1/2})$.
The second lemma we require is a result of Fox, Grinshpun and Pach <cit.> saying that the multicolour Erdős–Hajnal conjecture holds for $3$-colourings of $K_n$ with no rainbow triangle. The result we use is somewhat weaker than the main result in <cit.>, but will be more than sufficient for our purposes.
Suppose that the edges of the complete graph $K_n$ have been $3$-coloured, in red, blue and green, so that there are no rainbow triangles with one edge in each of red, blue and green. Then there is a clique of order $n^{1/3}$ containing at most two of the three colours.
We are now ready to prove Theorem <ref>(i), that $r_3(H_t;3) = O(t^4 F(t^3)^2)$.
Proof of Theorem <ref>(i):
Suppose that the edges of the complete $3$-uniform hypergraph on $n = c t^4 F(t^3)^2$ vertices have been $3$-coloured, in red, blue and green, where $c$ is a sufficiently large constant to be chosen later. We will $4$-colour the edges of the graph on the same vertex set as follows: if $u$ and $v$ are contained in fewer than $\binom{t}{2} + t$ triples of a given colour, then we give the edge $uv$ that colour, noting that an edge may receive more than one colour (but at most two). On the other hand, if an edge is not coloured with any of red, blue or green, we colour it yellow.
We claim that this colouring has at most $t^2 n^2$ triangles containing all three of the colours red, blue and green (where we include the possibility that two of these colours may appear on the same edge). To see this, note that there are at most $(\binom{t}{2} + t) \binom{n}{2}$ red triples containing a red edge. In particular, since the triangles we wish to count always contain a red edge, there are at most $(\binom{t}{2} + t) \binom{n}{2}$ of these triangles in the graph corresponding to a red triple. Since we may similarly bound the number of these triangles corresponding to blue or green triples, we see that, for $t \geq 3$, there are at most $3 (\binom{t}{2} + t) \binom{n}{2} \leq t^2 n^2$ triangles in the graph which contain all three of the colours red, blue and green, as required.
If we let $H$ be the $3$-uniform hypergraph on $n$ vertices whose edges correspond to triangles containg all three of the colours red, blue and green, Lemma <ref> now yields a subset $U$ of order $\Omega(n^{1/2}/t )$ containing no such triangle. By taking $c$ to be sufficiently large, we may assume that $U$ has order at least $t F(t^3)$.
We now consider the graph $G$ on vertex set $U$ whose edge set consists of all those edges which received two colours in the $4$-colouring defined above. If we fix a vertex $u \in U$, then each of the edges in $G$ that contain $u$ must have received the same two colours in the original colouring. Otherwise, we would have a triangle containing all three of the colours red, blue and green. Suppose, therefore, that every edge in $G$ that contains $u$ received the colours red and blue in the original colouring. Then, again using the property that every triangle contains at most two of the colours red, blue and green, we see that the neighbourhood of $u$ in $G$ contains no green edges. Therefore, if $u$ had $t$ neighbours in $G$, we could use this neighbourhood to find a green copy of $H_t$. Since a similar argument holds if the edges containing $u$ correspond to blue and green or to red and green, we may assume that every vertex $u \in U$ is contained in fewer than $t$ edges in the graph $G$.
By Brooks' theorem, it follows that $U$ contains a subset $V$ of order at least $|U|/t \geq F(t^3)$ containing no edges from $G$, that is, such that every edge received at most one colour in the original colouring. Since $V$ is a $4$-coloured graph of order at least $F(t^3)$ containing no rainbow triangle in red, blue and green, there is a subset of order at least $t^3$ with at most three colours. If the missing colour is red, we may easily find a red copy of $H_t$ and similar conclusions hold if the missing colour is either blue or green. On the other hand, if the missing colour is yellow, we have a $3$-colouring, in red, blue and green, of a set of order at least $t^3$ containing no rainbow triangle, so Lemma <ref> tells us that there is a subset of order at least $t$ with at most two colours. If we again consider the missing colour, it is easy to find a monochromatic copy of $H_t$ in that colour.
The lower bound, $r_3(H_t; 3) \geq F(t)$ follows from a simple adaptation of the proof of Theorem <ref>(ii).
Proof of Theorem <ref>(ii):
By the definition of $F(t)$, there exists a $4$-colouring $\chi$, in red, blue, green and yellow, say, of the edges of the complete graph on $F(t) - 1$ vertices containing no rainbow triangle with one edge in each of red, blue and green and such that every clique of order $t$ contains all $4$ colours.
We now $3$-colour the complete $3$-uniform hypergraph on the same vertex set in red, blue and green, colouring the triple $uvw$ with any colour which is not contained within the set $\{\chi(u, v), \chi(v,w), \chi(w,u)\}$. Since there are no rainbow triangles in red, blue and green, this colouring is well-defined. Suppose now that there is a monochromatic copy of $H_t$ in red, say, and let $u_1, u_2, \dots, u_t$ be the body of this copy. Then, in the original graph colouring $\chi$, none of the edges $u_i u_j$ with $1 \leq i < j \leq t$ are red. However, this contradicts the property that every set of order $t$ contains all $4$ colours.
§ CONCLUDING REMARKS
The results of this paper raise a number of interesting questions, some of which we describe below.
§.§ Higher uniformity hedgehogs
The $k$-uniform hedgehog $H_t^{(k)}$ is the hypergraph with vertex set $[t + \binom{t}{k-1}]$ such that for every $(k-1)$-tuple $(i_1, \dots, i_{k-1})$ with $1 \leq i_1 < \dots < i_{k-1} \leq t$ there is a unique vertex $i_k > t$ such that $i_1 \dots i_k$ is an edge. A straightforward generalisation of the proof of Theorem <ref>(i) gives the following result.
For every integer $k \geq 4$, there exists a constant $c_k$ such that if $H_t^{(k)}$ is the $k$-uniform hedgehog with body of order $t$, then
\[r_k(H_t^{(k)}) \leq t_{k-2} (c_k t),\]
where the tower function $t_i(x)$ is defined by $t_1(x) = x$ and $t_{i+1}(x) = 2^{t_i(x)}$.
A construction due to Kostochka and Rödl <cit.> shows that this result is tight for $k = 4$, that is, that there exists a positive constant $c$ such that $r_4(H_t^{(4)}) \geq 2^{ct}$. Since the construction is simple, we describe it in full. To begin, take a colouring of the edges of the complete graph on $2^{ct}$ vertices such that every set of order $t$ contains both a red triangle and a blue triangle. We then colour the edges of the $4$-uniform hypergraph on the same vertex set by colouring a $4$-tuple red if it contains a red triangle, blue if it contains a blue triangle and arbitrarily otherwise. It is easy to check that this $2$-colouring contains no monochromatic copy of $H_t^{(4)}$. Already for $k = 5$, we were unable to prove a matching lower bound, since it seems that one would first need to know how to prove a double-exponential lower bound for $r_3(K_t)$.
We were also unable to prove an analogue of Theorem <ref>(ii) for $k = 4$. Again, this is because of a basic gap in our understanding of hypergraph Ramsey problems. While we know that there are $4$-colourings of the $3$-uniform hypergraph on $2^{2^{ct}}$ vertices such that every subset of order $t$ receives at least two colours, we do not know if the following variant holds.
Is there an integer $q$, a positive constant $c$ and a $q$-colouring of the $3$-uniform hypergraph on $2^{2^{ct}}$ vertices such that every subset of order $t$ receives at least three colours?
A positive answer to the analogous question where we ask that every subset of order $t$ receives at least five colours would allow us to prove that there exists an integer $q$ such that $r_4(H_t^{(4)}; q) \geq 2^{2^{ct}}$. The proof of this statement is a variant of the proof of Theorem <ref>(ii). Indeed, suppose that we have a $q$-colouring $\chi$ of the edges of the $3$-uniform hypergraph $K_n^{(3)}$ such that every subset of order $t$ receives at least five colours. Then we define a colouring of the complete $4$-uniform hypergraph $K_n^{(4)}$ with at most $q + \binom{q}{2} + \binom{q}{3} + \binom{q}{4}$ colours by colouring the edge $uvwx$ with the set $\{\chi(uvw), \chi(vwx), \chi(wxu), \chi(xuv)\}$. It is now easy to check that if there is a monochromatic $H_t^{(4)}$ in this colouring, then, in the original colouring $\chi$, the body of the hedgehog is a subset of order $t$ which receives at most $4$ colours, contradicting our choice of $\chi$.
Our motivation for investigating higher uniformity hedgehogs was the hope that they might allow us to show that there are families of hypergraphs for which there is an even wider separation between the $2$-colour and $q$-colour Ramsey numbers. However, it seems likely that for hedgehogs the separation between the tower heights is at most one for any uniformity. This leaves the following problem open.
For any integer $h \geq 3$, do there exist integers $k$ and $q$ and a family of $k$-uniform hypergraphs for which the $2$-colour Ramsey number grows as a polynomial in the number of vertices, while the $q$-colour Ramsey number grows as a tower of height $h$?
§.§ Burr–Erdős in hypergraphs
The degeneracy of a graph $H$ is the minimum $d$ such that every induced subset contains a vertex of degree at most $d$. Building on work of Kostochka and Sudakov <cit.> and Fox and Sudakov <cit.>, Lee <cit.> recently proved the famous Burr–Erdős conjecture <cit.>, that graphs of bounded degeneracy have linear Ramsey numbers. That is, he showed that for every positive integer $d$ there exists a constant $c(d)$ such that the Ramsey number of any graph $H$ with $n$ vertices and degeneracy $d$ satisfies $r(H) \leq c(d) n$.
If we define the degeneracy of a hypergraph $H$ in a similar way, that is, as the minimum $d$ such that every induced subset contains a vertex of degree at most $d$, we may ask whether the analogous statement holds in hypergraphs. Unfortunately, as first observed by Kostochka and Rödl <cit.>, the $4$-uniform analogue of the Burr–Erdős conjecture is false, since $H_t^{(4)}$ is $1$-degenerate and $r_4(H_t^{(4)}) \geq 2^{ct}$.
Since $H_t$ is a $1$-degenerate hypergraph, the results of this paper show that the Burr–Erdős conjecture also fails for $3$-uniform hypergraphs and $3$ or more colours. For $4$ colours, this follows immediately from Theorem <ref>(ii). For $3$ colours, it follows from Theorem <ref>(ii) and the observation that $F(t) = \Omega(t^3/\log^6 t)$. To show this, we amend a construction of Fox, Grinshpun and Pach <cit.>, taking the lexicographic product of three $3$-colourings of the complete graph on $t/16 \log^2 t$ vertices, one for each triple of colours from the set $\{$red, blue, green, yellow$\}$ that contains yellow, each having the property that the union of any two colours contains no clique of order $4 \log t$. This colouring will contain no rainbow triangle with one edge in each of red, blue and green and no clique of order $t$ with at most $3$ colours. For further details, we refer the reader to Theorem 3.1 of <cit.>.
While it is also unlikely that an analogue of the Burr–Erdős conjecture holds in the $2$-colour case, it may still be the case that $r_3(H_t)$ is linear in the number of vertices, that is, that $r_3(H_t) = O(t^2)$. It would already be interesting to prove an approximate version of this statement.
Show that $r_3(H_t) = t^{2 + o(1)}$.
§.§ Multicolour Erdős–Hajnal
It is somewhat curious that our upper bound for $r_3(H_t; 3)$ mirrors the best known lower bound for $r_3(K_t;3)$, due to Conlon, Fox and Sudakov <cit.>, which says that there exists a positive constant $c$ such that
\[r_3(K_t;3) \geq 2^{t^{c \log t}}.\]
However, it seems likely that this is mere coincidence and that the function $F(t)$ defined in the introduction is polynomial in $t$. Phrasing the question in a more traditional fashion, we would very much like to know the answer to the following special case of the multicolour Erdős–Hajnal conjecture.
Show that there exists a positive constant $c$ such that if the edges of $K_n$ are $4$-coloured, in red, blue, green and yellow, so that there are no rainbow triangles with one edge in each of red, blue and green, then there is a clique of order $n^c$ containing at most three of the four colours.
That being said, if $F(t)$ were superpolynomial, it would not only disprove the multicolour Erdős–Hajnal conjecture, it would also strengthen the curious correspondence between the bounds for $r_3(H_t; q)$ and $r_3(K_t; q)$. This would certainly be the more interesting outcome.
S. A. Burr and P. Erdős, On the magnitude of generalized Ramsey numbers for graphs, in Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), 214–240, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
D. Conlon, J. Fox and B. Sudakov, Hypergraph Ramsey numbers, J. Amer. Math. Soc. 23 (2010), 247–266.
D. Conlon, J. Fox and B. Sudakov, Recent developments in graph Ramsey theory, in Surveys in Combinatorics 2015, London Math. Soc. Lecture Note Ser., Vol. 424, 49–118, Cambridge University Press, Cambridge, 2015.
P. Erdős and A. Hajnal, Ramsey-type theorems, Discrete Appl. Math. 25 (1989), 37–52.
P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196.
P. Erdős and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. 3 (1952), 417–439.
J. Fox, A. Grinshpun and J. Pach, The Erdős–Hajnal conjecture for rainbow triangles, J. Combin. Theory Ser. B 111 (2015), 75–125.
J. Fox and B. Sudakov, Two remarks on the Burr–Erdős conjecture, European J. Combin. 30 (2009), 1630–1645.
R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey theory, 2nd edition, John Wiley & Sons, 1990.
A. V. Kostochka and V. Rödl, On Ramsey numbers of uniform hypergraphs with given maximum degree, J. Combin. Theory Ser. A 113 (2006), 1555–1564.
A. V. Kostochka and B. Sudakov, On Ramsey numbers of sparse graphs, Combin. Probab. Comput. 12 (2003), 627–641.
C. Lee, Ramsey numbers of degenerate graphs, arXiv:1505.04773 [math.CO].
J. Spencer, Turán's theorem for $k$-graphs, Discrete Math. 2 (1972), 183–186.
|
1511.00498
|
On the $\pi\pi$ - scattering lengths in the theory with effective Lagrangian
E.P. Shabalin [e-mail: [email protected]]
Institute of Theoretical and Experimental Physics; National
Research Center Kurchatov Institute, Bol'shaya Cheremushkinskaya
ul., 25, Moscow, 117218 Russia.
Use of the effective Lagrangian incorporating both the scalar and pseudoscalar mesons gives a possibility
to calculate the $\pi\pi$-scattering lengths without attraction of the ChPT theory.
§ INTRODUCTION
The question about the numerical quantities of the $\pi\pi$ -scattering lengths engages the minds of theoreticians and the experimentalists so far. Nowdays, a number of the observed events of $\pi\pi$ - scattering increased many times, that gives a
possibility to verify the different approaches to work out a theory of the low-energy $\pi\pi$ - interaction.
It is clear, that an approach based on the use of the Lagrangian scheme looks as more convenient for deep understanding
of the theorems concerning the soft pions.
Just in this scheme, it is possible to obtain the precise results of the current algebra even on the "trees" level, that is, on the level of the diagrams without loops [1].
In this paper, this peculiarity of the Lagrangian scheme will be used for the calculation of the $\pi\pi$ scattering lengths. And besides, the conformity of the properties of the QCD objects and objects of the real world will be ensured also.
In QCD, the spinless flavoured objects are $\bar q_R t^a q_L$ and their Hermitian conjugate. These objects
are formed by opposite-parity components. For this property to be reproduced in the Lagrangian of real particles, it must be expressed in terms of the matrix
\begin{equation}
U = (\sigma_a + i\pi_a)t_a,
\label{1}
\end{equation}
where $t_0 = \sqrt{1/3} $ , $t_{1....8} = \sqrt{1/2}_{1,....8}$ and $\sigma_a$ and $\pi_a$ are nonets of,
respectively, scalar and pseudoscalar mesons.
This idea is not new and was used in $ [2] $ and $ [3 ]$, but we go over directly to the final form of the Lagrangian containing
all forms of breakdown of chiral symmetry [4], [5]:
\begin{equation}
\begin{array}{ll}
L = \frac{1}{2}Tr(\partial U_{\mu}\partial U^+ _{\mu}) - cTr(UU^+
- A^2t^2_0)^2 - c\xi( Tr(UU^+ - A^2t^2_0))^2 + \nonumber \\
\frac{F_{\pi}}{2\sqrt{2}}Tr{(M(U +U^+)} + \Delta L^{U(1)}_{PS}.
\end{array}
\label{2}
\end{equation}
These forms permit to express all properties of $\sigma$ - mesons using the properties of $\pi$ - mesons
and the parameters $R=F_K/F_{\pi}$ and $\xi$. The values of them can be found from the data on the decays
$K,\pi \to \mu \nu$ and identification of the scalar $\sigma_{\pi}$ - meson with the resonance $a_0(980)$ [4], [5], [6].
In a theory with the broken $U(3)_L\otimes U(3)_R$ chiral symmetry there are two isosinglet $\sigma$-particles having nonzero vacuum expectation values $<\sigma>$. As a result, the Lagrangian(2) contains
the vertices $<\pi\pi|\sigma>$. Then, the set of the pole diagrams with the intermediate $\sigma$ - meson appears. The amplitude of the $\pi\pi \to \pi\pi$ - scattering acquires the form:
\begin{equation}
\begin{array}{ll}
T_{\sigma} = <\pi_k(p'_1)\pi_l(p'_2)|\pi_i(p_1)\pi_j(p_2)> =
A_{\sigma}\delta_{ij}\delta_{kl} + B_{\sigma} \delta_{ik}\delta_{jl} + C_{\sigma} \delta_{il}\delta_{jk},
\label{3}
\end {array}
\end{equation}
\begin{equation}
\begin{array}{ll|}
A_{\sigma} = (s-\mu^2)\sum_{n=1,2} \frac{G_n}{m^2_{\sigma_n}-s}, \quad
B_{\sigma} = (t-\mu^2 )\sum_{n=1,2}
\frac{G_n}{m^2_{\sigma_n}-t},\nonumber \\
~~ \nonumber \\
C_{\sigma} = (u-\mu^2)\sum_{n=1,2} \frac{G_n}{m^2_{\sigma_n} - u}, \quad \mu \equiv m_{\pi},
\end{array}
\label{4}
\end{equation}
and where
\begin{equation}
\begin{array}{ll}
s = (p_1 + p_2)^2, \quad t = (p_1 - p'_1)^2, \quad u = (p_1 - p'_2)^2, \quad
G_n=\frac{g^2_n}{m^2_{\sigma_n} - \mu^2}.
\end{array}
\label{5}
\end{equation}
In the theory specified by the Lagrangian (2) , the following relation holds:
\begin{equation}
\frac{G_1}{m^2_{\sigma_1} - \mu^2} +\frac{G_2}{m^2_{\sigma_2} - \mu^2}=\frac{1}{F^2_{\pi} }, \quad
F_{\pi} = 93 \:{ MeV}.
\label{6}
\end{equation}
For the case of fixed total isospin of a system of initial pions, the expressions for the amplitudes are given in [7]:
\begin{equation}
T^{(0)} = 3A + B + C.
\label{7}
\end{equation}
\begin{equation}
T^{(1)} =B-C
\label{8}
\end {equation}
\begin{equation}
T^{(2)} = B +C.
\label{9}
\end{equation}
The decomposition of the isotopic amplitudes into amplitudes corresponding to fixed values of the orbital angular momentum is given by
\begin{equation}
T^{(I)} = 32\pi\sum_{l=0}^{\infty} (2l+1)t^{(I)}_l(s)P_l(\cos \theta).
\label{10}
\end{equation}
It follows from (10) that the partial-wave amplituda $t^{(I)}$ is:
\begin{equation}
t^{(I)}_l(s) = \frac{1}{64\pi} \int_{-1}^{1} T^{(I)}P_I(\cos\theta) d \cos\theta.
\label{11}
\end{equation}
The scattering lengths arise from the expansion
\begin{equation}
t^I_l(s)m^{-1}_{\pi} = q^{2l}[a^{(I)}_l + b^{(I)}_l q^2 + O(q^4)], \qquad q^2=\frac{s}{4}-\mu^2.
\label{12}
\end{equation}
To calculate $a^{(0)}_0$ and $a^{(2)}_0$, we make use of the relations (3,7,9,11). The scattering length
$a^{(1)}_1$ will be considered later, since besides the $\sigma$ - mesons, the $\rho$-mesons also
contribute into this scattering length. We begin calculations from $a^{(2)_0}$, because this scattering
according to (9) and (7),enters into $a^{(0)}_0$.
§ THE SCATTERING LENGTH $A^{(2)}_0$
According to (9)
\begin{equation}
T^{(2)}_{\sigma} = \sum_{n=1,2} G_n\left(\frac{t - \mu^2}{m^2_{\sigma_n} - t} +\frac{u - \mu^2}{m^2_{\sigma_n}
-u} \right).
\label{13}
\end{equation}
On the threshold $t$ and $u$ are equal to zero. Using the expressions for masses and coupling constants
of the $\sigma_n$ - mesons [5] and the last numerical data on their values [6], we obtain:
\begin{equation}
T^{(2)}_{\sigma} = - \left(\frac{2G_1\mu^2}{m^2_{\sigma_1|}} +\frac{2G_2\mu^2}{m^2_{\sigma_2}} \right) =
- 4.3171.
\label{14}
\end{equation}
According to (11) and (12)
\begin{equation}
a^{(2)}_0 = - 0.04294m^{-1}_{\pi}.
\label{15}
\end{equation}
This result is in agreement with the result of analysis of the $K^{\pm} \to \pi^{+}\pi^{-}e^{\pm}\nu$ decay,
based on the statistics of 1.13 million decays [8]:
\begin{equation}
a^{(2)}_0 = (- 0.0432 \pm 0.0086_{stat} \pm 0.0034_{syst} \pm 0.0028_{th})m^{-1}_{\pi}
\label{16}
\end{equation}
§ THE SCATTERING LENGTH $A^{(0)}_0$
In accordance with (7) , the amplitude of $S$-wave with the isospin 0 looks as
\begin{equation}
T^{(0)} = \sum_{n=1,2} G_n \left( 3\frac{s - \mu^2}{m^2_{\sigma_n} - s} \right) +T^{(2)}_{\sigma}.
\label{17}
\end{equation}
At the threshold, the first addendum in (17) performs into
\begin{equation}
9\mu^2 \sum_{n=1,2} G_n(m^2_{\sigma_n} - 4\mu^2)^{-1} = 23.3335.
\label{18}
\end{equation}
Adding the result (14) for $T^{(2)}_{\sigma}$, we come to
\begin{equation}
T^{(0)}_{\sigma} = 19.0164.
\label{19}
\end{equation}
And the final result for $a^{(0)}_0$ is:
\begin{equation}
(a^{(0)}_0)_{\sigma} = \frac{19.0164}{32\pi m_{\pi}} = 0.18916m^{-1}_{\pi}.
\label{20}
\end{equation}
This value agrees with the experimental one:
\begin{equation}
(a^{(0)}_0)_{exp} = (0.197\pm 0.010)m^{-1}_{\pi}
\label{21}
\end{equation}
obtained from the analysis of all data near threshold of the reaction $\pi N \to \pi\pi N$ [9].
The data obtained for $a^{(0)} _0$ by the collaboration E865 [10] depend from the Models used at analysis.
In particular, in the Model A $a^{(0)}_0 = (0.184 \pm 0.010)m^{-1}_{\pi}$, in the Model B $a^{(0)}_0=
(0.179 \pm 0.033)m^{-1}_{\pi}$ and in the Model C $a^{(0)}_0=(0.213 \pm 0.013)m^{(-1)}_{\pi}$. [See the Table 6 in [8]]
However, these results appear after taking into account the isospin corrections, also depending from the
Our result (20) does not demand any additional model corrections.
§ THE SCATTERING LENGTH $A^{(1)}_1$
In the used by us theory, besides the $\sigma$ - mesons, the intermediate $\rho$ - mesons also give
a contribution into $a^{(1)}_1$. In the present paper, a nature of the $\rho$ - mesons will be not associated with the vector quark current, as it was considered usually, but their nature will be associated with the
divergence of vector quark current [11 - 15].
A contibution of the $\sigma$ - mesons into isovector amplitude is:
\begin{equation}
T^{(1)}_{\sigma} = \sum_{n=1,2}G_n \left[\frac{t - \mu^2}{m^2_{\sigma_n} - t} - \frac{u - \mu^2}{m^2_{\sigma_n} -u} \right].
\label{22}
\end{equation}
According to the relations (11) and (12), the part of scattering length $a^{(1)}_1$ produced by $\sigma$ -
mesons is:
\begin{equation}
(a^{(1)}_1)_{\sigma} = \frac{1}{24\pi m_{\pi}} \left[ \frac{g^2_1}{m^4_{\sigma_1}} + \frac{g^2_2}{m^4_{\sigma_2}} \right].
\label{23}
\end{equation}
Using the numerical values of $m_{\sigma_{1,2}}$ and $g_{1,2}$, we find:
\begin{equation}
(a^{(1)}_1)_{\sigma} = 0.02744m^{(-3)}_{\pi}.
\label{24}
\end{equation}
The part of the isovector amplitude produced by the intermediate $\rho$ - mesons looks like the relation
(3), but with the differnt A, B and C [15]. Namely,
\begin{equation}
A_{\rho} = \frac{1}{M^2_{\rho}} \left(g^2(u)\frac{(s-t)u}{M^2_{\rho} -u} + g^2(t)\frac{(s-u)t}{M^2_{\rho} -t} \right).
\label{25}
\end{equation}
\begin{equation}
B_{\rho} = \frac{1}{M^2_{\rho}} \left(g^2(s)\frac{(t-u )s}{M^2_{\rho} -s} + g^2(u)\frac{(t-s)u}{M^2_{\rho} -u} \right).
\label{26}
\end{equation}
\begin{equation}
C_{\rho} = \frac{1}{M^2_{\rho}} \left(g^2(s)\frac{(u-t)s}{M^2_{\rho} -s} + g^2(t)\frac{(u-s)t}{M^2_{\rho} -t} \right).
\label{27}
\end{equation}
In accordance with (8)
\begin{equation}
T^{(1)}_{\rho} = B_{\rho} - C_{\rho}.
\label{28}
\end{equation}
As the explicit form of $g(x=s,t,u)$ is not specified by our theory, we are forced to resort to the
phenomenological relation elaborated in [15]:
\begin{equation}
g(x) = g_{\rho} exp \left(0.7855 \left[\frac{x}{2M^2_{\rho}} - \left(\frac{x}{2m^2_{\rho}}\right)^2 \right] \right),
\qquad x \le M^2_{\rho}.
\label{29}
\end{equation}
As we are interesting in behavior of the partial wave $t^{(1)}_1(s)$ near threshold, we obtain the following result:
\begin{equation}
(t^{(1)}_1)^{threshold}_{\rho} = \frac{4\mu^2}{3\pi M^2_{\rho}} \left(\frac{2g^2(4\mu^2)}{M^2_{\rho} - 4\mu^2} +\frac{g^2(0)}{M^2_{\rho}} \right)
\label{30}
\end{equation}
Using the experimental value of $g(M^2_{\rho})$=5.9764 and the formula (29), we find:
\begin{equation}
g(0) = 4.9108, \qquad g(4\mu^2) =5.5520.
\label{31}
\end{equation}
The part of the scatterinag length $a^{(1)}_1$ produced by the intermediate $\rho$ - mesons turns out to be:
\begin{equation}
(a^{(1)}_1)_{\rho} = 0.005918m^{(-3)}_{\pi}.
\label{32}
\end{equation}
Together with the part (24) produced by the $\sigma$ -mesons we get:
\begin{equation}
(a^{(1)}_1)^{total} =0.03336m^{-3}_{\pi}.
\label{33}
\end{equation}
§ CONCLUSION
The requirement of conformity between the properties of the QCD objects and objects of the real world gives rise to necessity of existance of the scalar mesons, that, as it turned out, play the principal role in the low-energy $\pi\pi$ - interactions.
Being the chiral partners of the pseudoscalar mesons, the scalar mesons possess quite definite properties, ascribed by the structure of the Lagrangian (2). Their masses and coupling constants depend only on
two parameters:$ R = F_{K}/F_{\pi}$ and $\xi$, the values of which are determined experimentally.
The found by us scattering lengths are:
\begin{equation}
a^{(0)}_0=0.18916m^{-1}_{\pi}, \; a^{(2)}_0=-0.04294m^{-1}_{\pi}, \;a^{(1)}_1=0.03336m^{(-3)}_{\pi}.
\label{33}
\end{equation}
And they did not require to attract the special models, concerning these scattering lengths.
In our theory, the current-algebra prediction [16]:
\begin{equation}
\frac{2a^{(0)}_0 - 5a^{(2)}_0}{18\mu^2{\pi}a^{(1)}_1} = 1
\label{34}
\end{equation}
is satisfied to within 1.24%.
1. V. de Alfaro, S.Fubini, G. Furlan, and C. Rossetti, Currents in Hadron
Physics (North-Holland, Essevier, Amsterdam,1973)
2. M. Levy, Nuovo Cimento A 52, 23 (1976); G. Cicongna,
Phys. Rev. D 1, 1786 (1970); J. Schechter and Y. Ueda, Phys.
Rev. D 3 168 (1971); A.S. Carrol et. al., Phys. Lett. B
96, 407 (1980).
3. C. Rosenzwe, J. Schechtor and C.G. Trachern, Phys. Rev. D 21, 3388 (1980); P. Di Vecchia and G. Veneziano, Nucl. Phys. B
171, 253 (1980); K. Kawarabyashi and N. Ohta, Prog. Theor.
Phys. 66, 1789 (1981); N. Ohta, Prog. Theor. Phys. 66, 1408 (1981); Prog. Theor. Phys. 67, 993 (E)(1982).
4. E.P. Shabalin, Sov. J. Nucl.Phys. 42, 260 (1985); Sov. J. Nucl. Phys. 46, 1312 (E) (1987).
5. E.P. Shabalin, Sov. J. Nucl. Phys. 49, 365 (1989).
6. E.P. Shabalin, Yad. Fiz. 77, 1547 (2014).
7. J.L. Petersen, Preprint No.77-04, CERN (Geneva, 1977).
8. J.R. Batley et al., Eur. Phys. J. C 70, 635 (2010).
9. H. Burkhardt and J. Lowe, Phys. Rev. Lett. 67 , 2622 (1991).
10. S. Pislak et al., Phys. Rev. Lett. 105, 019901 (2010).
11. W. Krolikowsky, Nuovo Cim. A 42, 435 (1966).
12. D. G. Caldi and H. Pagels, Phys. Rev. D 14, 809 (1976).
13. J. Gasser and H. Leutwyler, Ann. of Phys. 158, 142 (1984).
14. G. Ecker et al., Nucl. Phys. B 321, 311 (1989).
15. E. P. Shabalin, Yad. Fiz. 63, 659 (2000); Phys. At. Nucl. 63, 594 (2000).
16. S. Weinberg, Phys. Rev .Lett. 17, 616 (1966).
|
1511.00211
|
$^{1}$Physics Department, Tennessee Technological University, Cookeville, TN 38505, USA
$^{2}$Physics Department, Faculty of Sciences, Ankara University, 06100 Ankara, Turkey
$^{3}$Physics Department, Middle East Technical University, 06800 Ankara, Turkey
Nucleon exchange mechanism is investigated in the central collisions of ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems near the quasi-fission regime in the framework of the Stochastic Mean-Field (SMF) approach. Sufficiently below the fusion barrier, di-nuclear structure in the collisions is maintained to a large extend. Consequently, it is possible to describe nucleon exchange as a diffusion process familiar from deep-inelastic collisions. Diffusion coefficients for proton and neutron exchange are determined from the microscopic basis of the SMF approach in the semi-classical framework. Calculations show that after a fast charge equilibration the system drifts toward symmetry over a very long interaction time. Large dispersions of proton and neutron distributions of the produced fragments indicate that diffusion mechanism may help to populate heavy trans-uranium elements near the quasi-fission regime in these collisions.
24.10.Jv; 21.30.Fe; 21.65.+f; 26.60.+c
§ INTRODUCTION
Much experimental effort has been spent to synthesize super-heavy elements by fusion mechanism of heavy systems <cit.>. It is crucial to choose the right projectile-target combinations in order to have the largest probability for forming a compound nucleus that leads to production of the desired super-heavy element. Among different possibilities, formation of compound nucleus in collisions with deformed actinide targets with neutron rich projectile, so called hot fusion, provides a suitable choice for synthesizing of these elements <cit.>. However, in heavy systems compound nucleus formation is severely inhibited by the quasi-fission mechanism <cit.>. In quasi-fission, the colliding ions attach together for a long time, but separate without going through compound nucleus formation. During the long contact times many nucleon exchanges take place between projectile and target nuclei. A number of models are developed for a description of the reaction mechanism in the quasi-fission process <cit.>. In a recent work, the quasi-fission mechanism in ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U collisions was investigated in the mean-field approach of the time-dependent Hartree-Fock theory by Oberacker et al. <cit.>. The calculations carried out at bombarding energies around the fusion barrier exhibit an important difference between the collisions ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U. In the collisions with neutron rich isotope of calcium, for increasing bombarding energy quasi-fission mechanism diminishes. Hence the cross-section of the composite system formation goes up with bombarding energy as compared to the collisions with the stable calcium projectile. Calculations also show in the quasi-fission regime, in both collisions, large number of nucleon transfer takes place from heavy target to light projectile with increasing numbers as the bombarding energy goes up towards the fusion barrier. Slightly below the fusion barrier the mean values of the proton and neutron drift toward symmetry, and can reach large values of $\Delta Z\approx 10$, $\Delta N\approx 21$ in ${}^{40}$Ca induced collisions, and $\Delta Z\approx 6$, $\Delta N\approx 9$ in ${}^{48}$Ca induced collisions, respectively. The mean-field description of the TDHF can determine only the mean values of the proton and neutron drifts. On the other hand, it is very interesting and important to provide a description of the fragment mass and charge distributions in the quasi-fission reactions. As seen from Fig. 1, sufficiently below the fusion barrier, di-nuclear structure in the collisions is maintained to a large extend. This figure shows the density profiles in the reaction plane near maximum overlap obtained in TDHF calculations in the central collisions of ${}^{40}$Ca + ${}^{238}$U (a) and ${}^{48}$Ca + ${}^{238}$U (b) systems at energies $E_{\text{c.m.}} =202.0$ MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively. Red lines indicate the position of window planes. The windows are perpendicular to the symmetry lines and pass through the minimum density planes at each instant of the collision process. Consequently, it is possible to describe nucleon exchange as a diffusion process familiar from deep-inelastic collisions <cit.>. In this work, we investigate nucleon exchange mechanism in the quasi-fission reactions in head-on collisions of ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems at a bombarding energy slightly below the fusion barrier by employing the Stochastic Mean-Field (SMF) approach <cit.>.
The SMF approach gives rise to a Langevin description for neutron and proton exchanges characterized by diffusion and drift coefficients. We calculate these transport coefficients in the semi-classical framework in terms of the mean-field description of the TDHF solutions without any adjustable parameters. As a result of large contact times in the quasi-fission reaction, on the top of large drift toward symmetry, fragment mass and charge distributions have a very broad dispersions in both ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems. These results indicate that in the collisions of heavy systems, in addition to fusion, nucleon diffusion mechanism may help to populate heavy trans-uranium elements in the quasi-fission regime.
In section 2, we present a brief description of the nucleon diffusion mechanism based on the SMF approach. In section 3, we discuss transport coefficients for proton and neutron exchanges. In section 4, the result of calculations presented for central collisions of ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems. Conclusions are given in section 5.
§ DIFFUSION MECHANISM
In this work, we consider proton and neutron transfer mechanism in head-on collisions in the ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems at bombarding energies below the fusion barrier near the quasi-fission regime. Specifically we carry out calculations for ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems at $E_{\text{c.m.}} =202.0$ MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively. As seen from the TDHF calculations in <cit.>, near the quasi-fission regime, colliding ions stick together for a long time. At energies below the fusion barrier, as seen in Fig. 1, system maintains a binary structure to a large extent, and a visible window appears between the projectile-like and target-like partners. Therefore, we can analyze nucleon exchange mechanism, by employing nucleon diffusion concept based on the SMF approach. The phenomenological nucleon exchange model and diffusion description has been applied extensively for analyzing deep-inelastic heavy-ion collisions <cit.>. The SMF approach provides a more accurate microscopic framework for diffusion mechanism and extracting transport coefficients of relevant macroscopic variables without any adjustable parameters and taking the full collision geometry into account. In the SMF approach, the standard description is extended beyond the mean-field by incorporating the mean-field fluctuations in terms of generating an ensemble of events according to quantal and thermal fluctuations in the initial state (for details please refer to <cit.>). In extracting transport coefficients for nucleon exchange, we take the proton and neutron numbers of projectile-like fragments as independent variables. We can define the proton and neutron numbers $Z_{1}^{\lambda } (t)$, $N_{1}^{\lambda } (t)$ of the projectile-like fragments in each event by integrating over the nucleon density on the projectile side of the window as <cit.>,
\begin{equation} \label{Eq1}
\left(\begin{array}{c} {Z_{1}^{\lambda } (t)} \\ {N_{1}^{\lambda } (t)} \end{array}\right)=\int d^{3} r\theta \left[x-x_{0} \left(t\right)\right] \left(\begin{array}{c} {\rho _{p}^{\lambda } (\vec{r},t)} \\ {\rho _{n}^{\lambda } (\vec{r},t)} \end{array}\right).
\end{equation}
Here, $\lambda$ denotes the event label, $x_{0} \left(t\right)$ indicates the location of the window, and $\rho _{p}^{\lambda } (\vec{r},t)$, $\rho _{n}^{\lambda } (\vec{r},t)$ are the local densities of protons and neutrons. We take $x$-axis as the collision direction and the position $x_{0} \left(t\right)$ of the window plane is determined from the TDHF calculations. As described in <cit.>, the local density is projected on the reaction plane and the window is located at the lowest density plane on the neck at each time step.
(Color online) Density profiles in the reaction plane near maximum overlap in the central collisions of ${}^{40}$Ca + ${}^{238}$U (a) and ${}^{48}$Ca + ${}^{238}$U (b) systems at energies $E_{\text{c.m.}} =202.0$ MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively, obtained in TDHF calculations. Red lines indicate the position of window planes.
According to the SMF approach, the proton and neutron numbers of the projectile-like fragment follows a stochastic evolution according to the following Langevin equations,
\begin{eqnarray} \label{Eq2}
\frac{d}{dt} \left(\begin{array}{c} {Z_{1}^{\lambda } (t)} \\ {N_{1}^{\lambda } (t)} \end{array}\right)&=&\int dydz\left(\begin{array}{c} {j_{x,p}^{\lambda } (\vec{r},t)} \\ {j_{x,n}^{\lambda } (\vec{r},t)} \end{array}\right) _{x=x_{0} } \nonumber \\
&=&\left(\begin{array}{c} {v_{p}^{\lambda } (t)} \\ {v_{n}^{\lambda } (t)} \end{array}\right).
\end{eqnarray}
In this expression, the right hand side denotes the proton $v_{p}^{\lambda } (t)$ and neutron $v_{n}^{\lambda } (t)$ drift coefficients in the event $\lambda $, which are determined by the proton current $j_{x,p}^{\lambda } (\vec{r},t)$ and neutron current $j_{x,n}^{\lambda } (\vec{r},t)$ through the window. In the SMF approach, the fluctuating proton and neutron currents are defined as <cit.>,
\begin{eqnarray} \label{Eq3}
j_{x,a}^{\lambda } (\vec{r},t)=\frac{\hbar }{2im} \sum _{ij\in a}&&\left[\Phi _{j}^{\lambda *} (\vec{r},t)\nabla _{x} \Phi _{i}^{\lambda } (\vec{r},t)\right.\nonumber\\
&&-\left.\Phi _{i}^{\lambda } (\vec{r},t)\nabla _{x} \Phi _{j}^{\lambda *} (\vec{r},t)\right]\rho _{ji}^{\lambda},
\end{eqnarray}
where the summations $i$ and $j$ run over a complete set of single-particle states for protons and neutrons $a=p,n$. Drift coefficients fluctuate from event to event due to stochastic elements of the initial density matrix $\rho _{ji}^{\lambda } $ and also due to the different sets of the wave functions in different events. As a result, there are two sources for fluctuations of the nucleon current: (i) fluctuations that arise from the state dependence of the drift coefficients, which may be approximately represented in terms of fluctuations of proton and neutron partition of the di-nuclear system, and (ii) the explicit fluctuations $\delta v_{p}^{\lambda } (t)$ and $\delta v_{n}^{\lambda } (t)$ which arise from the stochastic part of proton and neutron currents. For small amplitude fluctuations, we can linearize the drift coefficients around their mean values $v_{p} (t)$ and $v_{n} (t)$ to obtain,
\begin{eqnarray} \label{Eq4}
\left(\begin{array}{c} {v_{p}^{\lambda } (t)} \\ {v_{n}^{\lambda } (t)} \end{array}\right)&=&\left(\begin{array}{c} {v_{p} \left(t\right)} \\ {v_{n} \left(t\right)} \end{array}\right)\nonumber\\
&&+\left(\begin{array}{c} {\frac{\partial v_{p} }{\partial Z_{1} } \left(Z_{1}^{\lambda } -\overline{Z_{1}^\lambda} \right)+\frac{\partial v_{p} }{\partial N_{1} } \left(N_{1}^{\lambda } -\overline{N_{1}^\lambda} \right)} \\ {\frac{\partial v_{n} }{\partial Z_{1} } \left(Z^{\lambda } -\overline{Z_{1}^\lambda} \right)+\frac{\partial v_{n} }{\partial N_{1} } \left(N_{1}^{\lambda } -\overline{N_{1}^\lambda} \right)} \end{array}\right)\nonumber\\
&&+\left(\begin{array}{c} {\delta v_{p}^{\lambda } (t)} \\ {\delta v_{n}^{\lambda } (t)} \end{array}\right).
\end{eqnarray}
Stochastic part of the fluctuations $\delta v_{p}^{\lambda } (t)$ and $\delta v_{n}^{\lambda } (t)$ are specified by uncorrelated Gaussian distributions. These distribution have zero mean values $\overline{\delta v_{p}^{\lambda }(t)}=0$, $\overline{\delta v_{n}^{\lambda }(t)}=0$ and their variances in Markovian approximation are determined by <cit.>
\begin{eqnarray} \label{Eq5}
\overline{\delta v_{p}^{\lambda } (t)\delta v_{p}^{\lambda } (t')}=2\delta (t-t')D_{ZZ} (t),
\end{eqnarray}
\begin{eqnarray} \label{Eq6}
\overline{\delta v_{n}^{\lambda } (t)\delta v_{n}^{\lambda } (t')}=2\delta (t-t')D_{NN} (t).
\end{eqnarray}
Here, $D_{ZZ} (t)$ and $D_{NN} (t)$ denote diffusion coefficients for proton and neutron exchange, and the mixed diffusion coefficient is zero $D_{NZ} (t)=0$.
By taking the average over the generated ensemble of the Langevin Eq. (<ref>), the mean values evolve according to,
\begin{eqnarray} \label{Eq7}
\frac{d}{dt} \left(\begin{array}{c} {\overline{Z_{1}^\lambda} (t)} \\ {\overline{N_{1}^\lambda} (t)} \end{array}\right)&=&\int dydz\left(\begin{array}{c} {j_{x,p}^{} (\vec{r},t)} \\ {j_{x,n}^{} (\vec{r},t)} \end{array}\right) _{x=x_{0} }\nonumber\\
&=&\left(\begin{array}{c} {v_{p} (t)} \\ {v_{n} (t)} \end{array}\right).
\end{eqnarray}
The mean values of drift coefficients are determined by the proton and neutron fluxes. These fluxes are calculated in the mean-field description of the TDHF equations with the mean values of proton and neutron currents,
\begin{eqnarray} \label{Eq8}
j_{x,a} (\vec{r},t)=\frac{\hbar }{2im} \sum _{j\in a}&&[\Phi _{j}^{*} (\vec{r},t)\nabla _{x} \Phi _{j} (\vec{r},t)\nonumber\\
&&-\Phi _{j} (\vec{r},t)\nabla _{x} \Phi _{j}^{*} (\vec{r},t)]n_{j},
\end{eqnarray}
where $n_{j} =1$ for occupied and $n_{j} =0$ unoccupied states.
In order to calculate the fluctuations of proton and neutron numbers, we use the stochastic part of Eq. (<ref>) around the mean evolution,
\begin{eqnarray} \label{Eq9}
\frac{d}{dt} \left(\begin{array}{c} {\delta Z_{1}^{\lambda } (t)} \\ {\delta N_{1}^{\lambda } (t)} \end{array}\right)&=&\left(\begin{array}{c} {\frac{\partial v_{p} }{\partial Z_{1} } \left(Z_{1}^{\lambda } -\overline{Z_{1}^\lambda} \right)+\frac{\partial v_{p} }{\partial N_{1} } \left(N_{1}^{\lambda } -\overline{N_{1}^\lambda} \right)} \\ {\frac{\partial v_{n} }{\partial Z_{1} } \left(Z^{\lambda } -\overline{Z_1^\lambda}\right)+\frac{\partial v_{n} }{\partial N_{1} } \left(N_{1}^{\lambda} -\overline{N_{1}^\lambda}\right)} \end{array}\right)\nonumber\\
&&+\left(\begin{array}{c} {\delta v_{p}^{\lambda } (t)} \\ {\delta v_{n}^{\lambda } (t)} \end{array}\right),
\end{eqnarray}
where the derivatives of drift coefficients are evaluated at the mean trajectory. The variances and the co-variance of neutron and proton distribution of projectile fragments are defined as $\sigma _{NN}^{2} (t)=\overline{\left(N_{1}^{\lambda } -\overline{N_{1}^\lambda} \right)^{2} }$, $\sigma _{ZZ}^{2} (t)=\overline{\left(Z_{1}^{\lambda } -\overline{Z_{1}^\lambda}\right)^{2} }$, and $\sigma _{NZ}^{2} (t)=\overline{\left(N_{1}^{\lambda } -\overline{N_{1}^\lambda} \right)\left(Z_{1}^{\lambda} -\overline{Z_{1}^\lambda} \right)}$. Multiplying both side of Langevin equations Eq. (<ref>) by $N_{1}^{\lambda } -\overline{N_{1}^\lambda}$ and $Z_{1}^{\lambda } -\overline{Z_{1}^\lambda}$, and taking the ensemble average, we find evolution of the co-variances are specified by the following set of coupled differential equations,
\begin{eqnarray} \label{Eq10}
\frac{\partial }{\partial t} \sigma _{NN}^{2} &=&2\frac{\partial v_{n} }{\partial N_{1} } \sigma _{NN}^{2} +2\frac{\partial v_{n} }{\partial Z_{1} } \sigma _{NZ}^{2} +2D_{NN}, \\
\label{Eq11}
\frac{\partial }{\partial t} \sigma _{ZZ}^{2} &=&2\frac{\partial v_{p} }{\partial Z_{1} } \sigma _{ZZ}^{2} +2\frac{\partial v_{p} }{\partial N_{1} } \sigma _{NZ}^{2} +2D_{ZZ}, \\
\label{Eq12}
\frac{\partial }{\partial t} \sigma _{NZ}^{2} &=&\frac{\partial v_{p} }{\partial N_{1} } \sigma _{NN}^{2} +\frac{\partial v_{n} }{\partial Z_{1} } \sigma _{ZZ}^{2} \nonumber\\
&&+\sigma _{NZ}^{2} \left(\frac{\partial v_{p} }{\partial Z_{1} } +\frac{\partial v_{n} }{\partial N_{1} } \right).
\end{eqnarray}
These co-variances describe a correlated Gaussian function for the proton and neutron distribution $P(N,Z,t)$ of the project-like or the target-like fragments,
\begin{eqnarray} \label{Eq12.1}
\end{eqnarray}
\begin{eqnarray} \label{Eq12.2}
\end{eqnarray}
with $\rho=\sigma^2_{NZ}/\sigma_{NN}\sigma_{ZZ}$. The mean values $\bar{N}$, $\bar{Z}$ denote the mean neutron and proton numbers of the target-like or projectile-like fragments. The set of coupled equations for co-variances are familiar from the phenomenological nucleon exchange model, and they were derived from the Fokker-Planck equation for the fragment neutron and proton distributions in the deep-inelastic heavy-ion collisions <cit.>.
§ TRANSPORT COEFFICIENTS
§.§ Nucleon diffusion coefficients
The proton and neutron diffusion coefficients $D_{ZZ}^{} $ and $D_{NN}^{} $, act as sources for determining co-variances in the coupled Eqs. (<ref>-<ref>). In earlier investigations, expressions these diffusion coefficients in the Markovian limit have been deduced from the SMF approach in the semi-classical framework. We present the results here, and for details we refer <cit.>. In the particular case of the head-on collisions, the expressions of proton and neutron diffusion coefficients are
\begin{eqnarray} \label{Eq13}
\left(\begin{array}{c} {D_{ZZ} } \\ {D_{NN} } \end{array}\right)&=&\int \frac{dp_{x} }{2\pi \hbar } \left|\frac{p_{x}}{m}\right|\nonumber\\
&&\left\{\begin{array}{c} {\bar{f}_{T}^{p} \left(x_{0} ,p_{x} ,t\right)\left[1-\frac{1}{\Omega } \bar{f}_{P}^{p} \left(x_{0} ,p_{x} ,t\right)\right]} \\ {\bar{f}_{T}^{n} \left(x_{0} ,p_{x} ,t\right)\left[1-\frac{1}{\Omega } \bar{f}_{P}^{n} \left(x_{0} ,p_{x} ,t\right)\right]} \end{array}\right\} .
\end{eqnarray}
Here, $\bar{f}_{T}^{p/n} \left(x,p_{x} ,t\right)|_{x=x_{0} } $and $\bar{f}_{P}^{p/n} \left(x,p_{x} ,t\right)|_{x=x_{0} } $ are the reduced Wigner functions in the collision direction for protons/neutrons, which are obtained by integrating coordinates and momenta over the window plane as discussed in Appendix of ref. <cit.>. The Wigner functions are calculated with the single-particle wave functions for protons and neutrons, which are originating from target $\left(T\right)$ and projectile $\left(P\right)$ nuclei, respectively. The quantity $\Omega$ denotes the volume of the phase space on the window plane.
§.§ Nucleon drift coefficients
In order to solve co-variances from Eqs. (<ref>-<ref>), in addition to the diffusion coefficients $D_{ZZ}^{}$ and $D_{NN}^{}$, we need to know the rate of change of drift coefficients in the vicinity of their mean values. According to the SMF approach, in order to calculate rates of the drift coefficients, we should calculate neighboring events in the vicinity of the mean-field event. Here, instead of such a detailed description, we employ the fluctuation-dissipation theorem, which provides a general relation between the diffusion and drift coefficients in the transport mechanism of the relevant collective variables as described in the phenomenological approaches <cit.>. Proton and neutron diffusions in the N-Z plane are driven in a correlated manner by the potential energy surface of the di-nuclear system. As a consequence of the symmetry energy, the diffusion in direction perpendicular to the beta stability valley takes place rather rapidly leading to a fast equilibration of the charge asymmetry, and diffusion continues rather slowly along the beta-stability valley. Fig. 2 illustrates very nicely the expected the mean-drift paths in the central collisions of ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems. The drift paths is obtained from the solution of the mean-field description of the TDHF equations. The di-nuclear system drifts towards symmetry during long contact time, but separates before reaching to the symmetry. Following this observation and borrowing an idea from references <cit.>, we parameterize the $N_{1} $ and $Z_{1} $ dependence of the potential energy surface of the di-nuclear system in terms of two parabolic forms,
\begin{eqnarray} \label{Eq14}
U(N_{1} ,Z_{1} )&=&\frac{1}{2} \alpha\left(z\cos \theta -n\sin \theta \right)^{2}\nonumber\\
&&+\frac{1}{2} \beta\left(z\sin \theta +n\cos \theta \right)^{2}.
\end{eqnarray}
Here, $z=Z_{0} -Z_{1} $, $n=N_{0} -N_{1} $ and $\theta $ denotes the angle between beta stability valley and the $N$ - axis in the $N-Z$ plane. We can determine these angles from the mean-drift paths in Fig. 2. The quantities $N_{0} $ and $Z_{0} $ denotes the equilibrium values of the neutron and proton numbers, which are approximately determined by the average values of the neutron and proton numbers of the projectile and target ions, $N_{0} =\left(N_{P} +N_{T} \right)/2$ and $Z_{0} =\left(Z_{P} +Z_{T} \right)/2$. The first term in this expression describes a strong driving force perpendicular to the beta stability valley, while the second term describes a relative weak driving force toward symmetry along the valley.
Mean drift paths of projectile-like fragments in $\left(N,Z\right)$ plane in the central collision of ${}^{40}$Ca + ${}^{238}$U (a) and ${}^{48}$Ca + ${}^{238}$U (b) systems at energies $E_{\text{c.m.}} =202.0$ MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively, obtained in TDHF calculations.
Following from the fluctuation-dissipation theorem, it is possible to relate the proton and neutron drift coefficients to the diffusion coefficients and the associated driving forces, in terms of the Einstein relations as follows <cit.>,
\begin{eqnarray} \label{Eq15}
v_{n}&=&-\frac{D_{NN} }{T} \frac{\partial U}{\partial N} =+\frac{D_{NN} }{T} \frac{\partial U}{\partial n} \nonumber\\
&=& D_{NN} \left[-\alpha \sin \theta \left(z\cos \theta -n\sin \theta \right)\right.\nonumber\\
&&\qquad\quad\left.+\beta \cos \theta \left(z\sin \theta +n\cos \theta \right)\right]
\end{eqnarray}
\begin{eqnarray} \label{Eq16}
v_{z}&=&-\frac{D_{ZZ} }{T} \frac{\partial U}{\partial Z} =+\frac{D_{ZZ} }{T} \frac{\partial U}{\partial z}\nonumber\\
&=&D_{ZZ} \left[+\alpha \cos \theta \left(z\cos \theta -n\sin \theta \right)\right.\nonumber\\
&&\qquad\quad\left.+\beta \sin \theta \left(z\sin \theta +n\cos \theta \right)\right].
\end{eqnarray}
Here, the temperature $T$ is absorbed into coefficients $\alpha$ and $\beta$, consequently temperature does not appear as a parameter in the description. We can determine $\alpha$ and $\beta$ by matching the mean values of neutron and proton drift coefficients obtained from the TDHF solutions. In this manner, microscopic description of the collision geometry and details of the dynamical effects are incorporated into the drift coefficients. In the liquid drop picture, the potential energy surfaces in perpendicular to the stability valley and along the stability valley have parabolic behaviors. Therefore we expect both coefficients $\alpha $ and $\beta $ to be positive. However, as a result of the quantal effects arising mainly from the shell structure, we observe that these coefficients exhibit fluctuations as a function of time, which can also be viewed as a function of the relative distance between ions. In Eqs. (<ref>-<ref>) for co-variances, we also need derivatives of drift coefficients with respect to proton and neutron numbers of projectile-like fragments. A great advantage of this approach, we can easily calculate these derivatives from drift coefficients to yield,
\begin{eqnarray} \label{Eq17}
\frac{\partial v_{n} }{\partial N_{1} } &=&-D_{NN} \left(\alpha \sin ^{2} \theta +\beta \cos ^{2} \theta \right),\\
\label{Eq18}
\frac{\partial v_{z} }{\partial Z_{1} } &=&-D_{ZZ} \left(\alpha \cos ^{2} \theta +\beta \sin ^{2} \theta \right),\\
\label{Eq19}
\frac{\partial v_{n} }{\partial Z_{1} } &=&-D_{NN} \left(\beta -\alpha \right)\sin \theta \cos \theta,\\
\label{Eq20}
\frac{\partial v_{z} }{\partial N_{1} } &=&-D_{ZZ} \left(\beta -\alpha \right)\sin \theta \cos \theta.
\end{eqnarray}
§ RESULTS
Employing the diffusion mechanism described in the previous section, we investigate nucleon exchange mechanism in central collisions of ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems near the quasi-fission region at bombarding energies $E_{\text{c.m.}} =202.0$ MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively. In the collision geometry, the elongation of deformed $^{238}$U target nucleus is taken as perpendicular direction to the beam (side collision). These energies are slightly below the fusion barriers. As a result, colliding ions stick together with a visible neck for a long time, and separate without forming a compound nucleus. Time dependent single-particle wave functions are determined from solutions of the TDHF equations by employing the code developed by P. Bonche et al. with the SLy4d Skyrme effective interactions <cit.>. Fig. 2 shows the mean-drift paths of the projectile-like fragments in $\left(N,Z\right)$ plane obtained in the TDHF calculations in the collisions of ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems. After a rapid charge equilibration, the system drift toward the symmetric fragmentation, which is specified by proton and neutron numbers $Z_{0} =56$ and $N_{0} =83$ for the ${}^{40}$Ca + ${}^{238}$U system and $Z_{0} =56$ and $N_{0} =87$ for the ${}^{40}$Ca + ${}^{238}$U system. The tangent of angle made by the mean-drift path with the $N$- axis is about $\tan \theta =2/3$ for both systems.
(Color online) Proton drift $v_{Z} $ (dashed line) and neutron drift $v_{N} $ (solid line) coefficients in the central collisions of ${}^{40}$Ca + ${}^{238}$U (a) and ${}^{48}$Ca + ${}^{238}$U (b) systems at energies $E_{\text{c.m.}} =202.0$MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively, obtained in TDHF calculations.
We use these values in parameterization of the driving potential energy in Eq. (<ref>) for both systems. In both collisions, the system stick together approximately from an initial touching time $t_{i} =8.0\times 10^{-22}$ seconds until separation time at $t_{f} =1.4\times 10^{-20}$ seconds. We observe that during the contact time, the mean number of proton and neutron drifts are about $z(t_{f})=10$, $n(t_{f} )=21$ in the ${}^{40}$Ca + ${}^{238}$U system, and $z(t_{f} )=6$, $n(t_{f} )=9$ in the ${}^{48}$Ca + ${}^{238}$U system, respectively. Even though the sticking time is about the same in both collisions, the smaller drift in ${}^{48}$Ca induced collision is due to the smaller bombarding energy by about $3.0$ MeV. Dashed and solid lines in Fig. 3 show the proton and neutron drift coefficients in the ${}^{40}$Ca + ${}^{238}$U system (a) and in the ${}^{48}$Ca + ${}^{238}$U system (b) as a function of collision time, which are obtained by the mean-field description of the TDHF. Because of small amplitude vibrations of the window positions, drift coefficients exhibit small fluctuations in time. This figure illustrates smoothed drift coefficients obtained by averaging over short time intervals. Probably due to shell effects, in ${}^{40}$Ca induced collision protons exhibit a rapid drift toward asymmetry during the initial phase of the collision, followed by persistent drift toward symmetry in both proton and neutron numbers in both systems. We determine the dimensionless parameters $\alpha (t)$ and $\beta (t)$ in Eqs. (<ref>,<ref>) by matching the proton and neutron drift coefficients to the results of obtained in TDHF calculations. In ${}^{40}$Ca induced collision, the parameter $\alpha (t)$ during the early times takes relatively large positive values around $\alpha (t)\approx 0.20$ until about $1.4\times 10^{-21} $ seconds while at later times it takes small fluctuating values around $\alpha (t)\approx \mp 0.05$. On the other hand parameter $\beta (t)$ take much smaller positive values $\beta (t)\approx 0.001$ as expected, also fluctuating in time. These coefficients exhibit similar behavior in the ${}^{48}$Ca induced collision. As noted above, we believe that fluctuations in these parameters are due to quantal effects arising mainly from the underlying shell structure.
(Color online) Proton diffusion $D_{ZZ} $ (dashed line) and neutron diffusion $D_{NN} $ (solid line) coefficients in the central collisions of ${}^{40}$Ca + ${}^{238}$U (a) and ${}^{48}$Ca + ${}^{238}$U (b) systems at energies $E_{\text{c.m.}} =202.0$ MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively, obtained in semi-classical framework of the SMF approach.
We calculate diffusions coefficients in Eq. (<ref>) for proton and neutron exchange in the semi-classical framework by employing the Wigner functions, which are determined in terms of the time-dependent single-particle wave functions of the TDHF solutions. The reduced Wigner functions are obtained by integrating over the phase-space volume on the window plane. The reduced Wigner functions exhibit fluctuations as a function of single-particle momentum and can take small negative values in classically forbidden regions. We eliminate these fluctuations and negative values of Wigner functions by performing a smoothing procedure as outline in <cit.>. Dashed and solid lines in Fig. 4 show the proton and neutron diffusion coefficients in ${}^{40}$Ca (a) and ${}^{48}$Ca (b) induced collisions as a function of time. Diffusion coefficients also exhibit small fluctuations in time due to small amplitude vibrations of the window positions. This figure illustrates smoothed diffusion coefficients obtained by averaging over short time intervals. Mainly as a result of the Coulomb barrier, the neutron diffusion coefficients are nearly twice as large as compared to the proton diffusion coefficients, in both systems.
(Color online) Proton dispersion $\sigma _{ZZ} $ (dashed line), neutron dispersion $\sigma _{NN}$ (solid line) and mixed dispersion $\sigma _{ZN} $ (dotted line) of the fragment distributions (projectile-like or target-like) in the central collisions of ${}^{40}$Ca + ${}^{238}$U (a) and ${}^{48}$Ca + ${}^{238}$U (b) systems at energies $E_{\text{c.m.}} =202.0$ MeV and $E_{\text{c.m.}} =198.7$ MeV, respectively.
We solve the coupled differential equations (10-12) for co-variances with the initial conditions
$\sigma _{ZZ}^{2} (t_{i} )=\sigma _{NN}^{2} (t_{i} )=\sigma _{ZN}^{2} (t_{i} )=0$. The results are plotted in Fig. 5 as a function time. Dashed and solid lines in the figure show proton and neutron dispersions $\sigma _{ZZ} (t)$, $\sigma _{NN} (t)$ and the square-root of co-variances $\sigma _{ZN} (t)$ in ${}^{40}$Ca (a) and ${}^{48}$Ca (b) induced collisions, respectively. The co-variances $\sigma _{ZN}^{2} (t)$ take negative values until $5.0\times 10^{-21} $ seconds and $3.2\times 10^{-21} $ seconds in ${}^{40}$Ca and ${}^{48}$Ca induced collisions, respectively.
Therefore, the square-root of co-variances are not shown in these time intervals. We observe that at the separation instant in the ${}^{40}$Ca + ${}^{238}$U collision, the dispersions of proton and neutron distributions are $\sigma _{ZZ} (t_{f} )=19$ and $\sigma _{NN} (t_{f} )=29$, and the co-dispersion is
$\sigma _{NZ} (t_{f} )=23$. The proton dispersion is nearly factor of 2.0 larger than the mean number of proton drift, while neutron dispersion is about 1.5 larger than the mean number of neutron drift. Target nucleus losses about 10 protons and 21 neutrons, but it can gain more protons and neutron by diffusion mechanism. On the other hand, at the separation instant in the ${}^{48}$Ca + ${}^{238}$U collision, the dispersions of proton and neutron distributions are $\sigma _{ZZ} (t_{f} )=14$ and $\sigma _{NN} (t_{f} )=21$, and the co-dispersion is $\sigma _{NZ} (t_{f} )=15$. The proton dispersion is nearly factor of 2.5 larger than the mean number of proton drift, while neutron dispersion is about 2.0 larger than the mean number of neutron drift. Target nucleus losses about 6 protons and 9 neutrons on the average, but it can gain more protons and neutron by diffusion mechanism. Therefore, in these collisions, diffusion mechanism near quasi-fission regime can help to populate elements heavier than uranium target nucleus. Probability distributions of the projectile-like or target-like fragments at the exit channel are determined by the correlated Gaussian of Eq. (<ref>), in which the magnitudes of co-variances and mean-values are taken at the separation instant of the collision. Fig. 6 shows equal probability lines for population of target-like fragments at the exit channel in the $\left(N,Z\right)$ plane in the ${}^{40}$Ca (a) and ${}^{48}$Ca (b) induced collisions. Probability of populating a fragment with neutron and proton numbers $\left(N_{2} ,Z_{2} \right)$ relative to populating the fragment with mean neutron and proton numbers is determined by $e^{-C}$, where $C$ indicate numbers on the equal probability lines in Fig. 6. In this figure dots at the centers of ellipses indicate the elements with the mean neutron and proton numbers at the exit channel. The mean values of neutron and proton numbers at the exit channel are $\left(\overline{N}_{2}=125,\overline{Z}_{2}=82\right)$ and $\left(\overline{N}_2=137,\overline{Z}_2=86\right)$ in ${}^{40}$Ca and ${}^{48}$Ca induced collisions, respectively. As an example, we can see from this figure, the probability of populating a heavy trans-uranium element with
$\left(N_{2}=155 ,Z_{2}=98 \right)$ relative to the populating the element with mean neutron and proton numbers is about $e^{-0.5}=0.6$ in the $^{40}$Ca + ${}^{238}$U collision. The relative population probability of the same element in the $^{48}$Ca + ${}^{238}$U collision has about the same magnitude. Fig. 7 illustrates the dispersion $\sigma _{AA} (t)$ of total mass number distributions of the projectile-like fragments or target-like fragments as a function of time in ${}^{40}$Ca (a) and ${}^{48}$Ca (b) induced collisions, respectively. The total dispersion is calculated from $\sigma _{AA}^{2} =\sigma _{ZZ}^{2} +\sigma _{NN}^{2} +2\sigma _{NZ}^{2} $.
(Color online) Equal probability lines for populating target-like elements with $C=0.5,1.0,1.5$ in the central collisions of ${}^{40}$Ca + ${}^{238}$U (a) and
${}^{48}$Ca + ${}^{238}$U (b).
Total mass dispersion $\sigma_{AA}$ of the fragment distribution of the fragment distributions (projectile-like or target-like) in the central collisions of ${}^{40}$Ca + ${}^{238}$U (a) and ${}^{48}$Ca + ${}^{238}$U (b) systems at energies $E_{\text{c.m.}}=202.0$ MeV and $E_{\text{c.m.}}=198.7$ MeV, respectively.
We note that the correlated Gaussian function of Eq. (<ref>), which is specified by the first two moments, provides an approximate description of the fragment population. The approximation is reasonable within the range of $\pm\sigma_{AA}$ around the center points, but becomes gradually unreasonable as we move out from the center points near to the tail of the distribution functions. For example, as seen from the upper ends of $C=1.5$ lines in Fig. 6, we observe finite but small probabilities for populating fragments even exceeding the total mass of the system. Therefore, in particular near the tail region, more accurate description of the fragment population probability is required.
In the present work, we do not discuss the energy dissipation and the excitation energy deposited in the populated fragments during nucleon diffusion process. However, we can provide an estimate of the excitation energy deposited in the fragments with mean values of protons and neutrons in the exit channel. It is possible to calculate the total excitation energy $E^*$ deposited in the mean fragments at the exit channel according to,
\begin{eqnarray}
\end{eqnarray}
where $Q$ denotes the $Q$-value and TKE is the asymptotic value of the total kinetic energy at the exit channel. We calculate the TKE for the mean-fragment exit channel by employing the TDHF [11]. Since the interaction time is very long, the initial relative kinetic energy totally dissipates and the TKE at the exit channel is essentially determined by the Coulomb repulsion. Because of very long interaction times, it is reasonable to consider the equilibration of the excitation energy. Under this circumstance, the excitation energy between the mean fragments is shared in proportion to their mass numbers,
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\end{eqnarray}
In ${}^{40}$Ca + ${}^{238}$U system, $\overline{A_1}=71$ and $\overline{A_2}=207$ with $A_{\text{tot}}=278$. In ${}^{48}$Ca + ${}^{238}$U system, $\overline{A_1}=63$ and $\overline{A_2}=223$ with $A_{\text{tot}}=286$. Calculation gives for the excitation energies $E^*_{1}=28.6$ MeV, $E^*_{2}=83.4$ MeV in the ${}^{40}$Ca + ${}^{238}$U system, and $E^*_{1}=11.2$ MeV, $E^*_{2}=39.8$ MeV in the ${}^{48}$Ca + ${}^{238}$U system.
§ CONCLUSIONS
We investigate nucleon exchange mechanism in the central collisions of ${}^{40}$Ca + ${}^{238}$U and ${}^{48}$Ca + ${}^{238}$U systems below the fusion barrier near the quasi-fission regime. Sufficiently below the fusion barrier, colliding system maintains a di-nuclear structure. As a result, it is possible to describe nucleon exchange as a diffusion mechanism which is familiar from the description of deep-inelastic heavy-ion collisions. The standard mean-field description based on the TDHF equations determines the mean drift path in the N-Z plane. In order to describe fluctuations around the drift path, we employ the microscopic basis of the SMF approach, which incorporates the mean-field fluctuations beyond the average description of the standard TDHF. We calculate diffusion coefficients for proton and neutron transfer mechanisms with the help of the SMF approach in the semi-classical framework. Proton and neutron diffusion occurs in the N-Z plane in a correlated manner according to the potential energy surface of the di-nuclear system. The potential energy surface along the beta-stability line and perpendicular to the stability line are parameterized in terms of two parabolic forms. Employing Einstein relations, we deduce simple analytical expressions for proton and neutron drift coefficients. Parameters of the drift coefficients are determined with help of the mean drift path obtained in the TDHF calculations. We determine the co-variances of the neutron and proton distributions of the projectile-like fragments. Calculations show that after a fast charge equilibration, large amount of mean drift in the numbers of protons and neutrons toward symmetry. We find large dispersions of the proton and neutron distributions of the projectile-fragments during very long interaction times. The mean numbers of proton and neutron of the target nucleus decrease due to drift toward symmetry. On the other hand, large values of proton and neutron dispersions indicate that diffusion mechanism helps to populate heavy trans-uranium elements near the quasi-fission regime.
S.A. gratefully acknowledges TUBITAK and the Middle East Technical University for partial support and warm hospitality extended to him during his visits. Authors gratefully acknowledge useful discussions with D. Lacroix and A. S. Umar. This work is supported in part by US DOE Grant No. DE-FG05-89ER40530, and in part by TUBITAK Grant No. 113F061.
R1 Y. T. Oganessian et al., Phys. Rev. Lett. 104, 142502 (2010).
R2 Y. T. Oganessian et al., Phys. Rev. Lett. 109, 162501 (2012).
R3 J. Khuyagbaatar et al., Phys. Rev. Lett. 112, 172501 (2014).
R4 S. Hofmann and G. Munzenberg, Rev. Mod. Phys. 72, 733 (2000).
R5 S. Hofmann et al., Eur. Phys. J. A 14, 147 (2002).
R6 C.-C. Sham et al., Z. Phys. A 319, 113 (1984).
R7 K.-H. Schmidt and Morawek, Rep. Prog. Phys. 54, 949 (1991).
R8 G. G. Adamian, N. V. Antonenko, and W. Scheid, Phys. Rev. C 68, 034601 (2003).
R9 V. Zagrebaev and W. Greiner. J. Phys. G 34, 2265 (2007).
R10 Y. Aritomo, Phys. Rev. C 80, 064604 (2009).
R11 V. E. Oberacker, A. S. Umar, and C. Simenel, Phys. Rev. C 90, 054605 (2014).
R12 J. Randrup, Nucl. Phys. A 327, 490 (1979).
R13 S. Ayik, Phys. Lett. B 658, 174 (2008).
R14 S. Ayik, K. Washiyama, and D. Lacroix, Phys. Rev. C 79, 054606 (2009).
R15 K. Washiyama, S. Ayik, and D. Lacroix, Phys. Rev. C 80, 031602(R) (2009).
R16 B. Yilmaz, S. Ayik, D. Lacroix, and K. Washiyama, Phys. Rev. C 83, 064615 (2011).
R17 B. Yilmaz, S. Ayik, D. Lacroix, and O. Yilmaz, Phys. Rev. C 90, 024613 (2014).
R18 S. Ayik et al., Phys. Rev. C 91, 054601 (2015).
R19 W. U. Schroder, J. R. Huizenga, and J. Randrup, Phys. Lett. B 98, 355 (1981).
R20 A. C. Merchant and W. Norenberg, Phys. Lett. B 104, 15 (1981).
R21 A. C. Merchant and W. Norenberg, Z. Phys. A 308, 315 (1982).
R22 K.-H. Kim, T. Otsuka, and P. Bonche, J. Phys. G 23, 1267 (1997).
|
1511.00270
|
This is a collection of open problems and conjectures from the seminars and problem sessions of the 2014
IML programme: Graphs, Hypergraphs, and Computing.
§ INTRODUCTION
This collection of problems and conjectures is based on a subset of the open problems from the seminar series and the problem sessions of the IML programme Graphs, Hypergraphs, and Computing. Each problem contributor has provided a write up of their proposed problem and the collection has been edited by Klas Markström.
§ SEMINAR JANUARY 16, 2014, JØRGEN BANG-JENSEN
§.§ Arc-disjoint spanning strong subdigraphs and disjoint Hamilton cycles
The arc set of every regular tournament can be decomposed into Hamilton cycles.
The Kelly Conjecture is true for tournaments on $n$ vertices where $n\geq M$ for some very large $M$.
As every $k$-regular tournament is
$k$-arc-strong, the following Conjecture implies the Kelly conjecture.
The arc set of every $k$-arc-strong tournament
$T=(V,A)$ can be decomposed into $k$ disjoint sets
$A_1,\ldots{},A_k$ such that each of the spanning subdigraphs $D_i=(V,A_i)$, $i=1,2,\ldots{},k$
is strongly connected.
Conjecture <ref> is true in the following cases:
* when $k=2$
* when every vertex of $T$ has in- and out-degree at least $37k$
* when there exists a non-trivial (both sides of the cut have at least 2 vertices) arc-cut of size $k$.
There exists a natural number $K$ such that every $K$-arc-strong digraph $D=(V,A)$ can be decomposed
into two arc-disjoint strong spanning subdigraphs $D_1=(V,A_1)$ and $D_2=(V,A_2)$.
The conjecture is true for tournaments with $K=2$ by Theorem <ref> and for locally semicomplete digraphs[a digraph is locally semicomplete if the out-neighbourhood and the in-neighbourhood of every vertex induces a semicomplete digraph. A digraph is semicomplete if it has no pair of non-adjacent vertices.]
with $K=3$ by a recent (non-trivial) result of Bang-Jensen and Huang (JCTB 2012).
For 2-regular digraphs (all in- and out-degrees equal to 2) the existence of arc-disjoint spanning
strong subdigraphs is equivalent to the existence of arc-disjoint Hamilton cycles. Hence, by the following theorem, it is NP-complete to decide whether a digraph has a pair of arc-disjoint strong spanning subdigraphs.
It is NP-complete to decide whether a 2-regular digraph contains two arc-disjoint Hamilton cycles.
There exists a natural number $C$ such that every $Ck^2\log^2{}k$-strong tournament contains $k$ arc-disjoint Hamilton cycles. This is best possible up to the log-factor.
Every 3-strong[A digraph $D$ is $k$-strong if it has at least $k+1$ vertices and $D-X$ is strongly connected for every subset $X$ of $V(D)$ of size at most $k-1$.] tournament has 2-arc-disjoint
Hamilton cycles.
§.§ Decompositions into vertex disjoint pieces/digraphs
There exists a naural number $C$ such that the vertex set of every $Ck^7t^4$-strong tournament
$T=(V,A)$ can be decomposed into disjoint subsets $V_1,V_2,\ldots{},V_t$ such that the tournaments
$T_i=\induce{T}{V_i}$ are $k$-strong for $i=1,2,\dots{},t$.
Can we also specify $t$ vertices $x_1,x_2,\ldots{},x_t$ and find $V_1,\ldots{},V_t$ as above such that
$x_i\in V_i$ holds for $i=1,2,\ldots{},t$?
Every digraph with minimum out-degree $2k-1$ contains $k$ disjoint directed cycles.
For $k=2$ this was verified in 1983 by Thomassen who also proved the existence of a function $f(k)$ such that every digraph out minimum out-degree at least $f(k)$ has $k$ disjoint cycles.
The bound $2k-1$ is best possible as seen by considering the complete digraph on $2k-2$ vertices.
Conjecture <ref> holds for tournaments
§.§ Further open problems that were mentioned
Is there a polynomial algorithm for deciding whether the underlying graph $UG(D)$ of
a digraph $D$ contains a 2-factor $C_1,C_2,\ldots{},C_k$ such that $C_1$ is a directed cycle in $D$,
while $C_i$, $i>1$ does not have to respect the orientations of arcs in $D$?
What is the complexity of the following problem: given a 2-edge-coloured bipartite graph $B=(U,V,E)$; decide whether $B$ has two edge-disjoint perfect matchings $M_1,M_2$ so that every edge of $M_1$ has colour 1, while $M_2$ may use edges of both colours?
§ SEMINAR JANUARY 16, 2014, OLEG PIKHURKO
Here are two open problems (as simplified as possible without losing their essence) that will be quite useful for measurable edge-colourings of graphings. –Oleg Pikhurko
§.§ An Open Question about Finite Graphs
Estimate the minimum $f=f(d)$ such that the following holds. Let $G$ be a (finite) graph of maximum degree at most $d$ with at most $d$ pendant edges pre-coloured (with no two incident pre-coloured edges having the same colour). Then this pre-colouring can be extended to a proper $(d+f)$-edge-colouring of $G$.
We can show $f=O(\sqrt d)$ suffices but it would be nice to prove that $f=O(1)$ is enough.
§.§ Towards a Measurable Local Lemma
For our purposes, it is enough to define a graphing $\C G$ as a graph whose
vertex set is the unit interval $I=[0,1]$ (with the Borel $\sigma$-algebra $\C B$ and
the Lebesgue measure $\mu$) and whose edge set $E$ can be represented as
E=\{\{x,y\}\mid x,y\in I,\ x\not=y,\ \exists i\in[k]\ \phi_i(x)=y\},
for some (finite) family of measure-preserving invertible maps $\phi_1,\dots,
\phi_k:I\to I$. The general definition (and an excellent introduction) to graphings can
be found in Lovász book <cit.>.
Let $d\to\infty$. Prove that there is $g(d)=o(d/\log d)$ such that any graphing $\C G$ of maximum degree at most $d$ admits a measurable
partition $I=A\cup B$ such that for every vertex $x$ is $(A,B)$-balanced, meaning that its degrees into $A$ and into $B$ differ
by at most $g(d)$.
Some remarks:
* The (finite) Local Lemma shows that the required partition $A\cup B$ exists for every finite graph
with $g(d)=O(\sqrt{d\log d})$. By the Compactness Principle, this extends to all countable graphs.
* In Problem <ref> it is enough to find a partition such that the measure of the set $X$ of $(A,B)$-unbalanced vertices $x$ is zero. Indeed, one can show that the measure of the union $Y$ of connectivity components of a graphing that intersect the null set $X$ is zero too. By the previous remark, we
can find a good partition of each component in $Y$; assuming the Axiom of Choice we can modify $A,B$
on the null set $Y$ to make every vertex $(A,B)$-balanced.
* Gabor Kun <cit.> proved some analytic version of the Local Lemma that in particular
implies for Problem <ref> that, for every $\varepsilon>0$, there is a measurable partition $I=A\cup B$ such that
the measure of $(A,B)$-unbalaned vertices is at most $\varepsilon$. But we do need the bad set to have measure zero in our application.
G. Kun.
A measurable version of the Lovász Local Lemma.
Talk at the 2-Day Combinatorics Colloquia, LSE/QMUL, London, 16 May,
L. Lovász.
Large Networks and Graph Limits.
Colloquium Publications. Amer. Math. Soc, 2012.
§ PROBLEM SESSION FEBUARY 6, 2014
§.§ Peter Allen
Let $R(G,G)$ denote the 2-colour Ramsey-number for $G$. It is known that there exists a constant $C$ such that if $G$ is a planar graph on $n$ vertices then $R(G,G)\leq C n$. It is also known that $C$ must be at least 4.
Is $C \leq 12$?
§.§ Hal Kierstead
An equitable coloring of a graph is a partiton of its
vertices into independent sets differing in size by at most one. In
1970 Hajnal and Szemerédi <cit.> proved that for every
graph $G$ and integer $k$, if $\Delta(G)<k$ then $G$ has an equitable
$k$-coloring. Their proof did not yield a polynomial algorithm. About
$35$ years later, Mydlarz and Szemerédi, and independently Kostochka
and I, found such algorithms, and then joined forces to produce an
$O(kn^{2})$ algorithm <cit.>.
The maximum Ore degree of a graph $G$ is $\theta(G):=\max\{d(x)+d(y):xy\in E(G)\}$.
Kostochka and I <cit.> proved that for every graph $G$ and integer
$k$, if $\theta(G)<2k$ then $G$ has an equitable $k$ coloring,
but the proof does not yield a polynomial algorithm.$ $
Is there a polynomial algorithm for constructing an equitable $k$-coloring
of any graph $G$ with $\theta(G)<2k?$
HSzA. Hajnal and E. Szemerédi, Proof of a conjecture of
Erdős, in: A. Rényi and V.T. Sós, eds. Combinatorial Theory
and Its Applications, Vol. II, North- Holland, Amsterdam, 1970, 601–623.
KKH. Kierstead and A. Kostochka, An Ore-type Theorem on
Equitable Coloring, Journal of Combinatorial Theory, Series
B 98 (2008), 226–234.
KKMSzH. Kierstead, A. Kostochka, M. Mydlarz, and E. Szemerédi,
A fast algorithm for equitable coloring, Combinatorica 30 (2010)
§.§ Jan van den Heuvel
§.§.§ Cyclic Orderings
The following is a very special case of a much more general conjecture
which appears in Kajitani et al. (1988).
Let $T_1,T_2,T_3$ be edge-disjoint spanning trees in a graph $G$ on $n$
vertices (so each tree has $n-1$ edges). Then there exists a cyclic
ordering of the edges in $E(T_1)\cup E(T_2)\cup E(T_3)$ such that every
$n-1$ cyclically consecutive edges in that ordering form a spanning tree.
In fact, the same question can be asked for any number of spanning trees.
For two trees the result is proved in Kajitani et al. (1988), who
in fact prove it in the stronger form according to Conjecture <ref>
Conjecture <ref> is really a problem about matroids. The following
appears in several places, including Gabov (1976),
Cordovil & Moreira (1993) and Wiedemann (2006).
Let $B=\{b_1,\dots,b_r\}$ and $B'=\{b'_1,\dots,b'_r\}$ be two disjoint
bases of a matroid. Then there is a permutation
$(b_{\pi(1)},\ldots,b_{\pi(r)})$ of the elements of $B$ and a permutation
$(b'_{\pi'(1)},\ldots,b'_{\pi'(r)})$ of the elements of $B'$ such that
the combined sequence
$(b_{\pi(1)},\ldots,b_{\pi(r)},b'_{\pi'(1)},\ldots,b'_{\pi'(r)})$ is a
cyclic ordering in which every $r$ cyclically consecutive elements form a
A weaker form of Conjecture <ref> is to just ask for a cyclic for a
cyclic ordering of $B_1\cup B_2$ (so we don't require that each base
appears as a consecutive part of the ordering). Even that conjecture is
open for matroids in general.
The most general conjecture in this area can be found in Kajitani
et al. (1988); partial and related results appear in van den
Heuvel & Thomassé (2012).
R. Cordovil and M.L. Moreira Bases-cobases graphs and
polytopes of matroids. Combinatorica 13 (1993), 157–165.
H. Gabow, Decomposing symmetric exchanges in matroid
bases. Math. Programming 10 (1976), 271–276.
J. van den Heuvel and S. Thomassé, Cyclic orderings and
cyclic arboricity of matroids. J. Combin. Theory Ser. B 102
(2012), 638-646.
Y. Kajitani, S. Ueno, and H. Miyano, Ordering of the
elements of a matroid such that its consecutive $w$ elements are
independent. Discrete Math. 72 (1988) 187–194.
D. Wiedemann, Cyclic base orders of
matroids. Manuscript, 2006. Retrieved 23 April 2007 from
Earlier version : Cyclic ordering of matroids. Unpublished
manuscript, University of Waterloo, 1984
§.§.§ Strong Colourings of Hypergraphs
All hypergraphs in this section are allowed to have multiple edges and
edges of any size. The rank $r(H)$ of a hypergraph $H$ is the size
of the largest edge.
The strong chromatic number $\chi_s(H)$ of a hypergraph $H$ is the
smallest number of colours needed to colour the vertices so that for every
edge the vertices in that edge all receive a different colour. (So this is
the same as the chromatic number of the graph obtained by replacing every
edge by a clique.)
Let's call a derived graph of a hypergraph $H$ a graph $G$ on the
same vertex set as $H$ where for each edge $e$ of $H$ of size at least two
we choose a pair $u,v\in e$ and add the edge $uv$ to $G$. And let's call
the following parameter the graph chromatic number of $H$:
\[\chi_d(H)=\max\{\,\chi(G)\mid\text{$G$ is a derived graph of
It is obvious that $\chi_s(H)\ge\max\{r(H),\chi_d(H)\}$, and it is not so
hard to prove that $\chi_s(H\le\chi_d(H)^{\binom{r(H)}{2}}$. A little bit
more thinking will give
\[\chi_s(H)\le\chi_d(H)^{r(H)-1}.\]
The question is the find better upper bounds of $\chi_s(H)$ in terms
of $\chi_d(H)$ and $r(H)$. It might even be true that there is an upper
bound that is linear in $r(H)$.
Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{R}_+$ so
that for every hypergraph $H$ we have
\[\chi_s(H)\le f(\chi_d(H))\cdot r(H)\,\text{?}\]
A neat argument, due to my former PhD student Alexey Pokrovskiy, gives a
proof that if $\chi_d(H)=2$, then $\chi_s(H)=r(H)$.
Other special classes of hypergraphs for which Question <ref> has a
positive answer can be found in Dvořák & Esperet (2013).
That paper was also the inspiration for starting to think about this type
of questions.
Z. Dvořák and L. Esperet, Distance-two coloring of
sparse graphs. arXiv:1303.3191 [math.CO],
<http://arxiv.org/abs/1303.3191> (2013), 13 pages.
§.§ Victor Falgas-Ravry
§.§.§ Largest antichain in the independence complex of a graph
Write $Q_n$ for the collection of all subsets of $[n]=\{1,2, \ldots n\}$. We denote by $Q_n^{(r)}$ the $r^{\textrm{th}}$ layer of $Q_n$, that is, the collection of all subsets of $[n]$ of size $r$.
A family $\mathcal{A}\subseteq Q_n$ is an antichain if for every pair of distinct elements $A,B \in \mathcal{A}$, $A$ is not a subset of $B$ and $B$ is not a subset of $A$. How large an antichain can we find in $Q_n$? Clearly each layer of $Q_n$ forms an antichain, and a celebrated theorem of Sperner from 1928 asserts that we cannot do better than picking a largest layer:
Let $\mathcal{A}\subseteq Q_n$ be an antichain. Then
\[\vert \mathcal{A} \vert \leq \max_r \vert Q_n^{(r)}\vert.\]
I am interested in a generalisation of Sperner's theorem where $\mathcal{A}$ is restricted to a subset of $Q_n$: suppose we are given a graph $G$ on $[n]$. The independence complex of $G$ is the collection of all independent sets from $V(G)=[n]$,
\[Q(G)=\{A \subseteq [n]: \ A \textrm{ independent in }G\}.\]
We write $Q^{(r)}(G)= \{A \in Q(G): \ \vert A\vert =r\}$ for the $r^{\textrm{th}}$ layer of $Q(G)$, and define the width of $G$, $s(G)$, to be the size of a largest antichain in $Q(G)$. Clearly we have
\begin{align}
s(G)&\geq \max_r \vert Q^{(r)}(G) \vert.\label{trivialbound}
\end{align}
In general, $s(G)$ can be much larger than this: it is not hard to construct examples of graph sequences $(G_n)_{n\in \mathbb{N}}$ for which $\max_r \vert Q^{(r)}(G_n\vert) =o\left(s(G_n)\right)$. However one would expect that if $G$ is reasonably homogeneous then ( <ref>) should be close to tight.
When do we have (almost) equality in ( <ref>) ? What conditions on $G$ are sufficient to guarantee (almost) equality ?
I am particularly interested in the cases where $G=C_n$, the cycle of length $n$, or where $G=P_n$, the path of length $n-1$. In this setting, an analogue of the Erdős̋–Ko–Rado theorem in $Q(G)$ was proved by Talbot <cit.>, using an ingenious compression argument. It is known <cit.> that the size of a largest antichain in a class of graphs including both $C_n$ and $P_n$ is of the same order as the size of a largest layer. However we really should have equality here:
\[s(C_n)= \max_r \vert Q(C_n)^{(r)} \vert \textrm{ and }s(P_n)= \max_r \vert Q(P_n)^{(r)}\vert.\]
It would also be interesting to know what happens in the case of random graphs:
Let $p=cn^{-1}$, for some constant $c>0$. Is it true with high probability that
\[s(G_{n,p})=(1+o(1)) \max_r \vert Q(G_{n,p})^{(r)} \vert? \]
V. Falgas-Ravry.
Sperner's problem for $G$-independent families.
Accepted, Combinatorics, Probability & Computing (2014). Arxiv ref: 1302.6039.
E. Sperner.
Ein Satz über Untermengen einer endlichen Menge (in
Mathematische Zeitschrift, 27(1):544–548, 1928.
J. Talbot.
Intersecting families of separated sets.
Journal of the London Mathematical Society, 68(1):37–51, 2003.
§ PROBLEM SESSION FEBUARY 19, 2014
§.§ Miklós Simonovits
[Paul Erdős, via M. Simonovits]
Is it true that if $G_n$ is a 4-chromatic $n$-vertex graph and
deleting any edge of it we get a 3-chromatic graph, then the
minimum degree of $G_n$ is $o(n)$ (as $n\to\infty$)?
Motivation, partial results: A graph $G$ is called $k$-color-critical if it is $k$-chromatic but deleting any edge of
$G$ we get a $k-1$-chromatic graph. (Actually, we could speak of
edge-critical and vertex-critical graphs, but we stick to the
edge-critical case.)
Bjarne Toft and myself, using a construction of Toft and a
transformation of mine constructed (infinitely many) 4-colour-critical graphs where the
minimum degree is $>c\root 3 \of n$. (Simonovits, M.: On
colour-critical graphs. Studia Sci. Math. Hungar. 7 (1972), 67–81, and
Toft, B. Two theorems on critical 4-chromatic graphs, Studia
Sci. Math. Hungar. 7 (1972), 83–89.) I do not know of anything with
higher minimum degree (though I may overlook some newer results?)
A trivial construction of G. Dirac, obtained by joining two odd
$n/2$-cycles completely shows that there exist $6$-critical graphs
with minimum degrees $n/2+2$. The difficulties occur for 4 and
5-critical graphs. (The 3-critical graphs are just the odd cycles.)
As I wrote, a basic ingredient of our construction was an earlier
construction of Bjarne Toft, a 4-critical graph with $\approx
{n^2\over 16}$ edges, where the vertices are in four groups of $n/4$
vertices, and (a) the first and last groups form two odd cycles, (b) the
second and third groups form a complete bipartite graphs, (c) the first
group is joined to the second class by a 1-factor and the third group
to the last group also by a 1-factor.
§.§ Fedor Fomin
For a given $n$-vertex planar graph $G$, is it possible to find in polynomial time (or to show that this is NP-hard) an independent set of size $\lfloor n/4 \rfloor +1$?
§.§ Carsten Thomassen
Smith's theorem says that, for every edge e in a cubic graph, there is in an even number of Hamiltonian cycles containing e. As a consequence, every cubic Hamiltonian graph has at least 3 Hamiltonian cycles.
If e is an edge of a Hamiltonian bipartite cubic graph G, then G has en even number of Hamiltonian cycles through e. Using Smith's theorem once more, also G-e han an even number of Hamiltonian cycles. Hence the total number of Hamiltonian cycles is even, in contrast to the situation for non-bipartite graphs where it is 3 for infinitely many graphs.
Does there exist a 3-connected cubic bipartite graph having an edge e such that there are precisely two Hamiltonian cycles containing e? (Otherwise, there will be at least 4 such cycles.)
Does there exist a 3-connected cubic bipartite graph having precisely 4 Hamiltonian cycles? (Otherwise, there will be at least 6 such cycles.)
It is an old problem whether a second Hamiltonian cycle in a Hamiltonian cubic graph can be found in polynomial time. Andrew Thomason”s lollipop method is a simple algorithm producing a second Hamiltonian cycle, but it may take exponential many steps. The known examples have many edge-cuts with three edges.
Does there exist a family of cubic, cyclically 4-edge-connected Hamiltonian graphs for which the lollipop method takes superpolynomially many steps?
If Problem <ref> has a negative answer, one can show that there exists a polynomially bounded algorithm for finding a second Hamiiltonian cycle in a cubic Hamiltonian graph.
§.§ Jörgen Backelin
The problem, stated briefly:
Determine the exact chromatic numbers for shift graphs with short vertices for cyclically ordered points.
Consider a finite set $C$, ordered cyclically. Define a shift graph by letting the vertices be all
$r$-subsets of $C$, for some small $r$ (I suggest doingt his for $r=2$ or 3, in the first place), and by
letting two vertices form an edge if their are disjoint, and intertwined in a prescribed manner.
Determine the exact chromatic numbers for these graphs.
Detailed explanation:
A cyclic order on a finite set
$C$ intuitively is what you think it should be, if you place the elements in a circle and follow it e. g.
clockwise: It is not meaningful to say that the element a precedes b; but it is meaningful to say
that starting from $a$ we pass $b$ before encountering $c$. Thus, a formal definition would have to deal with ternary rather than binary predicates (properties).
Cyclic orders was treated by P. J. Cameron 1978 (Math. Z. 148, pp. 127-139). One way to define them
formally is as a property P, which holds for some triples of different elements in C, such that for
any different $a, b, c, d$ in $C$ we have:
* Precisely one of $P(a,b,c)$ and $P(a,c,b)$ holds.
* If $P(a,b,c)$, then $P(b,c,a)$.
* If $P(a,b,c)$ and $P(a,c,d)$, then $P(a,b,d)$.
(Think of $P(a,b,c)$ as the statement "Starting from $a$, we pass $b$ before arriving at $c$".)
"Intertwining vertices" should be done analogously as for ordinary shift graphs. E. g., for $r = 2$, there are two possibilities for two disjoint
vertices $\{a,b\}$ and $\{c,d\}$: Either the pattern XXOO, which holds if either both $P(a,b,c)$ and $P(a,b,d)$ hold, or neither does, or the pattern XOXO. The
first case has a trivial chromatic number for ordinary shift graphs, but not self-evidently so in the cyclic order situation.
For $r=3$, there are essentially only three intertwinement patterns: XXXOOO, XXOXOO, and XOXOXO.
I do not know if there are any direct applications of this. In general, working with cyclic orders removes a kind of lack of balance between different parts
of a graph, which means that we often may construct more efficient examples of graphs with certain properties in this manner. Thus, I find this a more natural setting.
Note on generalisations:
There are cyclically ordered sets of any cardinality. I do not know if there is a theory for "cyclically well-ordered sets" of large cardinalities, though.
§.§ Andrzej Ruciński
For graphs $F$ and $G$ and a positive integer $r$, we write $F\to(G)_r^v$ if every $r$-coloring of the vertices of $F$ results in a monochromatic copy of $G$ in $F$ (not necessarily induced). E.g., $K_{r(s-1)+1}\to(K_s)^v_r$ by the Pigeon-hole Principle, where $K_n$ is the complete graph on $n$ vertices. Define
$$mad(F)=\max_{H\subseteq F}\frac{2|E(H)|}{|V(H)|}$$
$$m_{cr}(G,r)=\inf\{mad(F)\;:\;F\to (G)_r^v\}.$$
Determine or estimate $m_{cr}(G,r)$ for every graph $G$ and $r\ge2$.
Sasha Kostochka receiving the award
It is known <cit.> that
$$r\max_{H\subseteq G}\delta(H)\le m_{cr}(G,r)\le 2r\max_{H\subseteq G}\delta(H),$$
where $\delta(G)$ is the minimum vertex degree in $G$.
The lower bound is attained by complete graphs, that is, $m_{cr}(K_s,r)=mad(K_{r(s-1)+1})=r(s-1)$.
On the other hand, the upper bound is asymptotically achieved by large stars as it was proved in <cit.> that, in particular for $r=2$ colors,
$$4-\frac4{k+1}\le m_{cr}(S_k,2) \le 4-\frac{2(k+1)}{k^2+1},$$
where $S_k$ is the star with $k$ edges.
For $k=2$ this reads
$$\frac83\le m_{cr}(S_2,2) \le\frac{14}5.$$
In <cit.> I offered 400,000 zł (Polish currency in 1993, equivalent after denomination of 1995 to 40 PLN) for the determination of the exact value of
$ m_{cr}(S_2,2)$.
Recently, it was pointed out by A.Pokrovskiy that a result of Borodin, Kostochka, and Yancey <cit.> 1-improper 2-colorings of sparse graphs yields that
During the Open Problem session at the Mittag-Leffler Institute I handed to Sasha Kostochka an envelope with four 10 PLN bills so that he could share the award among his co-authors and A. Pokrovskiy as well.
Determine $m_cr(S_2,3)$ (no monetary award for that!) The current bounds, unimproved since 1994 are
$$\frac{18}5\le m_cr(S_2,3)\le \frac{22}5.$$
kr A. Kurek and A. Rucinski, Globally sparse vertex-Ramsey graphs, J.Graph Theory 18 (1994) 73-81.
sasha O. V. Borodin, A. Kostochka, and M. Yancey, On 1-improper 2-colorings of sparse graphs, Discrete Mathematics 313 (2013) 2638-2649.
§ SEMINAR FEBRUARY 20, 2014, ALLAN LO
For a graph $G$, we know that $\chi'(G) = \Delta (G)$ or $\chi'(G) = \Delta (G)+1$.
A subgraph $H$ of $G$ is overfull if $e(H) > \Delta (G) \lfloor |H|/2 \rfloor$ (note this requires $|H|$ to be odd).
Note that an overfull subgraph is a trivial obstruction for $\chi'(G) = \Delta(G)$.
In 1986, Chetwynd and Hilton <cit.> gave the following conjecture.
A graph $G$ on $n$ vertices with $\Delta (G) \geq n/3$ satisfies $\chi'(G)=\Delta(G)$ if and only if $G$ contains no overfull subgraph.
(Some remark about regular graphs)
The $1$-factorization conjecture is a special case of the overfull subgraph conjecture.
If $G$ is $d$-regular and contains no overfull subgraph, then ($|G|$ is even) and every odd cut has size at least $d$ edges.
So $G$ has a $1$-factor.
Meredith <cit.> showed that for all $d\ge 3$, there exists a $d$-regular graph $G$ on $20d-10$ vertices with $\chi'(G) = d+1$, which contains no overfull subgraph.
overfull A.G. Chetwynd and A.J.W. Hilton,
Star multigraphs with three vertices of maximum degree,
Math. Proc. Cambridge Philosophical Soc. 100 (1986), 303–317.
meredith G.H.J. Meredith,
Regular $n$-valent $n$-connected nonHamiltonian non-$n$-edge-colorable graphs,
J. Combinatorial Theory Ser. B, 14 (1973), 55–60.
§ PROBLEM SESSION MARCH 5, 2014
§.§ Jacques Verstraëte
§.§.§ A problem on Majority Percolation
Let $G$ be a finite graph, and let $p \in [0,1]$. Suppose that vertices of $G$ are randomly and independently infected with probability $p$ – this is the infection probability. Then consider the following deterministic rule: at any stage, an uninfected vertex becomes infected if strictly more than half of its neighbors are
infected. Let $A(G)$ be the event that in finite time every vertex of $G$ becomes infected with associated probability measure $P_p$.
Does there exist a sequence of graphs $(G_n)_{n \in \mathbb N}$ such that for every $p > 0$:
\[ \lim_{n \rightarrow \infty} P_p(A(G_n)) = 1 \quad \mbox{?}\]
I believe the answer is no. This process is a version of a process called bootstrap percolation.
The most studied case is the $n \times n$ grid $\Gamma_n$, with the rule that at least two infected neighbors of an uninfected vertex cause the vertex to become infected. In this case, a remarkable paper of Holroyd shows that for all $\varepsilon > 0$,
\[ P_p(A(\Gamma_n)) \rightarrow \left\{\begin{array}{ll}
1 & \mbox{ if } p > (1 + \varepsilon)\frac{\pi^2}{18\log n} \\
0 & \mbox{ if } p < (1 - \varepsilon)\frac{\pi^2}{18\log n}
\end{array}\right.
\]
Finer control of the relationship between $\varepsilon$ and $n$ was obtained by Graver, Holroyd and Morris.
§.§ Klas Markström
Given a matrix $m \in \mathrm{GL}(n,2)$, i.e an invertible matrix with entries 0/1, we let $\mathcal{D}(m)$ denote the smallest number of row-operations we can use in order to reduce $m$ to the identity matrix. Equivalently $\mathcal{D}(m)$ is the distance from $m$ to $I$ in the Cayely graph $\mathrm{Cay}(\mathrm{GL}(n,2),S)$, where $S$ is the set of elementary matrices. In <cit.> an algorithm was given which can row reduce a matrix $m$ to te identity using
$$\frac{n^2}{\log_{2} n} + o\left(\frac{n^2}{\log_{2} n}\right ) $$
row operations, and it was proven that the expected value of $\mathcal{D}(m)$ for a random matrix from $\mathrm{GL}(n,2)$ is not less than half of that. Equivalently, this shows that the diameter of $\mathrm{Cay}(\mathrm{GL}(n,2),S)$ is at most the first bound, and the average distance is at least the second.
* Give an explicit (non-random) example of an matrix $m \in \mathrm{GL}(n,2)$ such that $\mathcal{D}(m)\geq 100 n$
* Give an explicit example of an matrix $m \in \mathrm{GL}(n,2)$ such that $\mathcal{D}(m)\geq n\log n$
AHM D. Andrén, L. Hellström, K. Markstrom
On the complexity of matrix reduction over finite fields,
Advances in Applied Mathematics 39 (2007), 428–452.
§.§ Andrzej Ruciński
Given integers $1\leq \ell< k$, we define an $\ell$-overlapping cycle as a $k$-uniform
hypergraph (or $k$-graph, for short) in which, for some cyclic ordering of its vertices, every
edge consists of $k$ consecutive vertices, and every two consecutive edges (in the natural ordering
of the edges induced by the ordering of the vertices) share exactly $\ell$ vertices. If $H$
contains an $\ell$-overlapping Hamiltonian cycle then $H$ itself is called
A $k$-graph $H$ is
$\ell$-Hamiltonian saturated, $1\le \ell\le k-1$, if $H$ is not $\ell$-Hamiltonian but for
every $e\in H^c$ the $k$-graph $H+e$ is such. For $n$ divisible by $k-\ell$, let $sat(n,k,\ell)$ be
the smallest number of edges in an $\ell$-Hamiltonian saturated $k$-graph on $n$ vertices.
In the case of graphs, Clark and Entringer <cit.> proved in 1983 that $sat(n,2,1)=\lceil
\tfrac{3n}2\rceil$ for $n$ large enough.
A. Żak showed that for $k\ge2$, $sat(n,k,k-1)=\Theta(n^{k-1})$ <cit.>. Together, we proved
that for all $k\ge3$ and $\ell=1$, as well as for all $\tfrac45k\le\ell\le k-1$
\begin{equation}\label{1}
\end{equation}
and conjectured that (<ref>) holds for all $k$ and $1\le\ell\le k-1$ <cit.>. The smallest
open case is $k=4\,, \ell=2$. Recently, we have got some partial results: $sat(n,k,\ell)=O(n^{
(k+\ell)/2 })$ and $sat(n,4,2)=O( n^{14/5} )$.
CE L.Clark and R. Entringer, Smallest maximally non-Hamiltonian
graphs, Period. Math. Hungar., 14(1) (1983) 57-68.
RZ A. Rucinski, A. Żak, Hamilton Saturated Hypergraphs of Essentially Minimum Size,
Electr. J. Comb. 20(2) (2013) P25.
zak A. Żak, Growth order for the size of smallest hamiltonian chain saturated uniform hypergraphs,
Eur. J. Comb. 34(4) (2013) 724-735.
§ PROBLEM SESSION MARCH 19, 2014
§.§ Brendan McKay
How many $n/2$-cycles can a cubic graph have?
Given a simple cubic graph with $n$ vertices, what is a good upper bound on the number of cycles of length $n/2$ it can have?
A random cubic graph has $\Theta((4/3)n/n)$ cycles of length $n/2$. So do random cubic bipartite graphs. Also the whole cycle space has
size $2^{n/2+1}$, so twice that is a (silly) upper bound.
The actual maximums for $n=4,6,\ldots,24$ are: 0,2,6,12,20,20,48,48,132,118,312 (not in OEIS). All these are achieved uniquely except that for 20 vertices
there are two graphs with 132 10-cycles.
Maximum automorphism group for a 3-connected cubic graph
Let $a(n)$ be the greatest order of the automorphism group of a
3-connected cubic graph with $n$ vertices. I conjecture: for $n\ge 16$, $a(n)<n\, 2^{n/4}$.
There is a paper of Opstall and Veliche that finds the maximum over all
cubic graphs, but the maximum occurs for graphs very far from being 3-connected.
When $n$ is a multiple of 4 there is a vertex-transitive cubic graph
achieving half the conjectured bound, so if true the bound is pretty sharp.
A paper of Potočnik, Spiga and Verret (arxiv.org/abs/1010.2546), together with some computation, establishes the conjecture for vertex-transitive graphs, so the remaining problem is whether one can do better for non-transitive graphs. For 20, and all odd multiples of 2 vertices from 18 to at least 998 (but not for 4–16 or 24 vertices) the graph achieving the maximum is not vertex-transitive.
Probability that a random integer matrix is positive
Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, equivalently the probability that a random matrix from the uniform distribution on $M(n,k)$ has no zero entries.
Obviously $P(n,k)=0$ for $k<n$. It seems obvious that $P(n,k)$ should be increasing as a function of $k$ when $n$ is fixed and $k\ge n$, but can you prove it?
We know that $P(n,k)=|M(n,k)|/|M(n,k)|$. Also, note that $M(n,k)$ is the set of integer points in the $k$-dilated Birkhoff polytope, and $P(n,k)$
is the fraction of such points that don't lie on the boundary. Ehrhart theory tells us that $|M(n,k)|=H_n(k)$ where $H_n$ is a polynomial, and that
$P(n,k) = (-1)^{n+1} H_n(-k)/H_n(k)$. Does it help?
§.§ Andrew Thomason (by proxy)
Suppose you are given a set $E$ and a collection of finite sequences of
elements of $E$. We now wish to determine if there is a graph such that $E$ is the
edges of the graph and the sequences are (nice, simple) paths in the graph.
This can be done, the graph would have at most $2|E|$ vertices so you
can search all possibilities. So the question would be whether you can do
it efficiently.
§.§ Bruce Richter
Hajós conjectured that if the chromatic number $\chi(G)$ of a graph $G$ is at least $r$, then $G$ contains a subdivision of $K_r$. Hadwiger conjectured that $G$ has $K_r$ as a minor.
Albertson's Conjecture: If $\chi(G)\ge r$, then the crossing number cr$(G)\ge \textrm{cr}(K_r)$.
This is known for $r\le 16$, with the best result being that of J. Barát and G. Tóth, Towards the Albertson conjecture, Elec. J. Combin. 2010. The interesting thing here is that the crossing number of $K_r$ is only known when $r\le 12$.
The conjecture is easy for large $r$-critical graphs. The problem occurs when $|V(G)|$ is just a little larger than $r$.
§.§ Alexander Kostochka
§.§.§ Problem 1
For nonnegative integers $j,k$, a $(j,k)$-coloring of graph $G$ is a partition $V(G)=J\cup K$ such that
$\Delta(G[J])\leq j$ and $\Delta(G[K])\leq k$. An old result of Lovász implies that every graph $G$ with
$\Delta(G)\leq j+k+1$ has a $(j,k)$-coloring. The proof is short and one may wonder about possible
Brooks-type refinements of the result. In seeking such possibilities, Corréa, Havet, and Sereni [1],
conjectured that for sufficiently large $k$ (say, $k>10^6$), every planar graph $G$ with $\Delta(G)\leq 2k+2$ has a $(k,k)$-coloring.
[1] R. Corréa, F. Havet, and J.-S. Sereni,
About a Brooks-type theorem for improper colouring.
Australas. J. Combin. 43 (2009), 219–230.
§.§.§ Problem 2
A graph is a circle graph, if it is the
intersection graph of a family of chords of a circle. Circle graphs arise
in many combinatorial problems ranging from sorting problems to studying planar
graphs to continous fractions. In particular,
for a given permutation $P$ of $\{1,2,\ldots,n\}$,
the problem of finding the minimum number of stacks needed to
obtain the permutation $\{1,2,\ldots,n\}$ from $P$ reduces to finding
the chromatic number of a corresponding circle graph.
There are polynomial algorithms for finding the clique number and the independence number of a circle graph,
but finding the chromatic number of a circle graph is an NP-hard problem.
Let $f(k)$ denote the maximum chromatic number of a circle graph with clique number $k$.
Gyárfás <cit.>
proved that $f(k)$ is well defined and $ f(k)\leq 2^k(2^k-2)k^2$. The only known exact
value is $f(2)=5$. The best bounds known to me are
$$0.5k(\ln k-2)\leq f(k)\leq 50\cdot 2^k.$$
The lower bound is only barely superlinear, and the upper is very superlinear. It would be
interesting to improve any of them. More info on circle graphs and their colorings could
be found in <cit.>.
Gol M. Golumbic, Algorithmic graph theory and
perfect graphs.
Academic Press, 1980.
G1 A. Gyárfás,
Problems from the world surrounding perfect graphs,
Proceedings of the International Conference on Combinatorial
Analysis and its
Applications (Pokrzywna, 1985).
Zastos. Mat. 19 (1987), 413–441.
G2 A. Gyárfás,
On the chromatic number of multiple interval graphs and overlap
graphs, Discrete Math. 55 (1985), 161–166.
AK A. Kostochka,
Coloring intersection graphs of geometric figures with a given clique number.
Towards a theory of geometric graphs, 127–138,
Contemp. Math., 342, Amer. Math. Soc., Providence, RI, 2004.
§.§ David Conlon
Monochromatic cycle partitions in mean colourings
A well-known result of Erdős, Gyárfás and Pyber <cit.> says that there exists a constant $c(r)$, depending only on $r$, such that if the edges of the complete graph $K_n$ are coloured with $r$ colours, then the vertex set of $K_n$ may be partitioned into at most $c(r)$ disjoint monochromatic cycles, where we allow the empty set, single vertices and edges to be cycles. For $r = 2$, it is known <cit.> that two disjoint monochromatic cycles of different colours suffice, while the best known general bound <cit.> is $c(r) = O(r \log r)$.
With Maya Stein <cit.>, we recently considered a generalisation of this monochromatic cycle partition question to graphs with locally bounded colourings. We say that an edge colouring of a graph is an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Note that we do not restrict the total number of colours. Somewhat surprisingly, we prove that even for local colourings, a variant of the Erdős-Gyárfás-Pyber result holds.
The vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r^2 \log r)$ disjoint monochromatic cycles.
For $r = 2$, we have the following more precise theorem.
The vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours.
An edge colouring of a graph is said to be an $r$-mean colouring if the average number of colours incident to any vertex is at most $r$. We suspect that a theorem analogous to Theorem <ref> may also hold for $r$-mean colourings but have been unable to resolve this question in general.
Does there exist a constant $m(r)$, depending only on $r$, such that the vertex set of any $r$-mean coloured graph may be partitioned into at most $m(r)$ cycles?
We can show that the vertex set of any $2$-mean coloured graph may be partitioned into at most two cycles of different colours but the proof uses tricks which are specific to the case $r = 2$.
S. Bessy and S. Thomassé, Partitioning a graph into a cycle and an anticycle: a proof of Lehel's conjecture, J. Combin. Theory Ser. B 100 (2010), 176–180.
D. Conlon and M. Stein, Monochromatic cycle partitions in local edge colourings, submitted.
P. Erdős, A. Gyárfás and L. Pyber, Vertex coverings by monochromatic cycles and trees,
J. Combin. Theory Ser. B 51 (1991), 90–95.
A. Gyárfás, M. Ruszinkó, G. Sárközy and E. Szemerédi, An improved bound for the monochromatic cycle partition number,
J. Combin. Theory Ser. B 96 (2006), 855–873.
§ PROBLEM SESSION APRIL 2ND, 2014
§.§ Miklós Simonovits
The problem we discuss here is informally as follows:
Is it true that the Turán number of an infinite family of forbidden
(bipartite) graphs $\mathcal{L}$ can be approximated arbitrarily well in the
exponent by finite subfamilies?
More precisely, here we consider ordinary simple graphs: no loops or
multiple edges are allowed. ${\bf ext}(n,\mathcal{L})$ denotes the maximum number
of edges a graph $G_n$ on $n$ vertices can have without containing
subgraphs from $\mathcal{L}$. The problem below is motivated by the fact that
if $\mathcal{C}$ is the family of all cycles then ${\bf ext}(n,\mathcal{C})=n-1$, however
for any finite $\mathcal{C}^*\subset\mathcal{C}$, for some $\alpha=\alpha(\mathcal{C}^*)>0$,
and $c_1>0$, ${\bf ext}(n,\mathcal{C}^*)>c_1n^{1+\alpha}$. This means that some
kind of compactness is missing here. On the other hand, the continuity
of the exponent easily follows from Bondy-Simonovits theorem:
$${\bf ext}(n,C_{2k})=O(n^{1+{1/k}}).$$
This means a continuity in the exponent.
[Continuity of the exponent]
Let $\mathcal{L}$ denote an infinite family of bipartite (excluded) graphs and
$$\mathcal{L}_m:=\{L~:~L\in\mathcal{L},~ v(L)\le m\}.$$
Is it true that if for some $\alpha>0$ and $c>0$ we have ${\bf ext}t(n,\mathcal{L})=
O(n^{1+\alpha})$, then for any $\varepsilon>0$, we have
$${\bf ext}(n,\mathcal{L}_m)=O(n^{1+\alpha+\varepsilon}),\quad\mbox{as}\quad
if $m$ is large enough?
§.§ Erik Aas and Brendan McKay
Let $G = (V,E)$ be a connected simple graph, $k^E$ its edge space over the field $k$. We are interested in the subspace $C(G)$ spanned by the (characteristic vectors of the) cycles of $G$.
When $k$ is the field with two elements, it is a classical fact that the dimension of $C(G)$ is $|E(G)| - |V(G)| + 1$. This can be proved by providing an explicit basis of $C(G)$, as follows. Pick any spanning tree $T$ of $G$, and for each edge $e$ not in $T$ consider the unique cycle whose only edge not in $T$ is $e$. These cycles are linearly independent and thus span $C(G)$ in this case.
Now, when $k$ does not have characteristic $2$, the dimension is not a simple function of $|E(G)|$ and $|V(G)|$. However, in the case $G$ is $3$-edge-connected, it is not difficult to prove that in fact the dimension of $C(G)$ is $|E(G)|$.
Question: Is there a nice explicit basis for $C(G)$ consisting of cycles indexed by $E$, assuming $G$ is $3$-edge-connected?
§.§ András Gyárfás
A 3-tournament $T_n^3$ is the set of all
triples on vertex set $[n]=\{1,2,\dots,n\}$ such that in each triple
some vertex is designated as the root of the triple. A set
$X\subset [n]$ is a dominating set in a 3-tournament $T_n^3$
if for every $z\in [n]\setminus X$ there exist $x\in X, y\in [n]$
($y\ne z, y\ne x$) such that $x$ is the root of the triple
$(x,y,z)$. Let $dom(T_n^3)$ denote the cardinality of a smallest
dominating set of $T_n^3$.
There exists a 3-tournament $T_n^3$ such that $dom(T_n^3)\ge 2014$.
If any four vertices of a 3-tournament $T_n^3$ contain at least two triples with the same root then $dom(T_n)\le 2014$.
I already posed this pair of conjectures at the 2012 Prague
Midsummer Combinatorial Workshop (of course with 2012 in the role of
Note that the $2$-dimensional versions of the above conjectures are
true: there exist tournaments $T$ with $dom(T)\ge 2014$; if any
three vertices of a tournament $T$ contain two pairs with the same
root then $dom(T)=1$. Also, if three triples are required with
the same root in every four vertices of a $T_n^3$ then
$dom(T_n^3)=1$ follows easily (a remark with Tuza).
§.§ Klas Markström
If $G=(V,E)$ is an $n$-vertex graph then the strong chromatic number of $G$, denoted $s_\chi (G)$, is the minimum $k$ such that the
following hold: Any graph which is the union of $G$ and a set of $\lceil \frac{n}{k} \rceil$ vertex-disjoint $k$-cliques is $k$-colourable. Here we take the union
of edge sets, adding isolated vertices to G if necessary to make $n$ divisible by $k$.
It is easy to see that $\Delta{G}+1 \leq s_\chi (G)$, and Penny Haxell <cit.> has proven that $s_\chi (G) \leq c \Delta(G)$ for all $c>\frac{11}{4}$ if $\Delta$ is large enough, and <cit.> $s_\chi (G) \leq 3 \Delta(G) -1$ in general. The folklore conjecture here is that $s_\chi (G) \leq 2 \Delta(G)$. This is known to be true if $\Delta \geq \frac{n}{6}$ <cit.>.
Let us define the biclique number $\omega_b(G)$ to be the maximum $t$ such that there exists a $K_{a,b}\subset G$ with $t=a+b$.
A few years ago I made the following conjecture:
$\omega_b(G)\leq s_\chi (G) \leq \omega_b(G) +1 $
The lower bound is easily seen to be true so the conjecture really concerns the upper bound.
P. Haxell An improved bound for the strong chromatic number Journal of Graph Theory 58 (2008), 148–158.
P. Haxell On the strong chromatic number Combinatorics, Probability and Computing 13 (2004), 857–865.
A. Johansson, R. Johansson and K. Markström Factors of $r$-partite graphs and bounds for the strong chromatic number,
Ars Combinatoria 95 (2010), 277Đ287.
§.§ Benny Sudakov
How many edges do we need to delete to make a $K_r$-free graph $G$ of
order $n$ bipartite?
For $r=3, 4$ this was asked long time ago by P. Erdős. For triangle-free graphs he conjectured that deletion of
$n^2/25$ edges is always enough and that extremal example is a blow-up of a $5$-cycle. Sudakov answered the question for $r=4$ and proved
that the unique extremal construction in this case is a complete $3$-partite graph with equal parts. This result suggests that a complete
$(r-1)$-partite graph of order $n$ with equal parts is worst example also for all remaining values of $r$. Therefore we believe that it is enough to
delete at most $\frac{(r-2)^2}{4(r-1)^2}n^2$ edges for even $r\geq 5$ and at most $\frac{r-3}{4(r-1)}n^2$ edges for odd $r\geq 5$ to make bipartite
any $K_r$-free graph $G$ of order $n$.
§ PROBLEM SESSION APRIL 16TH, 2014
§.§ Dhruv Mubayi
Fix $k \ge 2$ and recall that the Ramsey number $r(G,H)$ is the minimum
$n$ such that every red/blue edge-coloring of the complete $k$-uniform
hypergraph on $n$ vertices yields either a red copy of $G$ or a blue
copy of $H$. A 3-cycle $C_3$ is the $k$-uniform hypergraph comprising
three edges $A, B, C$ such that every pair of them has intersection size
1 and no point lies in all three edges. Classical results of
Ajtai-Komlós-Szemerédi and a construction by Kim show that for
$k=2$, we have $r(C_3, K_t)=\Theta(t^2/\log t)$, where $K_t$ is the
complete graph on $t$ vertices. Kostochka, Mubayi, and Verstraëte
proved that for $k=3$, there are positive constants $a, b$ such that
$$at^{3/2}/(\log t)^{3/4}< r(C_3, K_t) < bt^{3/2}.$$
For $k=3$, we have $r(C_3, K_t)=o(t^{3/2})$.
§.§ Jørgen Bang-Jensen
§.§.§ Longest $(x,y)$-path in a tournament
A digraph on at least $k+1$ vertices is $k$-strong if it remains strongly connected after the deletion of any subset $X$ of at most $k-1$ vertices. An $(x,y)$-path is a directed path from $x$ to $y$.
A digraph is hamiltonian-connected if it contains a hamiltonian $(x,y)$-path for every choice of distinct vertices $x,y$.
Every 4-strong tournament is hamiltonian-connected and this is best possible.
There exists a polynomial algorithm for deciding whether a given tournament $T$ with specified vertices $x,y$ has an $(x,y)$-hamiltonian path.
[Conjecture 9.1]<cit.>
What is the complexity of finding the longest $(x,y)$-path in a tournament?
The algorithm of Theorem <ref> uses a divide and conquer approach to reduce a given instance into a number of smaller instances which can either be recursively solved or for which we have a theoretical result solving the problem. Thus the approach cannot be used to solve the case where we are not looking for hamiltonian paths.
§.§.§ Hamiltonian paths in path-mergeable digraphs
A digraph $D$ is path-mergeable if, for every choice of distinct vertices $x,y\in V(D)$ and internally disjoint (only end vertices in common) $(x,y)$-paths $P_1,P_2$ there is an $(x,y)$-path $P$ in $D$ such that $V(P)=V(P_1)\cup V(P_2)$. It was shown in <cit.> that one can recognize path-mergeable digraphs in polynomial time. A cutvertex in a digraph is a vertex whose removal
results in a digraph whose underlying undirected graph is disconnected.
A path-mergeable digraph has a hamiltonian cycle if and only if it is strongly connected and has no cutvertex. Furthermore, a hamiltonian cycle of each block of $D$ can be produced in polynomial time.
What is the complexity of the hamiltonian path problem for path-mergeable digraphs?
Note that the problem is easy if $D$ is not connected or has no cutvertex so the problem is easy when the block graph of $D$ is not a path (if there is just one block the digraph has a hamiltonian cycle, by Theorem <ref> and if the block graph is not a path there can be no hamiltonian path. However, when the block graph is a path, the fact that we have a hamiltonian cycle in each block does not help much. In fact for every internal block with connection to its sourrounding blocks through the vertices $x,y$, we need to check the existence of an $(x,y)$-hamiltonian path.
J. Bang-Jensen, Digraphs With the path merging property, J. Graph Theory
20 (1995) 255-265.
bang2009aJ. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications 2nd Ed., Springer Verlag, Londomn 2009.
bangJGT28J. Bang-Jensen and G. Gutin. Generalizations of tournaments: A survey.
J. Graph Theory 28 (1998) 171-202.
bangJA13J.Bang-Jensen, Y. Manoussakis and C.Thomassen, A polynomial algorithm for hamiltonian
connectedness in semicomplete digraphs, J.Algorithms 13 (1992) 114-127.
thomassenJCT28 C. Thomassen, Hamiltonian-connected tournaments, J. Combin. Theory Ser. B 28 (1980) 142-163.
§.§ Klas Markström
Let $A$ be an $n \times n$ array with entries from $\{0,\ldots,n\}$, such that each non-zero $x$ appears in at most $n-2$ positions in $A$. ( Each entry of $A$ is just a single number.)
For any $A$ there exists a latin square $L$, using the symbols $\{1,\ldots, n\}$ such $L_{i,j}\neq A_{i,j}$
In <cit.> it was proven that the conjecture holds when only two symbols appear in $A$, and a full characterization of unavoidable arrays with two symbols was given, and a complete list of small unavoidable arrays where each entry in $A$ can now be a list of numbers,. In <cit.> the conjecture was shown to hold if $n-2$ is replaced by $\frac{n}{5}$, and in <cit.> it was shown to hold if $A$ is a partial latin square.
C Casselgren, Carl Johan, On avoiding some families of arrays,
Discrete Math. 312(5) (2012) 963–972.
KO Markström, Klas and Öhman, Lars-Daniel, Unavoidable arrays
, Contrib. Discrete Math., 5(1) (2010) 90-106.
O Öhman, Lars-Daniel, Partial Latin squares are avoidable,
Annals of Combinatorics 15(3) (2011) 485–497.
§ PROBLEM SESSION APRIL 29TH, 2014
§.§ Jørgen Bang-Jensen
A digraph $D=(V,A)$ is $k$-arc-strong if $D-A'$ remains strongly connected for every subset $A'\subseteq A$ with $|A'|\leq k-1$. We denote by $\lambda{}(D)$ the maximum $k$ such that $D$ is $k$-arc-strong.
<cit.> Every $k$ arc-strong tournament $T$ on $n$ vertices contains a spanning
$k$-arc-strong subdigraph with at most $nk + 136k^2$ arcs.
Every $k$ arc-strong tournament $T=(V,A)$ on $n$ vertices contains a spanning subdigraph $D'=(V,A')$ such that every vertex in $D'$ has in- and out-degree at least $k$ and $|A'|\leq nk+\frac{k(k-1)}{2}$ and this is best possible.
For a given tournament $T$ let $\alpha_k(T)$ denote the minimum number of arcs in a spanning subdigraph of $T$ which has minimum in- and out-degree at least $k$. For given $k$-arc-strong tournament $T$ let $\beta_k(T)$ denote the
minimum number of arcs in a spanning $k$-arc-strong subdigraph of $T$.
For every $k$ arc-stroing tournament $T$ we have $\alpha_k(T)=\beta_k(T)$, in particular we have
$\beta_k(T)\leq nk+\frac{k(k-1)}{2}$.
There exists a polynomial algorithm for finding, in a given $k$-arc-strong tournament $T=(V,A)$ a minimum set of arcs $A'$ (of size $\beta_k(T)$) such that the subdigraph induced by $A'$ is already $k$-arc-strong.
Note that the following theorem shows that a similar property as that conjectured above holds when we consider the minimum number $r^{arc-strong}_k(T)$ of arcs whose reversal results in a $k$-arc-strong tournament. It shows that, except when the degrees are almost right already so that some cut needs more arcs reversed, we have equality between the numbers $r^{arc-strong}_k(T)$ and $r^{deg}_k(T)$, where the later is the minimum number of arcs whose reversal in $T$ results in a tournament $T''$ with minimum in- and out-degree at least $k$.
For every tournament $T$ on at least $2k+1$ vertices the number $r^{arc-strong}_k(T)$ is equal to the maximum of the numbers
$k-\lambda{}(T)$ and $r^{deg}_k(T)$. In particular, we always have $r^{arc-strong}_k(T)\leq\frac{k(k+1)}{2}$ (equality for transitive tournaments on at least $2k+1$ vertices).
bang2009J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications 2nd Ed., Springer Verlag, London 2009.
bangJGT46 J.Bang-Jensen, J. Huang and A. Yeo, Spanning $k$-arc-strong subdigraphs with few arcs in
$k$-arc-strong tournaments, J. Graph Theory 46 (2004) 265-284.
bangDAM136J.Bang-Jensen and A. Yeo, Making a tournament $k$-arc-strong by reversing or deorienting arcs, Discrete Appl. Math. 136 (2004) 161-171.
§.§ Matas Šileikis
A family $\mathcal{F}$ of subsets of $\left[n\right] = \left\{ 1, \dots, n \right\}$ is called
* $k$-intersecting if for all $A,B \in \mathcal{F}$ we have $\left|A\cap B\right|\geq k$,
* an antichain if for all $A,B \in \mathcal{F}$ such that $A \neq B$ we have $A \nsubseteq B$,
In 1964 Katona <cit.> (see also <cit.>) determined the least upper bound for the size of a $k$-intersecting family:
\begin{equation}\label{Kat}
|\mathcal{F}| \le
\begin{cases}
\sum_{j=t}^{n} \binom{n}{j}, \qquad &\text{\rm if}\;\; k+n = 2t, \\
\sum_{j=t}^{n} \binom{n}{j}+\binom{n-1}{t-1}, \qquad &\text{\rm if}\;\; k+n = 2t - 1,
\end{cases}
\end{equation}
with equality attained by the family consisting of all sets of size at least $t$ plus, when $k + n$ is odd, subsets of $[n-1]$ of size $t-1$.
In 1966 Kleitman <cit.> (see also <cit.>) observed that the bound (<ref>) remains true under a weaker condition that $\mathcal{F}$ has diameter at most $n-k$, that is, when for every $A, B \in \mathcal{F}$ we have $|A \bigtriangleup B| \le n - k$.
In 1968 Milner <cit.> determined the least upper bound for the size of a $k$-intersecting antichain (which generalizes Sperner's Lemma, when $k = 0$):
\begin{equation}\label{Mil}
|{\cal F}| \leq
\binom{n}{t}, \qquad t = \left\lceil \frac {n+k} 2 \right\rceil.
\end{equation}
Question. Does the bound (<ref>) still hold for antichains satisfying the weaker condition that the diameter of $\mathcal{F}$ is at most $n - k$?
B. Bollobás.
Combinatorics. Set systems, hypergraphs, families of vectors
and combinatorial probability.
Cambridge University Press, 1986.
G. Katona.
Intersection theorems for systems of finite sets.
Acta Math. Acad. Sci. Hung., 15:329–337, 1964.
D. J. Kleitman.
On a combinatorial conjecture of Erdős.
J. Combinatorial Theory, 1:209–214, 1966.
E. C. Milner.
A combinatorial theorem on systems of sets.
J. London Math. Soc., 43:204–206, 1968.
§ PROBLEM SESSION MAY 7TH, 2014
§.§ Imre Leader
A Ramsey Question in the Symmetric Group.
Given $k$ and $r$, does there exist $n$ such that whenever the
symmetric group $S_n$ is $k$-coloured there is a monochromatic copy of $S_r$?
To make sense of this, it is necessary to explain what a `copy of $S_r$'
means. We view $S_n$ as the set of all words of length $n$, on symbols
$1,...,n$, such that no symbol is repeated. Given words $x_1,...,x_r$ on
symbols $1,...,n$, such that the sum of the lengths of the $x_i$ is $n$ and no
symbol is repeated, a copy of $S_r$ means the set of all possible $r!$
concatenations (in any order) of $x_1,...,x_r$.
This is easy to check when $r=2$, but even for $r=3$ we do not know it. In fact,
we do not even know it in the case $r=3$ and $k=2$.
|
1511.00388
|
Enrico Fermi Institute and Department of Physics,
The University of Chicago, Chicago, IL 60637, USA
We derive the first law of black hole mechanics for physical theories based on a local, covariant and gauge-invariant Lagrangian where the dynamical fields transform non-trivially under the action of some internal gauge transformations. The theories of interest include General Relativity formulated in terms of tetrads, Einstein-Yang-Mills theory and Einstein-Dirac theory. Since the dynamical fields of these theories have some internal gauge freedom, we argue that there is no natural group action of diffeomorphisms of spacetime on such dynamical fields. In general, such fields cannot even be represented as smooth, globally well-defined tensor fields on spacetime. Consequently the derivation of the first law by Iyer and Wald cannot be used directly. Nevertheless, we show how such theories can be formulated on a principal bundle and that there is a natural action of automorphisms of the bundle on the fields. These bundle automorphisms encode both spacetime diffeomorphisms and internal gauge transformations. Using this reformulation we define the Noether charge associated to an infinitesimal automorphism and the corresponding notion of stationarity and axisymmetry of the dynamical fields. We first show that we can define certain potentials and charges at the horizon of a black hole so that the potentials are constant on the bifurcate Killing horizon, giving a generalised zeroth law for bifurcate Killing horizons. We further identify the gravitational potential and perturbed charge as the temperature and perturbed entropy of the black hole which gives an explicit formula for the perturbed entropy analogous to the Wald entropy formula. We then obtain a general first law of black hole mechanics for such theories. The first law relates the perturbed Hamiltonians at spatial infinity and the horizon, and the horizon contributions take the form of a “potential times perturbed charge" term. We also comment on the ambiguities in defining a prescription for the total entropy for black holes.
§ INTRODUCTION
In <cit.> Lee and Wald provided a construction of the phase space, Noether current and Noether charge for Lagrangian field theories where the dynamical fields can be viewed as tensor fields on spacetime (or more generally maps from spacetime into another finite-dimensional manifold). Using this construction, Iyer and Wald <cit.> have given a derivation of the first law of black hole mechanics for arbitrary (in particular non-stationary) perturbations off a stationary axisymmetric black hole in any diffeomorphism covariant theory of gravity where the gravitational dynamical field is a Lorentzian metric and the matter fields are smooth tensor fields on the spacetime manifold. They also identified the corresponding entropy as the integral over a horizon cross-section of the Noether charge associated with the action of diffeomorphisms generated by the horizon Killing field. This formulation of the Noether charges and first law has been useful in analysing solutions to Einstein-matter theories <cit.>, and stability of black holes <cit.> and perfect fluid stars <cit.>.
Even though the results of <cit.> encompass a wide variety of theories, there are situations of physical interest where their analysis cannot be directly applied. In particular, we would like to derive a first law of black hole mechanics for gravity formulated in terms of orthonormal coframes (i.e. vielbeins), Einstein-Yang-Mills theory and Einstein-Dirac theory. In all these cases the dynamical fields of the theory have some internal gauge freedom under the action of a group. As we will discuss below this internal gauge freedom is the main reason we cannot directly use the results of <cit.> to derive a first law. Since all the “fundamental" fields in the Standard Model of particle physics have such gauge freedom, it is of interest to formulate the first law when such dynamical fields are present in the theory. The main obstructions to using the formalism of <cit.> for charged fields are as follows.
The first obstruction is that fields with internal gauge transformations cannot, in general, be represented as globally smooth tensor fields on spacetime. A typical example from Maxwell electrodynamics is when the source is a magnetic monopole. In the presence of a magnetic monopole the Maxwell gauge field (or vector potential) \(A_\mu\) cannot be chosen to be smooth everywhere (in any choice of gauge; see Problem 2. Vbis <cit.>). As is well-known, this “singularity" in the Maxwell gauge field is an artefact of trying to make a global choice of gauge. Even in the absence of monopoles, the most “convenient" choice of gauge might make the Maxwell vector potential singular; see <cit.> for a discussion of the Reissner-Nordström black hole where, in the traditional choice of gauge, the vector potential is singular on the bifurcation surface. Similarly, in Yang-Mills theories the dyanmical gauge fields \(A_\mu^I\) might not be representable as smooth tensor fields on spacetime. The analysis of <cit.> assumes from the outset that a global choice of gauge can be made to represent Yang-Mills gauge fields as tensor fields on spacetime. To derive the first law for Yang-Mills theory, Sudarsky and Wald <cit.> also assume that a choice of gauge has been made so that the “gauge-fixed" fields are smooth everywhere, and moreover, are stationary (i.e. \(\Lie_{t} A_\mu^I = 0\)) in that choice of gauge. Similarly, for a coframe formulation of gravity (and also to describe spinor fields), one introduces a Lorentz gauge field (or spin connection) \(\omega_\mu{}^a{}_b\) which might not be smooth everywhere in some chosen gauge (see III. <cit.>). In fact, for non-parallelisable manifolds, there does not even exist a globally smooth choice of coframes \(e^a_\mu\). In all of these cases the obstruction is that a globally smooth choice of gauge can only be made under certain topological restrictions. Even when we can make a global gauge choice it is far from obvious that a gauge choice can be made such that the gauge-fixed fields are stationary in that gauge (see 4 <cit.> for a related discussion). Thus, it is of interest to have a formulation of the first law for theories like Yang-Mills and coframe gravity that does not require a choice of gauge a priori.
The second obstruction arises in defining the action of diffeomorphisms of spacetime on dynamical fields with internal gauge transformations. To formulate a first law of black hole mechanics, we need a notion of a stationary solution to the equations of motion. When the dynamical fields are usual tensor fields on the spacetime \(M\), there is a natural action of the diffeomorphism group of the spacetime \(M\) on tensor fields which can be used to define stationarity by the action of the Lie derivative with respect to the corresponding vector fields. However, when the dynamical fields transform under some internal gauge transformation, we cannot distinguish the action of a diffeomorphism of the spacetime manifold from the action of a diffeomorphism along with an arbitrary (spacetime dependent) internal gauge transformation. That is, we only have a notion of “diffeomorphisms up to a gauge transformations". Similarly in general, a stationary (axisymmetric) black hole will only be “stationary (axisymmetric) up to an internal gauge transformation" i.e. \(\Lie_{t} \psi = gauge\) where \(\psi\) denotes the charged dynamical fields.[Such a notion of “stationarity up to gauge" was already used in <cit.> where nevertheless, the dynamical fields are assumed to be tensor fields on spacetime.] Thus, the full group of transformations of the dynamical fields of the theory is not simply a product group of diffeomorphisms and internal gauge transformations. Without such a separation of diffeomorphisms and internal gauge transformations we have to consider the full group of transformations to define the appropriate notion of Noether charge to obtain a first law.
There have been numerous attempts to sidestep this problem. One approach is to use the usual Lie derivative on spacetime, ignoring the internal gauge structure of the fields <cit.>. A straightforward computation shows that such a Lie derivative (acting on the Yang-Mills gauge fields \(A_\mu^I\) for instance) depends on the gauge choice made. Thus any definition of Noether charges, stationarity and first law using such Lie derivatives on charged fields will also depend on the choice of gauge used. In <cit.> stationarity of the Yang-Mills gauge fields was defined by requiring a global choice of gauge so that the gauge-fixed fields \(A_\mu^I\) on spacetime are annihilated by the Lie derivative along the stationary Killing field. It is far from obvious that such a global choice of gauge exists, even when a global gauge choice can be made, and as we will show, this assumption is actually a restriction on the types of Yang-Mills fields considered in <cit.>.
One could alternatively attempt to define the infinitesimal action of a diffeomorphism of spacetime by a “gauge covariant Lie derivative" and use the vanishing of this Lie derivative to define stationarity and axisymmetry. For example in Ch.10 <cit.> the action of a diffeomorphism along a vector field \(\dfM X \equiv X^\mu \in TM\) is defined through a “gauge covariant Lie derivative" of a Yang-Mills gauge field \(A_\mu^I\) (here the \(I\) index only refers to the Yang-Mills Lie algebra) as
_X A^I_μX^νF_νμ^I
where \(F_{\mu\nu}^I\) is the curvature \(2\)-form for the gauge field \( A_\mu^I\). Similarly, to obtain a first law for a coframe formulation of gravity, Jacobson and Mohd <cit.> use a Lorentz-Lie derivative for the coframes by [<cit.> define the Lorentz-Lie derivative to act on arbitrary tensors, including the Lorentz connection, carrying a representation of the Lorentz group, but we only present the coframes to be brief. They also use the symbol \(\mc K\) to denote the Lorentz-Lie derivative in honour of Kosmann.] (see also <cit.> and references in <cit.>)
_X e^a_μ _X e^a_μ+ λ^a_b e^b_μ
with λ^ab E^μ[a_X e^b]_μ= X^μω_μ^ab + E^μ[a e^b]_ν∇_μX^νwhere the \(\Lie_{\dfM X}\) is computed ignoring the internal indices and \(\omega_\mu{}^{ab}\) is the Lorentz connection on spacetime. Likewise, there have been many attempts to define a Lie derivative for spinor fields (viewed as fields on spacetime). A definition of Lie derivative for spinors with respect to a Killing field of the metric was put forth by Lichnerowicz <cit.>, and then generalised for arbitrary vector fields by Kosmann <cit.> by prescribing that one use the same formula as that given by Lichnerowicz but for any vector field (see also <cit.> and Supplement 2. <cit.>). The Kosmann prescription can be formalised on a principal bundle through the notion of a Kosmann lift (see <cit.> and the references therein).[Also note that the “spinorial Lie derivative" prescription in <cit.> annihilates the metric for any vector field — making every vector field a “Killing field" and all spacetimes stationary — which is clearly not desirable.] The Lichnerowicz-Kosmann-Lie derivative acts on Dirac spinor fields \(\Psi\) according to
_X ΨX^μD_μΨ- 18 ∇_[μX_ν][γ^μ, γ^ν] Ψwhere \(D_\mu\) is the covariant spin derivative on Dirac spinor fields with respect to a torsionless spin connection. Even though these definitions of Lie derivative are gauge covariant, it is a straightforward computation to verify that
[_X, _Y] = _[X,Y] + gauge
and the \(gauge\)-term does not vanish except when
* \(\dfM{\df F}^I = 0\) for “Lie derivative" in <ref>
* either \(X^\mu\) or \(Y^\mu\) is a conformal Killing field of the metric for the Lorentz-Lie derivative <ref> and the Lichnerowicz-Kosmann-Lie derivative <ref> (see <cit.> for a proof).
Thus even though, the linear maps \(X^\mu \mapsto \hat{\Lie}_{\dfM X}\) in <ref>-<ref> are gauge-covariant, none of the above prescriptions for the Lie derivative implement the Lie algebra for the diffeomorphism group of spacetime and the Noether currents derived from these notions of a Lie derivative cannot be interpreted as Noether currents associated to diffeomorphisms.
Further, if the dynamical fields \(\psi\) of the theory are chosen to be stationary with respect to these modified Lie derivatives, i.e. \(\hat{\Lie}_{\dfM X}\psi = 0 \) then they are obviously “stationary up to internal gauge transformations". But it might not be possible to choose a globally smooth gauge representative of the dynamical fields so that they are stationary in this modified sense, particularly when we cannot make a global choice of gauge.
The main aim of this work is to address these issues directly, by formulating physical theories with charged dynamical fields on a principal bundle (see <ref> for details). All the charged fields are legitimate (globally well-defined and smooth) tensor fields on the principal bundle. A smooth global choice of gauge exists only when the principal bundle is trivial, and only in that case can we write the charged fields as smooth tensor fields on spacetime. However, working directly on a principal bundle avoids the issue of making any choice of gauge, and thus, we can handle theories defined on non-trivial principal bundles where there is no way to represent the fields as smooth tensor fields on spacetime. We shall similarly formulate the Lagrangian, Noether currents and charges directly on the principal bundle without making any gauge choices. The principal bundle also provides the necessary structure to consider the full group of transformations of the dynamical fields as the group of automorphisms of the bundle manifold. The automorphisms of the bundle then encode both diffeomorphisms and internal gauge transformations, and we will not need to “artificially" single out the action of just diffeomorphisms. Infinitesimal actions of these automorphisms are then generated by (standard) Lie derivatives with respect to vector fields on the bundle. Using this we define the appropriate notions of “stationarity (and axisymmetry) up to gauge transformations" for charged dynamical fields as automorphisms of the bundle that project down to the stationary (axisymmetric) diffeomorphisms of spacetime (see <ref>). We then generalise the constructions of <cit.> to define the symplectic form on arbitrary (non-stationary) perturbations of the dynamical fields, and the Noether current and Noether charge associated with any bundle automorphism. Next, we describe the main results of this paper, which include a derivation of a generalised zeroth law for bifurcate Killing horizons, and the first law of black hole mechanics for stationary and axisymmetric black holes which arise from solutions to the equations of motion, for theories with charged dynamical fields.
We are interested in static or stationary-axisymmetric, asymptotically flat black hole spacetimes with bifurcate Killing horizons determined by dynamical fields which have non-trivial internal gauge transformations. We expect our results can be generalised to spacetimes with different asymptotics but we stick to the asymptotically flat case. We refer the reader to <ref> for a more detailed description of the spacetimes under consideration.
As our first result we show in <ref> that on any bifurcate Killing horizon in spacetime (not necessarily a solution to any equations of motion) we can define certain potentials \(\ms V^\Lambda\) at the horizon which are constant along the entire bifurcate Killing horizon. This can be viewed as a generalised zeroth law for bifurcate Killing horizons. These potentials are defined solely in terms of the dynamical gauge fields of the theory (for instance, a Yang-Mills gauge field or the Lorentz gauge field for gravity) and get no direct contributions from any other matter fields. In <ref>, we show that the perturbed Hamiltonian \(\delta H_K\) associated to the horizon Killing field \(K^\mu\) at the bifurcation surface can be put into a “potential times perturbed charge" form where the charges are determined by the dependence of the Lagrangian on the curvatures of the dynamical gauge fields of the theory. Then, in <ref>, we provide a new perspective on the temperature \(T_{\ms H}\) and perturbed entropy \(\delta S\) of the black hole by identifying them with the potential and perturbed charge, respectively, corresponding to the Lorentz connection in a first-order formulation of gravity. Thus, the temperature and perturbed entropy can be viewed on the same footing as any other potentials and perturbed charges of any matter gauge fields (like Yang-Mills gauge fields) in the theory. This also gives us an explicit formula for the perturbed entropy in direct parallel with the Wald entropy formula <cit.>.
Our main result (<ref>) is a general formulation of the first law of black hole mechanics for theories with charged dynamical fields, where the dynamical fields solve the equations of motion obtained from a gauge-invariant Lagrangian. The first law is obtained as an equality between the perturbed boundary Hamiltonian \(\delta H_K\) associated to the horizon Killing field \(K^\mu\) evaluated at the bifurcation surface and at spatial infinity, and takes the form
T_HδS + V'^ΛδQ'_Λ= δE_can - Ω_H^(i) δJ_(i),can
for any perturbation which solves the linearised equations of motion off a stationary, axisymmetric (up to internal gauge transformations) black hole background which solves the equations of motion. The left-hand-side terms are the potentials and charges of the black hole on the bifurcation surface defined in <ref> and <ref>. The first term consists of the temperature and perturbed entropy of the black hole, identified with the gravitational potential and perturbed charge (<ref>), while the second term is the contribution of the non-gravitational gauge fields (such as Yang-Mills fields). The quantities on the right-hand-side are the perturbed canonical energy (associated to the stationary Killing field \(t^\mu\)) and angular momenta (associated to the axial Killing fields \(\phi^\mu_{(i)}\); here the index \((i)\) is used to denote the multiple axial Killing fields in greater than \(4\)-spacetime dimensions) defined at spatial infinity (<ref>). The form of the perturbed canonical energy and angular momenta at infinity depends on the theory under consideration and also the asymptotic fall-off conditions on the fields, and they contain contributions from both the gravitational dynamical fields and other matter dynamical fields in the theory. For instance, in Einstein-Yang-Mills theory, \(\delta E_{can}\) contains both the perturbed ADM mass and a “potential times perturbed charge" term from the Yang-Mills gauge field at infinity (see <ref>). Similarly, \(\delta J_{can}\) contains both the perturbed ADM angular momentum and the perturbed angular momentum of the Yang-Mills fields.
Note that for Einstein-Yang-Mills theory, Sudarsky and Wald <cit.> get a vanishing Yang-Mills potential term at the horizon because of their assumption that there exists a smooth choice of gauge such that the gauge-fixed Yang-Mills fields are stationary \(\Lie_{K} A_\mu^I = 0\). We will argue that in general such a gauge choice cannot be made, and the “potential times perturbed charge" at the horizon can be set to vanish only in special situations (at the cost of changing the contributions to perturbed canonical energy and angular momenta at infinity; see <ref>). The existence of this non-vanishing term at the horizon was also pointed out in <cit.>, though they could not write the term in terms of potentials and perturbed charges for non-abelian Yang-Mills fields.
We also show that the ambiguities in defining the Noether charge for a Lagrangian do not affect the first law and the perturbed entropy. We also discuss the ambiguities in defining a total entropy for a stationary axisymmetric black hole. We argue that a second law of black mechanics could fix at least some of these ambiguities in the total entropy. Since we do not know of a general derivation of the second law for arbitrary theories of gravity (except in the case of General Relativity), we do not make an attempt to define the total entropy or a notion of dynamical black hole entropy in this paper.
Even though we use a first-order coframe formulation for gravity, the general form of the first law described above is applicable to any Lagrangian theory for gravity where some fields with internal gauge freedom are considered as dynamical fields instead of the metric <cit.> including, higher-derivative theories of gravity <cit.> with Lagrangians depending on torsion, curvature and finitely many of their derivatives. One can also include “non-metricity", metric-affine theories of gravity <cit.> by a simple extension of the formalism. On the matter side, we can include all the charged matter fields in the Standard Model of particle physics. We also expect that our results can be generalised to include supersymmetric theories following, for instance, <cit.>.
Despite this generality, there remain some potentially physically interesting theories that are not covered by our formalism. These include higher \(p\)-form gauge theories in the presence of magnetic charges (see <cit.> for work in this direction) and Chern-Simons Lagrangians which are only gauge-invariant up to a total derivative term. The entropy contribution of gravitational Chern-Simons Lagrangians was computed from a spacetime point of view in <cit.> (using a “modified Lie derivative"), and in <cit.> (using a modification of the symplectic current). We shall defer the analysis of Chern-Simons theories from a bundle point of view to a forthcoming paper <cit.>.
The remainder of this work is organised as follows. We describe the principal bundle formulation of dynamical fields and gauge-invariant Lagrangians in <ref> and give a general form for the symplectic potential, symplectic current, and Noether charge for any bundle automorphism (i.e. combined diffeomorphisms and gauge transformations) for such theories. We define the horizon potentials and charges, and derive a generalised zeroth law for bifurcate Killing horizons in <ref>. In <ref> we give a formulation of the first law of black hole mechanics for the theories under consideration. In <ref> we use this formalism to derive a first law for General Relativity in a first-order tetrad formulation, Einstein-Yang-Mills theory and Einstein-Dirac theory. The appendix <ref> collects some technical definitions and formulae along new results which will be used in the main arguments of the paper.
§ NOTATION
We will use an abstract index notation for vector spaces and tensor fields whenever convenient. Tensor fields on spacetime (or some base space for a principal bundle) will be denoted by indices \(\mu,\nu,\lambda,\ldots\) from the middle of the lower case Greek alphabet, e.g. \(X^\mu\) is a vector field and \(\sigma_\mu\) is a covector field on spacetime. Similarly, lower case Latin indices \(m,n,l,\ldots\) denote tensors on a principal bundle, e.g. \(X^m\) is a vector field and \(\sigma_m\) is a covector field on a principal bundle.
It will be convenient to often use an index-free notation for differential forms and vector fields. We will use the factor and sign conventions of Wald <cit.> when translating differential forms to and from an index notation and use the symbol \(\equiv\) to denote such a translation. When using an index-free notation we denote differential forms by a bold-face symbol, for instance, a differential \(k\)-form on a principal bundle is denoted by \( \df\sigma \equiv \sigma_{m_1 \ldots m_k} = \sigma_{[m_1 \ldots m_k]} \) and for a vector field \(X^m\), we denote the interior product by \(X \cdot \df\sigma \equiv X^{m_1}\sigma_{m_1 \ldots m_k}\) and the Lie derivative by \(\Lie_X \df\sigma = X \cdot d\df\sigma + d (X\cdot \df\sigma)\). When using an index-free notation (and for scalars which have no indices) we will also use an underline to distinguish between functions, differential forms and vector fields on the base space from those on the principal bundle i.e. \(\dfM \varphi\) is a function, and \(\dfM{\df \sigma} \equiv \sigma_{[\mu_1\ldots \mu_k]}\) a \(k\)-form, respectively on \(M\), and \(\dfM X \cdot \dfM{\df \sigma}\) is the interior product, on the base space.
Upper case indices \(I,J,K,\ldots\) from middle of the Latin alphabet will denote elements of a finite-dimensional Lie algebra \(\mf g\) e.g. \(X^I\) is an element of a Lie algebra and \({c^I}_{JK} = {c^I}_{[JK]}\) denotes the structure constants. We write the Lie bracket on \(\mf g\) as \([X,Y]^K = {c^K}_{IJ}X^I Y^J\). The Killing form on \(\mf g \) is a bilinear, symmetric form defined by
k_IJ c^L_IKc^K_JL
The Killing form is invariant under the adjoint action of the group \(G\) on its Lie algebra \(\mf g\). It is non-degenerate if and only if \(\mf g\) is semisimple, and hence defines a metric on the Lie algebra. Further, when the group \(G\) is compact the Killing form is negative definite. Throughout the paper we stick to the semisimple case and use \(k_{IJ}\) and its inverse \(k^{IJ}\) to raise and lower the abstract indices on elements of \(\mf g\).
Upper case letters \(A,B,\ldots\) from the beginning of the Latin alphabet denote elements of a vector space \(\bb V\) with some representation \(R\) of a finite-dimensional group \(G\). The action of any \(g \in G\) on any element \(\varphi^A \in \bb V\) under the representation \(R\) is denoted by \(R(g)\varphi\); omitting the indices for simplicity. The corresponding action \(r\) of a Lie algebra element \(X^I \in \mf{g}\) is denoted (using the abstract index notation) by \(X^I {{r_I}^A}_B \varphi^B\). We shall also use \(\alpha,\beta,\ldots\) as indices to denote some collection of fields, each of which can be tensor fields valued in different vector spaces (for example in <ref>).
§ DYNAMICAL FIELDS AND LAGRANGIAN THEORIES ON A PRINCIPAL BUNDLE
Fields with internal gauge freedom under the action of a group \(G\) (which we assume to be semisimple) are usually written as tensor fields on spacetime \(M\) valued in some vector space which transform under a representation of \(G\). As noted in the Introduction, in general such fields can only be represented locally as smooth tensor fields. Since we are interested in the first law of black hole mechanics which is a global equality relating quantities defined at the horizon to those defined at spatial infinity, it will be very convenient to have globally well-defined smooth dynamical fields to describe the physical theory. Such fields with internal gauge transformations under some group \(G\) can be defined globally on a \(G\)-principal bundle \(P\). For details of principal bundle formalism we refer to the classic treatment of <cit.>.[Note that these references may use different conventions for numerical factors and signs for differential forms when converting to and from an index notation. Throughout this paper we use the conventions of <cit.>.] We briefly recall the essential concepts needed below, while some new technical results are collected in <ref>.
Let \(\pi : P \to M\) be a \(G\)-principal bundle over \(M\). Denote the space of vertical vector fields by \(VP\) containing vector fields \(X^m\) whose projection vanishes i.e. \((\pi_*)^\mu_m X^m = 0\). The space of horizontal \(k\)-forms is denoted \(\Omega^k_{hor}P\) containing differential forms \(\df\sigma\) such that \(X \cdot \df\sigma = 0\) for all vertical vector fields \(X^m\). Recall that differential forms on the base space that are invariant under internal gauge transformations are isomorphic to horizontal differential forms on the bundle i.e. \(\pi^*\Omega^kM \cong \Omega^k_{hor}P\) (see Example 5.1 in II.5 <cit.>). Similarly, the space of horizontal \(k\)-forms which are valued in a vector space \(\bb V\) transforming under a representation \(R\) of the group \(G\) is denoted \(\Omega^k_{hor}P(\bb V, R)\); these correspond to gauge-covariant differential forms on spacetime.
Since we are interested in theories with gravity described by some orthonormal coframes, we choose \(P\) to have the structure[Most of our results generalise straightforwardly to the more general case where the Lorentz bundle \(P_O\) is simply a subbundle of principal bundle \(P\).]
P = P_O ⊕P'
where \(P_O\) is a Lorentz bundle, and \(P'\) is a principal bundle with structure group \(G'\) corresponding to other internal gauge transformations of the matter fields; the structure group of \(P\) is then \(G = O(d-1, 1) \times G'\). As described in <ref>, the coframes \(\df e^a\) are horizontal forms on \(P\), while the frames \(E_a^m\) are represented by vector fields. The gauge fields of the theory are represented by a connection \(\df A^I\) on \(P\) which is a \(1\)-form valued in the Lie algebra \(\mf g\) transforming in the adjoint representation \(\rm{Ad}\). For the bundle \(P\) in <ref> it is of the form
A^I = ω^a_b, A'^I' ∈Ω^1P(g, Ad)
where \(\df \omega^a{}_b\) is a Lorentz \(SO(d-1,1)\)-connection on \(P_O\) and \(\df A'^{I'}\) is a \(G'\)-connection on \(P'\).
Now we describe our strategy to write charged fields with internal gauge transformations as tensor fields on the bundle \(P\) instead of the spacetime \(M\). First let us consider the case of a charged scalar field i.e. a (local) function \(\dfM \varphi^A\) on \(M\) valued in some vector space \(\bb V\) which has a representation \(R\) of the internal gauge group \(G\). The field \(\dfM \varphi^A\) is represented on the principal bundle \(P\) as an equivariant function valued in \(\bb V\) i.e. by \(\varphi^A \in \Omega^0P(\bb V, R) \). The main example of such a charged scalar field we'll consider is the Dirac spinor field in <ref> in which case the group \(G\) is the spin group \(Spin^0(3,1)\) and the corresponding bundle is a spin bundle \(P_{Spin}\) (described in <ref>). Another example of such a field is the Higgs field of the Standard Model where the group \(G\) is taken to be \(SU(2) \times U(1)\).
To write more general charged tensor fields on \(P\) it will be convenient to express them in terms of their frame components as follows. Let \(\sigma_\mu^A\) be a charged covector field and and \(\eta^\mu_A\) a charged vector field on \(M\) valued in some vector space \(\bb V\) with some internal gauge transformation. Locally choosing a set of frames and coframes, we write the frame components as
σ_a^A σ_μ^A E^μ_a η^a_A η^μ_A e_μ^a
Now we can view the frame components \(\sigma^A_a\) and \(\eta_A^a\) as scalar fields valued in \(\bb R^d \otimes \bb V\) with internal gauge transformations under both \(O(d-1,1)\) and \(G'\). Then, we can consider the frame components as globally smooth functions valued in \(\bb R^d \otimes \bb V\) on the bundle \(P\), in the same manner as the charged scalar field discussed above. Similarly, we can write any charged tensor field — with arbitrary tensor structure and with internal gauge transformations under the full structure group \(G = O(d-1,1) \times G'\) — on \(P\) in terms of its frame components, the frames and coframes. Henceforth we will always represent charged tensor fields defined on spacetime \(M\) by their frame components on the bundle \(P\) written as charged scalars \(\varphi^A\) where now the \(A\) index includes the frame component indices.
Using the covariant exterior derivative \(D\) associated to the connection \(\df A^I\) in <ref> we can similarly represent the covariant derivatives of any charged tensor field in terms of their frame components on the principal bundle. We write the frame components of the \(k\)-derivatives of \(\varphi^A\) on the bundle \(P\) using the shorthand \( \varphi^A_{a_1\ldots a_k} \) where
\[
\varphi^A_{a_1\ldots a_k} \defn E_{a_k}\cdot D (\ldots E_{a_1}\cdot D\varphi^A) = E_{a_k}\cdot D \varphi^A_{a_1\ldots a_{k-1}}
\]
Thus we can describe all the dynamical fields with internal gauge transformations and their covariant derivatives as globally smooth tensor fields on the bundle \(P\) without making any choice of gauge.
The other problem that arises if the dynamical fields have some internal gauge freedom is that we can only define a notion of “diffeomorphisms up to a gauge transformation", and consequently there is only notion of “stationarity up to internal gauge transformations". As discussed in the Introduction <ref>, if one uses the ordinary Lie derivative on spacetime (ignoring the internal gauge transformations), the result is not gauge-invariant. Also, the various attempts at defining “covariant Lie derivatives" (eq:YM-gauge-Lieeq:LK-Lie) do not implement the Lie algebra of diffeomorphisms of spacetime (see <ref>).
Since we have defined the dynamical fields on a principal bundle \(P\), the source of this problem becomes more apparent. For theories with dynamical tensor fields defined on spacetime \(M\), the group of transformations is the group of diffeomorphisms of spacetime i.e. Diff(M). Similarly, for theories with dynamical fields defined on the bundle \(P\) the group of transformations consists of automorphisms of the principal bundle (see I.5 <cit.>). The automorphism group of \(P\) has the semi-direct product structure \({\rm Aut}(P) \cong {\rm Diff}(M) \ltimes {\rm Aut}_V(P)\). Here \({\rm Aut}_V(P)\) is the normal subgroup of vertical automorphisms that do not move the points in the base space \(M\) i.e. these correspond to internal gauge transformations. Since, \({\rm Aut}_V(P)\) is a normal subgroup, the action of internal gauge transformations leaving the spacetime points fixed is well-defined. However, without picking a gauge choice there is only a notion of “diffeomorphisms up to internal gauge transformations" and any attempt to define an action of just diffeomorphisms of \(M\) on charged fields is doomed to fail. In fact from <ref> we see that even when one defines some “gauge covariant Lie derivative" one has to consider diffeomorphisms and internal gauge transformations simultaneously. Thus, again we are lead to work directly with fields defined on the principal bundle with \({\rm Aut}(P)\) acting as the full group of transformations. The corresponding Lie algebra of infinitesimal automorphisms \(\mf{aut}(P)\) consists of vector fields on the bundle which act on charged fields by the usual Lie derivative. We then define stationary (axisymmetric) charged fields as fields that are preserved under those automorphisms of the bundle \(P\) which project to stationary (axisymmetric) diffeomorphisms of spacetime \(M\) (see <ref>). Viewed from the base spacetime this gives the appropriate notion of dynamical fields being stationary (axisymmetric) up to gauge. This point of view has the further advantage that we can treat both diffeomorphisms and gauge transformations simultaneously using standard tools of differential calculus on the bundle.
Even though in general there is no unique way to associate a given diffeomorphism of spacetime to an automorphism of the bundle, if we require that the automorphism preserves a given connection on the bundle, we can prove that the non-uniqueness is given by a global symmetry (if any exist) which keeps the chosen connection fixed at every point (see <ref> and <ref>). As we will show, in Einstein-Yang-Mills theory, when such a global symmetry of the solution Yang-Mills connection does not exist we cannot set the Yang-Mills potential at the horizon to vanish and we get a new non-vanishing “potential times perturbed charge" term at the horizon for the first law, generalising the results of <cit.> on the first law for Einstein-Yang-Mills theory.
Similarly, if an automorphism of the Lorentz bundle \(P_O\) is required to preserve the coframes, then it is uniquely determined by the corresponding isometry of the spacetime metric (see <ref>). This uniqueness essentially implies that for a Killing field of the spacetime metric, the Lie derivative on the bundle coincides with the Lorentz-Lie derivative <ref> on coframes and the Lichnerowicz-Kosmann-Lie derivative <ref> on spinors (see <ref>). Thus, even though our Noether charges for arbitrary automorphisms differ from those derived by <cit.> for a coframe formulation of gravity, we get the same first law for stationary spacetimes for the first-order formulation of gravity.
Now that we have defined the dynamical fields (and the Lagrangian of the theory; see the next <ref>) on the bundle \(P\) instead of the spacetime \(M\), we can derive a first law of black hole mechanics. Note that we do not make any choice of gauge and consider the full group \({\rm Aut}(P)\) instead of just diffeomorphisms of \(M\). Most of the computations proceed in direct analogy to the computations of <cit.> except they are carried out on the bundle. The only additional task is to check that the relevant quantities are infact gauge-invariant (or covariant), which is easily done by verifying that the computations yield horizontal forms on the bundle.
§.§ The form of the gauge-invariant Lagrangian
On spacetime the Lagrangian is a \(d\)-form \(\dfM{\df L} \in \Omega^dM\) and we further assume that the Lagrangian is invariant under internal gauge transformations.[For instance, in this paper we do not consider theories with a Chern-Simons Lagrangian deferring their analysis to future work <cit.>.] Thus we can pullback the Lagrangian \(\dfM{\df L}\) from the spacetime \(M\) to the bundle \(P\) that is, we consider the Lagrangian of the theory as a real horizontal \(d\)-form on the \(P\) given by \(\df L \in \Omega^d_{hor}P\).
We will take the Lagrangian to depend on the frames \(E^m_a\), the coframes \(\df e^a\), the connection \(\df A^I\) <ref>, the frame components \(\varphi^A\) of charged tensor fields, and their finitely many covariant derivatives \(\varphi^A_{a_1\ldots a_k}\) (written as functions on \(P\); see the discussion above). We also allow dependence on the curvature \(\df F^I\) and the torsion \(\df T^a\), and finitely many of their covariant derivatives.
Any antisymmetrisation in the derivatives of the tensor fields \(\varphi^A\) can be converted to terms with lower order derivatives and torsion and curvature terms using
2 φ^A_[ab] = - T^c_ab φ^A_c + F^I_ab r_I^A_B φ^B
where \(T^c{}_{ab} = E_b \cdot E_a \cdot \df T^c\) and \(F^I{}_{ab} = E_b \cdot E_a \cdot \df F^I\) are the frame components of the torsion and curvature. Using this on higher order antisymmetrised derivatives we can write all derivative terms in terms of completely symmetrised derivatives and derivatives of torsion and curvature. Then in a similar manner we can eliminate any antisymmetrised derivatives of the torsion and curvature. Finally, using <ref> we eliminate any dependence of the Lagrangian on \(D \df T^a\) in favour of the Lorentz curvature \(\df R^a{}_b\) and the coframes \(\df e^a\). For later convenience we introduce the shorthand
χ^α{φ^A, T^c_ab, F^I_ab}
and the frame components of their completely symmetrised derivatives by \(\chi^\alpha_{a_1\ldots a_i}\).
Thus the dependence of the Lagrangian on the dynamical fields can be written as[Note that we simply assume that the Lagrangian is independent of any background fields and do not attempt to prove a “Thomas replacement theorem" as done in Lemma 2.1 <cit.>.]
L(E^m_a, e^a, A^I, {χ^α_a_1…a_i}) ∈Ω^d_horP
where \(0\leq i \leq k\) counts the number of completely symmetrised derivatives of the corresponding fields in <ref>. As discussed above the frames \(E^m_a\) on the bundle are only defined up to vertical vector fields, so we also demand that the Lagrangian depend on the frames so that \(\df L[E^m_a] = \df L[E'^m_a]\) whenever \(E'^m_a - E^m_a\) is a vertical vector field. The full set of dynamical fields of the theory then includes the coframes \(\df e^a\), the connection \(\df A^I\) and the frame components of the charged tensor fields \(\varphi^A\) which we collectively denote as a differential form on \(P\) valued in a collective vector space \(\bb V\)
ψ^α{ e^a, A^I, φ^A } ∈Ω^α P(V)
Here and henceforth, we use the notation \(\degf{\alpha}\) to denote the degree of the differential form corresponding to the dynamical field \(\df\psi^\alpha\) with an \(\alpha\) index i.e.
α = { a, I, A } = { 1, 1, 0 }
The Lagrangian is further required to be a local and covariant functional of the fields in the sense of <ref> i.e. for any automorphism of the bundle \(f \in {\rm Aut}(P)\) we have
f^* L[ψ] = L[f^*ψ] ∀f ∈Aut(P)
where it is implicit that on the right-hand-side that \(f\) also acts on the derivatives of \(\df\psi\). If \(X^m \in \mf{aut}(P)\) is the vector field generating the automorphism \(f\) then the above equation implies that
_X L[ψ] = L[_Xψ] ∀X^m ∈aut(P)
Note that since we assume that the Lagrangian is gauge-invariant, we have
_X L[ψ] = 0 ∀X^m ∈aut_V(P)
§.§ Equations of motion, the symplectic potential and symplectic current
With the above described Lagrangian the equations of motion of the theory are obtained by a variation of the Lagrangian with respect to the dynamical fields <ref>. To consider such variations, we take any smooth \(1\)-parameter family of dynamical fields \(\df\psi^\alpha(\lambda)\) with \(\df\psi^\alpha(0) = \df\psi^\alpha\) corresponding to the background dynamical fields of interest. Define the first variation or perturbation about \(\df\psi^\alpha\) by
δψ^α.d/dλψ^α(λ)|_λ= 0
We also use the symbol \(\delta\) to denote variations of any functional \(\mc F\) of the dynamical fields defined in the same way i.e.
δF[ψ] .d/dλF [ψ(λ)]|_λ= 0
Since the difference of two connections is horizontal and all the other dynamical fields are already horizontal the perturbations of the dynamical fields <ref> given by \(\delta \df\psi^\alpha \in \Omega_{hor}^{\degf{\alpha}} P(\mathbb V) \) are all horizontal forms on \(P\). Further, since \(E_a \cdot \df e^b = \delta^b_a\) holds at each \(\lambda\) of the \(1\)-parameter family of frames and coframes, we have \( \delta E_a \cdot \df e^b = - E_a \cdot \delta \df e^b \). Since the frames are considered equivalent if their difference is vertical we have
δE^m_a = - (E_a ·δe^b) E^m_b = - E^m_b E^n_a δe_n^b
and we can convert all variations of the Lagrangian of the form <ref> obtained from frame variations to variations of the dynamical coframe fields as
δL/δE^m_a δE^m_a = ( - δL/δE^n_b E^n_a E^m_b ) δe_m^a
The variation of the Lagrangian can be written in the form (we will prove this in <ref>):
δL = Ẽ^m_1 …m_α_αδψ_m_1 …m_α^α+ dθ(ψ;δψ)
where the equations of motion \(\tilde{\df{\mc E}}_\alpha : \Omega^{\degf{\alpha}}_{hor} P(\bb V) \to \Omega^d_{hor} P(\bb V^*)\) are the following functional derivative
Ẽ^m_1 …m_α_α= δL/δψ_m_1 …m_α^α
The symplectic potential \(\df \theta\) denotes the “boundary term" in the variation and depends locally and covariantly on the background \(\df\psi\) and linearly on the perturbation \(\delta\df\psi\) and its derivatives.
For subsequent computations it will be very convenient to express the equations of motion purely as differential forms rather than linear maps valued in differential forms as in <ref>. Defining the equations of motion \(\df{\mc E}_\alpha\) corresponding to the variation of each dynamical field \(\df\psi^\alpha \) (<ref>), with \(k \in \degf{\alpha} = \{ 1,1,0 \}\) (<ref>) by
E_α≡(E_α)_m_1…m_d-k = (d-k)!k!/d! δL_m_1…m_d-kl_1…l_k/δψ_n_1 …n_k^α δ^l_1_n_1⋯δ^l_k_n_k ∈Ω_hor^d-k P(𝕍^*)
a straightforward computation shows that the variational principle can be written in the more convenient form
δL = E_α(ψ) ∧δψ^α+ dθ(ψ;δψ)
This rewriting of the variational principle (as opposed to <ref>) will simplify a lot of the later computations as compared to similar ones in <cit.>. That the equations of motion <ref> are horizontal forms expresses the well-known fact that gauge-invariant Lagrangians give gauge-covariant equations of motion. The dynamical fields \(\df\psi^\alpha \) which satisfy the equations of motion \(\df{\mc E}_\alpha(\psi) = 0\) form the subspace of solutions. Given a solution \(\df\psi^\alpha\), any perturbation \(\delta\df\psi^\alpha\) is called a linearised solution if it satisfies the linearised equations of motion \(\delta\df{\mc E}_\alpha = 0\) at \(\df\psi^\alpha\).
The variational principle <ref> implies that \(d\df\theta\) is a horizontal form but we can show \(\df\theta\) itself can be chosen to be horizontal (i.e. gauge-invariant) in the following lemma which is an extension of Lemma 3.1 <cit.> to the bundle.
For any Lagrangian of the form specified in <ref>, the symplectic potential \(\df\theta(\psi;\delta\psi)\) can be chosen to be a horizontal form on \(P\) of the form
θ= (-)^d-2 Z_I ∧δA^I + (-)^d-2 Z_a ∧δe^a + θ' ∈Ω^d-1_horP
θ' = - ∑_i=1^k ( E_a_i ·Z_α^a_1…a_i ) δχ^α_a_1…a_i-1
\begin{align}
\df Z_I & \equiv (Z_I)_{m_1\ldots m_{d-2}} = \frac{(d-2)!2!}{d!} \frac{\delta L_{m_1\ldots m_{d-2}l_1l_2}}{\delta F_{n_1n_2}^I} \delta^{l_1}_{n_1} \delta^{l_2}_{n_2} \in \Omega^{d-2}_{hor}P(\mf g^*) \label{eq:ZI}\\
\df Z_a & \equiv (Z_a)_{m_1\ldots m_{d-2}} = \frac{(d-2)!2!}{d!} \frac{\delta L_{m_1\ldots m_{d-2}l_1l_2}}{\delta T_{n_1n_2}^a} \delta^{l_1}_{n_1} \delta^{l_2}_{n_2} \in \Omega^{d-2}_{hor}P({\bb R^d}^*) \label{eq:Za}
\end{align}
would be the equations of motion obtained if the curvature \(\df F^I\) and torsion \(\df T^a\), respectively, are viewed as an independent fields. Similarly
Z_α^a_1…a_i = δL/δχ^α_a_1…a_i
would be the equations of motion if the all the derivatives up to the \(i\)-th derivative of \(\chi^\alpha\) <ref> (but not higher derivatives) are viewed as independent fields.
The proof proceeds by varying the Lagrangian <ref> considering all the fields and their derivatives as independent and then “integrating by parts" the variations due to the derivatives. Write the variation of the Lagrangian as
\[
\delta \df L = \sum_{i=0}^k \df U_\alpha^{a_1\ldots a_i} \delta \chi^\alpha_{a_1\ldots a_i} + [\cdots]
\]
where here (and throughout the rest of this proof) \([\cdots]\) denotes terms proportional to \(\delta \df e^a\) and \(\delta \df A^I\) and
U_α^a_1…a_i ∂L/∂χ^α_a_1…a_i
is a horizontal \(d\)-form valued in the appropriate representation of the structure group and we fix the index permutation symmetries of \(\df U\) to be the same as the corresponding field \(\chi\). Note that we have used \(\partial\) in <ref> to emphasise that we have not performed any “integration by parts" yet. To get the form of the variational principle we have to rewrite the terms obtained by a variation of the derivatives of \(\chi^\alpha\) in terms of variations of \(\chi^\alpha\) by “integrating by parts". Consider the variation due to the \(i\)-th derivatives as
\[\begin{split}
\df U_\alpha^{a_1\ldots a_i} \delta \chi^\alpha_{a_1\ldots a_i} & = \df U_\alpha^{a_1\ldots a_i} \delta \lb( E_{a_i}\cdot D\chi^\alpha_{a_1\ldots a_{i-1}} \rb) \\
& = \df U_\alpha^{a_1\ldots a_i} \lb[ -(E_{a_i}\cdot\delta \df e^b) E_b\cdot D\chi^\alpha_{a_1\ldots a_{i-1}} + E_{a_i}\cdot \delta D\chi^\alpha_{a_1\ldots a_{i-1}} \rb] \\
& = (-)^{d+1}E_{a_i} \cdot \df U_\alpha^{a_1\ldots a_i} \wedge D\delta\chi^\alpha_{a_1\ldots a_{i-1}} + [\cdots] \\
& = \df Y_\alpha^{a_1\ldots a_{i-1}} \delta\chi^\alpha_{a_1\ldots a_{i-1}} + d\df \theta^{(i)} + [\cdots]
\end{split}\]
where the second line uses <ref>. The term \(Y\) then contributes to the variation with respect to the \((i-1)\)-th derivative term. Thus define recursively, for any \(0 \leq i \leq k\), the term obtained by a variation of the Lagrangian considering the derivatives up to the \(i\)-th derivative of \(\chi^\alpha\), but not higher derivatives, as independent
\[
\df Z_\alpha^{a_1\ldots a_i} \defn
\begin{cases}
\df U_\alpha^{a_1\ldots a_i} &\quad\text{for}\quad i = k \\
\df U_\alpha^{a_1\ldots a_i} + D\lb( E_{a_{i+1}} \cdot \df Z_\alpha^{a_1\ldots a_{i+1}} \rb) &\quad\text{for}\quad 0 \leq i < k
\end{cases}
\]
Using this the variation due to the \(i\)-th derivative term has the terms
\[\begin{split}
\df Y_\alpha^{a_1\ldots a_i} &= D\lb( E_{a_{i+1}} \cdot \df Z_\alpha^{a_1\ldots a_{i+1}} \rb) \\
\df\theta^{(i)} &= - \lb( E_{a_i} \cdot \df Z_\alpha^{a_1\ldots a_i} \rb) \delta \chi^\alpha_{a_1\ldots a_{i-1}}
\end{split}\]
Iterating the above computation for each derivative order we can write
\[
\delta \df L = \df Z_\alpha \delta\chi^\alpha + \sum_{i=1}^k d\df\theta^{(i)} + [\cdots]
\]
The second term above gives us the term in \(\df\theta'\) in <ref>. From the collective notation <ref>, the first term has variations of the charged tensor fields \(\varphi^A\) (which contribute to the equation of motion) as well as those of the curvature and torsion. We then convert the terms obtained from the variations of the curvature and torsion to variations of the connection and coframes.
Z_I^ab δF^I_ab = Z_I^ab δ(E_b ·E_a ·F^I) = (-)^d-2 d( Z_I ∧δA^I) + [⋯]
Z_c^ab δT^c_ab = Z_c^ab δ(E_b ·E_a ·T^c) = (-)^d-2 d( Z_a ∧δe^a) + [⋯]
where \(\df Z_I = E_b \cdot E_a \cdot \df Z_I^{ab}\) and \(\df Z_a = E_c \cdot E_b \cdot \df Z_a^{bc}\), and are explicitly given by <ref>.
Thus, in the total variation of the Lagrangian we can collect all terms proportional to \(\delta \df e^a\), \(\delta \df A^I\) and \(\delta\varphi^A\) into the respective equations of motion and an exact form
\[
\delta \df L = \df{\mc E}_\alpha \wedge \delta\df\psi^\alpha + d\df\theta
\]
with \(\df\theta\) given by the claim of the lemma.
The above algorithm for choosing a horizontal \(\df\theta\) has some ambiguities which we enumerate next.
* For some \(\df\mu(E^m_a, \df e^a, \df A^I, \{\chi^\alpha_{a_1\ldots a_i}\}) \in \Omega^{d-1}P\), we can add a exact form \(d\df\mu \in \Omega^d_{hor}P\) to the Lagrangian without changing the equations of motion (i.e. the dynamical content of the theory) as
L ↦L + dμSince we restrict to horizontal Lagrangians, we only consider \(d\df\mu\) that are horizontal forms, but we do not demand that \(\df\mu\) itself be horizontal i.e. gauge-invariant in contrast to <cit.>. For instance, \(d\df\mu\) could be the integrand of a topological invariant of the bundle (for example, the Euler density), in which case \(\df\mu\) itself would not be horizontal. This shifts the symplectic potential as
\[
\df\theta(\delta \psi) \mapsto \df\theta(\delta \psi) + \delta \df\mu
\]
Note that we can apply <ref> to the Lagrangian \(\df L + d\df\mu\) and conclude that \(\delta\df\mu\) is horizontal i.e. invariant under internal gauge transformations, even if \(\df\mu\) is not.
* Given a choice of Lagrangian, the variational principle <ref> only determines the symplectic potential up to the addition of a local, covariant and horizontal \((d-1)\)-form \(\df\lambda'(\psi;\delta\psi)\) which is linear in the perturbation \(\delta\df\psi\) and \(d\df\lambda' = 0 \). Using <ref> (with \(\df\psi^\alpha\) as the “background field" and \(\delta\df\psi^\alpha\) as the “dynamical field") we get, \(\df\lambda' = d\df\lambda\) for some local and covariant horizontal form \(\df\lambda(\psi;\delta\psi) \in \Omega^{d-2}_{hor}P\). Thus, this additional ambiguity in the symplectic current is
θ(δψ) ↦θ(δψ) + dλ(δψ)
Using the symplectic potential we define the symplectic current as an antisymmetric bilinear map on perturbations (see <cit.>)
ω(ψ;δ_1ψ, δ_2ψ) δ_1θ(δ_2ψ) - δ_2θ(δ_1ψ) - θ([δ_1,δ_2]ψ) ∈Ω^d-1_hor P
In the above definition we have considered the perturbations \(\delta\psi\) as vector fields on the space of field configurations evaluated at the background given by \(\psi\). The commutator \( [\delta_1,\delta_2]\psi \defn \delta_1\delta_2 \psi - \delta_2\delta_1 \psi \) depends on how one chooses to extend these vector fields away from the background \(\psi\) in configuration space, even though the symplectic current \(\df\omega\) at \(\psi\) is independent of this choice. If the variations \(\delta_1\) and \(\delta_2\) are extended to correspond to “independent" one-parameter families of dynamical fields then the commutator vanishes, which suffices for most situations. On the other hand, if the variations are extended to correspond to the same infinitesimal action of bundle automorphisms the commutator is non-vanishing in general (this is useful, for instance, when considering Einstein-fluid systems; see <cit.>).
Note that the symplectic current <ref> is a horizontal form and hence gauge-invariant, in the sense that it is invariant under any vertical automorphism \(f \in {\rm Aut}_V(P)\) of the bundle. However, it does not vanish if we substitute one of the perturbations, say \(\delta_2\df\psi\), by a perturbation \(\Lie_X\df\psi\) generated by an infinitesimal vertical automorphism \(X^m \in \mf{aut}_V(P)\) (see <ref>).
By taking a second variation of the Lagrangian the symplectic current can be shown to be a closed form on solutions (see <cit.>) that is
The symplectic current is closed when restricted to solutions and linearised solutions.
Consider a second variation of the Lagrangian <ref>
\[
\delta_1\delta_2\df L = \delta_1\df{\mc E}_\alpha \wedge \delta_2 \df\psi^\alpha + \df{\mc E}_\alpha \wedge \delta_1\delta_2 \df\psi^\alpha + d\delta_1\df\theta(\delta_2\psi)
\]
Using the identity \((\delta_1\delta_2 - \delta_2\delta_1 - [\delta_1,\delta_2])\df L = 0\) we have:
0 = δ_1E_α∧δ_2 ψ^α- δ_2E_α∧δ_1 ψ^α- E_α∧[δ_1,δ_2] ψ^α+ dωRestricting the above on solutions \(\df\psi\) and linearised solutions \(\delta\df\psi\) we have \( d\df\omega = 0 \).
Since the sympectic current \(\df\omega\) is a horizontal form on \(P\) there is a corresponding gauge-invariant form on spacetime which we denote by \(\dfM{\df\omega}\). Given a Cauchy surface \(\Sigma\), the symplectic current defines a symplectic form \(W_\Sigma\) on perturbations as
W_Σ(ψ;δ_1ψ, δ_2ψ) ∫_Σω(ψ;δ_1ψ, δ_2ψ)
From <ref> we can conclude that the symplectic form is conserved on linearised solutions i.e. if \(\Sigma_t\) is a time-evolved Cauchy surface then \(W_\Sigma(\psi;\delta_1\psi, \delta_2\psi) = W_{\Sigma_t}(\psi;\delta_1\psi, \delta_2\psi)\), whenever \(\psi\) is a solution, \(\delta_1\psi\) and \(\delta_2\psi\) are linearised solutions with boundary conditions such that there is no symplectic flux at infinity.
The \(\df\mu\)-ambiguity in the Lagrangian <ref> does not affect the symplectic current and from the \(\df\lambda\)-ambiguity <ref> we have
\[
\df\omega(\delta_1\psi, \delta_2 \psi) \mapsto \df\omega(\delta_1\psi, \delta_2 \psi) + d\lb[ \delta_1\df\lambda(\delta_2\psi) - \delta_2\df\lambda(\delta_1 \psi) - \df\lambda([\delta_1, \delta_2] \psi) \rb]
\]
which adds a boundary term to the symplectic form \(W_\Sigma\)
W_Σ↦W_Σ+ ∫_∂Σ [ δ_1λ(δ_2ψ) - δ_2λ(δ_1ψ) - λ([δ_1, δ_2] ψ) ]
where \(\dfM{\df\lambda}\) is the unique gauge-invariant differential form on \(M\) corresponding to \(\df\lambda\).
Following <cit.>, we will use the symplectic form to derive the first law of black hole mechanics and show that the above ambiguities do not effect the first law. At this point one can generalise the entire analysis of <cit.> to construct the phase space and Poisson brackets for such theories which is certainly of independent interest. Since we are primarily interested in the first law of black hole mechanics, we turn next to the definition of the Noether charge for any bundle automorphism. The first law then follows from the relation between the symplectic form defined above and the Noether charge (see <ref>).
§.§ Noether current, Noether charge and boundary Hamiltonians
As is well known, Noether's theorem associates gauge symmetries of a Lagrangian theory to conserved currents and charges (see <cit.> for instance). The Lagrangians we are considering are both covariant under diffeomorphisms of the base spacetime \(M\) as well as invariant under internal gauge transformations. Though, as discussed earlier, there is no natural group action of the diffeomorphisms of \(M\) on the dynamical fields with non-trivial internal gauge transformations and we only have a notion of “diffeomorphism up to internal gauge". Thus, we cannot separately define Noether currents associated to only diffeomorphisms and have to consider the full gauge group of the theory given by the group of automorphisms \({\rm Aut}(P)\) of the principal bundle \(P\). Thus, we will define Noether currents associated to any automorphism in \({\rm Aut}(P)\) by adapting the procedure used in <cit.> to work directly on the principal bundle instead of the base spacetime.
We denote the variation obtained by the bundle automorphism generated by a vector field \(X^m \in \mf{aut}(P)\) as \(\delta_X \df\phi \defn \Lie_X\df\phi \). Since we have assumed that the Lagrangian is covariant under such automorphisms, to each automorphism we can associate a Noether current as follows. For the gauge-invariant Lagrangians under consideration \(d\df L \in \Omega^{d+1}_{hor}P\), and hence \(d\df L = 0\). Then for the variation of the Lagrangian under an automorphism we have
δ_X L = _X L = X ·dL + d(X·L) = d(X ·L)
The Noether current corresponding to any \(X^m \in \mf{aut}(P) \) is defined by (see <cit.>):
J_X θ(δ_X ψ) - X ·L
Here we note that if \(X^m \in \mf{aut}_V(P)\) generates vertical automorphisms of the bundle i.e. internal gauge transformations we have \(X \cdot \df L = 0\).
We can define the Noether charge associated to the Noether current \(\df J_X\) as follows. Consider the following computation:
dJ_X = dθ(δ_X ψ) - d(X ·L) = δ_X L - E_α∧δ_X ψ^α- δ_X L
= - E_α∧_X ψ^α
By adapting the procedure in <cit.> to work on the bundle we can define a Noether charge without using the equations of motion (“off-shell"). To do this, we first define the following linear maps from infinitesimal automorphisms of the bundle to horizontal forms, which are generalised versions of the constraints and Bianchi identities (see <cit.>).
The constraints are linear maps \( \df{\mc C}:\mf{aut}(P) \to \Omega^{d-1}_{hor}P \) given by
C(X) (-)^d-α+1 E_α∧X ·ψ^α
= (-)^d [ E_a (X ·e^a) + E_I (X ·A^I) ]
and in the second line we have used <ref>. Since \(\df{\mc C}(X)\) is a horizontal form on the bundle we can consider the corresponding gauge-invariant \((d-1)\)-form \(\dfM{\df{\mc C}}(X)\) on spacetime. The pullback of \(\dfM{\df{\mc C}}(X)\) to any Cauchy surface then are the constraint equations that hold for any initial data for dynamical fields which correspond to a solution to the equations of motion. Note that none of the charged tensor fields \(\varphi^A\) and their equations of motion contribute to the constraints, since we consider the frame components of tensor fields as the dynamical fields. Further, if \(X^m \in \mf{aut}_V(P)\) then from <ref> only the connection \(\df A^I\) and its equation of motion \(\df{\mc E}_I\) contribute to the constraints corresponding to gauge transformations.
The Bianchi identities \(\df{\mc B}:\mf{aut}(P) \to \Omega^d_{hor}P \) are linear maps given explicitly by
B(X) - E_α∧X·dψ^α+(-)^d-α dE_α∧X·ψ^α
= - E_a ∧(X ·T^a) + (-)^d-1 DE_a (X ·e^a) + E_a ∧e^b (X ·ω^a_b)
- E_I ∧(X ·F^I) + (-)^d-1 D E_I (X ·A^I)
- E_A (X ·D φ^A) + E_A φ^B (X ·A^I)r_I^A_B
where in the second equality we have used <ref> and written all terms as manifestly horizontal forms. Note that the second line only depends on the gravitational connection \({\df \omega}^a{}_b\) while the rest depend on the full connection \(\df A^I\) on \(P\) (see <ref>). These Bianchi identities above are a generalisation of the ones for diffeomorphism covariant theories given in <cit.> to include all automorphisms of the bundle i.e. both gauge transformations as well as diffeomorphisms.
Using <ref>, we can rewrite <ref> in the following form
d[ J_X - C(X) ] = B(X)
Using the arguments in IV. <cit.>, we can show that the Bianchi identities \(\df{\mc B}(X)\) vanish identically on all dynamical fields even those that do not satisfy the equations of motion.
The Bianchi identities \(\df{\mc B}(X)\) vanish for any dynamical field \(\df\psi^\alpha\) for all \(X^m \in \mf{aut}(P)\).
Since \(\df{\mc B}(X)\) is a horizontal form denote the corresponding gauge-invariant form on spacetime by \(\dfM{\df{\mc B}}(X) \in \Omega^dM\). Then we can show that \(\dfM{\df{\mc B}}(X) = 0\) using same argument as in IV. <cit.>. Thus \(\df{\mc B}(X) = \pi^*\dfM{\df{\mc B}}(X) = 0\) for any \(\df\psi\) and all \(X \in \mf{aut}(P)\).
Using the above we can define the Noether charge without using any equations of motion.
For any infinitesimal automorphism \(X^m \in \mf{aut}(P)\) there exists a horizontal \((d-2)\)-form \(\df Q_X \in \Omega^{d-2}_{hor}P\) called the Noether charge, such that the Noether current \(\df J_X\) can be written in the form
J_X = dQ_X + C(X)
Since \(\df{\mc B}(X) = 0\), from <ref> we have \(d\left[ \df J_X - \df{\mc C}(X) \right] = 0\). Then using <ref> (with \(X^m\) as the “dynamical field" and \(\df\psi^\alpha\) as a “background field") we conclude that there exists a \(\df Q_X \in \Omega^{d-2}_{hor}P\) (which depends linearly on \(X^m\) and finitely many of it derivatives) such <ref> holds.
<ref> shows that the Noether charge exists but for any theory based on a Lagrangian of the form <ref> we can obtain an explicit useful expression for the Noether charge (this generalises Prop. 4.1 <cit.>) as follows
The Noether charge \(\df Q_X\) for \(X^m \in \mf{aut}(P)\) can be chosen to be of the form
Q_X = Z_I (X ·A^I) + Z_a (X·e^a) ∈Ω^d-2_horP
where \(\df Z_I\) and \(\df Z_a\) are as in <ref> and can be computed directly from the Lagrangian using <ref>.
To get an explicit form for the Noether charge we start with <ref> and use the form of the symplectic current given by <ref> to compute
\[
\df\theta(\delta_X\psi) = (-)^{d-2} \df Z_I \wedge \Lie_X \df A^I + (-)^{d-2}\df Z_a \wedge \Lie_X \df e^a + \df\theta'(\delta_X \psi)
\]
Using <ref> for the Lie derivatives of the connection and coframes, we can write the first two terms as
\[\begin{split}
& (-)^{d-2} \df Z_I \wedge \lb[X\cdot \df F^I + D(X \cdot \df A^I)\rb] + (-)^{d-2} \df Z_a \wedge \lb[ X \cdot \df T^a + D(X\cdot \df e^a) - (X\cdot {\df A^a}_b)\df e^b \rb] \\
= &~ d\lb[ \df Z_I (X \cdot \df A^I) + \df Z_a (X\cdot \df e^a) \rb] + [\cdots]
\end{split}\]
where thoughout this proof the \([\cdots]\) represents a local, covariant, and horizontal \(d-1\) form which is linear in \(X^m\) and independent of its derivatives. A similar computation for the \(\df\theta'\) term only gives \([\cdots]\)-type terms since the \(\chi^\alpha_{a_1\ldots a_i}\) are all functions i.e. \(0\)-forms. Thus, using <ref> the Noether current can be written as
\[
\df J_X = d\lb[ \df Z_I (X \cdot \df A^I) + \df Z_a (X\cdot \df e^a) \rb] + [\cdots]
\]
where we have again absorbed \(X \cdot \df L\) into the \([\cdots]\)-term. Adding the constraints \(\df{\mc C}(X)\) in <ref>, which are also of \([\cdots]\)-type, to both sides we get
\[
\df J_X - \df{\mc C}(X) - d\lb[ \df Z_I (X \cdot \df A^I) + \df Z_a (X\cdot \df e^a) \rb] = [\cdots]
\]
Since \(d\left[ \df J_X - \df{\mc C}(X) \right] = 0 \) from <ref> and <ref>, we get that the right-hand-side is a closed horizontal \((d-1)\)-form that does not depend on derivatives of \(X^m\). Using <ref> (with \(X^m\) as the “dynamical field" and \(\df\psi^\alpha\) as a “background field") we conclude that right-hand-side vanishes and we get
\[
\df J_X = \df{\mc C}(X) + d\lb[ \df Z_I (X \cdot \df A^I) + \df Z_a (X\cdot \df e^a) \rb]
\]
Thus the Noether charge can be chosen to be of the form in <ref>.
Note here that only the coframes and connection contribute explicitly to the form of the Noether charge since we have converted all other tensor fields and their derivatives into functions (using the coframe and frames). Consequently the form of the Noether charge given by <ref> is much simpler than the corresponding one in Prop. 4.1 <cit.>. Further the expression <ref> for the Noether charge is completely specified by the dependence of the Lagrangian on the curvature and torsion.
The ambiguities <ref> in the Lagrangian and the symplectic potential lead to the following ambiguities in the Noether current and Noether charge
J_X ↦J_X + d(X ·μ) + dλ(δ_Xψ)
Q_X ↦Q_X + X ·μ+ λ(δ_Xψ) + dρwhere \(\df\rho\) is an extra ambiguity in the Noether charge, since the charge is defined by the Noether current only up to a closed and hence exact form (see <ref>). We note that the form of the Noether charge given by <ref> is not unambiguous. If the terms \(\df\mu\) and \(\df\lambda\) have suitable dependence on the curvature and torsion they can contribute non-trivially to the Noether charge.
The utility of the above formalism in deriving a first law stems from the following relation between the symplectic current and the Noether charge. The proof follows by a simple computation on the bundle \(P\), in exact parallel to the ones on spacetime in <cit.>.
For any perturbation \(\delta\df\psi\) and \(X^m \in \mf{aut}(P)\), the symplectic current \(\df\omega(\delta\psi, \Lie_X\psi)\) is related to the Noether charge \(\df Q_X\) by
ω(δψ, _Xψ) = d [ δQ_X - X ·θ(δψ) ] + δC(X) + X·(E_α∧δψ^α)
Consider a variation of <ref> with a given fixed \(X^m\)
δJ_X = δθ(δ_Xψ) - X ·δL
= δθ(δ_Xψ) - X ·dθ(δψ) - X·(E_α∧δψ^α)
= δθ(_Xψ) - _X θ(δψ) + d(X ·θ(δψ) ) - X·(E_α∧δψ^α)
The first two terms on the right-hand-side can be rewritten in terms of the symplectic current using <ref> to get
ω(δψ, _Xψ) = δJ_X - d(X ·θ(δψ) ) + X·(E_α∧δψ^α)
= d [ δQ_X - X ·θ(δψ) ] + δC(X) + X·(E_α∧δψ^α)
From <ref> we see that for linearised solutions \(\delta\df\psi^\alpha\) we have
ω(δψ, _Xψ) = d [δQ_X - X ·θ(δψ) ]
and the corresponding symplectic form <ref> on a Cauchy surface \(\Sigma\) is an integral of a boundary term on \(\partial\Sigma\).
W_Σ(ψ;δψ,_Xψ) = ∫_∂Σ δQ_X - X ·θ(δψ)
Here we have used the fact that both \(\df Q_X\) and \(\df\theta\) are horizontal (i.e. gauge-invariant) forms on \(P\) and thus can be represented as gauge-invariant forms \(\dfM{\df Q}_X\) and \(\dfM{\df\theta}\) on spacetime \(M\). Further, since \(\df\theta\) is horizontal, only the projection \(\dfM X \equiv X^\mu = (\pi_*)^m_\mu X^\mu\) contributes in the second term, but the Noether charge \(\dfM{\df Q}_X\) depends on the full vector field \(X^m \in \mf{aut}(P)\).
As discussed in <cit.>, a boundary Hamiltonian for the dynamics generated by \(X^m\) exists if and only if there is a function \(H_X\) on the space of solutions such that its variation is given by
δH_X = W_Σ(ψ;δψ,_Xψ) = ∫_∂Σ δQ_X - X ·θ(δψ)
which is equivalent to the existence of \(\dfM{\df\Theta}(\psi) \in \Omega^{d-1}M\) so that
∫_∂Σ X ·θ(δψ) = δ∫_∂Σ X ·Θ(ψ)
Note here that \(\dfM{\df\Theta}\) need not be covariant or gauge-invariant in its dependence on the dynamical fields. Thus the boundary Hamiltonian becomes
H_X = ∫_∂ΣQ_X - X ·Θ(ψ)
The existence of the Hamiltonian \(H_X\) is intimately related to the boundary conditions at \(\partial\Sigma\) on the dynamical fields \(\df\psi\), the perturbation \(\delta\df\psi\), and the vector field \(X^m\); for general field configurations and arbitrary perturbations the Hamiltonian might not exist or may not be unique. Imposing boundary conditions so that the symplectic form \(W_\Sigma (\psi; \delta\psi, \Lie_X\psi)\) is finite ensures that the perturbed Hamiltonian \(\delta H_X\) is also well-defined. But, even if we choose boundary conditions so that \(\delta H_X\) is well-defined (it is manifestly covariant and gauge-invariant as it is defined in terms of horizontal forms on \(P\)) there still might not exist a unique, or covariant, or gauge-invariant Hamiltonian \(H_X\).
§ HORIZON POTENTIALS AND CHARGES, AND THE ZEROTH LAW FOR BIFURCATE KILLING HORIZONS
We describe next the spacetimes for which we will derive a zeroth law for bifurcate Killing horizons and a first law of black hole mechanics.
To formulate the zeroth law for bifucate Killing horizons, we will consider dynamical fields \(\df\psi^\alpha\) which determine a \(d\)-dimensional spacetime \(M\) with a bifurcate Killing horizon \( \ms H \defn \ms H^+ \union \ms H^- \) and let the bifurcation surface be \( B \defn \ms H^+ \inter \ms H^- \). The horizon Killing field \(K^\mu\) is null on \(\ms H\) and vanishes on \(B\). We denote the corresponding infinitesimal automorphism on the bundle by \(K^m\), which preserves the background dynamical fields i.e. \(\Lie_K\df\psi^\alpha = 0\). The possible ambiguity in the choice of such \(K^m\) is given in <ref>, <ref> and <ref>. For the zeroth law, we do not demand that the spacetime described above be determined by solutions to the equations of motion \(\df{\mc E}_\alpha = 0\) (<ref>) nor do we require any asymptotic conditions.
Carter-Penrose diagram of the black hole exterior spacetime \((M,g_{\mu\nu})\).
For the first law of black hole mechanics we will consider stationary and axisymmetric dynamical fields \(\df\psi^\alpha\) (<ref>) which determine a \(d\)-dimensional, asymptotically flat, stationary and axisymmetric black hole spacetime with Lorentzian metric \(g_{\mu\nu}\) with a bifurcate Killing horizon as described above. For the bundle \(P\) (<ref>) on which we have formulated the theory, we choose as the base space \(M\), the exterior (including the horizon) of the black hole (see <ref>). The metric \(g_{\mu\nu}\) on \(M\) is determined by the coframes \(\df e^a\) in the standard manner. Thus, we can now identify the abstract Lorentz bundle \(P_O\) in <ref> with the bundle \(F_OM\) of orthonormal frames determined by \(g_{\mu\nu}\).
Since, the dynamical fields \(\df\psi^\alpha\) are defined on the bundle \(P\) we define stationarity and axisymmetry of \(\df\psi^\alpha\) using <ref> which we summarise as follows. Let \(t^\mu\) denote the time translation Killing field, i.e. the Killing field that is timelike near infinity, and \(\phi^\mu_{(i)}\) the axial Killing fields (we use the index \({(i)}\) to account for more than one axial Killing fields at infinity in greater than \(4\) dimensions). Then, there exist infinitesimal automorphisms, \(t^m,\phi^m_{(i)} \in \mf{aut}(P;\df\psi)\) which preserve the dynamical fields i.e. \(\Lie_t\df\psi^\alpha = 0 = \Lie_{\phi_{(i)}}\df\psi^\alpha\) on the bundle \(P\), which project to the corresponding stationary and axial Killing fields \(t^\mu\) and \(\phi^\mu_{(i)}\) respectively. The ambiguity in the choice of such \(t^m\) and \(\phi^m_{(i)}\) is given in <ref> and <ref>. In this case, the horizon Killing field is \(K^\mu = t^\mu + \Omega_{\ms H}^{(i)} \phi^\mu_{(i)}\) (where \(\Omega_{\ms H}^{(i)}\) are constants representing the horizon angular velocities) is null on \(\ms H\) and vanishes on \(B\). We denote the corresponding infinitesimal automorphism on the bundle by \(K^m = t^m + \Omega_{\ms H}^{(i)} \phi^m_{(i)}\).
We will use the notion of asymptotic flatness defined as follows.[One could define asymptotic flatness for principal bundles more rigorously, generalising those in Ch. 11 <cit.>, however for our purposes it will suffice to specify the fall-off of the fields in terms of some asymptotic radial coordinate.] There exist asymptotically Minkowskian coordinates \((t, x^i)\) near spatial infinity, with \(r \defn \sqrt{\sum_i(x^i)^2}\) being the asymptotic radial coordinate and \(\eta_{\mu\nu} \equiv {\rm diag}(-1,1,\ldots,1)\) be the asymptotic flat metric in these coordinates. We require the asymptotic fall-off of the metric to be
g_μν = η_μν + O(1/r^d-3)
with each derivative of the metric falling-off faster by a factor of \(1/r\). To prescribe the fall-off conditions for the dynamical fields defined on the bundle, we lift the asymptotic radial coordinate \(r\) (viewed as a function on \(M\) near infinity) to the bundle near spatial infinity. Then, the fall-off conditions on dynamical fields on the bundle are prescribed as their behaviour in \(1/r\); the precise fall-off conditions are chosen depending on the equations of motion of the theory under consideration so that the solution metric behaves as in <ref>. We expect our results can be generalised to spacetimes with different asymptotics but we stick to the asymptotically flat case.
In the formulation of the first law we will also consider an asymptotically flat Cauchy surface \(\Sigma\) which smoothly terminates at the bifurcation surface \(B\). We further assume that the embeddings of \(\Sigma\) and \(B\) in \(M\) are regular i.e. they admit smooth no-where vanishing normals.
Next, we show that the black hole spacetime described above satisfies a generalisation of the zeroth law for bifurcate Killing horizons in the sense that, we can define certain potentials which are constant on the bifurcate Killing horizon \(\ms H\). The term “zeroth law" for this result is justified by <ref> (see <ref>), where we show that the horizon potential contributed by the gravitational Lorentz connection can be identified with the surface gravity of the black hole.
To prove the zeroth law we first show that the Lie-algebra-valued function \(K \cdot \df A^I\) is covariantly constant on the bundle over the bifurcate Killing horizon.
Let the principal bundle \(P\) restricted to the bifurcate Killing horizon \(\ms H\) be \(P_{\ms H}\). Then, the pullback of \(D(K\cdot \df A^I)\) to \(P_{\ms H}\) vanishes i.e.
. D(K·A^I) |_P_H = 0
First let \(P_B\) be the restriction of the principal bundle to the bifurcation surface \(B\). Since the horizon Killing field satisfies \(K^\mu\vert_{B} = 0\), the corresponding vector field \(K^m\) is vertical on \(P_B\). As a result we have \(\lb. K \cdot \df F^I \rb\vert_{P_B} = 0\), since \(\df F^I\) is a horizontal form.
Now, consider any cross-section \(B'\) of the bifurcate Killing horizon \(\ms H\). Along the flow generated by the vector field \(K^\mu\), \(B'\) limits to the bifurcation surface \(B\) (see 2 <cit.>). Similarly, the principal bundle \(P_{B'}\) limits to \(P_B\) under the flow along the integral curves of the corresponding vector field \(K^m\) on the bundle \(P_{\ms H}\). Clearly, the pullback of \(K \cdot \df F^I\) to the integral curves of \(K^m\) vanishes, and so consider then \(\lb. K \cdot \df F^I \rb\vert_{P_{B'}}\). Taking the limit of \(\lb. K \cdot \df F^I \rb\vert_{P_{B'}}\) along the flow of \(K^m\) as \(P_{B'} \to P_B\), we have \(\lb. K \cdot \df F^I \rb\vert_{P_{B'}} \to 0\). Since, \(K^m\) is an infinitesimal automorphism which preserves the connection \(\df A^I\), \(\Lie_K (\lb. K \cdot \df F^I) \rb\vert_{P_{\ms H}} = 0\) and since the curvature and \(K^m\) are smooth on the bundle \(P_{\ms H}\) we must have \(\lb. K \cdot \df F^I \rb\vert_{P_{B'}} = 0\), and thus
. K ·F^I |_P_H = 0
Thus, using <ref> for the Lie derivative along \(K^m\) of the connection we have
0 = . _K A^I |_P_H = . K ·F^I |_P_H + . D(K ·A^I) |_P_H
= . D(K ·A^I) |_P_H
To define the horizon potentials we will expand the covariantly constant Lie-algebra-valued function \(K \cdot \df A^I\) on \(P_{\ms H}\) in a suitable choice of basis of the Lie algebra \(\mf g\). To motivate the construction first consider the case where \(\mf g = \mf{su}(2)\) (which can be thought of as the Lie algebra of \(3\)-dimensional Euclidean rotations) and the basis of \(\mf{su}(2)\) given by the Pauli matrices \(\{\sigma^I_x, \sigma^I_y, \sigma^I_z\}\) where \(\sigma^I_z\) is diagonal. Now, we note that any given element of \(\mf{su}(2)\) can be aligned with \(\sigma^I_z\) by the adjoint action of some element in the group \(SU(2)\) (i.e. any direction in \(3\)-dimensional Euclidean space can be rotated to align with the \(Z\)-axis). In particular, we align \(K \cdot \df A^I\) with \(\sigma^I_z\) and define the corresponding “potential" \(\ms V\) by the expansion \(K \cdot \df A^I = \ms V \sigma^I_z\).
The above construction for \(\mf{su}(2)\) can be generalised to any semisimple Lie algebra \(\mf g\) by picking a maximal abelian subalgebra of diagonalisable elements called a Cartan subalgebra (for \(\mf{su}(2)\) the Cartan subalgebra can be chosen to be spanned by \(\sigma^I_z\) as done above) and the corresponding Weyl-Chevalley basis of \(\mf g\); the relevant properties of this construction are recalled in <ref>. In the following we assume that a Cartan subalgebra \(\mf h\) of \(\mf g\) has been picked and denote the basis of \(\mf h\) by \(h^I_\Lambda\) where the index \(\Lambda\) enumerates the chosen basis.
Using the properties of a Cartan subalgebra of \(\mf g\) (see <ref>) we define the horizon potentials \(\ms V^\Lambda\) at a point of the bifurcate Killing horizon \(\ms H\) as follows.[Our strategy to define the horizon potentials in terms of the Cartan subalgebra parallels the one used in V. <cit.> to define global charges in Yang-Mills theory.]
Let \(\mf h\) be some fixed choice of Cartan subalgebra of \(\mf g\) and let \(h^I_\Lambda\) given by the simple coroots be a choice of basis of \(\mf h\) (see <ref>). For any point \(x \in \ms H\) on the horizon there exists a point \(u \in P_{\ms H}\) such that \(\pi(u) = x\) and \(K \cdot \df A^I(u) \in \mf h\). Thus, in the chosen basis of \(\mf h\) we can write
K·A^I (u) = V^Λh^I_ΛThe set of coefficients \(\ms V^\Lambda\) is determined up to the action of the Weyl group of \(\mf g\) (see <ref>), irrespective of the chosen point \(u \in \pi^{-1}(x)\) and the chosen simple coroots \(h^I_\Lambda\). We call the coefficients \(\ms V^\Lambda\) in the above expansion, the horizon potentials.
For any point \(x \in \ms H\), the fibre of the principal bundle \(P_{\ms H}\) over \(x\) is \(\pi^{-1}(x) \cong G\). Note that \(K \cdot \df A^I\) is a \(\mf g\)-valued function with the adjoint representation of \(G\). Using <ref>, there exists some point \(u \in \pi^{-1}(x)\) in the fibre where \(K\cdot \df A^I(u) \in \mf h\). Then, using the chosen basis \(h^I_\Lambda\) of \(\mf h\) we can define the coefficients \(\ms V^\Lambda\) as in <ref> at the point \(u \in P_{\ms H}\).
Suppose there exists another point \(\tilde u \in \pi^{-1}(x)\) such that \(K \cdot \df A^I(\tilde u) \in \mf h\) and hence <ref> holds with some other set of coefficients \(\tilde{\ms V}^\Lambda\). From <ref>, the new set of potentials \(\tilde{\ms V}^\Lambda\) are related to the original set \(\ms V^\Lambda\) by the action of some element of the Weyl group on \(\mf h\).
Similarly by <ref>, the possible choice of simple coroots are given by the action of the Weyl group on the original choice \(h^I_\Lambda\).
Thus, the potentials \(\ms V^\Lambda\) are well-defined at any point on the horizon up to the action of the Weyl group of \(\mf g\).
By parallel-transporting the covariant constant \(K \cdot \df A^I\) (from <ref>) and using <ref> as the definition of the horizon potentials we show that the potentials can be consistently chosen to be constant over the entire horizon. This gives us the following generalised zeroth law for bifurcate Killing horizons.
The horizon potentials \(\ms V^\Lambda\) given by <ref> can be chosen to be constant on the horizon
. dV^Λ|_H = 0
Let \(x, x' \in \ms H\) be any two points on the horizon connected by a path \(\gamma\), and let <ref> hold at some choice of \(u \in \pi^{-1}(x)\) as discussed in <ref>. Let \(\Gamma\) be the unique path in \(P_{\ms H}\) starting at \(u\) which is horizontal with respect to the given connection \(\df A^I\) and projects to the path \(\gamma\) (see Prop. 3.1 II.3 <cit.>), and let \(u' \in \pi^{-1}(x')\) be the endpoint of \(\Gamma\). From <ref> we have \(\lb. D(K \cdot \df A^I) \rb\vert_{P_{\ms H}} = 0\). This implies that we can obtain \(K \cdot \df A^I(u')\) by parallel-transporting \(K\cdot \df A^I(u)\) along \(\Gamma\) to always point in the same Lie algebra direction, and further since \(K\cdot \df A^I\) is covariantly constant on \(P_{\ms H}\), the result is independent of the choice of path \(\gamma\) on the horizon. Thus, \(K \cdot \df A^I(u') \in \mf h\) and <ref> holds at the point \(u' \in \pi^{-1}(x')\) with the same set of potentials \(\ms V^\Lambda\). Since the chosen points \(x,x'\) are arbitrary and \(K\cdot \df A^I\) is smooth, the potentials \(\ms V^\Lambda\) must be constant on the entire horizon.
The first set of ambiguities in the horizon potentials arises due to our choice of a fixed Cartan subalgebra \(\mf h\). From <ref> we see that different choices of \(\mf h\) will lead to equivalent sets of horizon potentials.
The other ambiguity in the horizon potentials arises due to our choice of the vector field \(K^m\) on the bundle. From <ref>, given the horizon Killing field \(K^\mu\) on spacetime \(M\), the ambiguity in the corresponding vector field \(K^m\) on \(P\) is given by \(K^m \mapsto \tilde K^m = K^m + Y^m\) where \(Y^m\) is a vertical vector field so that \(Y\cdot \df A^I \in \mf g\) is covariantly constant everywhere (not just on the horizon) on \(P\). Further, if there are other dynamical charged tensor fields (such as the \(\varphi^A\) in <ref>) in the background, then \(Y^m\) is also required to preserve them i.e. \(Y^m\) must also satisfy <ref>. If a non-trivial \(Y^m\) exists for the given dynamical fields \(\df\psi^\alpha\) (<ref>), the new choice \(\tilde K^m\) will define a new set of potentials \(\tilde{\ms V}^\Lambda\) at the horizon. This ambiguity in the potentials does not affect the zeroth law <ref> since \(\tilde{\ms V}^\Lambda\) are also constant on the horizon. However, there might exist some \(Y^m\) so that we can reduce the number of linearly independent potentials. From <ref> we see that this ambiguity \(Y^m\) corresponds to a global symmetry of all the dynamical fields \(\df\psi^\alpha\). Thus, the number of linearly independent horizon potentials are ambiguous if the dynamical fields \(\df\psi^\alpha\) have a global symmetry on \(P\). In that case, we can use \(Y^m\) to redefine the vector field \(K^m\) on \(P\) so that some, or all, of the horizon potentials vanish. This redefinition of the horizon potentials also changes the terms at infinity in the first law (see <ref> for the case of Einstein-Yang-Mills theory). Note however, for some given choice of \(\df\psi^\alpha\) that there might not exist any global symmetries and in general one cannot set the horizon potentials to vanish.
Also from <ref>, \(K^m\) is uniquely determined on the Lorentz bundle part of \(P\) (<ref>) and so this ambiguity does not affect the potentials due to the gravitational Lorentz connection.
From <ref>, the maximum number of non-zero horizon potentials \(\ms V^\Lambda\) is the dimension of \(\mf h\) i.e. the rank of \(\mf g\). Thus, there are at most \(l\) non-zero potentials for each of the simple Lie algebras of rank \(l\) in Cartan's classification (see Theorem A 2.14 <cit.>). For \(SU(2)\)-Yang-Mills theory, this reduces to the case considered by Sudarsky and Wald <cit.>, where they find only one Yang-Mills potential. The analysis can be easily extended to include abelian Lie algebras (which are, by definition, neither simple nor semisimple) to find \(l\) horizon potentials for an abelian Lie algebra of dimension \(l\).
Using the horizon potentials defined in <ref> we show that the perturbed boundary Hamiltonian \(\delta H_K\) on \(B\) associated to the infinitesimal automorphism \(K^m\) can be put into a “potential times perturbed charge" form for any theory under consideration, when the dynamical fields \(\df\psi^\alpha\) satisfy the equations of motion and the perturbation \(\delta\df\psi^\alpha\) satisfies the linearised equations of motion.
The perturbed Hamiltonian on the bifurcation surface \(B\) associated to \(K^m\) can be written as a “potential times perturbed charge" term of the form
. δH_K|_B = V^ΛδQ_Λwhere the horizon potentials \(\ms V^\Lambda\) are as defined in <ref> and the charges \(\ms Q_\Lambda\) are defined by
Q_Λ∫_B Z_I h^I_Λ
where \(\dfM{ \df Z_I h^I_\Lambda }\) is the gauge-invariant \((d-2)\)-form on \(M\) such that \(\df Z_I h^I_\Lambda = \pi^* \lb( \dfM{ \df Z_I h^I_\Lambda } \rb) \), with \(\df Z_I\) given by <ref>.
We first evaluate the perturbed Hamiltonian on the bifurcation surface using <ref>. By <ref>, the symplectic potential \(\df\theta\) is horizontal and the second term in <ref> vanishes at the bifurcation surface since \(\lb.K^m\rb\vert_{P_B}\) is vertical. Similarly, the second term in the form of the Noether charge <ref> for \(K^m\) vanishes. Thus, the perturbed horizon Hamiltonian associated to \(K^m\) is
\[
\lb.\delta H_K\rb\vert_B = \int_B \delta \dfM{\df Q}_K = \int_B \dfM{ \delta \df Z_I ~( K \cdot \df A^I )}
\]
where in the last equality we again use the fact that \(K^m\) is vertical and \(\delta \df A^I\) is horizontal.
Then, using the definition of the horizon potentials (<ref>) and the zeroth law (<ref>) we get the form of the perturbed Hamiltonian in <ref> with the charges \(\ms Q_\Lambda\) given by <ref>.
We note, from <ref>, that only the projection of \(\df Z_I\) to the chosen Cartan subalgebra contributes to the charges in <ref>. Further, from <ref>, the maximum number of non-zero charges \(\ms Q_\Lambda\) is the dimension of \(\mf h\) i.e. the rank of \(\mf g\). For \(SU(2)\)-Yang-Mills theory there is only one Yang-Mills charge as found in <cit.>.
We can show that the ambiguities in the symplectic potential <ref> and the Noether charge <ref> do not affect the perturbed Hamiltonian \(\delta H_K\vert_B\) (the following argument also apply to the perturbed Hamiltonian at spatial infinity). These ambiguities give rise to the following change
\[\begin{split}
\delta \df Q_K - K \cdot \df\theta(\delta\psi) \mapsto & ~\quad \delta \df Q_K - K \cdot \df \theta(\delta\psi) + \delta \df\lambda(\delta_K\psi) + d\delta\df\rho - K \cdot d\df\lambda(\delta\psi) \\
& = \delta \df Q_K - K \cdot \df\theta(\delta\psi) + \delta \df\lambda(\delta_K\psi) - \Lie_K \df\lambda(\delta\psi) + d \lb[ K \cdot \df\lambda(\delta\psi) + \delta\df\rho \rb]
\end{split}\]
Since \(K^m\) is an infinitesimal automorphism which preserves the background dynamical fields \(\df\psi^\alpha\) we have
\[
\delta\df\lambda(\delta_K\psi) = \delta\df\lambda[\psi;\Lie_K\psi] = \df\lambda[\psi;\Lie_K\delta\psi] = \Lie_K \df\lambda(\delta\psi)
\]
and thus
\[
\delta \df Q_K - K \cdot \df\theta(\delta\psi) \mapsto \delta \df Q_K - K \cdot \df\theta(\delta\psi) + d \lb[ K \cdot \df\lambda(\delta\psi) + \delta\df\rho \rb]
\]
Since \(\df\lambda\) and \(\df\rho\) are local and covariant horizontal forms, the integral of \(\delta \dfM{\df Q}_K - \dfM{K} \cdot \dfM{\df\theta}(\delta\psi)\) over a closed surface (the bifurcation surface \(B\)), and consequently the perturbed boundary Hamiltonian \(\delta H_K\vert_B\) (from <ref>), is unambiguous. Since the potentials are defined independently of these ambiguities, from <ref> we see that the charges \(\ms Q_\Lambda\) are also unaffected by these ambiguities.
§.§ Temperature and entropy as the horizon potential and charge for gravity
In the following, we show that the gravitational potential and the perturbed gravitational charge corresponding to the Lorentz connection \(\df\omega^a{}_b\) can be identified (up to conventions of numerical factors) with the temperature and perturbed entropy of the black hole respectively. Thus, the first-order formulation of gravity in terms of the coframes \(\df e^a\) and the Lorentz connection \(\df\omega^a{}_b\) on the Lorentz bundle \(P_O\) gives a new point of view on the temperature and perturbed entropy of the horizon. In particular, the temperature and perturbed entropy can be viewed on the same footing as the potentials and perturbed charges of any matter gauge fields in the theory. Further, we give an explicit formula for the gravitational charge which is a direct parallel of the Wald entropy formula <cit.>.
The gravitational potential (corresponding to the Lorentz connection \(\df\omega^a{}_b\) on the Lorentz bundle \(P_O\) in <ref>) at the bifurcation surface \(B\) only has non-vanishing values in a \(1\)-dimensional vector space spanning the abelian Lie algebra \(\mf{so}(1,1)\) corresponding to local Lorentz boosts of the frames \(\tilde E^m_a\) normal to \(B\) in \(M\). Let \(\tilde\epsilon^{ab} = \tilde E^b \cdot \tilde E^a \cdot \df{\tilde\varepsilon}_2\) be the frame components of the binormal to \(B\) along normal frames \(\tilde E^m_a\). Then, \(-\tilde\epsilon^{ab}\) forms a basis of \(\mf{so}(1,1)\) so that the gravitational potential and charge can be written as
V_grav = Q_grav = -∫_B Z_ab ϵ̃^ab
where \(\surgrav\) is the surface gravity of \(B\). In <ref> \(\dfM{ \df Z_{ab} \tilde\epsilon^{ab}}\) is the unique gauge-invariant form on spacetime that pullsback to \(\df Z_{ab} \tilde\epsilon^{ab}\) on the bundle. \(\df Z_{ab}\) can be computed from <ref> for the Lorentz connection \(\df\omega^a{}_b\), and thus, the gravitational charge formula is in direct parallel to the Wald entropy formula <cit.>.
Thus, we can define the temperature and the perturbed entropy of the bifurcate Killing horizon as
T_H 1/2πV_grav δS 2πδQ_grav
Since, for a stationary axisymmetric black hole, \(K^m \in \mf{aut}(P;\df e^a)\) is an infinitesimal automorphism which preserves the coframes, we have from <ref>
\[
K \cdot \df \omega^{ab} = \tfrac{1}{2}E_a \cdot E_b \cdot d\df\xi + (K \cdot \df e^c) C_{cab}
\]
where \(\df\xi = (K\cdot \df e^a)\df e_a\) is the pullback to the bundle of the Killing form \(\dfM{\df \xi} \equiv \xi_\mu = g_{\mu\nu}K^\nu\) and the contorsion \(C_{abc}\) is defined in <ref>. The vector field \(K^m\) is vertical on \(P_B\) and using usual definition of the surface gravity \(\surgrav\) we can write the Killing form on the bifurcation surface as \(d\dfM{\df\xi}\vert_B = 2\surgrav \dfM{\df{\tilde\varepsilon}}_2\), where \(\dfM{\df{\tilde\varepsilon}}_2\) is the binormal to \(B\) (see 12.5 <cit.>). Thus we have
K ·ω^ab|_P_B = -E^b ·E^a ·ε̃_2
where \(\df{\tilde\varepsilon}_2\) is the binormal \(\dfM{\df{\tilde\varepsilon}}_2\) lifted to the bundle \(P_B\). Note that the torsion terms vanish due to \(K^m\) being a vertical vector field on \(P_B\).
The surface gravity is constant on a bifurcate Killing horizon (see <cit.> and also <ref> below) i.e. \(d\surgrav\vert_{P_B} = 0\), and using <ref> for the Lorentz connection, we get \(D (E^b \cdot E^a \cdot \df{\tilde\varepsilon}_2)\vert_{P_B} = 0\). Thus the Lorentz bundle \(P_O\vert_B\), identified with the orthonormal frame bundle \(F_OM\vert_B\) of the background solution metric, can be reduced to the bundle of adapted orthonormal frames (see VII.1 <cit.>) i.e. \(P_O\vert_B \cong F_OB \oplus F_NB \) where \(F_OB\) is the orthonormal frame bundle of \(B\) and \(F_NB\) is the \(O(1,1)\)-bundle of frames normal to \(B\) in \(M\). Denote the adapted normal frames in \(F_NB\) by \(\tilde E_a^m\).
From Prop. 1.4 in VII.1 <cit.> we see that the connection \({\df \omega^a}_b\vert_{P_B}\) can be written as a direct sum of a \(O(d-2)\)-connection on \(F_OB\) and an abelian \(O(1,1)\)-connection \(\df \omega_N\) on \(F_NB\). The invariant tensor \(\tilde\epsilon^{ab} = \tilde E^b \cdot \tilde E^a \cdot \df{\tilde\varepsilon}_2 \) acts as a projector to this abelian \(O(1,1)\)-connection as \(\df \omega_N \defn \tfrac{1}{2}\df \omega_{ab} \tilde\epsilon^{ab}\). Thus, <ref> becomes \(K \cdot \df \omega_N = \surgrav\) i.e. the surface gravity is the vertical part of \(K^m\) (with respect to the background connection) in the normal frame bundle of \(B\).
A choice of Cartan subalgebra \(\mf h\) of the Lorentz Lie algebra is spanned by boosts in \(\mf{so}(1,1)\) normal to \(B\) and some choice of commuting rotations in \(\mf{so}(d-2)\). Since, \(K\cdot \df \omega^{ab}\) only points in the \(\mf{so}(1,1)\)-part we have (choosing \(-\tilde\epsilon^{ab}\) as a basis of \(\mf{so}(1,1)\))
\[
\ms V_{grav} = K \cdot \df \omega_N = \surgrav
\]
and the corresponding charge, using <ref>
\[
\ms Q_{grav} = -\int_B \dfM{ \df Z_{ab} \tilde\epsilon^{ab}}
\]
and \(\df Z_{ab}\) is given by <ref> for the Lorentz connection \(\df\omega^a{}_b\). Thus, the gravitational charge is determined in direct parallel to the Wald entropy formula <cit.> for any theory of gravity formulated in terms of the coframes and a Lorentz connection.
Note that since the coframes completely fix the form of the infinitesimal automorphism \(K^m\) (see <ref>) we cannot eliminate the temperature and the perturbed entropy by a redefinition of \(K^m\) in any spacetime. For General Relativity, the gravitational charge can be computed to be \( \ms Q_{grav} = \tfrac{1}{8\pi}{\rm Area}(B) \) (see <ref>) and we get the usual notion of temperature and perturbed entropy for black holes.
The gravitational potential i.e. the surface gravity \(\surgrav\), was shown to be constant on bifurcate Killing horizons in <cit.> without using the Einstein equations. Thus, in view of <ref>, we can see that <ref> can be seen as a “generalised zeroth law for bifurcate Killing horizons" (analogous to the result of <cit.>) showing that the potentials defined in <ref> are always constant on a bifurcate Killing horizon without using any equations of motion.
Rácz and Wald <cit.> showed that if the surface gravity has non-vanishing gradient on any null geodesic generator of a (not necessarily bifurcate) Killing horizon then there necessarily exists a parallel-propagated curvature singularity on the horizon, without using the Einstein equations. If one uses the Einstein equations and the dominant energy condition on matter fields, then the surface gravity was shown to be constant on any (not necessarily bifurcate) Killing horizon in <cit.>. The results of <cit.> can also be viewed as different versions of “the zeroth law".
§ THE FIRST LAW FOR GAUGE-INVARIANT LAGRANGIANS
Next, we formulate the first law of black hole mechanics for a stationary-axisymmetric black hole solution described at the beginning of <ref>. The first law is obtained by evaluating the symplectic form on any Cauchy surface \(\Sigma\) for \(X^m = K^m\) where \(K^m\) is the infinitesimal automorphism that projects to the horizon Killing field \(K^\mu\). Since the black hole is stationary and axisymmetric \(\Lie_K\df\psi^\alpha = 0\), using <ref> for the symplectic form, the first law is an equality of the perturbed boundary Hamiltonians \(\delta H_K\) evaluated at the bifurcation surface and at spatial infinity.
The perturbed Hamiltonian on the bifurcation surface was already put into a “potential times perturbed charge" form in <ref>. Near spatial infinity, we can lift the asymptotic Minkowski radial coordinate \(r\) (viewed as a gauge-invariant function) to the bundle \(P\). We choose the dynamical fields and their perturbations to fall-off suitably in \(1/r\) so that the symplectic form \(W_{\Sigma}\) is finite. The particular choice of fall-off in general depends on the specific Lagrangian theory under consideration. For the infinitesimal automorphisms \(t^m\) and \(\phi^m_{(i)}\) we define the canonical energy and canonical angular momentum as the corresponding Hamiltonians (whenever they exist) at infinity (see <cit.>).
E_can . H_t|_∞= ∫_∞Q_t - t ·Θ
J_(i),can - . H_ϕ_(i)|_∞= -∫_∞Q_ϕ_(i) - ϕ_(i) ·Θ
where \(\dfM{\df\Theta}\) is as described by <ref>. Thus, the perturbed Hamiltonian at infinity associated to \(K^m = t^m + \Omega^{(i)}_{\ms H} \phi^m_{(i)}\) becomes \( \delta H_K\vert_\infty = \delta E_{can} - \Omega_{\ms H}^{(i)} \delta J_{(i),can} \). In all the examples we consider in <ref>, only the Einstein-Hilbert Lagrangian contributes a non-zero \(\dfM{\df \Theta}\) at infinity and for all other cases \(\dfM K \cdot \dfM{\df \theta}\) falls off fast enough that we can choose \(\dfM{\df \Theta} = 0\). Note that in \(4\)-dimensions, we might need to impose faster fall-off conditions or some suitable generalisation of the Regge-Teitelboim parity conditions <cit.> on the dynamical fields and their perturbations for the \(J_{(i),can}\) to be well-defined; we assume that such choices have been made to get a well-defined canonical angular momentum.
This leads to our main result in the following theorem which gives us a general formulation of the first law of black hole mechanics.
Consider any theory with a local, covariant and gauge-invariant Lagrangian of the form <ref>. Let \(\df\psi^\alpha\) be a solution corresponding to a stationary axisymmetric black hole with a bifurcate Killing horizon (described at the beginning of <ref>) and \(\delta\df\psi^\alpha\) be an arbitrary linearised solution. Then the first law of black hole mechanics takes the form
T_HδS + V'^ΛδQ'_Λ= δE_can - Ω_H^(i) δJ_(i),can
where, on left-hand-side the first term consists of the temperature and perturbed entropy of the black hole as described in <ref> and the second term is the potential and perturbed charge of the connection \(\df A'^{I'}\) (see <ref>) described in <ref> and <ref>, and the terms on the right-hand-side being defined at infinity by <ref>.
The first law is obtained by evaluating the expression <ref> with \(X^m = K^m\) on a hypersurface \(\Sigma\) which goes from the bifurcation surface \(B\) to spatial infinity. For a stationary axisymmetric black hole \(\Lie_K \df\psi^\alpha = 0\) and the left-hand-side of <ref> vanishes and then the first law equates the perturbed boundary Hamiltonian of \(K^m\) at \(B\) to the one at infinity.
δH_K |_B = δH_t |_∞- Ω^(i)_H δH_ϕ_(i)|_∞Using <ref>, <ref>, <ref> and <ref> we get the first law <ref>.
At this point, we emphasise that we have not assumed any choice of gauge in the above form of the first law, and in fact this form holds even when the principal bundle is non-trivial and no choice of gauge can be made.
As discussed after <ref>, the perturbed Hamiltonian at \(B\), and also at infinity is not affected by the ambiguities in the Lagrangian and the symplectic potential. However there is an ambiguity in choosing the vector field \(K^m\) on the bundle \(P\) (see <ref> and <ref>) corresponding to a global symmetry of the background dynamical fields \(\df\psi^\alpha\) if any exists. Note that the vector field \(K^m\) is uniquely determined over the Lorentz bundle \(P_O\) part (from <ref>) and thus, the temperature and perturbed entropy, as well as, the gravitational contributions to the canonical energy and canonical angular momentum are unambiguous. Thus, the possible ambiguity in choosing \(K^m\) leads to a simultaneous redefinition of the horizon potentials \(\ms V'^\Lambda\) and charges \(\ms Q'_\Lambda\), and the contributions to the canonical energy and angular momenta at infinity of any non-gravitational fields. One can use such an ambiguity to set some, or possibly all, of the horizon potentials at the horizon to vanish (see <ref>) at the cost of changing the contributions to the canonical energy and canonical angular momenta (see also <ref> for Einstein-Yang-Mills theory). Even though this ambiguity affects the individual terms, the form of the first law <ref> holds for any choice of \(K^m\) on the bundle.
Even though \(\delta H_K\) is unambiguously defined (given a choice of \(K^m\)), the Hamiltonian \(H_K\) (if it exists) is not. Consider first the Hamiltonian at the bifurcation surface \(B\) (the analysis proceeds similarly for spatial infinity). Since the \(\df\theta \) contribution vanishes at \(B\), the choice of \(\dfM{\df\Theta}\) has the ambiguity
\[
\int_B \dfM K\cdot \dfM{\df\Theta} \mapsto \int_B \dfM K\cdot \dfM{\df\Theta} - \int_B \dfM{K} \cdot \dfM{\df\Lambda}
\]
where \(\int_B \dfM{K} \cdot \dfM{\df\Lambda}\) is some topological invariant of \(B\) (possibly depending on the embedding of \(B\) in \(M\)) and does not change under variations. Similarly for the Noether charge at \(B\) we have the ambiguities
\[
\int_B \dfM{\df Q}_K \mapsto \int_B \dfM{\df Q}_K + \int_B \dfM{K \cdot \df\mu}
\]
Thus the ambiguity in the Hamiltonian \(H_K\) is of the form
\[
H_K \mapsto H_K + \int_B \dfM{K} \cdot \dfM{\df\Lambda} +\int_B \dfM{K \cdot \df\mu}
\]
But we have already shown that \(\delta H_K\) is unambiguous. Thus, any contribution of the ambiguities \(\df\mu\) and \(\dfM{\df\Lambda}\) to the boundary Hamiltonian can be considered as a topological charge. Similarly, we can have topological charge contributions to the Hamiltonian at spatial infinity. In the gravitational case, the topological charges at spatial infinity can be fixed by requiring that flat Minkowski spacetime have vanishing ADM mass and ADM angular momentum. Nevertheless, it is possible to have topologically non-trivial solutions to Yang-Mills theory which result in non-trivial topological charges (e.g. magnetic monopole charges) at the horizon. In fact such topological charges do arise in Yang-Mills theory when we add the \(\df\mu\)-ambiguity to the Lagrangian (<ref>). Similarly, the addition of the Euler density to the Einstein-Hilbert Lagrangian corresponds to the \(\df\mu\)-ambiguity <ref> where \(\df\mu\) is not horizontal[Iyer and Wald <cit.> only considered \(\df\mu\) that were gauge-invariant in which case the \(\df\mu\)-ambiguity does not affect the Hamiltonian at \(B\).] (see <ref>) and does contribute a topological term to the Noether charge at \(B\) <cit.>. Even though these topological charges do not affect the first law of black hole mechanics for stationary black holes, they do affect any attempt to define a total entropy and charge for stationary black holes purely from the first law, as we shall discuss later.
Since the perturbed entropy is given by the perturbed gravitational charge, Iyer and Wald <cit.> prescribe that the total entropy (known as the Wald entropy) for a stationary axisymmetric black hole be defined as the gravitational charge
S_Wald Q_grav = - ∫_B Z_ab ε̃^ab
This prescription has the advantage that the entropy \(S_{\rm Wald}\) satisfies the first law and is the same on any cross-section of the horizon of a stationary axisymmetric black hole since, \(\df J_K = 0\) on the horizon <cit.> . But this prescription is not unambiguous. For instance, an alternative definition of the entropy as \(S = S_{\rm Wald} + C(B)\) where \(C(B)\) is a topological invariant of the bifurcation surface \(B\) also satisfies the first law, since \(C(B)\) does not change under linearised variations <cit.>. In fact the \(\df\mu\)-ambiguity <ref> in the Lagrangian contributes a topological charge of precisely this nature. In the case of General Relativity, as pointed out by <cit.>, the gravitational charge does acquire such a topological contribution (the Euler number of \(B\)) when the Euler density is added to the Einstein-Hilbert Lagrangian in \(4\)-dimensions even though the equations of motion are unaffected (also see <ref>). The area theorem for General Relativity in \(4\)-dimensions (along with an energy condition) guarantees that the entropy defined as the area of a horizon cross-section always increases and thus satisfies the second law of black hole mechanics (see <cit.>, Prop. 9.2.7 <cit.> or Theorem 12.2.6 <cit.>). If one includes the topological charges of the horizon in the definition of the entropy then the second law is violated even for General Relativity (see <cit.>). Thus, it seems that a version of the second law is needed to fix at least some of the ambiguities in defining the total entropy even for a stationary black hole. To consider the second law one has to consider non-stationary black hole configurations where the entropy prescription <ref> can have more ambiguities which vanish only in the stationary case <cit.>. Unfortunately, a general formulation of the second law for an arbitrary theory of gravity including arbitrary matter fields remains out of reach; though it has been investigated in special situations for higher curvature gravity <cit.>. In the absence of a second law, the first law <ref> only determines the perturbed entropy \(\delta S\). In light of this we will refrain from giving a prescription for the total entropy \(S\) for stationary black holes or for a dynamical entropy for non-stationary ones.
§ EXAMPLES
In this section we use the formalism described above to derive the first law of black hole mechanics for the first-order coframe formulation of General Relativity, Einstein-Yang-Mills theory and Einstein-Dirac theory. The Lagrangians considered in this section are of the form \(\df L = \df L_{\rm grav} + \df L_{\rm matter}\), where the gravitational Lagrangian \(\df L_{\rm grav}\) only depends on the coframes \(\df e^a\) and a Lorentz connection \({\df \omega^a}_b\) on the Lorentz bundle \(P_O\) and not on the matter fields.[For the case of dilaton gravity considered in <cit.> the gravitational Lagrangian does depend on an additional scalar field. We will not consider such examples in this section but they are covered in the more general formulation in <ref>.] It will be convenient to work in the first-order formalism where we consider the coframes \(\df e^a\) and the Lorentz connection \(\df\omega^a{}_b\) as independent fields. We will write the equations of motion obtained by varying the Lagrangian with respect to the coframes and the Lorentz connection as
E_a - T_a = 0 E_ab - S_ab = 0
where the gravitational contributions to the equations of motion are
E_a ≡(E_a)_m_1…m_d-1 = 1/d δ(L_grav)_m_1…m_d-1l/δe_n^a δ^l_n
E_ab ≡(E_ab)_m_1…m_d-1 = 1/d δ(L_grav)_m_1…m_d-1l/δω_n^ab δ^l_n
and the matter contributions are
T_a ≡(T_a)_m_1…m_d-1
= - 1/d δ(L_matter)_m_1…m_d-1l/δe_n^a δ^l_n
S_ab ≡(S_ab)_m_1…m_d-1 = -1/d δ(L_matter)_m_1…m_d-1l/δω_n^ab δ^l_n
i.e. \(\df{\mc T}_a\) is the energy-momentum and \(\df{\mc S}_{ab}\) is the spin current of the matter fields, both written as \((d-1)\)-forms. We note here that when the matter Lagrangian depends on the gravitational Lorentz connection used, \(\df{\mc T}_a\) does not give the usual symmetric energy-momentum tensor <cit.>.
If one insists on having vanishing torsion from the outset (i.e. one works in the second-order formalism) then the Lorentz connection is completely determined by the coframes (see <ref>) and one can use <ref> (and <ref>) to convert all the variations of the torsionless connection to variation of the coframes. In that case, the second equation of motion in <ref> is deleted but the “new" energy-momentum tensor \(\df{\mc T}_a\) gets contributions from the spin current <cit.>.
We consider the first-order formulation of General Relativity with the gravitational Lagrangian to be given by the Palatini-Holst Lagrangian. We show that the gravitational charge <ref> is given by the area of the bifurcation surface and thus we reproduce the usual identification between the perturbed entropy and perturbed area of the bifurcation surface. Similarly, (up to terms involving torsion) <ref> reproduces the ADM mass and ADM angular momentum and we get the usual first law of black hole mechanics for General Relativity. For the matter Lagrangian we consider the two cases of Yang-Mills Lagrangian for gauge fields of any semisimple group, and the free Dirac Lagrangian for spinor fields. These examples can be generalised to include chiral spinor fields with Yang-Mills charge, charged scalar fields such as the Higgs field, and thus, the entire Standard Model of particle physics. We shall work out the details in \(4\)-spacetime dimensions but the computations can be easily generalised to other dimensions. We also illustrate the topological charge ambiguities that arise in the definitions of the Hamiltonian and the entropy.
§.§ Palatini-Holst
To start let us consider the first-order formulation of General Relativity in \(4\)-spacetime dimensions in vacuum. A derivation of the first law in this case was recently given in <cit.> by using a generalised notion of Lie derivatives of the coframes called the Lorentz-Lie derivative. As discussed in <ref> the Lorentz-Lie derivative defined in <cit.> depends non-linearly on the coframes and does not form a Lie algebra for diffeomorphisms of spacetime. Thus, the Lorentz-Lie derivative is not the generator of any group action on the coframes. We will write the first-order Palatini-Holst action on the oriented Lorentz bundle i.e. for this section \(P = P_{SO}\) over spacetime \(M\) and use the notion of the Lie derivative on the bundle to obtain a first law. When a vector field \(X^m\) is an automorphism which preserves the coframes the bundle notion of Lie derivative coincides with the Lorentz-Lie derivative defined by <cit.>. Thus, even though the Noether charge we define for arbitrary automorphisms of the frame bundle in not equivalent to that of <cit.> we get the same results for the first law for stationary axisymmetric black holes.
The dynamical fields for the first-order formulation of General Relativity are the coframes \(\df e^a \in \Omega^1_{hor}P(\bb R^4)\) and the Lorentz connection \({\df \omega^a}_b \in \Omega^1P(\mf{so}(3,1))\). The Palatini-Holst Lagrangian for General Relativity can be written as:
L_PH = 132πϕ_abcd e^a ∧e^b ∧R^cd ∈Ω^4_horP
where \(\phi_{abcd} \defn \epsilon_{abcd} + \tfrac{2}{\gamma}\eta_{a[c}\eta_{d]b} \) is an \(SO(3,1)\)-invariant tensor with \(\gamma\) being the Barbero-Immirzi parameter. This tensor satisfies the index symmetries \(\phi_{abcd} = \phi_{cdab}\) and \(\phi_{abcd} = - \phi_{bacd} = -\phi_{abdc}\). The \(\epsilon\)-term is the usual Palatini-Einstein-Hilbert Lagrangian and the \(\gamma\)-term corresponds to the Holst Lagrangian <cit.>. The corresponding Lagrangian form on spacetime is
\[
\dfM{\df L}_{PH} = \frac{1}{16\pi} \dfM{\df\varepsilon}_4 \left( R - \frac{1}{2\gamma} \varepsilon^{\mu\nu\lambda\rho} R_{\mu\nu\lambda\rho} \right)
\]
Note that the last term vanishes (using <ref>) when one restricts, a priori, to a torsionless connection.
A variation of the Lagrangian gives \( \delta \df L_{PH} = \df{\mc E}_a \wedge \delta \df e^a + \df{\mc E}_{ab} \wedge \delta \df \omega^{ab} + d\df\theta \) where:
E_a = -116π ϕ_abcd e^b ∧R^cd
E_ab = 116πϕ_abcd e^c ∧T^d
θ = 132πϕ_abcd e^a ∧e^b ∧δω^cd
while the symplectic current is:
ω= 116πϕ_abcd e^a ∧( δ_1 e^b ∧δ_2 ω^cd - δ_2 e^b ∧δ_1 ω^cd )
In vacuum the equation of motion \(\df{\mc E}_{ab} = 0\) implies that \(\df T^a = 0\) <cit.>. Since we define the Noether current and Noether charge “off-shell" we will not make use of this fact. Also, in <ref> the torsion can be sourced by the spin current of matter fields and so it will be useful to keep this term to analyse the effects of torsion.
A straightforward computation gives the Noether current <ref> for \( X^m \in \mf{aut}(P) \) as:
J_X = θ_X - X ·L = dQ_X + E_a (X ·e^a) + E_ab (X ·ω^ab)
with the Noether charge \(\df Q_X = \tfrac{1}{32\pi}\phi_{abcd}~ \df e^a \wedge \df e^b (X \cdot \df \omega^{cd}) \in \Omega^2_{hor}P\).
Now, consider \( X^m \in \mf{aut}(P; \df e^a, {\df \omega^a}_b)\) i.e. an infinitesimal automorphism that preserves the coframes and the connection. From <ref> such an automorphism is uniquely determined by a Killing field of the spacetime metric determined by the coframes. The Noether charge for such an \(X^m\) becomes:
Q_X = 1/16π1/4ϕ_ab^cd e^a ∧e^b ( E_c ·E_d ·dξ+ 2(X ·e^e) C_ecd )
= -1/16π ( ⋆[ dξ- 2(X ·e^a) C_a ] + 1/γ [ dξ- 2(X ·e^a) C_a ] )
where \(\df\xi = (X \cdot \df e^a) \df e_a\), \(\star\) is the horizontal Hodge dual operation on differential forms on the Lorentz bundle <ref> and we have written the contorsion <ref> as a horizontal \(2\)-form as \(\df C_a \defn \tfrac{1}{2}C_{abc} \df e^b \wedge \df e^c\).
To get the first law for a stationary axisymmetric black hole solution (see <ref>) we use the infinitesimal automorphism \(X^m = K^m = t^m + \Omega_{\ms H} \phi^m\) which projects to the horizon Killing field \(K^\mu\) on \(M\). Since \(K^\mu\) vanishes on the bifurcation surface the \(\df\theta\)-term and the contorsion terms do not contribute. Using \(*d\dfM{\df\xi}\vert_B = -2\surgrav\dfM{\df\varepsilon}_{2}\) where \(\dfM{\df\varepsilon}_{2}\) is the volume form on \(B\) we have:
∫_B Q_X = -1/16π ∫_B(* dξ + 1/γ dξ ) = /8π ∫_B ε_2 = /2π 1/4Area(B)
Thus, following <ref>, the perturbed entropy for General Relativity is
δS = 1/4δArea(B)
Next we compute the canonical energy and angular momentum using <ref> and show that they correspond to the ADM mass and ADM angular momentum up to torsion terms. Since the spacetime is asymptotically flat, near spatial infinity the spacetime is asymptotically Minkowskian, and the Lorentz bundle is asymptotically trivial. Then, near infinity there is a section (i.e. choice of gauge) \(s_\infty : M \to P\) such that the pullbacks \( \dfM{\df e}^a \equiv e^a_\mu = (s_\infty^*)^m_\mu e^a_m \) and \({\dfM{\df \omega}^a}_b \equiv {\omega^a{}_b}_\mu = (s_\infty^*)^m_\mu {\omega^a{}_b}_m \) satisfy the asymptotic conditions
e^a = e_M^a + O(1/r) ; ω^a_b = O(1/r^2)
where the asymptotic coframes \(\dfM{\df e}_{\bb M}^a\) are adapted to the asymptotic Minkowskian coordinates as
e_M^a = dt, dx, dy, dz
To compute the canonical energy <ref>, consider the pullback of \(t \cdot \df\theta(\delta\varphi)\) through the section \(s_\infty\) at infinity. Using the fall-off conditions on the pullback of the dynamical fields at infinity <ref> we can write \( s_\infty^*(t \cdot \df\theta) = \delta (\dfM t \cdot \dfM{\df\Theta} ) \) where:
t ·Θ = t·(1/32πϕ_abcd e^a ∧e^b ∧ω^cd) = 1/16πϕ_abcd (t·e^a) e^b ∧ω^cd + Q_X
Note, that \(\dfM{\df\Theta}\) is not gauge-invariant under the action of the Lorentz group \(O(d-1,1)\). Using <ref> The canonical energy at infinity is given by
\[
E_{can} = -\tfrac{1}{16\pi}\int_\infty \phi_{abcd} (\dfM t\cdot \dfM{\df e}^a) \dfM{\df e}^b \wedge \dfM{\df \omega}^{cd}
\]
Note here that the Noether charge contribution to the canonical energy actually cancels out. As discussed in <cit.> this accounts for the “factor of \(2\)" discrepancy in the Komar formula for the ADM mass relative to the one for ADM angular momentum. Also note that this cancellation is much more easily obtained in the computation presented here than the corresponding one in <cit.>. We next show that the above expression for the canonical energy reproduces the well-known ADM mass formula.
Let \(r_\mu\) be the outward pointing spatial conormal to a \(2\)-sphere at infinity with the volume form \(\dfM {\df\varepsilon}_2\) and \(h_{\mu\nu}\) be the asymptotic spatial metric on the Cauchy surface \(\Sigma\). The Einstein-Hilbert term gives
-116π∫_∞ϵ_abcd (t·e^a) e^b ∧ω^cd = -116π ∫_∞ε_2 ϵ_abcd t^σe^a_σe^b_μω^cd_νt_λr_ρε^λρμν
= -3!16π∫_∞ε_2 t^σt_λr_ρω^cd_νδ^[λ_σE^ρ_c E^ν]_d
= 18π∫_∞ε_2 r_ρE^ρ_c (E^dη∇̂_νe^c_η) ( t^λt_λE^ν_d - t^νt_λE^λ_d )
= 18π∫_∞ε_2 r_ρE^ρ_c (∇̂_νe^c_η) ( - g^νη - t^νt^η)
= -18π∫_∞ε_2 r^[λ h^ν]μ e_λa ( ∂_μe^a_ν- Γ^σ_μνe^a_σ)
where in the third line we written the Lorentz connection in terms of the derivatives of the coframes and in the last line in terms of the Christoffel symbols.
Using the asymptotic conditions <ref> the first term can be written as
-18π∫_∞ε_2 ∂_μ( r^λh_M^νμ e_M^a_[λe_ν]a) = -116π∫_∞d *_2[ r ·( e_M^a ∧e_a ) ] = 0
where \( h_{\bb M}^{\nu\mu}\) is the flat spatial Cartesian metric on the asymptotic Cauchy surface and \(*_2\) is the \(2\)-dimensional Hodge dual on the asymptotic sphere. The second term, depending on the Christoffel symbols, can be written as
18π∫_∞ε_2 r^λh^νμ Γ_[λν]μ = 116π ∫_∞ε_2 r^λh^νμ ( ∂_νh_λμ - ∂_λh_νμ - T_μνλ )
using the definition of the Christoffel symbols (with torsion)
\[
\Gamma_{\lambda\nu\mu} = \tfrac{1}{2} \left( \partial_\mu g_{\nu\lambda} + 2\partial_{[\nu}g_{\lambda]\mu} \right) + \tfrac{1}{2}\left( - T_{\mu\nu\lambda} + 2T_{(\nu\lambda)\mu} \right)
\]
Now computing the Holst-term contribution to the canonical energy we have
-18πγ∫_∞(t·e^a) e^b ∧ω_ab = -18πγ∫_∞(t·e^a) ( de_a - T_a)
= 116πγ ∫_∞ε_2 t^με^νλ T_μνλ
Thus we get the total canonical energy as:
E_can = 116π ∫_∞ε_2 r^λh^νμ ( ∂_νh_λμ - ∂_λh_νμ ) + 116π ∫_∞ε_2 (- r^λh^νμ +1γt^με^νλ ) T_μνλ
The first term can be recognised as the usual formula for the ADM mass \(M_{ADM}\), while the second is the canonical energy contributed by the presence of any torsion at infinity.
Now for the canonical angular momentum, the \(\phi\cdot\df\theta\)-term does not contribute when pulled back to the sphere at infinity and we get (here \(\dfM{\df \phi} \equiv \phi_\mu\))
J_can = - ∫_∞Q_ϕ= 116π∫_∞* [ dϕ - 2ϕ^μC_μ] + 1γ [ dϕ - 2ϕ^μC_μ]
= 116π∫_∞* dϕ - 116π∫_∞ε_2 [- ε̃^νλ + 1γ ε^νλ ] ϕ^μC_μνλ
where the first term is the Komar formula for the ADM angular momentum \(J_{ADM}\) and second is the angular mometum due to any torsion at infinity (\(\tilde{\varepsilon}^{\mu\nu}\) is the binormal to the \(2\)-sphere at infinity).
As noted before in vacuum, the equation of motion \(\df{\mc E}_{ab} = 0\) ensures that the torsion vanishes everywhere and the canonical energy and angular momentum are exactly the ones given by the ADM quantities. This is also the case if any matter sources for torsion fall-off suitably at infinity, as happens in the case of the Dirac field <ref>. We note that, when the torsion due to matter sources does not fall-off fast enough the Barbero-Immirizi parameter \(\gamma\) does contribute to the canonical energy and angular momentum at infinity.
Thus, we get the usual first law for vacuum General Relativity
/2π 1/4δArea(B) = δM_ADM - Ω_H δJ_ADM
To illustrate the \(\df\mu\)-ambiguity <ref>, we consider the following three topological terms, that can be added to the Palatini-Holst Lagrangian in \(4\) dimensions.
\[\begin{split}
\df L_{\rm E} & = \tfrac{1}{2}\epsilon_{abcd} \df R^{ab} \wedge \df R^{cd} \\
\df L_{\rm P} & = {\df R^a}_b \wedge {\df R^b}_a = -\df R^{ab} \wedge \df R_{ab} \\
\df L_{\rm NY} & = \df T^a \wedge \df T_a - \df e^a \wedge \df e^b \wedge \df R_{ab}
\end{split}\]
The first is the Euler character of the Lorentz bundle over \(M\) (also known as the Gauss-Bonnet invariant), the second is the corresponding Pontryagin character, and the third is the Nieh-Yan character <cit.>, which exists only for a connection with torsion. Each of these terms are exact forms \(\df L = d\df\mu\) where the corresponding \(\df\mu\)'s can be computed to be[The explicit expression for \(\df\mu_{\rm E}\) seems to be largely absent from the literature except in <cit.>, and in <cit.> where it is given in terms of the Dirac matrices.]
\[\begin{split}
\df\mu_{\rm E} & = \tfrac{1}{2} \epsilon_{abcd} \df \omega^{ab} \wedge \left( \df R^{cd} - \tfrac{1}{3} {\df \omega^c}_e \wedge \df \omega^{ed} \right) \\
\df\mu_{\rm P} & = {\df \omega^a}_b \wedge \left( {\df R^b}_a - \tfrac{1}{3} {\df \omega^b}_c \wedge {\df \omega^c}_a \right) \\
\df\mu_{\rm NY} & = \df e^a \wedge \df T_a
\end{split}\]
Note that \(\df\mu_{\rm P}\) is just the Chern-Simons term for the Lorentz bundle, \(\df\mu_{\rm E}\) is similar but with the “trace" taken with a \(\epsilon_{abcd}\), and \(\df\mu_{\rm NY}\) can be viewed as the Chern-Simons term for torsion. These terms contribute the following additional terms to the Noether charge at the bifurcation surface
\[\begin{split}
\lb. \df Q_K \rb\vert_{\rm E} & = \epsilon_{cdab} \df R^{cd} (K \cdot \df \omega^{ab}) = \tfrac{1}{2} \epsilon_{cdab} \df R^{cd} (E^a \cdot E^b \cdot d\df\xi) \\
\lb. \df Q_K \rb\vert_{\rm P} & = -2 \df R_{ab} (K \cdot \df \omega^{ab}) = -\df R_{ab} (E^a \cdot E^b \cdot d\df\xi)\\
\lb. \df Q_K \rb\vert_{\rm NY} & = - \df e_a \wedge \df e_b (K \cdot \df \omega^{ab}) = -\tfrac{1}{2} \df e_a \wedge \df e_b(E^a \cdot E^b \cdot d\df\xi)\\
\end{split}\]
and hence integrating the corresponding gauge-invariant forms over \(B\) gives
∫_B . Q_K |_E = - 2 ∫_B ⋆ϵ̃_ab R^ab = 2 ∫_B ϵ_ab R^ab
∫_B . Q_K |_P = 2 ∫_B ϵ̃_ab R^ab
∫_B . Q_K |_NY = ∫_B dξ = 0
The Euler contribution is the Euler class of the tangent bundle \(TB\) i.e. the Euler characteristic of \(B\). The Pontryagin contribution is, similarly, the Euler class of the normal bundle of \(B\) in \(M\). Since, we have smooth, no-where vanishing normals to \(B\), the Euler class of the normal bundle must vanish (see Prop. 11.17 in <cit.>).[This can also be shown by an explicit computation of this term, and the fact that the extrinsic curvature of \(B\) in \(M\) vanishes (see
<cit.>).] Further, we see that the Holst Lagrangian is \(\df L_{Holst} \sim -\df L_{NY} + \df T^a \wedge \df T_a\), and is thus an exact form up to terms not involving the curvature. This explains why the Holst term does not contribute to the Noether charge at the horizon.[See <cit.> for an Euclidean path integral approach to the Holst and Nieh-Yan terms in the Lagrangian.] The Noether charge contributions <ref> are purely topological and do not contribute to the perturbed entropy, and due to the asymptotic fall-off conditions they do not contribute to the canonical energy and angular momentum at infinity. Hence none of the above terms affect the first law. But they do affect a prescription for a total entropy as discussed towards the end of <ref>. As shown in <cit.>, the Euler term in the Noether charge leads to violations of the second law in General Relativity if one prescribes that the total entropy be given by the gravitational charge.
§.§ Yang-Mills
Next let us consider Einstein-Yang-Mills theory where the Yang-Mills connection \(\df A'^{I'}\) in <ref> is a dynamical field governed by the Yang-Mills Lagrangian <ref>. Since we have worked out the gravitational contribution in <ref> in this section we only deal with the Yang-Mills connection, and so, for simplicity omit the “primes" and write the Yang-Mills connection as \(\df A^I\).
The contribution of Yang-Mills fields to the first law was worked out by Sudarsky and Wald <cit.> under the assumption that one can pick a global choice of gauge i.e. an everywhere smooth section \(s : M \to P\) (see <ref>).[Sudarsky and Wald also assume that the Yang-Mills group \(G\) is compact, and is \(SU(2)\), but this restriction can be easily removed.] They then consider the pullback \(A^I_\mu = (s^*)^m_\mu A^I_m\) of the Yang-Mills connection as a tensor field on spacetime. They further assume that the section \(s\) can be chosen so that \(A^I_\mu\) is stationary with respect to the horizon Killing field \(K^\mu\).
As discussed in <ref>, these assumptions are too restrictive to cover all Yang-Mills fields that are of interest. As noted before, sections on a principal bundle exist if and only if the bundle is trivial. For a non-trivial bundle there exist no global sections and the Yang-Mills connection cannot be considered as a globally well-defined tensor field on spacetime. It is far from clear that even for a trivial bundle a section \(s\) can be chosen so that for a given connection \(\df A^I\), the pullback \(s^*\df A^I \equiv A_\mu^I\) is both smooth everywhere and is annihilated by the Lie derivative with the horizon Killing field \(K^\mu\). Using this assumption Sudarsky and Wald conclude that the Yang-Mills fields do not contribute to the first law at the bifurcation surface as the Yang-Mills potential on \(B\) vanishes since \(K^\mu A^I_\mu\vert_B = 0\). In fact, as argued by Gao <cit.>, the Maxwell gauge field on the Reissner-Norstörm spacetime, in the standard choice of gauge, is singular on the bifurcation surface. Working instead on some other cross-section of the horizon where the Maxwell vector potential is smooth, Gao finds a “potential times perturbed charge" term at the horizon for Maxwell fields. <cit.> also finds that Yang-Mills fields do contribute at the horizon but their contribution cannot be put into a “potential times perturbed charge" form without additional gauge choices which might be incompatible with the assumed stationarity of \(A_\mu^I\) (see 4 <cit.>).
Our formalism allows us to work on arbitrary non-trivial bundles for Yang-Mills fields without making any choice of gauge, and we only assume that the Yang-Mills connection on the principal bundle is Lie-derived up to a gauge transformation i.e. \(\Lie_K \df A^I = 0\) where \(K^m\) is an infinitesimal automorphism of the bundle which projects to the horizon Killing field \(K^\mu\). Using this, we find potentials for the Yang-Mills fields that are constant on the horizon (<ref>) and that the Yang-Mills fields do contribute a “potential times perturbed charge" term at the bifurcation surface (<ref>) without assuming any choice of gauge. Thus, the following will be a generalisation of the results of <cit.>.
The Yang-Mills Lagrangian can be written for any structure group \(G\) using a non-degenerate, symmetric bilinear form on its Lie algebra \(\mf g\), which is invariant under the adjoint action of \(G\) on its Lie algebra. Since we have assumed that the Lie algebra \(\mf g\) of the structure group is semisimple we will use the Killing form \(k_{IJ}\) (<ref>) as such a non-degenerate, symmetric, invariant bilinear form.[Our convention for the Killing form differs from that of <cit.> by a sign and a factor of \(2\).] Further, any semisimple Lie algebra can be decomposed uniquely into a direct sum of simple Lie algebras (see 1.10 <cit.> or 11.2 <cit.>), and thus the Yang-Mills Lagrangian for \(\mf g\) can be written as a sum of Yang-Mills Lagrangians of the same form for each simple factor (with possibly different coupling constants). We can also include abelian groups (which, by definition, are neither simple nor semisimple) into the theory by using the natural product on their Lie algebra \(\bb R^n\). For instance, to get Maxwell electromagnetism we can use \( k_{IJ} \to -2 \) and \(g^2 \to \mu_0 \) in <ref>.
With the above discussion, we write the Yang-Mills Lagrangian on the bundle as:
L_YM = 1/4g^2 (⋆F_I ) ∧F^I = 1/8g^2 ε_4 ( F^2 ) ∈Ω^4_horP
where \(F^2 \defn (E^a \cdot E^b \cdot \df F^I)(E_a \cdot E_b \cdot \df F_I)\) and \(g^2\) is the Yang-Mills coupling constant. On spacetime \(M\) we get the usual Lagrangian \(\dfM{\df L}_{YM} = \frac{1}{8g^2}\dfM{\df\varepsilon}_4 F^I_{\mu\nu} F_I^{\mu\nu} \).
The first variation gives \(\delta \df L_{YM} = -\df{\mc T}_a \wedge \delta \df e^a + \df{\mc E}_I \wedge \delta \df A^I + d\df\theta^{\rm (YM)}\) with:
E_I = -12g^2 D⋆F_I
θ^(YM) = 12g^2 ⋆F_I ∧δA^I
where we have used the first form of the Lagrangian. The symplectic current contribution takes the form \( \df\omega^{\rm (YM)} = \tfrac{1}{2g^2} \left[ \delta_1 (\star \df F_I) \wedge \delta_2 \df A^I - \delta_2 (\star \df F_I) \wedge \delta_1 \df A^I \right] \).
To compute the energy-momentum \(3\)-form \(\df{\mc T}_a\), it is convenient to use the second form of the Lagrangian <ref>. Varying with the tetrad we have
\[\begin{split}
\tfrac{1}{8g^2}\delta_e (\df\varepsilon_4 F^2) & = \tfrac{1}{8g^2}\tfrac{1}{3!} \epsilon_{abcd} \delta \df e^a \wedge \df e^b \wedge \df e^c \wedge \df e^d ~F^2 + \tfrac{1}{8g^2} \df\varepsilon_4 \delta_e \left[ (E^a \cdot E^b \cdot \df F^I) (E_a \cdot E_b \cdot \df F_I) \right] \\
& = \tfrac{1}{8g^2} \left[ -\tfrac{1}{3!} \epsilon_{abcd} \df e^b \wedge \df e^c \wedge \df e^d ~F^2 \right] \wedge \delta \df e^a + \df\varepsilon_4 \tfrac{1}{2g^2} (E^a \cdot E^b \cdot \df F^I) (\delta E_a \cdot E_b \cdot \df F_I)
\end{split}\]
The first term can be written in the form
\[
\tfrac{1}{8g^2} \left[ -\tfrac{1}{3!} \epsilon_{abcd} \df e^b \wedge \df e^c \wedge \df e^d ~F^2 \right] \wedge \delta \df e^a = \tfrac{1}{2g^2} \left[ \star \df e_a \left( -\tfrac{1}{4} F^2 \right) \right] \wedge \delta \df e^a
\]
and the second term as (using <ref>)
\[\begin{split}
\df\varepsilon_4 \tfrac{1}{2g^2} (E^a \cdot E^b \cdot \df F^I) (\delta E_a \cdot E_b \cdot \df F_I) &= - \df\varepsilon_4 \tfrac{1}{2g^2} (E^c \cdot E^b \cdot \df F^I) ( E_a \cdot E_b \cdot \df F_I) E_c \cdot \delta \df e^a \\
& = \tfrac{1}{2g^2} \left[ (E_c \cdot \df\varepsilon_4) (E^c \cdot E^b \cdot \df F^I) (E_a \cdot E_b \cdot \df F_I) \right] \wedge \delta \df e^a \\
& = \tfrac{1}{2g^2} \left[ \star \df e_c (E^c \cdot E^b \cdot \df F^I) (E_a \cdot E_b \cdot \df F_I) \right] \wedge \delta \df e^a
\end{split}\]
putting these together we have
\[
\df{\mc T}_a = - \tfrac{1}{2g^2} \star \df e^c \left[ (E_a \cdot E_b \cdot \df F_I) (E_c \cdot E^b \cdot \df F^I) - \tfrac{1}{4} \eta_{ac} F^2 \right]
\]
To find the Noether current consider the following computation for some \(X^m \in \mf{aut}(P)\):
X ·L_YM = 18g^2 X ·(ε_4 F^2) = 18g^2 13! ϵ_abcd (X ·e^a) e^b ∧e^c ∧e^d F^2
= -12g^2 [ ⋆e_a ( -14 F^2 ) ] (X·e^a)
θ_X^(YM) = 12g^2⋆F_I ∧_X A^I = 12g^2⋆F_I ∧( X ·F^I + D(X ·A^I ) )
= 12g^2⋆F_I ∧X ·F^I + 12g^2D( ⋆F_I (X ·A^I) ) - 12g^2 D ⋆F_I (X ·A^I)
The first term in \(\df\theta_X^{\rm (YM)}\) can be written as
\[\begin{split}
\tfrac{1}{2g^2}\star \df F_I \wedge X \cdot \df F^I & = \tfrac{1}{2g^2}\star \df F^I \wedge \tfrac{1}{2!} X \cdot (\df e^a \wedge \df e^b) (E_b \cdot E_a \cdot \df F_I) \\
& = \tfrac{1}{2g^2} \tfrac{1}{2! 2!} \epsilon_{efcd} \df e^c \wedge \df e^d (E^f \cdot E^e \cdot \df F^I) \wedge \df e^b (E_b \cdot E_a \cdot \df F_I) ( X \cdot \df e^a) \\
& = \tfrac{1}{2g^2} \star \df e^c (E^b \cdot E_c \cdot \df F^I) (E_b \cdot E_a \cdot \df F_I) ( X \cdot \df e^a)
\end{split}\]
This gives the Noether current:
J_X^(YM) = 12g^2 ⋆e^c [ (E^b ·E_c ·F^I) (E_b ·E_a ·F_I) -14 η_ac F^2 ] (X·e^a)
+ E_I (X ·A^I) + 12g^2D( ⋆F_I (X ·A^I) )
= dQ_X^(YM) + E_I (X ·A^I) - T_a (X ·e^a)
The terms with \(\df{\mc E}_I\) and \(\df{\mc T}_a\) contribute to the constraints of Einstein-Yang-Mills theory (see <ref>) and the Noether charge contribution is:
Q_X^(YM) = 12g^2⋆F_I (X ·A^I)
which is of the general form given in <ref>.
Now consider a stationary-axisymmetric connection \(\df A^I\) which satisfies the Einstein-Yang-Mills equations on a stationary axisymmetric black hole spacetime. This means that the horizon Killing field \(K^m\) is an infinitesimal automorphism that preserves the Yang-Mills conection \(\Lie_K \df A^I = 0\). The extent to which \(K^m\) is determined by its projection \(K^\mu\) on \(M\) is given by <ref>. Following the computations in <ref>–<ref> we can write the Noether charge of Yang-Mills fields at the horizon as
\[
\int_B \dfM{\df Q}_K^{\rm (YM)} = \ms V^\Lambda \ms Q_\Lambda
\]
where \(\ms V^\Lambda\) is the Yang-Mills potential and \(\ms Q_\Lambda \) is the Yang-Mills electric charge (also see <cit.>) given by
Q_Λ= 12g^2∫_B * F_I h^I_Λ
where the \(h^I_\Lambda\) are some fixed basis of a fixed choice of Cartan subalgebra on the horizon as defined in <ref>, and the contribution to the first law at \(B\) becomes
δ∫_B Q_K^(YM) = V^ΛδQ_Λ
To compute the contribution at infinity, first consider the vector field \(t^m\) which gives the canonical energy \(E_{can}\). We choose the Yang-Mills fields to satisfy the asymptotic conditions
\[
\df F^I = O(1/r^2)
\]
which also gives \(\delta \df A^I = O(1/r)\). We immediately see that \(\df\theta^{\rm (YM)} = \tfrac{1}{2g^2}\star \df F_I \wedge \delta \df A^I = O(1/r^3)\) and does not contribute to the first law.
Since \(t^m\) is an infinitesimal automorphism which preserves the connection we see that
\[\begin{split}
0 &= \left(\Lie_t \df F^I\right)\vert_{P_\infty} = D(t \cdot \df F^I) - {c^I}_{JK} (t \cdot \df A^J) \df F^K \\
& = - {c^I}_{JK} (t \cdot \df A^J) \df F^K + O(1/r^3) \\
0 & = (\Lie_t \df A^I)\vert_{P_\infty} = t\cdot \df F^I + D(t\cdot \df A^I) \\
& = D (t\cdot \df A^I) + O(1/r^2)
\end{split}\]
Up to higher order terms in \(1/r\), we can repeat the procedure in <ref>–<ref> to get contribution of Yang-Mills fields to the canonical energy as
\[
E_{can}^{\rm (YM)} = \int_\infty \dfM{\df Q}_t^{\rm (YM)} = \ms V^\Lambda \ms Q_\Lambda
\]
with the Yang-Mills electric charge <ref> but evaluated at infinity. Note that aymptotically the Yang-Mills equation of motion at infinity becomes
\[\begin{split}
0 = \df{\mc E}_I\vert_{P_\infty} = -\tfrac{1}{2g^2} D\star \df F_I & = O(1/r^3) \\
\implies D(\star \df F_I h^I_\Lambda) = d (\star \df F_I h^I_\Lambda) & = O(1/r^3)
\end{split}\]
where we used the basis \(h^I_\Lambda\) of the fixed Cartan subalgebra defined in <ref>. Thus, the Yang-Mills charge can be computed over any “sufficiently large" surface homologous to a sphere. Further
δ(t ·A^I)|_∞= (t ·δA^I)|_∞= O(1/r)
and the variation of the potential term falls off faster at infinity and we have the first law contribution at infinity as
δE_can^(YM) = δ∫_∞Q_t^(YM) = V^ΛδQ_Λ
In a similar manner (assuming faster fall-off for the asymptotic fields if necessary), the Yang-Mills contribution to the canonical angular momentum is
J_can^(YM) = -∫_∞Q_ϕ^(YM) = - 12g^2 ∫_∞(ϕ·A^I) * F_I
The first law of black hole mechanics in Einstein-Yang-Mills then can be written as
T_HδS + .(V^ΛδQ_Λ)|_B = δM_ADM + .(V^ΛδQ_Λ)|_∞- Ω_H ( δJ_ADM + δJ_can^(YM) )
When the Yang-Mills structure group is abelian the charge at \(B\) and infinity are equal (using the abelian Yang-Mills equation of motion). Thus, the abelian Yang-Mills term in the first law can be written as a “difference in potentials times perturbed charge" \(\lb( \lb.\ms V^\Lambda\rb\vert_\infty - \lb.\ms V^\Lambda\rb\vert_B \rb) \delta \ms Q_\Lambda\). Further, the ambiguity in the choice of the vector field \(K^m\) (for a given horizon Killing field \(K^\mu\)) is given by a \(\bb R^n\)-valued constant function \(\lambda\); here \(n\) is the dimension of the abelian structure group (see <ref>). By a suitable choice of \(\lambda\) we can always set the potentials at the horizon to vanish, while shifting the potentials at infinity by a constant.
Even for a non-abelian structure group, if there exists a vertical vector field \(Y^m \in VP\) corresponding to a global symmetry of \(\df A^I\) (which is a solution to the Einstein-Yang-Mills equations) as described in <ref>, we can use \(Y^m\) to redefine our choice of the horizon Killing field \(K^m\) on the bundle. Using, this freedom we can set some, or possibly all, of the horizon potentials to zero but at the cost of changing the potentials at infinity. The cases where we can set all potentials at the horizon to vanish correspond to the analysis of Sudarsky and Wald <cit.>. For arbitrary connections \(\df A^I\), which solve the Einstein-Yang-Mills equations, there may not exist any such global symmetry (which corresponds to the existence of global solutions to \(D\lambda^I = 0 \), see <ref>). Thus, it seems that the first law as derived in <cit.> applies only to special cases and in general, the first law for Einstein-Yang-Mills takes the form above (also see the related discussion in <cit.>).
As an illustration of the \(\df\mu\)-ambiguity (<ref>) we add a topological term to the Yang-Mills Lagrangian using the Chern character of the bundle as [In particle physics literature this is also known as the \(\theta\)-term; where the \(\theta\) refers to the conventional coupling constant in front of \(\df L_C\) and is unrelated to the symplectic potential.]
L_C 12 F_I ∧F^I = dμ_C
where \(\df\mu_{\rm C}\) is the Chern-Simons form
μ_C = 12(A_I ∧F^I - 16c_IJK A^I ∧A^J ∧A^K)
The corresponding Noether charge for \(K^m\) then gets an additional contribution
∫_B Q_K = V^ΛQ̃_Λwhere the Yang-Mills magnetic charge is
Q̃_Λ= ∫_B F_I h^I_Λ
The magnetic charge is purely a topological charge that does not vary under perturbations and hence does not contribute to the first law.
§.§ Dirac spinor
As the third case of interest we consider spinor matter fields in Einstein-Dirac theory. The principal bundle of interest in this case is \(P_{Spin}\) with the structure group \(Spin^0(3,1)\) (see <ref> for details). The Dirac Lagrangian can be written as:
L_Dirac ε_4 ( 1/2ΨΨ- 1/2ΨΨ- m ΨΨ)
where \(\df\varepsilon_4\) is the horizontal volume \(4\)-form on \(P_{Spin}\) <ref> and \(\dirac\) is the Dirac operator <ref>. Note that we have admitted a connection with torsion and so this Lagrangian is not equivalent to usual Dirac Lagrangian which uses the torsionless Levi-Civita connection <ref> (see V.B.4 <cit.>); the dynamics of the Dirac field does depend on the choice of spin connection used. If one assumes that the connection is torsionless from the outset, then one has to work in the “second-order formulation". Since the following computations are easier in the first-order formalism, we will continue to use an independent connection with torsion to obtain a first law for Einstein-Dirac theory. The computations in the second-order formalism can be performed in exactly the same manner (see <ref>).
To get the Dirac equation and the symplectic potential compute first the variation with the Dirac spinor fields:
ε_4 (ΨδΨ) = ε_4 ( Ψγ^a E_a ·DδΨ) = - ( E_a ·ε_4 ) Ψ∧γ^a DδΨ
= D [ (E_a ·ε_4) Ψγ^a δΨ] - D[ (E_a·ε_4) Ψγ^a ] δΨwhere the second equality in the first line uses the vanishing of a horizontal \(5\)-form. The first term in the last line contributes to the symplectic potential \(\df\theta\) while, using <ref> the second term can be written as:
- D[ (E_a·ε_4) Ψγ^a ] δΨ = - 1/3! D ( Ψγ^a ϵ_abcd e^b ∧e^c ∧e^d ) δΨ
= - ε_4 (ΨδΨ) - 1/3! Ψγ^a ϵ_abcd T^b ∧e^c ∧e^d δΨ
Similarly computing the variation with respect to the Dirac cospinor field \(\adj\Psi\) we have \( \delta_\Psi \df L_{Dirac} = \adj{\df{\mc E}} \delta\Psi + \delta\adj\Psi~ \df{\mc E} + d\df\theta^{\rm (Dirac)} \) with
E = [ (- m )Ψ- 13! T^b_ba (γ^a Ψ) ] ε_4
E = [ (- - m )Ψ+ 13! T^b_ba (Ψγ^a) ] ε_4
θ^(Dirac) = 12 ( E_a·ε_4) ( Ψγ^a δΨ- δΨγ^a Ψ)
= 12 ( ⋆e_a) ( Ψγ^a δΨ- δΨγ^a Ψ)
where we have used the frame components of the torsion \(T^c{}_{ab} \defn E^b \cdot E^a \cdot \df T^c\) and the horizontal Hodge dual <ref>.
On spacetime \(M\), the Dirac equation \(\dfM{\df{\mc E}}\) takes the form
\[
\dfM{\df{\mc E}} = \left[ (\dirac - m )\Psi - \tfrac{1}{3!} {T^\nu}_{\nu\mu}~ \gamma^\mu \Psi \right]\dfM{\df\varepsilon}_4
\]
We again, note that this is not equivalent to the usual Dirac equation since we chose a spin connection with torsion <cit.> — setting the torsion to vanish however does give us the standard Dirac equation.
For the energy-momentum form we have to compute the variation with the tetrad. For this we rewrite
\[\begin{split}
\df\varepsilon_4 \adj\Psi \dirac\Psi & = \tfrac{1}{4!} \epsilon_{abcd} \df e^a \wedge \df e^b \wedge \df e^c \wedge \df e^d \adj \Psi \gamma^e E_e \cdot D\Psi \\
& = - \tfrac{1}{3!} \epsilon_{abcd} \df e^b \wedge \df e^c \wedge \df e^d \wedge (\adj\Psi \gamma^a D \Psi)
\end{split}\]
\[
\delta_e (\df\varepsilon_4 \adj\Psi \dirac\Psi) = -\tfrac{1}{2} \epsilon_{abcd} \df e^b \wedge \df e^c \wedge (\adj\Psi \gamma^d D\Psi) \wedge \delta \df e^a
\]
\[
- \delta_e (\df\varepsilon_4 m \adj\Psi \Psi) = \tfrac{1}{3!} \epsilon_{abcd} \df e^b \wedge \df e^c \wedge \df e^d (m \adj\Psi \Psi) \wedge \delta \df e^a
\]
Thus, we have \(\delta_e \df L_{Dirac} = - \df{\mc T}_a \wedge \delta \df e^a\) with the energy-momentum
\[\begin{split}
\df{\mc T}_a &= \epsilon_{abcd} \df e^b \wedge \df e^c \wedge \left[ \tfrac{1}{4} \left( \adj\Psi \gamma^d D\Psi - D\adj\Psi \gamma^d \Psi \right) - \tfrac{1}{3!} \df e^d m \adj\Psi \Psi \right] \\
& = (\star \df e_a) \left( \tfrac{1}{2}\adj\Psi \dirac \Psi - \tfrac{1}{2}\dirac \adj\Psi \Psi - m \adj\Psi\Psi \right) - \tfrac{1}{2} (\star \df e_b) \left( \adj\Psi \gamma^b E_a \cdot D\Psi - E_a \cdot D\adj\Psi \gamma^b \Psi \right)
\end{split}\]
For the spin current compute the variation due to the connection:
δ_ω1/2( ε_4 ΨΨ) = -1/16 ε_4 ( Ψγ^c [γ_a, γ_b] ΨE_c ·δω^ab ) = 1/16 ( E_c ·ε_4 ) ( Ψγ^c [γ_a, γ_b] Ψ) ∧δω^ab
where the last equality uses the vanishing of a horizontal \(5\)-form. Thus we have \( \delta_\omega \df L = -\df{\mc S}_{ab} \wedge \delta \df \omega^{ab} \) where the spin current is:
S_ab -1/16 (E_c ·ε_4 ) ( Ψγ^c [γ_a, γ_b] Ψ+ Ψ[γ_a, γ_b]γ^c Ψ)
= -1/16 (⋆e_c) ( Ψγ^c [γ_a, γ_b] Ψ+ Ψ[γ_a, γ_b]γ^c Ψ)
To compute the Noether current and Noether charge we need a notion of a “Lie derivative" for spinor fields. As discussed in <ref> the prescriptions for defining a Lie derivative on spinors in the existing literature <cit.> are not satisfactory. Since we view spinors as fields defined on the spin bundle \(P_{Spin}\) we can use the natural notion of Lie derivatives with respect to a vector field \(X^m \in \mf{aut}(P)\) on the bundle which we will show (see <ref>) reduces to the definition given by Lichnerowicz <cit.> in the case \(X^m\) projects to a Killing vector field. Using <ref> the Lie derivative on spinor fields on the bundle can be written as \( \Lie_X\Psi \defn X \cdot d\Psi = X \cdot D\Psi + \tfrac{1}{8} X \cdot \df \omega_{ab}[\gamma^a, \gamma^b]\Psi \). The Noether current then is
J_X^(Dirac) = - T_a (X ·e^a) - S_ab (X ·ω^ab)
The energy-momentum and spin current terms on the right-hand-side contribute to the constraints. Thus the Noether charge contribution of the Dirac fields can be chosen to vanish i.e. \(\df Q_X^{\rm (Dirac)} = 0\), as we expect from the general formula in <ref>.
For the first law, stationary axisymmetric Dirac fields i.e. \(\Lie_K \Psi = 0 \) do not explicitly contribute to the black hole entropy since the Dirac field contribution to the Noether charge vanishes identically. Near infinity, the Dirac field \(\Psi\) falls off faster than \(1/r^{3/2}\), in which case \(t \cdot \df\theta^{\rm (Dirac)}\) falls-off faster than \(1/r^3\). Since the Noether charge contribution of the Dirac field vanishes, the Dirac field does not explicitly contribute to boundary integral defining the canonical energy (<ref>). These fall-offs also ensure that the torsion terms in the gravitational canonical energy in <ref> vanish. Similarly, there is no explicit Dirac contribution to the boundary integral for the canonical angular momentum and the torsion terms in the gravitational canonical angular momentum <ref> also vanish. Thus, the first law of black hole mechanics with spinor fields governed by the Einstein-Dirac Lagrangian <ref> retains the form <ref>.
§ ACKNOWLEDGEMENTS
I would like to thank Robert M. Wald for numerous insightful discussions throughout the course of this work, and Caner Nazaroglu for very helpful comments on Lie groups and Lie algebras. I would also like to thank Arif Mohd, Ted Jacobson, Igor Khavkine and Alexander Grant for comments on the initial draft of this work. This research was supported in part by the NSF grants PHY 12-02718 and PHY 15-05124 to the University of Chicago.
§ MATHEMATICAL ASIDE
In this section we collect some useful mathematical formulae and new results needed in the main arguments of the paper.
First we recall the definition of a Cartan subalgebra and the properties of the corresponding Weyl-Chevalley basis for a semisimple Lie algebra \(\mf g\), which are needed to define the horizon potentials in <ref>.
A Cartan subalgebra \(\mf h\) of a complex semisimple Lie algebra \(\mf g\) is a maximal abelian Lie subalgebra such that the adjoint action of \(\mf h\) on \(\mf g\) (given by the Lie bracket) is diagonalisable (see 2.1 <cit.>).
[Properties of a Cartan subalgebra] We list below the key properties of Cartan subalgebras we will need in the following.
* Any two Cartan subalgebras of a semisimple complex Lie algebra \(\mf g\) are isomorphic under the adjoint action of some element of the corresponding group \(G\) (Theorem F. 2.10 <cit.>); the dimension of the Cartan subalgebras is called the rank of \(\mf g\).
* The Killing form \(k_{IJ}\) is non-degenerate on any Cartan subalgebra of \(\mf g\) ( 2.3 <cit.>).
* For any given choice of Cartan subalgebra \(\mf h\) (of dimension \(l\)), there exists a choice of basis for \(\mf g\) (of dimension \(n\)) — the Weyl-Chevalley basis — given by \(\{h^I_\Lambda, a^I_i\}\). Here, \(h^I_\Lambda\) with \(\Lambda = 1,2,\ldots,l\) are a choice of simple coroots (see 2 <cit.>) and form a basis of \(\mf h\). The remaining basis elements \(a^I_i\) for \(i = 1,2,\ldots, n-l\) are orthogonal to \(\mf h\) with respect to the Killing form \(k_{IJ}\) (see 2.8 and 2.9 <cit.> for details).
* Given a choice of the simple coroots \(h^I_\Lambda\) any other choice can be obtained by the action of a finite subgroup of \(G\) — called the Weyl group of \(\mf g\) ( 2.11 <cit.>). The action of the Weyl group elements on the \(h^I_\Lambda\) is generated by certain permutations and sign changes (see 2.14 <cit.> for a description of the simple coroots and the Weyl group for simple Lie algebras).
* Any given element \(X^I \in \mf g\) can be mapped into a chosen Cartan subalgebra \(\mf h\) by the adjoint action of some element of the group \(G\) <cit.>. For a given \(X^I \in \mf g\), all possible choices of the corresponding element in \(\mf h\) under the above map, are related by the action of the Weyl group on \(\mf h\); since the Weyl group is a finite group there are only finitely many possible choices.[This last statement can be proved by writing an element of \(\mf h\) in a basis given by the simple coroots, and then applying the results of the first theorem in 10.3 <cit.>.]
The above properties of a Cartan subalgebra strictly hold for a complex semisimple Lie algebra. When, the Lie algebra \(\mf g\) of the theory under consideration is a real semisimple Lie algebra, we first take its complexification (which is also semisimple; see 11.3 <cit.>) to apply the Cartan subalgebra construction above, and then in the end take the real form corresponding to the original real Lie algebra \(\mf g\) (see 11.10 <cit.>).
§.§ Principal fibre bundles
Consider a \(G\)-principal bundle \(\pi : P \to M\). Let \(\df A^I \in \Omega^1P(\mf g, {\rm Ad})\) be a connection on \(P\) and let the corresponding covariant exterior derivative \(D : \Omega^kP(\bb V;R) \to \Omega^{k+1}_{hor}P(\bb V;R)\) on equivariant differential forms valued in a vector space \(\bb V\) on \(P\) (see II.5. <cit.> and Vbis.A.4 <cit.>). If \(\df\sigma^A \in \Omega^k_{hor}P(\bb V;R)\) is a horizontal equivariant form (corresponding to gauge covariant fields on spacetime) then
Dσ^A = dσ^A + ( A^I r_I^A_B )∧σ^B
where \(r\) is the representation of the Lie algebra \(\mf g\) on \(\bb V\). The covariant derivative \(D\) acting on the connection itself defines the curvature \(2\)-form \(\df F^I \in \Omega^2_{hor}P(\mf{g}, {\rm Ad})\) as
F^I DA^I = dA^I + 12c^I_JK A^J ∧A^K
where \({c^I}_{JK}\) are the structure constants of the Lie algebra. The curvature further satisfies the Bianchi identity
DF^I = dF^I + c^I_JK A^J ∧F^K = 0
which can be directly checked using <ref>.
If \(f : P \to P\) is an automorphism of the principal bundle \(P\) we denote the corresponding diffeomorphism of \(M\) by \(\dfM f\) so that \( \pi \circ f = \dfM f \circ \pi\). Let the group of automorphisms of \(P\) be \({\rm Aut}(P)\) and the corresponding Lie algebra of vector fields \(\mf{aut}(P) \subset TP\). A vertical automorphism is an \(f \in {\rm Aut}(P)\) which projects to the identity on the base space \(M\) i.e. \(\dfM f = \id_M\). The vector fields in the Lie algebra \(\mf{aut}(P)\) act on equivariant differential forms by the usual Lie derivative. Using <ref> we can write the Lie derivative of equivariant forms in terms of the covariant exterior derivative as
_X σ^A = X ·Dσ^A + D( X ·σ^A) - (X ·A^I) r_I^A_B σ^B
_X A^I = X ·F^I + D(X ·A^I)
for \(\df\sigma^A \in \Omega^k_{hor}P(\bb V;R)\) and the connection \(\df A^I\). Note both \(\Lie_X \df\sigma^A\) and \(\Lie_X \df A^I\) are horizontal forms (and hence gauge covariant). When \(X^m \in \mf{aut}_V(P)\) this corresponds to infinitesimal internal gauge transformations.
Given some equivariant differential form \(\df\sigma^A\) there might exist some bundle automorphisms which keep \(\df\sigma^A\) invariant.
An automorphism of the bundle \(f \in {\rm Aut}(P)\) is an automorphism which preserves some given equivariant differential form \(\df\sigma^A \in \Omega^kP(\bb V;R)\) if \(f^*\df\sigma^A = \df\sigma^A\). Similarly \(X^m \in \mf{aut}(P)\) is an infinitesimal automorphism which preserves \(\df\sigma^A\) if \(\Lie_X \df\sigma^A = 0\).
For a given \(\df\sigma^A\), denote the subgroup of the automorphisms preserving \(\df\sigma^A\) by \({\rm Aut}(P;\df\sigma^A) \subseteq {\rm Aut}(P)\), and the corresponding Lie subalgebra of infinitesimal automorphisms \(\mf{aut}(P;\df\sigma^A) \subseteq \mf{aut}(P)\).
Since for the first law we are interested in stationary and axisymmetric field configurations, we define a notion of stationarity (axisymmetry) for a charged field \(\df\sigma^A\) defined on a bundle \(P\) over a base space \(M\) with a stationary (axisymmetric) metric as bundle automorphisms which preserve \(\df\sigma^A\) and project to the stationary (axisymmetric) isometries of some given metric \(g_{\mu\nu}\) on \(M\).
For a stationary and/or axisymmetric spacetime \(M\) with a metric \(g_{\mu\nu}\), a charged field \(\df\sigma^A\) defined on a bundle \(P \to M\) is called stationary if there exists a one-parameter family \(f_t \in {\rm Aut}(P;\df\sigma^A) \) which projects to a one-parameter family \(\dfM f_t\) of stationary isometries of \(g_{\mu\nu}\). Similarly, \(\df\sigma^A\) is axisymmetric if there exists a one-parameter family \(f_\phi \in {\rm Aut}(P;\df\sigma^A) \) which projects to a one-parameter family \(\dfM f_\phi\) of axisymmetric isometries of \(g_{\mu\nu}\).
The corresponding stationary (and axial) infinitesimal automorphism vector field \(t^m\) (and \(\phi^m\)) projects to the stationary (and axial) Killing vector \(t^\mu\) (and \(\phi^\mu\)) of \(g_{\mu\nu}\), respectively.
A diffeomorphism of the base space does not uniquely determine an automorphism of the bundle. But if we require that the automorphism preserve the connection (for example, if the Yang-Mills connection is stationary) then we can classify this ambiguity as follows.
For a given connection \(\df A^I\) on \(P\), any \(X^m \in \mf{aut}(P;\df A^I)\) is uniquely determined by its projection \((\pi_*)^\mu_m X^m \in TM\) up to a vertical vector field \(Y^m \in \mf{aut}_V(P)\) such that
D(Y ·A^I) = 0
if any such non-trivial \(Y^m\) exists on \(P\).
Any \(X^m \in \mf{aut}(P;\df A^I)\) satisfies (using <ref>)
0 = _X A^I = X ·F^I + D(X·A^I)
If \(X^m \in VP\) i.e. \((\pi_*)^\mu_m X^m = 0\) this means \(X\cdot \df A^I\) is a covariantly constant \(\mf g\)-valued function (since \(\df F^I\) is a horizontal form). So if \(X^m\) and \(X'^m\) are such that \(X^\mu = (\pi_*)^\mu_m X^m = (\pi_*)^\mu_m X'^m\), then \(Y^m = X^m - X'^m \in VP\) and \(\lambda^I = Y \cdot \df A^I \) satisfies <ref>. Since the connection is an isomorphism between \(V P\) and \(\Omega^0P(\mf g)\), any such choice of \(\lambda^I\) uniquely fixes the ambiguity \(Y^m\).
The ambiguity \(Y^m\) in the above lemma corresponds to a global symmetry of the chosen connection in the following sense. Taking Lie derivative of the connection with respect to \(Y^m\) we have
\[
\Lie_Y \df A^I = D(Y\cdot \df A^I) = 0
\]
i.e. the gauge transformation \(f_Y \in {\rm Aut}_V(P)\) generated by \(Y^m\) keeps the connection invariant and hence \(\df A^I\) and \(f_Y^* \df A^I\) correspond to the same physical field configuration at every point. Connections with such a non-trivial automorphism \(f_Y\) are called reducible while connections for which no such non-trivial automorphism exists are called irreducible (see 4.2.2 <cit.>). For a compact structure group, the space of irreducible connections is known to be an open and dense subspace of the space of all connections <cit.>.
Consider the case where we have both a connection \(\df A^I\) and some charged field \(\sigma^A\) (we consider a scalar field for simplicity) which transforms under a representation \(r\) of the Lie algebra \(\mf g\) on the bundle \(P\). If \(X^m\) is an infinitesimal automorphism that preserves both \(\df A^I\) and \(\sigma^A\) then, in addition to <ref>, we have
0 = _X σ^A = X ·Dσ^A - (X ·A^I) r_I^A_B σ^B
This imposes a further restriction on the ambiguity \(Y^m \in \mf{aut}_V(P) \) in the choice of \(X^m\) given by <ref> i.e. \(Y^m\) has to satisfy the additional condition
0 = (Y ·A^I) r_I^A_B σ^B
i.e. the gauge transformation generated by \(Y^m\) keeps both the connection \(\df A^I\) and the field \(\sigma^A\) invariant at every point. The question of whether any non-trivial \(Y^m\) exists depends on the chosen connection \(\df A^I\) and field \(\sigma^A\). As we will see in <ref>, for the Lorentz connection \(\df\omega^a{}_b\) and the coframes \(\df e^a\), there is no non-trivial ambiguity.
To derive a first law for a coframe formulation of gravity we write the orthonormal coframes and the Lorentz gauge field on a principal bundle over spacetime \(M\). Since our treatment is a bit non-standard, we review the relevant constructions in this section.
Consider the linear frame bundle \(FM\) which is a \(GL(d, \bb R)\)-principal bundle over \(M\); the details can be found in I.5 <cit.>, III.B.2 and Vbis.A.5. <cit.>. The fibre of \(FM\) over any point \(x \in M\) is the set of all possible choices of linearly-independent frames \(E^\mu_a\) at \(x\). When writing Lagrangians that depend explicitly on the frames \(E^\mu_a\) it would be inconvenient to have an explicit dependence on the point in \(FM\). To avoid this, we consider the frames as vector fields (instead of points) on the frame bundle as follows. Locally, at any point \(u = (x, E^\mu_a) \in FM\) and for any \(X^a \in \bb R^d\) define a \({\bb R^d}^*\)-valued vector field \(E^m_a \in T_uFM({\bb R^d}^*)\) by
(π_*)^μ_m (X^a E^m_a) = X^a E^μ_a
This construction can be extended globally to define frames \(E^m_a\) as smooth vector fields on \(FM\). Note we consider two frames \(E^m_a\) and \(E'^m_a\) as equivalent (defined as vector fields on \(FM\)) iff \(E'^m_a - E^m_a \in VP({\mathbb R^d}^*)\) since they project to the same frame \(E^\mu_a\) on \(M\). The frames \(E^m_a\) are non-degenerate on horizontal forms in the sense
E_a ·σ= 0 σ= 0 ∀ σ∈Ω^k_horP
We define the coframes \(\df e^a\) on \(FM\) in the standard way as the canonical form or soldering form (see Prop. 2.1 3.2 <cit.>). The coframes are non-degenerate in the sense
X ·e^a = 0 X^m ∈VP
Choosing a preferred Lorentzian metric \(\eta_{ab} = {\rm diag}(-1,1,\ldots,1)\) on \(\bb R^d\) the bundle \(FM\) can be reduced to an orthonormal frame bundle \(F_OM\) with structure group \(O(d-1,1)\) i.e the Lorentz group. Any choice of a reduction of the frame bundle \(FM\) to an orthonormal frame bundle \(F_OM\) gives rise to a metric on \(M\) and conversely a choice of metric \(g_{\mu\nu}\) on \(M\) gives a reduction of the frame bundle to some subbundle of orthonormal frames (see Example 5.7 Ch. I <cit.>). The orthonormal frame bundle \(F^{(g)}_OM\) determined by \(g_{\mu\nu}\) then consists precisely of those frames that are orthonormal in the sense
g_μν E^μ_a E^ν_b = η_ab
Thus, to formulate a theory of gravity in terms of orthonormal coframes, it seems one should work on some choice of orthonormal frame bundle \(F^{(g)}_OM\), but such a choice will necessarily give us a fixed metric \(g_{\mu\nu}\) on spacetime. Since the orthonormal coframes \(e^a_\mu\) are dynamical fields of the theory, we do not have an a priori fixed metric on spacetime. Moreover, consider an automorphism \(f\) of the frame bundle \(FM\). In general, the corresponding projection \(\dfM f \in {\rm Diff}(M)\) need not preserve a given metric \(g_{\mu\nu}\) on \(M\). Thus, an arbitrary automorphism \(f \in {\rm Aut}(FM)\) will map the subbundle \(F^{(g)}_OM\) determined by \(g_{\mu\nu}\) to \(F^{(g')}_OM\) determined by \(g'_{\mu\nu} = ({\dfM f^*})^{-1} g_{\mu\nu}\). Thus, it will be problematic to pick a particular orthonormal frame bundle if one wants to consider the coframes as dynamical fields and the action of diffeomorphisms on the dynamical fields of the theory.
We circumvent this issue as follows. We will consider an abstract principal Lorentz bundle \(P_O\) with structure group \(O(d-1,1)\) which has globally well-defined coframes \(\df e^a = e^a_m\) and frames \(E^m_a\), which are non-degenerate in the sense of <ref>, similar to the ones defined on the frame bundle above. Once, we have some specific choice of coframes \(\df e^a\) obtained by solving the equations of motion of the theory, we can identify the abstract bundle \(P_O\) with the orthonormal frame bundle \(F^{(g)}_OM\) determined by the solution metric \(g_{\mu\nu} = \eta_{ab} e^a_\mu e^b_\nu\).
We can similarly construct the oriented Lorentz bundle \(P_{SO}\) (by choosing a preferred orientation \(\epsilon_{{a_1}\ldots {a_d}}\)) and the proper Lorentz bundle \(P_{SO}^0\) (by choosing a time-orientation) as abstract principal bundles with structure groups \(SO(d-1,1)\) and \(SO^0(d-1,1)\). Henceforth, we will always work on the above defined Lorentz bundle instead of the frame bundle. We will use the Lorentz bundle \(P_O\) to formulate general theories of gravity in terms of the coframes. For the Lagrangian of General Relativity (<ref>) we will need to introduce an orientation and hence we work on the oriented Lorentz bundle \(P_{SO}\). Similarly to define spinor fields we will need the proper Lorentz bundle \(P_{SO}^0\) (<ref>).
On the oriented Lorentz bundle \(P_{SO}\) we have a preferred orientation \(\epsilon_{a_1\ldots a_d}\) given by the completely anti-symmetric symbol with \( \epsilon_{01\ldots d-1} \defn 1 \). Using this orientation we can define the horizontal volume form on \(P_{SO}\) as
ε_d 1d!ϵ_a_1…a_d e^a_1 ∧…∧e^a_d ∈Ω^d_horP_SO
which is the lift through \(\pi\) of the volume form \(\dfM{\df\varepsilon}_d \equiv \varepsilon_{\mu_1 \ldots \mu_d}\) on \(M\). For a horizontal form \(\df\sigma^A \in \Omega^k_{hor}P_{SO}(\bb V, R) \) define the horizontal Hodge dual \(\star: \Omega^k_{hor}P_{SO}(\bb V, R) \to \Omega^{d-k}_{hor}P_{SO}(\bb V, R)\) as:
⋆σ^A 1/(d-k)!k!ϵ^a_1…a_k_b_1…b_d-k (E_a_k ·…·E_a_1 ·σ^A ) e^b_1∧…∧e^b_d-k
or in terms of the frame components as
(⋆σ)^A_b_1…b_d-k = 1/k!ϵ^a_1…a_k_b_1…b_d-k σ^A_a_1…a_k
Note, that the horizontal Hodge dual maps equivariant horizontal forms to equivariant horizontal forms. It is straightforward to verify that for an invariant form \(\df \sigma\), if \(\df\sigma = \pi^*\dfM{\df\sigma}\) then \(\star\df\sigma = \pi^*(*\dfM{\df\sigma})\) where \(*\) is the Hodge dual acting on differential forms on \(M\).
The \(O(d-1,1)\)-connection on the Lorentz bundle \(P_O\) is \( {\df \omega^a}_b \in \Omega^1P_O(\mf{g}) \) with \(\df \omega^{ab} = - \df \omega^{ba}\). The curvature and torsion of \({\df \omega^a}_b\) are defined by
\begin{align}
{\df R^a}_b & \defn D{\df \omega^a}_b = d{\df \omega^a}_b + {\df \omega^a}_c \wedge {\df \omega^c}_b \label{eq:F-frame-defn} \\
\df T^a & \defn D\df e^a = d\df e^a + {\df \omega^a}_b \wedge \df e^b \label{eq:torsion-defn}
\end{align}
and satisfy
DT^a = R^a_b ∧e^b
In the torsionless case \(\df T^a = 0\), there is a unique Lorentz connection, the Levi-Civita connection \(\tilde{\df \omega}^a{}_b\) which can be expressed as
ω̃^a_b = - E^[a·d e^b] + 1/2 ( E^a ·E^b ·d e^c ) e_c
Any Lorentz connection with torsion can be written in terms of the Levi-Civita connection as
ω^a_b = ω̃^a_b + C_c^a_b e^c
where contorsion \(C_{cab} = C_{c[ab]}\) is defined by
C_cab 12 (T_cab - T_abc - T_bca )
with \(T_{cab} = E_b \cdot E_a \cdot \df T_c\) being the frame components of the torsion form <ref>. One can work with coframes and the contorsion (or the torsion) as independent dynamical fields instead of the coframes and the connection \(\df\omega^a{}_b\). However, the computations are much simpler in the latter case. Thus, in the main body of the paper we will always consider the Lorentz connection \(\df\omega^a{}_b\) as independent of the coframes \(\df e^a\) i.e. we work in the first-order formalism, but we provide eq:LC-conneq:contorsion for readers interested in the second-order formalism for gravity.
Next we consider the possible automorphisms of the Lorentz bundle \(P_O\) that preserve the orthonormal coframes. From <ref> we know that an infinitesimal automorphism preserving the connection is determined only up to a covariantly constant function but we can show that if \(X^m\) is an infinitesimal automorphism that preserves the coframes, it is completely determined by a Killing vector field on \(M\) as follows (see <cit.> for a spacetime version of this result).
Given an automorphism which preserves some chosen orthonormal coframes \(X^m \in \mf{aut}(P_O; \df e^a)\) so that \(\Lie_X \df e^a = 0\), the projection \(X^\mu = (\pi_*)^\mu_m X^m\) is a Killing vector field for the metric \(g_{\mu\nu} = \eta_{ab}e^a_\mu e^b_\nu\) on \(M\) determined by the chosen coframes.
Further, given any connection \({\df \omega^a}_b\) on \(P_O\), \(X^m\) is uniquely determined by \(X^\mu\) as follows. Denote the Killing form corresponding to the Killing vector field \(X^\mu \) as \(\dfM{\df \xi} \equiv g_{\mu\nu}X^\nu\) and its lift to \(P_O\) as \(\df\xi \defn \pi^*\dfM{\df\xi} = (X \cdot \df e^a)\df e_a \) then the vertical part of \(X^m\) with respect to the chosen connection \({\df \omega^a}_b\) is given by
X ·ω_ab = 12E_a ·E_b ·dξ+ (X ·e^c) C_cab
where the contorsion \(C_{cab}\) is defined by <ref>.
For some chosen coframes \(\df e^a\) let, \(X^m \in \mf{aut}(P_O;\df e^a)\) be an infinitesimal automorphism preserving \(\df e^a\) i.e. \(\Lie_X \df e^a = 0\). It immediately follows that the projection \(\dfM X \equiv X^\mu = (\pi_*)^\mu_m X^m\) satisfies
_X (η_abe^a_μe^b_ν) = _X (g_μν) = 0
Thus, \(X^m\) projects to a Killing vector for \(g_{\mu\nu}\).
Using <ref> we have
0 = _X e^a = X ·T^a + D(X ·e^a ) - (X ·ω^a_b ) e^b
and taking the interior product of <ref> with \(E^m_c\) we immediately get
X ·ω^ab = E^[b ·D(X ·e^a]) + E^[b ·X ·T^a]
For the Killing form \(\df\xi = (X\cdot \df e^a)\df e_a\) we have
dξ= Dξ= D(X·e^a)∧e_a + (X ·e^a) T_a
E^a ·E^b ·dξ= 2 E^[b ·D(X ·e^a]) + (X ·e^c) E^a ·E^b ·T_c
Substituting this into <ref> we can write
X ·ω^ab = 12E^a ·E^b ·dξ- 12 (X ·e^c) E^a ·E^b ·T_c - E^[a ·X ·T^b]
Using the frame components of the torsion \(2\)-form and <ref>, we get <ref>. The right-hand-side depends only on \(X^\mu\) (and its first derivative) and the torsion of the chosen connection. So we see that any \(X^m \in \mf{aut}(P_O;\df e^a)\) is uniquely determined by its projection.
Using <ref> we show that for an automorphism preserving some orthonormal coframes, the Lie derivative on the bundle coincides with the Lorentz-Lie derivative of <cit.>. We consider a scalar field \(\sigma^a\) that transforms under the local Lorentz transformations for simplicity. The Lie derivative on the bundle with respect to \(X^m \in \mf{aut}(P_O; \df e^a)\) is then (using <ref>)
_X σ^a = X ·Dσ^a - (12E_a ·E_b ·dξ+ (X ·e^c) C_cab) σ^b
Picking a local section and denoting the projection \(\dfM{X} \equiv X^\mu = (\pi_*)^\mu_m X^m\) we get (in the torsionless case)
_X σ^a = X^μD_μσ^a + ( E^a μ E^b ν ∇_[μX_ν] ) σ_b
= _X σ^a + ( X^μω_μ^ab + E^μ[a e^b]_ν∇_μX^ν) σ_b
The first line is (up to differences in sign and factor conventions) the Lie derivative on Lorentz tensors defined in <cit.>. In the second line, \(\Lie_{\dfM X}\) is the Lie derivative computed by ignoring the internal index \(a\) and the second term coincides with \(\lambda^a{}_b\) (<ref>) used by <cit.>.
Next we collect our notational conventions for spinor fields (see <cit.>, Problem 4 of Vbis. <cit.>, and Ch.1 <cit.> for details of the construction of spinor fields). We note that these references use the “mostly minus" signature for the Lorentzian metric but to conform to the earlier sections we stick to the “mostly plus" signature making appropriate changes in signs according to Remark 3.8 <cit.>. We will stick to the case of a \(3+1\)-dimensional spacetime (for general dimensions and signature see <cit.>).
Consider the Clifford algebra generated by an identity element \(\id\) and the Dirac matrices \( \gamma^a \) which satisfy (see <cit.>):
{ γ^a, γ^b } γ^a γ^b + γ^b γ^a = -2η^ab𝕀The group \(Spin^0(3,1)\) is embedded in the Clifford algebra according to Definition 2.4 of <cit.>. The Dirac matrices also implement the double cover homomorphism \(Spin^0(3,1) \to SO^0(3,1)\) as detailed in Prop. 2.6 of <cit.>. The complex representation of the Clifford algebra as a matrix group on \(\bb C^4\) (see Theorem 2.2 of <cit.>), induces a representation of \(Spin^0(3,1)\) on \(\bb D \cong \bb C^4\) which is the vector space of Dirac spinors. Similarly, we denote the dual vector space of Dirac cospinors by \( \bb D^* \cong \bb C^4 \). To avoid a proliferation of indices, we will use the standard “matrix-type notation" for spinors.
We denote the Dirac adjoint map or Dirac conjugation by \(\adj{\phantom{x}}\), so that in our conventions for \(u \in \mathbb D^*\) and \(v \in \mathbb D\)
(uv) = v u (γ^a) = - γ^a (vv) = vv ∈ℝ
To consider spinor fields on spacetime we need the notion of a spin structure on \(M\) that is a \(Spin^0(3,1)\)-principal fibre bundle \(F_{Spin}M\) of spin frames together with a \(2\)-to-\(1\) bundle homomorphism to the proper orthonormal frame bundle \(F^0_{SO}M\) which is equivariant with respect to the double cover map on the respective structure groups. To formulate Dirac fields on a spacetime \(M\) with a fixed metric \(g_{\mu\nu}\) and orientation, we can choose some spin structure \(F_{Spin}M\) corresponding to the proper frame bundle \(F_{SO}^0M\) determined by the given metric and orientation on \(M\). However, as discussed before for the Lorentz bundle, this is problematic when considering theories where the metric (or the coframes) themselves are dynamical fields. As before we circumvent this, by considering a choice of spin bundle \(P_{Spin}\) corresponding to a proper Lorentz bundle \(P_{SO}^0\) in a manner similar to the spin structure \(F_{Spin}M\) corresponding to \(F_{SO}^0M\). Once we have solved the equations of motion to get a metric, we can identify \(P_{Spin}\) with a spin structure \(F_{Spin}M\) given by that metric.
On the spin bundle \(P_{Spin}\), we can lift the Lorentz connection \( {\df \omega^a}_b \) on \(P^0_{SO}\) to a connection on \(P_{Spin}\) as \( \df \omega_{spin} \defn -\frac{1}{8}\df \omega_{ab}[\gamma^a, \gamma^b] \). We can similarly lift other structures such as the coframes and frames, the horizontal volume form <ref>, and the horizontal Hodge dual <ref>.
A Dirac spinor field \(\Psi \in \Omega^0 P_{Spin}(\bb D)\) is a function on the spin bundle \(P_{Spin}\) valued in the Dirac spinor representation \(\bb D\) and similarly, a Dirac cospinor field is \(\Phi \in \Omega^0F_{Spin}M(\bb D^*)\). The spin covariant exterior derivative for \( \Psi \in \Omega^0 P_{Spin}(\mathbb D) \) and \( \Phi \in \Omega^0 P_{Spin}(\mathbb D^*) \) is given by:
DΨ= dΨ- 1/8(ω_ab[γ^a, γ^b]) ΨDΦ= dΦ+ 1/8Φ (ω_ab[γ^a, γ^b])
and the Dirac operator \(\dirac \) on Dirac spinor fields and cospinor fields is given by
Ψγ^a E_a ·DΨΦ(E_a ·DΦ)γ^a
Note that in our conventions \(\adj{\dirac\Psi} = - \dirac\adj\Psi\).
Any infinitesimal automorphism \(X^m \in \mf{aut}(P_{Spin})\) then acts on the Dirac spinor field through the Lie derivative
_XΨX ·dΨ= X ·DΨ+ 18 (X ·ω_ab)[γ^a, γ^b]Ψ
Consider now an \(X^m\) preserving some orthonormal frames. By <ref>, such an \(X^{m}\) always projects to a Killing field \(X^\mu\) of the metric on \(M\) determined by the chosen coframes, and is uniquely determined using <ref>. For such vector fields the Lie derivative <ref> on the bundle of the Dirac spinor field reads (in the torsionless case)
_X Ψ= X ·DΨ+ 18 ( 12E^a ·E^b ·dξ)[γ_a, γ_b]ΨViewed on spacetime with \(\dfM{X} \equiv X^\mu \) being a Killing field of the given metric, this becomes
_X Ψ= X^μD_μΨ- 18 ( ∇_[μX_ν] )[γ^μ, γ^ν]Ψwhich is <ref>, the Lie derivative of spinors with respect to the Killing field \(X^\mu\) as defined by Lichnerowicz <cit.> (see also <cit.>).
§.§ Local and covariant functionals on a principal bundle
In constructing physical theories on the bundle we will require that the Lagrangian be a locally and covariantly constructed functional of the dynamical fields on the bundle which we define as follows.
A functional \(\mc F[\Phi] \) on a \(G\)-principal bundle \(P\) depending on a set of fields \(\Phi\) and finitely many of its derivatives (with respect to an arbitrary derivative operator which is taken to be part of \(\Phi\)) is a local and covariant functional if for any \(f \in {\rm Aut}(P)\) we have
(f^* F)[Φ] = F[f^*Φ]
where it is implicit that on the right-hand-side \(f\) also acts on the derivatives of \(\Phi\). If \(X^m\) is the vector field generating the automorphism \(f\) then the above equation implies that
_X F[Φ] = F[_X Φ]
Each of \(\mc F\) and \(\Phi\) can have arbitrary tensorial structure on \(P\) and be valued in some representation of the structure group.
For many of the crucial results in the main body we will need to ensure that a closed differential form on \(P\) is in fact, (globally) exact. For instance, such a result is used in the classification of the ambiguities in the symplectic potential (<ref>), and is needed to ensure that a horizontal (i.e. gauge-invariant) Noether charge exists for any Noether current (<ref>). Under certain assumptions on a differential form \(\df\sigma\) and its dependence on certain fields \(\Phi\), which we detail next, we show that if \(\df \sigma\) is closed then it is exact. We shall show this in direct analogy to Lemma 1 <cit.> (this result can also be derived using jet bundle methods and the variational bicomplex; see Theorem 3.1 <cit.>) and the assumptions on \(\df\sigma\) given below are geared towards generalising the algorithm of Lemma 1 <cit.> to work with differential forms on the bundle \(P\).
[Assumptions for <ref>]
Let \(\Phi = \{ \phi, \psi \}\) be a collection of two types of fields — \(\phi\) are the “dynamical fields" and \(\psi\) are the “background fields" (distinguished by the assumptions listed below), and let \(\df\sigma[\Phi] \in \Omega^p P\) with \(p<d\) (where \(d\) is the dimension of the base space \(M\)) be a \(p\)-form on a principal bundle \(P\) so that
* \(\df\sigma\) is a horizontal form on \(P\) which is invariant under the action of the structure group \(G\) on \(P\) i.e. \(\df\sigma[\Phi] \in \Omega^p_{hor} P\).
* \(\df\sigma[\Phi]\) is a local and covariant functional of the fields \(\Phi = \{\phi,\psi\}\) as in <ref>.
* The “dynamical fields" \(\phi\) are sections of a vector bundle over \(P\) which is equivariant under the group action \(G\) on \(P\), and the action of any automorphism \(f \in {\rm Aut}(P)\) on \(\phi\) is linear (i.e. preserves the vector bundle structure).
* \(\df\sigma[\Phi]\) depends linearly on up to \(k\)-derivatives of the “dynamical fields" \(\phi\).
With the above assumptions on \(\df\sigma\) we can prove the following (generalising Lemma 1 <cit.>)
Let \(\df\sigma[\Phi] \in \Omega^p_{hor}P\) and \(\Phi = \{\phi,\psi\}\) be as assumed in <ref>. If \(d\df\sigma[\Phi] = 0\) for all “dynamical fields" \(\phi\) and any given “background fields" \(\psi\), then there exists (globally) a differential form \(\df\eta[\Phi] \in \Omega^{p-1}_{hor}P\) which, similarly satisfies <ref> but depends linearly on at most \((k-1)\)-derivatives of \(\phi\) such that \(\df\sigma = d\df\eta\).
Further, if \(k=0\) i.e. \(\df\sigma\) does not depend on derivatives of the “dynamical fields" \(\phi\), then \(\df\sigma=0\).
To begin, we note that connections on a principal bundle \(P\) are not sections of a vector bundle, (see Remark 1, Ch. IV <cit.>). Thus, any choice of connection on \(P\) would be part of the “background fields" \(\psi\).[Note, that even though <cit.> assumes in the beginning that the “background fields" \(\psi\) also are sections of a vector bundle, it is not required for the proof of Lemma 1 <cit.> as discussed towards the end of II <cit.>.] We assume that a choice of such a connection has been made and denote the corresponding (horizontal) covariant derivative operator on \(P\) by \(D_m\) and the covariant exterior derivative by \(D\).
By <ref>, the derivative operator \(D_m\) can be used to take horizontal derivatives of \(\phi\); the derivatives of \(\phi\) along the vertical directions are fixed by the equivariance requirement. Furthermore, any antisymmetric derivatives of \(\phi\) can be written in terms of lower order derivatives and the curvature (and possibly torsion on a Lorentz bundle) of the chosen connection; thus, we only need to consider totally symmetrised derivatives of \(\phi\). Thus, using <ref> and <ref> we can write the \(p\)-form \(\df\sigma\) as
σ≡σ_m_1…m_p = ∑_i=0^k S^(i)_m_1…m_p^n_1…n_i_A D_(n_1 ⋯D_n_i) ϕ^A
where, each of the tensors \(S^{(i)}\) are local and covariant functionals of the “background fields" \(\psi\) and we have used an abstract index notation on the “dynamical fields" \(\phi \equiv \phi^A\). <ref> is the direct analogue of Eq. 2 <cit.> on the principal bundle \(P\). Note here that, since all of the \(m\)-indices are horizontal and the \(n,A\)-indices are contracted away, \(\df\sigma\) tranforms as a horizontal form under the action of any automorphism \(f \in {\rm Aut}(P)\) as required in <ref>.
Again using <ref> we have
dσ= Dσ≡(p+1)∑_i=0^k D_[l { S^(i)_m_1…m_p]^n_1…n_i_A D_(n_1 ⋯D_n_i) ϕ^A }
Since, \(d\df\sigma = 0\) for all “dynamical fields" \(\phi\) and the horizontal symmetrised derivatives of \(\phi\) can be specified independently at any point of \(P\), we can directly apply the arguments of Lemma 1 <cit.> to <ref>. Note, that in each step of the algorithm of Lemma 1 <cit.>, the \(m\)-indices are always horizontal and the \(n,A\)-indices are contracted away. Thus the algorithm of <cit.> gives us a horizontal \((p-1)\)-form \(\df\eta\) on \(P\) so that \(\df\sigma = d\df\eta\) where \(\df\eta\) has an expansion similar to <ref> except that it depends linearly on at most, \((k-1)\)-derivatives of \(\phi\). Thus, the form \(\df\eta\) manifestly satisfies <ref> and the claim of this lemma.
The algorithm of <cit.> further shows that when \(k=0\) we have \(\df\sigma = 0\).
We point out that all of <ref> are crucial to prove <ref>. <ref> was used in the first equality in <ref> to convert the exterior derivative \(d\) to the covariant exterior derivative \(D\) (which does not hold if \(\df\sigma\) is, either not invariant under the \(G\)-action, or not horizontal) to get an expansion in terms of derivatives of \(\phi\) which is the first step in using the algorithm of <cit.>. As already noted, <ref> are needed to write down the expansion <ref>. Finally, <ref> is already used to formulate <ref>, since a vector bundle structure is necessary for the notion of \(\df\sigma\) depending linearly on \(\phi\) and its derivatives as in <ref>. Similarly, the assumptions of equivariance of \(\phi\) and linear action of \(f \in {\rm Aut}(P)\) on \(\phi\) in <ref>, are necessary to ensure that the expansion <ref> — and the corresponding expansion for \(\df\eta\) obtained by applying the algorithm of <cit.> — give us horizontal forms on \(P\). <ref> also ensures that we can use any connection, as part of the background fields \(\psi\), to define the horizontal covariant derivatives of \(\phi\). This is crucial since only the horizontal derivatives of \(\phi\) can be freely specified at any point of \(P\) and one needs a connection to define horizontal derivatives.
In our applications of <ref> in the main body of the paper, the “dynamical fields" \(\phi\) will either be
* the perturbations \(\delta\df\psi^\alpha\) of the dynamical fields \(\df\psi^\alpha\) (<ref>) of the theory which are equivariant horizontal forms valued in some vector space carrying a representation of \(G\), or
* infinitesimal automorphisms \(X^m \in \mf{aut}(P)\) of the principal bundle \(P\).
In both cases the “dynamical fields" \(\phi\) can be considered as sections of vector bundles over \(P\) in accordance with <ref>.
|
1511.00546
|
We consider a Degree-Corrected Planted-Partition model: a random graph on $n$ nodes with two equal-sized clusters. The model parameters are two constants $a,b > 0$ and an i.i.d. sequence of weights $(\phi_u)_{u=1}^n$, with finite second moment $\PHItwo$. Vertices $u$ and $v$ are joined by an edge with probability $\frac{\phi_u \phi_v}{n}a$ when they are in the same class and with probability $\frac{\phi_u \phi_v}{n}b$ otherwise.
We prove that it is information-theoretically impossible to estimate the spins in a way positively correlated with the true community structure when $(a-b)^2 \PHItwo \leq 2(a+b)$.
A by-product of our proof is a precise coupling-result for local-neighbourhoods in Degree-Corrected Planted-Partition models, which could be of independent interest.
§ INTRODUCTION
It is well-known that many networks exhibit a community structure. Think about groups of friends, web pages discussing related topics, or people speaking the same language (for instance, the Belgium population could be roughly divided into people speaking either Flemish or French). Finding those communities helps us understand and exploit general networks.
Instead of looking directly at real networks, we experiment first with models for networks with communities. One of the most elementary models is the Planted-Partition Model (PPM) <cit.>: a random graph on $n$ vertices partitioned into two equal-sized clusters such that vertices within the same cluster are connected with probability $p$ and between the two communities with probability $q$. Note that the PPM is a special case of the Stochastic Block Model (SBM). The question is now: given an instance of the PPM, can we retrieve the community-membership of its vertices?
Most real networks are sparse and a thorough analysis of the sparse regime in the PPM - i.e., $p = \frac{a}{n}$ and $q = \frac{b}{n}$ for some constants $a,b>0$ - will therefore lead to a better understanding of networks.
When the difference between $a$ and $b$ is small, the graph might not even contain enough information to distinguish between the two clusters. In <cit.> it was first conjectured that a detectability phase-transition exists in the PPM: detection would be possible if and only if $\lr{\frac{a-b}{2}}^2 > \frac{a+b}{2}$. The negative-side of this conjecture has been confirmed in <cit.>.
The positive side has been recently confirmed in <cit.> and <cit.> using sophisticated (but still running in polynomial time) algorithms designed for this particular problem.
In this paper we study an extension of the PPM: a Degree-Corrected Planted-Partition Model (DC-PPM), a special case of the Degree-Corrected Stochastic Block Model (DC-SBM) in <cit.>.
Because, although the PPM is a useful model due to its analytical tractability, it fails to accurately describe networks with a wide variety in their degree-sequences (because nodes in the same cluster are stochastically indistinguishable). Indeed, real degree distributions follow often, but not always, a power-law <cit.>. Compare this to fitting a straight line on intrinsically curved data, which is doomed to miss important information.
The DC-PPM is defined as follows: It is a random graph on $n$ vertices partitioned into two asymptotically equal-sized clusters by giving each vertex $v$ a spin $\sigma(v)$ drawn uniformly from $\spm$. The vertices have i.i.d. weights $\{\phi_u\}_{u=1}^n$ governed by some law $\nu$ with support in $W \subset [ \PHImin , \PHImax ],$ where $0 < \PHImin \leq \PHImax < \infty$ are constants. We denote the second moment of the weights by $\PHItwo$. An edge is drawn between nodes $u$ and $v$ with probability $\frac{\phi_u \phi_v}{n} a$ when $u$ and $v$ have the same spin and with probability $\frac{\phi_u \phi_v}{n} b$ otherwise. The model parameters $a$ and $b$ are constant.
In the underlying paper we extend results in <cit.> to the degree-corrected setting. More specifically, we prove that when $(a-b)^2 \PHItwo \leq 2(a+b)$, it is information-theoretically impossible to estimate the spins in a way positively correlated with the true community structure.
In a follow-up paper <cit.>, we show that above the threshold (i.e., $(a-b)^2 \PHItwo > 2(a+b)$), reconstruction is possible based on the second eigenvector of the so-called non-backtracking matrix. This is an extension of the results in <cit.> for the ordinary Stochastic Block Model.
We remark that there is an interpretation of the threshold in terms of eigenvalues of the conditional expectation of $A$. Indeed, if $A$ denotes the adjacency matrix and $f_1$ and $f_2$ are the vectors defined for $u \in V$ by $\psi_1(u) = \frac{1}{\sqrt{2}} \phi_u$ and $\psi_2(u) = \frac{1}{\sqrt{2}} \sigma_u \phi_u$ , then
\E{A|\phi_1, \ldots, \phi_n} = \frac{a+b}{n}\psi_1 \psi_1^* + \frac{a-b}{n}\psi_2 \psi_2^* - a \frac{1}{n} \text{diag}\{\phi_u^2 \}.
So that, in probability,
\E{A|\phi_1, \ldots, \phi_n} \psi_1 = \lr{ \frac{a+b}{2} \frac{1}{n} \sum_{u=1}^n \phi_u^2 } \psi_1
+ \lr{ \frac{a-b}{2} \frac{1}{n} \sum_{u=1}^n \sigma_u \phi_u^2 } \psi_2 + \bigO(1) \to \frac{a+b}{2} \PHItwo \psi_1,
\E{A|\phi_1, \ldots, \phi_n} \psi_2 = \lr{ \frac{a-b}{2} \frac{1}{n} \sum_{u=1}^n \phi_u^2 } \psi_2
+ \lr{ \frac{a+b}{2} \frac{1}{n} \sum_{u=1}^n \sigma_u \phi_u^2 } \psi_1 + \bigO(1) \to \frac{a-b}{2} \PHItwo \psi_2,
by the law of large numbers. Now, the condition $(a-b)^2 \PHItwo \leq 2(a+b)$ is equivalent to $\lr{\frac{a-b}{2} \PHItwo }^2 \leq \frac{a+b}{2} \PHItwo.$
§.§ Our results
It is a well-known fact that in the sparse graph, $\Theta(n)$ vertices are isolated whose type thus cannot be recovered by any algorithm. Therefore, the best we can ask for is that our reconstruction is positively correlated with the true partition:
Let $G$ be an observation of the degree-corrected planted partition-model, with true communities $\{\sigma_u\}_{u=1}^n$. Further, let $\{\widehat{\sigma}_u\}_{u=1}^n$ be a reconstruction of the communities, based on the observation $G$, such that $|\{ u: \widehat{\sigma}_u = + \}| = \frac{n}{2}$.
Then, we say that $\{\widehat{\sigma}_u\}_{u=1}^n$ is positively correlated with the true partition $\{\sigma_u\}_{u=1}^n$ if there exists $\delta > 0$ such that
\[\frac{1}{n} \s{u=1}{n} \indicator{ \sigma_u = \widehat{\sigma}_u} \geq \frac{1}{2} + \delta, \]
with high probability.
Our main result is as follows:
Let $G$ be an observation of the degree-corrected planted partition-model with $(a-b)^2 \Phi^2 \leq 2(a+b)$. Then, no reconstruction $\{\widehat{\sigma}_u\}_{u=1}^n$ based on $G$ is positively correlated with $\{\sigma_u\}_{u=1}^n$.
As mentioned above, this theorem is an extension of the result in <cit.>.
Their strategy, that we shall follow, invokes a connection with the tree reconstruction problem (see for instance <cit.>):
deducing the sign of the root based on all the spins at some distance $R \to \infty$ from the root.
Indeed, we shall see that the $R$-neighbourhood of a vertex looks like the following random $+/-$ labelled tree, that we denote by $T^{\text{Poi}}$:
We begin with a single particle, the root $o$, having spin $\sigma_o \in \spm$ and weight $\phi_o \in W \subset [\PHImin, \PHImax]$ (which we often take random). The root is replaced in generation $1$ by $\Pois{\frac{a}{2} \PHI \phi_o}$ particles of spin $\sigma_o$ and $\Pois{\frac{b}{2} \PHI \phi_o}$ particles of spin $-\sigma_o$. Further, the weights of those particles are i.i.d. distributed following law $\NUstar$, the size-biased version of $\nu$, defined for $x \in [\PHImin, \PHImax]$ by
(x) = 1/ ∫_^x y d ν(y).
For generation $t \geq 1$, a particle with spin $\sigma$ and weight $\phi^*$ is replaced in the next generation by $\Pois{\frac{a}{2} \PHI \phi^*}$ particles with the same spin and $\Pois{\frac{b}{2} \PHI \phi^*}$ particles of the opposite sign. Again, the weights of the particles in generation $t+1$ follow in an i.i.d. fashion the law $\NUstar$. The offspring-size of an individual is thus a Poisson-mixture.
There are equivalents descriptions of this branching process. In the sequel, we shall use the one that fits best our purposes:
The following are equivalent descriptions:
* The branching process in definition <ref>, with weights $\phi$ and spins $\sigma$. The type of a vertex $v$ is denoted by $x_v = \sigma_v \phi_v $.
* A discrete branching process such that a particle of type $x \in S$ is replaced in the next generation by a set of $N(x)=$ Poi$(\lambda_x(S))$ i.i.d. particles having types $\{y_i\}_{i=1}^{N(x)}$, with law $\nu_{\sigma_x}$.
* A copy of the branching process in definition <ref>, with the exception that the spins (which we shall denote by $\tau$) are replaced, independently of everything else, in the following way:
First, put $\tau_{\rho} = \sigma_{\rho}$. Then, conditionally independently given $\tau_{\rho}$, we take every child $u$ of $\rho$ and set $\tau_u = \tau_{\rho}$ with probability $\frac{a}{a+b}$ and $\tau_u = -\tau_{\rho}$ otherwise. We continue this construction recursively to obtain a labelling $\tau$ for which every vertex, independently, has probability $\frac{a}{a+b}$ of having the same label as its parent.
The result is a tree with weights $\phi$ and spins $\tau$. We denote the type of a vertex $v$ by $\widehat{x}_v = \tau_v \phi_v.$
Note that $(iii)$ is an extension of the definition of a Markov process on a tree as given in Section 4 of <cit.>.
The equivalence between $(i)$ and $(ii)$ stems from theorem $B.1$ in <cit.>. Indeed, the offspring distribution of a vertex with type $x \in S$ in both branching processes is completely determined by its void probabilities. We shall calculate them. For $B \subset S$, let $N_1(B)$ be the number of points that fall in $B$ during a realization of process $(i)$. Similarly, in process $(ii)$, let $N_2(B)$ be the number of points falling in $B$. Now,
\[ \P{N_1(B) = 0} = e^{-\lambda_x(B)}. \]
\[ \ba
\P{N_2(B) = 0} &= \s{n\geq 0}{} \P{N_2(B) = 0| N_2(S) = n} \P{\P{N_2(S) = n}} \\
&= \s{n\geq 0}{} (1-\nu(B))^n \frac{\lambda^n(S)}{n!} e^{-\lambda_x(s)} \\
&= e^{-\lambda_x(B)}.
\ea \]
Since both processes have the same offspring distribution, and all particles are replaced in subsequent generations independently, the branching processes are equal.
It remains to prove $(ii) \Leftrightarrow (iii)$. The branching process $(iii)$ can be constructed as follows: A vertex of type $x$ is replaced in the next generation by
\[ N = \text{Poi}\lr{(a+b)|x| \int_{S^+} y \mathrm{d} \mu(y)} \]
particles, call them $1, \ldots, N$. Their degree-coefficients are i.i.d. distributed with law $\nu_{|x|}$, where for $y \in S^+$,
\[ \mathrm{d} \nu_{|x|}(y) = \frac{y \mathrm{d} \mu(y)}{\int_{S^+} z \mathrm{d} \mu(z)}. \]
Further, independently of everything else we give every child spin sign$(x)$ with probability $\frac{a}{a+b}$ and the opposite sign otherwise. Since the offspring-size and their weight are independent of the parental spin, one can first construct the full tree including its weights. And then, independently of those weights, give each vertex a spin, according to $(iii)$.
Write $B \subset S$ as $B = -B^- \cup B^+$, where $-B^- \subset S^-$ and $B^+ \subset S^+$. Consider a particle of type $\widehat{x} > 0$. Each child $i$ has type $\widehat{x}_i = \tau_i \phi_i$, where $\tau_i$ is its spin and $\phi_i$ its weight, distributed as above.
\[ \ba \P{\widehat{x}_i \in B} &= \P{\tau_i = -} \P{\phi_i \in B^-} + \P{\tau_i = +} \P{\phi_i \in B^+} \\
&=\frac{b \int_{B^-} y \mathrm{d} \mu(y) + a \int_{B^+} y \mathrm{d} \mu(y)}{(a+b) \int_{S^+} y \mathrm{d} \mu(y)} \\
&= \frac{ \int_{B} \kappa(x,y) \mathrm{d} \mu(y) }{\int_{S} \kappa(x,y) \mathrm{d} \mu(y) }.
\ea. \]
Similarly for $\widehat{x} < 0$. Hence, the types are distributed precisely as in $(ii)$.
§.§ General proof idea
We first note that reconstruction is senseless when $\frac{a+b}{2}\PHItwo \leq 1$, because in this regime there is no giant component[Indeed, the main result in <cit.> concerns the existence, size and uniqueness of the giant component. In particular, in the setting considered here, a giant component emerges if and only if $\frac{a+b}{2} \Phi^2 >1$. We shall henceforth assume a giant component to emerge.]. Note further that $\frac{a + b}{2}\Phi^2 \leq 1$ already implies $(a-b)^2 \Phi^2 \leq 2(a+b)$.
To prove that detection is not possible when $\frac{a+b}{2}\PHItwo > 1$ and $(a-b)^2 \PHItwo \leq 2(a+b)$, we show in Theorem <ref> that for uniformly chosen vertices $u$ and $v$,
σ_u = + | σ_v , G ℙ →1/2, as $n \to \infty$.
I.e., it is already impossible to decide the sign of two random vertices, which is an easier problem than reconstructing the group-membership of all vertices (made precise in Lemma <ref>). To establish (<ref>), we condition on the boundary spins of an $R$ neighbourhood around $u$, where $R$ tends to infinity: it should make reconstruction easier. But, as we shall see, long-range correlations in this model are weak (Lemma <ref>). Hence, we can leave out the conditioning on the spin of $v$, so that we are precisely in the setting of a tree-reconstruction problem, see Section <ref>.
In fact, we shall prove (Theorem <ref>) that reconstruction of the sign of the root in a $T^{\text{Poi}}$ tree based on the spins at depth $R$ (where $R \to \infty$), is impossible when $(a-b)^2 \Phi^2 \leq 2(a+b)$.
§.§ Outline and differences with ordinary Planted-Partition model
Due to the presence of weights, the offspring in the branching process is governed by a Poisson-mixture. Section <ref> deals with these type of branching processes. The main theorem (i.e., Theorem <ref>) deals with a reconstruction problem on sequences of trees rather than a single random tree as in Theorem $4.1$ in <cit.>.
In Section <ref> we establish a coupling between the local neighbourhood and $T^{\text{Poi}}$. This result is different from the coupling in <cit.>, because we need the weights in the graph and their counterparts in the branching process to be exactly the same.
Finally, in Section <ref> we show that long-range interactions are weak. The proof of Lemma <ref> is based on an idea in the proof of Lemma $4.7$ in <cit.>. Note however that (besides the presence of weights) the statement of our Lemma <ref> is slightly stronger than Lemma $4.7$ in <cit.>, see below for details.
§.§ Background
Without the degree correction (i.e., $\phi_1 = \ldots = \phi_n = 1$), the authors of <cit.> where the first to conjecture a phase-transition for the ordinary planted partition model based on ideas from statistical physics: Clustering positively correlated with the true spins is possible if $(a-b)^2 > 2(a+b)$ and impossible if $(a-b)^2 <2(a+b)$. They conjectured further that using the the so-called belief propagation algorithm would establish the positive part.
In <cit.> 'spectral redemption conjecture' was made: detection using the second eigenvalue of the so called non-backtracking matrix would also establish the positive part.
The work <cit.> showed that positively correlated results exist in the sparse case, however not applying all the way down to the threshold. The remainder of the positive part in was established <cit.> by using a matrix counting the number of self-avoiding paths in the graph. The work <cit.> establishes independently of <cit.> the positive part of the conjecture in <cit.>. Further, the authors of <cit.> show impossibility in <cit.>. In fact they show a bit more, namely that for $(a-b)^2 \leq 2(a+b)$ reconstructions are never positively correlated. We shall here extend their results for the DC-PPM by relying on similar techniques.
Recovering the planted partition (without degree-corrections) often coincides with finding the minimum bisection in the same graph. That is, finding a partition of the graph such that the number of edges between separated components (the bisection width) is small. This problem is NP-hard <cit.>.
Graph bisection on random graphs has been studied intensively. For instance, <cit.> studies the collection of labelled simple random graphs that have $2n$ nodes, node-degree $d$ at least $3$ and bisection width $o(n^{1 - \lfloor (d+1)/2 \rfloor )})$. For these graphs the minimum bisection is much smaller than the average bisection. The main result is a polynomial-time algorithm based on the maxflow-mincut theorem, that finds exactly the minimum bisection for almost all graphs.
Another example is given in <cit.>. There, the authors consider the uniform distribution on the set of all graphs that have a small cut containing at most a fraction $1/2 - \epsilon$ of the total number of edges for some fixed $\epsilon > 0$. Those authors show that, if in the planted partition model $p > q$ are fixed (for $n \to \infty$), then the underlying community structure coincides with the minimum bisection and it can be retrieved in polynomial time. This result is improved in <cit.>.
In <cit.> the case of non-constant $p$ and $q$ is analysed. A spectral algorithm is presented that recovers the communities with probability $1-\delta$ if $p-q > c \lr{ \sqrt{p \frac{\text{log}(n/\delta)}{n}} }$. Here $c$ is a sufficiently large constant.
Positive results of spectral clustering in the DC-SBM have been obtained by various authors.
The work <cit.> introduces a reconstruction algorithm based on the matrix that is obtained by dividing each element of the adjacency matrix by the geometric mean of its row and column degrees.
The extended planted-partition is studied in <cit.>. In that model, an edge is present between $u$ and $v$ with probability $\lr{1_{\{\sigma_u = \sigma_v\}}a + 1_{\{\sigma_u \neq \sigma_v\}}b } \cdot(\phi_u \phi_v) / ( \bar{\phi} n),$ where $\bar{\phi} = \s{u=1}{n} \phi_u$, the average weight.
The main result is a polynomial time algorithm that outputs a partitioning that differs from the planted clusters on no more that $n \text{log}(\bar{\phi} ) / \bar{\phi}^{0.98}$ nodes. This recovery succeeds only under certain conditions: the minimum weight should be a fraction of the average weight and the degree of each vertex is $o(n)$.
The article <cit.> gives an algorithm based on the adjacency matrix of a graph together with performance guarantees. The average degree should be at least of order $\text{log} n$. However, since the spectrum of the adjacency matrix is dominated by the top eigenvalues <cit.>, the algorithm does a poor job when the degree-sequence is very irregular.
The authors of the underlying paper propose in <cit.> an algorithm that recovers consistently the block-membership of all but a vanishing fraction of nodes, even when the lowest degree is of order $\text{log} n$. It outperforms algorithms based on the adjacency matrix in case of heterogeneous degree-sequences.
§ BROADCASTING ON THE BRANCHING PROCESS
Here we repeat without changes the definition of a Markov broadcasting process on trees given in <cit.>.
Let $\calT$ be an infinite tree with root $\rho$. Given a number $0 \leq \epsilon < 1$, define a random labelling $\tau \in \{ \pm \}^{\calT}$. First, draw $\tau_{\rho}$ uniformly in $\{ \pm \}$. Then, conditionally independently given $\tau_{\rho}$, take every child $u$ of $\rho$ and set $\tau_u = \tau_{\rho}$ with probability $1 - \epsilon$ and $\tau_u = -\tau_{\rho}$ otherwise. Continue this construction recursively to obtain a labelling $\tau$ for which every vertex, independently, has probability $1-\epsilon$ of having the same label as its parent.
Suppose that the labels $\tau_{\partial \calT_m}$ at depth $m$ in the tree are known (here, $\tau_W = \{ \tau_i : i \in W \}$). The paper <cit.> gives precise conditions as to when reconstruction of the root label is feasible using the optimal reconstruction strategy (maximum likelihood), i.e., deciding according to the sign of $\E{\tau_{\rho} | \tau_{\partial \calT_m}}$.
Interestingly, this is completely decided by the branching number of $\mathcal{T}$.
The branching number of a tree $\mathcal{T}$, denoted by $\emph{Br}(\calT)$, is defined as follows:
* If $\calT$ is finite, then $\emph{Br}(\calT) = 0$;
* If $\calT$ is infinite, then
$ \emph{Br}(\calT) = \sup \{ \lambda \geq 1: \inf_{\Pi} \sum_{v \in \Pi} \lambda^{-|v|} > 0 \},$
where the infimum is taken over all cutsets $\Pi$.
Theorem $1.1$ in <cit.> reads tailored to our needs:
(Theorem $1.1$ in <cit.>)
Consider the problem of reconstructing $\tau_{\rho}$ from the spins $\tau_{\partial \calT_m}$ at the $m$th level of $\calT$. Define $\Delta_m$ as the difference between the probability of correct and incorrect reconstruction given the information at level $m$:
\[ \Delta_m:= \left| \P{\tau_{\rho} = +| \tau_{\partial \calT_m}} - \P{\tau_{\rho} = -| \tau_{\partial \calT_m}} \right|. \]
If $\emph{Br}(\calT)(1 - 2 \epsilon)^{2} > 1$ then $\lim_{m \to \infty} \E{\Delta_m} > 0$.
If, however, $\emph{Br}(\calT)(1 - 2 \epsilon)^{2} < 1$ then $\lim_{m \to \infty} \E{\Delta_m} = 0$.
Note that in this theorem the tree is fixed, compared to the setting in this paper where the multi-type branching process $T^{\text{Poi}}$ defined in Section <ref> is considered. But, it can be easily seen that the spins on a fixed instance $\calT$ of $T^{\text{Poi}}$ are distributed according to the above broadcasting process with error probability $\epsilon = \frac{b}{a+b}$ [Indeed, instances of the tree when ignoring spins are generated according to a Galton-Watson process where the number of offspring of a particle is an independent copy of Poi $\lr{\frac{a+b}{2} \PHI \phi^*}$, with $\phi^*$ governed by $\NUstar$. We obtain the spins by giving particles, independently, the same spin as its parent with probability $\frac{a}{a+b}$ and the opposite sign with probability $\frac{b}{a+b}$.
Thus, a particle gives birth to $\s{u=1}{\text{Poi}\lr{\frac{a+b}{2} \PHI \phi^*}} \IND{\frac{a}{a+b}} \overset{\text{d}}= \text{Poi}\lr{\frac{a}{2} \PHI \phi^*}$ particles of the same sign, and $\s{u=1}{\text{Poi}\lr{\frac{a+b}{2} \PHI \phi^*}} \lr{1 - \IND{\frac{a}{a+b}}} \overset{\text{d}}= \text{Poi}\lr{\frac{b}{2} \PHI \phi^*}$ particles of the opposite sign. Those numbers are seen to be independent. ].
We thus need to calculate the branching number of a typical instance $\calT$:
Assume that $\frac{a+b}{2} \PHItwo > 1$. Consider the multi-type branching process $T^{\emph{Poi}}$, where the root has spin drawn uniformly from $\{+,-\}$ and weight governed by $\nu$.
Then, given the event that the branching process does not go extinct, $\emph{Br}\lr{T^{\emph{Poi}}} \leq \frac{a+b}{2} \PHItwo$ almost surely.
Denote the multi-type branching process by $T$.
Assume w.l.o.g. that the root has $D \geq 1$ children denoted as $1, \ldots, D$. Denote by $T^{*}_u$ the subtree of all particles with common ancestor $u$.
We observe that $\text {Br}\lr{T} < c$ if and only if $\text{Br}\lr{T^{*}_u} < c$ for all $u$.
Now, conditional on the spin of the root, $\lr{T^{*}_u}_{u=1}^D$ are i.i.d. copies of $T^{\text {Poi}}$ with weight governed by the biased law $\NUstar$. The latter is a Galton-Watson process with offspring mean $\frac{a+b}{2} \PHItwo > 1$. Hence Proposition $6.4$ in <cit.> entails that $\text{Br}\lr{T^{*}_u} = \frac{a+b}{2} \PHItwo$ a.s.
Note that it can in fact be easily proved that $\text{Br}\lr{T^{\text{Poi}}} = \frac{a+b}{2} \PHItwo$ almost surely.
We conclude with the main theorem of this section. Note that we assume that $\frac{a+b}{2} \PHItwo > 1$, so that the branching process does not die out with non-zero probability. Remark further that the theorem is a bit more precise than Theorem $4.1$ in <cit.> (which deals with unweighed Poisson trees), in the sense that we need to re-sample the tree for each $n$. Indeed, in the coupling result Theorem <ref> below we re-sample for each $n$.
Assume that $\frac{a+b}{2} \PHItwo > 1$. Let $\{ T^n \}_{n=1}^{\infty}$ be a collection of i.i.d. copies of $T^{\text{Poi}}$. Denote for each tree $T^n$ its spins by $\tau^n$. Further, let $R$ be an unbounded non-decreasing function. Assume that $(a-b)^2 \Phi^2 < 2(a+b)$, then
\[ \P{\tau^n_{\rho} = + |T^n_{R(n)}, \tau^n_{\partial T^n_{R(n)}} } \overset{\mathbb{P}} \to \frac{1}{2}, \]
as $n \to \infty$.
We begin by describing the above broadcasting process on random trees more precisely. By the triple $(\Omega,\Sigma,\mathbb{P})$ we denote the underlying probability-space of the following stochastic process: Let $T$ be the branching process $T^{\text{Poi}}$ with root $\rho$ where we ignore all types on it (we denote the collection of its realizations by $\Omega'$, and we let $\Sigma'$ be a sigma-algebra on it). We define a new random labelling $\tau \in \{ \pm \}^{T}$ on every instance of $T$ by running the Markov broadcast process. I.e., $\tau_{\rho}$ is uniformly drawn from $\{ \pm \}$ and each child has the same spin as its parent with probability $\frac{a}{a+b}$.
Let, for each $n \in \mathbb{N}$, $(T^n, \tau^n)$ be an independent copy of $(T,\tau)$. Formally, the random variable $T^n$ is thus a mapping from $\Omega$ to $\Omega'$: it therefore makes sense to define the pull-back measure $\mathbb{P}_T: \Sigma' \to [0,1]$ for $B \in \Sigma'$ by $\mathbb{P}_T(B) = \P{(T^1)^{\leftarrow}(B)}$.
With this notation,
\[ \ba \E{ \left| \P{\tau^n_{\rho} = + |T^n_{R(n)}, \tau^n_{ \partial T^n_{R(n)} } } - \frac{1}{2} \right| } &=
\int_{\Omega'} \E{ \left| \left. \P{\tau^n_{\rho} = + |T^n_{R(n)}, \tau^n_{ \partial T^n_{R(n)} }} - \frac{1}{2} \right| \right| T^n = \cT } \mathrm{d} \mathbb{P}_T (\cT) \\
\int_{\Omega'} \E{ \left| \left. \P{\tau_{\rho} = + |T_{R(n)}, \tau_{ \partial T_{R(n)} }} - \frac{1}{2} \right| \right| T = \cT } \mathrm{d} \mathbb{P}_T (\cT) \\
&= \int_{\Omega'} \E{ \left| \P{\tau_{\rho} = + | T = \cT, \tau_{\partial \cT_{R(n)}} } - \frac{1}{2} \right| } \mathrm{d} \mathbb{P}_T (\cT). \\
\ea \]
Since Br$(T) \leq \frac{a+b}{2} \Phi^2$ almost surely, $\text{Br}(T)(1 - 2 \epsilon) < 1$ almost surely. Consequently,
\[f_n(\cT):= \E{ \left| \P{\tau_{\rho} = + |T = \cT, \tau_{\partial \cT_{R(n)}} } - \frac{1}{2} \right| } \to 0\]
for almost every realization $\cT$ of $T$. Because $f_n(\cdot) \leq 1/2$, it is the immediate consequence of Lebesgue's dominated convergence theorem that
\[ \P{\tau^n_{\rho} = + |T^n_{R(n)}, \tau^n_{\partial T^n_{R(n)}} } \to \frac{1}{2}, \]
in $L^1(\Omega,\Sigma,\mathbb{P})$
as $n \to \infty$.
Finally, it is a well-known fact that convergence in $L^1$ implies convergence in probability.
§ COUPLING OF LOCAL NEIGHBOURHOOD
This section has as its objective to establish a coupling between the local neighbourhood of an arbitrary fixed vertex in the DC-PPM and $T^{\text{Poi}}$. The main result is the following theorem, where we let $T$, $\tau$, and $\psi$ be random instances of $T^{\text{Poi}}$, its spins and its weights, respectively.
Let $R(n) = C \log(n)$, with $C < \frac{1 - \log(4/e)}{3 \log(2 \kappa_{\text{max}})}$. Let $\rho$ be a uniformly picked vertex in $V(G)$, where for each $n$, $G = G(n)$ is an instance of the DC-PPM. Let for each $n$, $(T^n,\tau^n,\psi^n)$ be an independent copy of $(T,\tau,\psi)$, then
\[\P{ \lr{G_{R(n)}(\rho), \sigma_{G_R}, \phi_{G_R}} =
\lr{ T_{R(n)}^n , \tau_{T_R}^n, \psi_{T_R}^n} }
= 1 - n^{- \frac{1}{2} \log (4/e)}.\]
We defer its proof to the end of this section. It uses an alternative description of the branching process in Section <ref>.
§.§ Alternative description of branching process
We obtain an alternative description of the graph by considering a particle $u$ with spin $\sigma_u$ and weight $\phi_u$ to be of type $x_u = \phi_u \sigma_u \in S = -W \cup W$. We denote the law of $x_u$ by $\mu$, i.e., for $A \subset S$, $\mu(A) = \int_A \frac{1}{2} \mathrm{d} \nu(|x|).$ Two distinct vertices $u$ and $v$ are then joined by an edge with probability $\frac{\kappa(x_u, x_v)}{n}$, where
$\kappa: S \times S \to \mathbb{R}$ is defined for $(x,y) \in S \times S$ by
κ(x,y) = |xy|1_{xy > 0}a + 1_{xy < 0}b .
Analogously, we obtain the following equivalent description of the branching process: We begin with a single particle $o$ of type $x_o$ governed by $\mu$, giving birth to Poi$(\lambda_{x_o}(S))$ children, where for $x \in S$, and $A \subset S$,
λ_x(A) = ∫_A κ(x,y) d μ(y). Conditional on $x_o$ the children have i.i.d. types governed by $\MUstar_{x_o}$ [Note that if $y$ has law $\MUstar_x$, then for any $A \subset W$, $\P{\text{sign}(y) = \text{sign}(x), |y| \in A} =\frac{a}{a+b} \int_A y \frac{\mathrm{d} y}{\PHI} = \P{\text{sign}(y) = \text{sign}(x)} \P{|y| \in A}$. Hence, we can identify sign$(y)$ with the particle's spin and $|y|$ with its independent weight.], where for $x \in S$, and $A \subset S$,
= λ_x(A)/λ_x(S)
= ∫_A a/a+bxy > 0 + b/a+bxy < 0 |y| d ν(|y|) /. For generation $t \geq 1$, all particles give birth independently in the following way: A particle with type $x^*$ is replaced in the next generation by Poi$(\lambda_{x^*}(S))$ children, again with i.i.d. types governed by $\MUstar_{x^*}$.
In <cit.> it is shown that local neighbourhoods of the graph are described by the above branching process, if we ignore the types. (To be precise: the equivalent description used in <cit.> is that a particle of type $x$ gives birth to Poi$(\lambda_{x}(A))$ children with type in $A$, for any $A \subset S$. Those numbers are independent for different sets $A$ and different particles.)
The coupling-technique in <cit.> uses a discretization of $\kappa$ as an intermediate step, thereby losing some information: types in the tree deviate slightly from their counterparts in the graph. We shall therefore use another coupling method, presented below, so that the types in graph and branching process are exactly the same.
§.§ Coupling
We use the following exploration process:
At time $m=0$, choose a vertex $\rho$ uniformly in $V(G)$, where $G$ is an instance of the DC-PPM. Initially, it is the only active vertex: $\mathcal{A}(0) = \{\rho\}$. All other vertices are neutral at start: $\mathcal{U}(0) = V(G) \setminus \{\rho\}$. No vertex has been explored yet: $\mathcal{E}(0) = \emptyset$. At each time $m \geq 0$ we arbitrarily pick an active vertex $u$ in $\mathcal{A}(m)$ that has shortest distance to $\rho$, and explore all its edges in $\{ uv: v \in \mathcal{U}(m) \}$: if $uv \in E(G)$ for $v \in \mathcal{U}(m)$, then we set $v$ active in step $m+1$, otherwise it remains neutral. At the end of step $m$, we designate $u$ to be explorated. Thus,
\[ \calE(m+1) = \calE(m) \cup \{u\}, \]
\[ \calA(m+1) = \lr{ \calA(m) \setminus \{u\} } \cup \lr{ \mathcal{N}(u) \cap \calU(m) }, \]
\[ \calU(m+1) = \calU(m) \setminus \mathcal{N}(u). \]
Our aim in this section is to show that the exploration process and the branching process are equal upto depth $R(n)$ (defined in Theorem <ref>) with probability tending to one for large $n$. We do this in two steps:
Firstly, we establish that the i.i.d. vertices in $\calU(m)$ follow a law $\mu^{(m)}$ such that
\[ \left| \left| \mu^{(m)} - \mu \right| \right|_{\text{TV}} = \bigO \lr{ \frac{m}{n} }. \]
This is the content of the following:
Let $1, \ldots, m$ be the vertices in $\calE(m)$, with types $X_1 = x_1, \ldots, X_m=x_m$. Then, the vertices in $\calU(m)$ are i.i.d. with law
$ \mu^{(m)} = \mu^{(m)}_{x_1, \ldots, x_m}, $
dμ^(m)(·) = g(·) d μ(·) /∫_S g(z) d μ(z) , with
g(·) = ∏_i=1^m 1 - κ(x_i, ·)/n. Further, in the regime $m = o(n)$, there exists $N_m$ such that for all $(x_1, \ldots, x_m)$:
\[ \left| \left| \mu^{(m)}_{x_1, \ldots, x_m} - \mu \right| \right|_{\text{TV}} \leq 2 \kappa_{\text{max}} \frac{m}{n}, \]
if $n \geq N_m$.
Secondly, if $u$ has type $X = x \in S$, then its $D$ neighbours in $\calU(m)$ (i.e., those vertices that will be added to $\calA(m+1)$) are i.i.d. with law $\mu^{*(m+1)}_x$, which is $\bigO \lr{\frac{m}{n}}$ away from $\MUstar_x$ in total variation distance. Further, we can approximate the number of neighbours $D$ by
$\text{Poi}\lr{ \lambda_x(S) }$ with error $\bigO \lr{\frac{|\calU(m)| - n}{n}} + \bigO \lr{\frac{m}{n}}$:
Assume $u$ has type $X=x$. Let $D$ be the number of neighbours $u$ has in $\calU(m)$. Then, the types of those neighbours are i.i.d. with law $\mu^{*(m)}_x$, where
d μ^*(m)_x(·) = κ(x,·) d μ^(m)(·) / ∫_S κ(x,y) d μ^(m)(y) . Recall $N_m$ from Lemma <ref>: if $n \geq N_m$ then
| | μ^*(m)_x - _x | |_TV ≤4 κ_max^3/κ_min^2 m/n. Further,
| | D - Poi λ_x(S) | |_TV ≤κ_max n - |(m)| /n + 3 κ_max^2 m/n.
To establish the desired coupling, let us give names to all good events:
\[ A_{r+1} = \{ \forall u \in \partial G_r : D_u = \widehat{D}_u \}, \]
\[ B_{r+1} = \{ \forall u \in \partial G_r, v \in \{1, \ldots, D_u \} : U_{uv} = \widehat{U}_{uv} \}, \]
\[ C_{r} = \{ |\partial G_s| \leq g(s) = 2^s M^s \text{log}(n) \ \forall s \leq r \}, \]
where, for $u \in \partial G_r$ (we identify $\partial G_r = \{ 1, \ldots, |\partial G_r| \}$),
* $D_u = |\mathcal{N}(u) \cap \mathcal{U}(|G_{r-1}| + u -1)|$;
and where, conditional that $u$ has type $X_u = x_u$,
* $\widehat{D}_u = \text{Poi}\lr{ \lambda_{x_u}(S) }$;
* moreover, for $v \in \{1, \ldots, D_u \} $:
* $U_{uv}$ denotes the type of child $v$ of vertex $u$;
* $\widehat{U}_{uv}$ is a random variable with law $\MUstar_{x_u}$.
The types attached to siblings are independent conditional on their parents type.
With the above lemma's established, the event
\[E_{r} = \bigcap_{s = 1}^r \{ A_{s} \cap B_{s} \cap C_{s} \} \]
happens indeed w.h.p.:
For any integer $r \leq R(n)$,
\[ \mathbb{P} \lr{E_{r+1}| E_{r}} \geq 1 - n^{3C \log(2\kappa_{\text{max}})-1} - n^{-\log (4/e)}, \]
for large enough $n$.
The events $A_{r}, B_{r}$ and $C_{r}$ alone do not contain enough information to completely reconstruct the neighbourhood of $\rho$: vertices in $\partial G_r$ might be merged among each other, or it is possible that they share a child in $V_r$. But, those events, are rare. Indeed,
let $K_r$ be the event that no vertex in $G_{r}$ has more than one neighbour outside $G_{r}$ and that there are no edges in $\partial G_r$. Then, we have the following:
Let $r \leq R$, then
\[ \P{K_r | C_{R}} \geq 1 - n^{3C \log (2\kappa_{\text{max}}) - 1} %\geq 1 - n^{- \frac{1}{3} \log (4/e)}
, \]
for large enough $n$.
Consider vertex $v \in \mathcal{U}(m)$ with type $Y$.
We show first that, conditional on $v \notin \mathcal{N}(1, \ldots, m)$ and $X_1 = x_1, \ldots, X_m = x_m$, $Y$ has law $\mu^{(m)}_{x_1, \ldots, x_m}$.
To this end we shall calculate for $y \in S$,
Y ≤y | v ∉𝒩(1, …, m) , X_1 = x_1, …, X_m = x_m
= Y ≤y v ∉𝒩(1, …, m) | Y ≤y, X_1 = x_1, …, X_m = x_m/v ∉𝒩(1, …, m) | X_1 = x_1, …, X_m = x_m,
since $\P{Y \leq y | X_1 = x_1, \ldots, X_m = x_m} = \P{Y \leq y }$.
Recall (<ref>) and observe that
\[ g(\cdot) = \P{v \notin \mathcal{N}(1, \ldots, m) | Y = \cdot, X_1 = x_1, \ldots, X_m = x_m}. \]
Hence, the denominator in (<ref>) is just
v ∉𝒩(1, …, m) | X_1 = x_1, …, X_m = x_m = ∫_S g(z) d μ(z). Evaluating the numerator yields,
Y ≤y v ∉𝒩(1, …, m) | Y ≤y, X_1 = x_1, …, X_m = x_m
= Y ≤y ∫_-^y g(z) d p(z) = ∫_-^y g(z) d μ(z),
where, we defined for $z \leq y$,
$ p(z) = \P{Y \leq z | Y \leq y, X_1 = x_1, \ldots, X_m = x_m}$.
By combining (<ref>) and (<ref>) we establish (<ref>), i.e., $Y$ has distribution $\mu^{(m)}$.
To see that $\mu^{(m)}$ indeed approximates $\mu$, observe that
\[ \lr{1 - \frac{\kappa_{\text{max}}}{n}}^m \leq \frac{g(\cdot) }{\int_S g(z) \mathrm{d} \mu(z) } \leq \frac{1}{\lr{1 - \frac{\kappa_{\text{max}}}{n}}^m}. \]
We use Taylor's theorem to appropriately bound both sides for large $n$:
\[ \lr{1 - \frac{\kappa_{\text{max}}}{n}}^m = 1 - \kappa_{\text{max}} \frac{m}{n} + \bigO \lr{ \frac{\kappa_{\text{max}}^2}{2} \lr{\frac{m}{n}}^2 }, \]
where $\left| \bigO \lr{ \frac{\kappa_{\text{max}}^2}{2} \lr{\frac{m}{n}}^2 } \right| \leq \frac{\kappa_{\text{max}}^2}{2} \lr{\frac{m}{n}}^2 $
\[ \frac{1}{\lr{1 - \frac{\kappa_{\text{max}}}{n}}^m} = 1 + \kappa_{\text{max}} \frac{m}{n} + \bigO \lr{ 4 \kappa_{\text{max}}^2 \lr{\frac{m}{n}}^2},\]
where $\left| \bigO \lr{ 4 \kappa_{\text{max}}^2 \lr{\frac{m}{n}}^2} \right| \leq 4 \kappa_{\text{max}}^2 \lr{\frac{m}{n}}^2$.
Consequently, there exists $N_m$ such that
\[ \ba
\left|\left|\mu^{(m)} - \mu \right|\right|_{\text{TV}} &\leq \int_S \left| \frac{g(y) }{\int_S g(z) \mathrm{d} \mu(z) } - 1\right| \mathrm{d} \mu(y)
% &\leq \kappa_{\text{max}} \frac{m}{n} + 4 \kappa_{\text{max}}^2 % \lr{\frac{m}{n}}^2 \\
\leq 2 \kappa_{\text{max}} \frac{m}{n},
\ea \]
for all $n \geq N_m$.
Put $n_m = |\mathcal{U}(m)|$ and let $ Y_1, \ldots, Y_D$ denote the types of the neighbours of $u$.
Let $f$ be an arbitrary measurable function. The first claim follows if we prove that
. e^ - j=1D f(Y_j) | D = d = ∫_S e^-f(y) d μ^*(m)_x(y) ^d Now,
\[ \ba
& \E{ \mathrm{e}^{ - \s{j=1}{D} f(Y_j) } 1_{D=d} } \\
&= \sum_{F \subset [n_m], |F| = d} \E{ \left. \mathrm{e}^{ - \s{j \in F}{} f(Y_j) } \right| F } \lr{1 - \frac{1}{n} \int_S \kappa(x,y) \mathrm{d} \mu^{(m)}(y)}^{n_m - d} \lr{\frac{1}{n} \int_S \kappa(x,y) \mathrm{d} \mu^{(m)}(y)}^{ d},
\ea \]
where, conditioning on $F$ means that $\mathcal{N}(u) \cap \mathcal{U}(m) = F$.
We have,
\[ \P{D = d} = {n_m \choose d} \lr{ 1 - \frac{1}{n} \int_S \kappa(x,y) \mathrm{d} \mu^{(m)}(y) }^{n_m - d} \lr{\frac{1}{n} \int_S \kappa(x,y) \mathrm{d} \mu^{(m)}(y) }^{d}. \]
\[ \E{ \left. \mathrm{e}^{ - \s{j=1}{D} f(Y_j) } \right| D = d } = \frac{1}{{n_m \choose d}} \sum_{F \subset [n_m], |F| = d} \E{ \left. \mathrm{e}^{ - \s{j \in F}{} f(Y_j) } \right| F }.\]
Conditional on $F \subset [n_m]$, the types $(Y_j)_{j \in F}$ are i.i.d., thus
\[ \ba
\E{ \left. \mathrm{e}^{ - \s{j \in F}{} f(Y_j) } \right| F } = \lr{ \frac{ \int_S \mathrm{e}^{-f(y)} \frac{\kappa(x,y)}{n} \mathrm{d} \mu^{(m)}(y) }{ \int_S \frac{\kappa(x,y)}{n} \mathrm{d} \mu^{(m)}(y) } }^d,
\ea \]
which combined with (<ref>) gives (<ref>), our first claim.
Further, an explicit calculation yields
\[ \int_S | \mathrm{d} \mu^{*(m)}_x - \mathrm{d} \MUstar_x | \leq 4 \frac{\kappa_{\text{max}}^3}{\kappa_{\text{min}}^2} \frac{m}{n}, \]
establishing (<ref>).
For the last claim, observe that $D = $ Bin$(n_m,p)$, where $p = \frac{1}{n} \int_S \kappa(x,y) \mathrm{d} \mu^{(m)}(y)$. Hence,
\[ \left| \left| \text{Bin}(n_m,p) - \text{Poi}\lr{ n_m p } \right| \right|_{\text{TV}} \leq \s{i=1}{n_m} p^2 \leq \frac{\kappa_{\text{max}}^2}{n}. \]
Standard bounds for Poisson random variables entail the existence of a constant $C_{\text{Poi}} \geq 1$ such that $||$Poi$(\mu) - $ Poi$(\lambda)||_{\text{TV}} \leq C_{\text{Poi}}|\mu - \lambda|$. Consequently,
\[ \ba \frac{1}{C_{\text{Poi}}} \left| \left| \text{Poi}(n_m p) - \text{Poi}\lr{ \lambda_x(S) } \right| \right|_{\text{TV}} &\leq |n_m-n|p + \left| \int_S \kappa(x,y) \mathrm{d} \mu^{(m)}(y) - \int_S \kappa(x,y) \mathrm{d} \mu(y) \right| \\
&\leq \kappa_{\text{max}} \frac{|n_m-n|}{n} + \kappa_{\text{max}} \int_S \left|\mathrm{d} \mu^{(m)}(y) - \mathrm{d} \mu(y) \right| \\
&\leq \kappa_{\text{max}} \frac{|n_m-n|}{n} + 2 \kappa_{\text{max}}^2 \frac{m}{n}.
\ea \]
\[ \left| \left| \text{Bin}(n_m,p) - \text{Poi}\lr{ \lambda_x(S) } \right| \right|_{\text{TV}} \leq C_{\text{Poi}} \lr{ \kappa_{\text{max}} \frac{|n_m-n|}{n} + 3 \kappa_{\text{max}}^2 \frac{m}{n}}. \]
Write $n_r = |\partial G_r|$.
We have
\[ \P{E_{r+1}| E_{r}} \geq \P{ B_{r+1}| E_{r}} - \P{\neg A_{r+1}| E_{r}} - \P{ \neg C_{r+1}| E_{r}} . \]
B_r+1| E_r,n_r
≥1 - ∑_u=1^n_r . B^(u)_r+1 | ⋂_v=1^u-1 B^(v)_r+1 ,E_r,
where $ B^{(u)}_{r+1} = \{ \forall w \in \{1, \ldots, D_u \} : U_{uw} = \widehat{U}_{uw} \}. $
Denote the already explored vertices by $1, \ldots, m$ (where $m = |G_{r-1}| + u - 1$) and their types as $X_1, \ldots, X_m$.
Conditional on those types, the vertices in $\mathcal{U}(m)$ are i.i.d. with distribution $\mu^{(m)}$. Hence:
. B^(u)_r+1 | ⋂_v=1^u-1 B^(v)_r+1 ,E_r, n_r, X_1, …, X_m
= . B^(u)_r+1 | X_1, …, X_m
≥. B^(u)_r+1 | D_u ≤(1 + log(n)) κ_max, X_1, …, X_m
D_u ≤(1 + log(n)) κ_max | X_1, …, X_m . .
Now, $D_u \overset{d} \leq \text{Bin}(n, \frac{\kappa_{\text{max}}}{n})$, regardless of $X_1, \ldots, X_m$. Consequently, due to a multiplicative Chernoff bound,
D_j ≤(1 + log(n)) κ_max | X_1, …, X_m .
≥1 - 1/n^3,
for large enough $n$. Lemma <ref> entails
B^(u)_r+1 | D_u ≤(1 + log(n)) κ_max, X_1, …, X_m .
≥1 - c_1 m log(n)/n . Then, (<ref>) - (<ref>) together give
\[ \P{ \left. B^{(u)}_{r+1} \right| \bigcap_{v=1}^{u-1} B^{(v)}_{r+1} ,E_{r}, X_1, \ldots, X_m} \geq 1 - c_2 \frac{m \log(n)}{n}. \]
Now, since $m \leq |G_r| \leq r g(r)$ and $n_r \leq g(r)$, (<ref>) gives
\[ \ba \P{ B_{r+1} | E_{r}} &\geq 1 - \frac{r g^2(r) \log^2(n)}{n}. \ea \]
We take a similar approach to quantify
A_r+1| E_r,n_r ≥1 - ∑_u=1^n_r . A^(u)_r+1 | ⋂_v=1^u-1 A^(v)_r+1 ,E_r,n_r,
$ A^{(u)}_{r+1} = \{ D_u = \widehat{D}_u, D_u \leq (1+ \text{log}(n)) \kappa_{\text{max}} \}. $
. A^(u)_r+1 | ⋂_v=1^u-1 A^(v)_r+1 ,E_r ≥
D_u = D_u | ⋂_v=1^u-1 A^(v)_r+1 ,E_r . - D_u > (1+ log(n)) | ⋂_v=1^u-1 A^(v)_r+1 ,E_r .
≥1 - r g(r) log^2(n)/n,
due to Lemma <ref>, since $n - |\mathcal{U} (m)| \leq |G_r| + (u-1)(1+ \text{log}(n))\kappa_{\text{max}}$.
Thus, (<ref>) gives
\[ \P{ A_{r+1}| E_{r}} \geq 1 - \frac{r g^2(r) \log^2(n)}{n}. \]
We finish by establishing the growth condition (i.e., $C_{r+1}$): On $C_{r}$, $|\partial G_r| \leq (2\kappa_{\text{max}})^r \text{log}(n)$, thus
\[ |\partial G_{r+1}| \leq Z:= \text{Bin}((2\kappa_{\text{max}})^r \text{log}(n) n, \frac{\kappa_{\text{max}}}{n}). \]
\[ \ba \P{\neg C_{r+1} | E_r} &= \P{|\partial G_{r+1}| > 2^{r+1} \kappa_{\text{max}}^{r+1} \text{log}(n) |E_r } \\
&\leq \P{Z > 2 \E{Z}} \\
&\leq \lr{\frac{e}{4}}^{\E{Z}},
\ea \]
by a multiplicative version of Chernoff's bound.
Now, $\E{Z} = 2^{r} \kappa_{\text{max}}^{r+1} \text{log}(n)$, hence,
\[ \P{ C_{r+1} | E_r} \geq 1 - \frac{1}{n^{ \text{log}(4/e) }}. \]
Fix $u,v \in \partial G_r$. The probability of having an edge between $u$ and $v$ is smaller than $\frac{\kappa_{\text{max}}}{n}$. For any $w \in V(G \setminus G_r)$, the probability that $(u,w)$ and $(v,w)$ both appear is smaller than $\frac{\kappa_{\text{max}}^2}{n^2}$. Now, Lemma <ref> implies that
\[ |G_r| \leq g(R)R(n) = n^{C \log (2\kappa_{\text{max}})} C \log^2(n) = o(n^{\beta}), \]
for all $\beta > C \log (2\kappa_{\text{max}})$. Hence, the result follows from a union bound over all triples $u,v,w$.
We have
≤r=1R E_r | E_r-1
≤R(n) n^3C log(2κ_max)-1 + n^-log(4/e) .
\[ \P{ \cap_{s=1}^R K_s | E_R } \leq R(n) n^{3C \log(2\kappa_{\text{max}})-1}. \]
Hence, due to the choice of $C$,
\[ \Pn{\cap_{s=1}^R K_s ,E_R} \geq 1 - n^{- \frac{1}{2} \log (4/e)} . \]
§ NO LONG-RANGE CORRELATION IN DC-PPM
In this section we establish (<ref>). To this end, we first condition on both the spins of $\partial G_{R(n)}$ and all weights in $G$. Lemma <ref> below shows that we then can remove the conditioning on $\sigma_v$ and the graph structure outside the $R$-neighbourhood (including the weights):
Var(σ_u | σ_∂G_R, σ_v , G, ϕ) = Var(σ_u | σ_∂G_R , G_R, ϕ_G_R) + o_n(1). We established in the previous section that a neighbourhood in $G$ looks like a $T^{\text{Poi}}$ tree with a Markov broadcasting process on it. Hence, the right-hand side of (<ref>) converges to $1$ in probability, establishing (<ref>). We show in Theorem <ref> below that this contradicts the existence of a reconstructed bisection that is positively correlated with the true type-assignment.
We begin by preparing an auxiliary lemma to prove (<ref>), it establishes that long-range interactions are sufficiently weak. Its proof is inspired by Lemma 4.7 in <cit.>. However (besides the additional complication of weights) the result stated here is stronger in the sense that the $o_n(1)$ terms converge uniformly to $0$ and that "conditioning on $G$" may now be replaced with "conditioning on $G_{A \cup B}$".
Let $G$ be an instance of the DC-PPM. Let $u$ be an uniformly picked vertex in $V(G)$.
Let $A = A(G)$, $B = B(G)$, $C = C(G) \subset V$ be a (random) partition of $V(G)$, with $u \in A$, such that $B$ separates $A$ and $C$ in $G$. If $|A \cup B| \leq n^{1/8}$ for asymptotically almost every realization of $G$, then
| σ_u =+ | σ_B ∪C , G,ϕ - σ_u =+ | σ_B , G_A ∪B, ϕ_A ∪B| = o_n(1), with probability $1 - o_n(1)$. Moreover, the $o_n(1)$ terms in (<ref>) converge uniformly to $0$.
For a fixed graph $g$, spin-configuration $\tau$ and degree-configuration $\psi$, we make a factorization of $\P{G=g,\sigma = \tau | \phi = \psi}$ into parts depending on $A,B$ and $C$.
We claim that the part that measures the interaction between $A$ and $C$ can be neglected with respect to the other parts.
\[ \Psi_{uv}(g,\tau, \psi) = \left\{
\begin{array}{l l}
a \frac{\psi_u \psi_v}{n} & \quad \text{ if } (u,v) \in E(g) \text{ and } \tau_u = \tau_v \\
b \frac{\psi_u \psi_v}{n} & \quad \text{ if } (u,v) \in E(g) \text{ and } \tau_u \neq \tau_v \\
1- a \frac{\psi_u \psi_v}{n} & \quad \text{ if } (u,v) \notin E(g) \text{ and } \tau_u = \tau_v \\
1 - b \frac{\psi_u \psi_v}{n} & \quad \text{ if } (u,v) \notin E(g) \text{ and } \tau_u \neq \tau_v. \\
\end{array} \right. \]
We define for arbitrary sets $U_1, U_2 \subset V$,
\[ Q_{U_1, U_2} = Q_{U_1, U_2}(g,\tau,\psi) = Q_{U_1, U_2}(g_{U_1 \cup U_2},\tau_{U_1 \cup U_2},\psi_{U_1 \cup U_2}) = \prod_{u \in U_1, v \in U_2} \Psi_{uv}(g,\tau, \psi), \]
where the subscript indicates restriction of the corresponding quantities to $U_1 \cup U_2$.
Then, we have,
G=g |σ=τ, ϕ= ψ = Q_A ∪B, A ∪B Q_B ∪C, C Q_A,C.
We begin by demonstrating that $Q_{A,C}$ is asymptotically independent of $\tau$: Write,
\[ Q_{A,C}(g,\tau,\psi) = \prod_{u \in A, v \in C: \tau_u = \tau_v} \lr{ 1- a \frac{\psi_u \psi_v}{n}} \prod_{u \in A, v \in C: \tau_u \neq \tau_v} \lr{ 1- b \frac{\psi_u \psi_v}{n}}, \]
since $A$ and $C$ are separated by $B$ (there are thus no edges between $A$ and $C$).
The first product may be rewritten as,
\[ \ba \prod_{u \in A, v \in C: \tau_u = \tau_v} \lr{ 1- a \frac{\psi_u \psi_v}{n}} &= \text{exp}\lr{\s{u \in A, v \in C: \tau_u = \tau_v}{} \text{log}\lr{1- a \frac{\psi_u \psi_v}{n}}} \\
&= \text{exp}\lr{\s{u \in A, v \in C: \tau_u = \tau_v}{} \lr{ - a \frac{\psi_u \psi_v}{n} + \bigO(1/n^2)}} \\
&= \text{exp}\lr{ - \frac{a }{n}\s{u \in A, v \in C: \tau_u = \tau_v}{} \psi_u \psi_v} \exp{ \lr{ \bigO(n_A n_C/n^2)} }.
\ea \]
Now, the sum $\s{u \in A, v \in C: \tau_u = \tau_v}{} \psi_u \psi_v$ tends to $\frac{\|A\| \|C\|}{2}$, if $(\tau,\psi) \in \Omega(n)$, where
\[ \|U\| = \s{u \in U}{} \psi_u, \quad (U \subset V), \]
and where,
Ω(n) = { (τ',ψ'): S_C(ψ',τ') ≤n^3/4 }, with, for $U \subset V$,
\[ S_U(\psi,\tau) = \s{u \in U}{} \psi_u \tau_u. \]
To prove that $\s{u \in A, v \in C: \tau_u = \tau_v}{} \psi_u \psi_v$ indeed converges to $\frac{\|A\| \|C\|}{2}$, we introduce the following quantities:
\[ |U^{\pm}|(\psi,\tau) = \s{u \in U: \tau_u = \pm}{} \psi_u, \quad (U \subset V), \]
to write,
\[ \|A\| \|C\| + S_A S_C = 2 (|A^+||C^+| + |A^-||C^-|). \]
We use the latter observation to rewrite
\[ \s{u \in A, v \in C: \tau_u = \tau_v}{} \psi_u \psi_v = |A^+||C^+| + |A^-||C^-| = \frac{\|A\| \|C\| + S_A S_C}{2}, \]
where $S_A S_C \leq n_A n^{3/4} = n^{\frac{7}{8}}$, for $(\tau,\psi) \in \Omega(n)$. As a consequence,
\[ \ba \prod_{u \in A, v \in C: \tau_u = \tau_v} \lr{ 1- a \frac{\psi_u \psi_v}{n}} &= \exp{ \lr{ \bigO \lr{\frac{n^{7/8}}{n}} }} \exp{ \lr{ \bigO\lr{ \frac{n_A n_C}{n^2}}} } \exp \lr{ - \frac{a \|A\| \|C\|}{2n} } \\
&= (1 + o_n(1))\exp{ \lr{ - \frac{a \|A\| \|C\|}{2n} } }, \ea \]
where the $o_n$ term is uniform for all $(\tau,\psi) \in \Omega(n)$. We carry out a similar calculation for the other product. Together we obtain
Q_A,C(g,τ,ψ) = (1 + o_n(1))exp - a + b/2 A C/n , uniformly for all $(\tau,\psi) \in \Omega(n)$. This proves that $Q_{A,C}(g,\tau,\psi)$ is indeed essentially independent of $\tau$ for most pairs $(\tau, \psi)$.
We use the above to prove that, for $u \in V$,
σ_u=τ_u | σ_B ∪C = τ_B ∪C, G=g, ϕ= ψ, ( ϕ, σ) ∈Ω(n)
= (1 + o_n(1)) σ_u=τ_u | σ_B = τ_B, G_A ∪B=g_A ∪B, ϕ_A ∪B = ψ_A ∪B, ( ϕ, σ) ∈Ω(n) +o_n(1). Fix $(\tau,\psi) \in \Omega(n)$. Then,
G=g,σ=τ| ϕ= ψ, ( ϕ, σ) ∈Ω(n) = G=g, σ=τ, ( ϕ, σ) ∈Ω(n) | ϕ= ψ/( ϕ, σ) ∈Ω(n) | ϕ= ψ
= G=g, σ=τ, ( ϕ, σ) ∈Ω(n) | ϕ= ψ/σ= τ| ϕ= ψ σ= τ| ϕ= ψ/( ϕ, σ) ∈Ω(n) | ϕ= ψ
= G=g | σ=τ, ϕ= ψ 2^-n/( ϕ, σ) ∈Ω(n) | ϕ= ψ
= G=g | σ=τ, ϕ= ψ f(ψ,n),
for some function $f$.
Hence, plugging (<ref>) and (<ref>) in (<ref>),
G=g,σ=τ| ϕ= ψ, ( ϕ, σ) ∈Ω(n)
= Q_A ∪B, A ∪B (g,τ,ψ) Q_B ∪C, C (g,τ,ψ) (1 + o_n(1))exp - a + b/2 AC/n f(ψ,n). Put, for $U \subset V$,
\[ \Omega_U(n) = \Omega_U( \psi, \tau_U, n) = \{ \tau': \tau'_U = \tau_U, (\tau',\psi) \in \Omega(n) \}, \]
then, invoking (<ref>),
G=g,σ_U=τ_U | ϕ= ψ, ( ϕ, σ) ∈Ω(n) = ∑_τ' ∈Ω_U(n) G=g,σ=τ' | ϕ= ψ, ( ϕ, σ) ∈Ω(n)
= ∑_τ' ∈Ω_U(n)
Q_A ∪B, A ∪B (g,τ',ψ) Q_B ∪C, C (g,τ',ψ)
·(1 + o_n(1)) exp - a + b/2 AC/n f(ψ,n)
= (1 + o_n(1)) exp - a + b/2 AC/n f(ψ,n)
·∑_τ' ∈Ω_U(n)
Q_A ∪B, A ∪B (g,τ',ψ) Q_B ∪C, C (g,τ',ψ),
where we could interchange the order $o_n(1)$ term and the sum because the former holds uniformly for all $(\phi,\sigma) \in \Omega(n)$.
We apply (<ref>) with $U=A$ and $U = A \cup B$, to rewrite the right hand side of
σ_A=τ_A | σ_B = τ_B, G=g, ϕ= ψ, ( ϕ, σ) ∈Ω(n) = G=g, σ_A ∪B=τ_A ∪B | ϕ= ψ, ( ϕ, σ) ∈Ω(n)/ G=g, σ_B=τ_B| ϕ= ψ, ( ϕ, σ) ∈Ω(n)
\[ \ba & \quad (1 + o_n(1)) \frac{\sum_{\tau' \in \Omega_{A \cup B}(n)}
Q_{A \cup B, A \cup B} (g,\tau',\psi) Q_{B \cup C, C} (g,\tau',\psi)}{\sum_{\tau' \in \Omega_{B}(n)}
Q_{A \cup B, A \cup B} (g,\tau',\psi) Q_{B \cup C, C} (g,\tau',\psi)} \\
&= (1 + o_n(1)) \frac{Q_{A \cup B, A \cup B} (g,\tau,\psi) \sum_{\tau' \in \Omega_{A \cup B}(n)}
Q_{B \cup C, C} (g,\tau',\psi)}{\sum_{\tau''' \in \Omega_{B \cup C}(n)}
Q_{A \cup B, A \cup B} (g,\tau''',\psi) \sum_{\tau'' \in \Omega_{A \cup B}(n)} Q_{B \cup C, C} (g,\tau'',\psi)}, \ea \]
where we used that $Q_{U_1,U_2}(\tau')$ depends on $\tau'$ only through $\tau'_{U_1 \cup U_2}$ to rewrite the numerator. Factorization of the denominator is justified as follows:
For an arbitrary $\tau' \in \Omega_B(n)$, put $\tau'' = (\tau_{A \cup B},\tau'_C) \in \Omega_{A \cup B}(n)$ and $\tau''' = (\tau'_A, \tau_{B \cup C}) \in \Omega_{B \cup C}(n)$. Then,
Q_A ∪B, A ∪B (g,τ',ψ) Q_B ∪C, C (g,τ',ψ) = Q_A ∪B, A ∪B (g,τ”',ψ) Q_B ∪C, C (g,τ”,ψ). This proves that the double summation is at least as large as the single sum. Equality follows upon putting $\tau' = (\tau'''_A,\tau_B,\tau''_C)$ for arbitrary $\tau'' \in \Omega_{A \cup B}(n)$ and $\tau''' \in \Omega_{B \cup C}(n)$: (<ref>) is then again satisfied. Hence, (<ref>) is equivalent to
σ_A=τ_A | σ_B = τ_B, G=g, ϕ= ψ, ( ϕ, σ) ∈Ω(n) = (1 + o_n(1)) Q_A ∪B, A ∪B (g,τ,ψ)/∑_τ”' ∈Ω_B ∪C(n)
Q_A ∪B, A ∪B (g,τ”',ψ) .
We shall rewrite the right hand side of (<ref>) to obtain on the one hand:
σ_u=τ_u | σ_B = τ_B, G=g, ϕ= ψ, ( ϕ, σ) ∈Ω(n) = (1 + o_n(1)) Fg_A ∪B,τ_u ∪B,ψ_A ∪B, for some function $\widehat{F}(\cdot) \leq 1$.
And, on the other hand:
σ_u=τ_u | σ_B = τ_B, G=g, ϕ= ψ, ( ϕ, σ) ∈Ω(n)
= (1 + o_n(1)) σ_u=τ_u | σ_B ∪C = τ_B ∪C, G=g, ϕ= ψ, ( ϕ, σ) ∈Ω(n). To do so, note that
\[ \sum_{\tau''' \in \Omega_{B \cup C}(n)} Q_{A \cup B, A \cup B} (g,\tau''',\psi) =
\sum_{\tau_A''' \in \{\pm \}^A } Q_{A \cup B, A \cup B} (g_{A \cup B},(\tau_A''',\tau_B),\psi_{A \cup B}).
\]
\[ \ba \frac{Q_{A \cup B, A \cup B} (g,\tau,\psi)}{\sum_{\tau''' \in \Omega_{B \cup C}(n)}
Q_{A \cup B, A \cup B} (g,\tau''',\psi) } &= \frac{Q_{A \cup B, A \cup B} (g_{A \cup B},\tau_{A \cup B},\psi_{A \cup B})}{\sum_{\tau_A''' \in \{\pm \}^A } Q_{A \cup B, A \cup B} (g_{A \cup B},(\tau_A''',\tau_B),\psi_{A \cup B}) } \\
&= F\lr{g_{A \cup B},\tau_{A \cup B},\psi_{A \cup B}}, \ea \]
for some function $F(\cdot) \leq 1$. Therefore, (<ref>) is equivalent to
\[ \P{\sigma_A=\tau_A | \sigma_B = \tau_B, G=g, \phi = \psi, ( \phi, \sigma) \in \Omega(n)} = (1 + o_n(1)) F\lr{g_{A \cup B},\tau_{A \cup B},\psi_{A \cup B}}. \]
If we fix $u \in A$ and integrate over all possible values of $\tau_{A \setminus u}$ while keeping $\tau_{B \cup C}$ and $\psi$ constant, we obtain (<ref>) (since the above formula holds for any $\tau$ such that $(\tau,\psi) \in \Omega(n)$ and the latter condition depends only on the value of $\tau_C $ and $\psi_C$).
To establish (<ref>), we multiply both denominator and enumerator of (<ref>) by $Q_{B \cup C, C} (g,\tau,\psi)$:
\[ \ba & \P{\sigma_A=\tau_A | \sigma_B = \tau_B, G=g, \phi = \psi, ( \phi, \sigma) \in \Omega(n)} \\
&= (1 + o_n(1)) \frac{Q_{A \cup B, A \cup B} (g,\tau,\psi) Q_{B \cup C, C} (g,\tau,\psi) }{\sum_{\tau' \in \Omega_{B \cup C}(n)}
Q_{A \cup B, A \cup B} (g,\tau',\psi) Q_{B \cup C, C} (g,\tau',\psi) } \\
&= (1 + o_n(1)) \frac{\P{G=g,\sigma =\tau | \phi = \psi, ( \phi, \sigma) \in \Omega(n)}}{\P{G=g,\sigma_{B \cup C}=\tau_{B \cup C} | \phi = \psi, ( \phi, \sigma) \in \Omega(n)}} \\
&= (1 + o_n(1)) \P{\sigma_A=\tau_A | \sigma_{B \cup C} = \tau_{B \cup C}, G=g, \phi = \psi, ( \phi, \sigma) \in \Omega(n)}. \ea \]
Integrating again over $\tau_{A \setminus u}$ gives (<ref>).
We use (<ref>) to obtain
σ_u=τ_u | σ_B = τ_B, G_A ∪B=g_A ∪B, ϕ_A ∪B = ψ_A ∪B, ( ϕ, σ) ∈Ω(n)
= g_C,ψ_C σ_u=τ_u | σ_B = τ_B, G=(g_A ∪B,g_C), ϕ= (ψ_A ∪B,ψ_C), ( ϕ, σ) ∈Ω(n)
·G_C = g_C, ϕ_C = ψ_C | σ_B = τ_B, G_A ∪B=g_A ∪B, ϕ_A ∪B = ψ_A ∪B, ( ϕ, σ) ∈Ω(n)
= g_C,ψ_C (1 + o_n(1)) Fg_A ∪B,τ_u ∪B,ψ_A ∪B
·G_C = g_C, ϕ_C = ψ_C | σ_B = τ_B, G_A ∪B=g_A ∪B, ϕ_A ∪B = ψ_A ∪B, ( ϕ, σ) ∈Ω(n)
= (1 + o_n(1)) Fg_A ∪B,τ_u ∪B,ψ_A ∪B +o_n(1)
= (1 + o_n(1)) σ_u=τ_u | σ_B = τ_B, G=g, ϕ= ψ, ( ϕ, σ) ∈Ω(n) +o_n(1). Combining (<ref>) and (<ref>) gives
\[ \ba &\P{\sigma_u=\tau_u | \sigma_{B \cup C} = \tau_{B \cup C}, G=g, \phi = \psi, ( \phi, \sigma) \in \Omega(n)} \\&= (1 + o_n(1)) \P{\sigma_u=\tau_u | \sigma_B = \tau_B, G=g, \phi = \psi, ( \phi, \sigma) \in \Omega(n)} \\ &= (1 + o_n(1)) \P{\sigma_u=\tau_u | \sigma_B = \tau_B, G_{A \cup B}=g_{A \cup B}, \phi_{A \cup B} = \psi_{A \cup B}, ( \phi, \sigma) \in \Omega(n)}, \ea \]
i.e., the claim (<ref>).
Our last step consists in removing the condition $(\sigma,\phi) \in \Omega(n)$:
Put $\epsilon(n) = 1 - \P{(\sigma,\phi) \in \Omega(n)}$ and note that $\lim_{n \to \infty} \epsilon(n) = 0$. Consider the random variable $\P{( \phi, \sigma) \in \Omega(n)| \sigma_B, G_{A \cup B}, \phi_{A \cup B} } = \E{ 1_{(\phi, \sigma) \in \Omega(n)} | \sigma_B, G_{A \cup B}, \phi_{A \cup B}} $. It has expectation $1 - \epsilon(n)$, so that
1_(ϕ, σ) ∈Ω(n) | σ_B, G_A ∪B, ϕ_A ∪B ≥1 - √(ϵ(n)) ≥1 - 2 √(ϵ(n)). Indeed, if contrary to our claim $f:= \E{ 1_{(\phi, \sigma) \in \Omega(n)} | \sigma_B, G_{A \cup B}, \phi_{A \cup B}} \geq 1 - \sqrt{\epsilon(n)}$ with probability at most $1 - 2 \sqrt{\epsilon(n)}$, then
\[ \ba \E{f} &\leq 1 \cdot (1 - 2 \sqrt{\epsilon(n)}) + (1 - \sqrt{\epsilon(n)})\cdot 2 \sqrt{\epsilon(n)} < 1 - \epsilon(n).
\ea \]
Similarly, for $B \cup C$,
1_(ϕ, σ) ∈Ω(n) | σ_B ∪C, G, ϕ ≥1 - √(ϵ(n)) ≥1 - 2 √(ϵ(n)). Because $(1+o_n(1))\lr{1 - \bigO \lr{\sqrt{\epsilon(n)}}} = (1+o_n(1))$, it follows that, with probability at least $1 - 4 \sqrt{\epsilon(n)}$,
\[ \ba \P{\sigma_u = +| \sigma_B , G_{A \cup B}, \phi_{A \cup B}} &= \lr{ 1 -\bigO \lr{ \sqrt{\epsilon(n)} }}\P{\sigma_u = + | \sigma_B, G_{A \cup B}, \phi_{A \cup B}, (\phi, \sigma) \in \Omega(n)} \\ & \quad+ \bigO \lr{ \sqrt{\epsilon(n)}} \P{\sigma_u = +| \sigma_B , G_{A \cup B}, \phi_{A \cup B}, (\phi, \sigma) \notin \Omega(n)} \\
&= (1 + o_n(1)) \P{\sigma_u = + | \sigma_{B \cup C} , G, \phi, (\phi, \sigma) \in \Omega(n)} + o_n(1)\\
&= (1 + o_n(1)) \P{\sigma_u = + | \sigma_{B \cup C} , G, \phi} + o_n(1),
\ea
\]
where we used (<ref>), (<ref>) and (<ref>) in the first, second, respectively last equality.
Assume that $\frac{a+b}{2 }\PHItwo \geq 1$ and $(a-b)^2 \PHItwo \leq 2(a+b)$. Let $G$ be an instance of the DC-PPM. Let $u$ and $v$ be uniformly chosen vertices in $G$. Then,
\[ \P{\sigma_u = + | \sigma_v , G} \overset{\mathbb{P}} \to \frac{1}{2},\]
as $n \to \infty$.
Put $A = G_{R - 1}$, $B = \partial G_R$ and $C = G \setminus G_R$. We use the monotonicity property of conditional variance:
\[ 1 \geq \text{Var}(\sigma_u | \sigma_v , G) \geq \text{Var}(\sigma_u | \sigma_{B \cup C} , G,\phi ). \]
Now, by using the partition $A \cup B \cup C$ of $V(G)$ in Lemma <ref>, we have, since $G_R \leq n^{1/8}$ w.h.p.,
\[ \P{\sigma_u = + | \sigma_{B \cup C} , G,\phi} \overset{w.h.p.} = \P{\sigma_u = + | \sigma_{\partial G_R} , G_R, \phi_{G_R}} + o_n(1) . \]
Theorem <ref> entails that the local neighbourhood is w.h.p. equal to $T^{\text{Poi}}$. Let $T^n$ be an independent copy of $T^{\text{Poi}}$ with root $\rho$, spins $\tau^n$ and weights $\psi^n$. Note that we stress the dependence on $n$, because the Poisson-tree is sampled again for each $n$.
σ_u = + | σ_∂G_R , G_R, ϕ_G_R + o_n(1) w.h.p. = τ^n_ρ = + | τ^n_∂T^n_R , T^n_R, ψ_T^n_R + o_n(1)
= τ^n_ρ = + | τ^n_∂T^n_R , T^n_R + o_n(1), due to the coupling from Theorem <ref>.
By Theorem <ref>, the right-hand side of (<ref>) tends to $1/2$ in probability. Hence $\text{Var}(\sigma_u | \sigma_v , G)$ tends to $1$ in probability.
Hence, if $\frac{a+b}{2} \PHItwo >1$ and $(a-b)^2 \PHItwo \leq 2(a+b)$, detection is not feasible:
Assume that $\frac{a+b}{2} \PHItwo >1$ and $(a-b)^2 \PHItwo \leq 2(a+b)$. Let $G$ be an observation of the degree-corrected planted partition-model, with true communities $\{\sigma_i\}_{i=1}^n$. Let $\{\widehat{\sigma}_i\}_{i=1}^n$ be a reconstruction of the communities, based on the observation $G$, such that $|\{ i: \widehat{\sigma}_i = + \}| = \frac{n}{2}$. Assume that there exists $\delta > 0$ such that
\[ f(n):= \frac{1}{n} \s{i=1}{n} \indicator{ \sigma_i = \widehat{\sigma}_i} \geq \frac{1}{2} + \delta, \]
with high probability. Then, $ \P{\sigma_u = + | \sigma_v = +, G}$ does not converge in probability to $1/2$.
Assume for a contradiction that $\P{\sigma_u = + | \sigma_v = +, G} \overset{\mathbb{P}} \to \frac{1}{2}$.
It suffices to analyse Var$(\indicator{\sigma_u = \sigma_v} | G).$ Indeed,
\[ \ba
\P{\sigma_u = \sigma_v | G} &= \frac{1/2 \cdot \P{\sigma_u = \sigma_v , G}}{1/2 \cdot \P{G}} = \frac{ \P{\sigma_u = +, \sigma_v = +, G}}{ \P{\sigma_v = +, G}} = \P{\sigma_u = + | \sigma_v = +, G},
\ea\]
so that
\[ \ba \text{Var}(\indicator{\sigma_u = \sigma_v} | G) &= \E{\indicator{\sigma_u = \sigma_v}^2|G} - \E{\indicator{\sigma_u = \sigma_v}|G}^2 \\
&= \P{\sigma_u = \sigma_v | G} - \P{\sigma_u = \sigma_v | G}^2 \overset{\mathbb{P}}\to 1/4,
\ea \]
as $n$ tends to infinity.
But, by the monotonicity of conditional variances,
\[ \text{Var}(\indicator{\sigma_u = \sigma_v} | G) \leq \text{Var}(\indicator{\sigma_u = \sigma_v} | \widehat{\sigma}_u = \widehat{\sigma}_v), \label{eq::Mon_Var_1} \]
for any estimator $(\widehat{\sigma_i})_{i=1}^n$ that is based on an observation of $G$. We shall show that this inequality is violated on the event that $\widehat{\sigma}_u = \widehat{\sigma}_v = +$.
Note that our reconstruction is an exact bisection, whereas the true community structure might be unevenly distributed. However, deviations are small (denote $n_{\pm} = | \{ i: \sigma_i = \pm \} |$) :
The event $E = \{ n_+ \in (\frac{n}{2} - n^{3/4} , \frac{n}{2} + n^{3/4}) \}$ happens with probability larger than $1 - 2$exp$(-n^{1/2})$.
Let $\epsilon > 0$ and condition on $E$. Assume that there are $(\frac{1}{2} + \epsilon) \frac{n}{2}$ vertices such that their true type and assigned type are both $+$. Then there are $\frac{n}{2} + \mathcal{O}(n^{3/4}) - (\frac{1}{2} + \epsilon) \frac{n}{2} = \frac{n}{4} - \epsilon \frac{n}{2} + \mathcal{O}(n^{3/4})$ vertices that have type $+$, but are assigned type $-$. Hence, $\frac{n}{2} - (\frac{n}{4} - \epsilon \frac{n}{2} + \mathcal{O}(n^{3/4}) ) = (\frac{1}{2} + \epsilon) \frac{n}{2} + \mathcal{O}(n^{3/4})$ vertices have true type and assigned type $-$.
\[ \P{\sigma_u = \sigma_v = + \left| \widehat{\sigma}_u = \widehat{\sigma}_v = +, f(n) = \frac{1}{2} + \epsilon, E \right.} = \frac{( \frac{1}{2} + \epsilon)\frac{n}{2}}{\frac{n}{2}} \cdot \frac{( \frac{1}{2} + \epsilon)\frac{n}{2} - 1}{\frac{n}{2} -1} + \mathcal{O}(n^{-1/4}) \to \lr{\frac{1}{2} + \epsilon}^2, \]
for large $n$. Similarly,
\[ \P{\sigma_u = \sigma_v = - \left| \widehat{\sigma}_u = \widehat{\sigma}_v = +, f(n) = \frac{1}{2} + \epsilon, E \right.} \to \lr{\frac{1}{2} - \epsilon}^2, \]
for large $n$.
\[ \P{\sigma_u = \sigma_v \left| \widehat{\sigma}_u = \widehat{\sigma}_v = +, f(n) = \frac{1}{2} + \epsilon, E \right.} \to \frac{1}{2} + 2 \epsilon^2, \]
uniformly for, say, all $\epsilon \geq \frac{\delta}{2}$. Consequently, there exists $N = N(\delta)$ such that for any $\epsilon \geq \frac{\delta}{2}$,
\[ \P{\sigma_u = \sigma_v \left| \widehat{\sigma}_u = \widehat{\sigma}_v = +, f(n) = \frac{1}{2} + \epsilon, E \right.} \geq \frac{1}{2} + \epsilon^2, \]
for all $n \geq N(\delta)$. Consequently,
\[ \ba
\P{\sigma_u = \sigma_v | \widehat{\sigma}_u = \widehat{\sigma}_v = +}
%&= \P{\sigma_u = \sigma_v \left| \widehat{\sigma}_u = %\widehat{\sigma}_v = +, f(n) \geq \frac{1}{2} + \frac{\delta}{2}, E %\right.} (1 - o_n(1))\\
&\geq \frac{1}{2} + \frac{\delta^2}{8},
\ea\]
for $n$ large.
Since $x \mapsto x - x^2$ is decreasing on $(1/2,1)$, we have
\[ \ba \text{Var}(\indicator{\sigma_u = \sigma_v} | \widehat{\sigma}_u = \widehat{\sigma}_v = +) &= \P{\indicator{\sigma_u = \sigma_v} | \widehat{\sigma}_u = \widehat{\sigma}_v = +} - \P{\indicator{\sigma_u = \sigma_v} | \widehat{\sigma}_u = \widehat{\sigma}_v = +}^2 \\ & \leq \frac{1}{4} - \frac{\delta^4}{64}, \ea\]
hereby indeed violating (<ref>) for large $n$ on an event that has probability
$ \frac{n/2}{n} \cdot \frac{n/2 - 1}{n - 1} \to \frac{1}{4}, $
for large $n$.
We summarize these results in the main theorem of this paper:
Combine Theorem <ref> and Lemma <ref>.
§ ACKNOWLEDGEMENT
The authors would like to thank Joe Neeman for an inspiring discussion.
|
1511.00397
|
§ INTRODUCTION
In our previous work <cit.>, we presented the results of
the wave function renormalization factor $Z_q$, mass renormalization
factor $Z_m$ and the complete set of renormalization factors for
bilinear operators obtained on the $20^3 \times 64$ MILC asqtad coarse
lattice at $a \approx 0.12$ fm with $am_{\ell}/am_s=0.01/0.05$.
In this proceeding, we analyse the $Z_q$ and $Z_m$ on the
$28^3\times96$ MILC asqtad fine lattices ($a\approx 0.09$ fm,
$am_\ell/am_s=0.0062/0.031$) and compare the results with those on the
coarse lattices.
§ RESULTS
We calculate the renormalization factors with Landau gauge fixing
using HYP-smeared staggered quarks.
To do the chiral extrapolation, we perform the measurements with 5
valence quark masses ($am = 0.0062, 0.0124, 0.0186, 0.0248, 0.031$)
on the MILC fine ensembles at $a\approx 0.09$ fm.
We also carry out the measurements for 20 external momenta given
in Table <ref>.
The measurements are done over 30 gauge configurations.
$n(x, y, z, t)$ $a|\wtd{p}|$ GeV
$n(x, y, z, t)$ $a|\wtd{p}|$ GeV
$n(x, y, z, t)$ $a|\wtd{p}|$ GeV
$(1,1,1,3)$ 0.4355 1.0197
$(1,1,1,4)$ 0.4686 1.0974
$(1,2,1,4)$ 0.6088 1.4257
$(1,2,1,6)$ 0.6755 1.5819
$(2,1,2,6)$ 0.7794 1.8250
$(2,2,2,7)$ 0.9023 2.1130
$(2,2,2,8)$ 0.9372 2.1947
$(2,2,2,9)$ 0.9753 2.2839
$(2,3,2,7)$ 1.0324 2.4177
$(2,3,2,8)$ 1.0631 2.4895
$(2,3,2,9)$ 1.0968 2.5684
$(3,2,3,8)$ 1.1756 2.7529
$(3,3,3,7)$ 1.2528 2.9337
$(3,3,3,8)$ 1.2782 2.9931
$(3,3,3,10)$ 1.3371 3.1312
$(3,4,3,9)$ 1.4349 3.3602
$(4,3,4,10)$ 1.5789 3.6973
$(4,4,4,10)$ 1.6868 3.9501
$(4,4,4,12)$ 1.7418 4.0788
$(4,4,4,14)$ 1.8046 4.2259
The list of momenta used for our analysis.
The first column is the four vectors in the units of
$(\dfrac{2\pi}{L_s}, \dfrac{2\pi}{L_s}, \dfrac{2\pi}{L_s},
\dfrac{2\pi}{L_t})$, where $L_s$ ($L_t$) is the number of sites in
the spatial (temporal) direction.
§.§ Wave Function Renormalization Factor $Z_q$
Let us consider the conserved vector current to obtain the wave function
renormalization factor $Z_q$.
We use the same method as in Ref. <cit.> to obtain the
First, we convert the raw data to the data defined at a common scale
(CS) $\mu_0 = 3$ GeV using the four-loop RG evolution equation in
Ref. <cit.>.
In Fig. <ref>, we present the raw data as the black circles
and CS data as blue diamonds as a function of the square of reduced
momentum $(a\wtd{p})^2$ at a fixed quark mass $(am = 0.0062)$.
$Z_q$ obtained from conserved vector current ($V \times S$)
at a fixed quark mass $(am = 0.0062)$. The black circles represent
raw data and blue diamonds are CS data at a CS $\mu_0=3\GeV$.
After converting the raw date to the CS data, we perform the fitting
with respect to quark masses at a fixed external momentum to the
following fitting function. We call this m-fit.
\begin{align}
f_{\text{m-fit}} = b_1 + b_2 \cdot am + b_3 \cdot (am)^2
\end{align}
The fitting results are presented in Table <ref> and the plot is
given in Fig. <ref>.
$b_1$ $b_2$ $b_3$ $\chi^2/\text{dof}$
0.84141(15) 0.0153(97) -0.31(17) 0.004(10)
m-fit results for $Z_q$ at $\mu_0 = 3\GeV$ for a fixed
external momentum $n=(3,3,3,7)$.
We take $b_1$ as the chiral limit values which are function of
external momentum $(a\wtd{p})^2$.
After m-fit, we fit $b_1$ to the following fitting function. We call
this p-fit.
\begin{align}
\label{eq:Z_q:p-fit}
f_{\text{p-fit}} = c_1 + c_2 (a\wtd{p})^2
+ c_3 \cdot ((a\wtd{p})^2)^2 + c_4 \cdot (a\wtd{p})^4
\end{align}
The fitting results are presented in Table <ref> and the
plot is presented in Fig. <ref>.
$c_1$ $c_2$ $c_3$ $c_4$ $\chi^2/\text{dof}$
1.0567(11) -0.1452(10) 0.00294(14) 0.0082(11) 0.13(26)
P-fit results for $Z_q$ at $\mu_0 = 3\GeV$.
The $\mathcal{O}((a\wtd{p})^2)$ and higher order terms correspond to
lattice artifacts.
Hence, we take $c_1$ as $Z_q$ value in RI-MOM scheme at $\mu_0 =
Using the four-loop RG running formula <cit.>, we convert the $Z_q$ from the RI-MOM scheme to the
$\MSb$ scheme.
[m-fit for $Z_q$]
[p-fit for $Z_q$]
sfig:Z_q:m-fit m-fit results for $Z_q$ at a reduced
momentum $n=(3,3,3,7)$ and sfig:Z_q:p-fit p-fit results for
$y_q$. Here, we use the conserved vector current at $\mu_0=3\GeV$.
$y_q \equiv Z_q(\mu_0,am=0) - \langle c_4 \rangle (a\wtd{p})^4$.
The blue circles are used for fitting.
We estimate the systematic error in two different ways.
One systematic error comes from truncation of four-loop RG running
factor which is used to convert the $Z_q$ from the RI-MOM scheme to
the $\MSb$ scheme.
Hence, we take five-loop uncertainty ($\sim\mathcal{O}(\alpha_s^4)$)
and define $E_t$ as follows.
\begin{align}
E_t = Z_q^{\text{RI-MOM}} \cdot (\alpha_s)^4
\end{align}
The other systematic error comes from the difference between the
conserved vector and axial currents.
Theoretically, $Z_q$ obtained from the conserved vector and axial
currents must be identical to each other. However, they are not same
in our study. Hence, we take the difference of them as the systematic
error and define $E_{\Delta}$ as follows.
\begin{align}
E_{\Delta} = |Z_q(V \otimes S) - Z_q(A \otimes P)|
\end{align}
The total error ($E_\text{tot}$) is obtained adding the statistical
error ($E_\text{stat}$) and the systematic errors in quadrature.
We present the final result of $Z_q$ in $\MSb$ scheme at $\mu_0 =
3\GeV$ and its statistical and systematic errors in Table
$Z_q^{\MSb}(\mu_0)$ $E_\text{stat}$ $E_t$ $E_{\Delta}$
1.0494 0.0011 0.0038 0.0099 0.0107
$Z_q$ in the $\MSb$ scheme at $\mu_0=3\GeV$ with statistical
and systematic errors.
§.§ Quark Mass Renormalization Factor $Z_m$
Quark mass renormalization factor $Z_m$ is obtained from the bilinear
operator $[S \otimes S]$.
Here, we use the same analysis method as in Ref. <cit.>.
Note that we analyse $Z_q \cdot Z_m$ instead of $Z_m$ directly.
After we obtain the $Z_q \cdot Z_m$ in RI-MOM scheme at $\mu_0=3\GeV$
through m-fit and p-fit, we divide by $Z_q$ obtained from the
conserved vector current.
First, we convert raw data to the CS data using the four-loop RG
running formula for $Z_q \cdot Z_m$.
We present the raw and CS data for $Z_q \cdot Z_m$ in
Fig. <ref>.
$Z_q \cdot Z_m$ obtained from $[S \times S]$ bilinear
operator at $am = 0.0062$. Here, $\mu_0 = 3$ GeV.
Using the CS data for $Z_q \cdot Z_m$, we carry out m-fit and p-fit.
The fitting functions for m-fit and p-fit are
\begin{align}
g_{\text{m-fit}} &= d_1 + d_2 \cdot am + d_3 \cdot \frac{1}{(am)^2}\\
\label{eq:Z_m:p-fit}
g_{\text{p-fit}} &= h_1 + h_2 (a\wtd{p})^2 + h_3 \cdot ((a\wtd{p})^2)^2 + h_4 \cdot (a\wtd{p})^4
\,.
\end{align}
We present fitting results of m-fit in Table <ref>
and in Fig. <ref> sfig:SxS_mass.
We show fitting results of p-fit in Table <ref> and in
Fig. <ref> sfig:SxS_mom.
$d_1$ $d_2$ $d_3$ $\chi^2/\text{dof}$
1.25664(60) -0.354(15) -0.000000019(51) 0.017(25)
Fitting results of $Z_q \cdot Z_m$ for m-fit. The reduced
momentum is fixed to $n=(3,4,3,9)$.
$h_1$ $h_2$ $h_3$ $h_4$ $\chi^2/\text{dof}$
1.3069(39) -0.0459(35) 0.00191(62) 0.0308(45) 0.37(32)
Fitting results of $Z_q \cdot Z_m$ for p-fit.
Fitting results of $Z_q \cdot Z_m$ for
sfig:SxS_mass m-fit and sfig:SxS_mom p-fit. For
the m-fit, the reduced momentum is fixed to $n=(3,4,3,9)$. For the
p-fit, $y_m \equiv (Z_q \cdot Z_m)(\mu_0, am=0) - \langle h_4
\rangle (a\wtd{p})^4$. The blue circle data are used for fitting.
$Z_m^{\MSb}(\mu_0)$ $E_{stat}$ $E_t$ $E_{\Delta}$ $E_{tot}$
1.0117 0.0032 0.0044 0.0005 0.0055
$Z_m$ in $\MSb$ scheme at $\mu_0=3\GeV$.
We determine $Z_m$ by dividing $Z_q \cdot Z_m$ by $Z_q$ obtained using
the conserved vector current.
Then, we convert $Z_m$ in the RI-MOM scheme into that in the $\MSb$ scheme
using the four-loop RG evolution formula.
\begin{align}
Z_m^{\MSb}(\mu_0) = U(\infty \to \mu_0, \MSb) \;
U(\mu_0 \to \infty, \text{RI-MOM}) \;
Z_m^\text{RI-MOM} (\mu_0) \,,
\end{align}
where $U(\mu_1 \to \mu_2, R)$ is the RG evolution matrix from the scale
$\mu_1$ to $\mu_2$ in the $R$ scheme.
The results are summarized in Table <ref>.
Here, the systematic errors are estimated in the same way as in $Z_q$.
§.§ Comparison of $Z_q$ and $Z_m$ between coarse and fine lattices
Now, we present the comparison of the $Z_q$ and $Z_m$ results at $\mu_0=3\GeV$
between coarse and fine lattices in Fig. <ref> and
Fig. <ref>.
[final results]
Comparison of $Z_q(\mu_0)$ on the coarse and fine lattices.
The red (blue) data represent results on the coarse (fine) lattice.
The results of $Z_q$ and $Z_m$ at $\mu_0=3\GeV$ on coarse and fine
lattices are presented in Table <ref>.
We find that the total errors of $Z_q^{\MSb}$ and $Z_m^{\MSb}$ on fine
lattice are reduced dramatically compared with those of coarse
coarse fine
$Z_q^{\MSb}$(3GeV) 1.0566(59)(231) 1.0494(11)(99)
$Z_m^{\MSb}$(3GeV) 0.865(21)(25) 1.0117(32)(44)
The comparison of $Z_q$ and $Z_m$ at $\mu_0 = 3\GeV$ between
coarse and fine lattices. The first error is statistical and the
second is systematic.
[final results]
Comparison of $Z_m(\mu_0)$ on the coarse and fine lattices.
The red (blue) data represent results on the coarse (fine) lattice.
§ CONCLUSION
Here, we present the results of the wave function renormalization
factor $Z_q$ and mass renormalization factor $Z_m$ for the staggered
bilinear operators defined in the $\MSb$ scheme at $\mu_0 = 3$ GeV.
We use the NPR method in the RI-MOM scheme as an intermediate scheme.
We use one of the MILC asqtad fine ($a\approx 0.09$ fm) ensembles to
calculate the matching factors.
By comparing results with those on the coarse ensembles, we find that
the statistical and systematic errors of $Z_q$ and $Z_m$ are reduced
dramatically on the fine lattice.
We plan to extend the calculation to the superfine ($a\approx
0.06$ fm) and ultrafine ($a\approx 0.045$ fm) ensembles in the
As a consequence, we will study on the scalability of $Z_q$ and $Z_m$.
We also plan to calculate the renormalization factors on the fine
ensembles with different sea quark masses, which will help us to
understand their dependence on sea quark masses.
J. Kim is supported by Young Scientists Fellowship through National
Research Council of Science & Technology (NST) of KOREA.
The research of W. Lee is supported by the Creative Research
Initiatives Program (No. 2015001776) of the NRF grant funded by the
Korean government (MEST).
W. Lee would like to acknowledge the support from the KISTI
supercomputing center through the strategic support program for the
supercomputing application research (No. KSC-2014-G3-002) with much
Computations were carried out on the DAVID GPU clusters at Seoul
National University.
|
1511.00167
|
Kaya et al.]Nucleation and high-density packing of $360^{\circ}$ domain walls on planar ferromagnetic nanowires by using circular magnetic fields
Department of Physics, Mount Holyoke College, South Hadley, MA, 01075, USA
Department of Physics, University of Massachusetts, Amherst, MA, 01003, USA
Department of Physics, Mount Holyoke College, South Hadley, MA, 01075, USA
We propose a mechanism for nucleation and high-density packing of $360^{\circ}$ domain walls (DWs) on planar ferromagnetic nanowires, of $100$ nm width, by using circular magnetic fields. The extent of the stray field from a $360^{\circ}$ DW is limited in comparison to $180^{\circ}$ DWs, which allows them to be packed more densely than $180^{\circ}$ DWs in a potential data storage device. We use micromagnetic simulations to demonstrate high-density packing of $360^{\circ}$ DWs, using a series of rectangular $16\times 16$ nm$^{2}$ notches to act as local pinning sites on the nanowires. For these notches, the minimum spacing between the DWs is $240$ nm, corresponding to a $360^{\circ}$ DW packing density of $4$ DWs per micron. Understanding the topological properties of the $360^{\circ}$ DWs allows us to understand their formation and annihilation in the proposed geometry. Adjacent $360^{\circ}$ DWs have opposite circulation, and closer spacing result in the adjacent walls breaking into $180^{\circ}$ DWs and annihilating.
Manipulating magnetic domain walls (DWs) in patterned ferromagnetic nanostructures and understanding their behavior are necessary to achieve proposed logic<cit.> and data storage devices.<cit.> Racetrack memory proposes the use of current driven transverse $180^{\circ}$ DWs, which interact over a range of about $2.5$ $\mu$m.<cit.> In contrast, $360^{\circ}$ DWs form an almost closed flux magnetic state, substantially reducing the interaction between neighboring DWs. For this reason, $360^{\circ}$ DWs have been proposed to serve as bits for data storage in a magnetic racetrack device.<cit.> A $360^{\circ}$ DW can be viewed as consisting of two $180^{\circ}$ DWs, and whether bringing together two transverse $180^{\circ}$ DWs results in annihilation or a $360^{\circ}$ DW depends on the topological edge charges of the $180^{\circ}$ DWs.<cit.> Current driven motion of $360^{\circ}$ DWs is predicted to be different from $180^{\circ}$ DWs,<cit.> but experimental confirmation has been challenging. Reliable nucleation and manipulation mechanisms are needed to study the properties of the $360^{\circ}$ DWs and to develop devices. Most proposals involve an injection pad with a rotating in-plane field,<cit.> with the exception of Gonzalez Oyarce et al.<cit.> Here, we propose a versatile technique to controllably nucleate $360^{\circ}$ DWs at arbitrary locations using a circular field centered in close proximity to a planar nanowire, allowing for the study of $360^{\circ}$ DWs in a wire and the potential to develop novel storage devices.
We perform micromagnetic simulations using the OOMMF<cit.> package to iteratively solve the Landau-Lifshitz-Gilbert equation. The nanowire dimensions used in the simulations are $10000\times 100$ nm$^{2}$ and the material parameters are for permalloy: $M_{s}=8\times 10^{5}$ A/m, $A=1.3\times 10^{-11}$ J/m. The cell size is $4$ nm along the three axes, there is no crystalline anisotropy, the damping parameter is 0.5, and simulations are run at $0$ K. The magnetization state evolves until structures reach an equilibrium state where $\frac{d\textbf{M}}{dt}<0.1$ deg/ns.
(a) Initialization of the nanowire and top-down view of the circular field. (b) Snapshot showing the nucleation of a $360^{\circ}$ DW and two $180^{\circ}$ DWs. (c) Relaxed state of a single $360^{\circ}$ DW. The color scale in (b) and (c) indicate the orientation of the moments along the $x$-axis: Red points to the right, blue to the left. Green and blue arrows help identify the topological winding of the DWs.
Figure <ref>a shows the mechanism for nucleating a $360^{\circ}$ DW. We initialize the nanowire by magnetizing it along the negative $x$-axis with a large in-plane magnetic field of $160~mT$ or higher. We apply a circular field, simulated as if from a current in an infinitely long wire that flows into the page, which produces a field that decreases as $1/r$, where $r$ is the distance from the center of the field. A current of $21$ mA, which is located at a distance of $r = 48$ nm from the nanowire along the positive $y$-axis, corresponds to a field of $87.5$ mT at the top edge of the wire. The circular field exerts a torque on the magnetic moments, whereby it nucleates a $360^{\circ}$ DW in the nanowire directly below the center of the circular field (Fig. <ref>b). Two $180^{\circ}$ DWs are created on either side of the $360^{\circ}$ DW. The moments directly below the center of the magnetic field experience the smallest torque (theoretically zero in a perfect structure at zero temperature, since they are aligned opposite to the applied field), while the other moments feel stronger torques to align with the field. Such circular fields can be experimentally implemented via the tip of an Atomic Force Microscope (AFM),<cit.>
which can be positioned at arbitrary locations to follow the pattern of fields described in this paper. For a scalable device, the procedure would presumably be realized by fabricating wires above each notch and by passing current through those wires perpendicular to the plane of the nanowire.
The simultaneous nucleation of two $180^{\circ}$ DWs on either side of the $360^{\circ}$ DW is a topological consequence, and is described in Bickel et al. for rings.<cit.> We characterize the $180^{\circ}$ DWs as “up" or “down" as conveniently revealed by whether the moments at their center are pointing in positive or negative y, indicated by the green or blue arrows in Figure $1b$. We use the same terminology for the $360^{\circ}$ DWs, which can be “up-down" or “down-up" depending on the constituent $180^{\circ}$ DWs (read from left to right). Figure <ref>b is a snapshot in time during the nucleation of a $360^{\circ}$ DW, while the circular field is still applied. The half integer winding numbers of the topological edge charges<cit.> are indicated as well. At the nucleation of the $360^{\circ}$ DW, two topological defects with charge $-1/2$ are created on the top edge of the nanowire (Fig. <ref>b), and two $+1/2$ charges are created at the bottom. Two switched (red) domains appear on either side of the $360^{\circ}$ DW, aligning with the applied field. Two $180^{\circ}$ DWs must also be created (at the farther edge of the switched domain), and these must carry the opposite topological charges, $+1/2$ on the top and $-1/2$ on the bottom. The total winding number of the wire is zero, as required.
Given our CCW field and the resulting down-up $360^{\circ}$ DW, the $180^{\circ}$ DW that emerges to the right of the $360^{\circ}$ DW is an up $180^{\circ}$ DW, while the one to the left is a down $180^{\circ}$ DW. If a down $180^{\circ}$ DW joins with another down $180^{\circ}$ DW, the topological edge charges sum up to zero on the top and the bottom, hence the DWs annihilate. Similarly, the joining of two up $180^{\circ}$ DWs results in annihilation.
When the applied circular field is removed, the wire in Figure <ref> relaxes to the state shown in Figure <ref>c. The $180^{\circ}$ DWs are pushed to the side until they encounter the end of the wire and annihilate. This is generally not the case for longer wires in which the $360^{\circ}$ DW slides towards one of the $180^{\circ}$ DWs and eventually annihilates into a single $180^{\circ}$ DW as a result of the summation of the topological charges.
The sequence of steps for packing $360^{\circ}$ DWs of opposite circulation at adjacent notches. Red doted lines indicate the center of the CCW circular field. (a) Initialization. (b) Resulting state after applying $21$ mA above notch I. (c) Resulting state after applying $21$ mA above notch III, creating a second $360^{\circ}$ DW directly below, and two $180^{\circ}$ DWs, one of which joins the $180^{\circ}$ DW at notch II to form a $360^{\circ}$ DW. (d) Resulting state after applying $21$ mA above notch V.
In order to pin $360^{\circ}$ DWs on the nanowire, a series of rectangular notches of $16\times 16$ nm$^{2}$ are introduced with an inter-notch distance of $280$ nm (Fig. <ref>). The length ($y$-axis) of the DW is reduced at the notches, thereby reducing the energy of the DWs and facilitating pinning at the notches. Figure $2$ demonstrates the sequence of steps required to generate a series of $360^{\circ}$ DWs with opposite circulation at adjacent notches. Once the nanowire is saturated along the negative $x$-axis as shown in Figure <ref>a, the first CW $360^{\circ}$ DW is nucleated at notch I by passing a current of $21$ mA vertically above the notch. As a result, an up $180^{\circ}$ DW pins at notch II, while the down $180^{\circ}$ DW slides to the end of the wire and annihilates due to the field gradient at the edge. The second $360^{\circ}$ DW at notch III is injected by following the same procedure. The simultaneously nucleated down $180^{\circ}$ DW to the left pairs with the up $180^{\circ}$ DW that was earlier nucleated and pinned at notch II, forming a CCW $360^{\circ}$ DW as shown in Figure <ref>c. Similarly, the circular field is positioned at notch V to nucleate the CW $360^{\circ}$ DW at V and form the CCW $360^{\circ}$ DW at IV. The magnetization circulation of the packed domain walls at notches I to V alternate between CW and CCW circulation, as shown in Figure <ref>d.
We have successfully simulated packing of $360^{\circ}$ DWs at adjacent notches with $260$ nm and $240$ nm inter-notch distances. As the notches are spaced more closely, the field strength is higher at notches adjacent to where the $360^{\circ}$ DW is nucleated. The $180^{\circ}$ DWs do not pin at the adjacent notch but are instead pinned two notches away. The second nucleated $360^{\circ}$ DW must also be formed an additional notch away. It is straightforward to push these nucleated DWs to neighboring notches, by effectively unravelling the $360^{\circ}$ into two $180^{\circ}$ DWs with the correct strength field, and then pushing the $180^{\circ}$ DWs with an appropriate strength field.
Failure mechanism when packing opposite circulation $360^{\circ}$ DWs at the inter-notch distance of $220$ nm. (a) $360^{\circ}$ DWs are formed at notches I and IV. The $180^{\circ}$ DWs interact but do not come together at a single notch. (b) Temporarily applying a CCW field above notch II pushes the two $180^{\circ}$ DWs into a tight $360^{\circ}$ DW at notch III. (c) When the field is removed, the constituent down DWs are sufficiently close to interact and annihilate, leaving the two up DWs at their respective notches.
The packing collapses if the distance between adjacent notches is $\leq 220$ nm. Figure <ref> shows why the previously described procedure fails when the notches are too close together, at $220$ nm. We first nucleate the down-up $360^{\circ}$ DW at I, creating an up $180^{\circ}$ DW at II. We nucleate the second $360^{\circ}$ DW at notch IV, as the $180^{\circ}$ DW at notch II is too close to notch III and prevents the nucleation of a down $180^{\circ}$ DW to the left of notch III. We instead nucleate the down-up $360^{\circ}$ DW at notch IV, and see that the down $180^{\circ}$ DW moves to notch III and the $180^{\circ}$ DWs at II and III are interacting, shown in Figure <ref>a. Figure <ref>b shows that by applying a CCW field at notch III, we can temporarily form a tight $360^{\circ}$ DW pinned at notch III. However, once the field is removed (Fig. <ref>c), the $360^{\circ}$ DWs at notch III and IV annihilate one another due to their topological charges. Effectively, the two down constituent $180^{\circ}$ DWs are adjacent, attract each other, and annihilate. The two up $180^{\circ}$ DWs remain at notches III and IV. Therefore, a distance of $\leq 220$ nm between notches prevents packing of $360^{\circ}$ DWs at adjacent notches on a nanowire, using this technique and geometry.
The minimum spacing between $360^{\circ}$ DWs is determined in part by the notch size and shape. Deeper notches allow closer packing but require stronger fields to de-pin the DWs. We have succeeded in simulating a dense packing of $360^{\circ}$ DWs at adjacent notches with $220$ nm spacing by using $16\times 32$ nm$^{2}$ rectangular notches. The procedure in this case differs slightly due to the stronger pinning of $180^{\circ}$ and $360^{\circ}$ DWs at deeper notches. Additionally, if we control the topology of the adjacent $360^{\circ}$ DWs so that they are all of the same circulation, the failure mechanism changes and we can pack the $360^{\circ}$ DWs more densely. This can be accomplished by annihilating the DW with the circulation that we do not want by using a strong local field above that DW. For example, a strong enough ($85$ mA) CCW field at notch I in Figure <ref>a annihilates the $360^{\circ}$ DW pinned at I. We can then shift the other $360^{\circ}$ DWs by unravelling them into two $180^{\circ}$ DWs and pushing the $180^{\circ}$ DWs. For $16\times 64$ nm$^{2}$ rectangular notches, we can successfully pack $360^{\circ}$ DWs with the opposite circulation at $180$ nm spacing between the notches. More work remains to be done to better understand the effects of the geometry of the notches and the circulation of adjacent $360^{\circ}$ DWs and their effects on the packing density.<cit.>
While these notched wires allow us to study the behavior of $360^{\circ}$ DWs, and using the tip of an AFM to manipulate the DWs provides flexibility in our experiments, a realistic device would have prefabricated wires positioned above each notch where we center the circular field in our simulations. The presence or absence of the $360^{\circ}$ DW could be used as the bit, or possibly the circulation of the $360^{\circ}$ DW. Geometry would be optimized to reduce the current density and power consumption while maintaining a close packing density. The readout might be similar to racetrack memory,<cit.> requiring a spin-polarized current to move the $360^{\circ}$ DWs. Generally, there will be a trade-off between strong pinning providing closer packing, and weak pinning requiring smaller fields and current to move the DWs.
In summary, we propose a mechanism to nucleate $360^{\circ}$ DWs at arbitrary locations determined by notches along an in-plane ferromagnetic nanowire. A circular field that decreases as $1/r$ and is centered directly above a notch along the y-axis will nucleate one $360^{\circ}$ DW and two $180^{\circ}$ DWs at that notch. Careful consideration of the series of circular fields allows us to nucleate $360^{\circ}$ DWs with opposite circulation at adjacent notches as close as $240$ nm, providing a packing density of about four DWs per micron in the permalloy nanowire simulated with $16\times 16$ nm$^{2}$ rectangular notches.
The authors acknowledged the support by NSF grants No. DMR 1208042 and 1207924. Simulations were performed with the computing facilities provided by the Center for Nanoscale Systems (CNS) at Harvard University (NSF award ECS-0335765), a member of the National Nanotechnology Infrastructure Network (NNIN).
|
1511.00561
|
We present a novel and practical deep fully convolutional neural network architecture for semantic pixel-wise segmentation termed SegNet. This core trainable segmentation engine consists of an encoder network, a corresponding decoder network followed by a pixel-wise classification layer. The architecture of the encoder network is topologically identical to the 13 convolutional layers in the VGG16 network <cit.>. The role of the decoder network is to map the low resolution encoder feature maps to full input resolution feature maps for pixel-wise classification. The novelty of SegNet lies is in the manner in which the decoder upsamples its lower resolution input feature map(s). Specifically, the decoder uses pooling indices computed in the max-pooling step of the corresponding encoder to perform non-linear upsampling. This eliminates the need for learning to upsample. The upsampled maps are sparse and are then convolved with trainable filters to produce dense feature maps. We compare our proposed architecture with the widely adopted FCN <cit.> and also with the well known DeepLab-LargeFOV <cit.>, DeconvNet <cit.> architectures. This comparison reveals the memory versus accuracy trade-off involved in achieving good segmentation performance.
SegNet was primarily motivated by scene understanding applications. Hence, it is designed to be efficient both in terms of memory and computational time during inference. It is also significantly smaller in the number of trainable parameters than other competing architectures and can be trained end-to-end using stochastic gradient descent. We also performed a controlled benchmark of SegNet and other architectures on both road scenes and SUN RGB-D indoor scene segmentation tasks. These quantitative assessments show that SegNet provides good performance with competitive inference time and most efficient inference memory-wise as compared to other architectures. We also provide a Caffe implementation of SegNet and a web demo at <http://mi.eng.cam.ac.uk/projects/segnet/>.
Deep Convolutional Neural Networks, Semantic Pixel-Wise Segmentation, Indoor Scenes, Road Scenes, Encoder, Decoder, Pooling, Upsampling.
§ INTRODUCTION
SegNet predictions on road scenes and indoor scenes. To try our system yourself, please see our online web demo at <http://mi.eng.cam.ac.uk/projects/segnet/>.
Semantic segmentation has a wide array of applications ranging from scene understanding, inferring support-relationships among objects to autonomous driving. Early methods that relied on low-level vision cues have fast been superseded by popular machine learning algorithms. In particular, deep learning has seen huge success lately in handwritten digit recognition, speech, categorising whole images and detecting objects in images <cit.>. Now there is an active interest for semantic pixel-wise labelling <cit.> <cit.>, <cit.>,<cit.>,<cit.>,<cit.>,<cit.>,<cit.>,<cit.>, <cit.>, <cit.>,<cit.>, <cit.>.
However, some of these recent approaches have tried to directly adopt deep architectures designed for category prediction to pixel-wise labelling <cit.>. The results, although very encouraging, appear coarse <cit.>. This is primarily because max pooling and sub-sampling reduce feature map resolution. Our motivation to design SegNet arises from this need to map low resolution features to input resolution for pixel-wise classification. This mapping must produce features which are useful for accurate boundary localization.
Our architecture, SegNet, is designed to be an efficient architecture for pixel-wise semantic segmentation. It is primarily motivated by road scene understanding applications which require the ability to model appearance (road, building), shape (cars, pedestrians) and understand the spatial-relationship (context) between different classes such as road and side-walk. In typical road scenes, the majority of the pixels belong to large classes such as road, building and hence the network must produce smooth segmentations. The engine must also have the ability to delineate objects based on their shape despite their small size. Hence it is important to retain boundary information in the extracted image representation. From a computational perspective, it is necessary for the network to be efficient in terms of both memory and computation time during inference. The ability to train end-to-end in order to jointly optimise all the weights in the network using an efficient weight update technique such as stochastic gradient descent (SGD) <cit.> is an additional benefit since it is more easily repeatable. The design of SegNet arose from a need to match these criteria.
The encoder network in SegNet is topologically identical to the convolutional layers in VGG16 <cit.>. We remove the fully connected layers of VGG16 which makes the SegNet encoder network significantly smaller and easier to train than many other recent architectures <cit.>. The key component of SegNet is the decoder network which consists of a hierarchy of decoders one corresponding to each encoder. Of these, the appropriate decoders use the max-pooling indices received from the corresponding encoder to perform non-linear upsampling of their input feature maps. This idea was inspired from an architecture designed for unsupervised feature learning <cit.>. Reusing max-pooling indices in the decoding process has several practical advantages; (i) it improves boundary delineation , (ii) it reduces the number of parameters enabling end-to-end training, and (iii) this form of upsampling can be incorporated into any encoder-decoder architecture such as <cit.> with only a little modification.
One of the main contributions of this paper is our analysis of the SegNet decoding technique and the widely used Fully Convolutional Network (FCN) <cit.>. This is in order to convey the practical trade-offs involved in designing segmentation architectures. Most recent deep architectures for segmentation have identical encoder networks, i.e VGG16, but differ in the form of the decoder network, training and inference. Another common feature is they have trainable parameters in the order of hundreds of millions and thus encounter difficulties in performing end-to-end training <cit.>. The difficulty of training these networks has led to multi-stage training <cit.>, appending networks to a pre-trained architecture such as FCN <cit.>, use of supporting aids such as region proposals for inference <cit.>, disjoint training of classification and segmentation networks <cit.> and use of additional training data for pre-training <cit.> <cit.> or for full training <cit.>. In addition, performance boosting post-processing techniques <cit.> have also been popular. Although all these factors improve performance on challenging benchmarks <cit.>, it is unfortunately difficult from their quantitative results to disentangle the key design factors necessary to achieve good performance. We therefore analysed the decoding process used in some of these approaches <cit.> and reveal their pros and cons.
We evaluate the performance of SegNet on two scene segmentation tasks, CamVid road scene segmentation <cit.> and SUN RGB-D indoor scene segmentation <cit.>. Pascal VOC12 <cit.> has been the benchmark challenge for segmentation over the years. However, the majority of this task has one or two foreground classes surrounded by a highly varied background. This implicitly favours techniques used for detection as shown by the recent work on a decoupled classification-segmentation network <cit.> where the classification network can be trained with a large set of weakly labelled data and the independent segmentation network performance is improved. The method of <cit.> also use the feature maps of the classification network with an independent CRF post-processing technique to perform segmentation. The performance can also be boosted by the use additional inference aids such as region proposals <cit.>, <cit.>. Therefore, it is different from scene understanding where the idea is to exploit co-occurrences of objects and other spatial-context to perform robust segmentation. To demonstrate the efficacy of SegNet, we present a real-time online demo of road scene segmentation into 11 classes of interest for autonomous driving (see link in Fig. <ref>). Some example test results produced on randomly sampled road scene images from Google and indoor test scenes from the SUN RGB-D dataset <cit.> are shown in Fig. <ref>.
The remainder of the paper is organized as follows. In Sec. <ref> we review related recent literature. We describe the SegNet architecture and its analysis in Sec. <ref>. In Sec. <ref> we evaluate the performance of SegNet on outdoor and indoor scene datasets. This is followed by a general discussion regarding our approach with pointers to future work in Sec. <ref>. We conclude in Sec. <ref>.
§ LITERATURE REVIEW
Semantic pixel-wise segmentation is an active topic of research, fuelled by challenging datasets <cit.>. Before the arrival of deep networks, the best performing methods mostly relied on hand engineered features classifying pixels independently. Typically, a patch is fed into a classifier e.g. Random Forest <cit.> or Boosting <cit.> to predict the class probabilities of the center pixel. Features based on appearance <cit.> or SfM and appearance <cit.> have been explored for the CamVid road scene understanding test <cit.>. These per-pixel noisy predictions (often called unary terms) from the classifiers are then smoothed by using a pair-wise or higher order CRF <cit.> to improve the accuracy. More recent approaches have aimed to produce high quality unaries by trying to predict the labels for all the pixels in a patch as opposed to only the center pixel. This improves the results of Random Forest based unaries <cit.> but thin structured classes are classified poorly. Dense depth maps computed from the CamVid video have also been used as input for classification using Random Forests <cit.>. Another approach argues for the use of a combination of popular hand designed features and spatio-temporal super-pixelization to obtain higher accuracy <cit.>. The best performing technique on the CamVid test <cit.> addresses the imbalance among label frequencies by combining object detection outputs with classifier predictions in a CRF framework. The result of all these techniques indicate the need for improved features for classification.
Indoor RGBD pixel-wise semantic segmentation has also gained popularity since the release of the NYU dataset <cit.>. This dataset showed the usefulness of the depth channel to improve segmentation. Their approach used features such as RGB-SIFT, depth-SIFT and pixel location as input to a neural network classifier to predict pixel unaries. The noisy unaries are then smoothed using a CRF. Improvements were made using a richer feature set including LBP and region segmentation to obtain higher accuracy <cit.> followed by a CRF. In more recent work <cit.>, both class segmentation and support relationships are inferred together using a combination of RGB and depth based cues. Another approach focuses on real-time joint reconstruction and semantic segmentation, where Random Forests are used as the classifier <cit.>. Gupta et al. <cit.> use boundary detection and hierarchical grouping before performing category segmentation. The common attribute in all these approaches is the use of hand engineered features for classification of either RGB or RGBD images.
The success of deep convolutional neural networks for object classification has more recently led researchers to exploit their feature learning capabilities for structured prediction problems such as segmentation. There have also been attempts to apply networks designed for object categorization to segmentation, particularly by replicating the deepest layer features in blocks to match image dimensions <cit.>. However, the resulting classification is blocky <cit.>. Another approach using recurrent neural networks <cit.> merges several low resolution predictions to create input image resolution predictions. These techniques are already an improvement over hand engineered features <cit.> but their ability to delineate boundaries is poor.
Newer deep architectures <cit.> particularly designed for segmentation have advanced the state-of-the-art by learning to decode or map low resolution image representations to pixel-wise predictions. The encoder network which produces these low resolution representations in all of these architectures is the VGG16 classification network <cit.> which has $13$ convolutional layers and $3$ fully connected layers. This encoder network weights are typically pre-trained on the large ImageNet object classification dataset <cit.>. The decoder network varies between these architectures and is the part which is responsible for producing multi-dimensional features for each pixel for classification.
Each decoder in the Fully Convolutional Network (FCN) architecture <cit.> learns to upsample its input feature map(s) and combines them with the corresponding encoder feature map to produce the input to the next decoder. It is an architecture which has a large number of trainable parameters in the encoder network (134M) but a very small decoder network (0.5M). The overall large size of this network makes it hard to train end-to-end on a relevant task. Therefore, the authors use a stage-wise training process. Here each decoder in the decoder network is progressively added to an existing trained network. The network is grown until no further increase in performance is observed. This growth is stopped after three decoders thus ignoring high resolution feature maps can certainly lead to loss of edge information <cit.>. Apart from training related issues, the need to reuse the encoder feature maps in the decoder makes it memory intensive in test time. We study this network in more detail as it the core of other recent architectures <cit.>.
An illustration of the SegNet architecture. There are no fully connected layers and hence it is only convolutional. A decoder upsamples its input using the transferred pool indices from its encoder to produce a sparse feature map(s). It then performs convolution with a trainable filter bank to densify the feature map. The final decoder output feature maps are fed to a soft-max classifier for pixel-wise classification.
The predictive performance of FCN has been improved further by appending the FCN with a recurrent neural network (RNN) <cit.> and fine-tuning them on large datasets <cit.>,<cit.>. The RNN layers mimic the sharp boundary delineation capabilities of CRFs while exploiting the feature representation power of FCN's. They show a significant improvement over FCN-8 but also show that this difference is reduced when more training data is used to train FCN-8. The main advantage of the CRF-RNN is revealed when it is jointly trained with an architecture such as the FCN-8. The fact that joint training helps is also shown in other recent results <cit.>, <cit.>. Interestingly, the deconvolutional network <cit.> performs significantly better than FCN although at the cost of a more complex training and inference. This however raises the question as to whether the perceived advantage of the CRF-RNN would be reduced as the core feed-forward segmentation engine is made better. In any case, the CRF-RNN network can be appended to any deep segmentation architecture including SegNet.
Multi-scale deep architectures are also being pursued <cit.>. They come in two flavours, (i) those which use input images at a few scales and corresponding deep feature extraction networks, and (ii) those which combine feature maps from different layers of a single deep architecture <cit.> <cit.>. The common idea is to use features extracted at multiple scales to provide both local and global context <cit.> and the using feature maps of the early encoding layers retain more high frequency detail leading to sharper class boundaries. Some of these architectures are difficult to train due to their parameter size <cit.>. Thus a multi-stage training process is employed along with data augmentation. The inference is also expensive with multiple convolutional pathways for feature extraction. Others <cit.> append a CRF to their multi-scale network and jointly train them. However, these are not feed-forward at test time and require optimization to determine the MAP labels.
Several of the recently proposed deep architectures for segmentation are not feed-forward in inference time <cit.>, <cit.>, <cit.>. They require either MAP inference over a CRF <cit.>, <cit.> or aids such as region proposals <cit.> for inference. We believe the perceived performance increase obtained by using a CRF is due to the lack of good decoding techniques in their core feed-forward segmentation engine. SegNet on the other hand uses decoders to obtain features for accurate pixel-wise classification.
The recently proposed Deconvolutional Network <cit.> and its semi-supervised variant the Decoupled network <cit.> use the max locations of the encoder feature maps (pooling indices) to perform non-linear upsampling in the decoder network. The authors of these architectures, independently of SegNet (first submitted to CVPR 2015 <cit.>), proposed this idea of decoding in the decoder network. However, their encoder network consists of the fully connected layers from the VGG-16 network which consists of about $90\%$ of the parameters of their entire network. This makes training of their network very difficult and thus require additional aids such as the use of region proposals to enable training. Moreover, during inference these proposals are used and this increases inference time significantly. From a benchmarking point of view, this also makes it difficult to evaluate the performance of their architecture (encoder-decoder network) without other aids. In this work we discard the fully connected layers of the VGG16 encoder network which enables us to train the network using the relevant training set using SGD optimization. Another recent method <cit.> shows the benefit of reducing the number of parameters significantly without sacrificing performance, reducing memory consumption and improving inference time.
Our work was inspired by the unsupervised feature learning architecture proposed by Ranzato et al. <cit.>. The key learning module is an encoder-decoder network. An encoder consists of convolution with a filter bank, element-wise tanh non-linearity, max-pooling and sub-sampling to obtain the feature maps. For each sample, the indices of the max locations computed during pooling are stored and passed to the decoder. The decoder upsamples the feature maps by using the stored pooled indices. It convolves this upsampled map using a trainable decoder filter bank to reconstruct the input image. This architecture was used for unsupervised pre-training for classification. A somewhat similar decoding technique is used for visualizing trained convolutional networks<cit.> for classification. The architecture of Ranzato et al. mainly focused on layer-wise feature learning using small input patches. This was extended by Kavukcuoglu et. al. <cit.> to accept full image sizes as input to learn hierarchical encoders. Both these approaches however did not attempt to use deep encoder-decoder networks for unsupervised feature training as they discarded the decoders after each encoder training. Here, SegNet differs from these architectures as the deep encoder-decoder network is trained jointly for a supervised learning task and hence the decoders are an integral part of the network in test time.
Other applications where pixel wise predictions are made using deep networks are image super-resolution <cit.> and depth map prediction from a single image <cit.>. The authors in <cit.> discuss the need for learning to upsample from low resolution feature maps which is the central topic of this paper.
§ ARCHITECTURE
SegNet has an encoder network and a corresponding decoder network, followed by a final pixelwise classification layer. This architecture is illustrated in Fig. <ref>. The encoder network consists of $13$ convolutional layers which correspond to the first $13$ convolutional layers in the VGG16 network <cit.> designed for object classification. We can therefore initialize the training process from weights trained for classification on large datasets <cit.>. We can also discard the fully connected layers in favour of retaining higher resolution feature maps at the deepest encoder output. This also reduces the number of parameters in the SegNet encoder network significantly (from 134M to 14.7M) as compared to other recent architectures <cit.>, <cit.> (see. Table <ref>). Each encoder layer has a corresponding decoder layer and hence the decoder network has $13$ layers. The final decoder output is fed to a multi-class soft-max classifier to produce class probabilities for each pixel independently.
Each encoder in the encoder network performs convolution with a filter bank to produce a set of feature maps. These are then batch normalized <cit.>,<cit.>). Then an element-wise rectified-linear non-linearity (ReLU) $max(0,x)$ is applied. Following that, max-pooling with a $2\times2$ window and stride $2$ (non-overlapping window) is performed and the resulting output is sub-sampled by a factor of $2$. Max-pooling is used to achieve translation invariance over small spatial shifts in the input image. Sub-sampling results in a large input image context (spatial window) for each pixel in the feature map. While several layers of max-pooling and sub-sampling can achieve more translation invariance for robust classification correspondingly there is a loss of spatial resolution of the feature maps. The increasingly lossy (boundary detail) image representation is not beneficial for segmentation where boundary delineation is vital. Therefore, it is necessary to capture and store boundary information in the encoder feature maps before sub-sampling is performed. If memory during inference is not constrained, then all the encoder feature maps (after sub-sampling) can be stored. This is usually not the case in practical applications and hence we propose a more efficient way to store this information. It involves storing only the max-pooling indices, i.e, the locations of the maximum feature value in each pooling window is memorized for each encoder feature map. In principle, this can be done using 2 bits for each $2\times2$ pooling window and is thus much more efficient to store as compared to memorizing feature map(s) in float precision. As we show later in this work, this lower memory storage results in a slight loss of accuracy but is still suitable for practical applications.
The appropriate decoder in the decoder network upsamples its input feature map(s) using the memorized max-pooling indices from the corresponding encoder feature map(s). This step produces sparse feature map(s). This SegNet decoding technique is illustrated in Fig. <ref>. These feature maps are then convolved with a trainable decoder filter bank to produce dense feature maps. A batch normalization step is then applied to each of these maps. Note that the decoder corresponding to the first encoder (closest to the input image) produces a multi-channel feature map, although its encoder input has 3 channels (RGB). This is unlike the other decoders in the network which produce feature maps with the same number of size and channels as their encoder inputs. The high dimensional feature representation at the output of the final decoder is fed to a trainable soft-max classifier. This soft-max classifies each pixel independently. The output of the soft-max classifier is a K channel image of probabilities where K is the number of classes. The predicted segmentation corresponds to the class with maximum probability at each pixel.
We add here that two other architectures, DeconvNet <cit.> and U-Net <cit.> share a similar architecture to SegNet but with some differences. DeconvNet has a much larger parameterization, needs more computational resources and is harder to train end-to-end (Table <ref>), primarily due to the use of fully connected layers (albeit in a convolutional manner) We report several comparisons with DeconvNet later in the paper Sec. <ref>.
As compared to SegNet, U-Net <cit.> (proposed for the medical imaging community) does not reuse pooling indices but instead transfers the entire feature map (at the cost of more memory) to the corresponding decoders and concatenates them to upsampled (via deconvolution) decoder feature maps. There is no conv5 and max-pool 5 block in U-Net as in the VGG net architecture. SegNet, on the other hand, uses all of the pre-trained convolutional layer weights from VGG net as pre-trained weights.
An illustration of SegNet and FCN <cit.> decoders. $a,b,c,d$ correspond to values in a feature map. SegNet uses the max pooling indices to upsample (without learning) the feature map(s) and convolves with a trainable decoder filter bank. FCN upsamples by learning to deconvolve the input feature map and adds the corresponding encoder feature map to produce the decoder output. This feature map is the output of the max-pooling layer (includes sub-sampling) in the corresponding encoder. Note that there are no trainable decoder filters in FCN.
4c 7cMedian frequency balancing 7|cNatural frequency balancing
Storage Infer 4c|Test 3c|Train 4c|Test 3cTrain
Variant Params (M) multiplier time (ms) G C mIoU BF G C mIoU G C mIoU BF G C 1cmIoU
16cFixed upsampling
Bilinear-Interpolation 0.625 0 24.2 77.9 61.1 43.3 20.83 89.1 90.2 82.7 82.7 52.5 43.8 23.08 93.5 74.1 59.9
16cUpsampling using max-pooling indices
SegNet-Basic 1.425 1 52.6 82.7 62.0 47.7 35.78 94.7 96. 2 92.7 84.0 54.6 46.3 36.67 96.1 83.9 73.3
SegNet-Basic-EncoderAddition 1.425 64 53.0 83.4 63.6 48.5 35.92 94.3 95.8 92.0 84.2 56.5 47.7 36.27 95.3 80.9 68.9
SegNet-Basic-SingleChannelDecoder 0.625 1 33.1 81.2 60.7 46.1 31.62 93.2 94.8 90.3 83.5 53.9 45.2 32.45 92.6 68.4 52.8
16cLearning to upsample (bilinear initialisation)
FCN-Basic 0.65 11 24.2 81.7 62.4 47.3 38.11 92.8 93.6 88.1 83.9 55.6 45.0 37.33 92.0 66.8 50.7
FCN-Basic-NoAddition 0.65 n/a 23.8 80.5 58.6 44.1 31.96 92.5 93.0 87.2 82.3 53.9 44.2 29.43 93.1 72.8 57.6
FCN-Basic-NoDimReduction 1.625 64 44.8 84.1 63.4 50.1 37.37 95.1 96.5 93.2 83.5 57.3 47.0 37.13 97.2 91.7 84.8
FCN-Basic-NoAddition-NoDimReduction 1.625 0 43.9 80.5 61.6 45.9 30.47 92.5 94.6 89.9 83.7 54.8 45.5 33.17 95.0 80.2 67.8
Comparison of decoder variants. We quantify the performance using global (G), class average (C), mean of intersection over union (mIoU) and a semantic contour measure (BF). The testing and training accuracies are shown as percentages for both natural frequency and median frequency balanced training loss function. SegNet-Basic performs at the same level as FCN-Basic but requires only storing max-pooling indices and is therefore more memory efficient during inference. Note that the theoretical memory requirement reported is based only on the size of the first layer encoder feature map. FCN-Basic, SegNet-Basic, SegNet-Basic-EncoderAddition all have high BF scores indicating the need to use information in encoder feature maps for better class contour delineation. Networks with larger decoders and those using the encoder feature maps in full perform best, although they are least efficient in terms of inference time and memory.
§.§ Decoder Variants
Many segmentation architectures <cit.> share the same encoder network and they only vary in the form of their decoder network.
Of these we choose to compare the SegNet decoding technique with the widely used Fully Convolutional Network (FCN) decoding technique <cit.>.
In order to analyse SegNet and compare its performance with FCN (decoder variants) we use a smaller version of SegNet, termed SegNet-Basic [SegNet-Basic was earlier termed SegNet in a archival version of this paper <cit.>], which has 4 encoders and 4 decoders. All the encoders in SegNet-Basic perform max-pooling and sub-sampling and the corresponding decoders upsample its input using the received max-pooling indices. Batch normalization is used after each convolutional layer in both the encoder and decoder network. No biases are used after convolutions and no ReLU non-linearity is present in the decoder network. Further, a constant kernel size of $7\times7$ over all the encoder and decoder layers is chosen to provide a wide context for smooth labelling i.e. a pixel in the deepest layer feature map (layer $4$) can be traced back to a context window in the input image of $106\times106$ pixels. This small size of SegNet-Basic allows us to explore many different variants (decoders) and train them in reasonable time. Similarly we create FCN-Basic, a comparable version of FCN for our analysis which shares the same encoder network as SegNet-Basic but with the FCN decoding technique (see Fig. <ref>) used in all its decoders.
On the left in Fig. <ref> is the decoding technique used by SegNet (also SegNet-Basic), where there is no learning involved in the upsampling step. However, the upsampled maps are convolved with trainable multi-channel decoder filters to densify its sparse inputs. Each decoder filter has the same number of channels as the number of upsampled feature maps. A smaller variant is one where the decoder filters are single channel, i.e they only convolve their corresponding upsampled feature map. This variant (SegNet-Basic-SingleChannelDecoder) reduces the number of trainable parameters and inference time significantly.
On the right in Fig. <ref> is the FCN (also FCN-Basic) decoding technique. The important design element of the FCN model is dimensionality reduction step of the encoder feature maps. This compresses the encoder feature maps which are then used in the corresponding decoders. Dimensionality reduction of the encoder feature maps, say of 64 channels, is performed by convolving them with $1\times 1\times 64\times K$ trainable filters, where $K$ is the number of classes. The compressed $K$ channel final encoder layer feature maps are the input to the decoder network. In a decoder of this network, upsampling is performed by inverse convolution using a fixed or trainable multi-channel upsampling kernel. We set the kernel size to $8\times8$. This manner of upsampling is also termed as deconvolution. Note that, in comparison, SegNet the multi-channel convolution using trainable decoder filters is performed after upsampling to densifying feature maps. The upsampled feature map in FCN has $K$ channels. It is then added element-wise to the corresponding resolution encoder feature map to produce the output decoder feature map. The upsampling kernels are initialized using bilinear interpolation weights <cit.>.
The FCN decoder model requires storing encoder feature maps during inference. This can be memory intensive for embedded applications; for e.g. storing 64 feature maps of the first layer of FCN-Basic at $180\times240$ resolution in 32 bit floating point precision takes 11MB. This can be made smaller using dimensionality reduction to the 11 feature maps which requires $\approx$ 1.9MB storage. SegNet on the other hand requires almost negligible storage cost for the pooling indices ($.17$MB if stored using 2 bits per $2\times2$ pooling window). We can also create a variant of the FCN-Basic model which discards the encoder feature map addition step and only learns the upsampling kernels (FCN-Basic-NoAddition).
In addition to the above variants, we study upsampling using fixed bilinear interpolation weights which therefore requires no learning for upsampling (Bilinear-Interpolation). At the other extreme, we can add 64 encoder feature maps at each layer to the corresponding output feature maps from the SegNet decoder to create a more memory intensive variant of SegNet (SegNet-Basic-EncoderAddition).
Here both the pooling indices for upsampling are used, followed by a convolution step to densify its sparse input. This is then added element-wise to the corresponding encoder feature maps to produce a decoders output.
Another and more memory intensive FCN-Basic variant (FCN-Basic-NoDimReduction) is where there is no dimensionality reduction performed for the encoder feature maps. This implies that unlike FCN-Basic the final encoder feature map is not compressed to $K$ channels before passing it to the decoder network. Therefore, the number of channels at the end of each decoder is the same as the corresponding encoder (i.e $64$).
We also tried other generic variants where feature maps are simply upsampled by replication <cit.>, or by using a fixed (and sparse) array of indices for upsampling. These performed quite poorly in comparison to the above variants. A variant without max-pooling and sub-sampling in the encoder network (decoders are redundant) consumes more memory, takes longer to converge and performs poorly.
Finally, please note that to encourage reproduction of our results we release the Caffe implementation of all the variants [See < http://mi.eng.cam.ac.uk/projects/segnet/> for our SegNet code and web demo.].
§.§ Training
We use the CamVid road scenes dataset to benchmark the performance of the decoder variants. This dataset is small, consisting of 367 training and 233 testing RGB images (day and dusk scenes) at $360\times480$ resolution. The challenge is to segment $11$ classes such as road, building, cars, pedestrians, signs, poles, side-walk etc. We perform local contrast normalization <cit.> to the RGB input.
The encoder and decoder weights were all initialized using the technique described in He et al. <cit.>. To train all the variants we use stochastic gradient descent (SGD) with a fixed learning rate of 0.1 and momentum of 0.9 <cit.> using our Caffe implementation of SegNet-Basic <cit.>. We train the variants until the training loss converges. Before each epoch, the training set is shuffled and each mini-batch (12 images) is then picked in order thus ensuring that each image is used only once in an epoch. We select the model which performs highest on a validation dataset.
We use the cross-entropy loss <cit.> as the objective function for training the network. The loss is summed up over all the pixels in a mini-batch. When there is large variation in the number of pixels in each class in the training set (e.g road, sky and building pixels dominate the CamVid dataset) then there is a need to weight the loss differently based on the true class. This is termed class balancing. We use median frequency balancing <cit.> where the weight assigned to a class in the loss function is the ratio of the median of class frequencies computed on the entire training set divided by the class frequency. This implies that larger classes in the training set have a weight smaller than $1$ and the weights of the smallest classes are the highest. We also experimented with training the different variants without class balancing or equivalently using natural frequency balancing.
§.§ Analysis
To compare the quantitative performance of the different decoder variants, we use three commonly used performance measures: global accuracy (G) which measures the percentage of pixels correctly classified in the dataset, class average accuracy (C) is the mean of the predictive accuracy over all classes and mean intersection over union (mIoU) over all classes as used in the Pascal VOC12 challenge <cit.>. The mIoU metric is a more stringent metric than class average accuracy since it penalizes false positive predictions. However, mIoU metric is not optimized for directly through the class balanced cross-entropy loss.
The mIoU metric otherwise known as the Jacard Index is most commonly used in benchmarking. However, Csurka et al. <cit.> note that this metric does not always correspond to human qualitative judgements (ranks) of good quality segmentation. They show with examples that mIoU favours region smoothness and does not evaluate boundary accuracy, a point also alluded to recently by the authors of FCN <cit.>. Hence they propose to complement the mIoU metric with a boundary measure based on the Berkeley contour matching score commonly used to evaluate unsupervised image segmentation quality <cit.>. Csurka et al. <cit.> simply extend this to semantic segmentation and show that the measure of semantic contour accuracy used in conjunction with the mIoU metric agrees more with human ranking of segmentation outputs.
The key idea in computing a semantic contour score is to evaluate the F1-measure <cit.> which involves computing the precision and recall values between the predicted and ground truth class boundary given a pixel tolerance distance. We used a value of $0.75\%$ of the image diagonal as the tolerance distance. The F1-measure for each class that is present in the ground truth test image is averaged to produce an image F1-measure. Then we compute the whole test set average, denoted the boundary F1-measure (BF) by average the image F1 measures.
We test each architectural variant after each $1000$ iterations of optimization on the CamVid validation set until the training loss converges. With a training mini-batch size of 12 this corresponds to testing approximately every 33 epochs (passes) through the training set. We select the iteration wherein the global accuracy is highest amongst the evaluations on the validation set. We report all the three measures of performance at this point on the held-out CamVid test set. Although we use class balancing while training the variants, it is still important to achieve high global accuracy to result in an overall smooth segmentation. Another reason is that the contribution of segmentation towards autonomous driving is mainly for delineating classes such as roads, buildings, side-walk, sky. These classes dominate the majority of the pixels in an image and a high global accuracy corresponds to good segmentation of these important classes. We also observed that reporting the numerical performance when class average is highest can often correspond to low global accuracy indicating a perceptually noisy segmentation output.
In Table <ref> we report the numerical results of our analysis. We also show the size of the trainable parameters and the highest resolution feature map or pooling indices storage memory, i.e, of the first layer feature maps after max-pooling and sub-sampling. We show the average time for one forward pass with our Caffe implementation, averaged over $50$ measurements using a $360\times480$ input on an NVIDIA Titan GPU with cuDNN v3 acceleration. We note that the upsampling layers in the SegNet variants are not optimised using cuDNN acceleration. We show the results for both testing and training for all the variants at the selected iteration. The results are also tabulated without class balancing (natural frequency) for training and testing accuracies. Below we analyse the results with class balancing.
From the Table <ref>, we see that bilinear interpolation based upsampling without any learning performs the worst based on all the measures of accuracy. All the other methods which either use learning for upsampling (FCN-Basic and variants) or learning decoder filters after upsampling (SegNet-Basic and its variants) perform significantly better. This emphasizes the need to learn decoders for segmentation. This is also supported by experimental evidence gathered by other authors when comparing FCN with SegNet-type decoding techniques<cit.>.
When we compare SegNet-Basic and FCN-Basic we see that both perform equally well on this test over all the measures of accuracy. The difference is that SegNet uses less memory during inference since it only stores max-pooling indices. On the other hand FCN-Basic stores encoder feature maps in full which consumes much more memory (11 times more). SegNet-Basic has a decoder with 64 feature maps in each decoder layer. In comparison FCN-Basic, which uses dimensionality reduction, has fewer (11) feature maps in each decoder layer. This reduces the number of convolutions in the decoder network and hence FCN-Basic is faster during inference (forward pass). From another perspective, the decoder network in SegNet-Basic makes it overall a larger network than FCN-Basic. This endows it with more flexibility and hence achieves higher training accuracy than FCN-Basic for the same number of iterations. Overall we see that SegNet-Basic has an advantage over FCN-Basic when inference time memory is constrained but where inference time can be compromised to some extent.
SegNet-Basic is most similar to FCN-Basic-NoAddition in terms of their decoders, although the decoder of SegNet is larger. Both learn to produce dense feature maps, either directly by learning to perform deconvolution as in FCN-Basic-NoAddition or by first upsampling and then convolving with trained decoder filters.
The performance of SegNet-Basic is superior, in part due to its larger decoder size. The accuracy of FCN-Basic-NoAddition is also lower as compared to FCN-Basic. This shows that it is vital to capture the information present in the encoder feature maps for better performance. In particular, note the large drop in the BF measure between these two variants. This can also explain the part of the reason why SegNet-Basic outperforms FCN-Basic-NoAddition.
The size of the FCN-Basic-NoAddition-NoDimReduction model is slightly larger than SegNet-Basic since the final encoder feature maps are not compressed to match the number of classes $K$. This makes it a fair comparison in terms of the size of the model. The performance of this FCN variant is poorer than SegNet-Basic in test but also its training accuracy is lower for the same number of training epochs. This shows that using a larger decoder is not enough but it is also important to capture encoder feature map information to learn better, particular the fine grained contour information (notice the drop in the BF measure). Here it is also interesting to see that SegNet-Basic has a competitive training accuracy when compared to larger models such FCN-Basic-NoDimReduction.
Another interesting comparison between FCN-Basic-NoAddition and SegNet-Basic-SingleChannelDecoder shows that using max-pooling indices for upsampling and an overall larger decoder leads to better performance. This also lends evidence to SegNet being a good architecture for segmentation, particularly when there is a need to find a compromise between storage cost, accuracy versus inference time. In the best case, when both memory and inference time is not constrained, larger models such as FCN-Basic-NoDimReduction and SegNet-EncoderAddition are both more accurate than the other variants. Particularly, discarding dimensionality reduction in the FCN-Basic model leads to the best performance amongst the FCN-Basic variants with a high BF score. This once again emphasizes the trade-off involved between memory and accuracy in segmentation architectures.
The last two columns of Table <ref> show the result when no class balancing is used (natural frequency). Here, we can observe that without weighting the results are poorer for all the variants, particularly for class average accuracy and mIoU metric. The global accuracy is the highest without weighting since the majority of the scene is dominated by sky, road and building pixels. Apart from this all the inference from the comparative analysis of variants holds true for natural frequency balancing too, including the trends for the BF measure. SegNet-Basic performs as well as FCN-Basic and is better than the larger FCN-Basic-NoAddition-NoDimReduction. The bigger but less efficient models FCN-Basic-NoDimReduction and SegNet-EncoderAddition perform better than the other variants.
We can now summarize the above analysis with the following general points.
* The best performance is achieved when encoder feature maps are stored in full. This is reflected in the semantic contour delineation metric (BF) most clearly.
* When memory during inference is constrained, then compressed forms of encoder feature maps (dimensionality reduction, max-pooling indices) can be stored and used with an appropriate decoder (e.g. SegNet type) to improve performance.
* Larger decoders increase performance for a given encoder network.
§ BENCHMARKING
We quantify the performance of SegNet on two scene segmentation benchmarks using our Caffe implementation [ Our web demo and Caffe implementation is available for evaluation at <http://mi.eng.cam.ac.uk/projects/segnet/>]. The first task is road scene segmentation which is of current practical interest for various autonomous driving related problems. The second task is indoor scene segmentation which is of immediate interest to several augmented reality (AR) applications. The input RGB images for both tasks were $360\times480$.
We benchmarked SegNet against several other well adopted deep architectures for segmentation such as FCN <cit.>, DeepLab-LargFOV <cit.> and DeconvNet <cit.>. Our objective was to understand the performance of these architectures when trained end-to-end on the same datasets. To enable end-to-end training we added batch normalization <cit.> layers after each convolutional layer. For DeepLab-LargeFOV, we changed the max pooling 3 stride to 1 to achieve a final predictive resolution of $45\times60$. We restricted the feature size in the fully connnected layers of DeconvNet to $1024$ so as to enable training with the same batch size as other models. Here note that the authors of DeepLab-LargeFOV <cit.> have also reported little loss in performance by reducing the size of the fully connected layers.
In order to perform a controlled benchmark we used the same SGD solver <cit.> with a fixed learning rate of $10^{-3}$ and momentum of $0.9$. The optimization was performed for more than 100 epochs through the dataset until no further performance increase was observed. Dropout of $0.5$ was added to the end of deeper convolutional layers in all models to prevent overfitting (see <http://mi.eng.cam.ac.uk/projects/segnet/tutorial.html> for example caffe prototxt). For the road scenes which have $11$ classes we used a mini-batch size of $5$ and for indoor scenes with $37$ classes we used a mini-batch size of $4$.
Results on CamVid day and dusk test samples.
SegNet shows superior performance, particularly with its ability to delineate boundaries, as compared to some of the larger models when all are trained in a controlled setting. DeepLab-LargeFOV is the most efficient model and with CRF post-processing can produce competitive results although smaller classes are lost. FCN with learnt deconvolution is clearly better. DeconvNet is the largest model with the longest training time, but its predictions loose small classes. Note that these results correspond to the model corresponding to the highest mIoU accuracy in Table <ref>.
§.§ Road Scene Segmentation
A number of road scene datasets are available for semantic parsing <cit.>. Of these we choose to benchmark SegNet using the CamVid dataset <cit.> as it contains video sequences. This enables us to compare our proposed architecture with those which use motion and structure <cit.> and video segments <cit.>. We also combine <cit.> to form an ensemble of 3433 images to train SegNet for an additional benchmark. For a web demo (see footnote <ref>) of road scene segmentation, we include the CamVid test set to this larger dataset.
Here, we would like to note that another recent and independent segmentation benchmark on road scenes has been performed for SegNet and the other competing architectures used in this paper <cit.>. However, the benchmark was not controlled, meaning that each architecture was trained with a separate recipe with varying input resolutions and sometimes with a validation set included. Therefore, we believe our more controlled benchmark can be used to complement their efforts.
1cMethod 1c90Building 1c90Tree 1c90Sky 1c90Car 1c90Sign-Symbol 1c90Road 1c90Pedestrian 1c90Fence 1c90Column-Pole 1c90Side-walk 1c90Bicyclist 1c90Class avg. 1c90Global avg. 1c90mIoU 1c90BF
SfM+Appearance <cit.> 46.2 61.9 89.7 68.6 42.9 89.5 53.6 46.6 0.7 60.5 22.5 53.0 69.1 2cn/a$^{*}$
Boosting <cit.> 61.9 67.3 91.1 71.1 58.5 92.9 49.5 37.6 25.8 77.8 24.7 59.8 76.4 2cn/a$^{*}$
Dense Depth Maps <cit.> 85.3 57.3 95.4 69.2 46.5 98.5 23.8 44.3 22.0 38.1 28.7 55.4 82.1 2cn/a$^{*}$
Structured Random Forests <cit.> 11c|n/a 51.4 72.5 2cn/a$^{*}$
Neural Decision Forests <cit.> 11c|n/a 56.1 82.1 2cn/a$^{*}$
Local Label Descriptors <cit.> 80.7 61.5 88.8 16.4 n/a 98.0 1.09 0.05 4.13 12.4 0.07 36.3 73.6 2cn/a$^{*}$
Super Parsing <cit.> 87.0 67.1 96.9 62.7 30.1 95.9 14.7 17.9 1.7 70.0 19.4 51.2 83.3 2cn/a$^{*}$
SegNet (3.5K dataset training - 140K) 89.6 83.4 96.1 87.7 52.7 96.4 62.2 53.45 32.1 93.3 36.5 71.20 90.40 60.10 46.84
15cCRF based approaches
Boosting + pairwise CRF <cit.> 70.7 70.8 94.7 74.4 55.9 94.1 45.7 37.2 13.0 79.3 23.1 59.9 79.8 2cn/a$^{*}$
Boosting+Higher order <cit.> 84.5 72.6 97.5 72.7 34.1 95.3 34.2 45.7 8.1 77.6 28.5 59.2 83.8 2cn/a$^{*}$
Boosting+Detectors+CRF <cit.> 81.5 76.6 96.2 78.7 40.2 93.9 43.0 47.6 14.3 81.5 33.9 62.5 83.8 2cn/a$^{*}$
Quantitative comparisons of SegNet with traditional methods on the CamVid 11 road class segmentation problem <cit.>. SegNet outperforms all the other methods, including those using depth, video and/or CRF's on the majority of classes. In comparison with the CRF based methods SegNet predictions are more accurate in 8 out of the 11 classes. It also shows a good $\approx 10\%$ improvement in class average accuracy when trained on a large dataset of 3.5K images. Particularly noteworthy are the significant improvements in accuracy for the smaller/thinner classes. * Note that we could not access predictions for older methods for computing the mIoU, BF metrics.
The qualitative comparisons of SegNet predictions with other deep architectures can be seen in Fig. <ref>. The qualitative results show the ability of the proposed architecture to segment smaller classes in road scenes while producing a smooth segmentation of the overall scene. Indeed, under the controlled benchmark setting,
SegNet shows superior performance as compared to some of the larger models. DeepLab-LargeFOV is the most efficient model and with CRF post-processing can produce competitive results although smaller classes are lost. FCN with learnt deconvolution is clearly better than with fixed bilinear upsampling. DeconvNet is the largest model and the most inefficient to train. Its predictions do not retain small classes.
We also use this benchmark to first compare SegNet with several non deep-learning methods including Random Forests <cit.>, Boosting <cit.> in combination with CRF based methods <cit.>. This was done to give the user a perspective of the improvements in accuracy that has been achieved using deep networks compared to classical feature engineering based techniques.
The results in Table <ref> show SegNet-Basic, SegNet obtain competitive results when compared with methods which use CRFs. This shows the ability of the deep architecture to extract meaningful features from the input image and map it to accurate and smooth class segment labels. The most interesting result here is the large performance improvement in class average and mIOU metrics that is obtained when a large training dataset, obtained by combining <cit.>, is used to train SegNet. Correspondingly, the qualitative results of SegNet (see Fig. <ref>) are clearly superior to the rest of the methods. It is able to segment both small and large classes well. We remark here that we used median frequency class balancing <cit.> in training SegNet-Basic and SegNet. In addition, there is an overall smooth quality of segmentation much like what is typically obtained with CRF post-processing. Although the fact that results improve with larger training sets is not surprising, the percentage improvement obtained using pre-trained encoder network and this training set indicates that this architecture can potentially be deployed for practical applications. Our random testing on urban and highway images from the internet (see Fig. <ref>) demonstrates that SegNet can absorb a large training set and generalize well to unseen images. It also indicates the contribution of the prior (CRF) can be lessened when sufficient amount of training data is made available.
In Table <ref> we compare SegNet's performance with now widely adopted fully convolutional architectures for segmentation. As compared to the experiment in Table <ref>, we did not use any class blancing for training any of the deep architectures including SegNet. This is because we found it difficult to train larger models such as DeconvNet with median frequency balancing. We benchmark performance at 40K, 80K and $>$80K iterations which given the mini-batch size and training set size approximately corresponds to $50,100$ and $>$100 epochs. For the last test point we also report the maximum number of iterations (here atleast 150 epochs) beyond which we observed no accuracy improvements or when over-fitting set in. We report the metrics at three stages in the training phase to reveal how the metrics varied with training time, particularly for larger networks. This is important to understand if additional training time is justified when set against accuracy increases. Note also that for each evaluation we performed a complete run through the dataset to obtain batch norm statistics and then evaluated the test model with this statistic (see <http://mi.eng.cam.ac.uk/projects/segnet/tutorial.html> for code.). These evaluations are expensive to perform on large training sets and hence we only report metrics at three time points in the training phase.
From Table <ref> we immediately see that SegNet, DeconvNet achieve the highest scores in all the metrics as compared to other models. DeconvNet has a higher boundary delineation accuracy but SegNet is much more efficient as compared to DeconvNet. This can be seen from the compute statistics in Table <ref>. FCN, DeconvNet which have fully connected layers (turned into convolutional layers) train much more slowly and have comparable or higher forward-backward pass time with reference to SegNet. Here we note also that over-fitting was not an issue in training these larger models, since at comparable iterations to SegNet their metrics showed an increasing trend.
For the FCN model learning the deconvolutional layers as opposed to fixing them with bi-linear interpolation weights improves performance particularly the BF score. It also achieves higher metrics in a far lesser time. This fact agrees with our earlier analysis in Sec. <ref>.
Surprisingly, DeepLab-LargeFOV which is trained to predict labels at a resolution of $45\times60$ produces competitive performance given that it is the smallest model in terms of parameterization and also has the fastest training time as per Table <ref>. However, the boundary accuracy is poorer and this is shared by the other architectures. DeconvNet's BF score is higher than the other networks when trained for a very long time. Given our analysis in Sec. <ref> and the fact that it shares a SegNet type architecture.
The impact of dense CRF <cit.> post-processing can be seen in the last time point for DeepLab-LargeFOV-denseCRF. Both global and mIoU improve but class average diminshes. However a large improvement is obtained for the BF score. Note here that the dense CRF hyperparameters were obtained by an expensive grid-search process on a subset of the training set since no validation set was available.
Network/Iterations 4c||40K 4c||80K 4c|$>$80K Max iter
G C mIoU BF G C mIoU BF G C mIoU BF
SegNet 88.81 59.93 50.02 35.78 89.68 69.82 57.18 42.08 90.40 71.20 60.10 46.84 140K
DeepLab-LargeFOV<cit.> 85.95 60.41 50.18 26.25 87.76 62.57 53.34 32.04 88.20 62.53 53.88 32.77 140K
DeepLab-LargeFOV-denseCRF<cit.> 8c|not computed 89.71 60.67 54.74 40.79 140K
FCN 81.97 54.38 46.59 22.86 82.71 56.22 47.95 24.76 83.27 59.56 49.83 27.99 200K
FCN (learnt deconv) <cit.> 83.21 56.05 48.68 27.40 83.71 59.64 50.80 31.01 83.14 64.21 51.96 33.18 160K
DeconvNet <cit.> 85.26 46.40 39.69 27.36 85.19 54.08 43.74 29.33 89.58 70.24 59.77 52.23 260K
Quantitative comparison of deep networks for semantic segmentation on the CamVid test set when trained on a corpus of 3433 road scenes without class balancing. When end-to-end training is performed with the same and fixed learning rate, smaller networks like SegNet learn to perform better in a shorter time. The BF score which measures the accuracy of inter-class boundary delineation is significantly higher for SegNet, DeconvNet as compared to other competing models. DeconvNet matches the metrics for SegNet but at a much larger computational cost. Also see Table <ref> for individual class accuracies for SegNet.
Network/Iterations 4c||80K 4c||140K 4c|$>$140K Max iter
G C mIoU BF G C mIoU BF G C mIoU BF
SegNet 70.73 30.82 22.52 9.16 71.66 37.60 27.46 11.33 72.63 44.76 31.84 12.66 240K
DeepLab-LargeFOV <cit.> 70.70 41.75 30.67 7.28 71.16 42.71 31.29 7.57 71.90 42.21 32.08 8.26 240K
DeepLab-LargeFOV-denseCRF <cit.> 8c|not computed 66.96 33.06 24.13 9.41 240K
FCN (learnt deconv) <cit.> 67.31 34.32 24.05 7.88 68.04 37.2 26.33 9.0 68.18 38.41 27.39 9.68 200K
DeconvNet <cit.> 59.62 12.93 8.35 6.50 63.28 22.53 15.14 7.86 66.13 32.28 22.57 10.47 380K
Quantitative comparison of deep architectures on the SUNRGB-D dataset when trained on a corpus of 5250 indoor scenes. Note that only the RGB modality was used in these experiments. In this complex task with $37$ classes all the architectures perform poorly, particularly because of the smaller sized classes and skew in the class distribution. DeepLab-Large FOV, the smallest and most efficient model has a slightly higher mIoU but SegNet has a better G,C,BF score. Also note that when SegNet was trained with median frequency class balancing it obtained 71.75, 44.85, 32.08, 14.06 (180K) as the metrics.
§.§ SUN RGB-D Indoor Scenes
SUN RGB-D <cit.> is a very challenging and large dataset of indoor scenes with $5285$ training and $5050$ testing images. The images are captured by different sensors and hence come in various resolutions. The task is to segment $37$ indoor scene classes including wall, floor, ceiling, table, chair, sofa etc. This task is made hard by the fact that object classes come in various shapes, sizes and in different poses. There are frequent partial occlusions since there are typically many different classes present in each of the test images. These factors make this one of the hardest segmentation challenges. We only use the RGB modality for our training and testing. Using the depth modality would necessitate architectural modifications/redesign <cit.>. Also the quality of depth images from current cameras require careful post-processing to fill-in missing measurements. They may also require using fusion of many frames to robustly extract features for segmentation. Therefore we believe using depth for segmentation merits a separate body of work which is not in the scope of this paper. We also note that an earlier benchmark dataset NYUv2 <cit.> is included as part of this dataset.
Qualitative assessment of SegNet predictions on RGB indoor test scenes from the recently released SUN RGB-D dataset <cit.>. In this hard challenge, SegNet predictions delineate inter class boundaries well for object classes in a variety of scenes and their view-points. Overall rhe segmentation quality is better when object classes are reasonably sized but is very noisy when the scene is more cluttered. Note that often parts of an image of a scene do not have ground truth labels and these are shown in black colour. These parts are not masked in the corresponding deep model predictions that are shown. Note that these results correspond to the model corresponding to the highest mIoU accuracy in Table <ref>.
Road scene images have limited variation, both in terms of the classes of interest and their spatial arrangements. When captured from a moving vehicle where the camera position is nearly always parallel to the road surface limiting variability in view points. This makes it easier for deep networks to learn to segment them robustly. In comparison, images of indoor scenes are more complex since the view points can vary a lot and there is less regularity in both the number of classes present in a scene and their spatial arrangement. Another difficulty is caused by the widely varying sizes of the object classes in the scene. Some test samples from the recent SUN RGB-D dataset <cit.> are shown in Fig. <ref>. We observe some scenes with few large classes and some others with dense clutter (bottom row and right). The appearance (texture and shape) can also widely vary in indoor scenes. Therefore, we believe this is the hardest challenge for segmentation architectures and methods in computer vision. Other challenges, such as Pascal VOC12 <cit.> salient object segmentation have occupied researchers more <cit.>, but we believe indoor scene segmentation is more challenging and has more current practical applications such as in AR and robotics. To encourage more research in this direction we compared well known deep architectures on the large SUN RGB-D dataset.
The qualitative results of SegNet on samples of indoor scenes of different types such as bedroom, living room, laboratory, meeting room, bathroom are shown in Fig. <ref>. We see that SegNet obtains reasonable predictions when the size of the classes are large under different view points. This is particularly interesting since the input modality is only RGB.
RGB images are also useful to segment thinner structures such as the legs of chairs and tables, lamps which is difficult to achieve using depth images from currently available sensors. This can be seen from the results of SegNet, DeconvNet in Fig. <ref>. It is also useful to segment decorative objects such as paintings on the wall for AR tasks. However as compared to outdoor scenes the segmentation quality is clearly more noisy. The quality drops significantly when clutter is increased (see the result sample in the middle column).
The quantitative results in Table <ref> show that all the deep architectures share low mIoU and boundary metrics. The global and class averages (correlates well with mIou) are also small. SegNet outperforms all other methods in terms of G,C, BF metrics and has a slightly lower mIoU than DeepLab-LargeFOV. As a stand alone experiment we trained SegNet with median frequency class balancing <cit.> and the metrics were higher (see Table <ref>) and this agrees with our analysis in Sec. <ref>. Interestingly, using the grid search based optimal hyperparameters for the dense-CRF worsened all except the BF score metric for DeepLab-LargeFOV-denseCRF. More optimal settings could perhaps be found but the grid search process was too expensive given the large inference time for dense-CRFs.
One reason for the overall poor performance is the large number of classes in this segmentation task, many of which occupy a small part of the image and appear infrequently. The accuracies reported in Table <ref> clearly show that larger classes have reasonable accuracy and smaller classes have lower accuracies. This can be improved with larger sized datasets and class distribution aware training techniques. Another reason for poor performance could lie in the inability of these deep architectures (all are based on the VGG architecture <cit.>) to large variability in indoor scenes . This conjecture on our part is based on the fact that the smallest model DeepLab-LargeFOV produces the best accuracy in terms of mIoU and in comparison, larger parameterizations in DeconvNet, FCN did not improve perfomance even with much longer training (DeconvNet). This suggests there could lie a common reason for poor performance across all architectures. More controlled datasets <cit.> are needed to verify this hypothesis.
§ DISCUSSION AND FUTURE WORK
Deep learning models have often achieved increasing success due to the availability of massive datasets and expanding model depth and parameterisation. However, in practice factors like memory and computational time during training and testing are important factors to consider when choosing a model from a large bank of models.
Training time becomes an important consideration particularly when the performance gain is not commensurate with increased training time as shown in our experiments. Test time memory and computational load are important to deploy models on specialised embedded devices, for example, in AR applications.
From an overall efficiency viewpoint, we feel less attention has been paid to smaller and more memory, time efficient models for real-time applications such as road scene understanding and AR. This was the primary motivation behind the proposal of SegNet, which is significantly smaller and faster than other competing architectures, but which we have shown to be efficient for tasks such as road scene understanding.
Segmentation challenges such as Pascal <cit.> and MS-COCO <cit.> are object segmentation challenges wherein a few classes are present in any test image. Scene segmentation is more challenging due to the high variability of indoor scenes and a need to segment a larger number of classes simultaneously. The task of outdoor and indoor scene segmentation are also more practically oriented with current applications such as autonomous driving, robotics and AR.
The metrics we chose to benchmark various deep segmentation architectures like the boundary F1-measure (BF) was done to complement the existing metrics which are more biased towards region accuracies. It is clear from our experiments and other independent benchmarks <cit.> that outdoor scene images captured from a moving car are easier to segment and deep architectures perform robustly. We hope our experiments will encourage researchers to engage their attention towards the more challenging indoor scene segmentation task.
An important choice we had to make when benchmarking different deep architectures of varying parameterization was the manner in which to train them. Many of these architectures have used a host of supporting techniques and multi-stage training recipes to arrive at high accuracies on datasets but this makes it difficult to gather evidence about their true performance under time and memory constraints. Instead we chose to perform a controlled benchmarking where we used batch normalization to enable end-to-end training with the same solver (SGD). However, we note that this approach cannot entirely disentangle the effects of model versus solver (optimization) in achieving a particular result. This is mainly due to the fact that training these networks involves gradient back-propagation which is imperfect and the optimization is a non-convex problem in extremely large dimensions. Acknowledging these shortcomings, our hope is that this controlled analysis complements other benchmarks <cit.> and reveals the practical trade-offs involved in different well known architectures.
For the future, we would like to exploit our understanding of segmentation architectures gathered from our analysis to design more efficient architectures for real-time applications. We are also interested in estimating the model uncertainty for predictions from deep segmentation architectures <cit.>, <cit.>.
1c|Wall 1c|Floor 1c|Cabinet 1c|Bed 1c|Chair 1c|Sofa 1c|Table 1c|Door 1c|Window 1c|Bookshelf 1c|Picture 1c|Counter 1cBlinds
83.42 93.43 63.37 73.18 75.92 59.57 64.18 52.50 57.51 42.05 56.17 37.66 40.29
Desk Shelves Curtain Dresser Pillow Mirror Floor mat Clothes Ceiling Books Fridge TV Paper
11.92 11.45 66.56 52.73 43.80 26.30 0.00 34.31 74.11 53.77 29.85 33.76 22.73
Towel Shower curtain Box Whiteboard Person Night stand Toilet Sink Lamp Bathtub Bag 2c2*
19.83 0.03 23.14 60.25 27.27 29.88 76.00 58.10 35.27 48.86 16.76 2c
Class average accuracies of SegNet predictions for the 37 indoor scene classes in the SUN RGB-D benchmark dataset. The performance correlates well with size of the classes in indoor scenes. Note that class average accuracy has a strong correlation with mIoU metric.
Network Forward pass(ms) Backward pass(ms) GPU training memory (MB) GPU inference memory (MB) Model size (MB)
SegNet 422.50 488.71 6803 1052 117
DeepLab-LargeFOV <cit.> 110.06 160.73 5618 1993 83
FCN (learnt deconv)<cit.> 317.09 484.11 9735 1806 539
DeconvNet <cit.> 474.65 602.15 9731 1872 877
A comparison of computational time and hardware resources required for various deep architectures. The caffe time command was used to compute time requirement averaged over 10 iterations with mini batch size 1 and an image of $360\times480$ resolution We used nvidia-smi unix command to compute memory consumption. For training memory computation we used a mini-batch of size 4 and for inference memory the batch size was 1. Model size was the size of the caffe models on disk. SegNet is most memory efficient during inference model.
§ CONCLUSION
We presented SegNet, a deep convolutional network architecture for semantic segmentation. The main motivation behind SegNet was the need to design an efficient architecture for road and indoor scene understanding which is efficient both in terms of memory and computational time. We analysed SegNet and compared it with other important variants to reveal the practical trade-offs involved in designing architectures for segmentation, particularly training time, memory versus accuracy. Those architectures which store the encoder network feature maps in full perform best but consume more memory during inference time. SegNet on the other hand is more efficient since it only stores the max-pooling indices of the feature maps and uses them in its decoder network to achieve good performance. On large and well known datasets SegNet performs competitively, achieving high scores for road scene understanding. End-to-end learning of deep segmentation architectures is a harder challenge and we hope to see more attention paid to this important problem.
[]Vijay Badrinarayanan obtained his Ph.D from INRIA Rennes, France in 2009. He was a senior post-doctoral research associate at the Machine Intelligence Laboratory, Department of Engineering, University of Cambridge, U.K. He currently works as a Principal Engineer, Deep Learning at Magic Leap, Inc. in Mountain View, CA. His research interests are in probabilistic graphical models, deep learning applied to image and video based perception problems.
[]Alex Kendall graduated with a Bachelor of Engineering with First Class Honours in 2013 from the University of Auckland, New Zealand. In 2014 he was awarded a Woolf Fisher Scholarship to study towards a Ph.D at the University of Cambridge, U.K. He is a member of the Machine Intelligence Laboratory and is interested in applications of deep learning for mobile robotics.
[]Roberto Cipolla obtained a B.A. (Engineering) degree from the University of Cambridge in 1984, an M.S.E. (Electrical Engineering) from the University of Pennsylvania in 1985 and a D.Phil. (Computer Vision) from the University of Oxford in 1991. from 1991-92 was a Toshiba Fellow and engineer at the Toshiba Corporation Research and Development Centre in Kawasaki, Japan. He joined the Department of Engineering, University of Cambridge in 1992 as a Lecturer and a Fellow of Jesus College. He became a Reader in Information Engineering in 1997 and a Professor in 2000. He became a Fellow of the Royal Academy of Engineering (FREng) in 2010. His research interests are in computer vision and robotics. He has authored 3 books, edited 9 volumes and co-authored more than 300 papers.
|
1511.00008
|
KASH$z$: the prevalence of ionised outflows]
The KMOS AGN Survey at High redshift (KASH$z$): the prevalence and
drivers of ionised outflows in the host galaxies of X-ray AGNBased on
observations obtained at the Very Large Telescope of the
European Southern Observatory. Programme IDs: 086.A-0518; 087.A-0711; 088.B-0316;
60.A-9460; 092.A-0884; 092.A-0144; 092.B-0538; 093.B-0106; 094.B-0061 and 095.B-0035.
C. M. Harrison et al.]
C. M. Harrison,$^{\! 1\, \dagger}$
D. M. Alexander,$^{\! 1}$
J. R. Mullaney,$^{\! 2}$
J. P. Stott,$^{\! 3,1}$
A. M. Swinbank,$^{\! 4,1}$
V. Arumugam,$^{\! 5}$
F. E. Bauer,$^{\! 6,7,8}$
R. G. Bower,$^{\! 4,1}$
A. J. Bunker,$^{\! 3,9}$
R. M. Sharples$^{\! 10,1}$
$^1$Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road,
Durham, DH1 3LE, U.K.
$^2$Department of Physics and Astronomy, University of
Sheffield, Sheffield, S3 7RH, U.K.
$^3$Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH, U.K.
$^4$Institute for Computational Cosmology, Department of Physics, Durham University, South Road,
Durham, DH1 3LE, U.K.
$^5$European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching, Germany
$^6$Instituto de Astrofísica, Facultad de Física,
Pontifica Universidad Católica de Chile, 306, Santiago 22,
$^7$Millennium Institute of Astrophysics, Vicuña
Mackenna 4860, 7820436 Macul, Santiago, Chile
$^8$Space Science Institute, 4750 Walnut Street, Suite 205, Boulder,
CO 80301, USA
$^9$Affiliate Member, Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced
Study, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Japan 277-8583
$^{10}$Centre for Advanced Instrumentation, Department of Physics, Durham University, South Road,
Durham, DH1 3LE, U.K.
$^{\dagger}$Email: [email protected]
We present the first results from the KMOS AGN Survey at High redshift
(KASH$z$), a VLT/KMOS integral-field spectroscopic (IFS) survey of
$z\gtrsim0.6$ AGN. We present galaxy-integrated spectra of 89
X-ray AGN ($L_{{\rm 2-10keV}}=10^{42}$–10$^{45}$ erg s$^{-1}$), for which
we observed [O iii] ($z$$\approx$1.1–1.7) or H$\alpha$ emission
($z$$\approx$0.6–1.1). The targets have X-ray luminosities representative of the parent
AGN population and we explore the emission-line luminosities as a function
of X-ray luminosity. For the [O iii] targets,
$\approx$50 per cent have ionised gas velocities indicative
of gas that is dominated by outflows and/or highly turbulent
material (i.e., overall line-widths $\gtrsim$$600$ km s$^{-1}$). The most
luminous half (i.e., $L_{X}>6\times10^{43}$ erg s$^{-1}$) have a $\gtrsim$2
times higher incidence of such velocities. On the basis of our results, we find no
evidence that X-ray obscured AGN are more likely to host extreme kinematics than unobscured AGN. Our KASH$z$ sample
has a distribution of gas velocities that is
consistent with a luminosity-matched sample of $z<0.4$ AGN. This implies little evolution in the prevalence of
ionised outflows, for a fixed AGN luminosity, despite an
order-of-magnitude decrease in average star-formation rates over
this redshift range. Furthermore, we compare our H$\alpha$ targets to a redshift-matched sample of star-forming
galaxies and despite a similar distribution of H$\alpha$ luminosities
and likely star-formation rates, we find extreme ionised gas
velocities are up to $\approx$10$\times$ more prevalent in the
AGN-host galaxies. Our results reveal a high prevalence of
extreme ionised gas velocities in high-luminosity X-ray AGN and imply that
the most powerful ionised outflows in high-redshift galaxies are driven by AGN activity.
galaxies: active; — galaxies: kinematics and dynamics; —
quasars: emission lines; — galaxies: evolution
§ INTRODUCTION
Massive galaxies are now known to host supermassive black
holes (SMBHs) at their centres. These SMBHs grow through mass accretion events,
during which, they become visible as active galactic nuclei
(AGN). A variety of indirect observational
evidence has been used to imply a connection
between the growth of SMBHs and the growth of the galaxies that they reside in. For
example: (1) the cosmic evolution of volume-averaged SMBH growth and
star formation look very similar; (2) growing SMBHs may be preferentially located in
star-forming galaxies and (3) SMBH mass is tightly correlated with galaxy bulge mass and stellar velocity dispersion
(see reviews in e.g., ;
). Suggestion of a more direct connection between SMBH
growth and galaxy growth largely comes from theoretical models of galaxy
formation. Most successful models propose that AGN are required to regulate the growth of
massive galaxies by injecting a fraction of their accretion
energy into the surrounding intergalactic medium (IGM) or interstellar
medium (ISM; e.g., ; ; ; ;
; ;
; ;
; ;
). Without this so-called “AGN feedback”, these models
fail to reproduce many key observables of the local Universe, such as
the observed SMBH mass-spheroid mass relationship (e.g., ); the sharp
cut-off in the galaxy mass function (e.g., ) and the
X-ray temperature-luminosity relationship observed in galaxy clusters and
groups (e.g., ).
In recent years there has been a large amount of observational work
searching for signatures of “AGN feedback” and to test theoretical predictions (see reviews in e.g.,
; ; ;
). One of the most promising candidates for a
universal feedback mechanism is AGN-driven outflows, which, if they
can be driven to galaxy-wide scales, could remove or heat cold gas that would otherwise form stars in the host
galaxy. There is now wide-spread observational evidence that galaxy-wide outflows exist
in both low- and high-redshift AGN-host galaxies, using tracers of atomic, molecular and ionised
gas (e.g., ; ;
; ;
; ;
Of specific relevance to our study are ionised outflows that have been
known for several decades to be identifiable using broad and
asymmetric emission-line profiles (e.g., ;
; ). However, it is now possible to use large optical spectroscopic samples, such as the Sloan Digital Sky
Survey (SDSS; ), to search for these signatures in hundreds to thousands of
$z\lesssim1$ AGN. By combining these spectroscopic surveys with multi-wavelength data sets, recent studies have provided excellent constraints on the
prevalence and drivers of ionised outflows in low-redshift AGN (e.g.,
; ; ). Follow-up integral-field spectroscopy (IFS) observations of objects drawn from these large samples,
have made it possible to constrain the prevalence of these outflows on
galaxy-wide scales, to explore their spatially-resolved characteristics as a function of AGN and host-galaxy properties
and to test theoretical predictions (e.g., ; ; ). These
studies have revealed that galaxy-wide ionised outflows are
common, perhaps ubiquitous, throughout the most optically-luminous
low-redshift AGN (i.e., with $L_{{\rm AGN}}\gtrsim10^{45}$ erg s$^{-1}$; although
also see ).
Despite the great insight provided by statistical IFS studies of outflows
in the low-redshift Universe, these studies do not cover the redshift
ranges during the peak epochs of SMBH and galaxy growth (i.e.,
$z\gtrsim$1; e.g., ; ) and
consequently the redshift ranges where AGN-driven outflows are
predicted to be most prevalent. Spatially-resolved spectroscopy of high-redshift AGN has been more
limited because the bright optical emission lines, that are excellent
traces of ionised gas kinematics, are redshifted
to the near infrared (NIR), which is much more challenging to observe in than optical wavelengths. Each study that has searched for galaxy-wide ionised
outflows at high redshifts has investigated only a small number of AGN, which were selected in a
variety of different ways (e.g., ; ;
; ;
; ;
; ; ); with the largest, to date, coming from
<cit.>, which presents 18 confirmed AGN identified using a combination of X-ray, infrared and
radio techniques. To make robust conclusions about the AGN population as a
whole, and to properly understand the role of ionised outflows in galaxy evolution, it is crucial to place
these observations into the context of the parent
population of AGN and galaxies. For example, it is particularly
important to assess how representative observations are if they have significant implications for our understanding of AGN feedback,
such as possible evidence that star formation has been suppressed by
ionised outflows in two high-redshift AGN (;
). There is clearly a need for IFS observations
of large samples of high-redshift AGN that are selected in a uniform way.
It is now possible to efficiently obtain large samples of NIR IFS data thanks to the commissioning of the K-band Multi Object Spectrograph (KMOS;
) on the European Southern Observatory's Very Large Telescope
(VLT). This instrument is ideal for systematic studies of
the rest-frame optical properties of high-redshift galaxies and AGN,
selected in the well studied extragalactic deep fields (e.g. ;
; ; Stott et al. 2015). In this
work we present the initial results from our survey of
high-redshift AGN: the KMOS AGN Survey at High-$z$
(KASH$z$). KASH$z$ is an ongoing guaranteed time project, led by
Durham University, to observe high-redshift ($z$$\approx$0.6–3.6) AGN
with KMOS. This survey will provide a huge leap
forward for our understanding of AGN outflows and host-galaxy
kinematics, by measuring the spatially-resolved kinematics of an order
of magnitude more sources that previous work. Furthermore,
this survey has been jointly run with the KMOS Redshift One
Spectroscopic Survey (KROSS) of high-redshift star-forming galaxies
(Stott et al. 2015), which makes it possible to place our observations
of AGN into the context of the galaxy population.
In this paper we present the first results
from KASH$z$. In Section <ref> we describe the survey and observations; in
Section <ref> we describe our data analysis and
comparison samples; in
Section <ref> we present our initial results and discuss their implications and in
Section <ref> we give our conclusions. Throughout, we
assume a Chabrier IMF () and assume $H_0 = 70$ Mpc$^{-1}$, $\Omega_{\rm{M}} = 0.30$ and $\Omega_{\Lambda}= 0.70$; in this cosmology, 1 arcsec corresponds to 7.5 kpc at
$z=0.8$ and 8.4 kpc at $z=1.4$.
§ SURVEY DESCRIPTION, SAMPLE SELECTION AND OBSERVATIONS
Hard-band (2–10 keV) X-ray luminosity versus redshift for
the sources in the four deep fields covered by KASH$z$ (see
Section <ref>). The dashed line indicates our
luminosity cut for target selection of $L_{{\rm 2-10keV}}>10^{42}$ erg s$^{-1}$. The large symbols indicate the 89 X-ray AGN that
have been observed for KASH$z$ so far. These filled and empty symbols indicate
emission-line detections (H$\alpha$ or [O iii]) and no
emission-line detections, respectively. Grey circles represent the 7 targets that are excluded from the
analyses (i.e., resulting in a final sample of 82 targets) because a lack of an emission-line
detection may be unphysical (Section <ref>). The
stars highlight targets that have $L_{{\rm 2-10\,keV}}$ values estimated from the soft band (see
Section <ref>). Overall the targets cover nearly three orders of
magnitude in X-ray luminosity (i.e., $L_{{\rm X}}$$\approx$10$^{42}$–10$^{45}$ erg s$^{-1}$).
KASH$z$ is designed to ultimately obtain spatially-resolved emission-line kinematics of $\approx$(100–200) high-redshift ($z$$\approx$0.6–3.6)
AGN. The overall aim of KASH$z$ is to provide insight into the feeding and feedback
processes occurring in the host galaxies of high-redshift AGN by
using IFS data to measure the ionised gas kinematics traced by the H$\alpha$, [O iii], H$\beta$, [N ii] and/or
[S ii] emission lines. The key aspect of KASH$z$ is to exploit
the unique capabilities of the multiple integral field units (IFUs) in
the KMOS instrument to perform such measurements on
larger, more uniformly selected samples of high-redshift AGN than was
possible in previous studies that used single IFU instruments. This
will make it possible to make conclusions on the overall high-redshift
AGN population and to place previous observations, of a few sources, into the
context of the parent population of AGN. Until now, this sort of
approach has only been possible for low-redshift AGN
(e.g., ; ; ).
In this paper we present the first results of KASH$z$,
which focuses on the galaxy-integrated emission-line profiles of $z$$\approx$0.6–1.7 X-ray
detected AGN. In future papers we will present results on the
spatially-resolved properties of the current sample, as well as
expand the sample size when more data are obtained during our ongoing
guaranteed-time KMOS observations.
§.§ Parent catalogues
For our target selection we make use of deep X-ray surveys performed in extragalactic
fields (COSMOS; CDF-S; UDS and SSA22). These surveys provide an efficient method for AGN selection that is largely free from host-galaxy
contamination (e.g., see review in ). The chosen X-ray fields are those visible
from the VLT with the deepest X-ray data available. These allow
us to uniformly select large samples of AGN that can be efficiently observed with KMOS (see
Section <ref>). The four deep fields are:
* Cosmic evolution survey field (COSMOS; see
). We obtain X-ray sources by combining
the 1.8 Ms Chandra catalogue that covers 0.9 deg$^{2}$
(C-COSMOS; ; ; ) with the wider and shallower
1.5 Ms XMM-Newton catalogue that covers 2.13 deg$^{2}$
(XMM-COSMOS; ; ). For sources detected in both
surveys we use the entries in the C-COSMOS catalogue.
* Chandra Deep Field South (CDF-S; see ). We
use the X-ray sources from the 4 Ms Chandra catalogue that covers 464.5 arcmin$^{2}$ ().
* The Subaru/XMM-Newton Deep Survey (SXDS: see
). We use the X-ray sources from the 400 ks
XMM-Newton catalogue that covers 1.14 deg$^{2}$
(). We only observe sources that are positioned inside the
central 0.8 deg$^{2}$ that are also covered by the near-IR Ultra Deep
Survey (UDS; see ). We use UDS as the name
of this field hereafter.
* The SSA22 $z=3.09$ protocluster field (see
). We use the X-ray sources from the 400 ks Chandra catalogue that covers 330 arcmin$^{2}$ ().
To define our sample we primarily make use of the hard-band fluxes ($F_{\rm 2-10keV}$) for each of
the X-ray sources. We also make use of the soft-band fluxes ($F_{{\rm
0.5-2keV}}$) when there is no hard-band detection and
for calculating the $F_{\rm 2-10keV}$/$F_{{\rm 0.5-2keV}}$ flux ratios to
select X-ray obscured candidates (see
Section <ref>). We note that throughout this work, $L_{X}$ refers to the
hard-band (2–10 keV) luminosities. To be consistent across the
four fields, we calculate fluxes making similar assumptions to the
C-COSMOS catalogue (see ); i.e., we convert quoted count
rates ($CR$) to fluxes, assuming a photon index of $\Gamma=1.4$ and
a typical galactic absorption of $N_{{\rm H}}\approx2\times10^{20}$ cm$^{-2}$. For the SSA22
and CDF-S Chandra catalogues, we use $CR$ to flux
conversion factors of 2.87$\times$10$^{-11}$ erg cm$^{-2}$ and 8.67$\times$10$^{-12}$ erg cm$^{-2}$ for
the hard and soft bands, respectively. We note that these conversion
factors also take into account the conversion between the quoted
2–8 keV energy-band values to 2–10 keV energy-band values. For the UDS XMM-Newton
catalogue, we use the conversion factors applicable for $\Gamma$=1.4 tabulated
by <cit.> and for C-COSMOS and XMM-COSMOS we use the quoted
flux values in <cit.> and <cit.>, respectively.[We note
that XMM-COSMOS catalogue assumes a power-law index of $\Gamma=1.7$ and
$\Gamma=2.0$ for the hard and soft bands, respectively. However,
this results in small differences in the calculated luminosity values from assuming $\Gamma=1.4$ (i.e.,
$\lesssim$20 per cent).] The X-ray fluxes are tabulated in
Table <ref>.
We obtained archival redshifts for the sources in the fields as follows: (1) for C-COSMOS we use the
compilation in <cit.>, using spectroscopic redshifts when
available and photometric redshifts otherwise; (2) for CDF-S we only target
sources with spectroscopic redshifts as compiled by <cit.>; (3) for UDS we used
the October 2010 spectroscopic redshift compilation provided by the UDS
(; ; Akiyama et al. in prep.) and (4) for SSA22 we use the spectroscopic redshifts compiled by
<cit.>.[For the target SSA22-39 we used the
spectroscopic redshift from .] For our observed
targets the redshifts used throughout this work, $z$, are those derived from our
measured emission-lines (i.e., $z_{\rm L}$; Section <ref>), in preference to the archival
redshifts (i.e., $z_{\rm A}$) described above. The archival redshifts
and our emission-line redshifts for the targets presented here are tabulated in Table <ref>.
The sources in our parent catalogues are plotted in the
$L_{{\rm X}}$–redshift plane in
Figure <ref>. We calculate X-ray luminosities following,
\begin{equation}
\label{eq:xraylum}
L = 4 \pi D_{L}^2 F (1+z)^{\Gamma-2},
\end{equation}
where $D_{L}$ is the luminosity distance, $F$ is
the X-ray flux in the relevant energy band, $z$ is the redshift (see above) and $\Gamma$ is the photon
index. As above, we assume $\Gamma=1.4$ for all sources. Some of the X-ray sources are detected in the soft band (0.5–2.0 keV) but not in the hard
band (2–10 keV) and for these sources we make use the catalogued
flux upper limits to plot them in Figure <ref>. However, for the targets that we observed in this work, that do not have
a hard-band detection, i.e., seven targets that are all from the COSMOS
field (see Table <ref>), we estimated hard-band fluxes by
extrapolating from the soft band assuming a power-law with
$\Gamma=1.4$ (see star symbols in Figure <ref>). In all
of these cases the extrapolated hard-band fluxes were consistent with the
measured upper limits and we verified that the X-ray emission was
AGN-dominated (see below).
§.§ Sample selection
Our KASH$z$ targets were selected using the archival redshifts
and X-ray luminosities described above. For this paper, we observed
targets in the NIR $J$-band, which covers a wavelength range of
$\lambda$$\approx$1.03–1.34 . Therefore, we selected targets with
redshifts in these two ranges ranges: (1) $z$$=$1.07–1.67, for which we could observe the [O iii]4959,5007 emission-line
doublet or (2) $z$$=$0.57–1.05, for which we could observe the
H$\alpha$ and [N ii]6548,6583 emission lines.
To avoid selecting non-AGN X-ray sources (i.e., extreme starburst galaxies), we select
sources with a measured hard-band luminosity of $L_{{\rm
2-10\,keV}}>10^{42}$ erg s$^{-1}$. We note that this X-ray
luminosity has little impact on the final selection because the X-ray
catalogues are not complete down to this luminosity for our
redshift ranges of interest (see Figure <ref>). Therefore, the majority of the
observed targets (i.e., all but four) have X-ray luminosities of $L_{{\rm
2-10\,keV}}>3\times10^{42}$ erg s$^{-1}$ and will have X-ray
emission that is AGN dominated; for X-ray emission of this luminosity to be produced by star-formation
processes alone, it would require extreme star-formation rates of $\gtrsim$1900 $M_{\odot}$ yr$^{-1}$,
based on empirical measurements of star-forming galaxies (following
Equation [4] of and converted to a Chabrier IMF; also see ). Four of our targets, all of which are in CDF-S, have $L_{{\rm
2-10\,keV}}=(1$–$3)\times10^{42}$ erg s$^{-1}$, which
could conceivably be produced by high levels of star formation; however, we
verified that the X-ray emission is AGN-dominated in these sources by finding that their
star-formation rates, taken from <cit.>, are too low by a factor of $>$30 to
produce the observed hard-band X-ray luminosities (following the same
procedure as above). We note that <cit.> also classify all of our CDF-S X-ray
targets as AGN, using a more comprehensive X-ray and
multi-wavelength identification procedure.
The initial KMOS data of KASH$z$ that are presented here, were obtained
during ESO periods P92–P95 (details provided in Section <ref>). During these observations we obtained
KMOS data for 79 X-ray AGN that met our selection
criteria. This programme was jointly observed with the KROSS
guaranteed time observing (GTO) programme (Stott et al. 2015) and the choice of these 79 AGN targets, drawn
from our parent sample, was dictated by those that could be observed inside the KMOS pointing
positions chosen by the KROSS team, with a slight preference to
selecting $z$$\approx$1.1–1.7 AGN for which we could observe [O iii]. In summary, the observed targets are effectively
randomly selected from the luminosity and redshift plane of the parent
sample (Figure <ref>; also see Section <ref> for more discussion on how representative our targets are).
We supplemented our sample of 79 KMOS targets with archival
observations taken with SINFONI (Spectrograph for INtegral Field
Observations in the Near Infrared; ). SINFONI
contains a single-object NIR IFU, similar to the individual KMOS IFUs (see
Section <ref>). We queried the ESO archive[http://archive.eso.org/eso/eso_archive_main.html]
for $J$-band SINFONI observations at the positions of the X-ray AGN in our
parent catalogues. This yielded 10 targets that met our selection criteria, to add to the $J$-band sample
presented in this paper.
Overall, our current KASH$z$ sample contains 89 targets. This sample consists of 54 $z$$\approx$1.1–1.7 AGN, for which we targeted the [O iii]4959,5007 emission-line doublet and 35 $z$$\approx$0.6–1.1 AGN, for which we targeted the H$\alpha$ and [N ii]6548,6583 emission lines (see Figure <ref>). The targets are listed in
Table <ref> and we have used a naming convention that combines their field names and corresponding X-ray ID
in the format: “field”-“X-ray ID”. For C-COSMOS and XMM-COSMOS the
field names are shortened to COS and XCOS, respectively.
Out of the 89 targets presented here, 83 have a spectroscopic archival redshift and 6 targets have a photometric
redshift. All of the photometric redshift targets are from the C-COSMOS field (Table <ref>). We note that two of the
spectroscopic targets are classified as having “insecure” redshifts (e.g.,
based on low signal-to-noise ratios or single-lines; see Table <ref>) by
<cit.>.[CDFS-549 is a third source with an “insecure”
spectroscopic redshift in <cit.> ($z_{\rm A}$$=$1.55); however, <cit.> detected H$\alpha$ at
$z$=1.553 so we re-classify it as secure.] For our emission-line
detected targets (see Section <ref>), our emission-line redshifts agree with
the archival redshifts, such that $|z_{L} - z_{A}|/(1+z_{L})\lesssim0.005$, except for the following
three exceptions: COS-44 ($z_{A}=1.51$; $z_{L}=0.80$) and COS-1199 ($z_{A}=0.77$; $z_{L}=0.85$), both
of which had photometric archival redshifts, and CDFS-561
($z_{A}$$=$0.80; $z_{L}$$=$0.98) which is a spectroscopic
archival redshift quoted by <cit.>. We discuss the detection rates
of our targets, and how they relate to the archival redshifts, in Section <ref>. For the emission-line
detected AGN, throughout this work, we define the redshift as
$z=z_{L}$ and for the undetected targets or non-targeted AGN in the
parent sample we use $z=z_{A}$.
§.§ The KASH$z$ sample in context
The 2–10 keV X-ray luminosity ($L_{{\rm X}}$) distributions
for the sources that met our selection criteria from the four deep fields described in Section <ref>
(solid histograms). Each field is shown in separate panels and we
overlay, as hashed histograms, the targets presented in
this work that were selected from these respective fields. With the exception of SSA22 (from which there are only two
targets presented here), the targets have a range of X-ray luminosities that
are representative of the parent catalogues from which they were
selected (see Section <ref>).
In Figure <ref> we show histograms of the X-ray
luminosities for the X-ray sources that met our selection criteria and
of the subset of these that were observed for this work (Table <ref>). This figure shows that our targets
represent the full luminosity range of the parent catalogues from
which they were selected. A two-sided Kolmogorov-Smirnov (KS)
test, yields probability values of 0.53, 0.08 and 0.96 that the two
distributions are drawn from the same distribution for COSMOS, CDF-S and UDS, respectively. Hence, there is
no evidence that the targets are not representative of the parent
population from which they were selected. For SSA22 there are only
two targets and hence it is not meaningful to perform a KS test. We
conclude that our targets are broadly representative of the X-ray AGN
population covered by the parent catalogues. It is worth noting
that, although X-rays surveys arguably provide the most uniform method
for selecting AGN, there will be some unknown fraction of luminous AGN
that are detected at other wavelengths, that are very heavily obscured
in X-rays and are not detected in these surveys
(e.g., ; ). We defer a comparison to AGN selected
using different observational methods to future work.
§.§ Selecting X-ray obscured candidates
For part of this work we compare the emission-line properties of the targets
that are most likely to be X-ray obscured (i.e., $N_{H}\gtrsim10^{22}$ cm$^{-2}$) with those that are X-ray unobscured. Due to the use of both XMM-Newton and Chandra catalogues in our AGN selection and due to the various depths of these catalogues, we opt to use
the simplest diagnostic possible to separate X-ray obscured and unobscured sources. We take the ratio of the hard-band
to soft-band X-ray fluxes (i.e., $F_{{\rm 2-10keV}}/F_{{\rm 0.5-2keV}}$; see
Table <ref>) as a proxy for an observed power-law index,
$\Gamma_{{\rm obs}}$. For the seven targets where there
are no direct hard-band detections, we
use the hard-band flux upper limits from the original X-ray
catalogues (see Section <ref>). We use a threshold of $\Gamma_{{\rm
obs}}<1.4$, or equivalently $F_{{\rm 2-10keV}}/F_{{\rm 0.5-2keV}}>3.03$,
to select our “obscured candidates”. Assuming a typical intrinsic
power-law index of $\Gamma=1.8$ (e.g., ;
), this threshold corresponds to an intrinsic column density of $N_{H}\gtrsim1\times$10$^{22}$ cm$^{-2}$, at the median redshift
of our H$\alpha$ targets (i.e., $z=0.86$), and $N_{H}\gtrsim2\times$10$^{22}$ cm$^{-2}$, at the median redshift
of our [O iii] targets (i.e., $z=1.4$). This criteria yields 32
“obscured candidates” and 52 “unobscured candidates” out of the
original sample of 89. The
other 5 targets have X-ray upper limit values such that they can not be
classified. We identify the classification of each target in
Table <ref>. Reassuringly, we find that only one of the obscured candidates has an identified broad-line region
component in our data (see Section <ref>), compared to 15 of the unobscured
candidates, as expected if X-ray obscured AGN are more likely to also have an
obscured broad line region. Furthermore, this one exception only just meets our
criteria for selecting obscured sources, i.e., it has a flux ratio of $F_{{\rm
2-10keV}}/F_{{\rm 0.5-2keV}}=3.1$. This provides extra support that our obscured AGN criterion is reliable.
§.§ Selecting “radio luminous” AGN
For part of this work we compare KASH$z$ targets that are
radio luminous with those that are not (Section <ref>). Therefore, we collated
the available 1.4 GHz radio catalogues for the COSMOS, CDF-S and UDS
fields (a suitable SSA22 catalogue is not available in the literature). For COSMOS we used the 5$\sigma$ VLA catalogue of
<cit.>; for CDF-S we use the 5$\sigma$ VLA catalogue presented in <cit.> and for UDS we use the 4$\sigma$
VLA catalogue in Arumugam et al. (2015). We match the positions of our targets to the radio
positions using a 2 arcsec matching radius. For the 87 targets in these three fields (i.e., excluding the
two SSA22 targets), we obtain 26 radio matches. We calculate radio
luminosities using the aperture-integrated flux densities and assume
a spectral index of $\alpha=-0.7$. All three catalogues have typical
sensitivities around $\approx$10 $\mu$Jy beam$^{-1}$ and therefore,
we are complete to 1.4 GHz radio luminosities of $L_{{\rm 1.4GHz}}\approx10^{24}$ W Hz$^{-1}$ at the redshift ranges of interest
in this work (see Figure <ref>). Therefore, for this
work we separate our targets which are “radio luminous", i.e., with
$L_{{\rm 1.4GHz}}>10^{24}$ W Hz$^{-1}$ (11 targets) from those with
$L_{{\rm 1.4GHz}}<10^{24}$ W Hz$^{-1}$ (76 targets). Table <ref>
highlights which targets fall into each category. Above this
luminosity threshold, sources are generally thought have AGN-dominated
radio emission and could be predominantly “radio-loud” sources
(e.g., ; ; ). Furthermore, we find that 13% of our targets are in our
“radio luminous” category, which is broadly consistent with the observed
radio-loud fraction of $\approx$10% for luminous AGN (e.g., ;
; ).
§.§ Observations
§.§.§ KMOS observations and data reduction
The majority of the KASH$z$ targets presented here (i.e., 79 out of the 89)
were observed using the KMOS instrument on the VLT
(). KMOS has 24 IFUs that operate simultaneously and can be
independently positioned inside a 7.2 arcmin diameter circular field. Each
IFU has a field of view of 2.8$^{\prime\prime}\times$2.8$^{\prime\prime}$ with a pixel scale of
0.2$^{\prime\prime}$. The 24 IFUs are fed to three spectrographs
(eight IFUs per spectrograph). For this initial KASH$z$ paper we present results of sources
observed with the $YJ$ grating that covers a wavelength range of 1.03–1.34 and has a band-centre
spectral resolution of R$\sim$3600. The KMOS observations were taken during ESO Periods 92–95,
sharing IFUs with the KROSS GTO programme,
which targeted high-redshift star-forming galaxies in the same fields
(see Stott et al. 2015; also see Section <ref>).
Observations were carried out using ABAABAAB sequences (where A frames
are on-source and B frames are on-sky), with 600 second integrations per position and up to
0.2 arcsec spatial dithering between on-source frames. The targets have total on-source exposure times
of 5.4–11.4 ks (i.e., 9–19 on-source exposures). The individual
exposure times were a result of the length of observing time with
acceptable weather conditions during the various observing runs
allocated to the GTO team and the final on-source exposure times are tabulated for individual targets
in Table <ref>. The median $J$-band seeing was
0.7$^{\prime\prime}$ with 90 per cent below 1.0$^{\prime\prime}$. The data were reduced using the esorex/spark pipeline () which flatfields,
illumination corrects, wavelength calibrates and uses observations of
standard stars, taken alongside the science frames, to flux
calibrate. Additional sky subtraction was performed using
dedicated sky IFUs. Repeated observations of targets were stacked into
the final fully reduced datacubes with a clipped
mean using the esorex pipeline. Finally, we rebinned the cubes onto a
0.1 arcsec pixel scale. For full details of the observations and data
reduction see Stott et al. (2015).
§.§.§ SINFONI observations and data reduction
In addition to the primary observations using KMOS, we supplement the
KASH$z$ sample with archival observations of 10 targets, that met our
selection criteria, taken with SINFONI (see Section <ref>). The observations presented here were all
observed using SINFONI's 8$\times$8 arcsec$^{2}$
field of view, which is divided into 32 slices of width 0.25 arcsec
with a pixel scale of 0.125 arcsec along the slices. The observations
were carried out using the $J$-band grating which has an approximate
resolution of $R$$\sim$3000. The observations used a variety of
observing strategies, but in all cases we were able to subtract on-sky frames from
on-target frames. We reduced the SINFONI data using the standard esorex pipeline
() that performs flat fielding, wavelength calibration and cube
re-construction. We flux calibrated individual data
cubes using standard star observations taken the same night as the
science observations at a similar airmass. We stacked individual cubes of the same
source by creating white-light (collapsed images), centroiding the
cubes based on these images and then performing a median stack with a
3$\sigma$ clipping threshold, rejecting cubes that could not be well centred. Three of the targets
(CDFS-51; CDFS-370 and CDFS-492) were not detected in any of the
individual cubes and for these targets, we used the offset pattern in
the headers to stack the cubes. The total on-source exposure times range
from 2.4–25.2 ks and are tabulated for the individual targets in Table <ref>.
§ ANALYSIS AND COMPARISON SAMPLES
In this section we describe our analyses of the galaxy-integrated
spectra for the targets presented in this work
(Section <ref>–<ref>) and we also describe our comparison samples of low-redshift AGN and high-redshift
star-forming galaxies (Section <ref>).
§.§ Galaxy-integrated spectra
We extracted a galaxy-integrated spectrum from the KMOS and SINFONI
data cubes using the methods described below. We show example spectra in
Figure <ref> and Figure <ref> and all
89 spectra are shown in Figure <ref>–Figure <ref>.
To obtain the galaxy-integrated spectra, we initially define a galaxy
“centroid” by creating wavelength-collapsed images from the data cubes, including both continuum and line-emission. For
three targets no emission lines or continuum were detected and we therefore assumed the centroid was at the centre of the
cube for these targets (we later exclude these three targets from our
analyses; see Section <ref>). For the targets CDFS-454 and CDFS-606 there are
bright continuum sources in the field of view that are spatially offset from the line-emitting
regions by $\approx$2 arcsec and $\approx$1.2 arcsec, respectively. For these two targets we centred on the
line emission. For each galaxy we summed the spectra
from all of the pixels within a circular aperture around the galaxy
centroids. We chose diameters to broadly match the physical
size scale of the spectra obtained with the SDSS fibres in our low-redshift
comparison sample (see Section <ref>). The median
redshift of the low-redshift SDSS comparison sample is $z$=0.14 which means
that the spectra are from a median 7.4 kpc diameter aperture due to the
3 arcsec fibres. For our KASH$z$ targets, the median redshift of the
[O iii] sample is $z$=1.4 (i.e., corresponding to 8.4 kpc arcsec$^{-1}$) and the median redshift of the H$\alpha$ sample
is $z$=0.86 (i.e., corresponding to 7.7 kpc arcsec$^{-1}$). Therefore, for the [O iii] targets we used a diameter of 0.9 arcsec and for the
H$\alpha$ targets we used a diameter of 1.0 arcsec. These correspond to
physical diameters of 7.6$\pm$0.8 kpc and 7.7$\pm$0.8 kpc,
respectively, where the uncertainties correspond to a 0.1 arcsec
pixel scale. To assess the effect of using alternative “galaxy wide”
apertures, which will cover any losses of flux due to seeing, we
also extracted spectra using 2$^{\prime\prime}$ diameter
apertures for all sources. We discuss the corrections required to the
emission-line luminosities in Section <ref>. For
the sources significantly detected in both apertures, we
find that our emission-line width measurements (Section <ref>) are
consistent between both apertures within 1$\times$ the errors for 77% of the targets and within
2$\times$ the errors for 97% of the targets. We defer discussion of
how the emission-line widths change as a function of
radius to future papers.
Three examples of our continuum-subtracted, high signal-to-noise
ratio, [O iii]4959,5007 emission-line profiles. The black curves show our fits to the emission-line profiles and the dashed curves
show the individual Gaussian components, where applicable, with an arbitrary
offset in the y-axis. The vertical dotted lines indicate the wavelengths of the
brightest sky lines (). These examples
demonstrate the diversity of emission-line profiles observed in the
sample. Top: a broad, highly asymmetric profile; Middle: a broad, almost symmetric profile and Bottom: a relatively narrow profile without a strong underlying
broad component. Figure <ref> presents the
[O iii]4959,5007 emission-line profiles for all of the targets.
Three examples of our continuum-subtracted
H$\alpha$+[N ii]6548,6583 emission-line
profiles. The black curves show our fits to the emission-line profiles and the dashed or dot-dashed curves
show the individual Gaussian components, where applicable, with an arbitrary
offset. The vertical dotted lines indicate the wavelengths of the
brightest sky lines (). These examples
represent the three different types of profile fits used in this work. Top: a
source with a BLR component in addition to the NLR component; Middle: a source with no BLR
component but two NLR components seen in H$\alpha$ and [N ii] and Bottom: a source with a single NLR component. Figure <ref> and Figure <ref> present the
H$\alpha$+[N ii]6548,6583 emission-line profiles for all of the targets.
§.§ Emission-line fitting
The emission-line profiles were fit with one or two Gaussian
components (with free centroids, line-widths and fluxes) and a
straight line to define the local continuum (with a free slope and normalisation). The continuum regions were defined to
be small wavelength regions each side of the emission lines being fitted. The
noise in the spectra were also calculated in these regions. The fits were performed using a
minimising-$\chi^2$ method, using the IDL routine MPFIT
() and we weighted against the wavelengths of the brightest
sky lines (taken from ; see
dotted lines in Figure <ref> and Figure <ref>). Quoted line widths have been corrected for
spectral resolution, where we measured the wavelength-dependant
spectral resolution using the emissions lines in sky
spectra. We provide details of how we modelled the emission-line profiles below and tabulate the parameters
of the fits in Table <ref>.
For the [O iii]5007,4959 emission line doublet (see examples in Figure <ref>), we simultaneously fit the
[O iii]5007 and [O iii]4959 emission lines using
the same velocity-widths and fixing the relative centroids, using the rest-frame
wavelengths of 5008.24$\AA$ and 4960.30$\AA$, respectively (i.e.,
their vacuum wavelengths). The flux ratio of the doublet was fixed to
be 2.99 (). We initially attempted to fit this
emission-line doublet with one Gaussian component (with three free
parameters) and then with two Gaussian components (with six free
parameters) per emission line. We accepted the two-component fit if there is a significant
improvement in the $\chi^2$ values. We required
$\Delta\chi^2>15$, where this threshold was chosen to provide
the best description of the emission-line profiles across the whole
sample.[We note that absolute $\Delta\chi^2$ values are
commonly used for model selection. For example, the Bayesian
Information Criterion (BIC; ) uses $\Delta\chi^2$ but penalises against
models with more free parameters. This is
defined as BIC=$\Delta\chi^2+k\ln(N)$, where N is the number of data
points and $k$ is the number of free parameters. For the fits where we
favoured two component models (including all of the H$\alpha$ fits) we
find a median of $\Delta $BIC=34 between the two and one component models, which corresponds to strong evidence in favour of the
two-component models (e.g. ).] We show the fits to the full set of [O iii]4959,5007
emission-line profiles in Figure <ref>.
For the H$\alpha$ emission-line profiles, along with the nearby [N ii]6548,6583 doublet, we follow a similar approach as for the
[O iii] emission-line profiles (see example fits in
Figure <ref>). Based on the atomic transition probabilities, we fix the flux
ratio of the [N ii] doublet to be 3.06 (). To reduce the degeneracy between parameters, we
couple the [N ii] and H$\alpha$ emission-line profiles with a
fixed wavelength separation between the three emission lines (i.e.,
by using the rest-frame vacuum wavelengths of 6549.86$\AA$,
6564.61$\AA$ and 6585.27$\AA$). We also fix the line-widths of the H$\alpha$, [N ii]6548 and [N ii]6583 emission-line components to be the
same. The common line-width, overall centroid and the individual fluxes of H$\alpha$ and the [N ii] doublet are free to vary. We note that the emission-line coupling is only applied for
narrow-line region (NLR) components and is not applicable for
broad-line region (BLR) components that are not observed in [N ii] emission. For these H$\alpha$ targets, we have two situations
to consider; those which exhibit a BLR component and whose which do not.
* For the sources where a broad component is seen in the H$\alpha$
emission-line profile but not in the [N ii] emission-line
profile, these are the BLR or “Type 1” sources. We note that all
seven BLR components that we identify have a full-width at half
maximum (FWHM) $>$2000 km s$^{-1}$ (see Table <ref>), further indicating
these are true BLRs. For these, we fit one Gaussian component to the
[N ii]6548,6583 emission-line doublet and two
Gaussian components to the H$\alpha$ emission line. The narrower of
the H$\alpha$ components is coupled to the [N ii] emission as described above. The
broader of these components; i.e., the BLR component, has a centroid, line-width and a flux that are
free to vary. Overall, for these sources, there are seven free
parameters in the fits and an example is given in the top panel of
Figure <ref>.
* For sources without a BLR component (i.e., the “Type 2”
sources) we first fit a single Gaussian component to the H$\alpha$
emission line and [N ii] doublet, which are coupled as
described above. We then add a second Gaussian
component to all three emission lines. This is coupled in the same
way, but with the additional constraint that we force the [N ii]/H$\alpha$ flux ratio
of the broader Gaussian components to be $\gtrsim$2$\times$ that of the
narrower component. This follows <cit.>, who found that this was typical for high-redshift galaxies and AGN. This extra
constraint is required to prevent considerable degeneracy in the
fits that can lead to unphysical results. As for the [O iii]
emission-line profiles, we accept the two component Gaussian fit if there is a significant
improvement in the $\chi^2$ values, i.e., $\Delta \chi^2>15$. Examples
of a one and two Gaussian component fit are
shown in Figure <ref>.
The coupling between the [N ii] and H$\alpha$
emission-line profiles is a requirement of the restricted
signal-to-noise of the observations, to avoid a high-level of degeneracy, and is not necessarily
physical. The same coupling approach is often followed for high-redshift
galaxies and AGN, that are inevitably subject to limited signal-to-noise
observations (e.g., ; ; Stott et al. 2015),
and makes the underlying assumption that the emission lines are being
produced by the same gas, undergoing the same kinematics (see
discussion on this in Section <ref>). However, our
emission-line profile models appear to be a good description of the
data and are sufficient for our purposes of measuring the H$\alpha$
emission-line widths and luminosities and the [N ii]/H$\alpha$ flux ratios. We show the full set of fits to the H$\alpha$
emission-line profiles in Figure <ref> and
Figure <ref> and tabulate the parameters in
Table <ref>.
To calculate the uncertainties on the derived parameters, we follow the
same procedure for both the [O iii] and H$\alpha$$+$[N ii] emission-line profiles. We use our best-fit models to
generate 1000 random spectra by adding random noise to these fits, at the level measured in
the original spectra, and then re-fit and re-derive the parameters for
these random spectra. The quoted uncertainties are from the average of the
16th and 84th percentiles in the distribution of the parameters of
these random fits. We add an additional uncertainty of 30% when plotting emission-line luminosities, to account for an estimated systematic
uncertainty on the absolute flux-calibration of the data cubes.
§.§ Measuring the overall emission-line velocity widths
A key aspect of this work is to characterise the overall velocity
widths of the emission-line profiles. Our spectra have a range in
emission-line profile shapes and signal-to-noise values (see
Figure <ref>–Figure <ref>). Therefore, it
is most applicable to characterise the velocity widths with
a single non-parametric measurement that is independent of the number
of Gaussian components used in the best-fit emission-line models (see Section <ref>). Furthermore, many
studies at low redshift use non-parametric definitions to describe the
very complex emission-line profiles (e.g., ; ; ) and it is useful to be able to
compare to these studies. Therefore, we use the line-width definition,
$W_{80}$, which is the velocity width that contains 80 per cent of
the emission-line flux; i.e., $W_{80}=v_{90}-v_{10}$, where $v_{10}$
and $v_{90}$ are the 10th and 90th percentiles, respectively. For a single Gaussian
component $W_{80}=1.09\times$FWHM. These $W_{80}$ values are measured from
the models of the emission-line profiles described in
Section <ref> and are tabulated in
Table <ref>. For this work, we are only interested in the kinematics of
the host-galaxy gas and therefore, for the 7 targets with a BLR component, the $W_{80}$ measurements only refer to the
NLR emission (i.e., only the NLR components were used to calculate $W_{80}$).
§.§ Comparison samples
A key aspect of this work is to compare our high-redshift
AGN targets to high-redshift star-forming galaxies and low-redshift
AGN. This enables us to assess the evolution in the prevalence of
outflows observed in AGN and also to compare high-redshift galaxies that do and do not
host X-ray detected AGN. In the following sub-sections we describe how we constructed
our comparison samples.
§.§.§ Low-redshift AGN comparison sample
For a low-redshift AGN comparison sample we make use of the catalogue provided by
<cit.>. This sample contains emission-line profile fits to
$\approx$24,000 $z<0.4$ optically-selected AGN from the SDSS
spectroscopic database. AGN are identified based on their [O iii]/H$\beta$ and H$\alpha$/[N ii] emission-line
ratios (following e.g., ) or the identification of a
BLR component. We take the 24,258 AGN in the <cit.>
catalogue, but reject the 37 sources where the [O iii]
emission-line profile fits have FWHM$=$4000 km s$^{-1}$, which
signifies that the fits failed for these sources (). This leads to a final sample of 24,221 $z<0.4$
optically selected AGN.
<cit.> fit the [O iii] emission-line
profiles of the low-redshift AGN with one or two Gaussian components, following a very similar
procedure to that adopted here (Section <ref>). Following Section <ref>,
we measure $W_{80}$ for these sources from the [O iii] emission-line
profile fits, correcting for the wavelength-dependent SDSS spectral
resolution. Due to the nature of the fitting routine in
<cit.>, the H$\alpha$ emission-line profile fits are coupled to the [O iii] emission-line profile
fits. We are unable to replicate this method for our KASH$z$ H$\alpha$
targets because we do not have simultaneous [O iii] and H$\alpha$ constraints. Therefore, to
avoid a biased comparison of line-width measurements, we do not use the H$\alpha$ emission-line fits provided by
<cit.>. Instead, we use the single Gaussian component emission-line fits, as
measured by the SDSS team for these targets (), to calculate the $W_{80}$ values (see
Section <ref>). These SDSS measurements do not separate NLR emission from BLR
emission; therefore, we are required to exclude the Type 1 AGN from
this comparison sample when comparing the H$\alpha$ emission-line widths
to the KASH$z$ targets (Section <ref>). However, we note that the comparison is
reasonable for our Type 2 KASH$z$ H$\alpha$ targets because all but
one of the KASH$z$ Type 2 AGN are fit with a single Gaussian
To obtain X-ray luminosities for the $z<0.4$ AGN sample described
above, we match the <cit.> sample to data release 5 of the XMM serendipitous survey
(), using a 1.5 arcsec matching radius. This
resulted in 554 matches. We calculate X-ray luminosities using the quoted 2–12 keV fluxes, which we convert to 2–10 keV fluxes using a
correction factor of 0.872 and convert to hard-band X-ray luminosities using
Equation <ref>.
For part of the analysis in this work (see Section <ref>
and Section <ref>) we are required to construct
low-redshift comparison samples that are luminosity-matched to our KASH$z$
samples. To do this, we randomly select sources from the sample
described above to construct the following three comparison samples: (1) a low-redshift
sample of $\approx$1000 AGN (both Type 1 and Type 2) that has an [O iii] luminosity distribution that is the same as our [O iii]-detected KASH$z$ targets (see Section <ref>); (2) a low-redshift sample of
$\approx$100 AGN (both Type 1 and Type 2) that has an X-ray luminosity distribution that is the same as our [O iii]-detected KASH$z$ targets and (3) a low-redshift sample of
$\approx$500 Type 2 AGN that has a H$\alpha$ luminosity distribution that is the same as our
H$\alpha$-detected KASH$z$ targets (see Section <ref>). It was not possible to construct an
low-redshift sample that was X-ray luminosity matched to our KASH$z$
H$\alpha$ targets, due to the lack of X-ray luminous Type 2 AGN in
<cit.>. However, we note that matching by [O iii]
luminosity compared to matching by X-ray luminosity
does not change the conclusions presented in Section <ref>.
§.§.§ High-redshift star-forming galaxy comparison sample
To construct a high-redshift galaxy comparison sample we
make use of the KROSS survey (Stott et al. 2015). This is a KMOS
GTO survey of $z$$\approx$0.6–1.1 star-forming galaxies, which was
observed simultaneously with KASH$z$ (see Section <ref>). These targets were
selected on the basis of their K-band magnitudes and $r$–$z$
colours, to create a sample of star-forming galaxies with stellar
masses of $\approx$$10^{9-11}$ M$_{\odot}$ (see Stott et al. 2015). This data set provides an ideal comparison sample
of galaxies for our H$\alpha$ sample of X-ray
detected AGN, at the same redshift.
The initial phases of the KROSS survey, i.e., the data obtained during ESO
periods P92–P94 and the KMOS commissioning run, contains 514 galaxies (Stott et al. 2015). For these
KROSS galaxies, we extract galaxy-integrated spectra from the KMOS data cubes
and perform emission-line profile fits following the same methods as
performed on the KASH$z$ sample (see Section <ref> and
Section <ref>). We require that H$\alpha$ emission is detected
at $\ge$3$\sigma$ and the emission-line profiles are well described using
one or two Gaussian components plus a straight-line local continuum
(i.e., following our methods described in Section <ref>). Based on these criteria, we end up
with $W_{80}$ H$\alpha$ measurements for 378 of the KROSS galaxies,
where we have also removed any X-ray AGN. In our analyses (Section <ref>),
we also make use of the stellar masses for these galaxies, as
described in Stott et al. (2015) and note that the average stellar mass of
the sample we have constructed is $\log (M_{\star}/M_{\odot})$=10.3.
§.§ Emission-line profile stacks
As part of our investigation we create average emission-line profiles by
using spectral stacking analyses on the KASH$z$
targets and comparison samples (see Section <ref> and Section <ref>). To create the emission-line profile stacks, we
de-redshift each continuum-subtracted spectrum to the
rest frame and then normalise each spectrum to the peak flux density
of the emission-line profile fits. We then construct the stacked
average spectra by taking a mean of the flux densities at each spectral
pixel, but removing the pixels affected by strong sky-line
residuals. An uncertainty on the average at each spectral pixel is
obtained by bootstrap resampling, with replacement, the stacks 1000
times and deriving the inner 68 per cent of these stacks. Our analysis is focused on the kinematics in the host
galaxies and therefore, when stacking the H$\alpha$ emission-line profiles, we do
not include any Type 1 sources (i.e., those containing an identified BLR component). We fit the stacked spectra following the
procedures described in Section <ref>; however, due to
the increased signal to noise we include one extra free parameter when
fitting the H$\alpha$ emission-line profiles which allows the flux
ratio of the H$\alpha$ and [N ii] emission lines to be free for
both Gaussian components. This means that is is possible for the
[N ii] and H$\alpha$ emission lines to have different line
widths (i.e., $W_{80}$ values; see discussion in Section <ref>).
§ RESULTS AND DISCUSSION
We present the first results from KASH$z$, which is an
ongoing programme that is utilising VLT/KMOS GTO to build up a large sample of IFS data of high-redshift
AGN (see Section <ref>). We have obtained new KMOS
observations and combined them with archival SINFONI observations, to
compile a sample of 89 X-ray detected AGN observed in the
$J$-band. These AGN have redshifts of $z$=0.6–1.7 and
hard-band (2–10 keV) X-ray luminosities in the range of
$L_{{\rm X}}=10^{42}$–10$^{45}$ erg s$^{-1}$ and are representative of the parent
population from which they were selected (see
Figure <ref> and Figure <ref>). Of the 89 targets presented here, 54 have $z$=1.1–1.7 and were targeted to
observe the [O iii]4959,5007 emission-line doublet and 35 have $z$=0.6–1.1 and
were targeted to observe the H$\alpha$+[N ii]6548,6583 emission
lines. In this paper, we present the
galaxy-integrated emission-line profiles for all 89 targets and these
are shown, along with their emission-line profile fits, in
Figure <ref>–Figure <ref>. In the
following sub-sections we present: (1) the
emission-line detection rates and definition of the final sample of 82
targets used for all further analyses (Section <ref>); (2)
the relationship between emission-line luminosities and X-ray luminosities (Section <ref>); and
(3) the prevalence and drivers of high-velocity ionised outflows
(Section <ref>).
§.§ Detection rates and defining the final sample
We detected continuum and/or emission lines in the IFS data for 86 out
of the 89 KASH$z$ targets (i.e., 97 per cent) (see
Table <ref> for details). Overall, 40 targets were detected in [O iii], 32
were detected in H$\alpha$ and 14 were detected in continuum only. One
of the reasons for a lack of an emission-line detection appears to be due to
inaccurate photometric redshifts (photometric redshifts were only used for six COSMOS
targets; see Section <ref>). Of the six targets with
photometric archival redshifts, only two resulted in emission-line detections and one of these was
detected in H$\alpha$ with $z_{{\rm L}}\approx1.5$, despite an archival redshift of
$z_{{\rm A}}\approx0.8$ (see Table <ref>). To ensure that a target is undetected in line emission because of an intrinsically low
emission-line flux, as opposed to an incorrect redshift or position,
for all further analyses we only consider non-detections if: (1) they
have a secure spectroscopic archival redshift (this excludes 5 non-detections) and (2) they were
detected in continuum so that we have a reliable source position to extract the
spectrum from the datacube (this excludes 2 further
non-detections). Based on these exclusions, for the remainder of this
work, we only consider 82 of the original 89 targets, which consists of: (a) 40 emission-line detected [O iii]
targets and 8 [O iii] targets that were only detected in continuum
(i.e., a 83 per cent emission-line detection rate) and (b) 32 emission-line detected H$\alpha$ targets and 2
H$\alpha$ targets that were detected only in continuum (i.e., a
94 per cent emission-line detection rate). Overall this results in an emission-line detection rate of
88 per cent for our final sample of 82 targets.
We note here that the non emission-line detected sources are not
only associated with the lowest X-ray luminosity sources (see
Figure <ref> and Figure <ref>). However, of the
10 targets from our final sample of 82, that were not detected in
emission lines, 5 of them are classified as X-ray obscured, 4 are classified as
unobscured and 1 is unclassified, which results in a 50–60% obscured fraction, compared
to 33–38% for the 72 emission-line detected targets (see
Section <ref> and Table <ref>). Although we can not draw
any firm conclusions from this, it is interesting to speculate that
this provides evidence that the material obscuring the X-rays may also be responsible for the lack
of emission-line luminosity and therefore that the obscuring material
may be associated with the host galaxy (i.e., galactic-scale dust; see
e.g., ; ; ).
Total [O iii] emission-line luminosity ($L_{{\rm [O~III]}}$)
versus hard-band (2–10 keV) X-ray luminosity ($L_{{\rm X}}$) for the
48 $z\approx1.1$–1.7 KASH$z$ targets. Non-detections are represented as hollow
symbols, which signify 3$\sigma$ upper limits, and stars are the same
as in Figure <ref>. The solid line shows
the median ratio of $\log(L_{{\rm [O~III]}}/L_{{\rm X}})=-2.1_{-0.5}^{+0.3}$ for these 48 KASH$z$ [O iii]
targets. The shaded region indicates the $\approx$1$\sigma$ scatter
on this ratio (see Section <ref>). The arrow shows the median aperture
correction when using a $\approx$2$\times$ larger aperture (see Section <ref>). The dot-dashed line is the
relationship for local Seyferts and QSOs presented in
<cit.>. We also show our $z<0.4$ optical AGN comparison sample
(see Section <ref>). The X-ray selected KASH$z$ sample is broadly consistent
with the low-redshift optically selected samples, but with the
expected tendency towards lower [O iii] luminosities (Section <ref>).
§.§ Emission-line luminosities compared to X-ray luminosities
§.§.§ [O iii] luminosity versus X-ray luminosity
Both the [O iii] emission-line luminosity and X-ray luminosity
have been used to estimate total AGN power (i.e., bolometric luminosities) and
therefore, the relationship between these two quantities and any
possible redshift evolution, is of fundamental
importance for interpreting observations (e.g., ; ;
; ; ;
; ; ; ). Until recently, there has been limited available
NIR spectroscopy and therefore limited measurements of the [O iii] emission-line luminosities for high-redshift, i.e.,
$z\gtrsim1$, X-ray detected AGN. In this sub-section we compare the [O iii] and
X-ray luminosities for our final KASH$z$ sample of 48 $z$$\approx$1.1–1.7
AGN that were targeted to observe [O iii] emission (see Section <ref>).
In Figure <ref> we show [O iii] luminosity
versus X-ray luminosity for the KASH$z$ targets. A correlation is
observed between these two quantities, although with a large scatter,
in qualitative agreement with studies of low-redshift AGN (e.g., ; ). We note
that we observe the same trend when plotting [O iii] flux versus
X-ray flux and therefore the correlation is not an artifact of flux limits. For the
48 targets, we find a median luminosity ratio of $\log(L_{{\rm [O~III]}}/L_{{\rm
X}})=-2.1^{+0.3}_{-0.5}$, where the quoted upper and lower
bounds contain the inner 66 per cent of the targets, i.e.,
roughly the 1$\sigma$ scatter.[We note that we quote the range on the [O iii] to X-ray
luminosity ratio for the inner 66 per cent, rather than the more
standard 68.3 per cent, applicable for 1$\sigma$, because 8 out of the 48 targets are undetected (i.e., 17%) and therefore
we can not constrain the 15.9 th percentile of the distribution.] To calculate the median ratio of [O iii]
to X-ray luminosity we have assumed that the non emission-line detected [O iii] targets fall into the bottom 50% of this distribution. Furthermore, to give the quoted range, we have assumed
that they intrinsically fall in the bottom 17 per cent of the $L_{{\rm
[O~III]}}/L_{{\rm X}}$ distribution. These are not unreasonable
assumptions given that all but one of these targets have 3$\sigma$ upper limits that fall very close to, or
below, this boundary (see Figure <ref>). We note
that, if we make our [O iii] luminosity measurements using a
2$^{\prime\prime}$ aperture (see Section <ref>) we find a median aperture correction
of 0.28 dex with a scatter of 0.09 dex. Therefore, using “total" luminosities could increase our
quoted $L_{{\rm [O~III]}}/L_{{\rm X}}$ ratio by $\approx$0.3 dex
(see Figure <ref>). Overall, our results
indicate that, typically, [O iii] luminosities are
$\approx$1% of the X-ray luminosities, with a factor of 2–3 scatter,
for $z\approx$1.1–1.7 X-ray detected AGN.
To compare to low-redshift AGN, in Figure <ref>, we show the objects from our $z<0.4$
AGN comparison sample (see Section <ref>) and the
relationship found for local Seyfert galaxies and QSOs by <cit.>. The KASH$z$
targets appear to broadly cover the same region of the $L_{{\rm [O
III]}}-L_{{\rm X}}$ parameter space as the $z<0.4$ AGN, but
with a slight tendency towards lower [O iii] luminosities. This
may be partly due to the different approaches used (or not used)
to aperture-correct the luminosities or a lack of correction for reddening to the
[O iii] emission-line measurements. Furthermore, the
$z<0.4$ AGN are initially drawn from an optically selected
SDSS sample, which is [O iii] flux limited (see
Section <ref>), while, in contrast, the KASH$z$ is
initially X-ray selected. Indeed, a plot of [O iii] flux versus X-ray flux
reveals that the deficit of low $L_{{\rm [O III]}}/L_{{\rm X}}$
luminosity ratios in the $z<0.4$ sample, observed in
Figure <ref>, is at least partly driven by the [O iii] flux
limit. <cit.> and <cit.> present further discussion on the
difference between optically selected and X-ray selected
The majority of the KASH$z$ [O iii] targets cover a narrow X-ray
luminosity range (i.e., $L_{{\rm X}}$=2$\times$10$^{43}$–2$\times$10$^{44}$ erg s$^{-1}$; see
Figure <ref>); therefore, we do not attempt to fit a luminosity-dependant relationship between
$L_{{\rm X}}$ and $L_{{\rm [O~III]}}$. However, we note that the local
$L_{{\rm X}}$–$L_{{\rm [O~III]}}$ relationship is found to be close to linear in log-log space, when
measured over five orders of magnitude in X-ray luminosity (i.e.,
$\log L_{{\rm [O~III]}} \propto (0.82\pm0.08)\log L_{{\rm X}}$;
; also see for an even more linear
relationship for a combined sample of optical and X-ray selected AGN). At the median
X-ray luminosity of our [O iii] KASH$z$ sample, i.e., $\log (L_{{\rm
X}}$/erg s$^{-1})=43.7$, the local relationship from
<cit.> results in a luminosity ratio of $\log(L_{{\rm [O~III]}}/L_{{\rm X}})=-1.86$, which is consistent with our KASH$z$ value of
$-2.1^{+0.3}_{-0.5}$. Therefore, based on these data, we have no
reason to conclude that there is any significant difference between the $L_{{\rm
X}}$–$L_{{\rm [O~III]}}$ relationship for our $z$$\approx$1.1–1.7 sample compared to local AGN.
In Figure <ref> we observe a large scatter of a factor of
$\approx$3. This is similar to the scatter seen in local X-ray selected AGN by <cit.>. The lack of a correction for X-ray obscuration may account for some
of this scatter; however, we note that the obscuration correction to hard-band (2-10 keV)
luminosities, at these redshifts, will be small except in the most
extreme cases (e.g., ; although see ). A more
significant factor on the amount of scatter will be the effect of various
amounts of dust reddening, in the host galaxy, that will affect the
[O iii] emission-line luminosities (see e.g.,
; ). Due to the lack of sufficient constraints on this
reddening effect across the sample, we do not attempt to correct for
this effect. A more interesting interpretation of the larger scatter in this
diagram is a possible lack of uniformity in the amount of gas inside the
AGN ionisation fields due to variations in opening angles and
inclinations, with respect to the host-galaxy gas. These effects will
lead to different volumes of gas being photoionised by the central
AGN. One further possible interpretation for the large
scatter observed in Figure <ref> is the different timescales of the accretion rates being
traced by X-ray versus [O iii] luminosity. Whilst the X-ray emission primarily traces nuclear
activity associated with the region very close to the accretion disk
(e.g., ), the [O iii] emission is found in NLRs, which can
extend on $\approx$0.1–10 kpc scales (e.g., ;
; ). Therefore, a NLR that is photoionised by an AGN could
act as an isotropic tracer of the time-averaged bolometric AGN
luminosity over $\approx$10$^{4}$ years, while in contrast, the X-ray
luminosity is an instantaneous measurement of the bolometric AGN
luminosity (see e.g., ; ; for further discussion on the
relative timescales of different AGN tracers).
H$\alpha$ emission-line luminosity ($L_{{\rm H}\alpha}$)
versus hard-band (2–10 keV) X-ray luminosity ($L_{{\rm X}}$) for the
34 $z$$\approx$0.6–1.1 KASH$z$ H$\alpha$ targets. In the top panel we show the total
emission-line luminosities, in the middle panel we show the
luminosities where the BLR components have been subtracted (i.e., the
NLR luminosities) and in the bottom panel we show the BLR
luminosities, where applicable. Non-detections are represented as hollow
symbols, which signify 3$\sigma$ upper limits, and the stars are the
same as in Figure <ref>. For clarity, the median
uncertainty on the line luminosities is shown at the bottom of each
panel. The arrow in the second panel shows the median aperture
correction when using a $\approx$2$\times$ larger
aperture (see Section <ref>). The dashed and
dot-dashed lines show the luminosity-dependent relationships for
Type 1 (T1) and Type 2 (T2) local AGN presented in
<cit.>. We also show our $z<0.4$ optical AGN
comparison sample (see Section <ref>).
§.§.§ H$\alpha$ luminosity versus X-ray luminosity
The H$\alpha$ emission-line luminosity is a very common tracer of
star-formation rates in local and high-redshift galaxies (e.g., see
; ). However, this is
complicated in AGN host-galaxies for two main reasons. Firstly, H$\alpha$
emission is also produced in the sub-parsec scale BLRs around SMBHs, which are most likely to be
photoionised by the central AGN and, secondly, the gas in the kpc-scale NLR of AGN
can also be photoionised by the AGN (e.g., ). The relative contribution of AGN versus
star-formation to producing the total H$\alpha$ luminosity ($L_{{\rm H\alpha}}$) will
significantly affect how well correlated $L_{{\rm H\alpha}}$ is with
tracers of bolometric AGN luminosity, such as the hard-band X-ray luminosity
($L_{{\rm X}}$). In this sub-section we compare the H$\alpha$ and
X-ray luminosities for our final KASH$z$ sample of 34 $z$$\approx$0.6–1.1
AGN that were targeted to observe H$\alpha$ emission (see Section <ref>).
In Figure <ref> we show $L_{{\rm H\alpha}}$ versus $L_{{\rm
X}}$ for our KASH$z$ targets. In the middle and bottom panels we
separate the H$\alpha$ luminosity into that associated with the NLRs
(i.e., $L_{{\rm H\alpha,NLR}}$) and that associated with the BLRs
(i.e., $L_{{\rm H\alpha,BLR}}$), respectively (see
Section <ref> for details of how we separate these
components). Following from our [O iii] analyses in the previous section, we do
not attempt to correct for X-ray obscuration or dust
reddening (although see discussion below). In Figure <ref> we also show
our $z<0.4$ AGN comparison sample (see Section <ref>)
and the relationships found for local AGN and QSOs by
<cit.>. We note
that, if we make our H$\alpha$ NLR luminosity measurements using a
2$^{\prime\prime}$ aperture (see Section <ref>) we find a median aperture correction
of 0.25 dex with a scatter of 0.08 dex.
In the bottom panel of Figure <ref> it can be seen
that there is a correlation between BLR region luminosity, $L_{{\rm H\alpha,BLR}}$, and $L_{{\rm
X}}$ for the $z<0.4$ AGN. These sources broadly follow the local
relationship observed for the Type 1 Seyferts and QSOs by
<cit.>, i.e., $\log L_{{\rm H\alpha,T1}} \propto (1.16\pm0.07)\log L_{{\rm X}}$. Although we are limited to seven BLR sources for the
KASH$z$ sample, our targets are found to be within the scatter of the
$z<0.4$ AGN, and therefore they qualitatively follow the same relationship
as low-redshift and local AGN (see Figure <ref>). It is not
surprising that the BLR luminosities are tightly correlated with
the X-ray luminosities (see Figure <ref>). The BLR is
directly illuminated by the central AGN, which is also responsible for the production of X-rays
around the accretion disk (e.g., ;
). Furthermore, due to the size scales of
the BLR that are typically on light-days to light-months (e.g.,
), the relative timescales of the accretion events
being probed by the X-ray emission and
BLR emission will be much closer compared to the
relative timescales between the X-ray emission and the kpc-scale NLR
emission (see discussion above for the [O iii] NLR emission).
In contrast to the BLR luminosities, there is little-to-no evidence for a correlation
observed between the NLR H$\alpha$ luminosity and X-ray luminosity for both the KASH$z$ sample and the $z<0.4$ comparison sample (see the middle panel of
Figure <ref>). In qualitative agreement with this for local AGN, the relationship is less steep for Type 2 AGN compared to
Type 1 AGN and QSOs by <cit.>; i.e.,
$\log L_{{\rm H\alpha,T2}} \propto (0.78\pm0.09)\log L_{{\rm
X}}$. However, some correlation does still exist in local
samples, whereas we do not see evidence for this in our current high-redshift sample. In addition to the timescale arguments already discussed,
this may be due to additional contributions to the NLR H$\alpha$
luminosities, in addition to photoionisation by AGN, such as by
star-formation processes (e.g., ). Indeed we observe that a significant
fraction of our sample have [N ii]/H$\alpha$ emission-line
ratios that could be produced by H ii regions (see
Section <ref>). The NLR region H$\alpha$ luminosities of the KASH$z$ targets
may follow an even shallower trend than the local relationship; however, the deviation is only observed at
the highest X-ray luminosities, where we are currently limited to a lower number
of sources (see Figure <ref>). We also note that the KASH$z$ AGN
are likely to be systematically low in $L_{{\rm H\alpha,NLR}}$ due to the lack of obscuration
correction (also see the discussion on aperture effects above). For example, the correction to the NLR
luminosities would be $\approx$0.7 dex, assuming the
median $A_{{\rm V,H\alpha}}=1.7$ for the star-forming galaxy comparison
sample at the same redshift (Stott et al. 2015; Section <ref>), while the average Balmer decrement of
the $z<0.4$ AGN comparison sample implies a median correction of only
$\approx$0.3 dex for the $z<0.4$ AGN (following ). Additionally, our X-ray selection compared to the optically selected comparison samples may
also provide a systematic effect towards lower line luminosities for
the high-redshift sources (see discussion above for the [O iii] targets).
Stacked [O iii]5007 emission-line
profiles for the 40 [O iii] detected KASH$z$ targets and the
X-ray luminosity matched $z<0.4$ AGN comparison sample (see Section <ref>). The
dotted curves show the stacked data and the dashed and solid curves are
fits to these stacks. The upward arrows show, from left-to-right,
the 5th, 10th, 90th and 95th percentile velocities of the KASH$z$
stack. The overall emission-line width of $W_{80}$=810 km s$^{-1}$ is also illustrated
(see Section <ref>). On average, the KASH$z$ AGN show
a broad and asymmetric emission-line profile, with velocities
reaching $\approx$1000 km s$^{-1}$. The low-redshift AGN
have a very similar average emission-line profile to the
high-redshift AGN, for these luminosity-matched samples.
§.§ The prevalence and drivers of ionised outflows
A key aspect of KASH$z$ is to constrain the prevalence of ionised
outflow features observed in the emission-line profiles of
high-redshift AGN. Additionally, KASH$z$ is designed to assess which AGN and host-galaxy properties are associated with the highest
prevalence of high-velocity outflows. One of the most common
approaches to search for ionised outflows is to look
for very broad and/or asymmetric emission-line profiles in the ionised gas
species such as [O iii] and non-BLR H$\alpha$ components (e.g.,
; ; ;
; ). For example, asymmetric emission-line profiles (most commonly a blue
wing) that reach high velocities (i.e., $\approx$1000 km s$^{-1}$) are
very difficult to explain other than through outflowing material
(e.g.,; ). Furthermore, extremely broad emission-line
profiles (i.e, $W_{80}\gtrsim600$ km s$^{-1}$) are very unlikely to be the result of
galaxy kinematics, but instead trace outflows or high levels of
turbulence (e.g., ; ; also see discussion
in Section <ref>) and studies of large
samples of low-redshift AGN have shown that the gas that is producing such broad emission-line
profiles is not in dynamical equilibrium with their host galaxies
(see and ).
In the following sub-sections we assess the
prevalence and drivers of ionised outflow features in the galaxy-integrated emission-line
profiles of our KASH$z$ AGN sample, following similar methods to
<cit.> who study $z<0.4$ AGN (see
Section <ref>). More specifically, we investigate the
distributions of emission-line velocity widths of individual sources, in combination with
emission-line profile stacks. For
clarity and ease of comparison to previous studies, we separate the discussion of the $z$$\approx$1.1–1.7 [O iii]
sample from the $z$$\approx$0.6–1.1 H$\alpha$ sample
(these are defined in Section <ref>). Furthermore, the [O iii] emitting gas is more
likely to be dominated by AGN illumination, while the H$\alpha$
emission may also have a significant contribution from star formation
(see discussion in Section <ref>). We defer a detailed comparison of these
two ionised gas tracers to future papers, which will be based on
spatially-resolved spectroscopy using both emission lines for the same
targets; however, see the works of <cit.> and <cit.> for
IFS data covering both H$\alpha$ and [O iii] measurements for two high-redshift AGN.
§.§.§ The distribution of [O iii] emission-line
velocity widths
In Figure <ref> we show the [O iii] emission-line
profiles, and our best-fitting solutions, for all the $z$$\approx$1.1–1.7 KASH$z$ targets. The parameters of all of the fits are provided in
Table <ref>. We identify secondary broad components (following the methods described in Section <ref>), with
FWHM$\approx$400–1400 km s$^{-1}$, in the emission-line profiles
for 14 out of the 40 [O iii] detected targets (i.e., 35 per cent). The velocity offsets of these broad components, with respect to the narrow
components, reach up to $|\Delta v|\approx500$ km s$^{-1}$. We note that <cit.> find that four out of
their eight $z\approx$1.5 X-ray luminous AGN identify a secondary
broad emission-line component at high-significance in their
[O iii] spectra, which is broadly consistent with our fraction given the low numbers
involved. While the fraction of broad emission-line components in our KASH$z$
sample already indicates a high prevalence of high-velocity ionised gas in high-redshift X-ray
AGN, these measurements do not provide a complete picture across all of the
targets. This is because it is very difficult to detect multiple
Gaussian components when the signal-to-noise ratio is modest; i.e., the detection of a second Gaussian component is limited to the highest
signal-to-noise ratio spectra. Therefore, in the following
discussion, we follow two methods to overcome these challenges. Firstly we assess the average emission-line profiles using stacking
analysis (see Section <ref>) and, secondly, we use a non-parametric definition to characterise the overall line width,
(i.e., $W_{80}$ which is the width that encloses 80 per cent of the
emission-line flux; see Section <ref>).
We show the stacked [O iii]5007 emission-line profile for our KASH$z$ targets in Figure <ref>. The overall-emission line width of this
average profile is $W_{80}$=810$^{+130}_{-220}$ km s$^{-1}$ (see Section <ref>), where the upper and
lower bounds indicate the 68 per cent range from bootstrap
resampling that stack. This indicates that, on average, the ionised
gas in these AGN have high-velocity kinematics, that are not
associated with the host galaxy dynamics. Furthermore, the average
emission-line profile clearly shows a luminous blueshifted broad
wing, which reveals high-velocity outflowing ionised gas out to velocities of
$\approx$1000 km s$^{-1}$. A preference for blueshifted broad
wings, compared to redshifted broad wings, has been previously been observed for both
high- and low-redshift AGN samples (e.g., ; ;
; ; ;
; ). This can be explained if the far-side of any outflowing gas, that is moving
away from the line of sight, is obscured by dust in the host galaxies (e.g.,
; ). To observe a
redshifted component, the outflow would need to be highly-inclined or extended beyond the obscuring
material (e.g., ; ; ;
While informative, the average emission-line profile shown in
Figure <ref> hides critical
information on the underlying distribution of ionised gas kinematics
in our sample. Therefore, in Figure <ref> we show the distribution of the [O iii] velocity-width values,
$W_{80{\rm ,[O~III]}}$, for the individually [O iii] detected targets. In the top panel we show the raw distribution and in
the bottom panel we show the cumulative distribution. 1$\sigma$ uncertainties
on the cumulative distribution have been calculated assuming Poisson
errors, suitable for small number statistics, following the analytical
expressions provided by <cit.>. We use the same uncertainty
calculations for all of the percentages presented for the remainder of
this section. We find that 50$^{+14}_{-11}$ per cent
of the [O iii] targets have velocity-widths indicative of tracing ionised outflows or highly turbulent material, i.e.,
$W_{80{\rm,[O~III]}}>600$ km s$^{-1}$ (e.g. ; ). This fraction ranges over
(42–58) per cent, when including the 8 non-detected targets (see
Section <ref>), for which we have no $W_{80{\rm
,[O~III]}}$ measurements.
Top: Histograms of the overall emission-line velocity width,
$W_{{\rm 80,[O III]}}$, for the 40 $z\approx$1.1–1.7 [O iii] detected
KASH$z$ targets (hatched) and a luminosity-matched
sample of 1000 $z<0.4$ optical AGN (filled; see Section <ref>). The KASH$z$ AGN show a very similar
distribution of velocities as the luminosity-matched
low-redshift AGN sample. This is further demonstrated in the bottom panel which shows the cumulative distributions. We split the KASH$z$ sample in
half, separating at $L_{X}=6\times10^{43}$ erg s$^{-1}$, and find that high-velocity
gas kinematics are more prevalent in higher luminosity AGN; for example,
it is $\gtrsim$2$\times$ more likely that the higher luminosity AGN have
$W_{80}\gtrsim$ 600 km s$^{-1}$, compared to the lower-luminosity
AGN (see vertical dotted line; Section <ref>).
To compare the prevalence of outflow features in our high-redshift AGN
sample, to low-redshift AGN, we show the distribution of
velocity-widths for our $z<0.4$ luminosity-matched comparison samples
in Figure <ref> (see Section <ref>). The
luminosity-matching is performed to account for the
observed correlation between luminosity and emission-line widths (; also see
Section <ref>). We note that we obtain the same conclusions if we match
by either $L_{{\rm [O III]}}$ or $L_{{\rm X}}$. In Figure <ref>, the [O iii] line-width distributions look indistinguishable between the KASH$z$ sample and the
luminosity-matched $z<0.4$ AGN samples. A two-sided KS test shows no
evidence that the two different redshifts have significantly different
velocity-width distributions (i.e., a $\approx$60 per cent chance
that the two redshift ranges have velocity-width values drawn from
the same distribution). In Figure <ref>, we further demonstrate that luminosity-matched AGN, from both redshift-ranges, have
similar [O iii] emission-line profiles by showing that the
stacked emission-line profiles look the same. This result implies that the prevalence of ionised outflows in low-redshift and
high-redshift AGN are very similar for AGN of the same luminosity. This is despite the fact that the average star-formation
rates of X-ray AGN are $\approx$10$\times$ higher at $z$$\approx$1
compared to $z\approx$0.2, irrespective of X-ray luminosity
(). Therefore, this result provides indirect
evidence that the prevalence of these high-velocity outflows is not
greatly influenced by the level of star formation. We draw similar conclusions
for the H$\alpha$ KASH$z$ sample in Section <ref>.
Using our representative targets (see Figure <ref> and
Section <ref>), it is now possible to place some previous observations of
high-redshift X-ray AGN into the context of the overall population. For example,
the SINFONI observations of XCOS-2028 were used by
<cit.> to present evidence of star-formation suppression by
the outflow observed in this source (see
our spectra for this source in Figure <ref>). This
conclusion was based on their observed deficit of NLR H$\alpha$ emission, a possible tracer of star
formation (see Section <ref> and
Section <ref>), in the location of the outflowing
[O iii] component. Based on our analyses, this source has an
emission-line velocity width of $W_{80}\approx730$ km s$^{-1}$. We find $\approx$30 per cent of the
overall X-ray AGN population have these emission-line widths or
greater. Therefore, this target does not have an exceptional outflow;
however, it is yet to be determined how common the observed deficit of
NLR H$\alpha$ emission in this source is in the
overall high-redshift AGN population.
Emission-line velocity width ($W_{80}$) versus [O iii] luminosity (left) and hard-band (2–10 keV) X-ray luminosity (right) for the $z\approx$1.1–1.7 KASH$z$ sample
(circles) and the $z<0.4$ AGN comparison sample (contours and crosses). The dashed lines show the $W_{80}$
value for the median spectral resolution for the $z<0.4$
sources. Although a wide distribution of velocities are observed, the most luminous AGN (either by line luminosity or X-ray
luminosity) preferentially host extreme gas velocities. For example, all
of the KASH$z$ AGN with $L_{{\rm[O~III]}}>10^{42.2}$ erg s$^{-1}$
have velocity-widths of $\gtrsim$600 km s$^{-1}$ (see
dotted line).
§.§.§ The physical drivers of the high-velocity outflows observed in [O iii]
It is of fundamental importance to constrain how AGN and host-galaxy properties are
related to the prevalence and properties of galaxy-wide outflows. For
example, in most cosmological models, the accretion rates of the AGN fundamentally determine the velocities and
energetics of the outflows (e.g., ), while some models, concentrating on individual sources, have invoked the mechanical output
from radio jets as a plausible outflow driving mechanism (e.g.,
). In this sub-section we investigate the role of
AGN luminosity, radio luminosity and X-ray obscuration on the
prevalence of ionised outflows in our KASH$z$ [O iii] sample.
In Figure <ref>, we plot the emission-line velocity width ($W_{{\rm 80,[O III]}}$) as a
function of [O iii] luminosity and X-ray luminosity for the
[O iii] detected KASH$z$ targets. Both [O iii] luminosity and X-ray luminosity may serve
as a tracer for the bolometric AGN luminosity, potentially on
different timescales (see Section <ref>). We find
that the most luminous AGN (both based on [O iii] and X-ray) are associated with the highest
velocities, although a large spread in velocities is seen at
lower luminosities. These same trends are also seen in our
low-redshift comparison sample (see crosses in
Figure <ref>). To quantify the effect of AGN luminosity on the prevalence
of ionised outflows, we split the [O iii] detected targets into two
halves by taking the 20 “lower” and 20 “higher” X-ray luminosity
targets, resulting in a luminosity threshold of $L_{{\rm
X}}=6\times10^{43}$ erg s$^{-1}$ between the two subsets. In
Figure <ref> we show the cumulative distributions of line-widths for
these two subsets. It can be seen that there is higher probability of
observing extreme ionised gas velocities in the “higher” luminosity sub-set. For example, 70$^{+24}_{-18}$ per cent of the “higher” luminosity
targets have line widths of $W_{80}>$600 km s$^{-1}$, while only
30$^{+18}_{-12}$ per cent of the “low” luminosity targets reach these
line widths. A two-sided KS test indicates
only a $\approx$2 per cent chance
that the two luminosity bins have velocity-width values drawn from
the same distribution. In future papers, we will be able to test this result to higher
significance as the KASH$z$ sample increases.
To first order, the results described above indicates that the highest
outflow velocities are associated with the most powerful AGN. This
result has been quoted throughout the literature for low-redshift AGN,
for both ionised outflows and for molecular outflows (e.g., ; ; ; ; ). However,
in their study of ionised ionised outflows of $\approx$24,000 AGN, <cit.>
found that the highest velocity outflows are more fundamentally driven
by the mechanical radio luminosity ($L_{{\rm 1.4\,GHz}}$) of the AGN, rather than the
radiative (i.e., [O iii]) luminosity. This result could either
be an indication that small-scale radio jets are driving high-velocity outflows, as observed in spatially-resolved studies of some low-redshift AGN (e.g.,
; ), or that radiatively-driven outflows are
producing shocks in the ISM which result in the production of radio
emission (; ; also see
). For our KASH$z$ targets, we are currently limited to only eight [O iii]
detected targets which we can define as “radio luminous” (i.e.,
$L_{{\rm 1.4\,GHz}}>10^{24}$ W Hz$^{-1}$; see
Section <ref>). A higher fraction of the radio luminous
sample have high velocity line widths of $W_{80}>$600 km s$^{-1}$ compared to the
non radio-luminous sources, i.e., 6/8 or 75$^{+25}_{-30}$ per cent
compared to 13/30 or 43$^{+16}_{-12}$ per cent (see Figure <ref>);[We note
that two of our [O iii] detected targets do not have the
required radio constraints to classify them as either radio luminous
or not (see Section <ref>).] however, the uncertainties
on these fractions are high. Furthermore, the radio
luminous sources are preferentially associated with higher X-ray
luminosity AGN and we do not currently have sufficient numbers of targets to control
for this (see Figure <ref>). We note that it was possible to control for AGN
luminosity, in this case [O iii] luminosity, for the
low-redshift study of <cit.>. In summary, based on the
current sample, we can not yet determine if the radio luminosity, or
the X-ray luminosity is more fundamental in driving the highest-velocity
outflows observed in high-redshift AGN.
Emission-line width ($W_{80}$) versus the ratio between
2–10 keV and 0.5–2 keV X-ray fluxes for the 40 [O iii]
detected KASH$z$ sample (circles). The X-ray axis here serves as a proxy for
observed photon index ($\Gamma_{obs}$), and also as an estimate for
obscuring column density (for an assumed intrinsic photon index and
known redshift). We classify the sources
with $F_{{\rm 2-10keV}}/F_{{\rm 0.5-2keV}}>3.03$ as “obscured
candidates” (corresponding to $N_{H}\gtrsim$(1–2)$\times10^{22}$ cm$^{-2}$;
see Section <ref>). We find no evidence that
high-velocity outflows are preferentially observed in the obscured X-ray AGN.
It has been suggested that massive galaxies may go through an
evolutionary sequence where obscured AGN reside in star-forming
galaxies during periods of rapid SMBH growth and galaxy growth during
which the AGN drive outflows that drive away the enshrouding material
to eventually reveal an unobscured AGN (e.g., ;
). Therefore, it may be expected that outflows are
more preferentially associated with X-ray obscured AGN. To test this, we compare the [O iii] emission-line velocities of our X-ray “obscured
candidates” to those “unobscured candidates”, which we separate at
an obscuring column density of $N_{H}\approx2\times10^{22}$ cm$^{-2}$
using a simple hard-band to soft-band flux ratio technique (see Section <ref>). In Figure <ref> we show
the emission-line velocity width, $W_{{\rm 80,[O III]}}$, as a function of this
flux ratio. We find no significant difference between the two
populations; that is, we find that 45$_{-9}^{+20}$ per cent of these sources have
$W_{{\rm 80,[O III]}}>600$ km s$^{-1}$, compared to a similar
fraction of 54$^{+15}_{-14}$ per cent for the unobscured
sources. This result implies that high-velocity ionised outflows are
certainly not uniquely, and do not appear to be preferentially,
associated with X-ray obscured AGN. This may be in
disagreement with some evolutionary scenarios for galaxy evolution; however, there may be a
population of heavily obscured AGN that are missed from even the deepest X-ray
surveys (see Section <ref>) that are not present in the current sample.
§.§.§ The prevalence and drivers of outflows observed in H$\alpha$
Stacked H$\alpha$+[N ii] emission-line
profiles for the 25 Type 2 H$\alpha$ detected KASH$z$ targets and the
comparison samples of: (1) star-forming galaxies at the same redshift
and (2) $z<0.4$ Type 2 AGN (see Section <ref>). The
dotted curves show the stacked data and the dashed and solid and
dashed curves are fits to the stacks. The KASH$z$ AGN clearly show
much broader emission-line profiles, and higher [N ii]/H$\alpha$ emission-line ratios than the star-forming
In Figure <ref> and Figure <ref> we show the H$\alpha$+[N ii] emission-line profiles, and our best-fitting solutions, for all targets
in the current $z$$\approx$0.6–1.1 KASH$z$ sample. The parameters for
all of the fits are provided in
Table <ref>. We detected 32 out of the 34 targets (see
Section <ref>) and of these 32, we identified a BLR in seven of the targets (see
Figure <ref>). For this study, we are interested in the kinematics of the host-galaxy, or equivalently the
NLR, and not the BLR. Therefore, for these seven BLR sources the
quoted emission-line velocity widths (i.e., the $W_{{\rm 80,H}\alpha}$), are for the NLR components only (see Section <ref>).
In Figure <ref> we show the stacked H$\alpha$+[N ii] emission-line profile for our 25 H$\alpha$ Type 2 AGN. The overall-emission line width of this
average profile is $W_{{\rm 80, H\alpha}}$=440$^{+50}_{-13}$ km s$^{-1}$ (see Section <ref>), where the upper and
lower bounds indicate the 68 per cent range from bootstrap
resampling that stack. This velocity width is lower than the
$W_{{\rm 80, [O III]}}$=810$^{+130}_{-220}$ km s$^{-1}$ observed in the stacked [O iii]
emission-line profile (Figure <ref>). This difference
is likely to be due, in part, to the almost order of magnitude difference in the average X-ray luminosity of the
H$\alpha$ sample compared to the [O iii] sample (i.e.,
2$\times$10$^{43}$ erg s$^{-1}$ compared to
1$\times$10$^{44}$ erg s$^{-1}$; see
Figure <ref>), which we have shown be a key driver for the
observed prevalence of high-velocity outflows (Section <ref>). However, as already briefly
mentioned, this may also be due to these two emission lines preferentially
tracing different regions of gas in the host galaxy, with H$\alpha$
likely to have a larger contribution from star-forming
regions. While we currently lack systematic
IFS studies of AGN covering both [O iii] and H$\alpha$ emission lines, observations of some
AGN have already indicated that the NLR H$\alpha$ emission has a contribution from star-forming regions
and is not necessarily dominated by the high-velocity outflows as
is the case for [O iii] emission (e.g., ;
). Furthermore, a strong blue wing can be observed
in the stacked KASH$z$ emission-line profile for the [N ii]
emission-line, which has a larger velocity
width than H$\alpha$ with $W_{{\rm 80, [N
II]}}$=590$^{+120}_{-50}$ km s$^{-1}$
(Figure <ref>). This high-ionisation line may be more
analogous to the [O iii] emission, and preferentially trace
outflowing/turbulent material compared to H$\alpha$. Indeed, the difficulty in
identifying broad-underling outflow components in H$\alpha$
emission could be that this emission line preferentially traces
rotation of the host galaxy and beam-smearing of this emission could dilute centrally-located broad components
(; ). We are unable to
de-couple the [N II] from the H$\alpha$ across the whole H$\alpha$ sample due
to limited signal-to-noise ratios (see Section <ref>) and therefore we
may be underestimating the outflow velocities in these sources. However, H$\alpha$ emission has been
used to identify outflows in AGN host-galaxies (e.g., ; ; ) and
analysis of this emission line using these methods provides an
informative comparison to other studies.
In Figure <ref> we show the distribution of the H$\alpha$
line-width values, $W_{80{\rm ,H\alpha}}$, for the 32 detected
targets, using the same format as above for the [O iii]
emission. We find that four of our KASH$z$ targets (i.e., 13$^{+9}_{-6}$ per cent) have $W_{{\rm
80,H}\alpha}>600$ km s$^{-1}$, indicative of emission that is
dominated by outflowing material (see discussion at the start of this
section). We note that the range on this percentage is 12–18 per cent if the two undetected targets are included, for which
we have no constraints on $W_{{\rm 80,H}\alpha}$. In agreement with the
[O iii] results (Section <ref>), we find no
appreciable difference between the distribution of line-widths between
the KASH$z$ sample and our low-redshift luminosity-matched comparison
sample (see Section <ref>).
Top: Histograms of the overall emission-line velocity width, $W_{{\rm 80,H\alpha}}$, for the 32
$z$$\approx$0.6–1.1, H$\alpha$ detected KASH$z$ targets (hatched) and the $z$$\approx$0.6–1.1 comparison
sample of star-forming galaxies from KROSS (filled; see
Section <ref>). The AGN preferentially have higher emission-line velocities
than the star-forming galaxies. This is further demonstrated in the
bottom panel which show the cumulative distributions. We also show
the KROSS sample after applying two stellar mass cuts, where the
the $M_{\star}>3\times10^{10}$ M$_{\odot}$ sub-sample is more comparable to the X-ray AGN host galaxies (Section <ref>). We also show the cumulative
distribution of the luminosity-matched comparison sample of $z<0.4$ AGN
(see Section <ref>) and find that the velocity distribution
is very similar to that of our high-redshift AGN.
In Figure <ref> we compare the H$\alpha$ emission-line
velocity distributions for our KASH$z$ targets with our comparison
sample of star-forming galaxies that are at the same redshift from KROSS (see
Section <ref>). It can clearly be seen that AGN
preferentially have higher emission-line widths than the star-forming
galaxies. For example, only 3/378 (i.e.,
0.8$_{-0.4}^{+0.8}$ per cent) of the star-forming galaxy sample
reach velocity-widths of $W_{{\rm 80,H}\alpha}>600$ km s$^{-1}$,
compared to 13$^{+9}_{-6}$ per cent found for the AGN. This is also demonstrated in
Figure <ref>, where we show the stacked emission-line
profiles for both samples. These results provide indirect evidence that
the high-velocity features we observe are not pre-dominantly
driven by star-formation. This is because X-ray AGN at these
redshifts have average star-formation rates that are consistent with the
global star-forming galaxy population of the same redshift (e.g.,
; ), and are possibly even distributed to lower
median star-formation rates (). We further test this conclusion by plotting the
velocity-width as a function of observed NLR H$\alpha$ luminosity
($L_{{\rm H\alpha,NLR}}$) for both the KASH$z$ sample and
star-forming comparison sample in Figure <ref>.
The observed NLR H$\alpha$ luminosity (i.e., excluding any BLR
components) is a tracer of the star-formation rates in
star-forming galaxies; however, for the
AGN-host galaxies these will, in general be relative over-estimates due to
the additional contribution of photoionisation by the central
AGN. We find that the KASH$z$ AGN
have a very similar distribution of $L_{{\rm H\alpha,NLR}}$ to our
star-forming galaxy comparison sample, indicating that they have similar, or possibly
lower star-formation rates. Despite this, the AGN-host galaxies preferentially have
larger velocity widths, and a higher fraction of sources indicative
of hosting high-velocity ionised outflows (Figure <ref>).
Emission-line velocity width, $W_{{\rm 80,H\alpha}}$ versus narrow-line region
H$\alpha$ luminosity ($L_{{\rm H\alpha,NLR}}$) for the KASH$z$ AGN targets (circles) and
the KROSS star-forming galaxies at the same redshift (squares). The AGN show a similar
distribution of $L_{{\rm H\alpha,NLR}}$, but preferentially have the
higher velocities. $L_{{\rm H\alpha,NLR}}$ is a tracer of the
star-formation rates for the galaxies and the AGN (although this
will be biased upwards for the AGN; see
Section <ref>). Therefore, the higher emission-line
velocities in the AGN are not the result of higher star-formation rates.
In addition to investigating the role of the star formation rates, it
is also important to consider the effect of
host galaxy masses on the H$\alpha$ emission-line widths. In the
cases where the H$\alpha$ emission-line profiles are
dominated by galaxy kinematics (which is the case for the
majority of the star-forming KROSS galaxies; e.g., Stott
et al. 2015; Swinbank et al. in prep), the line-widths will be driven to higher
values in galaxies with higher stellar masses due to the increased
velocity gradients across these galaxies. To demonstrate this,
in Figure <ref>, we also show the cumulative
distributions of line-widths for the star-forming galaxies when we apply increasing mass
cuts. As expected, the higher mass galaxies tend to have the broader
emission-line widths. We do not attempt to derive stellar masses for
our AGN, due to the variable quality of photometric data sets
available for our targets; however, we note that X-ray AGN appear to
typically have stellar masses of $\gtrsim3\times$10$^{10}$ M$_{\odot}$ (e.g.,
; ; ; ). Therefore,
in Figure <ref>, we compare the star-forming galaxy
sample, but limiting it to galaxies with stellar masses
$>3\times$10$^{10}$ M$_{\odot}$. We still find that the KASH$z$ AGN
have a higher
prevalence of the highest velocity widths, compared to this higher
mass subset of the star-forming galaxies, with only
3$^{+6}_{-1}$ per cent exhibiting emission-line
velocity widths of $W_{{\rm 80,H}\alpha}>600$ km s$^{-1}$.
To further test the possible role of mass in driving the high emission-line
velocity widths observed in our targets, in Figure <ref> we plot $W_{{\rm
80,H}\alpha}$ as a function of the emission-line ratio
$\log$([N ii]/H$\alpha$). This emission-line ratio is a tracer of
the metallicity of star-forming galaxies (e.g., ;
; ), as well as an indicator for the source of ionising
radiation (e.g. ; ). Furthermore,
there is an observed relationship between mass
and metallicity and therefore an expected
relationship between this emission-line ratio and stellar mass (e.g., ;
; ; ). We can also make a crude prediction for the
relationship between $W_{{\rm 80,H}\alpha}$ and mass, under the
assumption that the line-width is a tracer of the stellar
velocity dispersion. Therefore, for a given stellar mass, we predict
the position galaxies would be located in Figure <ref> by combining: (1) the
observed $z=0.7$ mass-metallicity relation (following ) and (2) the observed mass-velocity
dispersion relationship for massive galaxies (following
). The majority of the star-forming galaxies
appear to broadly follow the expected trend. Furthermore, the measurements from
the stacked average emission-line profile (Figure <ref>) are in agreement
with the rough mass-driven prediction for the average mass of these
galaxies (i.e., $\log (M_{\star}/M_{\odot})$=10.3; see Figure <ref>). In
contrast, the AGN typically have higher $\log$([N ii]/H$\alpha$)
emission-line ratios and higher velocity widths that the
star-forming galaxy sample (also visible in the stacked profiles; Figure <ref>) and a positive correlation is observed between
these two quantities. Such a positive correlation has been shown
to be a tracer of shocks and outflows in the ISM through
IFS observations of AGN and star-forming galaxies (e.g., ;
). Interestingly, the small number of star-forming galaxies with high line-widths (i.e., $W_{80}\gtrsim400$ km s$^{-1}$) appear to
follow the same relationship as the KASH$z$ AGN, which may
indicate that these galaxies also have a contribution from shocks
and/or host AGN that were not detected in the X-ray surveys. We clarify that some of the AGN targets have $\log$([N ii]/H$\alpha$) emission-line
ratios that could also be photoionised by H ii regions (see
Figure <ref>) and we will explore these ideas further when exploring the
spatially-resolved outflow kinematics and emission-line flux ratios of
the KASH$z$ AGN in future papers.
Emission-line velocity width, $W_{{\rm 80,H\alpha}}$ versus
$\log$([N ii]/H$\alpha_{\rm NLR}$) emission-line ratio for the KASH$z$ AGN targets (circles) and
the KROSS star-forming galaxies (squares). The larger symbols
containing the stars are measured from the stacked emission-line
profiles (Figure <ref>). The vertical dashed line indicates the
maximum emission-line ratio expected for photoionised H ii regions
(e.g., ). The green track shows the
predicted trends as a function of mass (following the
mass-metallicity and mass-velocity dispersion relations; see
Section <ref>), where the triangles highlight various $\log(M_{\star})$
values in half dex bins starting at 9.0. The positive correlation observed for most of the AGN (i.e., those with $\log$([N ii]/H$\alpha$)$\gtrsim$$-0.5$ may be indicative of
outflows/shocks driving the velocity widths (see Section <ref>). Most
of the star-forming galaxies roughly follow the predicted mass-driven trend
(Section <ref>) and the stacked average is
consistent with the mass-driven prediction at their average mass, i.e., $\log (M_{\star}/M_{\odot})$=10.3. However, they appear to follow the same trend as the AGN at the largest velocity widths ($W_{80}\gtrsim$400 km s$^{-1}$).
We conclude that the highest H$\alpha$ line-widths (i.e., those
with $W_{80}>600$ km s$^{-1}$ and possibly those with
$W_{80}\gtrsim400$ km s$^{-1}$) are at least partially driven by ionised
outflow kinematics and/or shocks in the ISM. Furthermore, there is a increased likelihood to find these
highest ionised gas velocities in X-ray identified AGN compared to star-forming
galaxies at the same redshift that have similar star-formation
rates and masses. This is in agreement with local IFS studies that
have found higher-velocity ionised outflows, traced by H$\alpha$
emission, in star-forming galaxies that host AGN compared to those which do not (e.g., ; ).
§ CONCLUSIONS
We have presented the first results of the ongoing KMOS AGN Survey at
High redshift (KASH$z$). The first 89 targets, which are presented
here, are high-redshift ($z$$\approx$0.6–1.7) X-ray detected AGN, with hard-band (2–10 keV) luminosities in
the range $L_{{\rm X}}=10^{42}$–10$^{45}$ erg s$^{-1}$. The
targets have a distribution of X-ray luminosities that are representative of
the parent X-ray AGN population. The majority of the targets (79) were
observed with KMOS, supplemented with archival SINFONI observations of
10 targets. All of the observations were carried out in the $J$-band
and we detected 86 (i.e., 97 per cent) of the targets in continuum and/or
emission lines. However, for the analyses in this work, we excluded
seven un-reliable non-detections (e.g., due to inaccurate photometric
redshifts; Section <ref>). This leaves a final sample of 82
targets, for which we have presented results and discussion based on their galaxy-integrated
emission-line profiles.
We detected 72 of the final sample (i.e., 88 per cent) in
emission lines, of which 40 out of 48 targets were detected in [O iii] (the $z$$\approx$1.1–1.7 targets) and 32 out of 34 targets were detected
in H$\alpha$ (the $z$$\approx$0.6–1.1 targets). We
have explored the emission-line luminosities as a function of X-ray
luminosity for our targets and by characterising the individual emission-line profiles
and using stacking analyses, we have investigated the
prevalence and drivers of the broad and asymmetric emission-line profiles that are indicative of ionised outflows. Our main conclusions
are listed below.
* We find a median X-ray luminosity to [O iii] luminosity
ratio of $\log(L_{{\rm [O~III}]}/L_{X})=-2.1_{-0.5}^{+0.3}$, where
the range is roughly the 1$\sigma$ scatter. The observed relationship between these
two quantities for our high-redshift sample is broadly consistent with that
found for low-redshift AGN and local Seyferts and QSOs (see Figure <ref>). Our
results indicate that the [O iii] luminosities are typically
$\approx$1% of the X-ray luminosities for $z\approx$1.1–1.7 X-ray
AGN. The large scatter of a factor of $\approx$3, may be due to several
observational effects including dust reddening, or due to intrinsic
physical effects, such as a higher level of variability in the
sub-pc-scale production X-ray emission compared to
the kpc-scale production of [O iii] emission (Section <ref>).
* Our seven Type 1 H$\alpha$ targets have broad-line
region luminosities ($L_{{\rm H\alpha,BLR}}$) that are broadly
consistent with the correlation observed between $L_{{\rm H\alpha,BLR}}$ and
$L_{X}$ for low-redshift and local AGN. Although limited by
small numbers of very luminous sources, we find no evidence for
a correlation between the narrow-line
region H$\alpha$ luminosity and $L_{X}$ (see
Figure <ref>; Section <ref>).
* High-velocity emission-line features are
common in our [O iii] sample, with $\approx$50 per cent of
the targets exhibiting velocities indicative of being dominated
by outflowing ionised gas or highly turbulent material (i.e., emission-line velocity widths of $W_{{\rm 80,[O
III]}}$$>$600 km s$^{-1}$; see
Figure <ref>; Figure <ref> and
Section <ref>). On average the emission-line
profiles have a prominent blue-shifted wing, implying outflowing material. Outflowing or highly turbulent
material that does not dominate the individual emission-line profiles could
be even more common.
* The high-velocity [O iii] kinematics are more prevalent
for targets with higher AGN luminosities. For example, $\approx$70 per cent of
the $L_{X}>6\times10^{43}$ erg s$^{-1}$ targets have [O iii]
line widths of $W_{{\rm 80,[O III]}}$$>$600 km s$^{-1}$, while only $\approx$30 per cent of the
$L_{X}<6\times10^{43}$ erg s$^{-1}$ targets reach these line
widths (see Figure <ref> and Figure <ref>). Using our current sample, we are unable to determine
the role of radio luminosity in driving this trend (Section <ref>).
* Based on our current X-ray detected sample, we find no evidence
that the highest ionised gas velocities are
preferentially associated with X-ray obscured AGN (i.e., those with
$N_{H}\gtrsim10^{22}$ cm$^{-2}$), compared to X-ray unobscured AGN (see
Figure <ref>), in contrast to the predictions of some evolutionary
scenarios (Section <ref>).
* We compared the emission-line widths of our H$\alpha$ detected
AGN targets, with a redshift matched sample of star-forming
galaxies. Despite a similar distribution of H$\alpha$ luminosities
(excluding the broad-line region components), and implied similar
star-formation rates, the AGN-host galaxies exhibit a much higher
prevalence of high ionised gas velocities (see
Figure <ref> and Figure <ref>). For example, $\approx$13 per cent of the AGN
have H$\alpha$ emission-line widths of $W_{80}>$600 km s$^{-1}$,
whilst only $\approx$1 per cent of the star-forming galaxy sample reach these line
widths (Section <ref>).
* For both the [O iii] and H$\alpha$ KASH$z$ targets, we find no significant difference between the distribution of
velocity widths for our high-redshift AGN sample and
luminosity-matched comparison samples of $z<0.4$ AGN. Under the
assumption that the most extreme ionised gas velocities are
associated with outflows, to first
order, this implies that it is just as likely to find an ionised
outflow of a certain velocity in low-redshift AGN as in
high-redshift AGN of the same luminosities (see Figure <ref> and
Figure <ref>). This is despite the order-of-magnitude
global decrease in average star-formation rates towards the lower
redshift sample (see Section <ref> and Section <ref>).
Based on our systematic study of a representative sample of
high-redshift X-ray detected AGN, we have evidence that high-velocity
ionised outflows are prevalent, in qualitative agreement with theoretical predictions
of galaxy formation models. These features appear to be most common in the most powerful
AGN, and are equally prevalent in the high-redshift Universe as in the
low-redshift Universe for AGN of the same luminosity. Our
analyses focused on searching for emission-line profiles which are
dominated by these extreme gas kinematics and lower level outflows
or highly turbulent material could be even more common. Due to a higher fraction of galaxies hosting the most luminous AGN at
higher redshifts, our results imply that the most extreme ionised outflows are more prevalent in
high-redshift galaxies. In future papers, we will present results
based on the spatially-resolved kinematics of multiple emission
lines which will reveal information on the sizes and
morphologies of the high-velocity gas, enable us to disentangle
outflows from galaxy kinematics, and to measure the energetics of
the outflows (following e.g., ;
; ;
). Furthermore, as our sample size grows, we will
be able to make more definitive tests on the drivers and impact of ionised outflows in large
samples of representative AGN.
§.§ Acknowledgements
We thank the referee for their constructive comments. We acknowledge the Science and Technology Facilities Council
(CMH, DMA, JPS, AMS, RGB and RMS through grant code ST/L00075X/1) and the Leverhulme Trust
(DMA). JRM acknowledges support from the University of Sheffield via
its Vice-Chancellor Fellowship scheme. JPS acknowledges support from a
Hintze Research Fellowship. FEB acknowledges
support from CONICYT-Chile (Basal-CATA PFB-06/2007, FONDECYT 1141218,
“EMBIGGEN” Anillo ACT1101), and Project IC120009 “Millennium
Institute of Astrophysics (MAS)” funded by the Iniciativa
Científica Milenio del Ministerio de Economía, Fomento y
Turismo. We thank the members of the
KROSS team for access to their data and for
assisting with KASH$z$ observations. We also thank Holly Elbert
and Timothy Green for carrying out some observations. We thank F. Stanley
for providing star formation rate measurements and Ian Smail for
useful discussions.
[Abazajian et al.,Abazajian
et al.2009]Abazajian09
Abazajian K. N. et al., 2009, , 182, 543
[Aird et al.,Aird et al.2013]Aird13
Aird J. et al., 2013, , 775, 41
[Aird et al.,Aird et al.2010]Aird10
Aird J. et al., 2010, , 401, 2531
[Alexander et al.,Alexander
et al.2008]Alexander08b
Alexander D. M. et al., 2008, , 687, 835
[Alexander & HickoxAlexander &
Alexander D. M., Hickox R. C., 2012, , 56, 93
[Alexander, Swinbank, Smail, McDermid
& NesvadbaAlexander et al.2010]Alexander10
Alexander D. M., Swinbank A. M., Smail I., McDermid R.,
Nesvadba N. P. H., 2010, , 402, 2211
[Alloin, Collin-Souffrin, Joly &
VigrouxAlloin et al.1979]Alloin79
Alloin D., Collin-Souffrin S., Joly M., Vigroux L., 1979, ,
78, 200
[Arribas, Colina, Bellocchi, Maiolino
& Villar-MartínArribas et al.2014]Arribas14
Arribas S., Colina L., Bellocchi E., Maiolino R.,
Villar-Martín M., 2014, , 568, A14
[Azadi et al.,Azadi
et al.2015]Azadi15
Azadi M. et al., 2015, , 806, 187
[Baldry et al.,Baldry
et al.2012]Baldry12
Baldry I. K. et al., 2012, , 421, 621
[Balmaverde et al.,Balmaverde
et al.2015]Balmaverde15
Balmaverde B. et al., 2015, arXiv:1506.05984
[Barth, Greene & HoBarth
et al.2008]Barth08
Barth A. J., Greene J. E., Ho L. C., 2008, , 136, 1179
[Benson, Bower, Frenk, Lacey, Baugh
& ColeBenson et al.2003]Benson03
Benson A. J., Bower R. G., Frenk C. S., Lacey C. G., Baugh C. M.,
Cole S., 2003, , 599, 38
[Berney et al.,Berney
et al.2015]Berney15
Berney S. et al., 2015, , 454, 3622
[Bezanson, Franx & van
DokkumBezanson et al.2015]Bezanson15
Bezanson R., Franx M., van Dokkum P. G., 2015, , 799, 148
[Bongiorno et al.,Bongiorno
et al.2012]Bongiorno12
Bongiorno A. et al., 2012, , 427, 3103
[Bonzini, Padovani, Mainieri,
Kellermann, Miller, Rosati, Tozzi & VattakunnelBonzini
et al.2013]Bonzini13
Bonzini M., Padovani P., Mainieri V., Kellermann K. I., Miller
N., Rosati P., Tozzi P., Vattakunnel S., 2013, , 436, 3759
[Boroson, Persson & OkeBoroson
et al.1985]Boroson85
Boroson T. A., Persson S. E., Oke J. B., 1985, , 293, 120
[Bower, Benson, Malbon, Helly, Frenk,
Baugh, Cole & LaceyBower et al.2006]Bower06
Bower R. G., Benson A. J., Malbon R., Helly J. C., Frenk C. S.,
Baugh C. M., Cole S., Lacey C. G., 2006, , 370, 645
[Bower, McCarthy & BensonBower
et al.2008]Bower08
Bower R. G., McCarthy I. G., Benson A. J., 2008, , 390, 1399
[Brandt & AlexanderBrandt &
Brandt W. N., Alexander D. M., 2015, , 23, 1
[Brusa et al.,Brusa
et al.2015]Brusa15
Brusa M. et al., 2015, , 446, 2394
[Brusa et al.,Brusa
et al.2010]Brusa10
Brusa M. et al., 2010, , 716, 348
Calzetti D., 2013, Star Formation Rate Indicators.
p. 419
[Calzetti, Armus, Bohlin, Kinney,
Koornneef & Storchi-BergmannCalzetti et al.2000]Calzetti00
Calzetti D., Armus L., Bohlin R. C., Kinney A. L., Koornneef J.,
Storchi-Bergmann T., 2000, , 533, 682
[Cano-Díaz, Maiolino, Marconi,
Netzer, Shemmer & CresciCano-Díaz
et al.2012]CanoDiaz12
Cano-Díaz M., Maiolino R., Marconi A., Netzer H., Shemmer
O., Cresci G., 2012, , 537, L8
[Cappelluti et al.,Cappelluti
et al.2009]Cappelluti09
Cappelluti N. et al., 2009, , 497, 635
[Carniani et al.,Carniani
et al.2015]Carniani15
Carniani S. et al., 2015, , 580, A102
Chabrier G., 2003, , 115, 763
[Churazov, Sazonov, Sunyaev, Forman,
Jones & BöhringerChurazov et al.2005]Churazov05
Churazov E., Sazonov S., Sunyaev R., Forman W., Jones C.,
Böhringer H., 2005, , 363, L91
[Cicone et al.,Cicone
et al.2014]Cicone14
Cicone C. et al., 2014, , 562, A21
[Civano et al.,Civano
et al.2012]Civano12
Civano F. et al., 2012, , 201, 30
[Collet et al.,Collet
et al.2015]Collet15
Collet C. et al., 2015, arXiv:1510.06631
[Crenshaw, Schmitt, Kraemer, Mushotzky
& DunnCrenshaw et al.2010]Crenshaw10
Crenshaw D. M., Schmitt H. R., Kraemer S. B., Mushotzky R. F.,
Dunn J. P., 2010, , 708, 419
[Cresci et al.,Cresci
et al.2015]Cresci15
Cresci G. et al., 2015, , 799, 82
[Davies et al.,Davies
et al.2013]Davies13
Davies R. I. et al., 2013, , 558, A56
[Del Moro et al.,Del Moro
et al.2015]DelMoro15
Del Moro A. et al., 2015, arXiv:1504.03329
[Del Moro et al.,Del Moro
et al.2013]DelMoro13
Del Moro A. et al., 2013, , 549, A59
[Denicoló, Terlevich &
TerlevichDenicoló et al.2002]Denicolo02
Denicoló G., Terlevich R., Terlevich E., 2002, , 330, 69
[Dimitrijević, Popović, Kovačević, Dačić & IlićDimitrijević
et al.2007]Dimitrijevic07
Dimitrijević M. S., Popović L. Č., Kovačević J.,
Dačić M., Ilić D., 2007, , 374, 1181
[Eisenhauer et al.,Eisenhauer
et al.2003]Eisenhauer03
Eisenhauer F. et al., 2003, in Iye M., Moorwood A. F. M., eds,
Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
Vol. 4841, Instrument Design and Performance for Optical/Infrared
Ground-based Telescopes. pp 1548–1561
[Elvis et al.,Elvis
et al.2009]Elvis09
Elvis M. et al., 2009, , 184, 158
Fabian A. C., 2012, , 50, 455
[Förster Schreiber et al.,Förster
Schreiber et al.2014]ForsterSchreiber14
Förster Schreiber N. M. et al., 2014, , 787, 38
[Furusawa et al.,Furusawa
et al.2008]Furusawa08
Furusawa H. et al., 2008, , 176, 1
[Gaspari, Melioli, Brighenti &
D'ErcoleGaspari et al.2011]Gaspari11
Gaspari M., Melioli C., Brighenti F., D'Ercole A., 2011, ,
411, 349
Gehrels N., 1986, , 303, 336
[Genzel et al.,Genzel
et al.2014]Genzel14
Genzel R. et al., 2014, , 796, 7
[Genzel et al.,Genzel
et al.2011]Genzel11
Genzel R. et al., 2011, , 733, 101
[George, Turner, Yaqoob, Netzer,
Laor, Mushotzky, Nandra & TakahashiGeorge
et al.2000]George00
George I. M., Turner T. J., Yaqoob T., Netzer H., Laor A.,
Mushotzky R. F., Nandra K., Takahashi T., 2000, , 531, 52
[Giacconi et al.,Giacconi
et al.2001]Giacconi01
Giacconi R. et al., 2001, , 551, 624
[Goulding & AlexanderGoulding &
Goulding A. D., Alexander D. M., 2009, , 398, 1165
[Granato, De Zotti, Silva, Bressan &
DaneseGranato et al.2004]Granato04
Granato G. L., De Zotti G., Silva L., Bressan A., Danese L.,
2004, , 600, 580
[Hainline, Hickox, Greene, Myers &
ZakamskaHainline et al.2013]Hainline13
Hainline K. N., Hickox R., Greene J. E., Myers A. D., Zakamska
N. L., 2013, , 774, 145
[Harrison, Alexander, Mullaney &
SwinbankHarrison et al.2014]Harrison14b
Harrison C. M., Alexander D. M., Mullaney J. R., Swinbank A. M.,
2014, , 441, 3306
[Harrison et al.,Harrison
et al.2012]Harrison12a
Harrison C. M. et al., 2012, , 426, 1073
[Harrison, Thomson, Alexander, Bauer,
Edge, Hogan, Mullaney & SwinbankHarrison
et al.2015]Harrison15
Harrison C. M., Thomson A. P., Alexander D. M., Bauer F. E., Edge
A. C., Hogan M. T., Mullaney J. R., Swinbank A. M., 2015, ,
800, 45
[Heckman & BestHeckman &
Heckman T. M., Best P. N., 2014, , 52, 589
[Heckman, Miley, van Breugel &
ButcherHeckman et al.1981]Heckman81
Heckman T. M., Miley G. K., van Breugel W. J. M., Butcher H. R.,
1981, , 247, 403
[Heckman, Ptak, Hornschemeier &
KauffmannHeckman et al.2005]Heckman05
Heckman T. M., Ptak A., Hornschemeier A., Kauffmann G., 2005,
, 634, 161
[Hickox, Mullaney, Alexander, Chen,
Civano, Goulding & HainlineHickox et al.2014]Hickox14
Hickox R. C., Mullaney J. R., Alexander D. M., Chen C.-T. J.,
Civano F. M., Goulding A. D., Hainline K. N., 2014, , 782, 9
[Hill & ZakamskaHill &
Hill M. J., Zakamska N. L., 2014, , 439, 2701
[Ho et al.,Ho et al.2014]Ho14
Ho I.-T. et al., 2014, , 444, 3894
[Hopkins, Hernquist, Cox, Di Matteo,
Robertson & SpringelHopkins et al.2006]Hopkins06
Hopkins P. F., Hernquist L., Cox T. J., Di Matteo T., Robertson
B., Springel V., 2006, , 163, 1
[Husemann, Wisotzki, Sánchez &
JahnkeHusemann et al.2013]Husemann13
Husemann B., Wisotzki L., Sánchez S. F., Jahnke K., 2013,
, 549, A43
[Juneau et al.,Juneau
et al.2013]Juneau13
Juneau S. et al., 2013, , 764, 176
[Kalfountzou et al.,Kalfountzou
et al.2014]Kalfountzou14
Kalfountzou E. et al., 2014, , 442, 1181
Kennicutt Jr. R. C., 1998, , 36, 189
[Kewley & DopitaKewley &
Kewley L. J., Dopita M. A., 2002, , 142, 35
[Kewley, Dopita, Leitherer, Davé,
Yuan, Allen, Groves & SutherlandKewley et al.2013]Kewley13
Kewley L. J., Dopita M. A., Leitherer C., Davé R., Yuan T.,
Allen M., Groves B., Sutherland R., 2013, , 774, 100
[Kewley, Groves, Kauffmann &
HeckmanKewley et al.2006]Kewley06
Kewley L. J., Groves B., Kauffmann G., Heckman T., 2006, ,
372, 961
[Kormendy & HoKormendy &
Kormendy J., Ho L. C., 2013, , 51, 511
[La Franca, Melini & FioreLa Franca
et al.2010]LaFranca10
La Franca F., Melini G., Fiore F., 2010, , 718, 368
[LaMassa, Heckman, Ptak, Hornschemeier,
Martins, Sonnentrucker & TremontiLaMassa et al.2009]LaMassa09
LaMassa S. M., Heckman T. M., Ptak A., Hornschemeier A., Martins
L., Sonnentrucker P., Tremonti C., 2009, , 705, 568
[Lamastra, Bianchi, Matt, Perola,
Barcons & CarreraLamastra et al.2009]Lamastra09
Lamastra A., Bianchi S., Matt G., Perola G. C., Barcons X.,
Carrera F. J., 2009, , 504, 73
[Lawrence et al.,Lawrence
et al.2007]Lawrence07
Lawrence A. et al., 2007, , 379, 1599
[Lehmer, Alexander, Bauer, Brandt,
Goulding, Jenkins, Ptak & RobertsLehmer et al.2010]Lehmer10
Lehmer B. D., Alexander D. M., Bauer F. E., Brandt W. N.,
Goulding A. D., Jenkins L. P., Ptak A., Roberts T. P., 2010,
, 724, 559
[Lehmer et al.,Lehmer
et al.2009]Lehmer09
Lehmer B. D. et al., 2009, , 691, 687
[Lequeux, Peimbert, Rayo, Serrano &
Torres-PeimbertLequeux et al.1979]Lequeux79
Lequeux J., Peimbert M., Rayo J. F., Serrano A.,
Torres-Peimbert S., 1979, , 80, 155
[Liu, Zakamska, Greene, Nesvadba &
LiuLiu et al.2013]Liu13b
Liu G., Zakamska N. L., Greene J. E., Nesvadba N. P. H., Liu
X., 2013, , 436, 2576
[Lusso et al.,Lusso
et al.2012]Lusso12
Lusso E. et al., 2012, , 425, 623
[Madau & DickinsonMadau &
Madau P., Dickinson M., 2014, , 52, 415
[Maiolino et al.,Maiolino
et al.2008]Maiolino08
Maiolino R. et al., 2008, , 488, 463
[Malkan, Gorjian & TamMalkan
et al.1998]Malkan98
Malkan M. A., Gorjian V., Tam R., 1998, , 117, 25
Markevitch M., 1998, , 504, 27
Markwardt C. B., 2009, in Bohlender D. A., Durand D., Dowler P.,
eds, Astronomical Society of the Pacific Conference Series Vol. 411,
Astronomical Data Analysis Software and Systems XVIII. p. 251
Martin C. L., 2005, , 621, 227
[McCarthy et al.,McCarthy
et al.2010]McCarthy10
McCarthy I. G. et al., 2010, , 406, 822
[McElroy, Croom, Pracy, Sharp, Ho &
MedlingMcElroy et al.2015]McElroy15
McElroy R., Croom S. M., Pracy M., Sharp R., Ho I.-T.,
Medling A. M., 2015, , 446, 2186
[McNamara & NulsenMcNamara &
McNamara B. R., Nulsen P. E. J., 2012, New Journal of Physics, 14, 055023
[Miller et al.,Miller
et al.2013]Miller13
Miller N. A. et al., 2013, , 205, 13
[Modigliani et al.,Modigliani
et al.2007]Modigliani07
Modigliani A. et al., 2007, arXiv:0701297
[Morganti, Frieswijk, Oonk, Oosterloo
& TadhunterMorganti et al.2013]Morganti13
Morganti R., Frieswijk W., Oonk R. J. B., Oosterloo T.,
Tadhunter C., 2013, , 552, L4
[Morganti, Oosterloo, Tadhunter, van
Moorsel & EmontsMorganti et al.2005]Morganti05b
Morganti R., Oosterloo T. A., Tadhunter C. N., van Moorsel G.,
Emonts B., 2005, , 439, 521
[Mukherjee, Feigelson, Jogesh Babu,
Murtagh, Fraley & RafteryMukherjee et al.1998]Mukherjee98
Mukherjee S., Feigelson E. D., Jogesh Babu G., Murtagh F., Fraley
C., Raftery A., 1998, , 508, 314
[Mulchaey, Koratkar, Ward, Wilson,
Whittle, Antonucci, Kinney & HurtMulchaey
et al.1994]Mulchaey94
Mulchaey J. S., Koratkar A., Ward M. J., Wilson A. S., Whittle
M., Antonucci R. R. J., Kinney A. L., Hurt T., 1994, , 436,
[Mullaney et al.,Mullaney
et al.2015]Mullaney15
Mullaney J. R. et al., 2015, , 453, L83
[Mullaney, Alexander, Fine, Goulding,
Harrison & HickoxMullaney et al.2013]Mullaney13
Mullaney J. R., Alexander D. M., Fine S., Goulding A. D.,
Harrison C. M., Hickox R. C., 2013, , 433, 622
[Mullaney et al.,Mullaney
et al.2012]Mullaney12a
Mullaney J. R. et al., 2012, , 419, 95
[Nandra & PoundsNandra &
Nandra K., Pounds K. A., 1994, , 268, 405
[Nesvadba, Lehnert, De Breuck, Gilbert
& van BreugelNesvadba et al.2008]Nesvadba08
Nesvadba N. P. H., Lehnert M. D., De Breuck C., Gilbert A. M.,
van Breugel W., 2008, , 491, 407
[Nesvadba, Lehnert, Eisenhauer,
Gilbert, Tecza & AbuterNesvadba et al.2006]Nesvadba06
Nesvadba N. P. H., Lehnert M. D., Eisenhauer F., Gilbert A.,
Tecza M., Abuter R., 2006, , 650, 693
[Netzer, Mainieri, Rosati &
TrakhtenbrotNetzer et al.2006]Netzer06
Netzer H., Mainieri V., Rosati P., Trakhtenbrot B., 2006, ,
453, 525
[Nims, Quataert &
Faucher-GiguèreNims et al.2015]Nims15
Nims J., Quataert E., Faucher-Giguère C.-A., 2015, , 447,
[Osterbrock & FerlandOsterbrock &
Osterbrock D. E., Ferland G. J., 2006, Astrophysics of gaseous nebulae
and active galactic nuclei
[Panessa, Bassani, Cappi, Dadina,
Barcons, Carrera, Ho & IwasawaPanessa et al.2006]Panessa06
Panessa F., Bassani L., Cappi M., Dadina M., Barcons X.,
Carrera F. J., Ho L. C., Iwasawa K., 2006, , 455, 173
[Peterson et al.,Peterson
et al.2004]Peterson04
Peterson B. M. et al., 2004, , 613, 682
[Popesso et al.,Popesso
et al.2009]Popesso09
Popesso P. et al., 2009, , 494, 443
Pringle J. E., 1981, , 19, 137
[Puccetti et al.,Puccetti
et al.2009]Puccetti09
Puccetti S. et al., 2009, , 185, 586
[Rawlings et al.,Rawlings
et al.2015]Rawlings15
Rawlings J. I. et al., 2015, , 452, 4111
[Rich, Kewley & DopitaRich
et al.2014]Rich14
Rich J. A., Kewley L. J., Dopita M. A., 2014, , 781, L12
[Rosario et al.,Rosario
et al.2012]Rosario12
Rosario D. J. et al., 2012, , 545, A45
[Rosen et al.,Rosen
et al.2015]Rosen15
Rosen S. R. et al., 2015, arXiv:1504.07051
[Rousselot, Lidman, Cuby, Moreels &
MonnetRousselot et al.2000]Rousselot00
Rousselot P., Lidman C., Cuby J.-G., Moreels G., Monnet G.,
2000, , 354, 1134
[Rupke, Veilleux & SandersRupke
et al.2005]Rupke05c
Rupke D. S., Veilleux S., Sanders D. B., 2005, , 632, 751
[Rupke & VeilleuxRupke &
Rupke D. S. N., Veilleux S., 2013, , 768, 75
[Saez et al.,Saez et al.2015]Saez15
Saez C. et al., 2015, , 450, 2615
[Sanders, Soifer, Elias, Neugebauer &
MatthewsSanders et al.1988]Sanders88
Sanders D. B., Soifer B. T., Elias J. H., Neugebauer G.,
Matthews K., 1988, , 328, L35
[Schawinski, Koss, Berney &
SartoriSchawinski et al.2015]Schawinski15
Schawinski K., Koss M., Berney S., Sartori L. F., 2015, ,
451, 2517
[Schaye et al.,Schaye
et al.2015]Schaye15
Schaye J. et al., 2015, , 446, 521
[Schinnerer et al.,Schinnerer
et al.2010]Schinnerer10
Schinnerer E. et al., 2010, , 188, 384
Schwarz G., 1978, Ann. Stat., 6, 461
[Scoville et al.,Scoville
et al.2007]Scoville07
Scoville N. et al., 2007, , 172, 1
[Sharples et al.,Sharples
et al.2013]Sharples13
Sharples R. et al., 2013, The Messenger, 151, 21
[Sharples et al.,Sharples
et al.2004]Sharples04
Sharples R. M. et al., 2004, in Moorwood A. F. M., Iye M., eds,
Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
Vol. 5492, Ground-based Instrumentation for Astronomy. pp 1179–1186
[Silk & ReesSilk &
Silk J., Rees M. J., 1998, , 331, L1
[Simpson et al.,Simpson
et al.2012]Simpson12
Simpson C. et al., 2012, , 421, 3060
[Smail, Sharp, Swinbank, Akiyama,
Ueda, Foucaud, Almaini & CroomSmail et al.2008]Smail08
Smail I., Sharp R., Swinbank A. M., Akiyama M., Ueda Y.,
Foucaud S., Almaini O., Croom S., 2008, , 389, 407
[Sobral et al.,Sobral
et al.2013]Sobral13
Sobral D. et al., 2013, , 779, 139
[Somerville, Hopkins, Cox, Robertson &
HernquistSomerville et al.2008]Somerville08
Somerville R. S., Hopkins P. F., Cox T. J., Robertson B. E.,
Hernquist L., 2008, , 391, 481
[Stanley, Harrison, Alexander,
Swinbank, Aird, Del Moro, Hickox & MullaneyStanley
et al.2015]Stanley15
Stanley F., Harrison C. M., Alexander D. M., Swinbank A. M., Aird
J. A., Del Moro A., Hickox R. C., Mullaney J. R., 2015, ,
453, 591
[Steidel, Adelberger, Dickinson,
Giavalisco, Pettini & KelloggSteidel et al.1998]Steidel98
Steidel C. C., Adelberger K. L., Dickinson M., Giavalisco M.,
Pettini M., Kellogg M., 1998, , 492, 428
Stockton A., 1976, , 205, L113
[Stott et al.,Stott
et al.2013]Stott13
Stott J. P. et al., 2013, , 436, 1130
[Stott et al.,Stott
et al.2014]Stott14
Stott J. P. et al., 2014, , 443, 2695
[Swinbank, Smail, Chapman, Blain,
Ivison & KeelSwinbank et al.2004]Swinbank04
Swinbank A. M., Smail I., Chapman S. C., Blain A. W., Ivison
R. J., Keel W. C., 2004, , 617, 64
[Symeonidis et al.,Symeonidis
et al.2011]Symeonidis11
Symeonidis M. et al., 2011, , 417, 2239
[Tadhunter, Morganti, Rose, Oonk &
OosterlooTadhunter et al.2014]Tadhunter14
Tadhunter C., Morganti R., Rose M., Oonk J. B. R., Oosterloo
T., 2014, , 511, 440
[Tremonti et al.,Tremonti
et al.2004]Tremonti04
Tremonti C. A. et al., 2004, , 613, 898
[Trouille & BargerTrouille &
Trouille L., Barger A. J., 2010, , 722, 212
[Ueda et al.,Ueda et al.2008]Ueda08
Ueda Y. et al., 2008, , 179, 124
[Vega Beltrán, Pizzella, Corsini,
Funes, Zeilinger, Beckman & BertolaVega Beltrán
et al.2001]VegaBeltran01
Vega Beltrán J. C., Pizzella A., Corsini E. M., Funes J. G.,
Zeilinger W. W., Beckman J. E., Bertola F., 2001, , 374, 394
Veilleux S., 1991, , 75, 383
[Veilleux et al.,Veilleux
et al.2013]Veilleux13
Veilleux S. et al., 2013, , 776, 27
[Vignali, Alexander &
ComastriVignali et al.2006]Vignali06
Vignali C., Alexander D. M., Comastri A., 2006, , 373, 321
[Vogelsberger et al.,Vogelsberger
et al.2014]Vogelsberger14
Vogelsberger M. et al., 2014, , 444, 1518
Vrtilek J. M., 1985, , 294, 121
[Wagner, Bicknell & UmemuraWagner
et al.2012]Wagner12
Wagner A. Y., Bicknell G. V., Umemura M., 2012, , 757, 136
Weedman D. W., 1970, , 159, 405
[Westmoquette, Clements, Bendo &
KhanWestmoquette et al.2012]Westmoquette12
Westmoquette M. S., Clements D. L., Bendo G. J., Khan S. A.,
2012, , 424, 416
[Williams, Maiolino, Santini, Marconi,
Cresci, Mannucci & LutzWilliams et al.2014]Williams14
Williams R. J., Maiolino R., Santini P., Marconi A., Cresci G.,
Mannucci F., Lutz D., 2014, , 443, 3780
[Wisnioski et al.,Wisnioski
et al.2015]Wisnioski15
Wisnioski E. et al., 2015, , 799, 209
[Xue et al.,Xue et al.2011]Xue11
Xue Y. Q. et al., 2011, , 195, 10
[York et al.,York et al.2000]York00
York D. G. et al., 2000, , 120, 1579
[Zakamska & GreeneZakamska &
Zakamska N. L., Greene J. E., 2014, , 442, 784
[Zakamska, Strauss, Heckman, Ivezić
& KrolikZakamska et al.2004]Zakamska04
Zakamska N. L., Strauss M. A., Heckman T. M., Ivezić Ž.,
Krolik J. H., 2004, , 128, 1002
§ EMISSION-LINE PROFILES AND TABULATED DATA
In this Appendix we provide the galaxy-integrated spectra,
emission-line profile fits and tabulated data for all 89 of the
KASH$z$ targets presented in this work.
Galaxy-integrated spectra, shifted to the
rest frame, around the [O iii]4959,5007 emission-line
doublet, in arbitary flux density units, for the $z\approx$ 1.1–1.7
targets. For descriptions of the different curves see
Figure <ref>. The vertical dotted lines indicate the wavelengths of the
brightest sky lines (). In each panel we
also identify the target name, the logarithm of the hard-band X-ray luminosities
(erg s$^{-1}$) and the redshifts (Table <ref>).
Galaxy-integrated spectra, shifted to the
rest frame, around the H$\alpha$ and [N ii]6548,6583
emission-lines, in arbitary flux density units, for the
$z\approx$ 0.6–1.1 targets that do not show a BLR component (i.e.,
only the Type 2 sources). For descriptions of the different curves see Figure <ref>. The vertical dotted lines indicate the wavelengths of the
brightest sky lines (). In each panel we identify the target
name, the logarithm of the X-ray luminosity (in units of erg s$^{-1}$) and
the redshift (see Table <ref>).
Same as Figure <ref> but for the targets with a
H$\alpha$ BLR component (i.e., the Type 1 sources). In each panel, the small thick lines show
the wavelengths of the [S ii]6716,6731 emission-line doublet.
KASH$z$ Target Properties
Name $z_{A}$ Type $F_{{\rm 0.5-2}}$ $F_{{\rm 2-10}}$ Obs. RL Inst. Line t$_{e}$ Note BLR $z_{L}$ $S_{A}$ FW$_{A}$ $S_{B}$ FW$_{B}$ $\Delta v$ $S_{{\rm [N II]}}$ $W_{80}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)
UDS-186 1.437 s 4.07 5.06 U K O 9.0 C - -
$<$0.7 - - - - - -
UDS-310 1.645 s 6.96 14.15 U K O 5.4 L - 1.645
1.8$\pm$0.7 176$\pm$86 - - - - 192$\pm$94
UDS-357 1.100 s 16.16 33.18 U K O 5.4 L $\beta$
1.098 4.5$\pm$1.0 779$\pm$154 - - - - 848$\pm$168
UDS-574 1.533 s 4.55 4.51 U K O 6.0 C - -
$<$0.9 - - - - - -
UDS-584 1.602 s 2.74 5.22 U K O 9.0 L - 1.605
4.5$\pm$0.8 107$\pm$26 25.0$\pm$1.1 611$\pm$28 -52$\pm$13 -
UDS-586 1.171 s 11.97 16.01 U K O 6.0 L -
1.171 2.3$\pm$0.3 575$\pm$83 - - - - 625$\pm$90
UDS-605 1.096 s 2.77 9.76 O K O 6.0 L - 1.097
1.7$\pm$0.7 696$\pm$253 - - - - 758$\pm$276
UDS-619 1.410 s 19.18 35.01 U K O 6.0 L $\beta$
1.401 21.2$\pm$4.9 242$\pm$29 33.8$\pm$4.3 666$\pm$68 -18$\pm$15
- 562$\pm$33
UDS-631 1.192 s 3.79 10.70 U K O 8.4 L - 1.190
1.6$\pm$0.4 318$\pm$95 - - - - 346$\pm$104
UDS-640 1.092 s 6.61 10.32 U K O 6.0 L - 1.093
5.4$\pm$0.3 523$\pm$36 - - - - 570$\pm$39
UDS-643 1.087 s 3.46 5.45 U K O 6.0 L $\beta$
1.088 2.4$\pm$0.7 146$\pm$66 2.8$\pm$0.9 565$\pm$149 -189$\pm$121 -
UDS-655 1.397 s 4.92 8.32 U K O 6.0 L - 1.397
1.5$\pm$0.6 272$\pm$114 - - - - 297$\pm$125
UDS-671 1.083 s 0.27 8.02 O K O 6.0 L - 1.084
0.8$\pm$0.5 77$\pm$66 3.6$\pm$0.8 712$\pm$176 -191$\pm$90 -
UDS-676 1.086 s 3.41 1.91 U K O 9.0 L - 1.086
0.7$\pm$0.3 389$\pm$215 - - - - 424$\pm$235
UDS-701 1.653 s 1.45 3.35 U K O 9.0 L - 1.649
2.1$\pm$1.1 520$\pm$268 - - - - 566$\pm$291
UDS-763 1.413 s 18.88 26.59 U K O 9.0 L $\beta$
1.412 5.4$\pm$3.0 261$\pm$78 12.3$\pm$3.3 705$\pm$142 -168$\pm$90
- 668$\pm$84
UDS-796 1.132 s 3.80 29.03 O K O 9.0 C - -
$<$1.7 - - - - - -
UDS-814 1.074 s 25.01 22.10 U K O 9.0 L -
1.077 3.1$\pm$0.5 735$\pm$133 - - - - 800$\pm$144
UDS-855 1.407 s 5.19 20.10 O K O 8.4 L - 1.407
11.4$\pm$1.1 475$\pm$45 - - - - 517$\pm$49
COS-20 1.156 s 1.34 7.01 O K O 9.6 L - 1.154
1.7$\pm$0.5 324$\pm$130 0.9$\pm$0.6 380$\pm$214 381$\pm$25 -
COS-27$^{\star}$ 1.510 p $<$0.73 2.59 O K O 5.4 C
- - - - - - - - -
COS-40 1.510 s 4.69 9.69 U K O 9.0 L - 1.504
2.8$\pm$1.3 485$\pm$171 2.8$\pm$1.6 1277$\pm$375 0$\pm$72 -
COS-61 1.478 s 1.45 4.41$\dagger$ ? K O 8.4 L -
1.478 3.5$\pm$0.6 225$\pm$45 - - - - 245$\pm$49
COS-108 1.253 s 4.39 16.10 O S O 12.0 L - 1.258
25.6$\pm$3.0 331$\pm$26 10.9$\pm$4.6 1360$\pm$409 -506$\pm$220 -
COS-178 1.347 s 1.47 8.92 O S O 12.0 C - -
$<$1.2 - - - - - -
COS-203 1.360 s 7.37 14.20 U K O 7.2 L - 1.359
5.1$\pm$0.6 337$\pm$49 - - - - 367$\pm$53
COS-206 1.483 s 5.89 10.40 U K O 7.2 L $\beta$
1.480 3.8$\pm$2.0 630$\pm$323 - - - - 685$\pm$352
COS-211 1.166 s 2.65 4.12 U K O 7.2 L - 1.167
2.4$\pm$0.5 535$\pm$113 - - - - 582$\pm$123
COS-453 1.625 s 3.01 5.70 U K O 9.0 C - -
$<$0.5 - - - - - -
COS-454 1.478 s 5.91 10.90 U K O 9.0 L $\beta$
1.484 2.5$\pm$1.3 250$\pm$110 3.9$\pm$1.9 827$\pm$415 -480$\pm$198
- 922$\pm$280
COS-499 1.459 s 27.40 57.80 U K O 9.0 L $\beta$
1.455 3.6$\pm$1.2 165$\pm$49 22.1$\pm$1.7 786$\pm$58 -109$\pm$28
- 796$\pm$52
COS-649 1.369 s $<$1.05 3.77 O K O 5.4 L -
1.367 2.0$\pm$0.7 415$\pm$166 - - - - 451$\pm$180
COS-832$^{\star}$ 1.471 p $<$0.86 2.71 O K O 8.4 C
- - - - - - - - -
COS-885$^{\star}$ 1.594 p 0.44 1.34$\dagger$ ? K O 9.0
C - - - - - - - - -
COS-1015 1.379 s 1.14 3.47$\dagger$ U K O 7.2 L -
1.377 1.4$\pm$0.6 281$\pm$160 - - - - 306$\pm$174
COS-1518$^{\star}$ 1.122 s $<$0.24 2.90 O K O 9.6 N
- - - - - - - - -
COS-3178 1.355 s $<$0.59 2.54 O K O 7.2 L -
1.354 3.6$\pm$2.3 882$\pm$547 - - - - 960$\pm$596
COS-3715 1.103 s 0.36 1.11$\dagger$ ? K O 5.4 C -
- $<$0.7 - - - - - -
XCOS-2028 1.592 s 33.50 75.70 U S O 25.2 L $\beta$
1.593 17.3$\pm$5.1 410$\pm$36 16.4$\pm$5.4 657$\pm$88 -332$\pm$76
- 734$\pm$17
XCOS-5627 1.337 s 7.25 15.40 U S O 12.6 L $\beta$
1.349 0.7$\pm$0.7 62$\pm$62 9.0$\pm$2.1 573$\pm$191 -5$\pm$14 -
CDFS-25 1.374 s $<$0.19 1.21 O K O 6.0 L -
1.377 3.5$\pm$2.1 184$\pm$51 9.7$\pm$2.4 518$\pm$94 -183$\pm$64 -
Continued over page.
KASH$z$ Target Properties (continued)
Name $z_{A}$ Type $F_{{\rm 0.5-2}}$ $F_{{\rm 2-10}}$ Obs. RL Inst. Line t$_{e}$ Note BLR $z_{L}$ $S_{A}$ FW$_{A}$ $S_{B}$ FW$_{B}$ $\Delta v$ $S_{{\rm [N II]}}$ $W_{80}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)
CDFS-88 1.616 s 3.07 4.67 U K O 9.0 C - -
$<$0.7 - - - - - -
CDFS-113$^{\star}$ 1.608 i 0.57 0.98 U K O 9.0 N -
- - - - - - - -
CDFS-166 1.605 s 8.14 12.39 U K O 9.0 L - 1.611
5.7$\pm$0.4 371$\pm$25 - - - - 404$\pm$27
CDFS-191 1.185 s $<$0.13 4.18 O K O 6.0 L -
1.184 1.8$\pm$0.6 297$\pm$125 - - - - 323$\pm$136
CDFS-243 1.097 s 0.71 12.29 O K O 6.0 L - 1.096
2.5$\pm$0.8 468$\pm$160 - - - - 510$\pm$174
CDFS-343 1.090 s $<$0.08 1.46 O K O 9.6 C - -
$<$0.4 - - - - - -
CDFS-344 1.617 s 3.64 3.89 U K O 7.8 L - 1.613
3.7$\pm$0.9 258$\pm$40 7.4$\pm$1.4 1089$\pm$149 -270$\pm$83 -
CDFS-518 1.603 s 9.74 13.30 U K O 9.6 L - 1.609
8.3$\pm$1.0 850$\pm$88 - - - - 926$\pm$96
CDFS-549$^{\star}$ 1.553 s 1.36 3.25 U K O 11.4 N -
- - - - - - - -
CDFS-619 1.178 s 0.16 2.17 O K O 7.8 L - 1.179
1.6$\pm$0.6 419$\pm$147 - - - - 457$\pm$160
CDFS-720 1.609 s 10.93 18.44 U K O 9.0 L -
1.610 2.2$\pm$0.5 742$\pm$173 - - - - 808$\pm$189
SSA22-39 1.397 s 7.19 9.08 U ? S O 3.0 L - 1.400
8.2$\pm$1.8 605$\pm$143 - - - - 658$\pm$156
SSA22-141 1.370 s 20.53 27.38 U ? S O 2.4 L - 1.367
5.4$\pm$2.2 541$\pm$209 - - - - 589$\pm$227
UDS-194 0.627 s 5.80 10.13 U K H 9.0 L $\alpha$
0.628 8.1$\pm$0.9 328$\pm$42 25.2$\pm$2.3 5385$\pm$604 258$\pm$190
0.93 357$\pm$46
UDS-275 0.883 s 7.27 52.96 O K H 5.4 L - 0.882
4.5$\pm$0.9 476$\pm$87 - - - 3.85 518$\pm$95
UDS-393 0.822 s 9.70 9.38 U K H 9.0 L $\alpha$
0.822 6.0$\pm$0.5 183$\pm$22 49.1$\pm$3.4 5811$\pm$457 176$\pm$164
3.18 199$\pm$24
UDS-403 1.021 s 2.30 13.81 O K H 9.0 L - 1.021
3.7$\pm$0.9 461$\pm$118 - - - 2.89 502$\pm$129
UDS-600 0.873 s 4.41 4.34 U K H 6.0 L $\alpha$
0.873 11.6$\pm$1.3 343$\pm$36 29.2$\pm$3.1 2401$\pm$216
-158$\pm$108 6.39 373$\pm$39
UDS-620 0.842 s 1.08 3.44 O K H 9.0 L - 0.843
2.3$\pm$0.4 311$\pm$75 - - - 1.21 338$\pm$81
UDS-818 0.928 s 14.55 19.28 U K H 8.4 L -
0.928 5.5$\pm$0.6 140$\pm$26 - - - $<$0.66 153$\pm$29
UDS-827 0.658 s 4.65 12.22 U K H 8.4 L $\alpha$
0.657 8.0$\pm$0.4 201$\pm$15 22.7$\pm$2.1 5129$\pm$548 682$\pm$196
4.02 219$\pm$17
UDS-862 0.589 s 1.35 8.50 O K H 8.4 L - 0.589
3.8$\pm$0.5 471$\pm$64 - - - 2.88 512$\pm$70
UDS-883 0.961 s 24.39 56.58 U K H 8.4 L $\alpha$
0.961 5.2$\pm$0.4 574$\pm$29 30.5$\pm$2.9 7970$\pm$795 -748$\pm$271
9.50 625$\pm$32
COS-44 1.513 p 1.23 3.56 U K H 9.0 L - 0.801
4.9$\pm$0.4 709$\pm$87 - - - 0.95 771$\pm$95
COS-401 0.969 s $<$1.14 13.30 O S H 2.4 L -
0.971 7.6$\pm$0.9 495$\pm$53 - - - 8.80 539$\pm$58
COS-724 0.906 s $<$0.51 4.04 O K H 6.0 C - -
$<$1.2 - - - - - -
COS-829 0.885 s 0.39 1.17$\dagger$ ? K H 8.4 L -
0.885 6.4$\pm$0.8 353$\pm$53 - - - 3.50 384$\pm$57
COS-932 0.975 s 0.71 2.17$\dagger$ U K H 8.4 L -
0.974 6.5$\pm$0.6 469$\pm$49 - - - 5.46 511$\pm$53
COS-1070$^{\star}$ 0.858 p 1.06 2.30 U K H 8.4 C -
- - - - - - - -
COS-1157 0.915 s 0.49 1.50$\dagger$ ? K H 9.6 L -
0.925 5.2$\pm$0.6 197$\pm$29 - - - 3.07 215$\pm$31
COS-1199 0.771 p 0.67 2.10 O K H 9.0 L - 0.850
9.5$\pm$0.7 255$\pm$23 - - - 4.05 277$\pm$25
CDFS-51 0.737 s 1.81 18.33 O S H 2.7 L - 0.737
11.9$\pm$3.5 481$\pm$211 - - - $<$3.79 523$\pm$230
CDFS-101 0.977 s 11.57 20.66 U K H 9.0 L $\alpha$
0.978 2.9$\pm$0.5 290$\pm$68 133.5$\pm$3.7 7150$\pm$202 -843$\pm$77
$<$2.78 315$\pm$74
CDFS-356 1.034 s 0.29 0.57 U K H 6.0 L - 1.034
3.3$\pm$0.2 223$\pm$17 - - - 0.55 243$\pm$19
CDFS-370 0.734 s 1.07 3.38 O S H 2.7 C - -
$<$3.8 - - - - - -
CDFS-433 0.617 s 1.04 2.03 U K H 9.6 L - 0.620
3.7$\pm$0.4 461$\pm$52 - - - 3.03 501$\pm$57
CDFS-454 0.952 i 1.42 3.17 U K H 11.4 L - 0.953
2.0$\pm$0.5 257$\pm$82 - - - 1.30 279$\pm$89
CDFS-480 0.839 s 2.48 4.08 U K H 6.0 L - 0.841
7.3$\pm$2.1 286$\pm$41 2.0$\pm$2.1 427$\pm$280 -367$\pm$169 4.80
CDFS-492 0.735 s 0.13 2.55 O S H 5.4 L - 0.735
11.2$\pm$0.6 334$\pm$22 - - - 7.13 363$\pm$24
CDFS-506 0.665 s 4.30 7.35 U K H 7.8 L - 0.666
1.8$\pm$0.4 430$\pm$109 - - - $<$0.46 468$\pm$119
Continued over page.
KASH$z$ Target Properties (continued)
Name $z_{A}$ Type $F_{{\rm 0.5-2}}$ $F_{{\rm 2-10}}$ Obs. RL Inst. Line t$_{e}$ Note BLR $z_{L}$ $S_{A}$ FW$_{A}$ $S_{B}$ FW$_{B}$ $\Delta v$ $S_{{\rm [N II]}}$ $W_{80}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)
CDFS-514 0.664 s 3.29 4.49 U K H 7.8 L - 0.667
4.2$\pm$0.8 265$\pm$81 - - - $<$0.99 289$\pm$88
CDFS-561 0.798 s 0.08 2.69 O K H 7.8 L - 0.980
1.4$\pm$0.4 239$\pm$91 - - - 1.11 261$\pm$99
CDFS-592 1.016 s 0.69 4.03 O K H 9.0 L - 1.015
5.3$\pm$0.9 601$\pm$89 - - - 6.13 655$\pm$97
CDFS-606 0.733 s 0.34 2.58 O K H 7.8 L - 0.733
1.7$\pm$0.2 151$\pm$17 - - - 0.29 164$\pm$19
CDFS-614 0.668 s 0.38 0.86 U K H 9.0 L - 0.667
4.7$\pm$0.5 642$\pm$84 - - - 4.09 699$\pm$92
CDFS-629 0.667 s 0.83 2.60 O K H 9.0 L $\alpha$
0.667 1.7$\pm$0.5 360$\pm$89 30.1$\pm$6.3 9647$\pm$1706
-846$\pm$549 2.04 392$\pm$97
CDFS-652 1.020 s 0.66 1.18 U K H 7.8 L - 1.022
8.0$\pm$0.7 115$\pm$21 - - - 2.40 125$\pm$22
CDFS-706 0.891 s $<$0.15 1.02 O K H 9.0 L -
0.890 13.7$\pm$0.4 409$\pm$16 - - - 6.63 445$\pm$18
Properties for the KASH$z$ targets observed so far. Notes:
(1) Source name (`field' – `X-ray ID'), those targets followed by a
$^{\star}$ are excluded from the analyses presented in this paper (see
Section <ref>); (2) archival redshift; (3) archival
redshift type (i.e., photometric [p], secure spectroscopic redshift
[s] or insecure spectroscopic redshift [i]); (4) and (5) soft-band
(0.5–2 keV) and hard-band (2–10 keV)
X-ray fluxes ($\times$10$^{-15}$ erg s$^{-1}$ cm $^{-2}$), where
the hard-band values followed by a $\dagger$ were
estimated from the soft-band fluxes (see Section <ref>);
(6) flag to indicate X-ray obscured candidates (O) and unobscured
candidates (U), those without sufficient constraints are labelled with
“?”; (7) flag to indicate the “radio luminous” targets with
$L_{{\rm 1.4GHz}}>10^{24}$ W Hz$^{-1}$ (Section <ref>); (8) instrument used for the observations (K$=$KMOS and S$=$SINFONI); (9) primary targeted
emission line (O$=$[O iii] and H$=$H$\alpha$); (10) on-source
exposure time (kilo-seconds); (11) note on data (L$=$line detected;
C$=$ continuum only detected; N$=$no detection); (12) note for the
identification of a BLR component ($\alpha$ for H$\alpha$ and $\beta$ for
H$\beta$); (13) redshift derived from the narrowest Gaussian component of the emission-line
profile fit; (14) and (15) Flux ($\times10^{-17}$ erg s$^{-1}$ cm$^{-2}$) and FWHM
(km s$^{-1}$) of the narrower Gaussian component; (16) and (17) Flux ($\times10^{-17}$ erg s$^{-1}$ cm$^{-2}$) and FWHM of
the broader Gaussian component (for sources with “$\alpha$” in
column 10, this is the BLR component); (18) velocity offset between
the two Gaussian components (km s$^{-1}$); (19) flux of [N ii]$\lambda$6583 ($\times10^{-17}$ erg s$^{-1}$ cm$^{-2}$) where
applicable; (20) overall emission-line width (km s$^{-1}$). All of
the quoted uncertainties are the random errors on the fits due to the
noise in the spectra; however, we note that throughout this work we
add an extra 30% systematic error to the emission-line fluxes to
account for the uncertainty in the flux calibration (see Section <ref>).
|
1511.00289
|
Context free languages allow one to express data with hierarchical structure,
at the cost of losing some of the useful properties of languages recognized by
finite automata on words. However, it is possible
to restore some of these properties by making the structure of the tree visible,
such as is done by visibly pushdown languages, or finite automata on trees.
In this paper, we show that the structure given by such approaches remains
invisible when it is read by a finite automaton (on word).
In particular, we show
that separability with a regular language
is undecidable for visibly pushdown languages, just as it is undecidable for
general context free languages.
§ INTRODUCTION
Finite automata are a well known formalism for describing the simplest formal languages.
Regular languages – ones which are recognized by finite automata – have very nice
closure properties, such as decidability of most problems such as universality or
disjointness, equivalence of deterministic finite automata (DFA) and non-deterministic
finite automata (NFA), and closure under complement.
However, most programming and natural languages have to describe a hierarchical (tree)
structure, and finite automata on words are no longer appropriate.
To capture such a hierarchical structure,
Noam Chomsky proposed the classic notion of context free languages.
Context free languages are recognized
by context free grammars (CFGs), or equivalently by pushdown automata (PDA).
However, context free languages do not have as good properties as regular ones –
for example, universality and disjointness are no longer decidable, deterministic
PDA are less powerful than non-deterministic ones, and they are not closed under
complement. These properties fail since, although
words from a context free language have an underlying tree structure, it is hard
to tell what this structure is just by looking at the word – two completely different
derivation trees can yield a very similar output, consider for example the English
sentences Time flies like an arrow and fruit flies like a banana,
or The complex houses married and single soldiers and their families – after
reading the four first words of the latter sentence, one could think that
the complex houses is the subject and married is the verb, while in fact,
the complex is the subject and houses is the verb. This is also a big
problem in practical computer science, since such a possibility of incorrect parsing leads to many errors
– one famous example is the SQL injection attack, which is based on fabricating
SQL queries which will be parsed incorrectly, allowing unauthorized access to a database.
There are two popular approaches to solve this. One of them is to use the
finite automata on trees (TFAs) <cit.>, which work on trees directly. Another one is to
use visibly pushdown automata (VPDAs), also known as languages of nested
words <cit.>, where every symbol in our alphabet has a fixed type
with respect to the stack – it either always pushes a new symbols, or always pops a
symbol, or it never pushes or pops symbols – this property allows the tree structure to be easily
read. When we safely flatten a language on trees into a language of words – XML is the
common and effective way to do that – these two approaches are seen to be equivalent
(in some sense), and most properties of regular languages of words are retained –
non-deterministic and deterministic VPDAs and finite automata on trees are equivalent,
and universality and intersection problems are decidable. Hence,
representing our data as trees, instead of forcing a linear word structure,
definitely solves many problems – both theoretical and practical – efficiently.
In this paper, we show that not all problems are solved by these approaches. In
particular, we show that, informally,
although (flattened) TFAs and VPDAs are successful at making
the structure visible to powerful computation models such as
Turing machines, the structure still remains invisible to the simple
ones, such as finite automata on words. We use our technique to show that the following
problem is undecidable, just as in the usual “invisible” context free case
given two VPDAs (or, equivalently, flattened TFAs) accepting languages $L_1$ and $L_2$
such that $L_1$ and $L_2$ are disjoint, is there a regular language $R$ such that
$R$ accepts all words from $L_1$, but no words from $L_2$?
A similar property
is also obtained for separating by other classes of languages, as long as the
corresponding problem for CFGs is undecidable, and the separating class has basic
closure properties and a pumping property – the precise conditions are listed in the
In <cit.> it is shown that the separability problem of
context free languages is undecidable for any class which includes all
definite languages. On the other hand,
it has been shown recently that the problem of separability of CFLs by
piecewise testable languages is decidable <cit.>.
Our method solves the following open problem,
which has appeared on Rajeev Alur's website in early 2013 <cit.>:
A Challenging Open Problem
Consider the following decision problem: given two regular languages $L_1$ and $L_2$ of nested
words, does there exist a regular language $R$ of words over the tagged alphabet such that
Intersection($R$,$L_1$) equals $L_2$? [...]
We say that $L_2$ is a regular restriction of $L_1$
iff the above holds.
Since disjoint languages $L_1$ and $L_2$ are separable iff $L_2$ is a regular
restriction of $L_1 \cup L_2$, and separability is undecidable,
restriction-regularity is undecidable too.
Rajeev Alur's question is inspired by <cit.>, where
it is shown that, for a fully recursive DTD, it is decidable whether
there is a regular language $R$ such that, for any valid XML document $w$,
it is decidable whether $w$ is valid with respect to $D$. This is a special case
of our result – we take fully recursive DTDs instead of arbitrary finite automata
on trees, and we want to separate $L$ from its complement. It is stated as an open
problem in <cit.> whether the problem is still decidable for arbitrary
finite automata on trees.
On the other hand, in <cit.> it is shown that it is decidable whether,
for a given visibly push-down language $L$, there exists a language $R$ such that
$R \cap F = L$, where $F$ is the language of all well matched words.
Acknowledgements. Thanks to Charles Paperman for introducing me to
the separability problem for VPDAs, and for helping me with the references.
§ PRELIMINARIES
We remind the basic notions of automata theory; see <cit.>.
For an alphabet $\Sigma$, $\Sigma^*$ denotes the set of words over $\Sigma$, and
$\epsilon$ denotes the empty word.
A deterministic finite automaton (DFA) is a tuple $A = (\Sigma, Q, q_I, F, \delta$),
where $\Sigma$ is the alphabet of $A$, $Q$ is the set fo states, $q_I \in Q$ is the
initial state, $F \subseteq Q$ is the final state, and $\delta: Q \times \Sigma \ra Q$
is the transition function. We extend $\delta$ to $\delta: Q \times \Sigma^* \ra Q$
in the following way: $\delta(q,\epsilon) = q$, $\delta(q,wx) = \delta(\delta(q,w),x)$.
A context free grammar (CFG) is a tuple $G = (V, \Sigma, R, S)$, where $V$ is the
set of non-terminal symbols, $\Sigma$ is the set of terminal symbols, $R$ is a set
of productions of form $N \ra X_1 \ldots X_k$ where $N$ is a non-terminal symbol
and and each $X_i$ is either a terminal or non-terminal symbol, and $S \in V$ is
the start symbol. The language accepted by $G$, $L(G) \subseteq \Sigma^*$, is the set of
words which can be obtained from the start symbol $S$ by replacing non-terminal symbols
with words, according to the productions.
A binary flattened tree grammar (BFG) over $\Sigma$ is a context free grammar, whose
set of terminal symbols is $\Sigma \cup \{\oa, \ca\}$, and every production is
of form $N \ra t$, $N \ra \oa\ca$ or $N \ra \oa N_1 N_2 \ca$, where
$N_1, \ldots, N_k$ are non-terminals, and $t$ is a terminal.
A BFG corresponds to the XML encoding of a
regular language of trees ($\oa$ and $\ca$ correspond to the opening
and closing tag, respectively), and languages recognized by
flattened tree grammars are visibly pushdown languages. These two facts are routine
to check – we omit this to avoid having
to state the definitions of VPDAs and TFAs; we have decided to use flattened tree
grammars in this paper since they are easier to define than both of these formalisms.
We say that two languages $L_1$ and $L_2$ are separable if there is a
regular language $R$ such that for each $w \in L(G_1)$, $w \in R$,
but for each $w \in L(G_2)$, $w \notin R$.
INPUT Two context tree grammars $G_1$ and $G_2$ such that $L(G_1)$ and $L(G_2)$
are disjoint
OUTPUT Are $L(G_1)$ and $L(G_2)$ separable?
The CFL separation problem is known to be undecidable <cit.>. For
convenience, we include the idea of the proof here. Encode configurations of a
Turing machine $M$ as words, and say that $w_1 \ra w_2$ iff a machine
in configuration $w_1$ reaches the configuration $w_2$ in next step. It can be
easily shown
that (for simple encodings)
the languages $\{w_1 \# w_2^R: w_1 \ra w_2\}$ and $\{w_1^R \# w_2: w_1 \ra w_2\}$
are context free, and thus the languages $L_1$ and $L_2$ below are also context free.
They are separable iff $M$ terminates from the initial configuration $w_I$.
\begin{eqnarray*}
L_1 &=& \{w_1 \# w_2 \# \ldots w_2k \# a^{2k} : w_1 = w_I, w_{2i-1} \ra w_{2i}^R \} \\
L_2 &=& \{w_1 \# w_2 \# \ldots w_2k \# a^{k} : w_1 = w_I, w^R_{2i} \ra w_{2i+1} \}
\end{eqnarray*}
INPUT Two BFGs $G_1$ and $G_2$ such that $L(G_1)$ and $L(G_2)$ are disjoint
OUTPUT Are $L(G_1)$ and $L(G_2)$ separable?
Our main result is the following:
The problem BFG-SEPARABILITY is undecidable.
§ PROOF
We will reduce CFG-SEPARABILITY to BFG-SEPARABILITY. To do this,
we will take two CFGs $G_1$ and $G_2$, and create two BFGs $G'_1$ and $G'_2$
such that $G'_1$ and $G'_2$ are separable iff $G_1$ and $G_2$ are.
Without loss of generality, we can assume that grammars $G_i$ do not accept the
empty word, or any word of length 1. We can also assume that these grammars
are in the Chomsky normal form, that is, each production is of form
$N \ra N_1 N_2$ or $N \ra t$, where $N$, $N_1$ and $N_2$ are non-terminals,
and $t$ is a terminal. It is well known that any context free grammar is effectively
equivalent to a grammar in Chomsky normal form <cit.>.
Given a context free grammar $G = (V, \Sigma, R, S)$
in Chomsky normal form, we will construct a
binary flattened
tree grammar $G' = (V', \Sigma', R', S')$, in the following way:
* For each $X \in V$, we have a non-terminal $X'$. We also
have one special non-terminal $E'$. The starting symbol of $G'$ is $S'$.
* For each production $N \ra t$ in $R$, we have the corresponding production in $R'$:
\begin{equation}
N' \ra t \label{gp_terminal}
\end{equation}
* For each production $N \ra N_1 N_2$ in $R$, we have the corresponding
bracketed production in $R'$:
\begin{equation}
N' \ra \oa N_1' N_2' \ca \label{gp_bracketed}
\end{equation}
* For each $X \in V$, we also have the following productions for $X'$:
\begin{eqnarray}
X' &\ra& \oa E' X' \ca \label{gp_leftempty} \\
X' &\ra& \oa X' E' \ca \label{gp_rightempty}
\end{eqnarray}
* Where the productions for $E'$ are as follows:
\begin{eqnarray}
E' &\ra& \oa \ca \label{gp_leaf} \\
E' &\ra& \oa E' E' \label{gp_fork} \ca
\end{eqnarray}
Consider $\pi: \Sigma' \ra \Sigma$, the homomorphism which simply
removes the structural symbols $\oa$ and $\ca$. By applying
$\pi$ to all the production rules for $G'$, we obtain a grammar $\pi(G')$ which
accepts exactly $\pi(L(G))$. It is straightforward to check that $\pi(G')$ is
in fact equivalent to $G$ – the only difference is that $E'$ is inserted in some
places, but all words generated by $E'$ reduce to the empty word after applying $\pi$.
Therefore, $\pi(L(G'_i))$ equals $L(G_i)$, which makes the following straightforward:
If $L(G_1)$ and $L(G_2)$ are separable, then so are $L(G_1')$ and $L(G_2')$.
$\pi^{-1}(R)$ is a regular language
which separates $L(G_1')$ and $L(G_2')$ – in other words, the automaton separating
these two languages works exactly as the one separating $L(G_1)$ and $L(G_2)$
(it just ignores all the closing and structural symbols).
The rest of this section will prove the other direction:
If $L(G'_1)$ and $L(G'_2)$ are separable, then so are $L(G_1)$ and $L(G_2)$.
Assume that $L(G'_1)$ and $L(G'_2)$ are separable. Therefore, there is a finite automaton
$A$ such that $R = L(A)$ accepts all words from $L(G'_1)$, but no words from $L(G'_2)$. We say
that two words $w_1, w_2 \in \Sigma'^*$ are syntactically equivalent with respect to $R$
iff for any words $v, x \in \Sigma'^*$, we have $vwx\in R$ iff $vw'x \in R$.
Syntactic equivalence is a congruence with respect to concatenation.
There is a number $\omega \in \bbN$ such that for any $w \in \Sigma'^*$,
$w^\omega$ is syntactically equivalent to $w^{2\omega}$ with respect to $R$.
The set $S$ of all the equivalence classes
is a semigroup with concatenation as the operation. This semigroup is called the
syntactic semigroup of A, and it is finite – if for
two words $w_1$ and $w_2$ we have $\delta(q,w_1) = \delta(q,w_2)$ for each $q \in Q$,
then they are syntactically equivalent.
For any finite semigroup $(S, \cdot)$, there is a number $\omega \in \bbN$ such that
for any $s \in S$, we have $s^{2\omega} = s^\omega$ – since $k \mapsto s^k$
yields an ultimately cyclic sequence with period at most $|S|$,
$\omega = |S|!$ will work.
We say that $T: \Sigma^* \ra \Sigma'^*$ is a padding iff
there exist $e_L, e, e_R \in (\Sigma'-\Sigma)^*$ such that,
for any $w=t_1 \ldots t_n \in \Sigma^*$,
$T(w)$ is the word $e_L t_1 e t_2 e \ldots e t_n e_R$.
For a padding $T$,
the languages $L(G_1)$ and $L(G_2)$ are separable, iff $T(L(G_1))$ and $T(L(G_2))$ are.
The forward direction is straightforward, and proven just as Lemma <ref>
– the automaton simply ignores all the symbols from $\Sigma'-\Sigma$ (in fact,
the stronger version of this lemma where $e_L, e, e_R \in \Sigma'^*$ is also true).
For the backward direction, we take the DFA $A' = (\Sigma', Q, q_I, F, \delta)$.
We can assume that there are no transitions to the initial state $q_I$ in $A'$ –
otherwise, we create a copy of $q_I$ and make it the new initial state.
We construct a new DFA $A'' = (\Sigma, Q, q_I, F', \delta')$ in the following way: take
$\delta'(q_I,t) = \delta(q_I, e_L t)$, and $\delta'(q,t) = \delta(q, et)$ for $q \neq q_I$.
For $F'$ we take the set of states $q$ such that $\delta(q,e_R) \in F$.
automaton $A''$ working on $w \in \Sigma^*$ simulates the automaton $A'$ working
on $T(w)$, hence it accepts $w$ iff $A'$ accepts $T(w)$.
There is a padding $T$ with the following property:
for each $w \in L(G_i)$, there is a word $w' \in L(G'_i)$
which is equivalent to $T(w)$ with respect to $R$.
This proves Theorem <ref> and thus Theorem <ref>. Indeed,
we will show that $T(L(G_1))$ and $T(L(G_2))$ are separated by $R$ – then, after
applying Lemma <ref>, we get our claim.
Consider $w \in L(G_i)$; we have to show that $L(A)$ accepts $T(w)$ iff $i=1$.
From Lemma <ref> we know that there is some $w' \in L(G'_i)$ which
is equivalent to $T(w)$ with respect to $R$. Therefore,
we know that $T(w) \in R$ iff
$w' \in R$, and since $w' \in L(G'_i)$, $T(w) \in R$ iff $i=1$.
Let $\ob = \oa \oa \ca$, $\cb = \oa \ca \ca$. Thus, for any
non-terminal $N' \in V'$, by applying one of productions (
<ref>, <ref> or <ref>) and then
(<ref>), we have $N' \ra^* \ob N' \ca$ and
$N' \ra^* \oa N' \cb$. Also, let $\nu = \omega-1$.
By applying the above many times to terminals $K'$ and $L'$, we get
$K' \ra^* b_1 K' b_2$ and $C' \ra^* c_1 L' c_2$, where:
\begin{eqnarray}
b_1 &=& (\oa^\nu \ob^\nu)^\omega \oa^\nu \\
b_2 &=& \cb^\nu (\ca^\nu \cb^\nu)^\omega \\
c_1 &=& \ob^\omega (\ob^\nu \oa^\nu)^\omega \\
c_2 &=& (\cb^\nu \ca^\nu)^\omega \ca^\nu
\end{eqnarray}
Now, whenever we have a production $N \ra KL$ in $R$, we can do the following in $R'$:
\begin{equation} \label{goodjob}
N' \ra \oa K'L' \ca \ra^* \oa b_1 K' b_2 c_1 L' c_2 \ca
\end{equation}
This can be written as $N' \ra e_L K' e L' e_R$, where:
\begin{eqnarray}
e_L &=& \oa b_1 \\
e &=& b_2 c_1 \\
e_R &=& c_2 \ca
\end{eqnarray}
We claim that the padding $T$ given by the words $e_L$, $e$, $e_R$ defined above
satisfies our claim.
Indeed, take $w \in L(G_i)$. Consider the derivation tree of $w$ in $G_i$;
repeat this derivation in $G'_i$, replacing each production $N \ra KL$ with
$N' \ra e_L K' e L' e_R$ according to
the chain of productions (<ref>) above, and each production $N \ra t$ with $N' \ra t$
In the end, for $w = t_1 \ldots t_n$, we obtain the word $w' \in L(G'_i)$, which
contains the symbols $t_1, \ldots, t_n$ separated with $e$, possibly accompanied
by $e_L$'s on the right side and $e_R$'s on the left side, and with at least one
$e_L$ before $t_1$ and at least one $e_R$ after $t_N$. In other words,
\[
w' = e_L^{l_1} t_1 e_R^{r_1} e e_L^{l_2} t_2 e_R^{r_2} e e_L^{l_3} \ldots
t_n e_R^{r_N} \]
where all $l_i$, $r_i$ are integers, and $l_1,r_n \geq 1$.
Remembering that $\oa$ is a left factor of $\ob$ and thus $\oa^{\omega+\nu} \ob \equiv
\oa^\nu \ob$, and similarly $\cb \ca^{\omega+\nu} \equiv \cb \ca^\nu$,
it can be checked that the following equivalences hold:
* $e_L e_L$ is equivalent to $e_L$:
\begin{eqnarray*}
&& e_L e_L = \oa b_1 \oa b_1 = \\ &=&
\oa (\oa^\nu \ob^\nu)^\omega \oa^\nu \oa (\oa^\nu \ob^\nu)^\omega \oa^\nu \equiv \\
&\equiv& \oa (\oa^\nu \ob^\nu)^\omega \oa^\omega (\oa^\nu \ob^\nu)^\omega \oa^\nu \equiv \\
&\equiv& \oa (\oa^\nu \ob^\nu)^\omega (\oa^\nu \ob^\nu)^\omega \oa^\nu \equiv
\oa (\oa^\nu \ob^\nu)^\omega \oa^\nu = \oa b_1 = e_L
\end{eqnarray*}
* $e_R e_R$ is equivalent to $e_R$:
\begin{eqnarray*}
&& e_R e_R = c_2 \ca c_2 \ca = \\ &=&
(\cb^\nu \ca^\nu)^\omega \ca^\nu \ca (\cb^\nu \ca^\nu)^\omega \ca^\nu \ca \equiv \\
&\equiv& (\cb^\nu \ca^\nu)^\omega \ca^\omega (\cb^\nu \ca^\nu)^\omega \ca^\nu \ca \equiv \\
&\equiv& (\cb^\nu \ca^\nu)^\omega (\cb^\nu \ca^\nu)^\omega \ca^\nu \ca \equiv
(\cb^\nu \ca^\nu)^\omega \ca^\nu \ca = c_2 \ca = e_R
\end{eqnarray*}
* $e e_L$ is equivalent to $e$:
\begin{eqnarray*}
&& e e_L = b_2 c_1 \oa b_1 = \\ &=&
b_2 \ob^\omega (\ob^\nu \oa^\nu)^\omega \oa (\oa^\nu \ob^\nu)^\omega \oa^\nu \equiv \\
&\equiv& b_2 \ob^\omega (\ob^\nu \oa^\nu)^\omega \oa \oa^\nu (\ob^\nu \oa^\nu)^\omega \equiv \\
&\equiv& b_2 \ob^\omega (\ob^\nu \oa^\nu)^\omega (\ob^\nu \oa^\nu)^\omega \equiv
b_2 \ob^\omega (\ob^\nu \oa^\nu)^\omega = b_2 c_1 = e
\end{eqnarray*}
* $e_R e$ is equivalent to $e$:
\begin{eqnarray*}
&& e_R e = c_2 \ca b_2 c_1 = \\ &=&
(\cb^\nu \ca^\nu)^\omega \ca^\nu \ca \cb^\nu (\ca^\nu \cb^\nu)^\omega c_1 \equiv \\
&\equiv& (\cb^\nu \ca^\nu)^\omega \cb^\nu (\ca^\nu \cb^\nu)^\omega c_1 \equiv \\
&\equiv& (\cb^\nu \ca^\nu)^\omega (\cb^\nu \ca^\nu)^\omega \cb^\nu c_1
\equiv (\cb^\nu \ca^\nu)^\omega \cb^\nu c_1 = b_2 c_1 = e
\end{eqnarray*}
These rules allow us to reduce all $l_i$ and $r_i$ to zeros (except $l_1$ and $r_n$,
which can be reduced to 1). Hence, the word $w'$ is equivalent to $T(w)$.
§ WHY BINARY TREES?
Our proof uses binary flattened tree grammars – that is,
productions of form $N \ra \oa N_1 N_2 \ldots N_k \ca$ are allowed only for $k=0$
or $k=2$.
If we allowed such rules for any $k$, a somewhat easier proof would work.
The general structure is the same, but
it is not required to have $G_i$'s in the Chomsky normal form,
and the grammar $G'$ allows inserting empty brackets with productions
$E' \ra \oa\ca | \oa E' \ca | \oa E'E' \ca$ before and after every symbol:
$N \ra \oa E' X_1 E' X_2 \ldots X_k E' \ca$, and $e_L=e=e_R=(\oa^\omega \ca\omega)^\omega$
can be obtained by pumping each pair of $\oa\ca$ $\omega$ times, and then
using the rule $E' \ra E'E'$ to make sure that $\oa^\omega \ca^\omega$ appears
$k\omega$ times between each two consecutive terminals.
While such a proof
was sufficient to solve the problem for visibly pushdown languages, and for
XML encoding of trees, it left us with a craving for more, for the following
First, if we consider languages of terms, it is natural to consider
different symbols for nodes with different arities (numbers of children) –
while the simpler proof above heavily uses the fact that the XML encoding cannot
tell whether $\oa$ comes from the structurally significant rule $N \ra \oa E' X_1 E' X_2 \ldots X_k E' \ca$,
or it is a fake inserted by the $X \ra \oa X \ca$ or $E' \ra \oa E' E' \ca$ rules.
Since the arities here are respectively $2k+1,$ 1 and 2, this fails if
the automaton sees them.
Moreover, if we consider the process of flattening a tree as the result of an
automaton which traverses the tree recursively and react to what it is seeing on its path,
it is natural to assume that such an automaton sees the arity, and moreover,
between returning from the $i$-th child of $v$ and progressing to the $(i+1)$-th child,
the automaton should see that $i$ of $n$ children are progressed. This corresponds
to flattening rules of form $N \ra \oa_0^k X_1 \oa_1^k X_2 \oa_2^k X_3 \ldots X_k \oa_k^k$.
Restricting ourselves to the binary case allows us to solve the problem in full
generality – while the encoding in the definition of BFG does not explicitely
say whether $\oa$ comes from
the “empty leaf” rule $E' \ra \oa\ca$ or from one of the binary branching rules,
a finite automaton can easily tell which one is the case by looking at the neighborhood.
Also, while we do not write the infix structural symbols $\oa_1^2$ explicitely,
the automaton can tell that it is at such a branching point iff the last symbol was $\ca$,
and the next one is $\oa$.
In the binary case, a computer program was used to find the
appropriate words $e,e_L,e_R$ which work in Lemma <ref>.
§ CONCLUSION
We can also consider the separation problem for other classes of languages –
that is, is it decidable whether languages accepted by two BFGs are separable
by a language of class $\calC$? From the proof above, this problem is undecidable
for class $\calC$, as long as the following conditions are satisfied:
* The respective problem for CFGs is undecidable.
* The class $\calC$ has the following pumping property: for any $L \in \calC$,
there exists some $\omega$ such that for any words $v$, $w$, $x$,
$v{w^\omega}x \in L$ iff $v{w^{2\omega}}x \in L$. (This is Lemma
<ref>, and it is used in the proof of Lemma <ref>.)
* If $\pi$ is a homomorphism which ignores a subset of symbols,
and $L \in \calC$, then $\pi^{-1}(L) \in \calC$. (Used in the proofs of Lemma
<ref> and <ref>.)
* If $T(t_1 \ldots t_k) = e_L t_1 e t_2 e \ldots e t_k e_R$
for some $s$, and $L \in \calC$, then $T(L) \in \calC$. (Used in the proof of <ref>.)
Hence, the separability problem is also undecidable for other classes of languages,
such as the class of languages recognizable by first-order logic.
|
1511.00294
|
Viehweg's hyperbolicity conjecture]Viehweg's hyperbolicity conjecture for families with maximal variation
Department of Mathematics, Northwestern University,
2033 Sheridan Road, Evanston, IL 60208, USA
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651
[2000]14D06, 14D07; 14F10, 14E30
We use the theory of Hodge modules to construct Viehweg-Zuo sheaves on base spaces of families with maximal
variation and fibers of general type; and, more generally, families whose geometric generic fiber has a good minimal model. Combining
this with a result of Campana-Păun, we
deduce Viehweg's hyperbolicity conjecture in this context, namely the fact that the base spaces of such families
are of log general type. This is approached as part of a general problem of identifying what spaces can support Hodge theoretic objects with certain positivity properties.
§ INTRODUCTION
§.§ Families of varieties
The main aim of this paper is to give a proof of Viehweg's hyperbolicity conjecture
for base spaces of families of varieties of general type with maximal variation, and
more generally, when assuming the conjectures
of the minimal model program, for arbitrary families with maximal variation.
Let $f: Y \rightarrow X$ be an algebraic fiber space between smooth projective
varieties, and let $D \subseteq X$ be any divisor containing the singular locus of $f$. Assume that
$f$ has maximal variation, in the sense that $\Var (f) = \dim X$. Then:
* If the general fiber of $f$ is of general type, then the pair $(X, D)$ is of
log-general type, meaning that $\omX (D)$ is big.
* More generally, the same conclusion holds if the geometric generic fiber of $f$ admits a good minimal model.
We obtain thm:hyperbolicity as a consequence of the main result we prove, regarding the
existence of what are sometimes called Viehweg-Zuo sheaves, stated below, combined
with a key theorem of Campana-Păun <cit.> on the pseudo-effectivity of quotients of powers of log-cotangent bundles.
Let $f: Y \rightarrow X$ be an algebraic fiber space between smooth projective
varieties, such that the $f$-singular locus $D_f \subseteq X$ is a simple normal crossing divisor.
Assume that for every generically finite $\tau: \tilde X \rightarrow X$ with $\tilde
X$ smooth, and for every resolution $\tilde Y$ of $Y\times_X \tilde X$, there is an
integer $m \geq 1$ such that
$\det {\tilde f}_* \omega_{\tilde Y/ \tilde X}^{\otimes m}$ is big. Then there exists a big coherent sheaf $\shH$ on $X$ and an integer
$s \ge 1$, together with an inclusion
$$\shH \hookrightarrow \big(\OmX^1 (\log D_f) \big)^{\otimes s}.$$
Going back to the statement of thm:hyperbolicity, the variation $\Var
(f)$ is an invariant introduced by Viehweg <cit.> in order to measure how
much the birational isomorphism class of the fibers of $f$ varies along $Y$. Maximal
variation $\Var(f) = \dim X$ simply means that the general fiber can only be
birational to countably many other fibers. The connection between the two theorems is made
via Viehweg's $Q_{n,m}$ conjecture, which states that if $f$ has maximal variation,
then $\det \fl \omYX^{\otimes m}$ is big for some $m \geq 1$.[By the
determinant of a torsion-free sheaf $\shF$ of generic rank $r$, we mean $(\bigwedge^r
\shF)^{\vee \vee}$.] This implies Viehweg's $C_{n,m}^+$ generalization of Iitaka's
conjecture, see <cit.> and <cit.>, and was
shown to hold when the fibers are of general type by Kollár <cit.> (see also
<cit.>). Moreover, Kawamata <cit.> proved that
that $Q_{n,m}$ holds for any morphism whose geometric generic fiber has a good
minimal model.
§.§ Previous results
Viehweg's original conjecture (see <cit.> or <cit.>)
is a generalization of Shafarevich's conjecture on non-isotrivial one-parameter
families of curves: it states that if $X^{\circ}$ is smooth and quasi-projective,
and $f^{\circ}: Y^{\circ} \rightarrow X^{\circ}$ is a smooth family of canonically
polarized varieties with maximal variation, then $X^\circ$ must be of log-general
type.[It is standard that $f^{\circ}$ can be compactified to a morphism $f:
X\rightarrow Y$ whose singular locus $D = X \setminus X^{\circ}$ is a divisor. In the paper we phrase things directly in this set-up, but note that the conclusions should be seen as properties of the original family $f^{\circ}$.] This is of course very much related to the study of subvarieties of moduli stacks.
In this setting, i.e. for families of canonically polarized varieties,
thm:hyperbolicity is by now fully known. This is due
to important work of many authors, all relying crucially on the existence of
Viehweg-Zuo sheaves <cit.> for such families; we briefly
review the main highlights, without providing an exhaustive list (but see also
<cit.> and <cit.> for comprehensive surveys and references to previous
work over one-dimensional bases). The result was shown by Kebekus-Kovács when $X$
is a surface in <cit.>, and then in <cit.> when $D= \emptyset$ assuming the
main conjectures of the minimal model program, while <cit.> contains more refined results in
dimension at most three. It was then deduced unconditionally by Patakfalvi
<cit.> from the results of <cit.>, when $D = \emptyset$ and when $X$ is
not uniruled. Finally, Campana-Păun obtained the result in general,
based on their bigness criterion <cit.> that we use here as well. Using further
results from <cit.>, it is possible to extend this argument to families of varieties with semiample
canonical bundle.
The work of Viehweg-Zuo <cit.>, <cit.> suggested however that Viehweg's conjecture should hold much more generally,
indeed for all families with maximal variation. Note that the full thm:hyperbolicity in the case $\dim X = 1$ was proved in <cit.>. Our work owes a lot to the general strategy introduced in their papers, as we will see below.
§.§ Kebekus-Kovács and Campana conjectures
In <cit.> Kebekus and Kovács proposed a natural extension of Viehweg's conjecture taking into account
families of canonically polarized varieties that are not necessarily of maximal variation. At least when $\kappa (X^\circ) \ge 0$ it predicts that one should have
$\kappa (X^\circ) \ge {\rm Var}(f)$; see generalized_conjecture for the precise statement and the results they obtained.
We remark in <ref> that, at least when $X^\circ$ is projective, the methods of this paper also apply to the extension of this conjecture to families whose geometric generic fiber has a good minimal model, assuming a positive answer to an abundance-type conjecture of Campana-Peternell.
A closely related problem is Campana's isotriviality conjecture, which predicts that
smooth families of canonically polarized varieties over a special base must be
isotrivial, and which implies the Kebekus-Kovács conjecture. This conjecture has recently been proved by Taji <cit.>,
It would be interesting to know whether our construction (based on Hodge modules) can be adapted
to the case of orbifolds.
§.§ Abstract results on Hodge modules and Higgs bundles
The main new ingredient for proving thm:VZ-sheaves is the construction in chap:Hodge
of certain Hodge modules, and of Higgs bundles derived from them. They are associated to morphisms whose relative canonical bundle satisfies a mild positivity condition; see ($\ref{eq:sections}$) below.
Such Hodge modules and Higgs bundles satisfy a “largeness" property for the first
step in the Hodge filtration, defined in <ref>. The existence of a
Viehweg-Zuo sheaf, meaning a big sheaf as in thm:VZ-sheaves, turns out to be an instance of an abstract result about Hodge modules with this property.
Oversimplifying in order to explain the main idea, we consider pure Hodge modules
with the property that there exist a big line bundle $A$ and a sheaf inclusion
\begin{equation}\label{eqn:first}
A \hookrightarrow F_{p(\Mmod)} \Mmod \otimes \shO_X(\ell D)
\end{equation}
where $F_{p(\Mmod)} \Mmod$ is the lowest non-zero term in the filtration
$F_{\bullet} \Mmod$ on the underlying $\Dmod$-module $\Mmod$, $D$ is a divisor away from which $M$ is a variation of Hodge structure, and $\ell \ge 0$ is an integer. This can be seen as an abstract version of the more familiar property of the period map being immersive at a point.
Let $X$ be a smooth projective variety, and $M$ a polarizable pure Hodge module with strict support $X$, extending a variation of Hodge structure of weight $k$ on a dense open subset $U = X \setminus D$, with $D$ a divisor. If its underlying filtered
$\Dmod_X$-module $(\Mmod, F_{\bullet}\Mmod)$ satisfies ($\ref{eqn:first}$), then at least one of the following holds:
* $D$ is big.
* There exists $1 \le s \le k$, $r \ge 1$, and a big coherent sheaf $\shG$ on $X$
such that
$$\shH \hookrightarrow (\OmX^1)^{\otimes s} \otimes \shO(rD).$$
Consequently, if $X$ is not uniruled, then $\omX (D)$ is big.
Therefore variations of Hodge structure can have such large extensions only if they
are supported on the complement of a sufficiently positive divisor.[Another example of this phenomenon is <cit.>, which says that $D$ must be ample when $X$ is an abelian variety.] When the base is not uniruled this implies by the result of <cit.> that it must actually be of log-general type. In this abstract context the non-uniruledness hypothesis is necessary; see ex:uniruled-counterex.
In reality, for applications like thm:VZ-sheaves a more refined setup is needed. Besides the polarizable Hodge module $M$, we also need to consider a graded $\Sym \shT_X$-submodule $\shGb \subseteq \gr_{\bullet}^F \! \Mmod$ of the associated graded of the underlying filtered $\Dmod_X$-module $(\Mmod, F_{\bullet} \Mmod)$. This is constructed in thm:VZ.
In the case when $X$ is an abelian variety, this type of construction was considered in <cit.>.
We prove and use a slightly stronger version of thm:HM-positivity involving such submodules; see
thm:submodule-positivity for the precise statement, which uses a recent weak positivity result for Hodge modules from <cit.>.
To deal with the case when $X$ is uniruled, an additional step is needed. Using the
pair $(M, \shGb)$ from above, we produce
in thm:Higgs another pair $(\shEb, \shFb)$, where $(\shEb, \theta)$ is a
(graded logarithmic) Higgs bundle with poles along a divisor
$D$ containing $D_f$, while $\shFb \subseteq \shEb$ is a subsheaf such that
\[
\theta(\shFb) \subseteq \Omega^1_X (\log D_f) \otimes \shF_{\bullet+1},
\]
and which again satisfies a largeness property. This uses some of the more technical
aspects of the theory of Hodge modules, especially the interaction between the Hodge filtration and the $V$-filtration along hypersurfaces. The corresponding version of thm:HM-positivity is thm:submodule-Higgs, which finally produces the Viehweg-Zuo sheaf best suited for our purposes.
§.§ Outline of the proof
We summarize the discussion above into a brief outline of the proof of thm:VZ-sheaves and thm:hyperbolicity.
* Due to results of Kawamata, Kollár, and Viehweg, families with maximal
variation whose geometric generic fiber has a good minimal model satisfy the
hypothesis of thm:VZ-sheaves.
* Given a big line bundle $L$ on $X$, we show that the
$m$-th power of the line bundle $\omYX \tensor \fu L^{-1}$ has a nontrivial global
section for some $m \ge 1$. This uses the bigness of $\det \fl \omega_{\tilde Y/ \tilde X}^{\tensor m}$
on generically finite covers of $X$, Viehweg's fiber product trick, and semistable reduction; in the process, we replace $Y$
by a resolution of singularities of a very large fibered product $Y \times_X \dotsm
\times_X Y$.
* We construct a polarizable Hodge module $M$ on $X$, together with a graded
$\Sym \shT_X$-submodule $\shGb \subseteq \grFMb$, such that $\shG_0 = L$ and $\Supp
\shG \subseteq S_f$, the set of singular cotangent vectors of $f$ in $T^* X$.
Both are obtained from a resolution of singularities of a
branched covering determined by the section in (2). After resolving
singularities, we may assume that $M$ restricts to a variation of Hodge structure
outside a normal crossing divisor $D \supseteq D_f$.
* The variation of Hodge structure on $X \setminus D$ determines a (graded
logarithmic) Higgs bundle $\shEb$ with Higgs field
\[
\theta \colon \shEb \to \OmX^1(\log D) \tensor \shE_{\bullet+1}.
\]
From $\shGb$ in (3), we construct
a subsheaf $\shFb \subseteq \shEb$ such that
$\shF_0$ is a big line bundle and $\theta(\shFb) \subseteq \OmX^1(\log D_f) \tensor
\shF_{\bullet+1}$. This uses the interaction between the Hodge filtration and the $V$-filtration on Hodge modules.
* We deduce that some large tensor power of $\OmX^1(\log D_f)$ contains a big subsheaf
(or Viehweg-Zuo sheaf), concluding the proof of thm:VZ-sheaves.
The main ingredient is a slight extension of a theorem of Zuo,
to the effect that the dual of the kernel of $\theta \colon \shEb \to \OmX^1(\log D) \tensor
\shE_{\bullet+1}$ is weakly positive.
* Finally, to deduce thm:hyperbolicity, we can assume after a birational modification that the $f$-singular locus
$D_f$ is a divisor with simple normal crossings. We apply thm:VZ-sheaves and a recent theorem by Campana and Păun to conclude that,
in this situation, the line bundle $\det \OmX^1(\log D_f) = \omega_X ( D_f)$ is big. This
proves that the pair $(X, D_f)$ is of log general type.
These steps are addressed throughout the paper, and are collected together in
<ref>. We also include in <ref> a substantially
simpler proof of thm:hyperbolicity in the case when $X$ is not uniruled.
It avoids many of the technicalities involved in dealing with the remaining case,
while still containing all the key ideas in a particularly transparent form; hence it
may help at a first reading. In brief, it only needs a less precise version of (1),
which does not use semistable reduction; it does not need (4), which is the most
technical part of the general proof, and it replaces (5) with similar results applied directly to the Hodge module construction in (2).
§.§ What is new
As mentioned above, our work owes to the beautiful approach of Viehweg and Zuo
<cit.> to the study of families with maximal variation, by means of constructing Viehweg-Zuo sheaves as a main step towards understanding the base space of such a family. For families of varieties with semiample canonical bundle they constructed
Higgs systems as in (4) above, using a very delicate analysis based on weak positivity, while dealing with mild enough singularities due to the semiampleness assumption. They pioneered the idea of using negativity results for Kodaira-Spencer
kernels to extract positivity from the Higgs systems thus constructed.
This paper offers two main new inputs. The first is to view the hyperbolicity problem as a special case of the study of spaces supporting abstract Hodge theoretic objects satisfying the largeness property described above, an interesting problem in its own right.
The second, and most significant, is the use of Hodge modules. Just as in <cit.>, we are able to address a more general situation due to the fact that Hodge modules provide higher flexibility in dealing with the singularities created by section produced in (2), and for applying positivity results. Even in the case of canonically polarized fibers this simplifies the argument in <cit.>, at least when the base is not uniruled. In general however, besides appealing to a Hodge module construction, providing a Higgs system with all the necessary properties requires the use of some quite deep input from Saito's theory.
§.§ Acknowledgement
The first author is grateful for the hospitality of the Department of Mathematics at
the University of Michigan, where he completed part of this work. Both authors thank Frédéric Campana,
Stefan Kebekus, Sándor Kovács, Mihai Păun, and Behrouz Taji for useful discussions. They also
thank the referee for suggestions that improved the exposition.
During the preparation of the paper, M. Popa was partially supported by NSF grant
DMS-1405516 and a Simons Fellowship, and Ch. Schnell by NSF grant DMS-1404947 and a
Centennial Fellowship of the American Mathematical Society.
§ CONSTRUCTION OF HODGE MODULES AND HIGGS BUNDLES
Let $f \colon Y \to X$ be a surjective morphism between two smooth projective
varieties. In this chapter, we describe a general method for obtaining information
about the $f$-singular locus $D_f \subseteq X$ of the morphism from positivity
assumptions on the relative canonical bundle $\omYX$.
§.§ Cotangent bundles
We begin by introducing a more refined measure, inside the cotangent bundle, for the
singularities of a morphism. Given a surjective morphism $f \colon Y \to X$ between
two smooth projective varieties, we use the following notation for the induced
morphisms between the cotangent bundles of $X$ and $Y$.
\begin{equation} \label{eq:Japanese}
\begin{tikzcd}
Y \dar{f} & T^{\ast} X \times_X Y \lar[swap]{p_2} \dar{p_1} \rar{\df} & T^{\ast} Y \\
X & T^{\ast} X \lar[swap]{p}
\end{tikzcd}
\end{equation}
Inside the cotangent bundle of $X$, consider the set of singular cotangent
\[
S_f = p_1 \bigl( \df^{-1}(0) \bigr) \subseteq T^{\ast} X.
\]
A cotangent vector $(x, \xi) \in T^{\ast} X$ belongs to $S_f$ if and
only if $\fu \xi$ vanishes at some point $y \in f^{-1}(x)$, or equivalently, if $\xi$
annihilates the image of $T_y Y$ inside the tangent space $T_x X$. If this happens
and $\xi \neq 0$, then $f$ is not submersive at $y$, which means that $x$ belongs to
the $f$-singular locus $D_f \subseteq X$. Consequently, $S_f$ is the union of the zero
section and a closed conical subset of $T^{\ast} X$ whose image under the projection
$p \colon T^{\ast} X \to X$ is equal to $D_f$.
The set of singular cotangent vectors $S_f$ has the following interesting property.
One has $\dim S_f \leq \dim X$, and every irreducible component of $S_f$ of dimension
$\dim X$ is the conormal variety of a subvariety of $X$.
Fix an irreducible component $W \subseteq S_f$, and denote by $Z = p(W)$ its image in
$X$; because $S_f$ is conical, this is a closed subvariety of $X$. Both
assertions will follow if we manage to show that $W$ is contained in the conormal
\[
T_Z^{\ast} X
= \text{closure in $T^{\ast} X$ of the conormal bundle to the smooth locus of $Z$.}
\]
By definition, for every cotangent vector $(x, \xi) \in W$, there is a point $y \in
f^{-1}(x)$ such that $\xi$ vanishes on the image of $T_y Y \to T_x X$. At a general
smooth point $x \in Z$, this image contains the tangent space $T_x Z$, and so
$(x, \xi) \in T_Z^{\ast} X$.
The zero section is clearly one of the irreducible components of $S_f$; for dimension
reasons, the conormal varieties of the divisorial components of $D_f$ are also
contained in $S_f$. Other irreducible components of dimension $\dim X$ are less easy to
come by.
Evidently, the morphism $f$ is smooth if and only if $S_f$ is equal to the zero
section. One method for getting a lower bound on the size of $S_f$ – and, therefore,
of $D_f$ – is to look for coherent sheaves on $T^{\ast} X$ whose support is contained in
the set $S_f$. In practice, it is better to work with sheaves of graded modules over
the symmetric algebra $\shA_X = \Sym \shT_X$, where $\shT_X$ is the tangent sheaf of
$X$. Recall that
\[
\shA_X \simeq \pl \shO_{T^{\ast} X},
\]
and that taking the direct image under $p \colon T^{\ast} X \to X$ gives an
equivalence of categories between (algebraic) coherent sheaves on the cotangent bundle
and coherent $\shA_X$-modules. For a coherent graded $\shA_X$-module
\[
\shGb = \bigoplus_{k \in \ZZ} \shG_k,
\]
we use the symbol $\shG$, without the dot, to denote the associated coherent sheaf on
$T^{\ast} X$; it has the property that $\pl \shG \simeq \shGb$ as modules over
$\shA_X$ (without the grading).
§.§ The main result
For the remainder of the chapter, let us fix a surjective morphism $f \colon Y \to X$
between two smooth projective varieties. We also fix a line bundle $L$ on $X$, and
consider on $Y$ the line bundle
\[
B = \omYX \tensor \fu L^{-1}.
\]
We assume that the following condition holds:
\begin{equation}\label{eq:sections}
H^0 (Y, B^{\otimes m}) \neq 0 \quad \text{for some $m \geq 1$.}
\end{equation}
Starting from this data, we construct a graded module over $\Sym \shT_X$ with the
following properties.
Assuming (<ref>), one can find a graded $\shA_X$-module $\shGb$
that is coherent over $\shA_X$ and has the following properties:
As a coherent sheaf on the cotangent bundle, $\Supp \shG \subseteq S_f$.
One has $\shG_0 \simeq L \tensor \fl \OY$.
Each $\shG_k$ is torsion-free on the open subset $X \setminus D_f$.
There exists a regular holonomic $\Dmod$-module $\Mmod$ with good filtration
$F_{\bullet} \Mmod$, and an inclusion of graded $\shA_X$-modules $\shGb \subseteq
\gr_{\bullet}^F \! \Mmod$.
The filtered $\Dmod$-module $(\Mmod, F_{\bullet} \Mmod)$ underlies a polarizable Hodge module $M$ on
$X$ with strict support $X$, and $F_k \Mmod = 0$ for $k < 0$.
Roughly speaking, $\shGb$ is constructed by applying results from the theory of Hodge
modules to a resolution of singularities of a branched covering of $Y$. This type of
construction was invented by Viehweg and Zuo, but it becomes both simpler and more
flexible through the use of Hodge modules. An introduction to Hodge modules that
covers all the results we need here, with references to the original work of Saito, can be found in <cit.>.
Despite the technical advantages of working with Hodge modules, the proof of
Viehweg's conjecture in the general case works more naturally in the context of
Higgs sheaves (with logarithmic poles along $D_f$). Since we do not have any control
over the zero locus of the section
in (<ref>), we do not attempt to construct such an object directly from
the branched covering. Instead, we use the local properties of Hodge modules to
construct a suitable Higgs sheaf from the graded module $\shGb$, at least on
some birational model of $X$. More precisely, we can perform finitely many blowups
with smooth centers to assume that the $f$-singular locus $D_f$ is a divisor, and moreover to
put ourselves in the following situation:
\begin{equation} \label{eq:NCD}
\text{The singularities of $M$ occur along a normal crossing
divisor $D \supseteq D_f$.}
\end{equation}
Concretely, this means that the restriction of $M$ to the open subset $X \setminus D$
is a polarizable variation of Hodge structure. We use this fact to construct from
$\shGb$ an $\OX$-module $\shFb$ with the structure of a graded Higgs sheaf.
Let $f \colon Y \to X$ be a surjective morphism with connected fibers between two
smooth projective varieties. Assuming (<ref>) and (<ref>), one
can find an $\OX$-module $\shFb$ with the following properties:
One has $L(-D_f) \subseteq \shF_0 \subseteq L$.
Each $\shF_k$ is a reflexive coherent sheaf on $X$.
There exists a (graded logarithmic) Higgs bundle $\shEb$ with Higgs field
\[
\theta \colon \shEb \to \OmX^1(\log D) \tensor \shE_{\bullet+1},
\]
such that $\shFb \subseteq \shEb$ and $\theta(\shFb) \subseteq \OmX^1(\log D_f)
\tensor \shF_{\bullet+1}$.
The pair $(\shEb, \theta)$ comes from a polarizable variation of Hodge structure on
$X \setminus D$ with $\shE_k = 0$ for $k < 0$.
Both theorems will be proved in the remainder of this chapter.
§.§ Constructing the Hodge module
From now on, we assume that the line bundle $B = \omYX \tensor \fu L^{-1}$ satisfies
the hypothesis in (<ref>).
For the sake of convenience, let $m \geq 1$ be the smallest integer with the property
that there is a nontrivial global section $s \in H^0 \bigl( Y, B^{\tensor m} \bigr)$.
Such a section defines a branched covering $\pi \colon Y_m \to Y$ of degree $m$,
unramified outside the divisor $Z(s)$; see <cit.> for details. Since $m$ is
minimal, $Y_m$ is irreducible; let $\mu \colon Z \to Y_m$ be a resolution of
singularities that is an isomorphism over the complement of $Z(s)$, and define
$\varphi = \pi \circ \mu$ and $h = f \circ \varphi$. The following commutative
diagram shows all the relevant morphisms:
\begin{equation} \label{eq:geometry}
\begin{tikzcd}
Z \rar{\mu} \arrow[bend right=20]{drr}{h} \arrow[bend left=40]{rr}{\varphi}
& Y_m \rar{\pi} & Y \dar{f} \\
& & X
\end{tikzcd}
\end{equation}
To simplify the notation, set $n = \dim X$ and $d = \dim Y = \dim Z$. Let $\shH^0 \hl
\QQ_Z^H \decal{d} \in \HM(X, d)$ be the polarizable Hodge module obtained by taking
the direct image of the
constant Hodge module on $Z$; restricted to the smooth locus of $h$, this is just the
polarizable variation of Hodge structure on the middle cohomology of the fibers. Let
$M \in \HM_X(X, d)$ be the summand with strict support $X$ in the decomposition of
$\shH^0 \hl \QQ_Z^H \decal{d}$ by strict support <cit.>. Let $\Mmod$
denote the underlying
regular holonomic left $\Dmod_X$-module, and $F_{\bullet} \Mmod$ its Hodge
filtration. Since $F_{\bullet} \Mmod$ is a good filtration, the associated graded
\[
\gr_{\bullet}^F \! \Mmod = \bigoplus_{k \in \ZZ} \gr_k^F \! \Mmod
\]
is coherent over $\shA_X = \Sym \shT_X$; for simplicity, we denote the corresponding
coherent sheaf on the cotangent bundle by the symbol $\grFM$.
One has the following more concrete description of $\grFMb$. On $Z$, consider the
complex of graded $\hu \shA_X$-modules
\[
C_{Z, \bullet} = \left\lbrack
\hu \shA_X^{\bullet-n} \tensor \bigwedge^d \shT_Z \to
\hu \shA_X^{\bullet-n+1} \tensor \bigwedge^{d-1} \shT_Z \to \dotsb \to
\hu \shA_X^{\bullet-n+d} \right\rbrack,
\]
placed in cohomological degrees $-d, \dotsc, 0$; the differential in the complex is
induced by the natural morphism $\shT_Z \to \hu \shT_X$.
In the category of graded $\shA_X$-modules, $\grFMb$ is a direct summand of
$R^0 \hl \bigl( \omZX \tensor C_{Z, \bullet} \bigr)$.
This is a special case of the following more general result: the complex
\begin{equation} \label{eq:complex}
\derR \hl \bigl( \omZX \tensor C_{Z, \bullet} \bigr)
\end{equation}
splits in the derived category of graded $\shA_X$-modules, and its $i$-th
cohomology module computes the associated graded of the Hodge module $\shH^i
\hl \QQ_Z^H \decal{d}$. The proof is an application of several results by Saito. The
underlying filtered $\Dmod$-module of the trivial Hodge module is $(\OZ, F_{\bullet}
\OZ)$, where the filtration is such that $\gr_k^F \OZ = 0$ for $k \neq 0$. As shown
in <cit.>, the direct image of $(\OZ, F_{\bullet} \OZ)$ in the
derived category of filtered $\Dmod$-modules satisfies
\begin{equation} \label{eq:Laumon}
\gr_{\bullet}^F \hp(\OZ, F_{\bullet} \OZ) \simeq
\derR \hl \left( \omZX \tensor_{\OZ} \gr_{\bullet+d-n}^F \! \OZ
\Ltensor_{\shA_Z} \hu \shA_X \right).
\end{equation}
Since the morphism $h \colon Z \to X$ is projective, the complex $\hp(\OZ,
F_{\bullet} \OZ)$ is strict and splits in the derived category <cit.>;
the same is therefore true for the complex of
graded $\shA_X$-modules on the left-hand side of (<ref>). To conclude the
proof, we only have to show that the complex on the right-hand side of
(<ref>) is quasi-isomorphic to the one in (<ref>). Now the
associated graded of the constant Hodge module is $\OZ$, in degree zero, with the
trivial action by $\shA_Z$. It is naturally resolved by the complex of graded
\begin{equation} \label{eq:resolution}
\left\lbrack
\shA_Z^{\bullet-d} \tensor \bigwedge^d \shT_Z \to
\shA_Z^{\bullet-d+1} \tensor \bigwedge^{d-1} \shT_Z \to \dotsb \to
\shA_Z^{\bullet} \right\rbrack,
\end{equation}
placed in cohomological degrees $-d, \dotsc, 0$. After shifting the grading by $d-n$
and tensoring over $\shA_Z$ by $\hu \shA_X$, we obtain the desired result.
The lemma gives some information about the individual $\OX$-modules $\gr_k^F \Mmod$.
One has $\gr_k^F \Mmod = 0$ for $k < n-d$, whereas
$\gr_{n-d}^F \Mmod \simeq \hl \omZX$.
The first assertion is clear because $C_{Z,k} = 0$ for $k < n-d$. To prove the second
assertion, recall that we have a canonical decomposition
\[
\shH^0 \hl \QQ_Z^H \decal{d} \simeq M \oplus M',
\]
where $M$ has strict support $X$, and $M'$ is supported in a union of proper
subvarieties. prop:Laumon shows that
\[
F_{n-d} \Mmod \oplus F_{n-d} \Mmod'
\simeq R^0 \hl \bigl( \omZX \tensor C_{Z, n-d} \bigr) \simeq \hl \omZX.
\]
But now $F_{n-d} \Mmod'$ is supported in a union of proper subvarieties, whereas
$\hl\omZX$ is torsion-free; the conclusion is that $F_{n-d} \Mmod' = 0$. This is a
special case of a much more general result by Saito <cit.>.
The complex $C_{Z, \bullet}$ is closely related to the set of singular cotangent
vectors $S_h \subseteq T^{\ast} X$ of the morphism $h \colon Z \to X$. Recall the
following notation:
\begin{equation} \label{eq:diag-dh}
\begin{tikzcd}
Z \dar{h} & T^{\ast} X \times_X Z \lar[swap]{p_2} \dar{p_1} \rar{\mathit{dh}} & T^{\ast} Z \\
X & T^{\ast} X \lar[swap]{p}
\end{tikzcd}
\end{equation}
Let us denote by $C_Z$ the complex of coherent sheaves on $T^{\ast} X \times_X Z$
associated with the complex of graded $\hu \shA_X$-modules $C_{Z, \bullet}$.
The support of $C_Z$ is equal to $\mathit{dh}^{-1}(0) \subseteq T^{\ast} X \times_X Z$.
The complex of graded $\shA_Z$-modules in (<ref>) is a resolution of
$\OZ$ as a graded $\shA_Z$-module, and so the associated complex of coherent sheaves
on $T^{\ast} Z$ is quasi-isomorphic to the structure sheaf of the zero section. The
proof of prop:Laumon shows that $C_Z$ is the pullback of this complex via
the morphism $\mathit{dh}$ in the diagram in (<ref>);
its support must therefore be equal to $\mathit{dh}^{-1}(0)$.
This result also implies the well-known fact that the characteristic variety of the
Hodge module $M$ is contained inside the set $S_h \subseteq T^{\ast} X$.
The support of $\grFM$ is a union of irreducible components of $S_h$.
According to prop:Laumon, one has
\[
\Supp \grFM \subseteq \Supp R^0 p_{1\ast} \bigl( p_2^{\ast} \omZX \tensor C_Z \bigr),
\]
and because $\Supp C_Z$ is equal to $\mathit{dh}^{-1}(0)$, it follows that $\Supp
\grFM$ is contained in $S_h = p_1 \bigl( \mathit{dh}^{-1}(0) \bigr)$. Now
the support of $\grFM$ is by definition the characteristic variety of the regular
holonomic $\Dmod$-module $\Mmod$, and therefore of pure dimension $n = \dim X$. It
must therefore be a union of irreducible components of $S_h$, because we know from
lem:Sf that $\dim S_h \leq n$.
Note that $S_h$ may very well have additional components of dimension $n$ that are
not accounted for by the Hodge module $M$. In any case, the existence of $M$ by
itself tells us nothing about the original morphism $f$.
§.§ Constructing the graded module
We now explain how to use the geometry of the branched covering in
(<ref>) to construct a graded $\shA_X$-submodule
\[
\shGb \subseteq \grFMb
\]
which, unlike $\grFMb$ itself, encodes information about the $f$-singular locus $D_f$
of the original morphism $f \colon Y \to X$.
In fact, the support of $\shGb$ will be contained in the set of singular cotangent vectors $S_f$; the point is that $S_f$ is typically
much smaller than $S_h$, because both the covering and its resolution create
additional singular fibers. This construction will allow us to use positivity properties of Hodge modules towards the study of $D_f$.
By construction, the branched covering $Y_m$ is embedded into the total space of the
line bundle $B = \omYX \tensor \fu L^{-1}$, and so the pullback $\varphiu B$ has a
tautological section; the induced morphism $\varphiu B^{-1} \to \OZ$ is an
isomorphism over the complement of $Z(s)$. After composing with $\varphiu \OmY^k
\to \OmZ^k$, we obtain for every $k = 0, 1, \dotsc, d$ an injective morphism
\begin{equation} \label{eq:morphism}
i_k \colon \varphiu \bigl( B^{-1} \tensor \OmY^k \bigr) \to \OmZ^k,
\end{equation}
that is actually an isomorphism over the complement of $Z(s)$.
There is a morphism of complexes of graded $\shA_X$-modules
\[
\derR \fl \bigl( B^{-1} \tensor \omYX \tensor C_{Y, \bullet} \bigr) \to
\derR \hl \bigl( \omZX \tensor C_{Z, \bullet} \bigr),
\]
induced by the individual morphisms in (<ref>).
By adjunction, it suffices to construct a morphism of complexes
\[
\varphiu \bigl( B^{-1} \tensor \omYX \tensor C_{Y, \bullet} \bigr) \to
\omZX \tensor C_{Z, \bullet}.
\]
Using the fact that $\shT_Y$ and $\OmY^1$ are dual to each other, we have
\[
B^{-1} \tensor \omYX \tensor C_{Y, \bullet}^{k-d} \simeq
\fu \omX^{-1} \tensor \fu \shA_X^{\bullet-n-k} \tensor B^{-1} \tensor \OmY^k,
\]
which gives us a natural isomorphism
\[
\varphiu \bigl( B^{-1} \tensor \omYX \tensor C_{Y, \bullet}^{k-d} \bigr) \simeq
\hu \omX^{-1} \tensor \hu \shA_X^{\bullet-n-k}
\tensor \varphiu \bigl( B^{-1} \tensor \OmY^k \bigr).
\]
By composing with (<ref>), we obtain an $\hu \shA_X$-linear morphism to
\[
\hu \omX^{-1} \tensor \hu \shA_X^{\bullet-n-k} \tensor \OmZ^k \simeq
\omZX \tensor C_{Z, \bullet}^{k-d}.
\]
It remains to verify that the individual morphisms are compatible with the
differentials in the two complexes. Since they are by construction $\hu
\shA_X$-linear, the problem is reduced to proving the commutativity of the diagram
\[
\begin{tikzcd}[column sep=large]
\varphiu \bigl( B^{-1} \tensor \OmY^k \bigr) \dar \rar{i_k} & \OmZ^k \dar \\
\varphiu \bigl( B^{-1} \tensor \OmY^{k+1} \bigr) \tensor \hu \shT_X
\rar{i_{k+1} \tensor \id} & \OmZ^{k+1} \tensor \hu \shT_X
\end{tikzcd}
\]
in which the vertical morphisms are induced respectively by $\shT_Y \to \fu \shT_X$
and $\shT_Z \to \hu \shT_X$. This is an easy exercise.
Now we can construct a graded $\shA_X$-module $\shGb$ in the following manner. By
composing the morphism from prop:morphism with the projection to $\grFMb$,
we obtain a morphism of graded $\shA_X$-modules
\begin{equation} \label{eq:image}
R^0 \fl \bigl( B^{-1} \tensor \omYX \tensor C_{Y, \bullet} \bigr)
\to R^0 \hl \bigl( \omZX \tensor C_{Z, \bullet} \bigr)
\to \grFMb.
\end{equation}
We then define $\shGb \subseteq \grFMb$ as the image of this morphism, in
the category of graded $\shA_X$-modules. Remembering that $B^{-1} \tensor \omYX = \fu
L$, we see that $\shGb$ is also a quotient of the graded $\shA_X$-module $L \tensor
R^0 \fl C_{Y, \bullet}$. We can use this observation to prove that $\shGb$ is
One has $\shG_k = 0$ for $k < n-d$, whereas $\shG_{n-d} \simeq L \tensor \fl \OY$.
We make use of cor:shGMmin. The first assertion is clear because $\gr_k^F \Mmod = 0$ for $k < n-d$. By
construction, $\shG_{n-d}$ is a quotient of the $\OX$-module
\[
L \tensor R^0 \fl C_{Y,n-d} \simeq L \tensor \fl \OY.
\]
Since we already know that $\gr_{n-d}^F \Mmod$ is isomorphic to $\hl \omZX$, it is
therefore enough to prove that the morphism in prop:morphism is injective
in degree $n-d$. The morphism in question is
\[
\fl \bigl( B^{-1} \tensor \omYX \bigr) \to \hl \omZX,
\]
and is induced by (<ref>) for $k = d$. Now $i_d \colon \varphiu
\bigl( B^{-1} \tensor \omY \bigr) \to \omZ$ is injective, and because $\OY$ injects
into $\varphil \OZ$, the adjoint morphism $B^{-1} \tensor \omY \to \varphil \omZ$
remains injective. The second assertion follows from this because $\fl$ is left-exact.
It is also not hard to show that the support of the associated coherent sheaf $\shG$
on the cotangent bundle is contained in the set $S_f$.
We have $\Supp \shG \subseteq S_f$.
By construction, $\shG$ is a quotient of the coherent sheaf $\pu L \tensor R^0 p_{1\ast}
C_Y$. But the complex $C_Y$ is supported on the set $\df^{-1}(0)$ by
prop:support, and so the support of $\shG$ is contained in $S_f = p_1
\bigl( \df^{-1}(0) \bigr)$.
§.§ Additional properties
Except in trivial cases, $\shGb$ is not itself the associated graded of a Hodge
module. Nevertheless, we shall see in this section that $\shGb$ inherits several good
properties from $\grFMb$.
Every irreducible component of $\Supp \shG$ is the conormal variety of some
subvariety of $X$.
By a theorem of Saito <cit.>, $\grFM$ is a
Cohen-Macaulay sheaf on $T^{\ast} X$ of dimension $n = \dim X$;
in particular, it is unmixed, and every associated subvariety of $\grFM$ has
dimension $n$. This property is inherited by the subsheaf $\shG \subseteq
\grFM$; in particular, every irreducible component of $\Supp \shG$ is $n$-dimensional.
Since $\Supp \shG \subseteq S_f$, we conclude from lem:Sf that every such
component is the conormal variety of some subvariety of $X$.
Recall our notation $D_f \subseteq X$ for the singular locus of the surjective
morphism $f \colon Y \to X$. Being part of a Hodge module, the coherent sheaves
$\gr_k^F \Mmod$ are locally free on the open subset $X \setminus D_h$ where $M$ is a
variation of Hodge structure. Surprisingly, the sheaves $\shG_k$ are torsion-free on
the much larger open set $X \setminus D_f$.
For every $k \in \ZZ$, the sheaf $\shG_k$ is torsion-free on $X \setminus D_f$.
After replacing $X$ by the open subset $X \setminus D_f$, we may assume that
$\Supp \shG$ is contained in the zero section; the reason is that $\Supp \shG
\subseteq S_f$, and that $D_f$ is the image of $S_f$ minus the zero section. What
we need to prove is that $\shG_k$ is a torsion-free sheaf on $X$. This is equivalent
to saying that
\[
\codim_X \Supp R^i \shHom_{\shO}(\shG_k, \omX) \geq i+1,
\]
for every $i \geq 1$; see for instance <cit.>. We can compute the dual of $\shG_k$ directly by applying
Grothendieck duality to the projection $p \colon T^{\ast} X \to X$. Note first that
\[
\pl \shG \simeq \bigoplus_{k \in \ZZ} \shG_k
\]
is $\OX$-coherent because the support of $\shG$ is contained in the zero section; in
particular, $\shG_k = 0$ for $k \gg 0$. Since the relative dualizing sheaf is
$\pu \omX^{-1}$, Grothendieck duality gives us
\[
\derR \shHom_{\shO} \bigl( \shGb, \omX \bigr)
\simeq \pl \derR \shHom_{\shO} \bigl( \shG, \shO \decal{n} \bigr),
\]
and so $\Supp R^i \shHom_{\shO}(\shG_k, \omX)$ is contained in the image under
$p$ of
\[
\Supp R^i \shHom_{\shO} \bigl( \shG, \shO \decal{n} \bigr).
\]
As the zero section has codimension $n$, this reduces the problem to proving that
\begin{equation} \label{eq:codim}
\codim_{T^{\ast} X} \Supp R^i \shHom_{\shO} \bigl( \shG, \shO \decal{n} \bigr)
\geq n+i+1
\end{equation}
for every $i \geq 1$. On $T^{\ast} X$, we have a short exact sequence of coherent sheaves
\[
0 \to \shG \to \grFM \to \grFM/\shG \to 0,
\]
from which we get a distinguished triangle
\[
\derR \shHom_{\shO} \bigl( \grFM/\shG, \shO \decal{n} \bigr) \to
\derR \shHom_{\shO} \bigl( \grFM, \shO \decal{n} \bigr)
\to \derR \shHom_{\shO} \bigl( \shG, \shO \decal{n} \bigr) \to \dotsb
\]
All three complexes are concentrated in nonnegative degrees, because the original
sheaves are supported on a subset of codimension $\geq n$; moreover, $\derR
\shHom_{\shO} \bigl( \grFM, \shO \decal{n} \bigr)$ is a sheaf, because $\grFM$
is Cohen-Macaulay of dimension $n$. We conclude that
\[
R^i \shHom_{\shO} \bigl( \shG, \shO \decal{n} \bigr)
\simeq R^{i+1} \shHom_{\shO} \bigl( \grFM/\shG, \shO \decal{n} \bigr)
\]
for $i \geq 1$. But $\grFM/\shG$ is a sheaf, and so the support of the
right-hand side has codimension at least $n+i+1$; this implies the desired inequality
§.§ Proof of Theorem <ref>thm:VZ
In this section, we prove thm:VZ by putting together the results about
the graded $\shA_X$-module $\shGb$ that we have established so far. To resolve the
minor discrepancy in the indexing, we replace $M \in \HM_X(X,d)$ by its Tate twist
$M(d-n) \in \HM_X(X,2n-d)$; this leaves the underlying regular holonomic
$\Dmod$-module $\Mmod$ unchanged, but replaces the filtration $F_{\bullet} \Mmod$ by
the shift $F_{\bullet+n-d} \Mmod$. Similarly, we replace $\shGb$ by the shift
$\shG_{\bullet+n-d}$. Then the assertions in <ref> and <ref> hold by
construction, and the assertion in <ref> follows from
cor:shGMmin. In prop:SuppG, we showed that $\Supp \shG
\subseteq S_f$, which proves <ref>. The remaining assertion in <ref>
has been established in prop:torsion.
At this point, the reader interested in starting with a technically simpler proof of
thm:hyperbolicity when $X$ is not uniruled can move directly to
big-HM and from there to scn:non-uniruled.
§.§ Constructing the Higgs bundle
From now on, we assume in addition that the hypothesis in (<ref>) is
satisfied – recall that we can always arrange this by blowing up $X$. We also assume
for simplicity that our morphism $f \colon Y \to X$ has connected fibers; this
implies that $\shG_0 \simeq L \tensor \fl \OY \simeq L$ is a line bundle. Denote by
$\shV$ the polarizable variation of Hodge structure obtained by restricting
$M$ to the open subset $X \setminus D$; for the sake of convenience, we shall
consider the Hodge filtration as an increasing filtration $F_{\bullet} \shV$, with
the convention that $F_k \shV = F^{-k} \shV$.
Let $\shVt$ be the canonical meromorphic extension <cit.> of
the flat bundle $(\shV, \nabla)$, and denote by $\shVt^{\geq \alpha}$ and
$\shVt^{>\alpha}$ Deligne's canonical lattices with eigenvalues contained in the
intervals $[\alpha, \alpha + 1)$ and $(\alpha, \alpha+1]$, respectively <cit.>. The flat connection on $\shV$ extends
uniquely to a logarithmic connection
\begin{equation} \label{eq:nabla}
\nabla \colon \shVg \to \OmX^1(\log D) \tensor \shVg.
\end{equation}
As a consequence of Schmid's nilpotent orbit theorem, the Hodge filtration
$F_{\bullet} \shV$ extends to a filtration of $\shVg$ with locally free subquotients;
see <cit.> for a discussion of this point. The extension is given
\begin{equation} \label{eq:Hodge}
F_k \shVg = \shVg \cap \jl F_k \shV,
\end{equation}
where $j \colon X \setminus D \into X$ is the inclusion. On the associated graded
with respect to the Hodge filtration, the connection then induces an $\OX$-linear operator
\begin{equation} \label{eq:Higgs}
\theta \colon \gr_{\bullet}^F \shVg
\to \OmX^1(\log D) \tensor \gr_{\bullet+1}^F \shVg
\end{equation}
with the property that $\theta \wedge \theta = 0$. Setting
\[
\shEb = \gr_{\bullet}^F \shVg,
\]
we therefore obtain the desired (graded logarithmic) Higgs bundle $(\shEb, \theta)$. Since
$F_k \shV = 0$ for $k < 0$, it is clear that $\shE_k = 0$ in the same range.
It turns out that we can find a copy of $\shEb$ inside the associated graded
$\grFMb$ of the Hodge module $M$, and that the Higgs field $\theta$
can also be recovered from the $\shA_X$-module structure. The connection on
$\shV$ induces on the meromorphic extension $\shVt$ the structure of a left
$\Dmod$-module. The formulas for the minimal extension in <cit.>
show that $\shVg \subseteq \Mmod \subseteq \shVt$.
For every $k \in \ZZ$, we have an inclusion $\shE_k = \gr_k^F \shVg \subseteq \gr_k^F
\Mmod$.
We observe that $\shVg \cap F_k \Mmod = F_k \shVg$. Indeed, the construction of the
Hodge filtration on $\Mmod$ in <cit.> is such that one has
\[
\shVt^{>-1} \cap \jl F_k \shV \subseteq F_k \Mmod \subseteq \jl F_k \shV,
\]
as subsheaves of $\shVt$. Now intersect with $\shVg$ and use (<ref>) to get
the desired identity. The inclusion $\shVg \subseteq \Mmod$ induces an
inclusion $F_k \shVg \subseteq F_k \Mmod$, and the identity we have just proved
implies that $\gr_k^F \shVg \to \gr_k^F \Mmod$ stays injective.
To recover the action of the Higgs field $\theta$, let us choose local coordinates
$x_1, \dotsc, x_n$ that are adapted to the normal crossing divisor $D$. We denote the
corresponding vector fields by the symbols $\partial_1, \dotsc, \partial_n$. Since
the connection in (<ref>) has logarithmic poles, the action by the vector
field $x_i \partial_i$ preserves the lattice $\shVg$, and therefore induces an
$\OX$-linear morphism
\[
x_i \partial_i \colon \gr_k^F \shVg \to \gr_{k+1}^F \shVg.
\]
Note that if $x_i = 0$ is not a component of $D$, the factor of $x_i$ is
not needed, because in that case, $\partial_i$ itself already maps $\gr_k^F \shVg$
into $\gr_{k+1}^F \shVg$.
Putting together the individual morphisms, we obtain an $\OX$-linear morphism
\begin{equation} \label{eq:Higgs-new}
\gr_k^F \shVg \to \OmX^1(\log D) \tensor \gr_{k+1}^F \shVg,
\quad s \mapsto \sum_{i=1}^n \frac{\mathit{dx}_i}{x_i} \tensor (x_i \partial_i s),
\end{equation}
which is exactly the Higgs field in (<ref>).
§.§ Constructing the Higgs sheaf
We can now construct a collection of subsheaves $\shF_k \subseteq \shE_k$ by
intersecting $\shG_k$ and $\shE_k = \gr_k^F \shVg$ inside the larger coherent sheaf
$\gr_k^F \Mmod$. Since the intersection may not be reflexive, we actually define
\[
\shF_k = (\shG_k \cap \shE_k)^{\vee\vee} \subseteq \shE_k
\]
as the reflexive hull of the intersection. Since $\shE_k = 0$ for $k < 0$, we
obviously have $\shF_k = 0$ in the same range. It is also not hard to see that
$\shFb \subseteq \shEb$ is compatible with the action of the Higgs
field in (<ref>). Indeed, using the notation introduced before
(<ref>), the vector field $x_i \partial_i$ maps $\shF_k$ into
$\shF_{k+1}$, due to the fact that $\shGb$ is a graded $\shA_X$-submodule of
$\grFMb$. But this means that the Higgs field takes $\shF_k$ into the subsheaf
$\OmX^1(\log D) \tensor \shF_{k+1}$.
As in the work of Viehweg and Zuo, the key property is that the action of the Higgs
field on $\shFb$ only creates poles along the smaller divisor $D_f$.
The Higgs field maps $\shF_k$ into $\OmX^1(\log D_f) \tensor \shF_{k+1}$.
The proof exploits the relationship between $F_{\bullet} \Mmod$ and the V-filtration
with respect to locally defined holomorphic functions on $X$. We briefly review the
relevant properties; for a more careful discussion of how the V-filtration enters
into the definition of Hodge modules, see <cit.>. Suppose then that $t
\colon U \to
\CC$ is a non-constant holomorphic function on an open subset $U \subseteq X$, with
the property that $t^{-1}(0)$ is a smooth divisor; also suppose that we have a
holomorphic vector field $\partial_t$ such that $\lie{\partial_t}{t} = 1$. To keep
the notation simple, we denote the restriction of $M$ and $(\Mmod, F_{\bullet}
\Mmod)$ to the open set $U$ by the same symbols.
Being part of a Hodge module, the $\Dmod$-module $\Mmod$ admits a rational
V-filtration $V^{\bullet} \Mmod$ with respect to the function $t$. As we are working
with left $\Dmod$-modules, this is a decreasing filtration, discretely indexed by
$\alpha \in \QQ$, with the following properties:
* One has $t \cdot V^{\alpha} \Mmod \subseteq V^{\alpha+1} \Mmod$, with
equality for $\alpha > -1$.
* One has $\partial_t \cdot V^{\alpha} \Mmod \subseteq V^{\alpha-1} \Mmod$.
* The operator $t \partial_t - \alpha$ acts nilpotently on $\gr_V^{\alpha}
\Mmod = V^{\alpha} \Mmod / V^{>\alpha} \Mmod$, where $V^{>\alpha} \Mmod$ is
defined as the union of those $V^{\beta} \Mmod$ with $\beta > \alpha$.
* Each $V^{\alpha} \Mmod$ is coherent over $V^0 \Dmod_X$, which is defined as the
$\OX$-subalgebra of $\Dmod_X$ that preserves the ideal $t \OX \subseteq \OX$.
More generally, every regular holonomic $\Dmod$-module with quasi-unipotent local
monodromy[This is a shorthand for saying that all eigenvalues of the
monodromy operator on the perverse sheaf of nearby cycles $\psi_t \DR(\Mmod)$ are
roots of unity.] around the divisor $t^{-1}(0)$ has a unique rational V-filtration; this
result is due to Kashiwara <cit.>. The uniqueness statement implies that if two
$\Dmod$-modules $\Mmod_1$ and $\Mmod_2$ admit rational V-filtrations, then any
morphism $f \colon \Mmod_1 \to \Mmod_2$ between them is strictly compatible with
these filtrations, in the sense that
\[
f(\Mmod_1) \cap V^{\alpha} \Mmod_2 = f \bigl( V^{\alpha} \Mmod_1 \bigr)
\]
for all $\alpha \in \QQ$.
Saito defines the category of Hodge modules by requiring, among other
things, that the Hodge filtration $F_{\bullet} \Mmod$ interacts well with the
rational V-filtration $V^{\bullet} \Mmod$. The first requirement in the definition is
\begin{equation} \label{eq:t}
t \colon F_k V^{\alpha} \Mmod \to F_k V^{\alpha+1} \Mmod
\end{equation}
must be an isomorphism for $\alpha > -1$; the second requirement is that
\begin{equation} \label{eq:partial}
\partial_t \colon F_k \gr_V^{\alpha} \Mmod \to F_{k+1} \gr_V^{\alpha-1} \Mmod
\end{equation}
must be an isomorphism for $\alpha < 0$. Here and in what follows, $F_{\bullet}
\gr_V^{\alpha} \Mmod$ means the filtration induced by $F_{\bullet} \Mmod$; thus
\[
F_k \gr_V^{\alpha} \Mmod =
\frac{V^{\alpha} \Mmod \cap F_k \Mmod + V^{>\alpha} \Mmod}{V^{>\alpha} \Mmod}.
\]
These two conditions together give us very precise information on how the two
operators $t$ and $\partial_t$ interact with the Hodge filtration $F_{\bullet}
\Mmod$.
Since $\shF_k$ and $\OmX^1(\log D_f) \tensor \shF_{k+1}$ are reflexive coherent
sheaves, we only need to prove the assertion outside a
subset of codimension $\geq 2$. After removing the singular locus of the normal
crossing divisor $D$, it is therefore enough to show that when the Higgs field in
(<ref>) is applied to a local section of $\shF_k$, it does not actually
produce any poles
along the components of $D$ that do not belong to $D_f$. Fix such a component, and on a
sufficiently small open neighborhood $U$ of its generic point, choose local
coordinates $x_1, \dotsc, x_n$ such that $D \cap U$ is defined by the equation $x_n =
0$. Because we can ignore what happens on a subset of codimension $\geq 2$, we may
assume that $\shF_k = \shG_k \cap \shE_k$ on $U$. Moreover, the component in question
does not belong to $D_f$, and so we may further assume that $D_f \cap U = \emptyset$;
the part of $S_f$ that lies over $U$ is then contained in the zero section of
the cotangent bundle. As $\shGb$ is coherent over $\shA_X$, this implies
that any section in $H^0(U, \shG_k)$ is annihilated by a sufficiently large power of
Let $V_i^{\bullet} \Mmod$ be the rational V-filtration with respect to the
function $x_i$. Since $\Mmod$ is a flat bundle outside the divisor $x_n = 0$, it is
easy to see from the definition that
\[
V_i^{\alpha} \Mmod = x_i^{\max(0, \lfloor \alpha \rfloor)} \Mmod \subseteq \Mmod
\]
is essentially the $x_i$-adic filtration for $i = 1, \dotsc, n-1$; the same is true also
on the larger $\Dmod$-module $\shVt$. The defining property of the canonical lattices
implies that
\[
\shVt^{\geq \alpha} = \bigcap_{i=1}^n V_i^{\alpha} \shVt
= V_n^{\alpha} \shVt
\]
for $\alpha < 1$, as noted for example in <cit.>. Because any
morphism of $\Dmod$-modules is strictly compatible with the rational V-filtration, we obtain
\[
V_n^{\alpha} \Mmod = \Mmod \cap V_n^{\alpha} \shVt
= \Mmod \cap \shVt^{\geq \alpha}
\]
as long as $\alpha < 1$; in particular, $V_n^0 \Mmod = \shVg$. Over the open set $U$, we
thus get
\[
\shE_k = \gr_k^F \shVg = V_n^0 \gr_k^F \Mmod \subseteq \gr_k^F \Mmod,
\]
where $V_n^{\bullet} \gr_k^F \Mmod$ again means the filtration induced by $V_n^{\bullet}
\Mmod$, that is to say,
\[
V_n^{\alpha} \gr_k^F \Mmod =
\frac{V_n^{\alpha} \Mmod \cap F_k \Mmod + F_{k-1} \Mmod}{F_{k-1} \Mmod}.
\]
Now given any section $s \in \Gamma(U, \shF_k)$, we need to argue that $\partial_n s
\in H^0(U, \shF_{k+1})$; this will guarantee that the Higgs field in
(<ref>) does not create a pole along $x_n = 0$ when applied to the
section $s$. Viewing $s$ as a section of the larger locally free sheaf $\shE_k = V_n^0
\gr_k^F \Mmod$, and remembering that the operator $\partial_n$ maps $F_k \Mmod$ into
$F_{k+1} \Mmod$ and $V_n^0 \Mmod$ into $V_n^{-1} \Mmod$, we obtain
\begin{equation} \label{eq:partial-s}
\partial_n s \in H^0 \bigl( U, V_n^{-1} \gr_{k+1}^F \Mmod \bigr).
\end{equation}
We shall argue that, in fact, $\partial_n s \in H^0 \bigl( U, V_n^0 \gr_{k+1}^F \Mmod
\bigr)$. This is the crucial step in the proof; it rests entirely on the
compatibility between the Hodge filtration and the rational V-filtration, in the form
of Saito's condition (<ref>).
Since $\shF_k \subseteq \shG_k$, we already know that $\partial_n^{\ell+1} s = 0$ for
some $\ell \geq 1$; in other words, our section $\partial_n s$ is annihilated by the operator
$\partial_n^{\ell}$. To make use of this fact, consider the following commutative
diagram with short exact columns:
\[
\begin{tikzcd}
F_k \gr_{V_n}^{\alpha} \Mmod \dar[hook]
\rar{\partial_n^{\ell}} & F_{k+\ell} \gr_{V_n}^{\alpha-\ell} \Mmod \dar[hook] \\
F_{k+1} \gr_{V_n}^{\alpha} \Mmod \dar[two heads]
\rar{\partial_n^{\ell}} & F_{k+1+\ell} \gr_{V_n}^{\alpha-\ell} \Mmod \dar[two heads] \\
\gr_{k+1}^F \gr_{V_n}^{\alpha} \Mmod
\rar{\partial_n^{\ell}} & \gr_{k+1+\ell}^F \gr_{V_n}^{\alpha-\ell} \Mmod
\end{tikzcd}
\]
The condition in (<ref>) relating $F_{\bullet} \Mmod$ and $V_n^{\bullet}
\Mmod$ tells us that the morphisms in the first and second row
are isomorphisms for $\alpha < 0$; of course, the morphism in the third row is then
also an isomorphism. Taking $\alpha = -1$, we see that
\[
\partial_n^{\ell} \colon \gr_{k+1}^F \gr_{V_n}^{-1} \Mmod \to
\gr_{k+1+\ell}^F \gr_{V_n}^{-1-\ell} \Mmod
\]
is an isomorphism; because the image of $\partial_n s$ belongs to the kernel, we
conclude that $\partial_n s$ is in fact a section of
\[
V_n^{>-1} \gr_{k+1}^F \Mmod = V_n^{\alpha} \gr_{k+1}^F \Mmod
\]
for some $\alpha > -1$. As long as $\alpha < 0$, we can repeat this argument and
further increase the value of $\alpha$; because the rational V-filtration is discretely
indexed, we eventually arrive at the conclusion that
\[
\partial_n s \in H^0 \bigl( U, V_n^0 \gr_{k+1}^F \Mmod \bigr)
= H^0(U, \shE_{k+1}).
\]
Since also $\partial_n s \in H^0(U, \shG_{k+1})$, we obtain $\partial_n s \in H^0(U,
\shF_{k+1})$, as needed.
We also need to have some information about the subsheaf $\shF_0 \subseteq F_0
\shVg$. From the definition of the Hodge filtration on $\Mmod$ in
<cit.>, we immediately get $F_0 \Mmod = F_0 \shVt^{>-1}$, and so
the problem is to compute the intersection
\[
\shG_0 \cap F_0 \shVg \subseteq F_0 \shVt^{>-1}.
\]
This turns out to be fairly subtle, and the answer depends on the local monodromy
around the components of the divisor $D_f$. Fortunately, the following rather weak
result is enough for our purposes.
We have $L(-D_f) \subseteq \shF_0 \subseteq L$.
It is again enough to prove this outside a closed subset of codimension $\geq 2$.
After removing the singular locus of the normal crossing divisor $D$, we may
therefore consider one component of $D$ at a time; as in the proof of
prop:Higgs, we choose local coordinates $x_1, \dotsc, x_n$ on an
open subset $U$ such that $D \cap U$ is defined by the equation $x_n = 0$. Let $s \in
H^0(U, \shG_0)$ be any section. By definition,
\[
s \in H^0(U, F_0 \shVt^{>-1}) = H^0(U, F_0 V_n^{>-1} \Mmod).
\]
Now there are two possibilities. If the component in question belongs to the divisor
$D_f$, we use the obvious fact that
\[
x_n s \in H^0(U, F_0 V_n^0 \Mmod) \subseteq H^0(U, \shF_0)
\]
to conclude that multiplication by a local equation for $D_f$ maps $\shG_0 \simeq L$
into the subsheaf $\shF_0$. If the component in question does not belong to the
divisor $D_f$, then we argue as in the proof of prop:Higgs. Namely,
the part of $S_f$ that lies over $U$ is contained in the zero section of the
cotangent bundle, which means that we have $\partial_n^{\ell} s = 0$ for $\ell \gg
0$. As before, we use (<ref>) to conclude that
\[
s \in H^0(U, F_0 V_n^0 \Mmod) = H^0(U, \shF_0),
\]
which leads to the desired conclusion also in this case.
§.§ Proof of Theorem <ref>thm:Higgs
We finish this chapter by proving thm:Higgs. As already
mentioned, the (graded logarithmic) Higgs bundle $\shEb = \gr_{\bullet}^F \shVg$, with the
Higgs field in (<ref>), comes from the polarizable variation of Hodge
structure $\shV$ on $X \setminus D$, and so <ref> is true by construction.
The graded submodule $\shFb \subseteq \shEb$ satisfies <ref> by
construction, <ref> because of prop:shFmin, and
<ref> because of prop:Higgs.
§ POSITIVITY FOR HODGE MODULES AND HIGGS BUNDLES
§.§ Background on weak positivity
In this paragraph we fix a smooth quasi-projective variety $X$, and a torsion-free coherent sheaf $\shF$ on $X$.
* We call $\shF$ weakly positive over an open set $U \subseteq X$ if for
every integer $\alpha > 0$ and every ample line bundle $H$ on $X$, there is an integer $\beta > 0$ such that
$$(S^{\alpha \beta} \shF)^{\vee \vee} \otimes H^{\otimes \beta}$$
is generated by global sections at each point of $U$. We say that $\shF$ is
weakly positive if such an open set $U \neq \emptyset$ exists.
* We call $\shF$ big (in the sense of Viehweg) if for any line bundle $L$ on $X$, there exists some
integer $\gamma > 0$ such that $(S^{\gamma} \shF)^{\vee \vee} \otimes L^{-1}$ is weakly positive.
We recall some basic facts needed in the next section; they are immediate applications of <cit.> and
Let $\shF$ and $\shG$ be torsion-free coherent sheaves on $X$. Then:
* If $\shF \rightarrow \shG$ is surjective over $U$, and if $\shF$ is weakly positive over $U$, then $\shG$ is weakly positive over $U$. Moreover, if $\shF$ is big, then $\shG$ is big.
* If $\shF$ is weakly positive and $A$ is a big line bundle, then $\shF \otimes A$ is big.
* If $\shF$ is big, then $\det \shF$ is a big line bundle.
§.§ Positivity for Hodge modules
Let $M$ be a pure Hodge module on a smooth projective variety $X$, with underlying filtered $\Dmod_X$-module
$(\Mmod, F_{\bullet} \Mmod)$.
For each $k$, the filtration induces Kodaira-Spencer type $\OX$-module homomorphisms
$$\theta_k: \gr_k^F \Mmod \longrightarrow \gr_{k+1}^F \Mmod \otimes \OmX^1,$$
and we shall use the notation
$$K_k (M): = \ker \theta_k.$$
Below we will make use of the following weak positivity statement, extending to Hodge modules results of <cit.> and <cit.>, which are themselves generalizations of the well-known Fujita-Kawamata semipositivity theorem.
If $M$ is a pure polarizable Hodge module with strict support $X$, then the reflexive sheaf $K_k (M)^\vee$ is weakly positive for any $k$.
We now give an ad-hoc definition, for repeated use in what follows. Recall that
given filtered $\Dmod_X$-module $(\Mmod, F_{\bullet} \Mmod)$, the associated graded $\gr_{\bullet}^F \Mmod$ is coherent
graded $\shA_X$-module, with $\shA_X = \Sym \shT_X$. Moreover, we denote
\begin{equation}\label{minimum}
p (\Mmod) : = \min ~\{ ~p ~|~F_p \Mmod \neq 0~\}.
\end{equation}
In other words,
$$F_{p(\Mmod)} \Mmod = \gr_{p(\Mmod)} ^F \Mmod$$
is the lowest nonzero graded piece in the filtration on $\Mmod$. If $(\Mmod,
F_{\bullet} \Mmod)$ underlies a Hodge module $M$, we also use the notation $p(M)$
instead of $p(\Mmod)$.
Let $M$ be a pure Hodge module with strict support $X$. A graded $\shA_X$-submodule
$$\shGb \subseteq \gr_{\bullet}^F \Mmod$$
is called large (with respect to $D$) if there exist a big line bundle $A$ and an effective divisor $D$ on $X$, together with an integer $\ell \ge 0$, such that:
* there is a sheaf inclusion $A (-\ell D) \hookrightarrow \shG_{p(M)} $.
* the support of the torsion of all $\shG_k$ is contained in $D$.[Typically $D$ may be the complement of the locus where $M$ is a variation of Hodge structure, though we will also
have to deal with the case when $D$ is strictly contained in that locus.]
For applications we need to prove the following stronger version of
thm:HM-positivity; the reason is that the locus where the Hodge module
we consider is not a variation of Hodge structure is usually bigger than the
singular locus of the family we are interested in.
Let $X$ be a smooth projective variety, and let $M$ be a pure Hodge module $M$ with strict support $X$ and underlying filtered
$\Dmod_X$-module $(\Mmod, F_{\bullet} \Mmod)$, which is generically a variation of
Hodge structure of weight $k$. Assume that there exists a graded $\shA_X$-submodule
$\shGb \subseteq \gr_{\bullet}^F \Mmod$ which is large with respect to a divisor $D$.
Then at least one of the following holds:
* $D$ is big.
* There exist $1 \le s \le k$, $r \ge 1$, and a big coherent sheaf $\shH$ on $X$,
such that
$$\shH \hookrightarrow (\OmX^1)^{\otimes s} \otimes \OX(rD)$$
Moreover, if $X$ is not uniruled, then $\omX (D)$ is big.
Note to begin with that $F_{p(M)} \Mmod$ is a torsion-free sheaf; see for instance <cit.>. It follows that $\shG_{p(M)}$ is
torsion-free as well. By assumption, there is a big line bundle $A$ on $X$ and an integer $\ell \ge 0$,
together with an injective sheaf morphism
$$A (- \ell D) \hookrightarrow \shG_{p(M)}.$$
Denoting for simplicity $p = p(M)$, the graded $\shA_X$-module structure induces a chain of homomorphisms of coherent
$$0 \longrightarrow \shG_p \overset{\theta_p}{\longrightarrow} \shG_{p+1} \otimes \OmX^1
\overset{\theta_{p+1} \circ \id}{\longrightarrow} \shG_{p+2} \otimes
(\OmX^1)^{\otimes 2} \longrightarrow \cdots$$
Just as with Hodge modules, we will denote
$$K_k = K_k (\shGb) : = \ker \big( \theta_k: \shG_k \longrightarrow \shG_{k+1} \otimes \OmX^1\big).$$
There are obvious inclusions
$$K_k \hookrightarrow K_k (M),$$
which by thm:WP and lemma:basic-WP(1) imply that $K_k^\vee$ are weakly positive for all $k$.
To start making use of this property, note to begin with that given the inclusion of $A ( - \ell D)$ into $\shG_p$, there are
two possibilities:
The first is that the induced homomorphism
$$A ( -\ell D) \longrightarrow \shG_{p+1} \otimes \OmX^1$$
is not injective, i.e. $A$ maps into the torsion of this sheaf, whose support is assumed to be contained in $D$. It follows that there exists a
non-trivial subscheme $Z \subset X$ such that $Z_{\mathrm{red}} \subseteq D$ and $A (-\ell D) \otimes {\mathcal I}_Z \subset K_p$. This implies that there exists an integer $r \ge 1$ and an inclusion $A (- rD) \hookrightarrow K_p$, which induces a non-trivial homomorphism
$$K_p^\vee \longrightarrow A^{-1} (rD).$$
Using the weak positivity of $K_p^\vee$ again, we get that $A^{-1} (rD)$ is pseudo-effective. Since $A$ is big, we get that $D$ must be big, i.e. the condition in (i).
The second possibility is that we have an inclusion
$$A (-\ell D) \hookrightarrow \shG_{p+1} \otimes \OmX^1.$$
We can then repeat the same argument via the morphisms $\theta_s \circ \id$ with $s \ge p+1$.
The next thing to note however is that there is an $s \le k$ where the inclusions will have to stop, i.e. such that
$$A \subseteq \shG_{p+s} \otimes (\OmX^1)^{\otimes s} \,\,\,\,\,{\rm and} \,\,\,\,\,
A \not \subseteq \shG_{p+s+1} \otimes (\OmX^1)^{\otimes s+1}.$$
Indeed, note that an inclusion $A \subseteq \gr_{p+t}^F \Mmod \otimes (\OmX^1)^{\otimes t}$ can only hold as long as
$\gr_{p+t}^F \Mmod$ is not a torsion sheaf. Recall however that $(\Mmod, F_{\bullet} \Mmod)$ underlies an extension of a
variation of Hodge structure ${\bf V}$ of weight $k$ on an open set $U \subset X$. Thus over $U$ the sheaves $\gr_{p+t}^F \Mmod$ coincide with Hodge bundles of ${\bf V}$, and therefore are non-zero only for $t \le k$.
As above, this implies that there exists some $r \ge 1$ such that
\begin{equation}\label{limit_case}
A (-rD) \subseteq K_{p+s} \otimes (\OmX^1)^{\otimes s} .
\end{equation}
We conclude that for $s$ as in ($\ref{limit_case}$), there exists a nontrivial homomorphism
$$K_{p+s}^\vee \otimes A \longrightarrow (\OmX^1)^{\otimes s} \otimes \OX(rD)$$
and, using the weak positivity of $K_{p+s}^\vee$ and lemma:basic-WP(1) and (2), taking its image we obtain an inclusion
$$\shH \hookrightarrow (\OmX^1)^{\otimes s} \otimes \OX(rD)$$
with $\shH$ a big sheaf on $X$, i.e. the condition in (ii).
Let us now assume that $X$ is not uniruled. By <cit.> it follows that $\omX$ is pseudo-effective. If $D$ is big, then we immediately get the conclusion. Otherwise we employ the standard argument based on the pseudo-effectivity of quotients by Viehweg-Zuo type sheaves, inspired by an idea in <cit.> (see also <cit.>): using (ii), we have a short exact sequence
$$0 \longrightarrow \shH \longrightarrow (\OmX^1)^{\otimes s} \otimes \OX(rD) \longrightarrow \shQ \longrightarrow 0,$$
and passing to the saturation of $\shH$ we can assume that $\shQ$ is torsion-free.
Using that $X$ is not uniruled, a special case of <cit.> says that every torsion-free quotient of
$(\OmX^1)^{\otimes s}$ has pseudo-effective determinant; it follows that $\det \big( \shQ (-rD) \big)$ is pseudo-effective, which implies that $\det \shQ$ is pseudo-effective as well. Since $\shH$ is big, its determinant is also big by lemma:basic-WP(3), and one obtains by passing to determinants in the sequence above that
$$\omX^{\otimes s} \otimes \OX(nrs D),$$
is big, with $n= \dim X$. Finally, $\omX$ is pseudo-effective, and so multiplying by its suitable power implies that $\omX(D)$ is big.
The last statement of thm:submodule-positivity may fail if $X$ is
uniruled. For example, consider the double covering of $\PP^1$ branched at the two
points $0$ and $\infty$. Let $M$ be the direct image of the constant Hodge module,
and denote by $(\Mmod, F_{\bullet} \Mmod)$ the underlying filtered $\Dmod$-module.
Then cor:shGMmin shows that $F_0 \Mmod$ is the direct image of the
relative canonical bundle, which equals $\shO_{\PP^1} \oplus \shO_{\PP^1}(1)$. Even
though $F_0 \Mmod$ contains an ample line bundle, $\PP^1 \setminus \{0, \infty\}$ is
of course not of log general type. This phenomenon is partly explained by
prop:shFmin: in the process of constructing the Higgs subsheaf
$\shFb$, the fact that the local system corresponding to the summand
$\shO_{\PP^1}(1)$ has nontrivial monodromy of order $2$ around each of the two
points means that we end up with $\shF_0 = \shO_{\PP^1}(-1)$, which is no longer
§.§ Positivity for Higgs bundles
We will also need a version of thm:submodule-positivity in the case of
(graded logarithmic) Higgs bundles. This will
allow us later on to deal with the a priori possibility of $X$ being uniruled.
The set-up is as follows: $X$ is a smooth projective variety, and $D$ a simple normal crossings divisor on $X$.
We consider a (graded logarithmic) Higgs bundle
\[
\theta_p \colon \shE_p \to \OmX^1(\log D) \tensor \shE_{p+1}
\]
extending a polarizable variation of Hodge structure of weight $\ell$ on $X \setminus D$; up to Tate twist, we can make the convention that $\shE_p \neq 0$ if only if $0 \le p \le \ell$. We also consider a graded submodule
$\shFb \subset \shEb$ having the property that
$$\theta_p (\shF_p) \subseteq \OmX^1(\log B) \tensor \shF_{p+1}$$
for some divisor $B \subseteq D$. Note that since $\shE_p$ are vector bundles, the sheaves $\shF_p$ are automatically torsion-free. By analogy with the previous section, we say that $\shFb$ is large if there exists
a big line bundle $A$ such that $A\subseteq \shF_0$.
The first part of the next theorem is essentially due to Viehweg-Zuo <cit.>, at least in the geometric case; it can be proved
completely analogously to thm:submodule-positivity, replacing the chain of
coherent $\shO_X$-module homomorphisms there with
$$0 \longrightarrow \shF_0 \overset{\theta_0}{\longrightarrow} \shF_{1} \otimes \OmX^1 (\log B)
\overset{\theta_{1} \circ \id}{\longrightarrow} \shF_{2} \otimes
\big(\OmX^1 (\log B)\big)^{\otimes 2} \longrightarrow \cdots$$
The argument is in fact simpler, as no torsion issues arise. The weak positivity of $K_k (\shEb)^\vee$, with
$$K_k (\shEb) : = \ker \big( \theta_k: \shE_k \longrightarrow \OmX^1 (\log D) \otimes \shE _{k+1} \big),$$
is deduced in <cit.> (see Theorem 4.9 and its proof, a step towards the proof of thm:WP)
as a quick corollary of the results of <cit.> and <cit.>.
Assume that $X$ is endowed with large submodule of a (graded logarithmic) Higgs bundle, as above.
Then there exist a big coherent sheaf $\shH$ on $X$ and an integer $1 \le s \le \ell$, together with an inclusion
$$\shH \hookrightarrow \big(\OmX^1(\log B)\big)^{\otimes s}.$$
In particular, $(X, B)$ is of log-general type, i.e. $\omega_X(B)$ is big.
The last part of the theorem is due to Campana-Păun; once we have the existence of a Viehweg-Zuo sheaf $\shH$ as in the statement, it follows from:
Let $X$ be a smooth projective variety and $B$ a simple normal crossings divisor on $X$. Assume that there for some $s \ge 1$
there is an inclusion
$$\shH \hookrightarrow \big(\OmX^1(\log B)\big)^{\otimes s},$$
where $\shH$ is a sheaf whose determinant is big. Then $\omega_X (B)$ is a big line bundle.
Note that in <cit.> the result is stated when $\shH$ is a line bundle, but the
proof works identically for any subsheaf such that
$\det \shH$ is big. Moreover, what we state here is only a special case of their theorem; in fact, the possible non-pseudoeffectivity of
$\omega_X$ is very cleverly dealt with in <cit.> by proving a more general theorem that applies to the orbifold setting as well.
§ FAMILIES OF VARIETIES
To show the statement in thm:hyperbolicity, it is immediate that after a birational modification we can assume that the $f$-singular locus is a divisor $D_f$, and hence it is also enough to just take $D = D_f$. We will always assume that this is the case in what follows.
§.§ Non-uniruled case
In this section we prove thm:hyperbolicity under the extra assumption that the base space $X$ is not uniruled. This is
included since at a first reading it allows to avoid many of the technicalities involved in dealing with the remaining case, while containing all the key ideas. The proof in the general case is given in the next section.
As recalled in the introduction, Viehweg's $Q_{n,m}$ conjecture states that if $f$ is a fiber space with maximal variation,
then $\det \fl \omYX^{\otimes m}$ is big for some $m > 0$. It is known to hold when the fibers are of general type by <cit.>, and more generally (nowadays) when they have good minimal models by <cit.>. Thus thm:hyperbolicity in the non-uniruled case is a consequence of the following statement and thm:submodule-positivity.
Let $f: Y \rightarrow X$ be an algebraic fiber space between smooth projective varieties, with branch locus a divisor
$D_f \subset X$. Assume that there is an integer $m > 0$ such that $\det \fl \omYX^{\otimes m}$ is big. Then there exists a pure Hodge module $M$ with strict support $X$ and underlying filtered
$\Dmod_X$-module $(\Mmod, F_{\bullet} \Mmod)$, together with a graded $\shA_X$-submodule
$\shGb \subseteq \gr_{\bullet}^F \Mmod$ which is large with respect to $D_f$.
Note that since the conclusion is purely on $X$, we are allowed to change the domain $Y$ as necessary.
Step 1.
We reduce to the following situation: given a fiber space over $X$ as in the statement, and given any ample line bundle $A$ on $X$, we can modify the picture to a new family $f': Y' \rightarrow X$ satisfying the property that there exists an integer $k_0 \ge 0$ such that:
$$B = \omega_{Y'/X} \otimes {f'}^*L^{-1} \,\,\,\,{\rm satisfies} \,\, (\ref{eq:sections}) \,\,{\rm with} \,\,\,\, L = A (-k_0 D_f).$$
To this end, fix an $m > 0$ such that
$$L_m : = \det \fl \omYX^{\otimes m}$$
is a big line bundle. Given any ample line bundle $M$ on $X$, we will produce a new family
$f': Y' \rightarrow X$, smooth over $U = X \setminus D_f$, such that
\begin{equation}\label{wish}
H^0 \big(Y', \omega_{Y'/X}^{\otimes m} \otimes {f'}^* M^{-1}(kD_f)\big) \neq 0
\end{equation}
for some integer $k \ge 0$.
In particular, we can take $M = A^{\otimes m}$; also, perhaps by increasing it, we can assume $k = k_0 \cdot m$ for some $k_0\ge 0$ in order to obtain the reduction step.
To prove the existence of such a family $f'$, note first that for $N$ sufficiently large we can write
$$L_m^{\otimes N} \simeq M \otimes \OX(B),$$
where $B$ is an effective divisor. Denote by $r_0$ the rank of $\fl \omYX^{\otimes m}$ over $U$, where it is a locally free sheaf by Siu's invariance of plurigenera, and define $r := N\cdot r_0$. Then there is an inclusion of sheaves
$$L_m^{\otimes N} \hookrightarrow (\fl \omYX^{\otimes m})^{\otimes r}$$
(which is split over the locus where $\fl \omYX^{\otimes m}$
is locally free).
Now we take advantage of Viehweg's fiber product trick. Consider a resolution of
singularities $Y^{(r)}$ of the main component of the $r$-fold fiber
product $Y \times_{X} \cdots \times_{X} Y$, with its induced morphism
$f^{(r)}: Y^{(r)} \rightarrow X$. Note that this morphism is smooth over $U$ as well;
moreover, it is well known (see <cit.> or <cit.>) that there exists a morphism
$$f^{(r)}_* \omega_{Y^{(r)}/X}^{\otimes m} \longrightarrow \big( (\fl \omYX^{\otimes m})^{\otimes r}\big)^{\vee \vee},$$
which is an isomorphism over $U$. Since $L_m^{\otimes N}$ injects into the right hand side, and the
morphism $f^{(r)}$ degenerates at most over $D_f$, it follows
that there exists an inclusion
$$L_m^{\otimes N} (- kD_f) \hookrightarrow f^{(r)}_* \omega_{Y^{(r)}/X}^{\otimes m}$$
for some integer $k \ge 0$. This implies in particular that on $Y^{(r)}$ we have
$$H^0 \big(Y^{(r)}, \omega_{Y^{(r)}/X}^{\otimes m} \otimes {f^{(r)}}^* M^{-1} (kD_f) \big) \neq 0.$$
Thus we can take $(Y^{(r)} , f^{(r)})$ to play the role of $(Y', f')$ in ($\ref{wish}$).
Step 2.
Fix now an ample line bundle $A$ on $X$. The considerations above show that, in order
to prove the theorem, we can assume that there exists an integer
$k_0 \ge 0$ such that the condition in (<ref>) is satisfied for $f'$ with respect to the line bundle
$$L = A (-k_0D_f).$$
But then thm:VZ provides a graded $\Sym \shT_X$-module $\shGb$ as in its statement, which
in particular is large with respect to $D_f$.
§.§ General case
This section contains the proof of thm:VZ-sheaves, and explains how to deduce thm:hyperbolicity in
the general case.
The main reason thm:VZ-sheaves is better suited for the argument is that the integer $r$ in the
statement of thm:submodule-positivity could be very large, precluding its use in the uniruled case in a similar way to the previous section.
Step 1.
We refine Step $1$ in the proof of thm:submodule-existence to obtain a stronger statement under the assumption that the bigness of the determinant of some pluricanonical image holds on any generically finite cover. We claim that for any line bundle $A$ on $X$ one can modify the picture to a new family $f': Y' \rightarrow X$ such that:
$$B = \omega_{Y'/X} \otimes {f'}^*A^{-1} \,\,\,\,{\rm satisfies} \,\, (\ref{eq:sections}).$$
This requires using an extra semistable reduction in codimension one procedure, following work of Viehweg. Indeed, for instance <cit.> says that there is a commutative diagram
\begin{tikzcd}
Y \dar{f}
& \tilde Y \lar \dar{\tilde f} \\
X & \tilde X \lar{\tau}
\end{tikzcd}
with $\tau$ generically finite, $\tilde X$ and $\tilde Y$ smooth and projective, and
after removing a closed subset $Z$ of codimension at least $2$ in $X$, $\tau$ is finite and flat and $f^\prime$ is semistable.[Although not strictly necessary here, note that an even stronger construction based on the Abramovich-Karu weak semistable reduction can be considered; see e.g. <cit.>.] Since our goal is to check the inclusion of line bundles into various torsion-free sheaves, we can ignore $Z$ and assume that these properties hold for the full diagram above. (However
$\tilde f$ may have a bigger degeneracy locus than $\tau^{-1} (D_f)$.)
For any given $r$, we again consider a resolution of singularities $Y^{(r)}$ of the main component of the $r$-fold fiber
product, with its induced morphism $f^{(r)}: Y^{(r)} \rightarrow X$, and similarly for
$\tilde f: \tilde Y \rightarrow \tilde X$, chosen in such a way that we have a commutative diagram
\begin{tikzcd}
Y^{(r)} \dar{f^{(r)}}
& {\tilde Y}^{(r)} \lar \dar{{\tilde f}^{(r)}} \\
X & \tilde X \lar{\tau}.
\end{tikzcd}
Pick now a line bundle on $X$ of the form $L = A^{\otimes m} \otimes M$, where $M$ is an ample line
bundle chosen such that the sheaf $M \otimes (\tau_* \shO_{\tilde X})^\vee$ is globally generated.
By hypothesis there exists an integer $m > 0 $ such that
$\det {\tilde f}_* \omega_{\tilde Y/ \tilde X}^{\otimes m}$ is big. Therefore exactly as in
Step 1 in the proof of thm:submodule-existence, for some sufficiently large $r$ we obtain an inclusion
$$\tau^*L \hookrightarrow \big({\tilde f}_* \omega_{\tilde Y/ \tilde X}^{\otimes m}\big)^{\otimes r}.$$
But since $\tilde f$ is semistable, by <cit.> (see also <cit.>) the natural morphism
$${\tilde f}^{(r)}_* \omega_{{\tilde Y}^{(r)}/\tilde X}^{\otimes m} \longrightarrow \big(
({\tilde f}_* \omega_{\tilde Y/ \tilde X}^{\otimes m})^{\otimes r}\big)^{\vee \vee}$$
is in fact an isomorphism. On the other hand, since $\tau$ is flat, by <cit.> (see also <cit.>) we have an inclusion
$${\tilde f}^{(r)}_* \omega_{{\tilde Y}^{(r)}/ \tilde X}^{\otimes m} \hookrightarrow \tau^* f^{(r)}_* \omega_{Y^{(r)}/ X}^{\otimes m}.$$
Putting everything together, we obtain an inclusion
$$\tau^*L \hookrightarrow \tau^* f^{(r)}_* \omega_{Y^{(r)}/ X}^{\otimes m},$$
and consequently an inclusion
$$L \hookrightarrow f^{(r)}_* \omega_{Y^{(r)}/ X}^{\otimes m} \otimes \tau_* \shO_{\tilde X}.$$
Because of the global generation of $M \otimes (\tau_* \shO_{\tilde X})^\vee$, this in turn induces a sequence of inclusions
$$A^{\otimes m} \hookrightarrow f^{(r)}_* \omega_{Y^{(r)}/ X}^{\otimes m} \otimes \tau_* \shO_{\tilde X} \otimes M^{-1}
\hookrightarrow \bigoplus f^{(r)}_* \omega_{Y^{(r)}/ X}^{\otimes m},$$
Finally, this provides a non-trivial homomorphism
$A^{\otimes m} \rightarrow f^{(r)}_* \omega_{Y^{(r)}/ X}^{\otimes m}$, which means that we can take
$(Y^{(r)} , f^{(r)})$ to play the role of $(Y', f')$ as desired.
Step 2.
Fix now a line bundle $A$ on $X$ such that $A (-D_f)$ is ample.
Using Step $1$, and switching back to the notation $f: Y \rightarrow X$, we can now assume that condition ($\ref{eq:sections}$) is satisfied for $f$ with respect to $L = A$, so that thm:VZ applies to this set-up. By passing to a birational model of the base, and therefore assuming
only that $A (-D_f)$ is big and nef, we can arrange in addition that the singularities of the Hodge module $M$ constructed in thm:VZ occur along a simple normal crossings divisor $D$ which contains $D_f$.
This is the context where thm:Higgs applies; we use it to obtain a graded
submodule $\shFb \subseteq \shEb$ of a (graded logarithmic) Higgs bundle
\[
\theta \colon \shEb \to \OmX^1(\log D) \tensor \shE_{\bullet+1},
\]
such that
\[\theta(\shFb) \subseteq \OmX^1(\log D_f) \tensor \shF_{\bullet+1}
\]
and moreover
$$A(-D_f) \subseteq \shF_0 \,\,\,\,\,\, {\rm and} \,\,\,\,\,\, \shF_k = 0 ~~{\rm for } ~ ~k < 0.$$
We can then apply thm:submodule-Higgs to obtain the desired conclusion.
If $f$ has maximal variation, then clearly so does any $\tilde f: \tilde Y \rightarrow \tilde X$ obtained from $f$ by
a base change $\tau: \tilde X \rightarrow X$ followed by a desingularization of $Y\times_X \tilde X$. Under our
hypotheses it follows that thm:VZ-sheaves applies, which gives the conclusion
in combination with thm:CP.
§.§ On the Kebekus-Kovács conjecture
Kebekus and Kovács have proposed a natural conjecture generalizing Viehweg's hyperbolicity conjecture to families that are not necessarily of maximal variation.
If $X^{\circ}$ is smooth and quasi-projective, and $f^{\circ}: Y^{\circ} \rightarrow X^{\circ}$ is a smooth family of canonically
polarized varieties with maximal variation, then either $\kappa (X^{\circ}) = - \infty$ and $\dim X^\circ > {\rm Var}(f)$, or
$\kappa (X^\circ) \ge {\rm Var} (f)$.
In <cit.> and <cit.> they showed that this conjecture holds when $X^\circ$ has dimension two and three, respectively, and provided a beautiful structural analysis according to the different possible values of the variation in these cases.
In <cit.> they showed that it holds when $X^\circ$ is projective of arbitrary dimension if the conjectures of the minimal model program, including abundance, are assumed for all varieties of dimension at most $\dim X^\circ$. The conjecture is now known to
hold in general due to work of Taji <cit.>, who proved Campana's isotriviality conjecture, which in turn implies the Kebekus-Kovács conjecture.
We note for completeness that, again when $ X = X^\circ$ is projective, a small variation of the methods in this paper leads to a proof of generalized_conjecture for the more general types of families of varieties considered here as well, provided that a statement along the lines of the abundance conjecture were known for $K_X$. Such a statement was conjectured by Campana and Peternell; the case $A = 0$ is a famous special case of the abundance conjecture.
If $X$ is a smooth projective variety and
$$K_X \sim_{\QQ} A + B,$$
where $A$ and $B$ are an effective and a pseudo-effective $\QQ$-divisor respectively, then $\kappa (X) \ge \kappa (A)$.
Consider now a smooth family $f \colon Y \to X$, with $X$ and $Y$ smooth and projective. We assume that the fibers are of general type, or more generally that the geometric generic fiber has a good minimal model.
Claim: generalized_conjecture holds for $f$ assuming that CP_conjecture holds for $X$.
To see this, we can assume that $\kappa (X) \ge 0$,
since in the uniruled case thm:hyperbolicity already implies that ${\rm Var}(f) < \dim X$.
In any case, for all families with fibers as assumed, Kawamata <cit.> has shown a stronger version of the $Q_{n,m}$ conjecture: for $m\ge 1$ sufficiently large one has
$$\kappa ( \det f_* \omega_{Y/X}^{\otimes m}) \ge {\rm Var} (f).$$
This implies via an argument similar to that in thm:submodule-existence (and in fact simpler, since now
$D = \emptyset$) that there exists a graded submodule
$\shF_{\bullet}$ of a graded Higgs bundle $\shE_{\bullet} \to \Omega_X^1 \otimes \shE_{\bullet + 1}$,
and a line bundle with $\kappa (A) \ge {\rm Var}(f)$, such that
$$A \subseteq \shF_0 \,\,\,\,\,\, {\rm and} \,\,\,\,\,\, \shF_k = 0 ~~{\rm for } ~ ~k < 0.$$
This in turn produces for some $s \ge 1$ a subsheaf $\mathcal{H} \subseteq (\Omega_X^1)^{\otimes s}$ as in
thm:submodule-Higgs, only this time $\mathcal{H}$ is not big, but rather only has the property that $\mathcal{H} \simeq \mathcal{\shG} \otimes A$,
where $\shG$ is a weakly positive sheaf. Indeed, repeating the argument used in the proof of thm:submodule-positivity and thm:submodule-Higgs, we obtain a morphism
$$K_{p+s}^\vee \otimes A \longrightarrow (\Omega_X^1)^{\otimes s},$$
where $K_{p+s}^\vee$ is weakly positive, and again we take $\shH$ to be its image. In particular we have
$\det \mathcal{H} \simeq \det \shG \otimes B$ with $\det \shG$ pseudo-effective, and $\kappa (B) \ge {\rm Var}(f)$.
Moreover, the inclusion of $\shH$ induces an exact sequence
$$0 \longrightarrow \mathcal{H} \longrightarrow (\Omega_X^1)^{\otimes s} \longrightarrow \shQ \longrightarrow 0.$$
As at the end of the proof of thm:submodule-positivity, we can assume that $\shH$ is saturated, and therefore
by the same result of Campana-Păun that $\det \shQ$ is pseudoeffective. Passing to determinants we obtain
$$\omega_X^{\otimes s} \simeq \det \shQ \otimes \det \shG \otimes B,$$
where the first two line bundles on the right are pseudo-effective. At this stage CP_conjecture implies that
$$\kappa (X) \ge \kappa (B) \ge {\rm Var}(f),$$
as predicted by generalized_conjecture.
|
1511.00489
|
In this paper we consider a Lorentz-breaking scalar field theory within the Horava-Lifshtz
approach. We investigate the changes that a space-time anisotropy produces in the Casimir
effect. A massless real quantum scalar field is considered in two distinct situations:
between two parallel plates and inside a rectangular two-dimensional box. In both cases we
have adopted specific boundary conditions on the field at the boundary. As we shall see,
the energy and the Casimir force strongly depends on the parameter associated with the
breaking of Lorentz symmetry and also on the boundary conditions.
Keywords: Horava-Lifshtz; Casimir effects
PACS numbers: 03.70.+k, 11.10.Ef
§ INTRODUCTION
A free quantum field theory (QFT) can be treated as an infinite quantum system of simple harmonic oscillators,
with its fundamental excitations interpreted as the associated particles. Thus, the vacuum in QFT is the state
in which all quantum oscillators are in its ground state. But, as we know, the energy of the ground state of a
quantum harmonic oscillator is not zero. Consequently, the vacuum energy, being the sum of the energies of the ground
states of these oscillators, is infinite. The QFT offers several examples which show that this vacuum plays
a fundamental role not only in the physics of microscopic phenomena, but also the physics of
macroscopic phenomena. One of these phenomena the Casimir effect <cit.>.
The Casimir effect is one of the most notable consequences of vacuum quantum fluctuations.
In its most general description, the Casimir effect is a consequence of the changes caused
in the vacuum energy due to the presence of boundary conditions imposed on the fields.
The effect was first predicted theoretically by H. B Casimir in 1948 <cit.>,
and experimentally confirmed ten years later by M. J. Sparnnaay <cit.>.
In the 90s, experiments have confirmed the Casimir effect with high degree of accuracy <cit.>.
In his original work Casimir predicted that due to quantum fluctuations of the electromagnetic field,
two parallel flat neutral (grounded) plates attract each other with a force given by:
\begin{equation}
F = - A \frac{\pi^2 \hbar c}{240 a^4}\ ,
\end{equation}
where $A$ is the area of plates and $a$ is the distance between them. The Casimir effect
is traditionally studied by changing up the idealized effects of borders by boundary conditions.
Once the theory of relativity is the basis for QFT, the Lorentz symmetry is a fully conserved symmetry
in this theory. However, other theories include models where the Lorentz symmetry is violated.
In the quantum gravity, Hor̆ava-Lifshitz (HL) theory is a theory where the Lorentz symmetry is
broken in a strong manner. The space-time anisotropy caused by the breaking of Lorentz symmetry,
occurs due to different properties of scales in which coordinates space and time are set, so that
the theory is invariant under the rescaling $x\to bx$, $t\to b^{\xi}t$, where $\xi$ is a number
called the critical exponent <cit.>. The space-time anisotropy in a given field
theory model should certainly modify the spectrum of the Hamiltonian operator given.
The HL approach, or, as is the same, the idea of the space-time anisotropy, clearly can be
applied not only to gravity, but also to other field theory models, including scalar, spinor
and gauge theories. One of the main reasons for interest to HL-like generalizations of these
theories consists in possible improvement of convergence of quantum corrections. Among the most
important results achieved within their studies, one can emphasize calculation of the one-loop
effective potential in HL-like QED and HL-like Yukawa model <cit.> and study of different
issues related to renormalization of these theories <cit.>. Therefore, the problem of study
of the Casimir effect in HL-like generalizations of different field theories seems to be quite
natural. Some preliminary studies in this direction, for a very particular case, were performed in <cit.>.
In this paper we intend to generalize the results obtained in <cit.>
through the analysis the influence of the combination of the two above
mentioned effects on the vacuum energy associated with a real quantum scalar field, i.e.,
by the imposition of specific boundary conditions on the fields and the Lorentz-breaking symmetry.
In fact, we want to study the quantum scalar field in two different configurations:
between two parallel plates and inside a rectangular two-dimensional box. In section <ref>
we briefly derive the equation of motion obeyed by the field in a Lorentz symmetry breaking
context. In section <ref> we explicitly develop the calculations considering the
quantum system confined between two parallel plates and a two-dimensional rectangular box.
In both situations we impose on the field Dirichlet, Neumann and mixed boundary conditions on the
boundaries. Finally we leave for Conclusions <ref> our most relevant remarks found in this paper.
§ KLEIN-GORDON EQUATION WITH THE LORENTZ SYMMETRY BREAKING
The Lorentz invariance, known as a cornerstone of the quantum field theory, began to be intensively questioned
in recent decades, both within theoretical and experimental contexts. In 1989, V. A. Kostelecky and S.
Samuel <cit.> described a mechanism in string theory which allows violation of Lorentz
symmetry at the Planck energy scale. This mechanism is based on a spontaneous breaking of the Lorentz
symmetry implemented through acquiring non-zero vacuum expectation values by some vectors
or tensors, which implies privileged directions and hence anisotropy in space-time. This effect
is called condensation of tensors in vacuum. The breaking happened at the Planck energy
scale in a more fundamental theory, and the effects of this break manifest themselves for other energy scales in
different field theory models, for example the Standard Model, were not detected up to now,
because such effects are suppressed by powers of the Planck mass. Kostelecky and Samuel
also evaluated that this idea of Lorentz symmetry breaking should be incorporated into
the Standard Model (SM), thus giving rise to the Standard Model Extended (SME).
The proposed Lorentz symmetry breaking is intensively tested through observations and experiments.
So, astronomic observations in the star spectrum, show an evidence that the fine structure constant
$\left( \alpha = \frac{e^2}{\bar{h} c}\right)$, which is a measure of intensity of the electromagnetic
interaction between photons and electrons, slowly varies <cit.>.
Later studies indicated that other mechanisms for breaking of Lorentz symmetry are also possible,
such as space-time noncommutativity <cit.>, variation of coupling constants <cit.> and modifications of quantum gravity <cit.>.
Further, in the paper <cit.>, the concept of large Lorentz symmetry breaking,
or, as it is mostly called, space-time asymmetry, has been introduced. Following this idea,
the time and space coordinates, and derivatives with respect to them, enter in the action
in different degrees, so, the action continues to be quadratic in time derivatives
to avoid arising the ghosts, but involves higher spatial derivatives,
whose order is $2\xi$. Namely these theories will be the main object of our study here.
§.§ Modified Klein-Gordon Equation
In the seminal work <cit.> Hor̆ava called a great attention to theories with space-time
asymmetry, since it revitalized a hope to solve the key problem of quantum gravity, that is,
to find a renormalizable and ghost-free gravity theory. Indeed,
the four-dimensional HL gravity is power-counting renormalizable for $\xi=3$.
Using the HL approach, in the next section we will discuss how the Lorentz symmetry
violation interferes in the vacuum structure of a quantum scalar field theory. We
will study the Casimir effect which has been well studied in the usual field theories.
Before that, we first need to see how the HL theory modifies the Klein-Gordon equation.
We work with a theory which space-time coordinates no longer have the same weight,
as in the case where Lorentz symmetry is preserved. In this scenario, we consider
the theory of a massless real scalar field, which is the simplest case. The action
associated with this system is given by <cit.>:
\begin{equation}
\label{S}
S = \frac{1}{2} \int \mathrm{d}t \mathrm{d}^d x \left( \partial_{0}\phi\partial_{0}\phi
- l^{2(\xi-1)}\partial_{i_{1}}\partial_{i_{2}}...\partial_{i_{\xi}}
\phi\partial_{i_{1}}\partial_{i_{2}}...\partial_{i_{\xi}}\phi \right) \ .
\end{equation}
In the case (3 + 1)-dimensions the equation of motion is:
\begin{equation}
\label{kgmodific}
[\partial_{0}^2 + l^{2(\xi -1)}(-1)^{\xi}
\partial_{i_{1}}...\partial_{i_{\xi}}\partial_{i_{1}}...\partial_{i_{\xi}}]\phi = 0 \ .
\end{equation}
The equation (<ref>) is the modified Klein-Gordon equation within the HL-like theory.
It will be used in next section where we will study the changes that the breaking of Lorentz
symmetry implies on the Casimir effect.
§ VIOLATION OF LORENTZ SYMMETRY WITHIN THE CASIMIR EFFECT
In this section we will study how the space-time anisotropy generated by
the HL theory modifies the results of the Casimir effect associated with a
massless scalar quantum field. In what follows, we study two distinct cases.
In the first one we consider this effect between two parallel plates and the
second inside a two-dimensional rectangular box. In both cases we deal with
three kinds of boundary conditions.
§.§ Parallel Plates
We consider a massless scalar field inside two parallel plates, as it is shown in figure <ref>.
As we saw in the previous section, the equation that a massless real scalar field must satisfy
in the HL theory is given by:
Two parallel plates with area $L^2$ separated by a distance $d<<L$.
\begin{equation}
\label{kgmodific1}
[\partial_{0}^2 + l^{2(\xi -1)}(-1)^{\xi} \partial_{i_{1}}...
\partial_{i_{\xi}}\partial_{i_{1}}...\partial_{i_{\xi}}]\phi(x) = 0 \ .
\end{equation}
First we must obtain the solution for (<ref>) by imposing on the fields
specific boundary conditions and thus obtain the Hamiltonian $\hat{H}$ operator.
Afterwords, we can calculate the total vacuum energy of the system and then
determine the Casimir energy. Subsequently we will obtain Casimir force. In the
$(3 + 1)$-dimensional case, the term $\partial_{i_{1}}...
\partial_{i_{\xi}}\partial_{i_{1}}...\partial_{i_{\xi}}$ takes the following form:
\begin{equation}
\partial_{i_{1}}...\partial_{i_{\xi}}\partial_{i_{1}}...\partial_{i_{\xi}}
= (\partial_{x}^2+ \partial_{y}^2+ \partial_{z}^2)^{\xi} \ ,
\end{equation}
so the equation for $\phi(x)$ reads,
\begin{equation}
\label{eqmovql}
\left[\partial_{0}^2 + l^{2(\xi -1)}(-1)^{\xi} (\partial_{x}^2+ \partial_{y}^2+ \partial_{z}^2)^{\xi}\right]\phi(x) = 0.
\end{equation}
§.§.§ Dirichlet Condition
Now we must solve (<ref>) requiring that the solution must satisfies
the Dirichlet boundary conditions given below,
\begin{equation}
\phi(x)_{z=0} =\phi(x)_{z=d} = 0 \ .
\end{equation}
Adopting the standard procedure <cit.>, we write the field operator as
\begin{equation}
\label{ocdh}
\hat{\phi}(x) = \int \mathrm{d}^2\textbf{k} \sum_{n=1}^{\infty}
\frac{1}{[(2\pi)^2 d\, k_{0}]^{1/2}} \sin\left(\dfrac{n\pi}
{d}z\right)[a_{\textbf{k},n}e^{-i k x} + a^{\dagger}_{\textbf{k},n}e^{i k x}] \ ,
\end{equation}
where $a_{\textbf{k},n}$ and $a^{\dagger}_{\textbf{k},n}$ correspond to the
annihilation and creation operators, respectively, characterized by the set of quantum numbers
$\sigma=\{k_x, k_y, n\}$. These operators satisfy the algebra
\begin{equation}
\label{algebra}
\begin{split}
[a_{\textbf{k},n}, a^{\dagger}_{\textbf{k}',n'}] & = \delta_{n,n'}\delta^2(\textbf{k}-\textbf{k}') , \\
[a_{\textbf{k},n}, a_{\textbf{k}',n'}] = & [a^{\dagger}_{\textbf{k},n}, a^{\dagger}_{\textbf{k}',n'}] = 0 \ .
\end{split}
\end{equation}
In (<ref>) we have defined $kx \equiv k_{0}x_{0} - k_{x}x - k_{y}y$, being
\begin{equation}
k_{0} = l^{\xi-1}\omega_{\textbf{k},n}^{\xi} \ ,
\end{equation}
with $\omega_{\textbf{k},n}$ obeying the dispersion relation,
\begin{equation}
\omega_{\textbf{k},n}^2 = k_{x}^2 + k_{y}^2 + \left(\frac{n\pi}{d}\right)^2 \ .
\end{equation}
The Hamiltonian operator, $\hat{H}$, for this case is given by:
\begin{equation}
\hat{H} = \frac{l^{\xi-1}}{2}\int \mathrm{d}^2\textbf{k} \sum_{n=1}^{\infty}
\omega_{\textbf{k},n}^{\xi}\left[2 a^{\dagger}_{\textbf{k,n}} a_{\textbf{k,n}} +
\frac{L^2}{(2\pi)^2}\right] \ .
\end{equation}
Consequently the vacuum energy is obtained by taking the vacuum expectation value of $\hat{H}$:
\begin{equation}
\label{vacuum}
E_{0} = \bra{0}\hat{H}\ket{0} = \frac{l^{\xi-1} L^2}{8 \pi^2}\int \mathrm{d}^2\textbf{k}
\sum_{n=1}^{\infty} \omega_{\textbf{k},n}^{\xi} \ .
\end{equation}
In order to develop the summation on the quantum number $n$, we shall use the Abel-Plana formula <cit.>,
\begin{equation}
\label{Abel}
\sum_{n=0}^{\infty}F(n) = \frac{1}{2}F(0) + \int_{0}^{\infty} F(t)\mathrm{d}t +
i \int_{0}^{\infty}\frac{\mathrm{d}t}{e^{2\pi t}-1}[ F(it) - F(-i t)] \ .
\end{equation}
Performing in (<ref>) a change of coordinates in the plane $(k_{x},k_{y})$
to polar coordinate, we get
\begin{equation}
\label{vacuum1}
E_{0} = \frac{l^{\xi-1} L^2}{4\pi}\int_{0}^{\infty}
\mathrm{d}k k \left[- \frac{1}{2}F(0) + \int_{0}^{\infty}
F(t)\mathrm{d}t + i\int_{0}^{\infty} \frac{F(it) - F(-it)}{e^{2 \pi t} - 1}\mathrm{d}t\right] \ ,
\end{equation}
\begin{equation}
F(n) = \left[k^2 + \left(\frac{n\pi}{d}\right)^2\right]^{\xi/2}.
\end{equation}
Note that the first term on the right-hand side of (<ref>)
refers to vacuum energy in the presence of only one plate, and the
second one is connected with vacuum energy without boundary. Both
terms do not contribute to the Casimir energy. As a
result, the Casimir energy per unit area of the planes is given by
\begin{equation}
\label{E-C0}
\frac{E_{C}}{L^2} = i \frac{l^{\xi-1}}{4\pi}\int_{0}^{\infty}
\mathrm{d}k k \int_{0}^{\infty}\mathrm{d}t \frac{[k^2 +
(\frac{i t \pi}{d})^2]^{\xi/2} - [k^2 + (-\frac{i t \pi}{d})^2]^{\xi/2}}{e^{2 \pi t} - 1} \ .
\end{equation}
Performing a change of variable, where $\frac{t\pi}{d} = u$, we get
\begin{equation}
\frac{E_{C}}{L^2} = i \frac{l^{\xi-1} d}{4\pi^2}\int_{0}^{\infty}
\mathrm{d}k k \int_{0}^{\infty}\mathrm{d}u \frac{[k^2 + (i u)^2]^{\xi/2} -
[k^2 + (-i u)^2]^{\xi/2}}{e^{2 d u} - 1}.
\end{equation}
The integral over the $u$ variable must be considered in two cases,
for $k>u$ and $k<u$. Thus integrating can provide two different values.
* For $k>u$:
In this range we have,
\begin{equation}
\label{k>u}
[k^2 + (\pm i u)^2]^{\xi/2} = [k^2 - u^2]^{\xi/2}.
\end{equation}
* For $k<u$:
In this range we have,
\begin{equation}
\label{k<u}
\begin{split}
[k^2 + (\pm i u)^2]^{\xi/2} = e^{\pm i\xi \frac{\pi}{2}} [u^2 - k^2]^{\xi/2}.
\end{split}
\end{equation}
Consequently the integral $u$ in the interval $[0,k]$ vanishes. So, we get:
\begin{equation}
\begin{split}
\frac{E_{C}}{L^2} &= i \frac{l^{\xi-1} d}{4\pi^2}\int_{0}^{\infty}
\mathrm{d}k k \int_{k}^{\infty}\mathrm{d}u \frac{(u^2-k^2)^{\xi/2}}{e^{2 d u} - 1}
(e^{i \xi \frac{\pi}{2}} - e^{ -i \xi \frac{\pi}{2}}), \\ \\ &= -
\sin\left(\frac{\pi \xi}{2}\right) \frac{l^{\xi-1} L^2 d }
{2 \pi^2}\int_{0}^{\infty} \mathrm{d}k k \int_{k}^{\infty}\mathrm{d}u
\frac{(u^2-k^2)^{\xi/2}}{e^{2 d u} - 1}.
\end{split}
\end{equation}
To solve the integral in $u$ we introduced a new variable $t=u/k$, so
\begin{equation}
\frac{E_{C}}{L^2} = - \sin\left(\frac{\pi \xi}{2}\right)
\frac{l^{\xi-1} d }{2 \pi^2}\int_{1}^{\infty}(t^2 - 1)^{\xi/2}\mathrm{d}t
\int_{0}^{\infty}\mathrm{d}k \frac{k^{\xi + 2}}{e^{2 d k t} - 1}\ .
\end{equation}
Using the expression below <cit.>,
\begin{equation}
\int_{0}^{\infty}\mathrm{d}x \frac{x^{\nu - 1}}{e^{\mu x} -1} = \frac{1}{\mu^{\nu}}\Gamma(\mu)\zeta(\mu),
\end{equation}
where $\Gamma(\mu)$ and $\zeta(\mu)$ are the gamma and the Riemann zeta functions, respectively, we get,
\begin{equation}
\frac{E_{C}}{L^2} = - \sin\left(\frac{\pi \xi}{2}\right) \frac{l^{\xi-1} d }
{2 \pi^2} \frac{\Gamma(\xi+3)\zeta(\xi+3)}{(2 d)^{\xi+3}}\int_{1}^{\infty}\mathrm{d}t\frac{(t^2 - 1)^{\xi/2}}{t^{\xi+3}}.
\end{equation}
Therefore Casimir energy per unit area is expressed by:
\begin{equation}
\label{E-C}
\frac{E_{C}}{L^2} = - \sin\left(\frac{\pi \xi}{2}\right) \frac{ l^{\xi-1}
\Gamma(\xi+2)\zeta(\xi+3)}{2^{\xi+4} \hspace{0.2cm} \pi^2 d^{(\xi+2)}} \ .
\end{equation}
From (<ref>) the Casimir pressure between two
parallel planes due to scalar field oscillations takes the
\begin{equation}
P(d) = - \sin\left(\frac{\pi \xi}{2}\right) \frac{ l^{\xi-1}
\Gamma(\xi+3)\zeta(\xi+3)}{2^{\xi+4} \hspace{0.2cm} \pi^2 d^{(\xi+3)}} \ .
\end{equation}
An important remarks about the above result is that the Casimir pressure
depends on the critical exponent through the term $\sin(\frac{\pi \xi}{2})$.
For even value of this exponent the pressure vanishes; however, for odd values
it changes its sign. It can be positive, which corresponds to a repulsive force,
or negative, which corresponds to an attractive force.
For $\xi=1$, the Casimir pressure reproduces the usual one
given in <cit.>:
\begin{equation}
\label{C-Press}
P(d) = - \frac{\pi^2}{480 d^4}.
\end{equation}
§.§.§ Neumann Condition
Now we want to obtain solutions of Eq. (<ref>) which obey the
boundary condition below,
\begin{equation}
\frac{\partial \phi(\textbf{x})}{\partial z}\Big |_{z=0} =
\frac{\partial \phi(\textbf{x})}{\partial z}\Big |_{z=d} = 0.
\end{equation}
After some intermediate steps, we can say that for this case the field operator reads,
\begin{equation}
\label{ocnh}
\hat{\phi}(x) = \int \mathrm{d}^2\textbf{k} \sum_{n=0}^{\infty} c_{n}
\cos\left(\dfrac{n\pi}{d}z\right)[a_{\textbf{k},n}e^{-i k x} + a^{\dagger}_{\textbf{k},n}e^{i k x}] \ ,
\end{equation}
\begin{equation}
c_{n} = \begin{cases}
\dfrac{1}{\sqrt{2(2\pi)^2 d k_{0}}} & \text{for $n = 0$}, \\
\dfrac{1}{\sqrt{(2\pi)^2 d k_{0}}} & \text{for $n \ge 0$},
\end{cases}
\end{equation}
and $k_{0} = l^{\xi-1}\omega^{\xi}_{\textbf{k},n}$. Here $\omega_{\textbf{k},n}$
obeys the same dispersion relation as in the previous case:
\begin{equation}
\omega_{\textbf{k},n}^2 = k_{x}^2 + k_{y}^2 + \left(\frac{n \pi}{d}\right)^2.
\end{equation}
Using field operator and the commutation relation (<ref>), the operator
$\hat{H}$ takes the form
\begin{equation}
\hat{H} = \frac{l^{\xi-1}}{2}\int\mathrm{d}^2\textbf{k}
\sum_{n=0}^{\infty}\text{'} \omega_{\textbf{k},n}^{\xi}
\left[2 a^{\dagger}_{\textbf{k},n} a_{\textbf{k},n} + \frac{L^2}{(2\pi)^2}\right] \ ,
\end{equation}
where the prime in the summation symbol means that the term with $n=0$ should be
divided by two.
The vacuum energy is given by
\begin{equation}
E_{0} = \bra{0}\hat{H}\ket{0} = \frac{l^{\xi-1} L^2}{8 \pi^2}
\int\mathrm{d}^2\textbf{k}\sum_{n=0}^{\infty}\text{'} \omega_{\textbf{k},n}^{\xi} \,\ .
\end{equation}
Using again the Abel-Plana summation formula and rewriting the integral
on the plane $(k_{x},k_{y})$ in polar coordinates, we obtain
\begin{equation}
\label{E-C1}
E_{0} = \frac{l^{\xi-1} L^2}{4 \pi} \int_{0}^{\infty}\mathrm{d}k k
\left[\int_{0}^{\infty}F(t)\mathrm{d}t + i \int_{0}^{\infty} \mathrm{d}t \frac{F(it) - F(-it)}{e^{2 \pi t} - 1}\right],
\end{equation}
where here
\begin{eqnarray}
F(n) = \omega_{\textbf{k},n}^{\xi} = \left[k^2 + \big(\frac{n\pi}{a}\big)^2\right]^{\xi/2} \ .
\end{eqnarray}
As in previous cases, the Casimir energy per unit
area is given by the second term on the right side of (<ref>). So, we get
\begin{equation}
\frac{E_{C}}{L^2} = i \frac{l^{\xi-1}}{4 \pi} \int_{0}^{\infty}
\mathrm{d}k k\int_{0}^{\infty}\mathrm{d}t
\frac{[k^2 + (\frac{i t \pi}{d})^2]^{\xi/2} - [k^2 + (-\frac{i t \pi}{d})^2]^{\xi/2}}{e^{2 \pi t} - 1}.
\end{equation}
This is the same expression obtained for the Dirichlet case, Eq. (<ref>).
Consequently the Casimir energy per unit area is given by:
\begin{equation}
\frac{E_{C}}{L^2} = - \sin\left(\frac{\pi \xi}{2}\right) \frac{ l^{\xi-1}
\Gamma(\xi+2)\zeta(\xi+3)}{2^{\xi+4} \hspace{0.2cm} \pi^2 d^{(\xi+2)}} \ .
\end{equation}
The Casimir pressure reads,
\begin{equation}
P(d)= - \sin\left(\frac{\pi \xi}{2}\right) \frac{ l^{\xi-1}
\Gamma(\xi+3)\zeta(\xi+3)}{2^{\xi+4} \hspace{0.2cm} \pi^2 d^{(\xi+3)}}.
\end{equation}
As in the previous case, the Casimir pressure is zero for a even $\xi$ and
change its signal for odd values of this parameter.
§.§.§ Mixed Condition
Now let us consider a scalar field which obeys a Dirichlet boundary condition on one plane,
and a Neumann boundary condition on the other. Two different configurations take place:
* First configuration,
\begin{equation}
\phi(z=0) = \frac{\partial\phi(\textbf{x})}{\partial z} |_{z=d}=0 \ .
\end{equation}
* Second configuration,
\begin{equation}
\frac{\partial\phi(\textbf{x})}{\partial z} |_{z=0} = \phi(z=d) =0 \ .
\end{equation}
After solving Eq. (<ref>) with these conditions, the obtained fields operators
can be shown to look like:
\begin{equation}
\hat{\phi}_{(a)}(x) = \int \mathrm{d}^2 \textbf{k} \sum_{n = 0}^{\infty} \frac{1}
{\sqrt{(2 \pi)^2 d k_{0}}} \sin\left((n+1/2)\frac{ \pi}{d}z\right)
[ a_{\textbf{k},n}e^{-ikx} + a_{\textbf{k},n}^{\dagger} e^{ikx}]
\end{equation}
for the first configuration and
\begin{equation}
\hat{\phi}_{(b)}(x) = \int \mathrm{d}^2 \textbf{k} \sum_{n = 0}^{\infty}
\frac{1}{\sqrt{(2 \pi)^2 d k_{0}}} \cos\left((n+1/2)\frac{ \pi}
{d}z\right)[ a_{\textbf{k},n}e^{-ikx} + a_{\textbf{k},n}^{\dagger} e^{ikx}] \ ,
\end{equation}
for the second configuration. In both cases $k_{0}= l^{\xi-1}\omega_{\textbf{k},n}^{\xi}$
and $\omega_{\textbf{k},n}$ satisfies the dispersion relation,
\[\omega_{\textbf{k},n}^{2} = k_{x}^2 + k_{y}^2 + \big[(n+1/2)\frac{\pi}{d}\big]^2.\]
Both field operators, $\hat{\phi}{^a}(x)$ and $\hat{\phi}^{b}(x)$, provide the same
Hamiltonian operator,
\begin{equation}
\hat{H} = \frac{l^{\xi-1}}{2}\int\mathrm{d}^2\textbf{k}\sum_{n=0}^{\infty}
\omega_{\textbf{k},n}^{\xi} \left[2 a^{\dagger}_{\textbf{k},n} a_{\textbf{k},n} +
\frac{L^2}{(2\pi)^2}\right] \ .
\end{equation}
The vacuum energy of the scalar field is expressed as
\begin{equation}
E_{0} = \bra{0}\hat{H}\ket{0} = \frac{l^{\xi-1} L^2}{8 \pi^2}
\int\mathrm{d}^2\textbf{k}\sum_{n=0}^{\infty} \omega_{\textbf{k},n}^{\xi} .
\end{equation}
Changing the coordinates of the plane $(k_ {x},k_ {y})$ to polar ones,
and using the Abel-Plana summation formula for half-integer numbers <cit.>, we get:
\begin{equation}
E_{0} = \frac{l^{\xi-1} L^2}{4 \pi} \int_{0}^{\infty}\mathrm{d}k k \left\lbrace
\int_{0}^{\infty}F(t)\mathrm{d}t - i \int_{0}^{\infty}\mathrm{d}t\frac{F(it) -
F(-it)}{e^{2 \pi t} + 1}\right\rbrace \ .
\end{equation}
Again, the Casimir energy is given by the second term of the above expression.
The first one refers to the free energy vacuum. Then the Casimir energy is given by
\begin{equation}
E_{C} = - i \frac{l^{\xi-1} L^2}{4 \pi} \int_{0}^{\infty}\mathrm{d}k k
\int_{0}^{\infty}\mathrm{d}t\frac{[k^2 + \left(\frac{i t \pi}{d}\right)^2]^{\xi/2}
+ [k^2 + \left(-\frac{i t \pi}{d}\right)^2]^{\xi/2}}{e^{2 \pi t} + 1} \ .
\end{equation}
After performing a change of variable, $\frac{t\pi}{d}=u$, the Casimir energy by unit area reads,
\begin{equation}
\label{E-C-M}
\frac{E_{C}}{L^2} = - i \frac{l^{\xi-1} d}{4 \pi^2}
\int_{0}^{\infty}\mathrm{d}k k \int_{0}^{\infty}\mathrm{d}u
\frac{[k^2 + (iu)^2]^{\xi/2} + [k^2 +(-iu)^2]^{\xi/2}}{e^{2 a u} + 1}.
\end{equation}
Again, we must consider the integral over the variable $u$ in two sub-intervals:
The first one is $[0, k]$ and the second is $[k, \infty)$.
From (<ref>) we have to the integral in the interval $[0,k]$ vanishes,
so it remains only the integral in the second
interval. By using (<ref>), we get:
\begin{eqnarray}
\frac{E_{C}}{L^2} &=& - i \frac{l^{\xi-1} d}{4\pi^2}\int_{0}^{\infty}
\mathrm{d}k k \int_{k}^{\infty}\mathrm{d}u \frac{(u^2-k^2)^{\xi/2}}{e^{2 d u} + 1}
(e^{i \xi \frac{\pi}{2}} - e^{ -i \xi \frac{\pi}{2}})\ , \nonumber\\
&=&{\mathrm{sin}}\left(\frac{\pi \xi}{2}\right) \frac{l^{\xi-1} d }{2 \pi^2}\int_{0}^{\infty}
\mathrm{d}k k \int_{k}^{\infty}\mathrm{d}u \frac{(u^2-k^2)^{\xi/2}}{e^{2 d u} + 1}\ .
\end{eqnarray}
Introducing a new variable $t$, where $u=kt$, we get
\begin{equation}
E_{C} = {\mathrm{sin}}\left(\frac{\pi \xi}{2}\right)
\frac{l^{\xi-1} L^2 d }{2 \pi^2} \int_{1}^{\infty}\mathrm{d}t (t^2 - 1)^{\xi/2}
\int_{0}^{\infty} \mathrm{d}k \frac{k^{\xi+2}}{e^{2 d k t} + 1} \ .
\end{equation}
Using <cit.> one finds that the Casimir energy per unit area is given by
\begin{equation}
\frac{E_{C}}{L^2} = {\mathrm{sin}}\left(\frac{\pi \xi}{2}\right)
(1-2^{-(\xi+2)})\frac{l^{\xi-1} \Gamma(\xi+2)\zeta(\xi+3)}{2^{\xi+4}\hspace{0.2cm}
\pi^2 \hspace{0.2cm} d^{\xi+2} } \ .
\end{equation}
For this case, the Casimir pressure reads,
\begin{equation}
\frac{F_{C}}{L^2} = (1-2^{-(\xi+2)}) {\mathrm{sin}}\left(\frac{\pi \xi}{2}\right)
\frac{l^{\xi-1} \Gamma(\xi+3)\zeta(\xi+3)}{2^{\xi+4}\hspace{0.2cm} \pi^2 \hspace{0.2cm} d^{\xi+3} }.
\end{equation}
We again have here the same general behavior: for $\xi$ even the Casimir force vanishes,
while for $\xi$ odd the Casimir force switches between an attractive and repulsive one.
We can also see that for $\xi=$1 this force in fact recovers the usual Casimir pressure,
\begin{equation}
P(d) = \frac{7}{3840}\frac{\pi^2}{d^4} \ .
\end{equation}
This result differs from (<ref>) by the factor a numerical factor and
the sign.
§.§ The Casimir Effect In Rectangular Boxes
Now we will consider a massless scalar quantum field, $\phi(x)$, within a two-dimensional
rectangular boxes defined by $0 \le x \le d$ and $0\le y \le b$, as shown in figure <ref>.
We will disregard the $z$ coordinate, since it does not cause any
influence on the Casimir energy. Thus, the field equation $\phi(x)$ must satisfy is
\begin{equation}
\label{emret}
[\partial_{0}^2 + l^{2(\xi-1)}(-1)^\xi (\partial_{x}^2 + \partial_{y}^2)^\xi]\phi(x) = 0 \ .
\end{equation}
Rectangular boxe, with edges $b$ and $d$.
Again, in this section we are interested to calculate the Casimir
energy and Casimir force by imposing three different boundary
conditions on the field as shown below.
§.§.§ Dirichlet
First let us obtain the solution of Eq. (<ref>) satisfying the condition below:
\begin{equation}
\begin{cases}
\phi(t,0,y) = \phi(t,d,y) = 0\ ,\\
\phi(t,x,0) = \phi(t,x,b) = 0\ .
\end{cases}
\end{equation}
The resulting field operator, $\hat{\phi}(x)$ compatible with these condition is:
\begin{equation}
\label{phi2}
\hat{\phi}(x) = \sum_{n,m=1}^{\infty} \sqrt{\frac{2}{d b k_{0}}}
\sin\left(\frac{n \pi}{d} x\right) \sin\left(\frac{n \pi}{b} y\right)
[a_{n,m}e^{-i k_{0}t} + a^{\dagger}_{n,m}e^{i k_{0}t}] \ ,
\end{equation}
where $k_{0} = l^{\xi-1} \omega^{\xi}_{n,m}$ and $\omega_{n,m}$ obey the dispersion relation,
\begin{equation}
\omega_{n,m} =\sqrt{ \left(\frac{n\pi}{d}\right)^2 + \left(\frac{m\pi}{b}\right)^2} \ .
\end{equation}
In (<ref>) $a_{n,m}$ and $a^{\dagger}_{n,m}$ are the annihilation and
creation operators respectively satisfying the following commutation relations:
\begin{equation}
\label{rcr}
\begin{split}
&[a_{n,m},a^{\dagger}_{n',m'}] = \delta_{n,n'}\delta_{m,m'} \ , \\
&[a_{n,m},a_{n',m'}] = [a^{\dagger}_{n,m},a^{\dagger}_{n',m'}] = 0 \ .
\end{split}
\end{equation}
Thus the Hamiltonian operator $\hat{H}$ for this case is given by
\begin{equation}
\hat{H} = l^{\xi-1} \sum_{n,m=1}^{\infty} \omega^{\xi}_{n,m}
\left[a^{\dagger}_{n,m}a_{n,m} +\frac{1}{2}\right] \ ,
\end{equation}
So the vacuum energy is:
\begin{equation}
\label{ECDiric}
E_{0} = \bra{0}\hat{H}\ket{0} = \frac{l^{\xi-1}}{2}\sum_{n,m=1}^{\infty}
\omega^{\xi}_{n,m}\ .
\end{equation}
To develop the sums over the quantum numbers $n$ and $m$ we will again use the Abel-Plana
summation formula separately. First we will develop the sum over $m$ considering $F(m) = [\left(\frac{n\pi}{d}\right)^2 +
\left(\frac{m\pi}{b}\right)^2]^{\xi/2}$. Doing that we get:
\begin{equation}
\begin{split}
E_{0} = \frac{l^{\xi-1}}{2}\sum_{n=1}^{\infty} \Big[ &
-\frac{1}{2}\left(\frac{n\pi}{d}\right)^{\xi} + \int_{0}^{\infty}
\mathrm{d}t\left[\left(\frac{n\pi}{d}\right)^2 + \left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} + \\
& i \int_{0}^{\infty}\mathrm{d}t\frac{[\left(\frac{n\pi}{d}\right)^2 +
\left(\frac{it\pi}{b}\right)^2]^{\xi/2} - \left(\frac{n\pi}{d}\right)^2 +
\left(\frac{-it\pi}{b}\right)^2]^{\xi/2}}{e^{2 \pi t}-1}\Big]\ .
\end{split}
\end{equation}
The last integral should be divided in two parts, as shown below:
* For $\frac{t}{b}<\frac{n}{d}$ we have:
\begin{equation}
\label{sum1}
\left[\left(\frac{n\pi}{d}\right)^2 + \left(\pm i\frac{t\pi}{b}\right)^2\right]^{\xi/2}
= \left[\left(\frac{n\pi}{d}\right)^2 - \left(\frac{t\pi}{b}\right)^2\right]^{\xi/2}\ .
\end{equation}
* For $\frac{t}{b}>\frac{n}{d}$ we have:
\begin{equation}
\label{sum2}
\left[\left(\frac{n\pi}{d}\right)^2 + \left(\pm i\frac{t\pi}{b}\right)^2\right]^{\xi/2}
= e^{\pm i \xi \frac{\pi}{2}}\left[\left(\frac{t\pi}{b}\right)^2 -
\left(\frac{n\pi}{d}\right)^2\right]^{\xi/2} \ .
\end{equation}
Then, from (<ref>) and (<ref>) we have
\begin{equation}
\begin{split}
\label{E-C-box}
E_{0} = \frac{l^{\xi-1}}{2}\sum_{n=1}^{\infty} \Big[ &
\underbrace{-\frac{1}{2}\left(\frac{n\pi}{d}\right)^{\xi}}_{I} +
\underbrace{\int_{0}^{\infty}\mathrm{d}t\left[\left(\frac{n\pi}{d}\right)^2 +
\left(\frac{t\pi}{b}\right)^2\right]^{\xi/2}}_{II} -\\ & \underbrace{2
\sin(\xi\frac{\pi}{2})\int_{nb/d}^{\infty}\mathrm{d}t\frac{[\left(\frac{t\pi}{b}\right)^2 -
\left(\frac{n\pi}{d}\right)^2]^{\xi/2} }{e^{2 \pi t}-1}}_{III} \Big].
\end{split}
\end{equation}
Let us perform the sum in $n$ proceeding with each term separately. Applying the Abel-
Plana formula in the term $I$ of (<ref>), we find
\begin{eqnarray}
\label{I}
I&=& -\frac{1}{2}\left\lbrace \int_{0}^{\infty}\mathrm{d}v
\left(\frac{v \pi}{d}\right)^\xi + i\int_{0}^{\infty}\mathrm{d}v
\frac{(\frac{i v \pi}{d})^\xi - (-\frac{i v \pi}{d})^\xi}{e^{2 \pi v}-1} \right\rbrace \nonumber\\
&=& -\frac{1}{2}\int_{0}^{\infty}\mathrm{d}v \left(\frac{v \pi}{d}\right)^\xi
+ \sin\left(\xi\frac{\pi}{2}\right) \left(\frac{\pi}{d}\right)^\xi \int_{0}^{\infty}\mathrm{d}v \frac{v^\xi}{e^{2 \pi v}-1}.
\end{eqnarray}
The integral of the second term of (<ref>) is obtained from <cit.>.
Then, the term $I$ from (<ref>) is given by:
\begin{equation}
-\frac{1}{2}\sum_{n=1}^{\infty} \left(\frac{n\pi}{d}\right)^{\xi} =
-\frac{1}{2}\int_{0}^{\infty}\mathrm{d}v \left(\frac{v \pi}{d}\right)^\xi +
\sin\left(\xi\frac{\pi}{2}\right) \left(\frac{\pi}{d}\right)^\xi \frac{\Gamma(\xi+1)\zeta(\xi+1)}{(2 \pi)^{\xi+1}}.
\end{equation}
Performing the sum for the term $II$ of (<ref>) we have:
\begin{equation}
\begin{split}
II = \int_{0}^{\infty}\mathrm{d}t \Bigg\{& -\frac{1}{2}\left(\frac{t\pi}{b}\right)^\xi
+ \int_{0}^{\infty}\mathrm{d}v \left[\left(\frac{v\pi}{d}\right)^2 +
\left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} \\ &
- i \int_{0}^{\infty}\mathrm{d}v \frac{\left[\left(\frac{i v\pi}{d}\right)^2 +
\left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} - \left[\left(\frac{- i v\pi}{d}\right)^2 +
\left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} }{e^{2 \pi v}-1}\Bigg\}.
\end{split}
\end{equation}
The last term of this equation must also be split into two parts,
for $v<\frac{ta}{b}$ and $v>\frac{ta}{b}$, thus we have:
\begin{eqnarray}
II&= & -\frac{1}{2}\left(\frac{\pi}{b}\right)^\xi\int_{0}^{\infty}t^\xi\mathrm{d}t +
\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}v \left[\left(\frac{v\pi}{d}\right)^2 +
\left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} \nonumber\\
&-&2 \sin\left(\xi\frac{\pi}{2}\right)\int_{0}^{\infty} dt\int_{td/b}^{\infty}dv
\frac{\left[\left(\frac{v \pi}{d}\right)^2 - \left(\frac{t \pi}{b}\right)^2\right]^{\xi/2}}{e^{2 \pi x}-1} \ .
\end{eqnarray}
The integral of the last term of $II$ can be obtained by using <cit.>, so that $II$ reads,
\begin{equation}
\begin{split}
II = & -\frac{1}{2}\left(\frac{\pi}{b}\right)^\xi\int_{0}^{\infty}t^\xi\mathrm{d}t
+ \int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}v \left[\left(\frac{v\pi}{d}\right)^2
+ \left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} \\ &-
\sin\left(\xi\frac{\pi}{2}\right)\frac{d}{b}\left(\frac{\pi}{b}\right)^\xi
\frac{\Gamma(\xi+2)\zeta(\xi+2)\mathrm{B}(\frac{1}{2},\frac{\xi}{2}+1)}{(2 \pi d/b)^{\xi+2}},
\end{split}
\end{equation}
where $B(x,y)$ is the beta function <cit.>.
Developing the term $III$ of (<ref>), we have:
\begin{equation}
III = - 2 \sin\left(\xi\frac{\pi}{2}\right)\sum_{n=1}^{\infty}
\int_{nb/d}^{\infty}\mathrm{d}t \frac{[(\frac{t \pi}{b})^2 -
(\frac{n\pi}{d})^2]^{\xi/2}}{e^{2 \pi t}-1} \ .
\end{equation}
By using the identity $\sum\limits_{m=1}^{\infty}e^{- n m} = \frac{1}{e^n -1}$, we can rewrite $III$ as:
\begin{equation}
III = - 2 \sin\left(\xi\frac{\pi}{2}\right)\sum_{n,m=1}^{\infty}
\int_{nb/d}^{\infty}\mathrm{d}t \left[\left(\frac{t \pi}{b}\right)^2 -
\left(\frac{n\pi}{d}\right)^2\right]^{\xi/2}e^{- 2 \pi t m}.
\end{equation}
Defining a new variable $u=\frac{td}{n b}$, the above expressions is rewritten as:
\begin{equation}
III = - 2 \sin\left(\xi\frac{\pi}{2}\right)\left(\frac{\pi}{d}\right)^\xi
\left(\frac{b}{d} \sum_{n=1}^{\infty} n^{\xi+1}\sum_{m=1}^{\infty}
\int_{1}^{\infty}\mathrm{d}u[u^2-1]^{\xi/2} e^{- 2 \pi u n m b/d}\right) \ .
\end{equation}
Finally using the integral representation of the modified Bessel function <cit.>,
\begin{equation}
K_\nu(z)=\frac{(z/2)^\nu\Gamma(1/2)}{\Gamma(\nu+1/2)}\int_1^\infty dt (t^2-1)^{\nu-1/2}
\end{equation}
the term $III$ is given by
\begin{equation}
III = - 2 \sin\left(\xi\frac{\pi}{2}\right)\left(\frac{\pi}{d}\right)^\xi
\frac{\Gamma(\frac{\xi}{2}+1)}{\pi^{\frac{\xi+2}{2}}}
\left(\frac{b}{d}\right)^{\frac{1-\xi}{2}}\sum_{n,m=1}^{\infty}
\left(\frac{n}{m}\right)^{\frac{\xi+1}{2}}K_{\frac{\xi+1}{2}}(2 \pi n m b/d).
\end{equation}
Therefore, substituting the terms $I$, $II$ and $III$ into (<ref>), we have:
\begin{equation}
\label{E0RD}
\begin{split}
E_{0} = \frac{l^{\xi-1}}{2} & \Bigg\{ -\frac{\pi^\xi}{2}\left(\frac{1}{d^\xi}+
\frac{1}{b^\xi}\right)\int_{0}^{\infty}\mathrm{d}t t^\xi +\sin\left(\xi\frac{\pi}{2}\right)
\left(\frac{\pi}{d}\right)^\xi \frac{\Gamma(\xi+1)\zeta(\xi+1)}{(2 \pi)^{\xi+1}} + \\
\\
\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}&\mathrm{d}v \left[\left(\frac{v\pi}{d}\right)^2
+ \left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} - \sin\left(\xi\frac{\pi}{2}\right)
\frac{d}{b}\left(\frac{\pi}{b}\right)^\xi \frac{\Gamma(\xi+2)\zeta(\xi+2)\mathrm{B}
(\frac{1}{2},\frac{\xi}{2}+1)}{(2 \pi d/b)^{\xi+2}} \\
\\
- 2 \sin\left(\xi\frac{\pi}{2}\right)&\left(\frac{\pi}{d}\right)^\xi
\frac{\Gamma(\frac{\xi}{2}+1)}{\pi^{\frac{\xi+2}{2}}} \left(\frac{b}{d}\right)^{\frac{1-\xi}{2}}
\sum_{n,m=1}^{\infty} \left(\frac{n}{m}\right)^{\frac{\xi+1}{2}}K_{\frac{\xi+1}{2}}(2 \pi n m b/d)\Bigg\} \ .
\end{split}
\end{equation}
The first term of (<ref>) is proportional to the perimeter of rectangle with sides $b^\xi$ and
$d^\xi$.[In the case $\xi=1$ it is easier to visualize that this energy
refers to the energy in case where we have a rectangle with sides $b$ and $d$ <cit.>.]
The third term of (<ref>) refers to the free vacuum energy of the area bounded
by rectangle $bd$. Because we are interested to obtain vacuum energy arising due to the
imposition of boundary conditions on all sides of the rectangle $bd$, we can omit these
terms. This process is equivalent to a renormalization of the above result.
So Casimir energy for this case is given by:
\begin{equation}
\begin{split}
E_{C} = \frac{l^{\xi-1} \pi^\xi }{2} \sin\left(\xi\frac{\pi}{2}\right)\Big
\{& \frac{\Gamma(\xi+1)\zeta(\xi+1)}{d^\xi (2 \pi)^{\xi+1}} -
\frac{b}{d^{\xi+1} (2\pi)^{\xi+2}}\Gamma(\xi+2)\zeta(\xi+2)\mathrm{B}\Big(1/2,\frac{\xi}{2} +1\Big) \\
&- 2 \frac{\Gamma(\frac{\xi}{2}+1)}{d^{\frac{\xi+1}{2}} b^{\frac{\xi-1}{2}}
\pi^{\frac{\xi}{2}+1}}\sum_{n,m=1}^{\infty} \left(\frac{n}{m}\right)^{\frac{\xi+1}{2}}
K_{\frac{\xi+1}{2}}(2 \pi n m b/d)\Big\} \ .
\end{split}
\end{equation}
We can see that for $\xi$ even the Casimir energy vanishes. However, for odd values of this
parameter, the Casimir energy change the sign. In special cases $\xi=1$ and $\xi=3$ we have:
* For $\xi=1$
\begin{equation}
\label{ECBox1}
E^{\xi=1}_{C} = \frac{\pi}{2}\left\lbrace\frac{1}{24 d} - \frac{b}{8 \pi^2 d^2}\zeta(3) -
\frac{1}{d \pi}\sum_{n,m=1}^{\infty}\frac{n}{m}K_{1}(2\pi n m b/d)\right\rbrace.
\end{equation}
* For $\xi=3$
\begin{equation}
\label{ECbox2}
E^{\xi=3}_{C} = - \frac{l^{2} \pi^{3}}{2}\left\lbrace\frac{1}{240 d^3} -
\frac{9 b}{32 \pi^4 d^4}\zeta(5) - \frac{3}{2 b d^2 \pi^2}
\sum_{n,m=1}^{\infty}\frac{n^2}{m^2}K_{2}(2\pi n m b/d)\right\rbrace.
\end{equation}
So, we see that (<ref>) reproduces the usual result given in <cit.>.
From the above results we can obtain the Casimir force acting on the edges of the boxes:
* For $\xi=1$:
\begin{equation}
\begin{split}
F^{1}_{d} = - \frac{\partial E^{\xi=1}_{C}}{\partial d} = -\frac{\pi}{2}
\Bigg\{ -\frac{1}{24 d^2} + \frac{b}{4 \pi^2 d^3}\zeta(3) +
\frac{1}{\pi d^2}\sum_{n,m=1}^{\infty}\frac{n}{m}K_{1}(2 \pi n m b/d)
\\ - \frac{1}{\pi d}\sum_{n,m=1}^{\infty}\frac{n}{m}\frac{\partial K_{1}(2 \pi n m b/d)}{\partial d}\Bigg\}
\end{split}
\end{equation}
is the force acting on the edges located in $x=0$ and $x=d$.
\begin{equation}
F^{1}_{b} = - \frac{\partial E^{\xi=1}_{C}}{\partial b} =
-\frac{\pi}{2}\left\lbrace - \frac{\zeta(3)}{8 \pi^2 d^2} -
\frac{1}{\pi d}\sum_{n,m=1}^{\infty}\frac{n}{m}\frac{\partial
K_{1}(2 \pi n m b/d)}{\partial b}\right\rbrace
\end{equation}
is the force acting on the edges in $y=0$ and $y=b$.
For both results we have,
\begin{equation}
\frac{\partial K_{1}(F(x))}{\partial x} = -
\frac{1}{2}[K_{0}(F(x))+K_{2}(F(x))]\frac{\partial F(x)}{\partial x}.
\end{equation}
* For $\xi=3$, we have:
\begin{equation}
\begin{split}
F^{3}_{d} = - \frac{\partial E^{\xi=3}_{C}}{\partial d} =
\frac{l^2 \pi^3}{2}\Bigg\{ - \frac{1}{80 d^4} + \frac{9 b
\zeta(5)}{8 d^5 \pi^4}+ \frac{3}{b \pi^2 d^3}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2}
K_{2}(2\pi n m b/d) \\
-\frac{3}{2 b d^2 \pi^2}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2}
\frac{\partial K_{2}(2\pi n m b/d)}{\partial d} \Bigg\}
\end{split}
\end{equation}
is the force acting on the edges $x=0$ and $x=d$.
\begin{equation}
\begin{split}
F^{3}_{b} = - \frac{\partial E^{\xi=3}_{C}}{\partial b} =
\frac{l^2 \pi^3}{2}\Bigg\{ - \frac{9 \zeta(5)}{32 d^4 \pi^4} +
\frac{3}{2 b^2 \pi^2 d^2}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2} K_{2}(2\pi n m b/d) \\
-\frac{3}{2 b d^2 \pi^2}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2}
\frac{\partial K_{2}(2\pi n m b/d)}{\partial b} \Bigg\}
\end{split}
\end{equation}
is the force acting on the edges $y=0$ and $y=b$.
For this case we have,
\begin{equation}
\frac{\partial K_{2}(F(x))}{\partial x} = -\frac{1}{2}[K_{1}(F(x))+K_{3}(F(x))]\frac{\partial F(x)}{\partial x}.
\end{equation}
§.§.§ Neumann
In this case we must solve (<ref>) by imposing the conditions,
\begin{equation}
\begin{cases}
\frac{\partial\phi}{\partial x}\Big |_{x=0} = \frac{\partial\phi}{\partial x}\Big |_{x=d} = 0\ ,\\
\frac{\partial\phi}{\partial y}\Big |_{y=0} = \frac{\partial\phi}{\partial y}\Big |_{y=b} = 0 \ .
\end{cases}
\end{equation}
Under this condition the corresponding field operator is written by,
\begin{equation}
\hat{\phi}(x) = \sum_{n,m=0}^{\infty} c_{n,m} \cos\left(\frac{n \pi}{d} x\right)
\cos\left(\frac{n \pi}{b} y\right)[a_{n,m}e^{-i k_{0}t} + a^{\dagger}_{n,m}e^{i k_{0}t}]\ ,
\end{equation}
\begin{equation}
\label{cnm}
c_{n,m} = \begin{cases}
\sqrt{\frac{1}{2 d b k_{0}}}\,\ \mathrm{for} \,\ m=n=0\ , \\
\sqrt{\frac{1}{ d b k_{0}}} \,\ \mathrm{for} \,\ m \,\ or \,\ n=0 \\ \ ,
\sqrt{\frac{2}{ d b k_{0}}} \,\ \mathrm{for} \,\ m \,\ \mathrm{and} \,\ n\ge1 \ .
\end{cases}
\end{equation}
Also in this case, we have, $k_{0} = l^{\xi-1}\omega^{\xi}_{n,m}$ and
$\omega_{n,m} = \sqrt{(\frac{n \pi}{d})^2 + (\frac{m \pi}{b})^2}$.
The Hamiltonian operator for this field, is
\begin{equation}
\hat{H} = \frac{b d}{2}\sum_{n,m=0}^{\infty} C^{2}_{n,m}
k_{0}^{2}(a^{\dagger}_{n,m}a_{n,m} + 1/2) \ ,
\end{equation}
so that the vacuum energy is given by:
\begin{equation}
\label{ECboxDiric}
E_{0} = \frac{l^{\xi-1}}{4}\left(\frac{\pi}{d}\right)^{\xi}\sum_{n=1}^{\infty} n^{\xi} +
\frac{l^{\xi-1}}{4}\left(\frac{\pi}{b}\right)^{\xi}\sum_{m=1}^{\infty} m^{\xi} +
\frac{l^{\xi-1}}{2}\sum_{n,m=1}^{\infty}\omega^{\xi}_{n,m} \ .
\end{equation}
The last term of (<ref>) is the same vacuum energy for the Dirichlet case
(<ref>), so we need only to calculate the first two terms. Using the Abel-Plana
formula for the sum and results of <cit.>, we get:
\begin{equation}
\label{F1}
\frac{l^{\xi-1}}{4}\left(\frac{\pi}{d}\right)^{\xi}\sum_{n=1}^{\infty} n^{\xi} =
\frac{l^{\xi-1}}{4}\left(\frac{\pi}{d}\right)^{\xi}\left\lbrace\int_{0}^{\infty}t^{\xi}\mathrm{d}t -
2 \sin\left(\xi\frac{\pi}{2}\right)\frac{\Gamma(\xi+1)\zeta(\xi+1)}{(2\pi)^{\xi+1}}\right\rbrace
\end{equation}
\begin{equation}
\label{F2}
\frac{l^{\xi-1}}{4}\left(\frac{\pi}{b}\right)^{\xi}\sum_{m=1}^{\infty} m^{\xi} =
\frac{l^{\xi-1}}{4}\left(\frac{\pi}{b}\right)^{\xi}\left\lbrace\int_{0}^{\infty}t^{\xi}\mathrm{d}t -
2 \sin\left(\xi\frac{\pi}{2}\right)\frac{\Gamma(\xi+1)\zeta(\xi+1)}{(2\pi)^{\xi+1}}\right\rbrace \ .
\end{equation}
Then, substituting (<ref>), (<ref>) and (<ref>) into (<ref>), we find:
\begin{equation}
\label{E0RN}
\begin{split}
E_{0} = \frac{l^{\xi-1}}{2} & \Bigg\{ \int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}v
\left[\left(\frac{v\pi}{d}\right)^2 + \left(\frac{t\pi}{b}\right)^2\right]^{\xi/2} -
\sin\left(\xi\frac{\pi}{2}\right) \left(\frac{\pi}{b}\right)^\xi \frac{\Gamma(\xi+1)\zeta(\xi+1)}{(2 \pi)^{\xi+1}} \\
\\
& - \sin\left(\xi\frac{\pi}{2}\right)\frac{d}{b}\left(\frac{\pi}{b}\right)^\xi
\frac{\Gamma(\xi+2)\zeta(\xi+2)\mathrm{B}(\frac{1}{2},\frac{\xi}{2}+1)}{(2 \pi d/b)^{\xi+2}} \\
\\
- 2 & \sin\left(\xi\frac{\pi}{2}\right)\left(\frac{\pi}{d}\right)^\xi
\frac{\Gamma(\frac{\xi}{2}+1)}{\pi^{\frac{\xi+2}{2}}} \left(\frac{b}{d}\right)^{\frac{1-\xi}{2}}
\sum_{n,m=1}^{\infty} \left(\frac{n}{m}\right)^{\frac{\xi+1}{2}}K_{\frac{\xi+1}{2}}(2 \pi n m b/d)\Bigg\} \ .
\end{split}
\end{equation}
The first integral above refers to the free vacuum energy, as has been said before.
It is subtracted from the renormalization process. Thus Casimir energy for this case is given by:
\begin{equation}
\begin{split}
E_{C} = -\frac{l^{\xi-1} \pi^\xi }{2} \sin\left(\xi\frac{\pi}{2}\right)\Big\{&
\frac{\Gamma(\xi+1)\zeta(\xi+1)}{b^\xi (2 \pi)^{\xi+1}} + \frac{b}{d^{\xi+1}
(2\pi)^{\xi+2}}\Gamma(\xi+2)\zeta(\xi+2)\mathrm{B}\Big(1/2,\frac{\xi}{2} +1\Big) \\
&+ 2 \frac{\Gamma(\frac{\xi}{2}+1)}{d^{\frac{\xi+1}{2}} b^{\frac{\xi-1}{2}}
\pi^{\frac{\xi}{2}+1}}\sum_{n,m=1}^{\infty} \left(\frac{n}{m}\right)^{\frac{\xi+1}{2}}
K_{\frac{\xi+1}{2}}(2 \pi n m b/d)\Big\} \ .
\end{split}
\end{equation}
Again, for $\xi$ even the Casimir force vanishes. Let us look at the
two particular cases $\xi=1$ and $\xi=3$:
* $\xi=1$
\begin{equation}
E^{\xi=1}_{C} = -\frac{\pi}{2}\left\lbrace \frac{1}{24 b} + \frac{b}{8 \pi^2 d^2}\zeta(3) +
\frac{1}{d \pi}\sum_{n,m=1}^{\infty}\frac{n}{m}K_{1}(2\pi n m b/d)\right\rbrace \ .
\end{equation}
* $\xi=3$
\begin{equation}
E^{\xi=3}_{C} = \frac{l^{2} \pi^{3}}{2}\left\lbrace \frac{1}{240 b^3} + \frac{9 b}{32 \pi^4 d^4}\zeta(5) +
\frac{3 b}{2 d^2 \pi^2}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2}K_{2}(2\pi n m b/d)\right\rbrace \ .
\end{equation}
We can now calculate the Casimir force for these two cases:
* For $\xi=1$:
\begin{equation}
\begin{split}
F^{1}_{d} = - \frac{\partial E^{\xi=1}_{C}}{\partial d} = -\frac{\pi}{2}\Bigg\{ \frac{b}{4 \pi^2 d^3}\zeta(3) +
\frac{1}{\pi d^2}\sum_{n,m=1}^{\infty}\frac{n}{m}K_{1}(2 \pi n m b/d) \\ - \frac{1}{\pi d}
\sum_{n,m=1}^{\infty}\frac{n}{m}\frac{\partial K_{1}(2 \pi n m b/d)}{\partial d}\Bigg\}
\end{split}
\end{equation}
\begin{equation}
F^{1}_{b} = - \frac{\partial E^{\xi=1}_{C}}{\partial b} = -\frac{\pi}{2}\left\lbrace \frac{1}{24 b} -
\frac{\zeta(3)}{8 \pi^2 d^2} - \frac{1}{\pi d}\sum_{n,m=1}^{\infty}\frac{n}{m}
\frac{\partial K_{1}(2 \pi n m b/d)}{\partial b}\right\rbrace,
\end{equation}
where $F^{1}_{d}$ is the force acting on the edges $x=0$, $x=d$ and $F^{1}_{b}$
is the force acting on the edges $y=0$, $y=b$. In the above expressions
\begin{equation}
\frac{\partial K_{1}(F(x))}{\partial x} = -\frac{1}{2}[K_{0}(F(x))+
K_{2}(F(x))]\frac{\partial F(x)}{\partial x} \ .
\end{equation}
* For $\xi=3$:
\begin{equation}
\begin{split}
F^{3}_{d} = - \frac{\partial E^{\xi=3}_{C}}{\partial d} =
\frac{l^2 \pi^3}{2}\Bigg\{\frac{9 b \zeta(5)}{8 d^5 \pi^4}+
\frac{3b}{ \pi^2 d^3}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2} K_{2}(2\pi n m b/d) \\
-\frac{3b}{2 d^2 \pi^2}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2}
\frac{\partial K_{2}(2\pi n m b/d)}{\partial d} \Bigg\}
\end{split}
\end{equation}
is the force acting on the edges $x=0$, $x=d$, and
\begin{equation}
\begin{split}
F^{3}_{b} = - \frac{\partial E^{\xi=3}_{C}}{\partial b} = \frac{l^2 \pi^3}{2}\Bigg\{\frac{1}{80 b^4} -
\frac{9 \zeta(5)}{32 d^4 \pi^4} - \frac{3}{2 \pi^2 d^2}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2} K_{2}(2\pi n m b/d) \\
-\frac{3 b}{2 d^2 \pi^2}\sum_{n,m=1}^{\infty}\frac{n^2}{m^2} \frac{\partial K_{2}(2\pi n m b/d)}{\partial b} \Bigg\}
\end{split}
\end{equation}
is the force acting on the edges $y=0$, $y=b$. Also we have
\begin{equation}
\frac{\partial K_{2}(F(x))}{\partial x} = -
\frac{1}{2}[K_{1}(F(x))+K_{3}(F(x))]\frac{\partial F(x)}{\partial x}.
\end{equation}
§.§.§ Mixed Condition
We impose mixed boundary conditions to solve (<ref>):
\begin{equation}
\begin{split}
i) \begin{cases} \,\ \phi(x=0) = \frac{\partial\phi}{\partial x} |_{x=d}=0 \ , \\
\phi(y=0) = \frac{\partial\phi}{\partial y} |_{y=b}=0 \end{cases} \ . \\
ii) \begin{cases} \,\ \frac{\partial\phi}{\partial x} |_{x=0}= \phi(x=d) = 0 \ , \\
\frac{\partial\phi}{\partial y} |_{y=0}= \phi(y=b) =0 \end{cases}\ .
\end{split}
\end{equation}
The field operators consistent with these boundary conditions are given by,
\begin{equation}
\hat{\phi}_{i}(x) = \sum_{n,m=o}^{\infty} \sqrt{\frac{2}{d b k_{0}}}
\mathrm{sin}\left[(n+1/2)\frac{\pi}{d} x\right] \mathrm{\sin}
\left[(m+1/2)\frac{\pi}{b} y\right][a_{n,m}e^{-i k_{0}t} + a^{\dagger}_{n,m}e^{i k_{0}t}]
\end{equation}
\begin{equation}
\hat{\phi}_{ii}(x) = \sum_{n,m=o}^{\infty} \sqrt{\frac{2}{d b k_{0}}}
\cos\left[(n+1/2)\frac{\pi}{d} x\right] \cos\left[(m+1/2)\frac{\pi}{b} y\right]
[a_{n,m}e^{-i k_{0}t} + a^{\dagger}_{n,m}e^{i k_{0}t}] \ ,
\end{equation}
where $\omega_{n,m} = \sqrt{\left[(n+1/2)\frac{\pi}{d}\right]^2 +
\left[(m+1/2)\frac{\pi}{b}\right]^2}$.
The Hamiltonian operators obtained for both fields are the same:
\begin{equation}
\hat{H} = \frac{l^{\xi-1}}{2}\sum_{n,m=0}^{\infty}\omega^{\xi}_{n,m}[2a^{\dagger}_{n,m}a_{n,m} + 1].
\end{equation}
So the vacuum energy is given by:
\begin{equation}
E_{0} = \frac{l^{\xi-1}}{2}\pi^{\xi}\sum_{n,m=0}^{\infty}
\left[\left(\frac{n+1/2}{d}\right)^2 + \left(\frac{m+1/2}{b}\right)^2\right]^{\xi/2}.
\end{equation}
Using the Abel-Plana formula and analyzing the integration interval in the second term of the
sum, as it has been done several times during this work, we get:
\begin{equation}
\begin{split}
\label{ECBoxMix}
E_{0} = \frac{l^{\xi-1}}{2}\pi^{\xi}\sum_{n=0}^{\infty}
\Bigg\{ \underbrace{\int_{0}^{\infty}\mathrm{d}t\left[\left(\frac{t}{b}\right)^2 +
\left(\frac{n+1/2}{d}\right)^2\right]^{\xi/2}}_{I} +\\ \underbrace{2 \sin\left(\xi\frac{\pi}{2}\right)
\int_{(n+1/2)b/d}^{\infty}\mathrm{d}t\dfrac{[\left(\frac{t}{b}\right)^2 -
\left(\frac{n+1/2}{d}\right)^2]^{\xi/2}}{e^{2 \pi t}+1}}_{II}\Bigg\}\ .
\end{split}
\end{equation}
Using the Abel-Plana formula to perform the sum of $n$ for term $I$ of (<ref>), we have:
\begin{equation}
I = \int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}v\left[\left(\frac{t}{b}\right)^2 +
\left(\frac{v}{d}\right)^2\right]^{\xi/2} + 2 \sin\left(\xi\frac{\pi}{2}\right)
\int_{0}^{\infty}\mathrm{d}t\int_{td/b}^{\infty}\mathrm{d}t\dfrac{[\left(\frac{v}{d}\right)^2 -
\left(\frac{t}{b}\right)^2]^{\xi/2}}{e^{2 \pi t}+1} \ .
\end{equation}
The integral of the second term of $I$ is obtained from <cit.>, resulting in:
\begin{equation}
\begin{split}
\label{F1Mix}
I = &\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}v\left[\left(\frac{t}{b}\right)^2 +
\left(\frac{v}{d}\right)^2\right]^{\xi/2} + \\
&(1-2^{-(\xi-1)})\frac{b}{d^{\xi+1}} \sin\left(\xi\frac{\pi}{2}\right)
\frac{B \Big(1/2, \frac{\xi}{2}+1\Big)\Gamma(\xi+2)\zeta(\xi+2)}{(2\pi)^{\xi+2}} \ .
\end{split}
\end{equation}
The term $II$ of (<ref>) is given by:
\begin{equation}
II = 2 \sin\left(\xi\frac{\pi}{2}\right) \sum_{n=0}^{\infty}\int_{(n+1/2)b/d}^{\infty}
\mathrm{d}t\dfrac{[\left(\frac{t}{b}\right)^2 - \left(\frac{n+1/2}{d}\right)^2]^{\xi/2}}{e^{2 \pi t}+1}.
\end{equation}
\[\frac{1}{e^{m}+1} = -\sum_{l=1}^{\infty}(-1)^le^{-nl},\]
we can rewrite $II$ as:
\begin{equation}
II = - 2 \sin\left(\xi\frac{\pi}{2}\right) \sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m
\int_{(n+1/2)b/d}^{\infty}\mathrm{d}t\left[\left(\frac{t}{b}\right)^2 -
\left(\frac{n+1/2}{d}\right)^2\right]^{\xi/2}e^{-2 \pi t m }\ .
\end{equation}
Finally by using again <cit.>, we find:
\begin{equation}
\begin{split}
\label{F2Mix}
II = - 2 \sin\left(\xi\frac{\pi}{2}\right)& \frac{b^{\frac{1-\xi}{2}}}
\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m \left(\frac{n+1/2}{m}\right)^{\frac{\xi+1}{2}}
\\ & \times K_{\frac{\xi+1}{2}}(2 \pi(n+1/2)m b/d)\ .
\end{split}
\end{equation}
Substituting the results (<ref>) and (<ref>) into (<ref>), we have:
\begin{equation}
\begin{split}
E_{0} = \frac{l^{\xi-1}\pi^\xi}{2} \Bigg\{ &\int_{0}^{\infty}\mathrm{d}t
\int_{0}^{\infty}\mathrm{d}v\left[\left(\frac{t}{b}\right)^2 +
\left(\frac{v}{d}\right)^2\right]^{\xi/2} + \\
&(1-2^{-(\xi-1)})\frac{b}{d^{\xi+1}} \sin\left(\xi\frac{\pi}{2}\right)
\frac{B \Big(1/2, \frac{\xi}{2}+1\Big)\Gamma(\xi+2)\zeta(\xi+2)}{(2\pi)^{\xi+2}} \\
-&2 \sin\left(\xi\frac{\pi}{2}\right) \frac{b^{\frac{1-\xi}{2}}}{d^{\frac{\xi+1}{2}}}
\frac{\Gamma(\frac{\xi}{2}+1)}{\pi^{\frac{\xi}{2}+1}} \sum_{n=0}^{\infty}
\sum_{m=1}^{\infty}(-1)^m \left(\frac{n+1/2}{m}\right)^{\frac{\xi+1}{2}}
\\ &\times K_{\frac{\xi+1}{2}}(2 \pi(n+1/2)m b/d)\Bigg\} \ .
\end{split}
\end{equation}
Again, the first integral refers to the free vacuum energy in an area bounded
by the "rectangle" $ab$, however without the contours, so this term is subtracted
from the renormalization process. Consequently, the Casimir energy for this boundary condition is given by:
\begin{equation}
\begin{split}
\label{ECMix}
E_{C} = \frac{l^{\xi-1}\pi^\xi}{2}\sin\left(\xi\frac{\pi}{2}\right) \Bigg\{&(1-2^{-(\xi-1)})
\frac{b}{d^{\xi+1}} \frac{B \Big(1/2, \frac{\xi}{2}+1\Big)\Gamma(\xi+2)\zeta(\xi+2)}{(2\pi)^{\xi+2}} \\
- \,\ 2 \frac{b^{\frac{1-\xi}{2}}}{d^{\frac{\xi+1}{2}}}\frac{\Gamma(\frac{\xi}{2}+1)}
{\pi^{\frac{\xi}{2}+1}}& \sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m
\Big(\frac{n+1/2}{m}\Big)^{\frac{\xi+1}{2}}K_{\frac{\xi+1}{2}}(2 \pi(n+1/2)m b/d)\Bigg\}.
\end{split}
\end{equation}
As we see again, for $\xi$ even the Casimir energy is zero. So, let us present two particular cases, for
$\xi=1$ and $\xi=3$:
* $\xi=1$
\begin{equation}
E^{\xi=1}_{C} = \frac{\pi}{2}\left\lbrace \frac{3}{32}\frac{ b \zeta(3)}{d^2 \pi^2} -
\frac{1}{d\pi}\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)
K_{1}(2 \pi(n+1/2)m b/d)\right\rbrace \ .
\end{equation}
* $\xi=3$
\begin{equation}
E^{\xi=3}_{C} = - \frac{l^{2}\pi^{3}}{2}\left\lbrace \frac{135}{512}
\frac{ b \zeta(5)}{d^4 \pi^4} - \frac{3}{2 b d^2\pi^2}\sum_{n=0}^{\infty}
\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)^2K_{2}(2 \pi(n+1/2)m b/d)\right\rbrace \ .
\end{equation}
We can now calculate the Casimir forces:
* For $\xi=1$:
\begin{equation}
\begin{split}
F^{1}_{d} = -\frac{\pi}{2}\Big\{ - \frac{3}{16}\frac{ b \zeta(3)}{d^3 \pi^2} +
\frac{1}{d^2\pi}\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)
K_{1}(2 \pi(n+1/2)m b/d) \\ - \frac{1}{d\pi}\sum_{n=0}^{\infty}
\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)\frac{\partial K_{1}(2 \pi(n+1/2)m b/d)}{\partial d} \Big\} \ ,
\end{split}
\end{equation}
that is the force acting on the edges $x=0$, $x=d$, and
\begin{equation}
F^{1}_{b} = -\frac{\pi}{2}\left\lbrace \frac{3}{32}\frac{ \zeta(3)}{d^2 \pi^2} -
\frac{1}{d\pi}\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)
\frac{\partial K_{1}(2 \pi(n+1/2)m b/d)}{\partial b}\right\rbrace \ ,
\end{equation}
that is the force acting on the edges $y=0$, $y=b$.
In the above expressions, we can write,
\begin{eqnarray}
\frac{\partial K_{1}(F(x))}{\partial x} = -\frac{1}{2}[K_{0}(F(x))+K_{2}(F(x))]\frac{\partial F(x)}{\partial x} \ .
\end{eqnarray}
* For $\xi=3$:
\begin{equation}
\begin{split}
F^{3}_{d} = \frac{l^2\pi^3}{2}\Big\{ - \frac{135}{128}\frac{ b \zeta(5)}{d^5 \pi^4} +
\frac{3}{b d^3\pi^2}\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)^2
K_{2}(2 \pi(n+1/2)m b/d) \\ - \frac{3}{2 b d^2\pi2}\sum_{n=0}^{\infty}
\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)^2\frac{\partial K_{2}(2 \pi(n+1/2)m b/d)}{\partial d} \Big\} \ .
\end{split}
\end{equation}
\begin{equation}
\begin{split}
F^{3}_{b} = \frac{l^2\pi^3}{2}\Big\{ \frac{135}{512}\frac{ \zeta(5)}{d^4 \pi^4} + \frac{3}{2 b^2 d^2\pi^2}
\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)^2\ K_{2}(2 \pi(n+1/2)m b/d) \\ -
\frac{3}{2 b d^2\pi2}\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^m \Big(\frac{n+1/2}{m}\Big)^2\frac{\partial
K_{2}(2 \pi(n+1/2)m b/d)}{\partial b}\Big\} \ .
\end{split}
\end{equation}
These forces act on the edges, $x=0$, $x=d$, and $y=0$, $y=b$, respectively. Moreover, we can write,
\begin{equation}
\frac{\partial K_{2}(F(x))}{\partial x} = -\frac{1}{2}[K_{1}(F(x))+K_{3}(F(x))]\frac{\partial F(x)}{\partial x} \ .
\end{equation}
We have seen then that the Lorentz symmetry breaking, employed by HL theory, modifies the
vacuum structure, resulting in a change in the Casimir effects results. The HL theory parameter $\xi$
is the one that dictates what the outcome of the effect. We have seen that for $\xi$ even, the Casimir force is always
zero, while for $\xi$ odd the Casimir force switches between an attractive and a repulsive ones.
§ CONCLUDING REMARKS
In this paper we have investigated the Casimir effects associated to a massless scalar
real quantum field in the theory with a space-time asymmetry. To do it, we used the
modified Klein-Gordon equation to study the Casimir effect in the HL-like theory.
Two different situations are considered: The field confined between two
parallel plates, and the field confined in a two-dimensional rectangular boxes.
In this case we have admitted that the field obeys specific boundary condition.
First we study the Casimir effect for two parallel plates of area $L^2$ separated by a distance $d$.
For this case, we imposed three different types of boundary conditions on field $\phi(x)$ at
the boundary: Dirichlet, Neumann and mixed conditions. In each case we obtained the Casimir energy,
and found that in all cases, it is given by a divergent sum. We evaluated this sum by
using the Abel-Plana summation formulas. Considering the Dirichlet and Neumann conditions,
we observed that this sum has a three-term contribution: the free vacuum energy contribution
(borderless), vacuum energy contribution in the presence of only one plate, and finally the
vacuum energy contribution in the presence of two plates. At the same time, for the mixed conditions,
we saw that the sum is contributed by only two terms: vacuum energy contribution to the presence of
only one plate and the vacuum energy contribution in the presence of two plates.
Both contributions, free vacuum energy and the vacuum energy in the presence of only one plate
are infinite terms that are subtracted by the renormalization process, thus we obtained the
finite Casimir energy. After that we calculate the Casimir pressures. As in the usual
cases where the Lorentz symmetry is preserved, the Casimir pressure to the Dirichlet and Neumann
conditions of are equal and differ from the one to the mixed condition. In all three
cases we have seen that the Casimir pressure depends on the HL theory parameter $\xi$.
For $\xi=1$ we recover the usual results where the Lorentz symmetry is preserved,
for $\xi$ even we saw that the Casimir pressure is always zero, while for
$\xi$ odd the Casimir pressure switches corresponding to an attractive or repulsive forces.
We also consider the Casimir effect in a two-dimensional rectangular box with edges
“$b$” and “$d$”. Again we impose three different types of boundary conditions to the
field $\phi(x)$: Dirichlet, Neumann and mixed one. We found that the Casimir energy
for the three cases is given in terms of infinite sums. Using the Abel-Plana formula
to develop these sums, we saw that the Casimir energy is again given by three
contributions: the free vacuum energy, the vacuum energy at the presence only two edges
of the rectangle and the vacuum energy in the presence of the rectangle. We have
seen that the free vacuum energy and the vacuum energy at presence of only two edges
of the rectangle are terms infinite, and these are subtracted in the renormalization
process, so that we obtained the finite Casimir energy. Thus we calculate the Casimir forces and
see that they depend on the HL theory parameter $\xi$, where for $\xi$ even the
forces vanishes, while for $\xi$ odd, we have the forces that switches the signal,
which can be an attractive or repulsive force depending on the value of $\xi$.
In general, we can affirm that the HL-like modification change
the equation of motion that a field must satisfy in a theory, and this change has a crucial
implication on a very well known effect in the literature, the Casimir one.
Moreover, the Casimir force also depends also on the types of conditions
and also depend on HL theory parameter $\xi$.
Where once the Casimir force depended only on the type of condition imposed on the
field, we have that in this context the Casimir force also depends on a theory parameter of in question.
The natural application of the results we obtained can be the following one. Since the Casimir
effect now can be measured in a very precise manner, our corrections to the Casimir effect arising
from the Lorentz-breaking modifications of the field theory action, calculated theoretically
and then compared with the experiment, can serve for estimating the values of the Lorentz-breaking
parameters in the corresponding theory.
A natural continuation of this study could consist, first, in consideration of Casimir effect for other
fields; second, in studying of the Casimir effect for small Lorentz-breaking additive
corrections whose actions are discussed in <cit.>, <cit.> and <cit.>.
§ ACKNOWLEDGMENT
IJMU thanks the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). ERBM and
AYP thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).
Casimir H. G. B. Casimir; Proc. K. Ned. Akad. Wet. 51, 793 (1948).
Sparnaay M. J. Sparnaay; Physica 24, 751 (1958).
Lamoureux S. K. Lamoureux; Phys. Rev. Lett. 28, 5 (1997).
Moh U. Mohideen and A. Roy; Phys. Rev. Lett. 81, 21 (1998).
Horava P. Hor̆ava, Phys. Rev. D 79, 084008 (2009).
EP J. Alexandre, K. Farakos, A. Tsapalis, Phys. Rev. D81, 105029 (2010).
FarC. F. Farias, M. Gomes, J.R. Nascimento, A. Yu. Petrov, A. J. da Silva, Phys. Rev. D85,
127701 (2012).
Lima A. M. Lima, J. R. Nascimento, A. Yu. Petrov, R. F. Ribeiro, Phys. Rev. D91, 025027 (2015).
ren R. Iengo, J. Russo, M. Serone, JHEP 0911, 020 (2009).
Iengo R. Iengo, M. Serone, Phys. Rev. D81, 125005 (2010).
Gomes P. R. S. Gomes, M. Gomes, Phys. Rev. D85, 085018 (2012).
12 P. R. S. Gomes, M. Gomes, Phys. Rev. D85, 065010 (2012).
Petrov A. F. Ferrari, H. O. Girotti, M. Gomes, A. Yu. Petrov and
A. J. da Silva, Mod. Phys. Lett. A 28, 1350052 (2013).
KSamuel V. A. Kostelecky and S. Samuel. Phys. Rev. D 39, 683685 (1989).
Songaila A. Songaila and L. L. Cowie, Nature 398, 667 (1999).
Dav P. C. W. Davies, T. M. Davies and C. H. Lineweaver, Nature 418, 602 (2002).
Songaila2 A. Songaila and L. L. Cowie, Nature 428, 132 (2004).
Carroll S. M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001).
Anis A. Anisimov et al., Phys. Rev. D 65, 085032 (2002).
Carl C. E. Carlson et al.,Phys. Lett. B 518, 201 (2001).
Hew J. L. Hewett, F. J. Petriello, T. G. Rizzo, Phys. Rev. D 64, 075012 (2001).
Bert O. Bertolami and L. Guisado, JHEP 0312 (2003) 013.
Kost V. A. Kostelecky, R. Lehnert, M. Perry, Phys. Rev. D 68, 123511 (2003).
Bert2 O. Bertolami, R. Lehnert, R. Potting, A. Ribeiro, Phys. Rev. D 68, 083513 (2004).
Bert3 O. Bertolami, Class. Quant. Grav. 14, 2785 (1997).
Alfaro J. Alfaro, H. A. Morales-Tecotl, L. F. Urrutia, Phys. Rev. Lett. 84, 2318 (2000).
Alfaro2 J. Alfaro, H. A. Morales-Tecotl, L. F. Urrutia, Phys. Rev. D 65, 103509 (2002).
Anselm1 D. Anselmi, Ann. Phys. 324, 874 (2009).
Anselm2 D. Anselmi, Ann. Phys. 324, 1058 (2009).
Mandl F. Mandl, S. Shaw, Quantum Field Theory, 2nd ed., Wiley, 2010.
Adv.Cas M. Bordag, G. L. Klimchitskaya, U. Mohideen, V. M. Mostepanenko;
Advances in the Casimir Effect; Oxford Science Publications (2009).
Ford L. H. Ford, Phys. Rev. D 21, 933 (1980)
Grad I. S. Gradshteyn and I. M. Ryzhik; Table of Integrals, Series, and Products; Elsevier, 7th ed., 2007.
|
1511.00392
|
National Astronomical Observatories, Chinese Academy
of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012,
China. [email protected], [email protected]
We calculate the correlation function of 79,091 galaxy clusters in the
redshift region of $z \leq 0.5$ selected from the WH15 cluster
catalog. With a weight of cluster mass, a significant baryon acoustic
oscillation (BAO) peak is detected on the correlation function with a
significance of $3.7 \sigma$. By fitting the correlation function with
a $\Lambda$CDM model curve, we find $D_v(z = 0.331) r_d^{fid}/r_d =
1261.5 \pm 48$ Mpc which is consistent with the Planck 2015
cosmology. We find that the correlation function of the higher mass
sub-sample shows a higher amplitude at small scales of $r < 80\hMpc$,
which is consistent with our previous result. The 2D correlation
function of this large sample of galaxy clusters shows a faint BAO
ring with a significance of $1.8\sigma$, from which we find that the
distance scale parameters on directions across and along the
line-of-sight are $\alpha_{\sigma} = 1.02 \pm 0.06$ and $\alpha_{\pi}
= 0.94 \pm 0.10$, respectively.
§ INTRODUCTION
The matter distribution in the Universe is homogeneous and isotropic
on large scales. However, large-scale structures start to emerge from
the matter distribution on smaller scales <cit.>. The Baryon
Acoustic Oscillations (BAO) are an imprint of the oscillations in the
early Universe when baryons and photons were tightly coupled
<cit.>. The scale of the BAO can be used as
a `standard ruler' to measure cosmological distances. For example,
the reduced distance $D_v(z)$ was firstly introduced by
\begin{equation}
\label{eq:Dv}
D_v(z) = \left[ (1+z)^2 D_A(z)^2 \frac{cz}{H(z)}\right]^{1/3},
\end{equation}
where $H(z)$ is the Hubble parameter and $D_A(z)$ is the
comoving angular diameter distance. The measurement of the BAO
signals provide a powerful tool to constrain the cosmology parameters
which determine $D_v(z)$.
The BAO signal was firstly detected by <cit.> and
<cit.> using galaxy redshift data from the Sloan Digital Sky
Survey <cit.> and the 2dF Galaxy Redshift Survey
<cit.>. After that, similar measurements of
the BAO were confirmed by later SDSS data releases <cit.> and other
galaxy surveys <cit.>. In addition to
galaxies, the Ly$\alpha$ forests were used as a tracer to search the
BAO signal at higher redshifts. For example, the BAO feature was
detected clearly by using SDSS Ly$\alpha$ forest samples at redshift
$z \sim 2.3$ <cit.>.
Galaxy clusters are the largest gravitationally bound systems in the
Universe, trace the higher density peaks in the matter distribution
field than galaxies, which makes them a great probe for BAO
detection. By calculating the 2-point correlation function and power
spectrum of maxBCG clusters <cit.>,
<cit.> and <cit.> reported weak detections of
BAO signature. <cit.> extracted a spectroscopic sample of
13,904 clusters from <cit.> in the redshift region of $z \leq
0.4$, and detected the BAO signature from the cluster correlation
function with a significance of $\sim 1.9
\sigma$. <cit.> further improved this result with a
significance of $\sim 2.5 \sigma$ by using an updated cluster catalog
of <cit.>.
In this paper, we calculate and analyze the correlation function of
79,091 clusters from <cit.> with the
spectroscopic redshift information updated to SDSS Data Release 12
<cit.>. The cluster sample is described in
Section <ref>. The method to calculate the two-point
correlation function and the theoretical model to analyze the function
are introduced in Section <ref>, and we present the
correlation function results for the whole sample and 6 sub-samples in
subsections. The 2D correlation function of this currently largest
sample of galaxy clusters are presented and discussed in
Section <ref>. Conclusions are given in
Section <ref>.
Sky distribution of 79,091 clusters in our sample, with an
Aitoff projection centered at (RA, DEC) $= (6h, 0^\circ)$. There are
57,647 clusters in the Northern Galactic Cap and 21,444 clusters in
the Southern Galactic Cap.
Throughout this paper, we adopt a flat $\Lambda$CDM cosmology
following Planck 2015 results <cit.>, with $h=0.68$,
$\Omega_{m}=0.31$, $\Omega_{\Lambda}=0.69$, $\sigma_{8}=0.81$, where
$h\equiv H_{0}/100~{\rm kms^{-1}Mpc^{-1}}$.
§ DATA
Using the photometric data from SDSS-III, <cit.> identified
132,684 galaxy clusters with a redshift range of $z < 0.8$. All these
clusters have a richness of $R_{L*} \geq 12$ and more than 8 member
galaxies within $r_{200}$. Monte Carlo simulations give a false
detection rate of less than 6% for the whole catalog. The
completeness is more than 95% in the redshift range of $z < 0.42$ for
massive clusters with $M_{200} > 1 \times 10^{14} \Msolar$. By
applying a new richness estimation together with the latest SDSS DR12
spectroscopic data <cit.>, WH15 detected 25,000 high
redshift clusters which helps to get a high completeness in the region
of $z < 0.6$ for clusters of $M_{500} > 1 \times 10^{14} \Msolar$.
Although the photometric redshift is good enough for identifying the
galaxy clusters, its large uncertainties will affect correlation
function calculations and hence obstruct the detection of BAO
signature <cit.>. For this work, we use a sample
of 79,091 clusters derived from the WH15 cluster catalog, which have a
spectroscopic redshift from SDSS DR12 data <cit.>, including
57,647 clusters from the Northern Galactic Cap and 21,444 clusters
from the Southern Galactic Cap, as shown in Figure <ref>.
The whole sample covers a sky region of $\sim 11,000$ square degree in
total. To make sure our sample has a high completeness, we only use
the spectroscopic clusters within the redshift range of $z \leq 0.5$
with a mean redshift $\overline{z} = 0.331$ (see
Figure <ref>).
Redshift distribution of 79,091 clusters in our sample as
indicated by black solid line. The dashed line and dotted line
indicate the distributions of clusters in the Northern Cap and
Southern Cap, respectively.
§ THE TWO-POINT CORRELATION FUNCTIONS
We calculate the 2-point correlation function $\xi(r)$ of cluster
samples using the Landy-Szalay estimator <cit.>:
\begin{equation}
\label{eq:LS}
\xi(r)=\left[DD(r)\frac{N_{RR}}{N_{DD}}-2\;DR(r)\frac{N_{RR}}{N_{DR}}+RR(r)\right]/RR(r),
\end{equation}
where $DD(r)$, $DR(r)$ and $RR(r)$ stand for the weighted number of
data-data pairs, data-random pairs and random-random pairs within a
separation annulus of $r\pm\Delta r/2$, respectively. $N_{DD}$,
$N_{DR}$ and $N_{RR}$ are the weighted normalization factors. The
random sample used here is 16 times larger than the data sample, which
minimizes the shot noise effect during the calculations. The random
sample shares the same sky area and the same redshift distribution as
the real cluster sample.
More massive galaxy clusters trace more massive dark matter halos,
which should reflect large-scale structures with a larger weight. To
reveal the BAO feature from the complex matter distribution
background, the more massive clusters should have higher weights than
low mass ones. The galaxy clusters in this sample have a mass in the
range from $10^{13.5} \Msolar$ to $10^{15} \Msolar$ as shown in
Figure <ref>. <cit.> have related the cluster mass
with the $r$-band optical luminosity or the richness in their paper.
Here, we adopt a linear weight for cluster mass as
\begin{equation}
\label{eq:weight_mass}
\end{equation}
where $M_{500}$ is the cluster mass within the radius where the mean
density is 500 times of the critical density of the Universe (see WH15
for more details).
The completeness of clusters in the sample depends on the mass of
cluster. The completeness can reach 100% in the high mass end of the
sample distribution, but only about 50% in the low mass end. To
correct the effect of the detection rate, we apply a weight of
$w_{\textrm{completeness}}$ as the reciprocal of the mass-dependent
detection rate provided in the Figure 6 of <cit.>. The total
weight of the $i^{th}$ cluster for the 2-point correlation function is
thus taken as the combination of the above two weights:
\begin{equation}
\label{eq:weight}
w_i = w_{\textrm{mass}} \times w_{\textrm{completeness}}.
\end{equation}
Mass distribution of clusters in our cluster sample.
The error covariance of the correlation function is estimated by using
the log-normal mock catalogs. The log-normal error estimation method
was introduced by <cit.>, and adopted by several BAO
analysis works <cit.>. We create the
log-normal realizations using a model power spectrum:
\begin{equation}
\label{eq:model_power_log}
P(k) = b^2(1+\frac{2}{3}\beta+\frac{1}{5}\beta^2)P_{lin}(\overline{z},k),
\end{equation}
where $b$ is the bias measured by fitting the cluster correlation
function using the model curve with the covariance matrix estimated by
the jackknife method, $\beta = \Omega_m^{0.55}/b$,
$P_{lin}(\overline{z},k)$ is the linear power spectrum obtained from
the camb package <cit.> at the mean redshift
$\overline{z} = 0.331$. In total, 100 log-normal mock catalogs are
generated in the boxes of $3000 \times 3000 \times 3000\hMpc$ with
$600 \times 600 \times 600$ cells. The large box size makes sure that
the mock catalogs can cover the whole survey volume of the cluster
catalog, the cell size of $5\hMpc$ is a half of the bin size of our
correlation function measurements. The log-normal mock distribution is
smooth at scales smaller than the cell size. Correlation functions are
calculated for every mock catalog, the covariance matrix is then
generated by:
\begin{equation}
\label{eq:err}
\end{equation}
where $N=100$ is the number of mock catalogs, $\xi^{k}_{i}$ is the
correlation function value of the $k^{th}$ mock at the $i^{th}$ bin of
$r$ values, and $\overline{\xi}_{i}$ represents the mean value of the
all 100 mock catalogs at the $i^{th}$ bin. The error bars of $\xi(r)$
are given by the diagonal elements as $\sigma_{i}=\sqrt{C_{ii}}$.
The jackknife error estimation method is adopted for the mass weight
comparison. The details about the jackknife method and the comparison
between log-normal and jackknife covariance matrices are discussed in
the appendix.
We calculate the correlation function and the uncertainty in 18 bins
from $20\hMpc$ to $200\hMpc$. The analysis are made not only on the
whole sample of 79,091 clusters, but also on six sub-samples divided
according to sky region (Northern Cap and Southern Cap), or the
redshift ranges ($z \leq 0.35$ and $0.35 < z \leq 0.5$) or the
different cluster mass ($\textrm{M}_{500} \leq 1 \times 10^{14}
\textrm{M}_\odot$ and $\textrm{M}_{500} > 1 \times 10^{14}
\textrm{M}_\odot$).
After that, we analyze the correlation function of galaxy clusters
with a $\chi^2$ fitting to a $\Lambda$CDM model. First, the linear
matter power spectra $P_{\textrm{lin}}(z, k)$ are computed at each
central value of redshift bin shown in the Figure <ref>
using camb package <cit.>. The no-wiggle
approximation of the linear matter power spectrum $P_{\textrm{nw}}(z,
k)$ is generated by fitting the matter power spectrum with the model
described in <cit.>. The template power spectrum with
non-linear evolution effects is <cit.>
\begin{equation}
\begin{split}
P_{\textrm{template}}(z, k) =& \left(P_{\textrm{lin}}(z, k) -
P_{\textrm{nw}}(z, k)\right)\exp \left(-\frac{k^2
\Sigma_{\textrm{nl}}^2}{2}\right)\\
&+P_{\textrm{nw}}(z, k),
\end{split}
\end{equation}
where $\Sigma_{\textrm{nl}}$ is a parameter modeling the non-linear
degradation <cit.>, we
choose $\Sigma_{\textrm{nl}} = 8\hMpc$ in the analysis. The template
correlation function with damped BAO at each redshift is then given by
\begin{equation}
\xi_{\textrm{template}}(z, r) =\int\frac{k^2
dk}{2\pi^2}P_{\textrm{template}}(z, k) j_0(kr)
\exp\left(-k^2a^2\right),
\end{equation}
where $j_0(kr)$ is the zeroth-order spherical Bessel function, the
Gaussian term gives a high-$k$ damping during the transformation with
$a = 1\hMpc$, which is significantly smaller than the scale of the
structure we are interested in. The “averaged” template correlation
function $\xi_{\textrm{template}}(r)$ is then generated by weighting
the template correlation functions at each redshift using the
corresponding number counts $n(z)$ in the redshift bins. Finally, we
fit the cluster correlation function using a model form of
\begin{equation}
\xi_{\textrm{model}}(r) = b^2\xi_{\textrm{template}}(\alpha r)+A(r),
\end{equation}
\begin{equation}
\end{equation}
$b^2$, $\alpha$, $a_1$, $a_2$ and $a_3$ are free parameters, $b^2$,
$a_1$, $a_2$ and $a_3$ are marginalized finally. The $\chi^2$ fitting
runs in the parameter space of $0.80 \leq \alpha \leq 1.20$, where we
fix the other cosmological parameters to the Planck 2015 values of
$\Omega_b= 0.0484$, $n_s = 0.97$, $\sigma_8 = 0.81$, $\Omega_m =
0.31$, $\Omega_\Lambda = 0.69$ and $h = 0.68$. In this fiducial
cosmology, the distance parameter $D_v$ at redshift $z = 0.331$ is
$D_v^{fid} (z = 0.331) = 1301.9~\textrm{Mpc}$.
[]BAO fitting results of the cluster sample and sub-samples
Sample N $\alpha$ $\sigma$
Whole sample 79091 0.969 $\pm$ 0.037 3.7
North cap 57647 0.979 $\pm$ 0.058 2.2
South cap 21444 $\sim 0.939^{\star}$ 0.7
$\textrm{M}_{500} > 1 \times 10^{14} \textrm{M}_{\odot}$ 49207 0.979 $\pm$ 0.058 2.3
$\textrm{M}_{500} \leq 1 \times 10^{14} \textrm{M}_{\odot}$ 29884 $0.960^{\star}$ 0.6
$z \leq 0.35$ 40873 0.938 $\pm$ 0.041 3.3
$0.35 < z \leq 0.50$ 38218 1.020 $\pm$ 0.065 2.2
we cannot
provide an effective error estimation for the $\alpha$ value for the
`Southern Cap' and `low mass' sub-samples
because of very weak signals.
§.§ Results of the whole sample
Correlation function of 79,091 clusters plotted by black
squares with error bars. The solid line and dashed line indicate the
best-fit $\Lambda$CDM model with and without acoustic feature. In
the inset $\xi(r) r^2$ is plotted to show the BAO feature more
clearly. The error bars are estimated via the log-normal method.
The correlation function of all 79,091 clusters is shown in
Figure <ref>. We adopt a weight to correct the
selection bias of the sample and cluster mass. The BAO feature appears
at $r \sim 105\hMpc$ clearly. We do the $\chi^2$ fitting using with
the whole covariance matrix, and find the best-fit $\chi^2 = 6.77$ on
13 degrees of freedom, and the reduced $\chi^2 = 0.52$. A pure CDM
model without the BAO feature is also adopted to fit the correlation
function, which presents a $\chi^2 = 20.29$ and is rejected at $3.7
\sigma$. This is the first time of detecting the BAO signal from a
galaxy cluster sample with a confidence larger than $3 \sigma$. The
best-fit $\Lambda$CDM model offers a constraint on the parameter
$\alpha = 0.969 \pm 0.037$, which gives a constraint on the distance
parameter $D_v$ by $D_v (z=0.331) r_d^{fid}/r_d= 1261.5 \pm
48~\textrm{Mpc}$. See Table <ref> for a summary.
We compare the correlation function of the whole cluster sample
without weighting (i.e. all clusters share the same weight
equals to 1), and compare the result in Figure <ref>.
Since the log-normal method could not provide the cluster mass for the
mock catalogs, so we use the jackknife method which employs the
original cluster mass of the data catalog in this comparison. Because
the net effect of the weighting algorithm is giving higher weights to
more massive clusters, the weighting pulls the correlation function up
to the higher amplitude with a detection confidence of $3.9\sigma$,
while the non-weighted calculation gives a confidence of
$3.1\sigma$. We conclude that the mass weight can help the BAO
detection and dose not move the BAO signal position. The best fitted
$\alpha$ value is $\alpha = 0.972$ with the mass weight, compared with
$\alpha = 0.971$ without the mass weight. Therefore, the weightings
are used in all following calculations for sub-samples.
The correlation functions of the whole sample with (squares)
and without (circles, shifted to right by $2\hMpc$ for clarity)
weights during the calculations. The error bars are estimated by the
jackknife method. The solid lines and dashed lines indicate the
best-fit $\Lambda$CDM curves with and without acoustic feature.
§.§ Results for two sky regions
The correlation function for BAO detection from galaxy
clusters in the Northern Cap (squares) and the Southern Cap
(circles, shifted to right by $2\hMpc$ for clarity), as we do in
Figure <ref>.
We also calculate the correlation function with the weights for
clusters in the Northern Cap and Southern Cap separately. The
correlation functions are shown in the Figure <ref>
with the best-fit model lines. The BAO feature on the Northern Cap is
clear, with a detection confidence of $ 2.2 \sigma$. Due to the
smaller sample size, the BAO signal on the Southern Cap is week, which
has a confidence of $ 0.7 \sigma$.
We notice that the correlation function of the Southern Cap sample has
a higher amplitude comparing with the correlation function of the
Northern Cap sample. The BAO bump on the correlation function of the
Southern Cap sub-sample also shows a shift towards to the larger scale
direction, the model fitting reports a central value of the distance
parameter of $\alpha = 0.939$, which deviates from the model
prediction, but the low signal-to-noise ratio of the BAO bump leads a
difficulty to estimate the measuring accuracy, the flat $\chi^2$
distribution makes the attempts of determining the $1 \sigma$ error
bar failed.
§.§ Results for different mass ranges
The same as Figure <ref> but for the high
mass sample (squares) and the low mass (circles, shifted to right by
$2\hMpc$ for clarity) clusters.
The same as Figure <ref> but for clusters at
high redshift (squares) and low redshift (circles, shifted to right by
$2\hMpc$ for clarity).
To compare the correlation function of clusters with different masses,
we make two sub-samples. The high mass sub-sample contains 49,207
clusters with a mass of $\textrm{M}_{500} > 1 \times 10^{14}
\textrm{M}_{\odot}$, the low mass sub-sample has 29,884 clusters with
a mass of $\textrm{M}_{500} \leq 1 \times 10^{14}
\textrm{M}_{\odot}$. The correlation functions of these two
sub-samples are presented in the Figure <ref>. A clear
BAO signal is detected in the high mass sub-sample with a confidence
of $2.3 \sigma$, while the BAO bump of low mass sub-sample is very
week, only $0.6 \sigma$. Like the `Southern Cap' sub-sample, we cannot
provide $1 \sigma$ error for the low-mass sub-sample because of the
low signal-to-noise ratio.
In small scales of $r < 80\hMpc$, we note the amplitude of correlation
function for high mass clusters is systematically higher than the low
mass ones. <cit.> analyzed the correlation functions in
small scales of sub-samples with different cluster richness, found
that the correlation length and then the amplitude of the correlation
function are proportional to the cluster richness. It is expected that
clusters with high masses trace the more massive halos, which leads a
stronger correlation than the low mass sub-sample. Therefore the
result here is consistent with our previous conclusion.
§.§ Results for different redshift ranges
The whole sample is split into two sub-samples by the redshift. The
low redshift sub-sample contains 40,873 clusters with the redshift of
$z \leq 0.35$, the high redshift sub-sample contains 38,218 clusters
in the redshift region of $0.35 < z \leq 0.5$. The correlation
functions of high and low redshift sub-samples are shown in the
Figure <ref>. Both of the correlation functions show
BAO signals at the scale of $r \sim 105\hMpc$, the BAO peak detection
confidence is $3.3 \sigma$ and $2.2 \sigma$ on the low and high
redshift sub-samples, respectively.
The correlation amplitude is also found to be different for these two
sub-samples with the scales of $r < 80\hMpc$. The difference is due to
the different cluster mass distributions in the two samples. In the
higher redshift region, luminous and massive galaxies have larger
chances to be spectroscopically observed, which makes our high
redshift sub-sample contains relatively more massive clusters than the
low redshift sample. The mean mass of the high redshift sample is
$M_{500} = 1.42 \times 10^{14} \textrm{M}_{\odot}$, while, the mean
mass of the low redshift sample is $M_{500} = 1.24 \times 10^{14}
\textrm{M}_{\odot}$.
§.§ Discussions
<cit.> calculated both the correlation function and power
spectrum of 313,780 galaxies from SDSS DR11 over 7,341 square degrees,
in the redshift range of $0.15 < z < 0.43$ with a mean redshift
$\overline{z}= 0.32$. By fitting the BAO feature, they provided a
distance measurement of $D_V(0.32)=1264 \pm 25 (r_d/r_{d,fid})$, with
a measuring accuracy of 1.9%. In comparison, our cluster sample has a
similar redshift coverage and contains 79,091 clusters, only about
25% of the galaxy sample size used by <cit.>. We detect
the BAO signal by $3.7\sigma$ and get a distance measurement of $D_v
(z=0.331) r_d^{fid}/r_d = 1261.5 \pm 48~\textrm{Mpc}$ with a measuring
accuracy of 3.8%. This implies a potential economical way to study
the large-scale structures in the future. Spectroscopic observations
are very time consuming especially for faint galaxies. When doing the
large-scale structure studies using clusters, spectroscopic redshifts
are not necessary for every galaxy. We can identify galaxy clusters
from photometry survey data first, and does the spectroscopic
follow-up for BCGs which are bright and can be easily observed. A
much smaller sample of clusters can provide a fairly accurate
measurement to the cosmological parameters too. We noticed that after
this paper was submitted, <cit.> calculated the
2-point correlation function using the cluster catalog presented by
<cit.> and got a distance measurement consistent with ours.
§ THE 2D CORRELATION FUNCTION
We calculate the 2D correlation function of the 79,901 clusters
following the same estimator and same weighting method described by
Equation <ref> and Equation <ref>. The result is
shown in Figure <ref>, where $\pi$ is the separation between
two clusters along the line-of-sight and $\sigma$ is the separation
across the line-of-sight. The faint BAO ring appears at the scale of
$r \sim 105\hMpc$.
The 2D correlation function of the 79,901 clusters. The
correlation function is binned in 2$\hMpc$ bins.
2D correlation function with the best-fit theoretical
correlation function as white contours.
Following <cit.> and <cit.>
we build a theoretical 2D correlation
\begin{equation}
\begin{split}
\xi_{\textrm{template}}(\sigma,\pi) =&\xi_0^{\textrm{template}}(r)P_0(\mu)+
\xi_2^{\textrm{template}}(r)P_2(\mu)\\
\end{split}
\end{equation}
\begin{equation}
\label{eq:theory-2d_1}
\xi_0^{\textrm{template}}(r)=\left(1+\frac{2\beta}{3}+\frac{\beta^2}{5}\right)\xi(r),
\end{equation}
\begin{equation}
\xi_2^{\textrm{template}}(r)=\left(\frac{4\beta}{3}+\frac{4\beta^2}{7}\right)[\xi(r)-\overline{\xi}(r)],
\end{equation}
\begin{equation}
\xi_4^{\textrm{template}}(r)=\frac{8\beta^2}{35}\left[\xi(r)+\frac{5}{2}\overline{\xi}(r)-\frac{7}{2}\overline{\overline{\xi}}(r)\right],
\end{equation}
where $r=\sqrt{\sigma^2+\pi^2}$, $\mu$ is the cosine of the angle
between the direction of cluster and the LOS,
$\beta=\Omega_m^{0.55}/b$. $P_0(\mu)= 1$,
$P_2(\mu) = \frac{1}{2}\left(3\mu^2-1\right)$ and $P_4(\mu) = \frac{1}{8}
\left(35\mu^4-30\mu^2+3\right)$ are the Legendre polynomials and
\begin{equation}
\overline{\xi}(r)=\frac{3}{r^3}\int^r_0{\xi(r')r'^2dr'},
\end{equation}
\begin{equation}
\label{eq:theory-2d_2}
\overline{\overline{\xi}}(r)=\frac{5}{r^5}\int^r_0{\xi(r')r'^4dr'},
\end{equation}
where $\xi(r)$ is the theoretical correlation function generated from
the matter power spectrum provided by the camb package using
the same cosmological parameters adopted by the theoretical two-point
correlation function. Finally, the model correlation function is
given by:
\begin{equation}
\label{eq:theory-2d}
\xi_{\textrm{model}}(\sigma, \pi)=b^2\xi_{\textrm{template}}(\alpha_{\sigma}\sigma, \alpha_{\pi}\pi)+\frac{a_1}{r^2}+\frac{a_2}{r}+a_3,
\end{equation}
where $\alpha_\sigma, \alpha_\pi, a_1, a_2, a_3$ and the bias $b$ are
free parameters, $a_1, a_2, a_3$ and $b$ are marginalized.
By fitting this model to the result in Figure <ref>, we neglect
the component of `Finger-of-God' <cit.> which arises at
the small scales <cit.>,
and focus on the feature of BAO ring at the scale range of $40\hMpc
\leq r \leq 150\hMpc$ in the parameter space of $0.80 \leq
\alpha_\sigma \leq 1.20$ and $0.80 \leq \alpha_\pi \leq 1.20$. We
find the best-fit scale parameters of $\alpha_{\sigma} = 1.02 \pm
0.06$ and $\alpha_{\pi}=0.94 \pm 0.10$, respectively. By replacing
the theoretical correlation function $\xi(r)$ with a no-wiggle
correlation function $\xi^{nw}(r)$ in the
Equations <ref> to <ref>, we build a
no-wiggle 2D correlation function model, and find the difference of
the fitting $\chi^2$ between the models with and without baryon
feature is $\Delta \chi^2 = 3.4$ which provides a BAO ring detection
confidence of $1.8\sigma$. The best-fit model correlation function is
plotted as contours with the cluster correlation function in
Figure <ref>.
§ CONCLUSIONS
We build a galaxy cluster sample based on the updated cluster catalog
published by <cit.>, which contains 79,901 clusters in the
redshift range of $z \leq 0.5$ with a mean redshift $\overline{z} =
0.331$. All these clusters have spectroscopic redshift measurements
from the SDSS DR12 data <cit.>.
We calculate the 2-point correlation function of the cluster sample
with a weight of cluster mass and sample completeness. The weighting
algorithm not only corrects the selection bias introduced by the
cluster identifying process but also enhances the BAO signal on the
final correlation function. A baryon acoustic peak is detected at the
scale of $r \sim 105\hMpc$, with a detection confidence of $3.7
\sigma$. This is the first time to detect a significant BAO signal
using a galaxy cluster sample. By fitting the observed correlation
function using a $\Lambda$CDM model, we find a constraint of $\alpha =
0.969 \pm 0.037$ and $D_v (z=0.331) r_d^{fid}/r_d = 1261.5 \pm
48~\textrm{Mpc}$, which show a great consistency with the fiducial
cosmology obtained by the Planck 2015 data.
We also calculate the 2D correlation function of the cluster
sample. The faint BAO ring emerges at the scale of $r \sim
105\hMpc$. By fitting the correlation function using a theoretical 2D
correlation function, we detect the BAO ring with a detection
confidence of $1.8\sigma$. Though it is not good enough to detect the
BAO feature in the separated two directions, we get the constraint on
the distance parameters of $\alpha_{\sigma} = 1.02 \pm 0.06$ and
$\alpha_{\pi} = 0.94 \pm 0.10$.
We conclude that the BAO detection via spectroscopically observed BCGs
can easy the survey job, because one can find galaxy clusters first
via photometric data, and then do spectroscopic observations for a
much smaller sample of galaxies.
We thank X. Y. Gao, G. B. Zhao, Y. T. Wang and F. Beutler for useful
discussions and comments. The authors are supported by the National
Natural Science Foundation of China (11473034). TH and ZLW are also
supported by the Young Researcher Grant of National Astronomical
Observatories, Chinese Academy of Sciences.
§ A COMPARISON OF COVARIANCE MATRICES ESTIMATED BY THE LOG-NORMAL AND JACKKNIFE METHODS
The jackknife method estimates the covariance matrix by making
sub-samples based on the original data catalog (internal estimate),
which is somehow different from the external estimate based on
N-body simulations or the log-normal realizations. By
comparing the covariance matrix estimated of the internal and external
methods on the scales of $0.1 - 40\hMpc$, <cit.> found
the jackknife method overestimates the variance on small scales of
$\lesssim 2-3\hMpc$, but it works fine on larger scales of $\gtrsim
10\hMpc$. On the BAO scales of $\sim 100\hMpc$, <cit.>
concluded that the jackknife error is noisier and larger than the
log-normal error for the 6dFGS galaxy sample. Here we compare the
covariance matrices of the correlation function estimated by the
log-normal and jackknife methods.
The correlation function errors estimated by the log-normal
method (solid line) and the jackknife method (dashed line)
The correlation matrix of the log-normal errors and jackknife
We obtain the jackknife covariance matrix by dividing the sky area
into 32 disjoint sub-regions, each sub-region has approximately the
same area with others. The jackknife method is found to be robust when
changing the number of jackknife sub-samples
<cit.>. The 32 jackknife sub-samples are built by
removing the clusters in one sub-region, ensuring that each sub-region
is removed in one sub-sample only. The correlation function is
calculated for each sub-sample following Equation <ref>. The
covariance matrix is then built up as:
\begin{equation}
\label{eq:err_jack}
\end{equation}
where $N=32$ is the number of sub-samples, $\xi^{k}_{i}$ is the
correlation function value of the $k^{th}$ sub-sample at the $i^{th}$
bin of $r$ values, and $\overline{\xi}_{i}$ represents the mean value
of the all 32 sub-samples at the $i^{th}$ bin.
We show the jackknife error in Figure <ref> together with
the log-normal error as a comparison. The jackknife error is found to
be larger than the log-normal error in most of the bins, it is also
noisier than the log-normal error. Besides the diagonal term of the
covariance matrix, we also show the full matrix in
Figure <ref>. The covariance estimated by the
log-normal method is much smoother than the one estimated by the
jackknife method. The elements plotted in
Figure <ref> are defined as:
\begin{equation}
\end{equation}
where $C$ is the covariance matrix.
We noticed that <cit.> also compared the covariance
matrices of the jackknife method and the log-normal method using a
galaxy cluster sample with SDSS III spectroscopic redshift, and they
found a similar conclusion as ours.
|
1511.00324
|
By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. By extending some results for relative differential cohomology we prove an excision theorem for differential cohomology. We further establish Pontryagin duality for differential cohomology: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology.
*Report no.: EMPG–15–16
*Keywords: Cheeger-Simons differential characters, differential cohomology, relative differential cohomology, differential cohomology with compact support, smooth Pontryagin duality
*MSC 2010: 53C08, 55N20, 57R19
§ INTRODUCTION AND SUMMARY
The aim of this paper is to develop a notion of differential
cohomology with compact support and to discuss smooth Pontryagin
dualities for differential cohomology. Differential cohomology is a family of
functors $\dH^k(-;\bbZ): \Man^\op \to \Ab$, for $k \geq 0$,
from the category of smooth manifolds to the category of Abelian groups that comes together with four natural
transformations comparing it with certain groups of differential forms and
also with smooth singular cohomology (with coefficients in $\bbZ$ and $\bbT=\bbR/\bbZ$).
The differential cohomology groups $\dH^k(M;\bbZ)$ of
a smooth manifold $M$ may thus be regarded as a refinement of the smooth
singular cohomology groups $\H^k(M;\bbZ)$ with integer coefficients
by differential forms. As a classical geometric example, the second differential cohomology group $\dH^2(M;\bbZ)$
is a refinement of the Picard group of isomorphism classes of
Hermitean line bundles on $M$ to the group of isomorphism classes of Hermitean line bundles with connection.
From this perspective, it seems natural to expect differential cohomology
with compact support to be a family of functors $\dH^k_{\rm c}(-;\bbZ):\Man_{m,\emb} \to \Ab$, for
$k\geq 0$, from the category of $m$-dimensional manifolds with smooth open embeddings as morphisms
to the category of Abelian groups that comes together with natural transformations
comparing it to compactly supported differential forms and cohomology with compact support.
The notion of compactly supported differential cohomology that we develop in this paper
is easily seen to satisfy this property: It is functorial with respect to smooth open embeddings and it comes
together with four natural transformations similar to the ones of ordinary differential cohomology.
Thus the compactly supported differential cohomology groups $\dH_{\rm c}^k(M;\bbZ)$
of a manifold refine the compactly supported cohomology groups $\H^k_{\rm c}(M;\bbZ)$
with $\bbZ$-coefficients by compactly supported differential forms.
In the classical geometric example, the group $\dH_{\rm c}^2(M;\bbZ)$ describes
isomorphism classes of Hermitean line bundles with connection on $M$ and
a parallel section outside some compact subset.
By now there exist several different models for differential cohomology,
i.e. several different constructions of the functors $\dH^k(-;\bbZ): \Man^\op \to \Ab$,
given by differential characters <cit.>, smooth Deligne cohomology <cit.>,
differential cocycles <cit.> and de Rham-Federer currents <cit.>.
These models are known to be related by unique natural equivalences <cit.>.
In the present paper we use the original model of differential characters as
established by Cheeger and Simons in <cit.>, which are $\bbT$-valued group characters on the Abelian group $Z_{k-1}(M)$
of singular cycles in $M$ that satisfy a certain smoothness condition.
An interesting alternative perspective would be to use the
more abstract homotopy theoretical approach to differential cohomology,
see e.g. <cit.>. We expect that some of our results,
e.g. on functorial properties of various constructions and the excision theorem for differential cohomology,
are intrinsic properties of this framework and could be shown with less arguments.
However, we decided to use the more traditional Cheeger-Simons approach <cit.>
in order to avoid the rather technical tools of homotopy theory.
To introduce differential characters with compact support, we follow the
well-known construction of compactly supported cohomology as the colimit
of the relative cohomology functor $\H^\sharp(M,M\setminus -;G)$, with $G$ an Abelian group,
over the directed set $\mcK_M$ of compact subsets of $M$.
For this construction we need an appropriate notion of differential cohomology relative to a
smooth submanifold $S \subseteq M$ (possibly with boundary).
As explained in <cit.>, there are two different such notions in the realm of
differential characters: They arise from the two different ways to define relative
(de Rham and singular) cohomology as the cohomology of the mapping cone complex
of the inclusion $i_S:S \hookrightarrow M$ or as the cohomology of the subcomplex of
forms (or cochains) vanishing outside $S \subseteq M$.
Thus relative differential cohomology may be defined as the group of differential
characters on either cycles of the mapping cone complex or on relative cycles.
Some confusion might arise from the fact that what are called
relative differential characters in the literature <cit.>
are not differential characters on relative cycles but characters on mapping cone cycles.
In <cit.> it is shown that the group of differential characters on relative cycles
$\dH^k(M,S;\bbZ)$ is a subgroup of the group of relative differential characters.
Elements of this subgroup are called parallel relative differential characters
in <cit.> for geometric reasons. To avoid confusion, in the present paper we will
only use the groups $\dH^k(M,S;\bbZ)$ of differential characters on relative cycles.
Part of the present paper generalizes some results from <cit.> for differential characters on relative cycles
to a less restrictive setting: We allow arbitrary embedded submanifolds (possibly with boundary),
whereas in <cit.> only closed submanifolds are taken into account. Restricting the consideration to properly
embedded submanifolds, we immediately recover the results from <cit.> with
basically the same arguments.
In the course of introducing differential cohomology with compact support we also
establish the excision property for relative differential cohomology:
For an open subset $O \subseteq M$ and a closed subset $C \subseteq M$ such that
$C \subseteq O$, the morphism $(O,O \setminus C) \to (M,M \setminus C)$ in
the category of pairs induces an isomorphism
\begin{equation*}
\dH^k(M,M \setminus C;\bbZ) \simeq \dH^k(O,O \setminus C;\bbZ)
\end{equation*}
in differential cohomology. To the best of our knowledge, this property has
not been discussed rigorously in the literature so far, although it might have been
conjectured by experts in the field.
This isomorphism is important for establishing functoriality $\dH^k_{\rm c}(-;\bbZ):\Man_{m,\emb} \to \Ab$
of compactly supported differential cohomology with respect to open embeddings of $m$-dimensional manifolds.
This paper is organized as follows: In Section <ref>
we review some requisite preliminaries
on smooth singular (co)homology groups and their relative versions, compactly supported cohomology
and the Cheeger-Simons model for differential cohomology.
In Section <ref> we introduce and study
differential characters on relative cycles, which provide the model for relative differential cohomology used in this paper.
We prove in Theorem <ref> that, for any submanifold $S\subseteq M$ (possibly with boundary),
the group of differential characters on relative cycles $\dH^k(M,S;\bbZ)$ fits
into a commutative diagram involving a short exact sequence. For generic submanifolds $S$,
this diagram is an incomplete analog of the usual diagram for
(absolute) differential cohomology. In Theorem <ref> we shall prove that
for the case where $S\subseteq M$ is properly embedded, the incomplete diagram
can be extended to a full diagram of short exact sequences.
The incomplete diagram in Theorem <ref> is however enough to
prove an excision theorem for differential cohomology in Subsection <ref>.
We then show in Subsection <ref> that the graded group of differential characters on relative cycles
$\dH^\sharp(M,S;\bbZ)$ is a module over the differential cohomology ring $\dH^\sharp(M;\bbZ)$.
In Section <ref> we define the groups $\dH^k_{\rm c}(M;\bbZ)$ of differential characters
with compact support as a colimit of the relative differential cohomology functor.
In Theorem <ref> we obtain an analogue of the usual differential cohomology diagram for the compactly supported case.
We further prove that compactly supported differential cohomology forms a family of
functors $\dH^k_{\rm c}(-;\bbZ):\Man_{m,\emb} \to \Ab$, for
$k\geq 0$, from the category of $m$-dimensional manifolds with smooth open embeddings as morphisms
to the category of Abelian groups and that $\dH^\sharp_{\rm c}(M;\bbZ)$ is a module
over the differential cohomology ring $\dH^\sharp(M;\bbZ)$.
We compare our construction of compactly supported differential characters
with earlier results in <cit.>, see in particular Remark <ref>.
In Section <ref> we establish smooth Pontryagin duality for differential cohomology.
Similar results were proven in <cit.> by using (compactly supported) de Rham-Federer characters.
The main results here are the following Pontryagin dualities which are proven in Theorem <ref>:
For any oriented $m$-dimensional manifold $M$, we obtain a natural isomorphism
\dH^{m-k+1}(M;\bbZ) \stackrel{\simeq}{\longrightarrow} \dH^k_{\rm c}(M;\bbZ)^\star_\infty~
between the differential cohomology group $\dH^{m-k+1}(M;\bbZ)$
and the smooth Pontryagin dual $\dH^k_{\rm c}(M;\bbZ)^\star_\infty$ of the compactly supported
differential cohomology group $\dH^k_{\rm c}(M;\bbZ)$.
If moreover $M$ is of finite-type, we can also interchange
the roles of ordinary and compactly supported differential cohomology to get isomorphisms
\dH^{m-k+1}_{\rm c}(M;\bbZ) \stackrel{\simeq}{\longrightarrow} \dH^k(M;\bbZ)^\star_\infty~.
In <cit.> these results are applied to analyze
dualities in (higher) quantum Abelian gauge theories and to the quantization of self-dual fields.
§ COHOMOLOGICAL PRELIMINARIES
We briefly recall some background material which is used extensively throughout the rest of this paper, including smooth singular (co)homology together with their relative versions,
a colimit prescription for defining cohomology with compact support from relative cohomology,
and Cheeger-Simons differential characters.
In the following all manifolds will be assumed to be finite-dimensional, smooth,
paracompact and Hausdorff. Sometimes we shall also consider manifolds with a (smooth) boundary.
For some of our constructions and results in Section <ref>
we demand further conditions (such as connectedness, orientability or existence of finite good covers),
which will be stated explicitly where needed.
§.§ Smooth singular (co)homology and its relative version
Let $M$ be a manifold (possibly with boundary).
We denote by $C_\sharp(M)$ the chain complex of smooth singular chains on $M$ with $\bbZ$-coefficients.
The boundary homomorphism is denoted by $\del_k: C_k(M) \to C_{k-1}(M)$ and we shall frequently omit
the subscript $_{k}$ as it will be clear from the context.
The Abelian groups of $k$-cycles and $k$-boundaries on $M$ are given by
$Z_k(M) := \ker \del_k$ and $B_k(M) := \im \del_{k+1}$, respectively.
The smooth singular $k$-th homology group of $M$ is then defined as the quotient
$\H_k(M):=Z_k(M) / B_k(M)$.
For our purposes, we also have to consider homology on $M$
relative to a submanifold $S \subseteq M$ (possibly with boundary),
which is obtained by identifying all chains with support inside $S$ with $0$.
More precisely, the inclusion $S \subseteq M$ allows us to consider
the chain complex $C_\sharp(S)$ of smooth singular chains on $S$
as a subcomplex of $C_\sharp(M)$.
The complex of smooth singular chains on $M$ relative to $S$ is then defined
as the quotient $C_\sharp(M,S) := C_\sharp(M) / C_\sharp(S)$.
The boundary homomorphism is denoted by $\del_k: C_k(M,S) \to C_{k-1}(M,S)$.
Notice that any $C_k(M,S)$ is a free Abelian group, even though
it is defined as a quotient.[
To prove this statement, we observe that the short exact sequence
$0 \to C_k(S) \to C_k(M) \to C_k(M,S) \to 0$ can be split
by the unique homomorphism $C_k(M)\to C_k(S)$
which sends any $k$-simplex $\sigma \in C_k(M)$ with image contained in $S$
to the corresponding $\sigma\in C_k(S)$ and any other $k$-simplex to $0$.
It follows that $C_k(M,S)$ is a direct summand of the free Abelian group $C_k(M)$
and hence a free Abelian group as well.
Similarly to above, the Abelian groups of relative $k$-cycles $Z_k(M,S)$
and relative $k$-boundaries $B_k(M,S)$ are respectively given by
the kernel and image of the boundary homomorphism in $C_\sharp(M,S)$.
The relative $k$-th homology group is defined as the quotient
$\H_k(M,S):= Z_k(M,S) / B_k(M,S)$.
Let $G$ be an arbitrary Abelian group. The cochain complex $C^\sharp(M;G)$ of
$G$-valued smooth singular cochains on $M$ is defined by $C^k(M;G) := \Hom(C_k(M),G)$
together with the coboundary homomorphism $\cdel_k := \Hom(\del_{k+1},G): C^k(M;G) \to C^{k+1}(M;G)$.
The Abelian groups of $G$-valued $k$-cocycles and $k$-coboundaries on $M$ are given by
$Z^k(M;G) := \ker \cdel_k$ and $B^k(M;G) := \im \cdel_{k-1}$, respectively.
The $G$-valued smooth singular $k$-th cohomology group of $M$ is then defined as the quotient
$\H^k(M;G) := Z^k(M;G) / B^k(M;G)$. The relative version $C^\sharp(M,S;G)$ of the cochain complex
is defined analogously.
We set $C^k(M,S;G) := \Hom(C_k(M,S),G)$ and
$\cdel_k := \Hom(\del_{k+1},G): C^k(M,S;G) \to C^{k+1}(M,S;G)$.
Moreover, $G$-valued relative $k$-cocycles are defined as
$Z^k(M,S;G) := \ker \cdel_k$, $G$-valued relative $k$-coboundaries
as $B^k(M,S;G):= \im \cdel_{k-1}$ and the
$G$-valued relative smooth singular $k$-th cohomology group as
the quotient $\H^k(M,S;G) := Z^k(M,S;G) / B^k(M,S;G)$.
Given any chain complex $C_\sharp$ of free Abelian groups
and a short exact sequence $0 \to F \to G \to H \to 0$ of Abelian groups,
there exists a short exact sequence of cochain complexes
0[r] (C_♯,F) [r] (C_♯,G) [r] (C_♯,H) [r] 0 .
By a well-known result in homological algebra, see e.g. <cit.>,
the cohomology groups of these cochain complexes then fit into a long exact sequence.
Applying this result to the chain complexes $C_\sharp(M)$ and
$C_\sharp(M,S)$ we obtain the long exact sequences
\begin{align}\label{eqLESH}
\cdots \ar[r]
& \H^{k-1}(M;H) \ar[r]^-\beta
& \H^k(M;F) \ar[r]
& \H^k(M;G) \ar[r]
& \H^k(M;H) \ar[r]^-\beta & \cdots~~,
\\[4pt]
\label{eqLESRelH}
\cdots \ar[r]
& \H^{k-1}(M,S;H) \ar[r]^-\beta
& \H^k(M,S;F) \ar[r]
& \H^k(M,S;G) \ar[r]
& \H^k(M,S;H) \ar[r]^-\beta & \cdots~~,
\end{align}
where $\beta$ denotes the connecting homomorphisms.
In the following we will often refer to the functorial behavior of the absolute and relative
(co)homology groups. Let us introduce the relevant categories:
The objects are manifolds and the morphisms are smooth maps.
$\Man_{m,\emb}$: The subcategory of $\Man$ whose objects are $m$-dimensional manifolds
and morphisms are open embeddings.
The objects are pairs $(M,S)$ consisting of an object $M$ in $\Man$ and a submanifold $S \subseteq M$
(possibly with boundary) and the morphisms $f: (M,S) \to (M^\prime,S^\prime\,)$
are those morphisms $f: M \to M^\prime$ in $\Man$ which satisfy $f(S) \subseteq S^\prime$.
$\PePair$: The full subcategory of $\Pair$ whose objects
$(M,S)$ are such that $S \subseteq M$ is a properly embedded submanifold (possibly with boundary).
The objects are directed sets and the morphisms are functions preserving the preorder relation.
Alternatively, interpreting a directed set as a (small) category (with morphisms specified by the preorder relation),
we can interpret $\DSet$ as the full subcategory of the category
of small categories $\Cat$ whose objects are directed sets.
The objects are Abelian groups and the morphisms are group homomorphisms.
The objects are chain complexes of Abelian groups and the morphisms are chain maps.
Interpreting cochain complexes $C^\sharp$ canonically as chain
complexes via the reflection $C^k \to C^{-k}$, we observe that
absolute and relative smooth singular (co)chain complexes are functors
\begin{align}
C_\sharp(-): \Man \longrightarrow \Ch(\Ab)~, && C_\sharp(-): \Pair \longrightarrow \Ch(\Ab)~, \\[4pt]
C^\sharp(-;G): \Man^\op \longrightarrow \Ch(\Ab)~, && C^\sharp(-;G): \Pair^\op \longrightarrow \Ch(\Ab)~.
\end{align}
In fact, simplices in $M$ can be pushed forward along $f: M \to M^\prime$ and such a push-forward
along a morphism $f: (M,S) \to (M^\prime,S^\prime\,)$ in $\Pair$ sends simplices in $S$ to simplices in $S^\prime$.
Absolute and relative (co)homology inherit their functorial behavior from these functors, i.e.
\begin{align}
\H_k(-): \Man \longrightarrow \Ab~, && \H_k(-): \Pair\longrightarrow \Ab~, \\[4pt]
\H^k(-;G): \Man^\op \longrightarrow \Ab~, && \H^k(-;G): \Pair^\op \longrightarrow \Ab~.
\end{align}
§.§ Smooth singular cohomology with compact support
Following <cit.>, we introduce smooth singular cohomology with compact support
by means of a colimit prescription. Let
\begin{equation}\label{eqCompSubsetFunctor}
\mcK: \Man \longrightarrow \DSet~
\end{equation}
be the functor which assigns to any manifold $M$ the directed set $\mcK_{M} := \{K\subseteq M \text{ compact}\}$
(with preorder relation given by subset inclusion) and to any smooth map
$f: M \to M^\prime$ the morphism $\mcK_{f}: \mcK_{M} \to \mcK_{M^\prime}\,,~ K \mapsto f(K)$ of directed sets.
Interpreting $\mcK_{M}$ as a (small) category, we obtain a functor
\begin{equation}\label{eqCompSubsetToPairFunctor}
(M,M \setminus -): \mcK_M \longrightarrow \Pair^\op~,
\end{equation}
where the target category is $\Pair^\op$ because we take complements of subsets.
Given an Abelian group $G$, we can precompose the
relative smooth singular cohomology functor
$\H^k(-;G): \Pair^\op \to \Ab$ with the functor (<ref>)
and obtain $\H^k(M,M \setminus -;G): \mcK_M \to \Ab$.
We then define the $G$-valued compactly supported smooth singular
cohomology group of $M$ as the colimit of this functor, i.e.
\begin{equation}\label{eqHcdef}
\H^k_{\rm c}(M;G) := \colim \big(\H^k(M,M \setminus -;G): \mcK_M \to \Ab\big)~.
\end{equation}
Similarly to $\mcK_M$, we can define the directed set
$\mcO_M^{\rm c} := \{ O \subseteq M \text{ open}~:~\overline O \in \mcK_M\,,~\del \overline O \text{ smooth}\}$
of relatively compact open subsets with smooth boundary.
As for $\mcK_M$, the preorder relation on $\mcO_M^{\rm c}$ is given by subset inclusion.
By construction, the complement $M \setminus O$ of any $O \in \mcO_M^{\rm c}$ is a properly embedded submanifold
and thus the assignment $O \mapsto (M,M\setminus O)$ defines a functor
\begin{equation}\label{eqCompSubsetToPairFunctor_2}
(M,M \setminus -): \mcO_M^{\rm c} \longrightarrow \Pair^\op_{\mathsf{pe}}~.
\end{equation}
Composing this functor with the embedding $\Pair^\op_{\mathsf{pe}}\to \Pair^\op$
and the relative cohomology functor $\H^k(-;G): \Pair^\op \to \Ab$,
we obtain $\H^k(M,M \setminus -;G): \mcO_M^{\rm c} \to \Ab$.
We shall now show that the colimit of this functor provides an equivalent
definition of compactly supported cohomology:
Introducing the directed subset $\mcU_M := \mcK_M \cup \mcO_M^{\rm c}$
of the power set of $M$ yields another functor $\H^k(M,M \setminus -;G): \mcU_M \to \Ab$.
Because both $\mcK_M$ and $\mcO_M^{\rm c}$ are cofinal in $\mcU_M$
we obtain the chain of isomorphisms
\begin{equation}
\H^k_{\rm c}(M;G) \simeq \colim \big(\H^k(M,M \setminus -;G): \mcU_M \to \Ab\big)
\simeq \colim \big(\H^k(M,M \setminus -;G): \mcO_M^{\rm c} \to \Ab\big)~,
\end{equation}
which provides alternative definitions of $\H^k_{\rm c}(M;G) $.
We now prove that $G$-valued compactly supported smooth singular
cohomology is a functor
\begin{equation}
\H^k_{\rm c}(-;G): \Man_{m,\emb} \longrightarrow \Ab~.
\end{equation}
Given an open embedding $f: M \to M^\prime$ between $m$-dimensional manifolds,
there is a factorization $f = i \circ g$ into the inclusion $i : f(M) \to M^\prime$
and a diffeomorphism $g: M \to f(M)$.
Since $f(M) \setminus K \subseteq M^\prime\setminus K$, for any $K\in \mcK_{f(M)}$,
the inclusion $i$ defines a natural transformation
$i^\ast: \H^k(M^\prime,M^\prime \setminus -;G) \circ \mcK_i \Rightarrow \H^k(f(M),f(M) \setminus -;G)$
between functors from $\mcK_{f(M)}$ to $\Ab$.
By a standard excision argument for cohomology,
we find that $i^\ast$ is a natural isomorphism and
we denote its inverse by
\begin{equation}
(i^\ast)^{-1}: \H^k(f(M),f(M) \setminus -;G) \Longrightarrow \H^k(M^\prime,M^\prime \setminus -;G) \circ \mcK_i~.
\end{equation}
Furthermore, since $g$ is a diffeomorphism, we immediately get a natural isomorphism
$g^\ast: \H^k(f(M),f(M) \setminus -;G) \circ \mcK_g \Rightarrow \H^k(M,M \setminus -;G)$
between functors from $\mcK_{M}$ to $\Ab$ and we denote its inverse by
\begin{equation}
(g^\ast)^{-1}: \H^k(M,M \setminus -;G) \Longrightarrow \H^k(f(M),f(M) \setminus -;G) \circ \mcK_g~.
\end{equation}
We have thereby shown that $f^\ast = g^\ast \circ (i^\ast\, \mcK_g):
\H^k(M^\prime,M^\prime \setminus -;G) \circ \mcK_f \Rightarrow \H^k(M,M \setminus -;G)$
is a natural isomorphism with inverse denoted by
\begin{equation}
(f^\ast)^{-1} := ((i^\ast)^{-1}\, \mcK_g) \circ (g^\ast)^{-1}:
\H^k(M,M \setminus -;G) \Longrightarrow \H^k(M^\prime,M^\prime \setminus -;G) \circ \mcK_f~.
\end{equation}
By the universal property of the colimit, the natural transformation $(f^\ast)^{-1}$
induces a unique homomorphism $f_\ast: \H^k_{\rm c}(M;G) \to \H^k_{\rm c}(M^\prime;G)$.
It is easy to check that $\H^k_{\rm c}(-;G): \Man_{m,\emb} \to \Ab$ defined in this way is a functor.
We recall that assigning to diagrams in $\Ab$
over a directed set their colimits is an exact functor because $\Ab$ is a Grothendieck category.
Applying this observation to (<ref>), with $S=M\setminus K$ running over $K\in \mcK_M$,
we obtain a long exact sequence
\begin{equation}\label{eqLESHc}
\xymatrix{
\cdots \ar[r]
& \H^{k-1}_{\rm c}(M;H) \ar[r]^-\beta
& \H^k_{\rm c}(M;F) \ar[r]
& \H^k_{\rm c}(M;G) \ar[r]
& \H^k_{\rm c}(M;H) \ar[r]^-\beta & \cdots~
\end{equation}
for compactly supported cohomology.
§.§ Cheeger-Simons differential characters
The starting point for our investigations is the graded commutative ring
of Cheeger-Simons differential characters <cit.>,
which was later recognized as a model for differential cohomology <cit.>.
Different yet equivalent models have been developed in terms of
smooth Deligne cohomology <cit.>, differential cocycles <cit.>
and de Rham-Federer currents <cit.>.
It was proven in <cit.> that differential cohomology is uniquely determined up to unique natural equivalences.
See also <cit.> for a more abstract homotopy theoretic
approach to (generalized) differential cohomology theories.
A degree $k$ Cheeger-Simons differential character on a manifold $M$
is a homomorphism $h : Z_{k-1}(M)\to \bbT$ into the circle group $\bbT := \bbR/\bbZ$
for which there exists a differential form $\omega_h\in\Omega^k(M)$ such that
h(γ) = ∫_γ ω_h , ∀γ∈C_k(M) .
We denote the Abelian group of Cheeger-Simons differential characters by $\dH^k(M;\bbZ)$.
Given any $h \in \dH^k(M;\bbZ)$, it is easy to show that
$\omega_h \in \Omega^k(M)$ is uniquely specified by
(<ref>) and that it has integral periods.
Introducing the Abelian group of $k$-forms with integral periods
\begin{equation}
\OmegaZ^k(M)
:= \left\{\omega \in \Omega^k(M)~:~ \int_z\, \omega \in \bbZ\,,\; \forall z \in Z_k(M)\right\}~,
\end{equation}
we obtain the curvature homomorphism
\begin{equation}
\cu: \dH^k(M;\bbZ) \longrightarrow \OmegaZ^k(M)~,\qquad h \longmapsto \omega_h~.
\end{equation}
Any $u \in \H^{k-1}(M;\bbT)$ may be interpreted
as an element of $\dH^k(M;\bbZ)$ via the homomorphism
\begin{equation}\label{eqInclFlat}
\kappa: \H^{k-1}(M;\bbT) \longrightarrow \dH^k(M;\bbZ)
\end{equation}
called the “inclusion of flat fields”. It is constructed as follows:
Since $\bbT$ is divisible, the universal coefficient theorem for cohomology implies
that there exists a natural isomorphism $\H^{k-1}(M;\bbT) \simeq \Hom(\H_{k-1}(M),\bbT)$.
Given any $u \in \H^{k-1}(M;\bbT)$, via this isomorphism we regard it
as a homomorphism $u: \H_{k-1}(M) \to \bbT$ and define $\kappa\, u : Z_{k-1}(M)\to \bbT$
by precomposition with the quotient map $Z_{k-1}(M) \to \H_{k-1}(M)$.
By definition, the curvature of $\kappa\, u$ is $0$.
Because $Z_{k-1}(M)$ is a free Abelian group,
we can lift any $h \in \dH^k(M;\bbZ)\subseteq \Hom(Z_{k-1}(M),\bbT)$
to $\tilde h \in \Hom(Z_{k-1}(M),\bbR)$ along the quotient $\bbR \to \bbT$.
As a consequence of (<ref>), the cochain
$\int_\cdot\, \omega_h - \tilde h \circ \del\in C^k(M;\bbR)$
factors through the inclusion $\bbZ \to \bbR$
and hence there exists a unique integral cochain $c_h \in C^k(M;\bbZ)$ satisfying
$\tilde h \circ \del = \int_\cdot\, \omega_h - c_h$.
One easily checks that $\cdel c_h = 0$ and that the
cohomology class $[c_h] \in \H^k(M;\bbZ)$ is uniquely determined by $h$.
This defines the characteristic class homomorphism
\begin{equation}
\ch: \dH^k(M;\bbZ) \longrightarrow \H^k(M;\bbZ)~, \qquad h \longmapsto [c_h]~.
\end{equation}
Any differential form $A \in \Omega^{k-1}(M)$
defines a differential character $h_A \in \dH^k(M;\bbZ)$ by setting
\begin{equation}
h_A(z) = \int_z\, A \mod \bbZ~,
\end{equation}
for all $z \in Z_{k-1}(M)$. By Stokes' theorem
we observe that $h_A(\del \gamma) = \int_\gamma \, \dd A \mod \bbZ$, for all $\gamma \in C_k(M)$,
and hence that the curvature of $h_A$ is $\dd A$.
We further observe that $h_A$ has trivial characteristic class because
$\int_\cdot\, A \in \Hom(Z_{k-1},\bbR)$ is a lift of $h_A$ and that
$h_A$ is trivial for $A \in \OmegaZ^{k-1}(M)$.
This defines the topological trivialization homomorphism
\begin{equation}\label{eqRelTopTriv}
\iota: \frac{\Omega^{k-1}(M)}{\OmegaZ^{k-1}(M)} \longrightarrow \dH^k(M;\bbZ)~, \qquad [A] \longmapsto h_A~.
\end{equation}
It is shown in <cit.> that the group of
Cheeger-Simons differential characters
fits into the commutative diagram
\begin{equation}\label{eqDiffCharDia}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0\ar[d] \\
0 \ar[r] & \frac{\H^{k-1}(M;\bbR)}{\Hf^{k-1}(M;\bbZ)} \ar[r] \ar[d]
& \frac{\Omega^{k-1}(M)}{\OmegaZ^{k-1}(M)} \ar[r]^-\dd \ar[d]_-\iota
& \dd\Omega^{k-1}(M) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \H^{k-1}(M;\bbT) \ar[r]^-\kappa \ar[d]
& \dH^k(M;\bbZ) \ar[r]^-\cu \ar[d]_-\ch
& \OmegaZ^k(M) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \Ht^k(M;\bbZ) \ar[r] \ar[d]
& \H^k(M;\bbZ) \ar[r] \ar[d]
& \Hf^k(M;\bbZ) \ar[r] \ar[d] & 0 \\
& 0 & 0 & 0
\end{equation}
with all rows and columns short exact sequences.
The short exact sequences in the left column and in the bottom row
are obtained from the natural long exact sequence (<ref>) for cohomology
associated to the coefficient sequence $0 \to \bbZ \to \bbR \to \bbT \to 0$.
In the top row and in the right column, one uses the natural isomorphism
$\OmegaZ^k(M) / \dd \Omega^{k-1}(M) \simeq \Hf^k(M;\bbZ)$ which is a by-product of de Rham's theorem.
The assignment of the Abelian group $\dH^k(M;\bbZ)$ to each manifold $M$ defines a functor
\begin{equation}\label{eqDiffCharFunctor}
\dH^k(-;\bbZ): \Man^\op \longrightarrow \Ab~.
\end{equation}
In particular, any smooth map $f: M \to M^\prime$
induces a pull-back $f^\ast: \dH^k(M^\prime;\bbZ) \to \dH^k(M;\bbZ)$
by dualizing the push-forward $f_\ast: Z_{k-1}(M) \to Z_{k-1}(M^\prime\,)$ for cycles.
For any $h \in \dH^k(M^\prime;\bbZ)$, the homomorphism $f^\ast h : Z_{k-1}(M)\to\bbT$ fulfills
the condition (<ref>) with $\omega_{f^\ast h} = f^\ast \omega_h$.
Moreover, all homomorphisms in diagram (<ref>) are
natural transformations between functors from $\Man^\op$ to $\Ab$. In particular, we have
the natural transformations
\begin{align}
\cu: \dH^k(-;\bbZ) \Longrightarrow \OmegaZ^k(-)~, && \ch: \dH^k(-;\bbZ)\Longrightarrow \H^k(-;\bbZ)~, \\[4pt]
\iota: \frac{\Omega^{k-1}(-)}{\OmegaZ^{k-1}(-)} \Longrightarrow \dH^k(-;\bbZ)~,
&& \kappa: \H^{k-1}(-;\bbT) \Longrightarrow \dH^k(-;\bbZ)~.
\end{align}
The following examples are explained in detail in <cit.>:
For $k=1$, the Abelian group of differential characters $\dH^1(M;\bbZ)$
is canonically isomorphic to the Abelian group of circle-valued functions $C^\infty(M,\bbT)$;
for any $h\in C^\infty(M,\bbT)$, the curvature is $\cu(h) = \dd\log(h)$ and the
characteristic class is $\ch(h) = h^*[\bbT]$, where $[\bbT] \in \H^1(\bbT;\bbZ)$ denotes the fundamental class.
For $k=2$, the Abelian group of differential characters $\dH^2(M;\bbZ)$ describes
isomorphism classes of Hermitean line bundles with connection on $M$.
For $k=3$, the Abelian group of differential characters $\dH^3(M;\bbZ)$ describes
isomorphism classes of Abelian gerbes with connection on $M$.
Another source of examples of differential characters is provided
by a refinement of classical Chern-Weil theory <cit.>:
Let $G$ be a Lie group with finitely many connected components and $P \to M$
a principal $G$-bundle with connection $\theta$.
Denote by $F_\theta$ the curvature form of $\theta$.
Associated to an invariant polynomial $\lambda: \mathfrak g^k \to \bbR$ on the
Lie algebra and a corresponding universal characteristic class $u \in \H^{2k}(BG;\bbZ)$
is a differential character in $\dH^{2k}(M;\bbZ)$ with characteristic class $u(P)$
and curvature the Chern-Weil form $\lambda(F_\theta)$.
The differential character is uniquely determined by these properties and the
requirement of being natural with respect to connection preserving bundle morphisms.
The construction is reviewed in detail in <cit.> and also refined to relative
differential characters by taking into account the Chern-Simons
form of $(P,\theta)$ associated with $\lambda$.
*Ring structure:
The Abelian groups $\dH^\sharp(M;\bbZ)$ can be endowed with a
natural graded commutative ring structure
\begin{equation}\label{eqDiffCharRing0}
\cdot \,:\, \dH^k(M;\bbZ) \times \dH^l(M;\bbZ) \longrightarrow \dH^{k + l}(M;\bbZ)~, \qquad (h,h'\,) \longmapsto h \cdot h'\,~.
\end{equation}
Following <cit.>, the construction is as follows:
Choose a family of natural cochain homotopies
$B: \Omega^k(M) \times \Omega^l(M) \to C^{k+l-1}(M;\bbR)$,
for all $k,l\geq 0$, between the wedge product for differential forms
and the cup product for the corresponding cochains
such that $B(\omega,\theta) = (-1)^{k \,l}\, B(\theta,\omega)$,
for all $\omega \in \Omega^k(M)$ and $\theta \in \Omega^l(M)$.
An example of such a family of cochain homotopies is given in <cit.>
by means of iterated subdivisions, and the chain homotopy between subdivision and identity.
Two different choices of $B$ turn out to be naturally cochain homotopic,
see <cit.>.
Given $h \in \dH^k(M;\bbZ)$ and $h'\, \in \dH^l(M;\bbZ)$,
one defines their product $h \cdot h'\, \in \dH^{k+l}(M;\bbZ)$
as the homomorphism $Z_{k+l-1}(M)\to \bbT$ given by
\begin{align}
h \cdot h'\, & := \tilde h \smile \int_\cdot\, \cu\, h'\, + (-1)^k \, c_h \smile \tilde h'\, + B(\cu\, h,\cu\, h'\,) \mod \bbZ \\[4pt]
& \hphantom{:}= \tilde h \smile c_{h'}\, + (-1)^k \, \int_\cdot\, \cu\, h \smile \tilde h'\, + B(\cu\, h,\cu\, h'\,)
\mod \bbZ~,
\end{align}
where $\tilde h \in C^{k-1}(M;\bbR)$ and $\tilde h'\, \in C^{l-1}(M;\bbR)$
lift and extend $h$ and $h'\,$, respectively, while $c_h \in C^k(M;\bbZ)$ and $c_{h'}\, \in C^l(M;\bbZ)$
are cocycles such that $\cdel \tilde h = \int_\cdot\, \cu\, h - c_h$ and $\cdel \tilde h'\, = \int_\cdot\, \cu\, h'\, - c_{h'}$.
A proof that the map (<ref>) specified by (<ref>)
defines an associative and graded-commutative ring structure
can be found in <cit.> or in <cit.>.
Since both the cup product $\smile$ and the cochain homotopy $B$ are natural,
the expression (<ref>) defines a natural ring structure, i.e.
the pull-back of differential characters along a smooth map is a ring homomorphism.
The four natural transformations displayed in (<ref>)
are compatible with the ring structure, i.e.
\begin{align}
\cu(h \cdot h'\,) & = \cu\, h \wedge \cu\, h'\,~, & \ch(h \cdot h'\,) & = \ch\, h \smile \ch\, h'\,~,\\[4pt]
\iota\, [A] \cdot h'\, & = \iota([A \wedge \cu\, h'\,])~, & \kappa\, u \cdot h'\, & = \kappa(u \smile \ch\, h'\,)~,
\end{align}
for all $h \in \dH^k(M;\bbZ)$, $h'\, \in \dH^l(M;\bbZ)$,
$[A] \in \Omega^{k-1}(M) / \OmegaZ^{k-1}(M)$ and $u \in \H^{k-1}(M;\bbT)$.
The ring structure on $\dH^\sharp(M;\bbZ)$ provides
a construction of an isomorphism class of Hermitean line bundles with connection on $M$
out of two circle-valued functions $h_1, h_2 \in C^\infty(M,\bbT)$.
This bundle can be described explicitly as the pull-back along the product map
$(h_1, h_2): M \to \bbT^2$ of a universal line bundle with connection on the $2$-torus,
called the Poincaré bundle, see <cit.> and <cit.> for further details.
The construction of differential characters from classical Chern-Weil theory in Example <ref>
is multiplicative: Given two invariant polynomials $\lambda_1,\lambda_2$
on $\mathfrak g$ and corresponding universal characteristic classes
$u_1,u_2 \in \H^\sharp(BG;\bbZ)$, the differential character associated with
$\lambda_1 \cdot \lambda_2$ coincides with the product of the differential characters
from Example <ref>. Its characteristic class is the cup product $u_1(P) \smile u_2(P)$,
while its curvature is the wedge product $\lambda_1(F_\theta) \wedge \lambda_2(F_\theta)$
of the corresponding Chern-Weil forms.
§ DIFFERENTIAL CHARACTERS ON RELATIVE CYCLES
In this section we review a version of relative differential cohomology
which is used later to introduce compactly supported differential cohomology.
Recall that there are two different ways to define the smooth singular cohomology
of a manifold $M$ relative to a submanifold $S \subseteq M$ (possibly with boundary):
The first option is as the cohomology of the mapping cone complex of the inclusion $i_S:S \hookrightarrow M$
and the second option is as the cohomology of the cochain complex $C^\sharp(M,S;G) := \Hom(C_\sharp(M,S);G)$,
where $C_\sharp(M,S) := C_\sharp(M)/C_\sharp(S)$ is the quotient complex.
The latter version was discussed in more detail in Subsection <ref>.
A similar point of view can be taken for relative de Rham cohomology: It may be defined
as the cohomology of the mapping cone complex $\Omega^\sharp(i_S)$ of the inclusion $i_S:S \hookrightarrow M$ as in <cit.>
or as the cohomology of the subcomplex $\Omega^\sharp(M,S)$
of forms vanishing on $S$ as in <cit.>.
For a properly embedded submanifold $S \subseteq M$ (possibly with boundary),
taking into account the long exact cohomology sequences
arising from the relative/absolute exact sequences for $\Omega^\sharp(i_S)$ and for $\Omega^\sharp(M,S)$,
one concludes that both approaches give the same cohomology groups, although the complexes are different; explicitly, a five lemma argument shows that $\H^\sharp_\dR(M,S) \to \H^\sharp_\dR(i_S)$,
$[\omega] \mapsto [\omega,0]$ is an isomorphism.
Since differential cohomology is a refinement of smooth singular cohomology by differential forms,
the question arises whether to refine the relative cohomology $\H^\sharp(M,S;\bbZ)$ by the
mapping cone de Rham complex $\Omega^\sharp(i_S)$ or by the relative de Rham complex $\Omega^\sharp(M,S)$.
Differential characters based on the mapping cone complex were first introduced in <cit.>
and they were called relative differential characters.
Differential characters on relative cycles were first introduced in
<cit.>[In this reference it is assumed that $S \subseteq M$
is a closed submanifold, but the constructions and results directly generalize
to the case of properly embedded submanifolds.] and they
were called parallel relative differential characters.
These two versions of relative differential cohomology fit into diagrams
similar to (<ref>), provided that one considers properly embedded submanifolds $S\subseteq M$.
They also fit into long exact sequences relating absolute and relative differential cohomology groups,
see <cit.> and below. See <cit.> also for a comparison
between relative and parallel relative differential characters.
§.§ Definition and first properties
Let us begin by fixing
our notation for relative de Rham cohomology. Let $M$ be a manifold
and $S\subseteq M$ a submanifold (possibly with boundary).
We denote by
\begin{equation}\label{eqRelForms}
\Omega^k(M,S) := \{\omega \in \Omega^k(M)\,:\; \omega\vert_S = 0\}
\end{equation}
the Abelian group of $k$-forms vanishing on $S\subseteq M$ and by
\begin{equation}\label{eqRelFormsZ}
\Omega^k_\bbZ(M,S) := \Big\{\omega \in \Omega^k(M,S)\,:\;
\int_z\, \omega \in \bbZ\,,\; \forall z \in Z_k(M,S)\Big\}
\end{equation}
its subgroup of relative $k$-forms with integral periods.
The natural homomorphism $Z_k(M) \to Z_k(M,S)$ implies
that $\Omega^k_\bbZ(M,S)$ is a subgroup of $\Omega^k_\bbZ(M)$.
Since the exterior derivative $\dd$ preserves the
subgroup $\Omega^\sharp(M,S)\subseteq \Omega^\sharp(M)$,
the relative de Rham complex $(\Omega^\sharp(M,S),\dd)$
is a subcomplex of the usual de Rham complex $(\Omega^\sharp(M),\dd)$.
We denote the corresponding relative de Rham cohomology
by $\H^\sharp_\dR(M,S)$.
A degree $k$ differential character on relative cycles on a manifold $M$ with respect to $S\subseteq M$
is a homomorphism $h : Z_{k-1}(M,S)\to \bbT$
for which there exists a differential form $\omega_h\in\Omega^k(M)$ such that
h(γ) = ∫_γ ω_h , ∀γ∈C_k(M) .
We denote the Abelian group of differential characters on relative cycles by $\dH^k(M,S;\bbZ)$.
As in the case for (absolute) differential characters,
the form $\omega_h$ is uniquely determined by $h\in \dH^k(M,S;\bbZ)$.
Evaluating (<ref>) on $\gamma \in C_k(S)$
we obtain $\omega_h\vert_S = 0$ and evaluating
on $\gamma =z \in Z_k(M,S)$ it follows that $\int_z \, \omega_h \in \bbZ$.
This yields the curvature homomorphism for differential characters on relative cycles
\begin{equation}
\cu: \dH^k(M,S;\bbZ) \longrightarrow \OmegaZ^k(M,S)~, \qquad h \longmapsto \omega_h~.
\end{equation}
Any $u \in \H^{k-1}(M,S;\bbT)$ may be interpreted
as an element of $\dH^k(M,S;\bbZ)$ with vanishing curvature.
The argument is exactly the same as for the absolute case (see the text following (<ref>))
and we just have to replace all absolute (co)homology groups
with their relative analogues. The corresponding homomorphism
\begin{equation}\label{eqRelInclFlat}
\kappa: \H^{k-1}(M,S;\bbT) \longrightarrow \dH^k(M,S;\bbZ)
\end{equation}
is called the “inclusion of flat fields (on relative cycles)”.
Recalling that relative chains (and hence relative cycles)
form a free Abelian group, as argued in Subsection <ref>,
we can lift any $h \in \dH^k(M,S;\bbZ)$ to an element
$\tilde h \in \Hom(Z_{k-1}(M,S),\bbR)$ along the quotient $\bbR \to \bbT$.
As a consequence of (<ref>),
the relative cochain $\int_\cdot \, \omega_h - \tilde h \circ \del \in C^k(M,S;\bbR)$
factors through the inclusion $\bbZ \to \bbR$ and hence there exists a
unique integral relative cochain $c_h \in C^k(M,S;\bbZ)$ satisfying $\tilde h \circ \del = \int_\cdot\, \omega_h - c_h$.
One easily checks that $\cdel c_h = 0$
and that the relative cohomology class $[c_h] \in \H^k(M,S;\bbZ)$ is uniquely determined by $h$.
This defines the relative counterpart of the characteristic class homomorphism
\begin{equation}\label{eqRelCh}
\ch: \dH^k(M,S;\bbZ) \longrightarrow \H^k(M,S;\bbZ)~, \qquad h \longmapsto [c_h]~.
\end{equation}
Any relative differential form $A \in \Omega^{k-1}(M,S)$
defines a differential character on relative cycles $h_A \in \dH^k(M,S;\bbZ)$
by setting
\begin{equation}\label{eqn:hAdef}
h_A(z) = \int_z\, A \mod \bbZ~,
\end{equation}
for all $z \in Z_{k-1}(M,S)$. The curvature of $h_A$ is $\dd A$
and the relative characteristic class is $0$.
Notice further that for $A \in \OmegaZ^{k-1}(M,S)$
integration over relative $(k-1)$-cycles takes values in $\bbZ$, i.e. $h_A =0$.
This defines the relative version of the topological trivialization homomorphism
\begin{equation}
\iota: \frac{\Omega^{k-1}(M,S)}{\OmegaZ^{k-1}(M,S)} \longrightarrow
\dH^k(M,S)~, \qquad [A] \longmapsto h_A~.
\end{equation}
Let $M$ be a manifold and $S \subseteq M$ a submanifold (possibly with boundary).
Then all squares in the diagram
\begin{equation}\label{eqRelDiffCharDia}
\xymatrix{
& 0 \ar[d] \\
& \frac{\H^{k-1}(M,S;\bbR)}{\Hf^{k-1}(M,S;\bbZ)} \ar[d]
& \frac{\Omega^{k-1}(M,S)}{\OmegaZ^{k-1}(M,S)} \ar[r]^-\dd \ar[d]_-\iota
& \dd \Omega^{k-1}(M,S) \ar[d] \\
0 \ar[r] & \H^{k-1}(M,S;\bbT) \ar[r]^-\kappa \ar[d]
& \dH^k(M,S;\bbZ) \ar[r]^-\cu \ar[d]_-\ch
& \OmegaZ^k(M,S) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \Ht^k(M,S;\bbZ) \ar[r] \ar[d]
& \H^k(M,S;\bbZ) \ar[r]
& \Hf^k(M,S;\bbZ) \ar[r] & 0 \\
& 0
\end{equation}
commute. The left column, the middle row and the bottom row are short exact sequences.
The middle and right columns form sequences starting with injections
and the homomorphism in the top row is surjective.
As in the absolute case, the exact sequences in the left column and in the bottom row
are obtained from the long exact sequence (<ref>) for relative cohomology
associated to the coefficient sequence $0 \to \bbZ \to \bbR \to \bbT \to 0$.
In the right column, the homomorphism $\dd \Omega^{k-1}(M,S) \to \OmegaZ^k(M,S)$
is just the inclusion and the homomorphism $\OmegaZ^k(M,S) \to \Hf^k(M,S;\bbZ)$
is obtained by identifying $\Hf^k(M,S;\bbZ)$ with $\Hom(\H_k(M,S);\bbZ)$
and mapping $\omega \in \OmegaZ^k(M,S)$
to $\int_\cdot\, \omega \in \Hom(\H_k(M,S),\bbZ)$.
The right column forms a sequence because of Stokes' theorem. In the top row,
$\dd: \Omega^{k-1}(M,S) / \OmegaZ^{k-1}(M,S) \to \dd \Omega^{k-1}(M,S)$ is surjective by definition.
Commutativity of the top right square follows immediately from (<ref>).
To show commutativity of the bottom left square,
observe that the composition of the left and bottom arrows
is the connecting homomorphism $\beta: \H^{k-1}(M,S;\bbT) \to \H^k(M,S;\bbZ)$ in (<ref>).
For $u \in \H^{k-1}(M,S;\bbT)$, $\beta\, u$ is defined by choosing a representative
$\bar u \in Z^{k-1}(M,S;\bbT)$ of $u$,
lifting $\bar u$ to $\tilde{\bar u} \in C^{k-1}(M,S;\bbR)$, taking the unique $c_u \in Z^k(M,S;\bbZ)$
such that $c_u = \cdel \tilde{\bar u}$ and setting $\beta\, u = [c_u]$.
Assigning to $u$ the homomorphism $ \kappa\, u : Z_{k-1}(M,S)\to \bbT$
is by definition the same as restricting $\bar u$ to relative cycles.
Hence the restriction to relative cycles of $\tilde {\bar u} \in C^{k-1}(M,S;\bbR)$
provides a lift $\widetilde{\kappa\, u} \in \Hom(Z_{k-1}(M,S),\bbR)$ of $\kappa\, u \in \Hom(Z_{k-1}(M,S),\bbT)$ along the
quotient $\bbR \to \bbT$.
Clearly $\widetilde{\kappa \, u} \circ \del = \tilde{\bar u} \circ \del$,
therefore $\ch\, \kappa\, u = [c_u] = \beta\, u$ and the bottom left square is commutative as claimed.
Let us now show commutativity of the bottom right square. Given any $h \in \dH^k(M,S;\bbZ)$
and exploiting divisibility of $\bbT$, we choose an extension $\bar h \in C^{k-1}(M,S;\bbT)$ of $h$
and a lift $\tilde{\bar h} \in C^{k-1}(M,S;\bbR)$ of $\bar h$ along the quotient $\bbR \to \bbT$.
By definition $\ch\, h = [c_h] \in \H^k(M,S;\bbZ)$,
for $c_h \in Z^k(M,S;\bbZ)$ such that $\cdel \tilde{\bar h} = \int_\cdot\, \cu\, h - c_h$,
i.e. the class in $\Hf^k(M,S;\bbZ)$ represented by $c_h$
is the same as the one represented by $\int_\cdot \, \cu\, h$.
It is straightforward to prove that the middle column forms a sequence,
i.e. $\ch \circ \iota = 0$. Furthermore, the first arrow is injective
by (<ref>) and the definition of $\OmegaZ^{k-1}(M,S)$, cf. (<ref>).
It remains to show that the middle row is a short exact sequence.
First, let us notice that $\cu \circ \kappa = 0$ since $u \in \H^{k-1}(M,S;\bbT)$
vanishes when evaluated on relative boundaries.
Furthermore, if $u \in \H^{k-1}(M,S;\bbT)$ is such that $\kappa\, u = 0$
then $u$ vanishes on all relative cycles, i.e. $u = 0$ and hence $\kappa$ is injective.
To show that $\cu$ is surjective, we exploit exactness of the bottom row.
Given any $\omega \in \OmegaZ^k(M,S)$,
we find a preimage $[c] \in \H^k(M,S;\bbZ)$ of $\int_\cdot\, \omega \in \Hf^k(M,S;\bbZ)$.
Hence there exists $\tilde h \in C^{k-1}(M,S;\bbR)$
such that $\int_\cdot\, \omega = c + \cdel \tilde h \in C^k(M,S;\bbR)$,
where $c \in Z^k(M,S;\bbZ)$ denotes a representative of $[c]$.
Let us denote by $h \in \Hom(Z_{k-1}(M,S),\bbT)$ the restriction of $\tilde h \mod \bbZ$.
For each $\gamma \in C_k(M)$ we find that $h(\del \gamma) = \int_\gamma\, \omega \mod \bbZ$,
which implies $h \in \dH^k(M,S;\bbZ)$ and $\cu\, h = \omega$ according to Definition <ref>.
This shows that $\cu$ is surjective and we are left with proving that $\ker \cu = \im \kappa$.
Let $h \in \dH^k(M,S;\bbZ)$ be such that $\cu\, h = 0$.
Then $h: Z_{k-1}(M,S) \to \bbT$ vanishes on $B_{k-1}(M,S)$ and it
descends to $\underline h \in \Hom(\H_{k-1}(M,S),\bbT)$.
Recalling that $\H^{k-1}(M,S;\bbT) \simeq \Hom(\H_{k-1}(M,S),\bbT)$,
we find $u \in \H^{k-1}(M,S;\bbT)$ corresponding to $\underline h$.
The definition of $\kappa$ discussed above (<ref>) then shows that $\kappa\, u = h$.
The assignment of the Abelian groups $\dH^k(M,S;\bbZ)$
to objects $(M,S)$ in the category $\Pair$ (see Subsection <ref>)
defines a functor
\begin{equation}
\dH^k(-;\bbZ): \Pair^\op \longrightarrow \Ab~.
\end{equation}
Given any morphism $f: (M,S) \to (M^\prime,S^\prime\, )$ in $\Pair$,
we define for any $h\in \dH^k(M^\prime,S^\prime;\bbZ)$
the homomorphism $f^\ast h := h \circ f_\ast : Z_{k-1}(M,S)\to \bbT$
by exploiting functoriality of relative chains, see (<ref>).
Then $f^\ast h \in \dH^k(M,S;\bbZ)$ because of
$h(f_\ast \del \gamma) = \int_\gamma \, f^\ast \cu\, h$,
for all $\gamma \in C_k(M)$.
This also shows that the curvature $\cu$ is a natural transformation
for differential characters on relative cycles.
One can also easily show that $\kappa$, $\ch$ and $\iota$
are natural transformations for differential characters on relative cycles, i.e.
\begin{align}
\cu: \dH^k(-;\bbZ) \Longrightarrow \OmegaZ^k(-)~, && \ch: \dH^k(-;\bbZ) \Longrightarrow \H^k(-;\bbZ)~, \\[4pt]
\iota: \frac{\Omega^{k-1}(-)}{\OmegaZ^{k-1}(-)} \Longrightarrow \dH^k(-;\bbZ)~,
&& \kappa: \H^{k-1}(-;\bbT) \Longrightarrow \dH^k(-;\bbZ)~,
\end{align}
are natural transformations between functors from $\Pair^\op$ to $\Ab$.
Moreover, all the arrows in the diagram displayed in Theorem <ref>
are (the components of) natural transformations,
which follows from naturality of the long exact sequence
*The natural homomorphism $\boldsymbol I$:
Since the quotient map $C_\sharp(M) \to C_\sharp(M,S)$ preserves
the boundary homomorphisms $\del$, it maps cycles to relative cycles.
Precomposing differential characters on relative cycles with this quotient map
thus defines a natural homomorphism
\begin{equation}\label{eqI}
I:\dH^k(M,S;\bbZ) \longrightarrow \dH^k(M;\bbZ)~.
\end{equation}
Naturality of $I$ is expressed by commutativity of the diagram
\begin{equation}
\xymatrix{
\dH^k(M^\prime,S^\prime;\bbZ) \ar[r]^-I \ar[d]_-{f^\ast} & \dH^k(M^\prime;\bbZ) \ar[d]^-{f^\ast} \\
\dH^k(M,S;\bbZ) \ar[r]_-I & \dH^k(M;\bbZ)
\end{equation}
for all morphisms $f: (M,S) \to (M^\prime,S^\prime\, )$ in $\Pair$,
which follows from the commutative diagram
\begin{equation}
\xymatrix{
Z_{k-1}(M) \ar[r] \ar[d]_{f_\ast} & Z_{k-1}(M,S) \ar[d]^-{f_\ast} \\
Z_{k-1}(M^\prime\, ) \ar[r] & Z_{k-1}(M^\prime,S^\prime\, )
\end{equation}
for the quotient maps. In addition the natural diagrams
[d]_-^k(M,S;) [r]^-I ^k(M;) [d]^- [d]_-^k(M,S;) [r]^-I ^k(M;) [d]^-
Ω^k_(M,S) [r] Ω^k_(M) ^̋k(M,S;) [r] ^̋k(M;)
[d]_-κ ^̋k-1(M,S;) [r] ^̋k-1(M;)[d]^-κ [d]_-ιΩ^k-1(M,S)/Ω^k-1_(M,S) [r] Ω^k-1(M)/Ω^k-1_(M)[d]^-ι
^k(M,S;) [r]_-I ^k(M;) ^k(M,S;) [r]_-I ^k(M;)
commute, for all objects $(M,S)$ in $\Pair$.
The unlabeled arrows involving differential forms are induced by the inclusions $\Omega^p(M,S)\to \Omega^p(M)$
and the unlabeled arrows involving cohomology groups are
the canonical homomorphisms from relative to absolute cohomology.
In general, the homomorphism $I:\dH^k(M,S;\bbZ) \to \dH^k(M;\bbZ)$ fails to be injective.
To illustrate this fact, consider the commutative diagram
\begin{equation}
\xymatrix{
0 \ar[r] & \H^{k-1}(M,S;\bbT) \ar[r]^-\kappa \ar[d]
& \dH^k(M,S;\bbZ) \ar[r]^-\cu \ar[d]^-I
& \OmegaZ^k(M,S) \ar[d] \ar[r] & 0 \\
0 \ar[r] & \H^{k-1}(M;\bbT) \ar[r]_-\kappa
& \dH^k(M;\bbZ) \ar[r]_-\cu
& \OmegaZ^k(M) \ar[r] & 0
\end{equation}
with both rows being short exact sequences.
Since the right vertical arrow is injective (because it is a subset inclusion),
the middle vertical arrow is injective if and only if so is the left one. We now construct examples of pairs $(M,S)$
for which $\H^{k-1}(M,S;\bbT) \to \H^{k-1}(M;\bbT)$ is not injective: Let $m \geq 2 $ and $k \in \{2,\ldots,m+1\}$.
Consider $M = \bbR^m$ and $S = M \setminus (\bbR^{m-k+1} \times \bbB^{k-1})$,
where $\bbB^p$ is a closed $p$-ball in $\bbR^p$. Observe that
$M$ is homotopic to a point and $S$ is homotopic to the $(k-2)$-sphere $\bbS^{k-2}$.
Using the long exact sequence relating the relative cohomology of the
pair $(M,S)$ to the cohomologies of $M$ and $S$, cf. <cit.>,
we obtain the exact sequence
\begin{equation}
\xymatrix{
\H^{k-2}(M;\bbT) \ar[r]
& \H^{k-2}(S;\bbT) \ar[r]
& \H^{k-1}(M,S;\bbT) \ar[r]
& \H^{k-1}(M;\bbT)~.
\end{equation}
Since $\H^{k-1}(M;\bbT)$ is trivial by construction, $\H^{k-1}(M,S;\bbT)$ can be computed
as the quotient of $\H^{k-2}(S;\bbT)$ by the image of $\H^{k-2}(M;\bbT)$.
Specifically, one finds that $\bbT\simeq \H^{k-1}(M,S;\bbT) \to \H^{k-1}(M;\bbT) \simeq 0$
is not injective.
As in Example <ref>, the relative differential cohomology group
$\dH^2(M,S;\bbZ)$ in degree $k=2$ has an immediate geometric interpretation. It is
canonically isomorphic (by the holonomy map) to the group of isomorphism classes of triples $(L,\nabla,\sigma)$
consisting of a Hermitean line bundle $L \to M$ with connection
$\nabla$ and a $\nabla$-parallel section $\sigma:S \to L|_S$ of the restricted bundle
$L\vert_S \to S$. The homomorphism $I: \dH^2(M,S;\bbZ) \to \dH^2(M;\bbZ)$ is
then induced by the forgetful map from triples $(L,\nabla,\sigma)$ to pairs $(L,\nabla)$
that ignores the section, see <cit.> for details.
Since there may be inequivalent parallel sections on the same pair $(L,\nabla)$,
this gives a geometric interpretation of the non-injectivity of $I$, cf. Remark <ref>.
*Properly embedded $\boldsymbol{S\subseteq M}$:
In the following we shall specialize to the case where $S \subseteq M$
is a properly embedded submanifold (possibly with boundary).
In this case, differential forms on $S$ can be extended to differential forms on $M$,
see e.g. <cit.>.
In particular, we obtain the short exact sequence of de Rham complexes
\begin{equation}
\xymatrix{
0 \ar[r] & \Omega^\sharp(M,S) \ar[r] & \Omega^\sharp(M) \ar[r] & \Omega^\sharp(S) \ar[r] & 0~.
\end{equation}
Regarding differential forms as cochains via integration
over smooth singular chains, we obtain a commutative diagram of cochain
complexes of Abelian groups
\begin{equation}
\xymatrix{
0 \ar[r] & \Omega^\sharp(M,S) \ar[r] \ar[d]_{\int_\cdot}
& \Omega^\sharp(M) \ar[r] \ar[d]_{\int_\cdot}
& \Omega^\sharp(S) \ar[r] \ar[d]_{\int_\cdot} & 0 \\
0 \ar[r] & C^\sharp(M,S;\bbR) \ar[r]
& C^\sharp(M;\bbR) \ar[r]
& C^\sharp(S;\bbR)\ar[r] & 0
\end{equation}
and the corresponding commutative diagram of the long exact cohomology sequences
\begin{equation}
\xymatrix@C=12pt{
\cdots \ar[r] & \H^{k-1}_\dR(M) \ar[r] \ar[d]_\simeq
& \H^{k-1}_\dR(S) \ar[r] \ar[d]_\simeq
& \H^k_\dR(M,S) \ar[r] \ar[d]
& \H^k_\dR(M) \ar[r] \ar[d]_\simeq
& \H^k_\dR(S) \ar[r] \ar[d]_\simeq
& \cdots \\
\cdots \ar[r] & \H^{k-1}(M;\bbR) \ar[r]
& \H^{k-1}(S;\bbR) \ar[r]
& \H^k(M,S;\bbR) \ar[r]
& \H^k(M;\bbR) \ar[r]
& \H^k(S;\bbR) \ar[r]
& \cdots\,
\end{equation}
By de Rham's theorem,[The de Rham theorem also holds for manifolds with boundary;
the well-known proof due to A. Weil using Čech-de Rham and Čech-singular double complexes
can easily be adapted to the case of manifolds with boundary.
Alternatively, one may also argue with homotopy invariance of de Rham cohomology
using the fact that the inclusion $M \backslash \partial M \hookrightarrow M$ is a homotopy equivalence.]
the vertical arrows between absolute cohomology groups are isomorphisms
and hence by the five lemma we conclude that also $\H^k_\dR(M,S) \to \H^k(M,S;\bbR)$ is an isomorphism.
This provides us with a relative version of de Rham's theorem for the case of $S\subseteq M$ being
properly embedded. Using this result we can refine the statement
of Theorem <ref> to obtain the full commutative diagram for relative differential cohomology.
Let $M$ be a manifold and $S \subseteq M$ a properly embedded submanifold (possibly with boundary).
Then the diagram
\begin{equation}\label{eqRelDiffCharDiaAlt}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\
0 \ar[r] & \frac{\H^{k-1}(M,S;\bbR)}{\Hf^{k-1}(M,S;\bbZ)} \ar[r] \ar[d]
& \frac{\Omega^{k-1}(M,S)}{\OmegaZ^{k-1}(M,S)} \ar[r]^-\dd \ar[d]_-\iota
& \dd \Omega^{k-1}(M,S) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \H^{k-1}(M,S;\bbT) \ar[r]^-\kappa \ar[d]
& \dH^k(M,S;\bbZ) \ar[r]^-\cu \ar[d]_-\ch
& \OmegaZ^k(M,S) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \Ht^k(M,S;\bbZ) \ar[r] \ar[d]
& \H^k(M,S;\bbZ) \ar[r] \ar[d]
& \Hf^k(M,S;\bbZ) \ar[r] \ar[d] & 0 \\
& 0 & 0 & 0
\end{equation}
commutes and its rows and columns are short exact sequences.
This theorem is proven in <cit.> for closed submanifolds $S \subseteq M$.
The proof carries over to the more general setting of properly embedded submanifolds (possibly with boundary).
For completeness, we briefly outline the proof.
We first show that the right column is a short exact sequence.
For this, we only have to prove that the morphism
$\OmegaZ^k(M,S) \to \Hf^k(M,S;\bbZ)$ is surjective and that its kernel coincides
with $\dd \Omega^{k-1}(M,S)$. Both statements follow from de Rham's
theorem for relative cohomology
by taking into account the inclusion $\Hf^k(M,S;\bbZ) \subseteq \H^k(M,S;\bbR)$.
Using this result we can also complete the diagram in Theorem <ref> by defining the missing horizontal
arrow in the top row: With $\Omega^{p}_\dd(M,S)$ denoting the closed relative $p$-forms we have
\begin{equation}
\frac{\H^{k-1}(M,S;\bbR)}{\Hf^{k-1}(M,S;\bbZ)} \simeq \frac{\Omega^{k-1}_\dd(M,S)}{\OmegaZ^{k-1}(M,S)}
\subseteq \frac{\Omega^{k-1}(M,S)}{\OmegaZ^{k-1}(M,S)}~.
\end{equation}
This map is injective and its image agrees with the
kernel of $\dd: \Omega^{k-1}(M,S) / \OmegaZ^{k-1}(M,S) \to \dd \Omega^{k-1}(M,S)$.
Hence the top row is a short exact sequence.
To show that the top left square commutes, it is sufficient to represent
cohomology classes in $\H^{k-1}(M,S;\bbR)$
by means of closed $(k-1)$-forms according to the isomorphism displayed above.
Exactness of the middle column follows from exactness of the
other sequences.
In the present case of properly embedded submanifolds $S\subseteq M$
we can strengthen Remark <ref> on the
non-injectivity of the homomorphism $I:\dH^k(M,S;\bbZ) \to \dH^k(M;\bbZ)$
by fitting it into a long exact sequence connecting absolute
and relative (differential) cohomology groups.
By similar arguments as in <cit.>,
there exists a long exact sequence
\begin{equation}\label{eq:long_ex_sequ_par}
\xymatrix@C=44pt{
\cdots \ar[r]
& \H^{k-2}(M,S;\bbT) \ar[r] & \H^{k-2}(M;\bbT) \ar[r] & \H^{k-2}(S;\bbT) \ar`[r]`d[lll]`[llld]`[llldr][lld]^(0.3){\kappa \circ \beta} & \\
& \dH^k(M,S;\bbZ) \ar[r]^-{I} & \dH^k(M;\bbZ) \ar^{i_S^*}[r] & \dH^k(S;\bbZ) \ar`[r]`d[lll]`[llld]`[llldr][lld]^(0.3){\beta \circ \ch} & \\
& \H^{k+1}(M,S;\bbZ) \ar[r] & \H^{k+1}(M;\bbZ) \ar[r] & \H^{k+1}(S;\bbZ) \ar[r] & \cdots
\end{equation}
of Abelian groups, where $\beta: \H^k(S;G) \to \H^{k+1}(M,S;G)$ denotes the connecting
homomorphism (for $G = \bbT$ or $G=\bbZ$).
From (<ref>) it follows that $h \in \dH^k(M;\bbZ)$ descends to a differential character
on relative cycles if and only if it vanishes upon pull-back to $S$.
§.§ Excision theorem
We now prove a version of the excision theorem for differential characters
on relative cycles. This result will be used later in Section <ref>
to define the push-forward of compactly supported differential characters and
hence to understand their functorial behavior.
Let $M$ be a manifold. Consider $O \subseteq M$ open and
$C \subseteq M$ closed such that $C \subseteq O$.
Then the morphism $i : (O,O \setminus C) \to (M,M \setminus C)$
in $\Pair$ induces an isomorphism
\begin{equation}
i^\ast : \dH^k(M,M \setminus C;\bbZ) \longrightarrow \dH^k(O,O \setminus C;\bbZ)~.
\end{equation}
Consider the central row of diagram (<ref>)
and recall that it is a natural short exact sequence.
Hence the morphism $i: (O,O \setminus C) \to (M,M \setminus C)$ in $\Pair$
induces the commutative diagram
\begin{equation}
\xymatrix{
0 \ar[r] & \H^{k-1}(M,M \setminus C;\bbT) \ar[r]^-\kappa \ar[d]_-{i^\ast}
& \dH^k(M,M \setminus C;\bbZ) \ar[r]^-\cu \ar[d]_-{i^\ast}
& \OmegaZ^k(M,M \setminus C) \ar[r] \ar[d]_-{i^\ast} & 0 \\
0 \ar[r] & \H^{k-1}(O,O \setminus C;\bbT) \ar[r]_-\kappa
& \dH^k(O,O \setminus C;\bbZ) \ar[r]_-\cu
& \OmegaZ^k(O,O \setminus C) \ar[r] & 0
\end{equation}
Excision for ordinary cohomology implies that the left vertical arrow is an isomorphism.
By the five lemma, it is sufficient to show that also the right vertical arrow is an isomorphism
in order to complete the proof.
We will now construct an inverse of the homomorphism
$i^\ast: \OmegaZ^k(M,M \setminus C) \to \OmegaZ^k(O,O \setminus C)$.
First, notice that $i^\ast: \Omega^k(M,M \setminus C) \to \Omega^k(O,O \setminus C)$
(without the restriction to integral periods) is an isomorphism because
forms on $O$ that vanish on $O\setminus C$ can be extended by zero.
It remains to prove that the extension by zero homomorphism
$(i^\ast)^{-1}: \Omega^k(O,O \setminus C) \to \Omega^k(M,M \setminus C)$
preserves integral periods. Let $\omega \in \OmegaZ^k(O,O \setminus C)$ and $z \in Z_k(M, M \setminus C)$. Choosing a representative $\tilde z \in C_k(M)$ of $z$,
we find $\del \tilde z \in C_{k-1}(M \setminus C)$.
Because $\{O,M \setminus C\}$ is an open cover of $M$, there exists an integer $j \geq 0$
such that the $j$-th iterated subdivision $S^j \tilde z$ is a combination
of simplices which are supported either in $O$ or in $M \setminus C$.
Let $a \in C_k(O)$ denote the combination of those simplices in
$S^j \tilde z$ whose support intersects $C$.
By construction $b := S^j \tilde z - a \in C_k(M \setminus C)$
and $\del a = S^j \del \tilde z - \del b \in C_{k-1}(O \setminus C)$.
In particular, $a$ represents an element of $Z_k(O,O \setminus C)$.
Recalling that there exists a natural chain homotopy $D_j: C_p(M) \to C_{p+1}(M)$
between identity and $j$-th iterated subdivision,
we conclude that $\tilde z = a + b - D_j \del \tilde z - \del D_j \tilde z$.
By naturality, the chain homotopy $D_j$ preserves the supports of chains,
in particular $D_j \del \tilde z \in C_k(M \setminus C)$.
Since $\omega$ is closed, so is its extension by zero $(i^\ast)^{-1} \omega$.
This implies that
\begin{equation}
\int_z\, (i^\ast)^{-1} \omega
= \int_a\, \omega + \int_{b - D_j \del \tilde z} \, (i^\ast)^{-1} \omega - \int_{D_j \tilde z} \, \dd (i^\ast)^{-1} \omega
= \int_a\, \omega \in \bbZ~,
\end{equation}
where we have also used Stokes' theorem.
§.§ Module structure
We show that relative differential cohomology $\dH^\sharp(M,S;\bbZ)$ is a module
over the differential cohomology ring $\dH^\sharp(M;\bbZ)$, see Subsection <ref>.
Let $(M,S)$ be an object in $\Pair$. In the following we shall explain how
(<ref>) may be used to define a bihomomorphism
\begin{equation}\label{eqRelDiffCharModule}
\cdot \,:\, \dH^k(M;\bbZ) \times \dH^l(M,S;\bbZ) \longrightarrow \dH^{k + l}(M,S;\bbZ)~, \qquad (h,h'\,) \longmapsto h \cdot h'\,~,
\end{equation}
which provides us with the desired module structure.
Let $h \in \dH^k(M;\bbZ)$ and $h'\, \in \dH^l(M,S;\bbZ)$.
Let $\tilde h \in C^{k-1}(M;\bbR)$ be such that $\tilde h \mod \bbZ = h$ on $Z_{k-1}(M)$
and $c_h \in Z^k(M;\bbZ)$ such that $\cdel \tilde h = \int_\cdot\, \cu\, h - c_h \in C^k(M;\bbR)$.
The pair $(\tilde h,c_h)$ is unique up to a
term of the form $(\Delta + \cdel \Gamma,- \cdel \Delta)$,
where $\Delta \in C^{k-1}(M;\bbZ)$ and $\Gamma \in C^{k-2}(M;\bbR)$.
Similarly, let $\tilde h'\, \in C^{l-1}(M,S;\bbZ)$ be such that $\tilde h'\, \mod \bbZ = h'\,$
on $Z_{l-1}(M,S)$ and $c_{h'} \in Z^l(M,S;\bbZ)$
such that $\cdel \tilde h'\, = \int_\cdot\, \cu\, h'\, - c_{h'} \in C^l(M,S;\bbR)$.
The pair $(\tilde h'\,, c_{h'})$ is unique up to a term of
the form $(\Delta + \cdel \Gamma,-\cdel\Delta)$,
where $\Delta \in C^{l-1}(M,S;\bbZ)$ and $\Gamma \in C^{l-2}(M,S;\bbR)$.
As in Subsection <ref>, we choose a family of natural cochain homotopies
$B: \Omega^k(M) \times \Omega^l(M) \to C^{k+l-1}(M;\bbR)$,
for all $k,l\geq 0$, between the wedge product for differential forms
and the cup product for the corresponding cochains
such that $B(\omega,\theta) = (-1)^{k \,l}\, B(\theta,\omega)$,
for all $\omega \in \Omega^k(M)$ and $\theta \in \Omega^l(M)$.
These cochain homotopies imply the identities
\begin{equation}\label{eqWedgeCupHomotopy}
\int_\cdot\, \omega \wedge \theta - \int_\cdot\, \omega \smile \int_\cdot\, \theta
= \cdel B(\omega,\theta) + B(\dd \omega,\theta) + (-1)^k\, B(\omega,\dd \theta)~,
\end{equation}
for all $\omega \in \Omega^k(M)$ and $\theta \in \Omega^l(M)$.
Recall that an example of such a family of cochain homotopies is given
in <cit.>. It is obtained by iterated subdivisions,
the natural chain homotopy between subdivision and identity,
and by exploiting a result due to Kervaire <cit.>.
With this choice one explicitly observes that $B$ preserves supports, i.e.
$B(\omega,\theta) \in C^{k+l-1}(M,S)$ where $S= M\setminus (\supp\, \omega \cap \supp\, \theta)$
is the complement of the intersection of the supports of $\omega$ and $\theta$.
More abstractly, this fact follows from naturality of $B$.
As stressed in <cit.>, two different choices of $B$ are naturally cochain homotopic.
This result is crucial in showing that the ring structure on differential characters
does not depend on the choice of a natural cochain homotopy $B$. Similarly,
it will allow us to show that the module structure of differential characters on relative cycles
over the ring of differential characters does not depend on the choice of $B$.
Recalling that the cup product between cochains preserves supports,
we define $h \cdot h'\,$ in (<ref>) as the homomorphism
$Z_{k + l - 1}(M,S)\to\bbT$ given by
\begin{align}\label{eqRelModuleAction1}
h \cdot h'\, & := \tilde h \smile \int_\cdot \, \cu\, h'\, + (-1)^k\, c_h \smile \tilde h'\, + B(\cu\, h,\cu\, h'\,) \mod \bbZ \\[4pt]
& \phantom{:}= \tilde h \smile c_{h'} + (-1)^k\, \int_\cdot\, \cu\, h \smile \tilde h'\, + B(\cu\, h,\cu\, h'\,)
\mod \bbZ~,
\label{eqRelModuleAction2}
\end{align}
where $(\tilde h,c_h)$ and $(\tilde h'\,,c_{h'})$ were introduced above.
The two expressions in (<ref>) differ by the exact term $(-1)^{k+1}\, \cdel(\tilde h \smile \tilde h'\,) \in B^{k+l-1}(M,S;\bbR)$,
which is of course trivial when evaluated on $Z_{k + l - 1}(M,S)$.
While (<ref>) shows that $h \cdot h'\,$ does not depend on the choice of $(\tilde h'\,,c_{h'})$, the expression
(<ref>) shows independence with respect to the choice of $(\tilde h,c_h)$.
The fact that different choices of $B$ are naturally cochain homotopic entails that
$h \cdot h'\,$ does not depend on this choice as well.
It is easy to check that $h \cdot h'\,$ as defined above is an element of $\dH^{k+l}(M,S;\bbZ)$: using (<ref>) one finds that
\begin{equation}
(h \cdot h'\,)(\del \gamma) = \int_\gamma \, \cu\, h \wedge \cu\, h'\, \mod \bbZ~,
\end{equation}
for all $\gamma \in C_{k+l}(M)$.
Because the formula for the module structure on relative differential cohomology
is exactly the same as the one for the ring structure on differential cohomology,
one can easily adapt the arguments in <cit.> to the present case
and show that (<ref>) structures
$\dH^\sharp(M,S;\bbZ)$ into a module over $\dH^\sharp(M;\bbZ)$.
For an alternative approach see <cit.>.
Directly from (<ref>), one can prove the identities
\begin{align}
\cu(h \cdot h'\,) & = \cu\, h \wedge \cu\, h'\,~,
& \ch(h \cdot h'\,) & = \ch\, h \smile \ch\, h'\,~, \\[4pt]
h \cdot \kappa\, \upsilon & = (-1)^k \,\kappa(\ch\, h \smile \upsilon)~,
& h \cdot \iota\, [\alpha] & = (-1)^k \, \iota\, [\cu\, h \wedge \alpha]~, \\[4pt]
\kappa\, u \cdot h'\, & = \kappa(u \smile \ch\, h'\,)~,
& \iota\, [A] \cdot h'\, & = \iota\, [A \wedge \cu\, h'\,]~,
\end{align}
for all $h {\in} \dH^k(M;\bbZ)$, $h'\, {\in} \dH^l(M,S;\bbZ)$, $[A] \in \Omega^{k-1}(M) / \Omega^{k-1}_\bbZ(M)$,
$[\alpha] \in \Omega^{l-1}(M,S) / \Omega^{l-1}_\bbZ(M,S)$, $u \in \H^{k-1}(M;\bbT)$ and
$\upsilon \in \H^{l-1}(M,S;\bbT)$,
which express the compatibility of the module structure for differential characters on relative cycles
with respect to the natural homomorphisms $\cu$, $\ch$, $\iota$ and $\kappa$.
We conclude by noticing that the $\dH^\sharp(M;\bbZ)$-module structure on
$\dH^\sharp(M,S;\bbZ)$ is natural with respect to morphisms
$f:(M,S) \to (M^\prime,S^\prime\, )$ in the category $\Pair$, i.e.the pull-back $f^\ast: \dH^k(M^\prime,S^\prime;\bbZ) \to \dH^k(M,S;\bbZ)$ is a module homomorphism
with underlying ring homomorphism $f^\ast: \dH^k(M^\prime;\bbZ) \to \dH^k(M;\bbZ)$.
For this, let $h \in \dH^k(M^\prime;\bbZ)$ and $h'\, \in \dH^l(M^\prime,S^\prime;\bbZ)$
and choose cochains $\tilde h \in C^{k-1}(M^\prime;\bbR)$ and $\tilde h'\, \in C^{l-1}(M^\prime,S^\prime;\bbR)$
which extend and lift $h$ and $h'\,$, respectively. Take
$c_h \in Z^k(M^\prime;\bbZ)$ and $c_{h'} \in Z^l(M^\prime,S^\prime;\bbZ)$
such that $\cdel \tilde h = \int_\cdot\, \cu\, h - c_h$ and $\cdel \tilde h'\, = \int_\cdot\, \cu\, h'\, - c_{h'}$.
Then $f^\ast \tilde h \in C^{k-1}(M;\bbR)$ and $f^\ast \tilde h'\, \in C^{l-1}(M,S;\bbR)$
extend and lift $f^\ast h\in \dH^k(M;\bbZ)$ and $f^\ast h'\, \in \dH^l(M,S;\bbZ)$, respectively.
Furthermore, $f^\ast c_h \in Z^k(M;\bbZ)$ and $f^\ast c_{h'} \in Z^l(M,S;\bbZ)$
satisfy $\cdel f^\ast \tilde h = \int_\cdot\, \cu\, f^\ast h - f^\ast c_h$
and $\cdel f^\ast \tilde h'\, = \int_\cdot\, \cu\, f^\ast h'\, - f^\ast c_{h'}$.
Computing $f^\ast h \cdot f^\ast h'\,$ using (<ref>),
we conclude that $f^\ast h \cdot f^\ast h'\, = f^\ast(h \cdot h'\,)$
because the cup product $\smile$ and the cochain homotopy $B$ are natural.
We observe that (<ref>) also makes sense
when both factors are differential characters on relative cycles: For
$S$, $S^\prime$ and $S \cup S^\prime$ submanifolds (possibly with boundary) of $M$,
the same arguments would show that
\begin{equation}\label{eqRelDiffCharProd}
\cdot \, :\, \dH^k(M,S;\bbZ) \times \dH^l(M,S^\prime;\bbZ)
\longrightarrow \dH^{k+l}(M,S \cup S^\prime;\bbZ)~, \qquad
(h,h'\,) \longmapsto h \cdot h'\,
\end{equation}
is a well-defined bihomomorphism.
For $S = S^\prime$, (<ref>) defines
a graded-commutative ring structure (without unit if $S$ is non-empty)
on differential characters on relative cycles.
This ring structure coincides with the usual ring structure on differential characters
for $S = S^\prime = \emptyset$. Furthermore, when $S \subseteq S^\prime$,
we can interpret $\dH^l(M,S^\prime;\bbZ)$ as a module
over the ring $\dH^k(M,S;\bbZ)$ (without unit for $S \neq \emptyset$).
When $S = \emptyset$, this coincides
with the module structure introduced in (<ref>).
§ DIFFERENTIAL CHARACTERS WITH COMPACT SUPPORT
To introduce differential characters with compact support,
we follow an approach similar to the one used in Subsection <ref>
to define ordinary cohomology with compact support.
Let $M$ be a manifold. Consider its associated directed set $\mcK_M$ of compact subsets $K\subseteq M$
and introduce the functor $(M,M \setminus -): \mcK_M \to \Pair^\op$ as in (<ref>).
Composing this functor with the relative differential cohomology functor
$\dH^k(-;\bbZ): \Pair^\op \to \Ab$ results in the functor
\begin{equation}\label{eqn:dHdiagram}
\dH^k(M,M \setminus -;\bbZ): \mcK_M \longrightarrow \Ab~.
\end{equation}
The Abelian group of differential characters with compact support
is the colimit
\begin{equation}\label{eqCDiffChar}
\dH^k_{\rm c}(M;\bbZ) := \colim\big(\dH^k(M,M \setminus -;\bbZ): \mcK_M \to \Ab\big)~
\end{equation}
of the functor (<ref>) over the directed set $\mcK_M $.
As for ordinary cohomology (cf. Remark <ref>),
the Abelian groups $\dH^k_{\rm c}(M;\bbZ)$ can also be computed
as a colimit over the directed set $\mcO^{\rm c}_M$ instead of
$\mcK_M$ (since both $\mcO^{\rm c}_M$ and $\mcK_M$ are cofinal in a larger directed set $\mcU_{M}$).
Composing the functor $(M,M \setminus -): \mcO^{\rm c}_M \to \PePair^\op$
with the embedding $\PePair^\op\to \Pair^\op$ and
$\dH^k(-;\bbZ): \Pair^\op \to \Ab$, we obtain another functor
$\dH^k(M,M \setminus -;\bbZ): \mcO^{\rm c}_M \to \Ab$
whose colimit is isomorphic to $\dH^k_{\rm c}(M;\bbZ)$, i.e.
\begin{equation}\label{eqn:alternativedHc}
\dH^k_{\rm c}(M;\bbZ)\simeq \colim\big(\dH^k(M,M \setminus -;\bbZ): \mcO^{\rm c}_M \to \Ab\big)~.
\end{equation}
A technical advantage of this alternative point of view
is that $\dH^k(M,M \setminus O;\bbZ)$, for any $O\in \mcO^{\rm c}_M$,
fits into the full commutative diagram
of short exact sequences (<ref>)
while $\dH^k(M,M \setminus K;\bbZ)$, for $K\in \mcK_M$,
in general just fits into the incomplete diagram (<ref>).
We next define the Abelian group
\begin{equation}\label{eqColimFormsZ}
\Omega^k_{{\rm c},\bbZ}(M) := \colim \big(\Omega^k_\bbZ(M,M \setminus -): \mcK_M \to \Ab\big)
\end{equation}
of $k$-forms with compact support having integral periods on cycles relative to the complement of their support.
Recalling that $\colim$ is an exact functor for diagrams in Abelian groups over directed sets, it follows that
\begin{equation}
\colim \Big(\, \frac{\Omega^k(M,M \setminus -)}{\OmegaZ^k(M,M
\setminus -)}: \mcK_M \to \Ab \, \Big)
= \frac{\Omega^k_{\rm c}(M)}{\Omega^k_{{\rm c},\bbZ}(M)}~.
\end{equation}
Let $M$ be a manifold. Then the diagram
\begin{equation}\label{eqCDiffCharDia}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\
0 \ar[r] & \frac{\H^{k-1}_{\rm c}(M;\bbR)}{\H_{{\rm c},\free}^{k-1}(M;\bbZ)} \ar[r] \ar[d]
& \frac{\Omega^{k-1}_{\rm c}(M)}{\Omega^{k-1}_{{\rm c},\bbZ}(M)} \ar[r]^-\dd \ar[d]_-\iota
& \dd \Omega^{k-1}_{\rm c}(M) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \H^{k-1}_{\rm c}(M;\bbT) \ar[r]^-\kappa \ar[d]
& \dH^k_{\rm c}(M;\bbZ) \ar[r]^-\cu \ar[d]_-\ch
& \Omega^k_{{\rm c},\bbZ}(M) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \H^k_{{\rm c},\tor}(M;\bbZ) \ar[r] \ar[d]
& \H^k_{\rm c}(M;\bbZ) \ar[r] \ar[d]
& \H^k_{{\rm c},\free}(M;\bbZ) \ar[r] \ar[d] & 0 \\
& 0 & 0 & 0
\end{equation}
commutes and its rows and columns are short exact sequences.
This result follows immediately from (<ref>)
and Theorem <ref> because
the complement $M \setminus O$ of any $O \in \mcO^{\rm c}_M$ is properly embedded
and $\colim$ is an exact functor for diagrams in Abelian groups over directed sets.
We show that
\begin{equation}
\dH^k_{\rm c}(-;\bbZ): \Man_{m,\emb} \longrightarrow \Ab
\end{equation}
is a functor. Given an open embedding $f: M \to M^\prime$ between $m$-dimensional manifolds,
one can use the same arguments as in Subsection <ref> and the Excision Theorem <ref>
for differential characters on relative cycles to conclude that
$f^\ast: \dH^k(M^\prime,M^\prime \setminus -;\bbZ) \circ \mcK_f \Rightarrow \dH^k(M,M \setminus -;\bbZ)$
is a natural isomorphism. Denoting its inverse by
\begin{equation}
(f^\ast)^{-1}: \dH^k(M,M \setminus -;\bbZ) \Longrightarrow \dH^k(M^\prime,M^\prime \setminus -;\bbZ) \circ \mcK_f~,
\end{equation}
the universal property of the colimit allows us to define a
canonical homomorphism $f_\ast: \dH^k_{\rm c}(M;\bbZ) \to \dH^k_{\rm c}(M^\prime;\bbZ)$,
which we call the push-forward of differential characters with compact support.
One can then easily check that $\dH^k_{\rm c}(-;\bbZ): \Man_{m,\emb} \to \Ab$ is a functor
and that naturality of $\cu$, $\ch$, $\iota$ and $\kappa$ for differential characters on relative chains
carries over to the compactly supported case via our colimit prescription, i.e.
\begin{align}
\cu: \dH^k_{\rm c}(-;\bbZ) \Longrightarrow \Omega^k_{{\rm c},\bbZ}(-)~, && \ch: \dH^k_{\rm c}(-;\bbZ) \Longrightarrow \H^k_{\rm c}(-;\bbZ)~, \\[4pt]
\iota: \frac{\Omega^{k-1}_{\rm c}(-)}{\Omega^{k-1}_{{\rm c},\bbZ}(-)} \Longrightarrow \dH^k_{\rm c}(-;\bbZ)~,
&& \kappa: \H_{\rm c}^{k-1}(-;\bbT) \Longrightarrow \dH^k_{\rm c}(-;\bbZ)~,
\end{align}
are natural transformations between functors from $\Man_{m,\emb}$ to $\Ab$.
As explained in Examples <ref> and <ref>,
(relative) differential cohomology groups in degree $k=2$ are canonically isomorphic
to isomorphism classes of Hermitean line bundles with connection (and parallel section).
By passing to the colimit over the directed set $\mcO^{\rm c}_M$ we obtain a
canonical identification of the compactly supported differential cohomology group $\dH^2_{\rm c}(M;\bbZ)$
with the group of isomorphism classes of Hermitean line bundles with connection
and a parallel section outside some relatively compact open subset $O \subseteq M$.
Two such triples with sections $\sigma:M \setminus O \to L$ and $\sigma^\prime : M \setminus O^\prime \to L^\prime$
are isomorphic if and only if there exists a bundle isomorphism $\psi:L \to L^\prime$,
a subset $\tilde O \in \mcO^{\rm c}_M$ with $\tilde O \subset O \cap O^\prime$, and a section $\tilde\sigma : M \setminus \tilde O \to L$
such that
$\tilde \sigma|_{M\setminus O} = \sigma$ and
$\psi \circ \tilde\sigma|_{M \setminus O^\prime} = \sigma^\prime$.
*Module structure:
Using the natural module structure for differential characters on relative cycles
developed in Subsection <ref>, our colimit
prescription for differential characters with compact support
given in Definition <ref> yields a natural module
structure for $\dH^\sharp_{\rm c}(M;\bbZ)$ over the ring $\dH^\sharp(M;\bbZ)$.
We denote this module structure by
\begin{equation}\label{eqn:modcomp}
\cdot\,:\, \dH^k(M;\bbZ) \times \dH^l_{\rm c}(M;\bbZ)
\longrightarrow \dH^{k + l}_{\rm c}(M;\bbZ)~, \qquad (h,h'\,) \longmapsto h \cdot h'\,~.
\end{equation}
The bihomomorphism (<ref>) is obtained by taking the
colimit over $K \in \mcK_M$ in (<ref>) for $S = M \setminus K$.
Given any morphism $f:M \to M^\prime$ in $\Man_{m,\emb}$,
the diagram
\begin{equation}\label{eqCDiffCharModuleNat}
\xymatrix@C=40pt{
\dH^k(M^\prime;\bbZ) \times \dH^l_{\rm c}(M;\bbZ) \ar[r]^-{f^\ast \times \id} \ar[d]_-{\id \times f_\ast}
& \dH^k(M;\bbZ) \times \dH^l_{\rm c}(M;\bbZ) \ar[r]^-{\cdot}
& \dH^{k+l}_{\rm c}(M;\bbZ) \ar[d]^-{f_\ast} \\
\dH^k(M^\prime;\bbZ) \times \dH^l_{\rm c}(M^\prime;\bbZ) \ar[rr]_-{\cdot}
& & \dH^{k+l}_{\rm c}(M^\prime;\bbZ)
\end{equation}
commutes by construction of $f_\ast$, which implies that the $\dH^\sharp(M;\bbZ)$-module
structure on $\dH^\sharp_{\rm c}(M;\bbZ)$ is natural.
The homomorphisms $\cu$, $\ch$, $\iota$ and $\kappa$
for compactly supported differential characters are compatible with the module structure, i.e.
\begin{align}
\cu(h \cdot h'\,) & = \cu\, h \wedge \cu\, h'\,~,
& \ch(h \cdot h'\,) & = \ch\, h \smile \ch\, h'\,~, \\[4pt]
h \cdot \kappa\, \upsilon & = (-1)^k\, \kappa(\ch\, h \smile \upsilon)~,
& h \cdot \iota\, [\alpha] & = (-1)^k\, \iota\, [\cu\, h \wedge \alpha]~, \\[4pt]
\kappa\, u \cdot h'\, & = \kappa(u \smile \ch\, h'\,)~,
& \iota\, [A] \cdot h'\, & = \iota\, [A \wedge \cu\, h'\,]~,
\end{align}
for all $h \in \dH^k(M;\bbZ)$, $h'\, \in \dH^l_{\rm c}(M;\bbZ)$,
$[A] \in \Omega^{k-1}(M) / \Omega^{k-1}_\bbZ(M)$, $[\alpha] \in \Omega^{l-1}_{\rm c}(M) / \Omega^{l-1}_{{\rm c},\bbZ}(M)$,
$u \in \H^{k-1}(M;\bbT)$ and $\upsilon \in \H^{l-1}_{\rm c}(M;\bbT)$.
This again follows from similar results for the module structure of differential characters on relative cycles,
see Subsection <ref>.
*The natural homomorphism $\boldsymbol I$:
Applying our colimit prescription to the homomorphism displayed in (<ref>)
defines a natural homomorphism
\begin{equation}\label{eqCDiffCharToDiffChar}
I: \dH^k_{\rm c}(M;\bbZ) \longrightarrow \dH^k(M;\bbZ)~.
\end{equation}
Naturality means that for any morphism $f : M\to M^\prime$
in $\Man_{m,\emb}$ the diagram
\begin{equation}\label{eqCDiffCharToDiffCharNat}
\xymatrix{
\dH^k_{\rm c}(M;\bbZ) \ar[r]^-I \ar[d]_-{f_\ast} & \dH^k(M;\bbZ) \\
\dH^k_{\rm c}(M^\prime;\bbZ) \ar[r]_-I & \dH^k(M^\prime;\bbZ) \ar[u]_-{f^\ast}
\end{equation}
commutes. As for differential characters on relative cycles, this homomorphism
is compatible with curvature, characteristic class, topological trivialization
and inclusion of flat characters. It is furthermore multiplicative with respect to the
$\dH^\sharp(M;\bbZ)$-module structure of $\dH^\sharp_{\rm c}(M;\bbZ)$.
As for ordinary cohomology, the natural homomorphism
$I: \dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$ is in general neither surjective nor injective.
Its kernel and cokernel may be characterized by applying the colimit over the directed
set $\mcO^{\rm c}_M$ to the long exact sequence (<ref>).
We observe that injectivity of $I: \dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$
is equivalent to injectivity of $\H^{k-1}_{\rm c}(M;\bbT) \to \H^{k-1}(M;\bbT)$.
This follows from the colimit of the diagram in Remark <ref>
and observing that $\Omega^k_{{\rm c},\bbZ}(M) \to \OmegaZ^k(M)$ is an injection.
We provide below some examples for which $\H^{k-1}_{\rm c}(M;\bbT) \to \H^{k-1}(M;\bbT)$
fails to be injective, and therefore $I: \dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$ too:
Let $m \geq 3$, $k \in \{2, \ldots, m\}$ with $2k \neq m+2, m+3$
and take $M = \bbR^{k-1} \times \bbS^{m-k+1}$.
Observe that the collection of compact subsets $\mathcal B = \{B_r \times \bbS^{m-k+1}: r>0\}$,
where $B_r$ is the closed ball in $\bbR^{k-1}$ of radius $r$ centered at the origin, is cofinal in $\mcK_M$.
Therefore, recalling (<ref>), we find
$\H^{k-1}_{\rm c}(M;\bbT) \simeq \colim(\H^{k-1}(M, M \setminus -;\bbT): \mathcal B \to \Ab)$.
The long exact sequence relating relative and absolute cohomology groups,
see e.g. <cit.>, provides the exact sequence
\begin{equation}
\xymatrix{
\H^{k-2}(M;\bbT) \ar[r] & \H^{k-2}(M \setminus K;\bbT) \ar[r]
& \H^{k-1}(M, M \setminus K;\bbT) \ar[r] & \H^{k-1}(M;\bbT)
\end{equation}
for each $K \in \mathcal B$. Since $2k \neq m+2$, $\H^{k-1}(M;\bbT)$ is trivial;
therefore $\H^{k-1}(M, M \setminus K;\bbT)$ is isomorphic to the quotient of $\H^{k-2}(M \setminus K;\bbT)$
by the image of $\H^{k-2}(M;\bbT)$.
Taking also $2k \neq m+3$ into account, one concludes that $\H^{k-2}(M \setminus K;\bbT) \simeq \bbT$.
Since this result is the same for all $K \in \mathcal B$, the homomorphism
$\bbT \simeq \H^{k-1}_{\rm c}(M;\bbT) \to \H^{k-1}(M;\bbT) \simeq 0$ is not injective.
In <cit.> the authors develop
another approach to differential cohomology with compact support
which is based on de Rham-Federer currents.
In particular, <cit.> is the analogue of our Theorem <ref>.
However, in <cit.> it is stated that their notion of compactly supported differential cohomology
corresponds to the subgroup of Cheeger-Simons differential characters on $M$
which vanish upon restriction to the complement of some compact subset $K \subseteq M$.
This is in contrast with our results, see also Remark <ref> below.
Since differential cohomology groups are usually regarded as a refinement
of smooth singular cohomology groups by differential forms, it seems reasonable
to introduce compactly supported differential cohomology groups as a refinement of
compactly supported cohomology groups by compactly supported differential forms.
This is exactly the case for our notion of compactly supported differential
cohomology $\dH^\sharp_{\rm c}(M;\bbZ)$.
Actually, assuming that an Abelian group $\dH^k_{\rm c}(M;\bbZ)$ fits into the commutative diagram
\begin{equation}
\xymatrix{
0 \ar[r] & \H^{k-1}_{\rm c}(M;\bbT) \ar[r] \ar[d] & \dH^k_{\rm c}(M;\bbZ) \ar[r] \ar[d]_-{I} & \Omega^k_{{\rm c},\bbZ}(M) \ar[r] \ar[d] & 0 \\
0 \ar[r] & \H^{k-1}(M;\bbT) \ar[r] & \dH^k(M;\bbZ) \ar[r] & \OmegaZ^k(M) \ar[r] & 0
\end{equation}
with exact rows, the non-injectivity of the homomorphism $\H^{k-1}_{\rm c}(M;\bbT) \to \H^{k-1}(M;\bbT)$ implies that $\dH^k_{\rm c}(M;\bbZ)$ cannot be a subgroup of $\dH^k(M;\bbZ)$ in general.
On the other hand, from the long exact sequence (<ref>) we conclude that the inclusion of the subgroup of Cheeger-Simons characters on $M$ which vanish upon restriction to the complement of some relatively compact open subset $O \in \mcO^{\rm c}_M$ into $\dH^k(M;\bbZ)$ factorizes the map $I:\dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$.
In other words, the subgroup of $\dH^k(M;\bbZ)$ considered in <cit.> coincides with the image of the homomorphism $I:\dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$.
Another source of examples of compactly supported differential
characters is obtained by classical Chern-Weil theory.
As in Example <ref>, let $G$ be a Lie group with finitely many
connected components and $\lambda: \mathfrak g^k \to \bbR$ an invariant polynomial.
Let $P \to M$ be a principal $G$-bundle with connection $\theta$
and suppose that there exists an isomorphism $(P,\theta)|_{M \setminus O} \xrightarrow{\simeq} ((M\setminus O) \times G,\dd)$
to the trivial bundle with trivial connection outside some relatively compact open
subset $O \in \mcO^{\rm c}_M$.[Since $\theta$ is equivalent to the trivial connection outside $O$,
the constant maps in the trivialization are parallel sections.]
Mimicking in this setting the construction from <cit.>
of differential characters with curvature the Chern-Weil form $\lambda(F_\theta)$,
we obtain an element of $\dH^{2k}(M;\bbZ)$ that lies in the image of
the natural homomorphism $I: \dH^{2k}_{\rm c}(M;\bbZ) \to \dH^{2k}(M;\bbZ)$.
Since $I$ is in general not injective,
the question arises whether this character has a canonical preimage.
Taking into account the identification of $\dH^{2k}(M,M\setminus O;\bbZ)$
with the group of parallel relative differential characters from <cit.>,
we obtain a canonical character from the Cheeger-Chern-Simons character
associated with $(P,\theta)$, see <cit.> for details of the construction.
§ SMOOTH PONTRYAGIN DUALITY
The aim of this section is to establish a version of Pontryagin duality for Cheeger-Simons differential characters.
This should be compared to <cit.>, where a similar result is obtained
in a different model for differential cohomology based on de Rham-Federer currents.
In this section all manifolds are implicitly assumed to be connected, $m$-dimensional, and oriented.
This in particular allows us to define the integration homomorphism $\int_M : \Omega_{\rm c}^m(M)\to \bbR$.
We denote by $\oMan_{m,\emb}$ the corresponding category of
oriented and connected $m$-dimensional manifolds with morphisms given by orientation-preserving
open embeddings. Some of the following results are proven under the additional
(sufficient but not necessary) hypothesis that the manifold $M$ is of finite-type, which
implies that all (co)homology groups are finitely generated and in particular allows us to
interpret cohomology with compact support as the dual of ordinary cohomology,
see e.g. Poincaré duality for de Rham cohomology in <cit.>.
We will clearly indicate which statements rely on this
assumption. Some of the technical details required in
Section <ref> are delegated to
Appendix <ref> at the end of the paper.
§.§ Definitions
For any Abelian group $G$, we denote its Pontryagin dual
(also called the character group) by $G^\star:=\Hom(G,\bbT)$.
We define the smooth Pontryagin dual $\Omega^k_{\rm c}(M)^\star_\infty\subseteq \Omega^k_{\rm c}(M)^\star$
of $\Omega^k_{\rm c}(M)$ as the subgroup of elements $\varphi\in \Omega^k_{\rm c}(M)^\star$
which are smooth in the sense that there exists $\omega \in \Omega^{m-k}(M)$
such that
\begin{equation}
\varphi(\alpha) = \int_M \, \omega \wedge \alpha \mod \bbZ~,
\end{equation}
for all $\alpha \in \Omega_{\rm c}^k(M)$. Similarly, we define the smooth Pontryagin dual $\Omega^k(M)^\star_\infty\subseteq \Omega^k(M)^\star$
of $\Omega^k(M)$ as the subgroup of elements $\psi\in \Omega^k(M)^\star$
which are smooth in the sense that there exists $\alpha \in \Omega^{m-k}_{\rm c}(M)$
such that
\begin{equation}
\psi(\omega) = \int_M \, \omega \wedge \alpha \mod \bbZ~,
\end{equation}
for all $\omega \in \Omega^k(M)$. Introducing the (weakly non-degenerate) $\bbT$-valued pairing
\begin{equation}\label{eqPairingFormPre}
\ips{\cdot}{\cdot}_{\Omega}^{} : \Omega^k(M)\times \Omega^{m-k}_{\rm c}(M)\longrightarrow \bbT~, \qquad
(\omega,\alpha) \longmapsto (-1)^k\,\int_M\, \omega\wedge\alpha \mod \bbZ~,
\end{equation}
partial evaluation provides us with the two homomorphisms
Ω^k(M) ⟶Ω^m-k_c(M)^⋆ , ω⟼ω·_Ω^ ,
Ω^m-k_c(M) ⟶Ω^k(M)^⋆ , α⟼·α_Ω^ .
It is clear from these definitions that (<ref>) induces isomorphisms
\begin{equation}
\Omega^k(M) \simeq \Omega^{m-k}_{\rm c}(M)^\star_\infty~, \qquad \Omega^{m-k}_{\rm c}(M)\simeq \Omega^k(M)^\star_\infty~.
\end{equation}
We further define the smooth Pontryagin duals of the quotients
$\Omega^k_{\rm c}(M) / \Omega^k_{{\rm c},\bbZ}(M)$
and $ \Omega^k(M) / \OmegaZ^k(M)$ by
\begin{align}
\Big(\, \frac{\Omega^k_{\rm c}(M)}{\Omega^k_{{\rm c},\bbZ}(M)}\, \Big)^\star_\infty
& := \big\{\varphi \in \Omega^k_{\rm c}(M)^\star_\infty\,:\;
\varphi\big(\Omega^k_{{\rm c},\bbZ}(M)\big) = \{0\} \big\}\,,\\[4pt]
\Big(\, \frac{\Omega^k(M)}{\OmegaZ^k(M)}\, \Big)^\star_\infty
& := \big\{\psi \in \Omega^k(M)^\star_\infty\,:\; \psi\big(\OmegaZ^k(M)\big) = \{0\} \big\}\,.
\end{align}
Notice that the smooth Pontryagin dual
$(\Omega^k_{\rm c}(M) / \Omega^k_{{\rm c},\bbZ}(M))^\star_{\infty}$
can be identified with a subgroup of $(\Omega^k_{\rm c}(M) / \Omega^k_{{\rm c},\bbZ}(M))^\star$
and similarly that $(\Omega^k(M) / \OmegaZ^k(M))^\star_{\infty}$ can
be identified with a subgroup of $(\Omega^k(M) / \OmegaZ^k(M))^\star$.
We also define the smooth Pontryagin duals of the subgroups
$\Omega^k_{{\rm c},\bbZ}(M)\subseteq \Omega^k_{\rm c}(M)$ and $\OmegaZ^k(M)\subseteq \Omega^k(M)$ by
Ω^k_c,(M)^⋆_∞ := Ω^k_c(M)^⋆_∞/{φ∈Ω^k_c(M)^⋆_∞ :
φ(Ω^k_c,(M)) = {0} } = Ω^k_c(M)^⋆_∞/( Ω^k_c(M)/Ω^k_c,(M) )^⋆_∞ ,
^k(M)^⋆_∞ := Ω^k(M)^⋆_∞/{ψ∈Ω^k(M)^⋆_∞ : ψ(^k(M)) = {0} } = Ω^k(M)^⋆_∞/( Ω^k(M)/^k(M) )^⋆_∞ .
Notice that the smooth Pontryagin dual
$\Omega^k_{{\rm c},\bbZ}(M)^\star_\infty$
can be identified with a subgroup of $\Omega^k_{{\rm c},\bbZ}(M)^\star$
and similarly that $\OmegaZ^k(M)^\star_\infty$ can
be identified with a subgroup of $\OmegaZ^k(M)^\star$.
To characterize (<ref>) and (<ref>) more explicitly,
we notice that Lemma <ref> implies that
\begin{equation}\label{eqFormsZ}
\OmegaZ^k(M)
= \Big\{\omega \in \Omega^k(M)\,:\; \int_M\, \omega \wedge \Omega^{m-k}_{{\rm c},\bbZ}(M) \subseteq \bbZ\Big\}~.
\end{equation}
Hence (<ref>) induces the $\bbT$-valued pairing
\begin{equation}\label{eqPairingFormRightQuotient}
\ips{\cdot}{\cdot}_{\Omega}^{} : \OmegaZ^k(M)\times \frac{\Omega^{m-k}_{\rm c}(M)}{\Omega^{m-k}_{{\rm c},\bbZ}(M)}
\longrightarrow \bbT~, \qquad (\omega,[\alpha]) \longmapsto (-1)^k\,\int_M\, \omega\wedge\alpha \mod \bbZ
\end{equation}
and by partial evaluation the two homomorphisms
^k(M) ⟶( Ω^m-k_c(M)/Ω^m-k_c,(M) )^⋆ , ω⟼ω·_Ω^ ,
Ω^m-k_c(M)/Ω^m-k_c,(M) ⟶^k(M)^⋆ , [α] ⟼·[α]_Ω^ .
Moreover, again because of (<ref>), the pairing (<ref>) induces another $\bbT$-valued pairing
\begin{equation}\label{eqPairingFormLeftQuotient}
\ips{\cdot}{\cdot}_{\Omega}^{} : \frac{\Omega^k(M)}{\OmegaZ^k(M)} \times \Omega^{m-k}_{{\rm c},\bbZ}(M)
\longrightarrow \bbT~, \qquad ([\omega], \alpha) \longmapsto (-1)^k\,\int_M\, \omega\wedge\alpha \mod \bbZ
\end{equation}
and by partial evaluation the two homomorphisms
Ω^k(M)/^k(M) ⟶Ω^m-k_c,(M)^⋆ , [ω] ⟼[ω]·_Ω^ ,
Ω^m-k_c,(M) ⟶( Ω^k(M)/^k(M) )^⋆ , α⟼·α_Ω^ .
For manifolds $M$ of finite-type, Lemma <ref> implies that
\begin{equation}\label{eqCFormsZ}
\Omega^{m-k}_{{\rm c},\bbZ}(M)
= \Big\{\alpha \in \Omega^{m-k}_{\rm c}(M)\, :\; \int_M\, \Omega^{k}_\bbZ(M)\wedge \alpha \subseteq \bbZ\Big\}~.
\end{equation}
We are now ready to demonstrate
The homomorphisms (<ref>) and (<ref>) induce isomorphisms
\begin{equation}
\OmegaZ^k(M) \simeq \Big(\, \frac{\Omega^{m-k}_{\rm c}(M)}{\Omega^{m-k}_{{\rm c},\bbZ}(M)}\, \Big)^\star_\infty~,\qquad
\frac{\Omega^k(M)}{\OmegaZ^k(M)} \simeq \Omega^{m-k}_{{\rm c},\bbZ}(M)^\star_\infty~.
\end{equation}
For $M$ of finite-type, the homomorphisms (<ref>) and
(<ref>) induce isomorphisms
\begin{equation}
\frac{\Omega^{m-k}_{\rm c}(M)}{\Omega^{m-k}_{{\rm c},\bbZ}(M)} \simeq \OmegaZ^k(M)^\star_\infty ~,\qquad
\Omega^{m-k}_{{\rm c},\bbZ}(M)\simeq \Big(\, \frac{\Omega^k(M)}{\OmegaZ^k(M)}\, \Big)^\star_\infty~.
\end{equation}
The first part follows from (<ref>) by a straightforward calculation and the second part
similarly by using also (<ref>).
Following <cit.>, we finally define the smooth Pontryagin duals of
(compactly supported) differential cohomology.
(i) The smooth Pontryagin dual $\dH^k_{\rm c}(M;\bbZ)^\star_\infty \subseteq \dH^k_{\rm c}(M;\bbZ)^\star$
of $\dH^k_{\rm c}(M;\bbZ)$ is the preimage
\begin{equation}
\dH^k_{\rm c}(M;\bbZ)^\star_\infty
:= (\iota^\star)^{-1}\Big(\, \frac{\Omega_{\rm c}^{k-1}(M)}{\Omega^{k-1}_{{\rm c},\bbZ}(M)}\, \Big)^\star_\infty~,
\end{equation}
where $\iota^\star := \Hom(\iota,\bbT) : \dH^k_{\rm c}(M;\bbZ)^\star \to (\Omega_{\rm c}^{k-1}(M) / \Omega^{k-1}_{{\rm c},\bbZ}(M))^\star$
is the Pontryagin dual of the topological trivialization $\iota: \Omega_{\rm c}^{k-1}(M) / \Omega^{k-1}_{{\rm c},\bbZ}(M) \to \dH^k_{\rm c}(M;\bbZ)$.
(ii) The smooth Pontryagin dual $\dH^k(M;\bbZ)^\star_\infty\subseteq \dH^k(M;\bbZ)^\star$
of $\dH^k(M;\bbZ)$ is the preimage
\begin{equation}
\dH^k(M;\bbZ)^\star_\infty
:= (\iota^\star)^{-1}\Big(\, \frac{\Omega^{k-1}(M)}{\OmegaZ^{k-1}(M)}\, \Big)^\star_\infty~,
\end{equation}
where $\iota^\star := \Hom(\iota,\bbT) : \dH^k(M;\bbZ)^\star \to (\Omega^{k-1}(M) / \Omega^{k-1}_{\bbZ}(M))^\star$
is the Pontryagin dual of the topological trivialization $\iota: \Omega^{k-1}(M) / \Omega^{k-1}_{\bbZ}(M) \to \dH^k(M;\bbZ)$.
We show that the smooth Pontryagin duals of (compactly supported) differential characters
define functors
\begin{equation}
\dH^k(-;\bbZ)^\star_\infty: \oMan_{m,\emb} \longrightarrow \Ab~,\qquad
\dH^k_{\rm c}(-;\bbZ)^\star_\infty: \oMan_{m,\emb}^\op \longrightarrow \Ab~.
\end{equation}
These functors are subfunctors
of $\dH^k(-;\bbZ)^\star := \Hom(\dH^k(-;\bbZ),\bbT): \oMan_{m,\emb} \to \Ab$
and $\dH^k_{\rm c}(-;\bbZ)^\star := \Hom(\dH^k_{\rm c}(-;\bbZ),\bbT): \oMan_{m,\emb}^\op \to \Ab$.
For any morphism $f: M \to M^\prime$ in $\oMan_{m,\emb}$,
the corresponding push-forward $f_\ast := \Hom(f^\ast,\bbT): \dH^k(M;\bbZ)^\star \to \dH^k(M^\prime;\bbZ)^\star$
maps smooth group characters $\psi \in \dH^k(M;\bbZ)^\star_\infty$
to smooth group characters $f_\ast \psi \in \dH^k(M^\prime;\bbZ)^\star_\infty$
because $(\Omega^{k-1}(-)/\OmegaZ^{k-1}(-))^\star_\infty : \oMan_{m,\emb} \to \Ab$
is a functor (this follows from (<ref>)) and $\iota^\star$ is by construction
a natural transformation.
Similarly, the pull-back $f^\ast := \Hom(f_\ast,\bbT): \dH^k_{\rm c}(M^\prime;\bbZ)^\star \to \dH^k_{\rm c}(M;\bbZ)^\star$
maps smooth group characters $\varphi^\prime \in \dH^k_{\rm c}(M^\prime;\bbZ)^\star_\infty$ to smooth group
characters $f^\ast\varphi^\prime \in \dH^k_{\rm c}(M;\bbZ)^\star_\infty$ because
$(\Omega_{\rm c}^{k-1}(-) / \Omega^{k-1}_{{\rm c},\bbZ}(-))^\star_\infty :
\oMan_{m,\emb}^\op \to \Ab$ is a functor (this follows again from (<ref>))
and $\iota^\star$ is by construction a natural transformation.
§.§ Duality theorem
We will characterize $\dH^k_{\rm c}(M;\bbZ)^\star_\infty$
in terms of differential characters and $\dH^k(M;\bbZ)^\star_\infty$
in terms of differential characters with compact support. This will establish
a version of smooth Pontryagin duality for
(compactly supported) differential characters.
The module structure on compactly supported differential characters
given in Section <ref> defines a bihomomorphism
$\cdot : \dH^k(M;\bbZ) \times\dH^{m-k+1}_{\rm c}(M;\bbZ) \to \dH^{m+1}_{\rm c}(M;\bbZ)$.
Using the diagram in Theorem <ref>,
we observe that $\dH^{m+1}_{\rm c}(M;\bbZ)$ is canonically isomorphic
to $\H^m_{\rm c}(M;\bbR) / \H^m_{{\rm c},\free}(M;\bbZ)$
and, since we assume $M$ to be connected,[
The following constructions can be extended to disconnected manifolds by treating each connected component separately.
For the sake of simplicity we do not consider this more general scenario.
] we obtain an isomorphism $\H^m_{\rm c}(M;\bbR) / \H^m_{{\rm c},\free}(M;\bbZ) \simeq \bbT$.
Hence this bihomomorphism maps to the circle group and defines a $\bbT$-valued pairing
\begin{equation}\label{eqPairingDiffChar}
\ips{\cdot}{\cdot}_{\rm c}:\dH^k(M;\bbZ) \times \dH^{m-k+1}_{\rm c}(M;\bbZ) \longrightarrow \bbT~, \qquad
(h,h'\,) \longmapsto (h \cdot h'\,) \mu~.
\end{equation}
Here we have also given an explicit expression for the isomorphisms above,
which should be interpreted as follows: For $h'\, \in \dH^{m-k+1}_{\rm c}(M;\bbZ)$
we choose a representative in the colimit (denoted with abuse of notation by the same symbol)
$h'\,\in \dH^{m-k+1}(M,M \setminus K;\bbZ)$ for some compact $K\subseteq M$.
Then $h \cdot h'\, \in \dH^{m+1}(M,M \setminus K;\bbZ)$ is a differential character on relative
cycles and $(h \cdot h'\,)\mu$ denotes its evaluation on (some representative of)
the unique relative homology class $\mu \in \H_m(M,M \setminus K)$
which restricts to the orientation of $M$ for each point of $K$, cf. <cit.>.
The pairing (<ref>) is natural in the sense that
for all morphisms $f:M \to M^\prime$ in $\oMan_{m,\emb}$
the diagram
\begin{equation}\label{eqPairingDiffCharNat}
\xymatrix@C=45pt{
\dH^k(M^\prime;\bbZ) \times \dH^{m-k+1}_{\rm c}(M;\bbZ) \ar[r]^-{f^\ast \times \id} \ar[d]_-{\id \times f_\ast}
& \dH^k(M;\bbZ) \times \dH^{m-k+1}_{\rm c}(M;\bbZ) \ar[d]^-{\ips{\cdot}{\cdot}_{\rm c}} \\
\dH^k(M^\prime;\bbZ) \times \dH^{m-k+1}_{\rm c}(M^\prime;\bbZ) \ar[r]_-{\ips{\cdot}{\cdot}_{\rm c}}
& \bbT
\end{equation}
commutes; this is a consequence of naturality of the module structure
(<ref>) and uniqueness of the relative homology
class representing the orientation, see <cit.>.
By partial evaluation, the pairing (<ref>) defines the two homomorphisms
^k(M;) ⟶^m-k+1_c(M;)^⋆ , h ⟼h·_c ,
^m-k+1_c(M;) ⟶^k(M;)^⋆ , h' ⟼·h' _c .
Let us introduce another $\bbT$-valued pairing
\begin{equation}
\label{eqPairingCoho} \ips{\cdot}{\cdot}_{\H}^{} \,:\, \H^k(M;\bbT) \times \H^{m-k}_{\rm c}(M;\bbZ) \longrightarrow \bbT~, \qquad
(u,c) \longmapsto (u \smile c)\mu ~,
\end{equation}
between cohomology and compactly supported cohomology, which
is defined as in (<ref>)
by choosing a representative of the colimit $c \in \H^{m-k}(M,M \setminus K;\bbZ)$, for some $K\subseteq M$ compact,
and evaluating $u \smile c\in \H^{m}(M,M \setminus K;\bbT)$
on the unique element $\mu \in \H_m(M,M \setminus K)$
which restricts to the orientation of $M$ at each point of $K$.
Partial evaluation provides the homomorphisms
^̋k(M;) ⟶^̋m-k_c(M;)^⋆ , u⟼u·_^ ,
^̋m-k_c(M;) ⟶^̋k(M;)^⋆ , c⟼·c_^ .
Let $M$ be an object of $\oMan_{m,\emb}$. Then the homomorphism (<ref>) is an isomorphism.
For $M$ of finite-type, the homomorphism (<ref>) is also an isomorphism.
The first statement follows from Poincaré duality $\H_{\rm c}^{m-k}(M;\bbZ)\simeq \H_k(M)$,
see e.g. <cit.>, and the fact that
\begin{equation}\label{eqn:tmp}
\H^k(M;\bbT) \simeq \Hom(\H_k(M),\bbT)=\H_k(M)^\star~,
\end{equation}
which is a consequence of the universal coefficient theorem for cohomology and divisibility of $\bbT$.
The second statement follows by taking the Pontryagin dual of (<ref>) and recalling that Pontryagin
duality is reflexive for finitely generated Abelian groups, in particular
on all (co)homology groups of manifolds $M$ of finite-type.
Let $M$ be an object of $\oMan_{m,\emb}$. Then the diagram
\begin{equation}
\xymatrix@C=35pt{
0 \ar[r] & \H^{m-k}(M;\bbT) \ar[r]^-\kappa \ar[d]_\simeq
& \dH^{m-k+1}(M;\bbZ) \ar[r]^-\cu \ar[d]_\simeq
& \OmegaZ^{m-k+1}(M) \ar[r] \ar[d]^\simeq & 0 \\
0 \ar[r] & \H^k_{\rm c}(M;\bbZ)^\star \ar[r]_-{\ch^\star}
& \dH^k_{\rm c}(M;\bbZ)^\star_\infty \ar[r]_-{\iota^\star}
& \Big(\, \frac{\Omega^{k-1}_{\rm c}(M)}
{\Omega^{k-1}_{{\rm c},\bbZ}(M)}\, \Big)^\star_\infty \ar[r] & 0
\end{equation}
commutes, its rows are short exact sequences and the vertical arrows are natural isomorphisms.
For $M$ of finite-type, the diagram
\begin{equation}
\xymatrix@C=35pt{
0 \ar[r] & \frac{\Omega^{m-k}_{\rm c}(M)}{\Omega^{m-k}_{{\rm c},\bbZ}(M)} \ar[r]^-\iota \ar[d]_\simeq
& \dH^{m-k+1}_{\rm c}(M;\bbZ) \ar[r]^-\ch \ar[d]_\simeq
& \H^{m-k+1}_{\rm c}(M;\bbZ) \ar[r] \ar[d]^\simeq & 0 \\
0 \ar[r] & \OmegaZ^k(M)^\star_\infty \ar[r]_-{\cu^\star}
& \dH^k(M;\bbZ)^\star_\infty \ar[r]_-{\kappa^\star}
& \H^{k-1}(M;\bbT)^\star \ar[r] & 0
\end{equation}
commutes, its rows are short exact sequences and the vertical arrows are natural isomorphisms.
In both diagrams the vertical arrows are the partial evaluations given in (<ref>),
(<ref>) and (<ref>).
The top row in the first diagram is the middle row of (<ref>)
and the top row in the second diagram is the middle column in
(<ref>). Hence they are short exact sequences.
The bottom row in the second diagram is a short exact sequence by
<cit.>. We show that the bottom row
in the first diagram is a short exact sequence. Now $\ch^\star : \H^k_{\rm c}(M;\bbZ)^\star \to \dH^k_{\rm c}(M;\bbZ)^\star$
maps (injectively) to smooth group characters as a consequence of $\ch \circ \iota = 0$
and $\iota^\star: \dH^k_{\rm c}(M;\bbZ)^\star \to (\Omega^{k-1}_{\rm c}(M) / \Omega^{k-1}_{{\rm c},\bbZ}(M))^\star$
maps (surjectively because of
Definition <ref>) smooth group characters to smooth group characters.
Exactness at the middle object follows from the fact that
$\Hom(-,\bbT): \Ab^\op \to \Ab$ is an exact functor as $\bbT$ is divisible.
By Lemma <ref> and Lemma <ref>,
the left and right vertical arrows in both diagrams are isomorphisms.
Commutativity of both diagrams can be shown by using the properties (<ref>)
of the module structure on compactly supported differential characters.
Using the five lemma, we conclude that the middle vertical
arrow in each diagram is also an isomorphism.
This theorem immediately implies
The partial evaluations in (<ref>) define a natural isomorphism
between the functors
^m-k+1(-;) : _m,^⟶ , ^k_c(-;)^⋆_∞: _m,^⟶ ,
and a natural isomorphism between the functors
^m-k+1_c(-;) : _m,,ft ⟶ , ^k(-;)^⋆_∞: _m,,ft ⟶ ,
where $\oMan_{m,\emb,\mathrm{ft}}$ is the full subcategory of $\oMan_{m,\emb}$
whose objects are manifolds of finite-type.
Let $M$ be an oriented and connected $m$-dimensional manifold of finite-type. Then the pairing
\begin{equation}
\ips{\cdot}{\cdot}_{\rm c}:\dH^k(M;\bbZ) \times \dH^{m-k+1}_{\rm c}(M;\bbZ) \longrightarrow \bbT
\end{equation}
introduced in (<ref>) is weakly non-degenerate.
By introducing suitable pairings, one can easily extend Theorem <ref>
to the full diagrams for (compactly supported) differential characters
given in (<ref>) and (<ref>).
For example, the second diagram in Theorem <ref> (for $M$ of finite-type)
extends to the three-dimensional commutative diagram
\begin{equation}
\xymatrix@C=0pt@R=1pt{
&& 0 \ar@{.>}[dd] && 0 \ar@{.>}[dd] && 0 \ar@{.>}[dd] \\
& && 0 \ar[dd] && 0 \ar[dd] && 0 \ar[dd] \\
0 \ar@{.>}[rr] && \frac{\H^{m-k}_{\rm c}(M;\bbR)}{\H_{{\rm c},\free}^{m-k}(M;\bbZ)} \ar@{.>}'[r][rr] \ar@{.>}'[d][dd] \ar[rd]^-\simeq
&& \frac{\Omega^{m-k}_{\rm c}(M)}{\Omega^{n-k}_{{\rm c},\bbZ}(M)}
\ar@{.>}'[r][rr] \ar@{.>}'[d][dd]^-\iota \ar[rd]^-\simeq
&& \dd \Omega^{m-k}_{\rm c}(M) \ar@{.>}'[r][rr] \ar@{.>}'[d][dd] \ar[rd]^-\simeq
&& 0\\
& 0 \ar[rr] && \Hf^k(M;\bbZ)^\star \ar[rr] \ar[dd]
&& \Omega^k_\bbZ(M)^\star_\infty \ar[rr] \ar[dd]^(.3){\cu^\star}
&& (\dd \Omega^{k-1}(M))^\star_\infty \ar[rr] \ar[dd] && 0\\
0 \ar@{.>}[rr] && \H^{m-k}_{\rm c}(M;\bbT) \ar@{.>}'[r][rr]_-\kappa \ar@{.>}'[d][dd] \ar[rd]^-\simeq
&& \dH^{m-k+1}_{\rm c}(M;\bbZ) \ar@{.>}'[r][rr]_-\cu \ar@{.>}'[d][dd]^-\ch \ar[rd]^-\simeq
&& \Omega^{m-k+1}_{{\rm c},\bbZ}(M)
\ar@{.>}'[r][rr] \ar@{.>}'[d][dd] \ar[rd]^-\simeq && 0\\
& 0 \ar[rr] && \H^k(M;\bbZ)^\star \ar[rr]_(.4){\ch^\star} \ar[dd]
&& \dH^k(M;\bbZ)^\star_\infty
\ar[rr]_(.4){\iota^\star} \ar[dd]^(.3){\kappa^\star}
&& \left(\frac{\Omega^{k-1}(M)}{\Omega^{k-1}_\bbZ(M)}
\right)^\star_\infty \ar[rr] \ar[dd] && 0\\
0 \ar@{.>}[rr] && \H^{m-k+1}_{{\rm c},\tor}(M;\bbZ) \ar@{.>}'[r][rr] \ar@{.>}'[d][dd] \ar[rd]^-\simeq
&& \H^{m-k+1}_{\rm c}(M;\bbZ) \ar@{.>}'[r][rr] \ar@{.>}'[d][dd] \ar[rd]^-\simeq
&& \H^{m-k+1}_{{\rm c},\free}(M;\bbZ)
\ar@{.>}'[r][rr] \ar@{.>}'[d][dd] \ar[rd]^-\simeq && 0\\
& 0 \ar[rr] && \Ht^k(M;\bbZ)^\star \ar[rr] \ar[dd]
&& \H^{k-1}(M;\bbT)^\star \ar[rr] \ar[dd]
&& \left(\frac{\H^{k-1}(M;\bbR)}{\Hf^{k-1}(M;\bbZ)}
\right)^\star \ar[rr] \ar[dd] && 0\\
&& 0 && 0 && 0 \\
& && 0 && 0 && 0
\end{equation}
where all diagonal arrows are isomorphisms, the foreground face is the smooth Pontryagin dual of
(<ref>) and the background face is given by (<ref>).
We compare our results to <cit.>.
As mentioned in Remark <ref>, de Rham-Federer characters
with compact support are introduced in <cit.> by means of
de Rham-Federer currents. The group of de Rham-Federer characters
with compact support is isomorphic to the smooth Pontryagin dual
of the group of de Rham-Federer characters, cf. <cit.>.
According to <cit.>, the latter is isomorphic to the group
$\dH^\sharp(M;\bbZ)$ of Cheeger-Simons differential characters, therefore
we deduce from Theorem <ref> that the group of
de Rham-Federer characters with compact support is isomorphic to the
group $\dH^\sharp_{\rm c}(M;\bbZ)$ of compactly supported differential characters
introduced in Definition <ref>.
Together with Remark <ref>, this contradicts <cit.>,
which states that the group of de Rham-Federer characters with compact
support is isomorphic to a subgroup of the group of
Cheeger-Simons differential characters.
§.§ Pairing between differential characters with compact support
We conclude by defining a $\bbT$-valued pairing on
differential characters with compact support and describe its properties.
Let $M$ be any object of $\oMan_{m,\emb}$.
Using the homomorphism $I :\dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$ defined in
(<ref>), we introduce the pairing
\begin{equation}\label{eqPairingCDiffChar}
\ips{I \cdot}{\cdot}_{\rm c}: \dH^k_{\rm c}(M;\bbZ) \times \dH^{m-k+1}_{\rm c}(M;\bbZ)
\longrightarrow
\bbT~,
\qquad
(h,h'\,) \longmapsto \ips{I h}{h'\,}_{\rm c}~
\end{equation}
on compactly supported differential characters.
The pairing (<ref>) is graded symmetric, i.e.
I hh' _c = (-1)^k (m-k+1) I h' h_c ,
for all $h\in \dH^k_{\rm c}(M;\bbZ)$ and $h'\,\in \dH^{m-k+1}_{\rm c}(M;\bbZ)$.
This result is a consequence of the graded commutative (possibly non-unital) ring structure
on relative differential cohomology, see Remark <ref>:
Given $h \in \dH^k_{\rm c}(M;\bbZ)$ and $h'\, \in \dH^{m-k+1}_{\rm c}(M;\bbZ)$
there exists $K\subseteq M$ compact such that $h \in \dH^k(M,M \setminus K;\bbZ)$
and $h'\, \in \dH^{m-k+1}(M,M \setminus K;\bbZ)$ are representatives in the
corresponding colimits.
Using (<ref>) one shows the identities
$I h \cdot h'\, = h \cdot h'\,$ and $I h'\, \cdot h = h'\, \cdot h$
of elements in $\dH^{m+1}(M,M \setminus K;\bbZ)$,
where on the left hand sides $\cdot$ denotes the $\dH^\sharp(M;\bbZ)$-module structure
on $\dH^\sharp(M,M\setminus K;\bbZ)$ (see (<ref>))
and on the right hand sides $\cdot$ denotes the ring structure
on $\dH^\sharp(M,M\setminus K;\bbZ)$
(see (<ref>) for $S = S^\prime = M \setminus K$).
As a consequence of graded commutativity of the ring structure
on $\dH^\sharp(M,M\setminus K;\bbZ)$,
we obtain $I h \cdot h'\, = h \cdot h'\, = (-1)^{k\, (m-k+1)}\, h'\,
\cdot h = (-1)^{k\, (m-k+1)}\, I h'\, \cdot h$
and the result follows by recalling (<ref>).
Unlike $\ips{\cdot}{\cdot}_{\rm c}$, the pairing $\ips{I \cdot}{\cdot}_{\rm c}$ given in
(<ref>) might be degenerate, even if $M$ is of finite-type.
This is because the homomorphism $I: \dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$ in general
fails to be injective, cf. Remark <ref>.
For $M$ of finite-type, Corollary <ref> implies that
the degeneracy of the pairing (<ref>) coincides precisely with $\ker I$.
We finish by proving that the pairing (<ref>) is natural.
For any morphism $f: M \to M^\prime$ in $\oMan_{m,\emb}$ the diagram
\begin{equation}
\xymatrix{
\dH^k_{\rm c}(M;\bbZ) \times \dH^{m-k+1}_{\rm c}(M;\bbZ) \ar[rd]_-{\ips{I \cdot}{\cdot}_{\rm c}} \ar[rr]^-{f_\ast \times f_\ast} &&
\dH^k_{\rm c}(M^\prime;\bbZ) \times \dH^{m-k+1}_{\rm c}(M^\prime;\bbZ) \ar[ld]^-{\ips{I \cdot}{\cdot}_{\rm c}} \\
\end{equation}
This is a direct consequence of naturality of
$\ips{\cdot}{\cdot}_{\rm c}: \dH^k(M;\bbZ) \times \dH^{m-k+1}_{\rm c}(M;\bbZ) \to \bbT$,
cf. (<ref>),
and naturality of $I: \dH^k_{\rm c}(M;\bbZ) \to \dH^k(M;\bbZ)$,
cf. (<ref>): The short calculation
\begin{equation}
\ips{I \, f_\ast \cdot}{f_\ast \cdot}_{\rm c} = \ips{f^\ast \, I \, f_\ast \cdot}{\cdot}_{\rm c} = \ips{I\cdot}{\cdot}_{\rm c}~
\end{equation}
proves the claim.
§ ACKNOWLEDGMENTS
It is a pleasure to thank Christian Bär for very helpful discussions
and Ulrich Bunke for his valuable comments on the first version of this paper.
This work was supported in part by the Action MP1405 QSPACE from the
European Cooperation in Science and Technology (COST).
The work of C.B. is partially supported by the Collaborative Research Center (SFB)
“Raum Zeit Materie”, funded by the Deutsche Forschungsgemeinschaft (DFG, Germany).
The work of M.B. is supported partly by a Research Fellowship of the Della Riccia Foundation (Italy)
and partly by a Postdoctoral Fellowship of the Alexander von Humboldt Foundation (Germany).
The work of A.S. is supported by a Research Fellowship of the Deutsche Forschungsgemeinschaft (DFG, Germany).
The work of R.J.S. is partially supported by the Consolidated Grant ST/L000334/1
from the UK Science and Technology Facilities Council.
§ TECHNICAL DETAILS FOR SECTION <REF>
We prove two lemmas which are used in Section <ref>
to obtain the explicit characterizations (<ref>) and (<ref>)
of (compactly supported) forms with integral periods.
For the first lemma we consider manifolds of finite-type only,
while in the second lemma there is no such restriction.
Let $M$ be an oriented $m$-dimensional manifold of finite-type.
Let $\alpha \in \Omega^k_{\rm c}(M)$ and recall the definition (<ref>).
Then the following conditions are equivalent:
* There exists $K \subseteq M$ compact such that $\alpha \in \OmegaZ^k(M,M \setminus K)$;
* $\int_M\, \omega \wedge \alpha \in \bbZ$, for each $\omega \in \OmegaZ^{m-k}(M)$.
$1 \Rightarrow 2$: Let $\alpha \in \Omega^k_\bbZ(M,M \setminus K)$ for
a compact subset $K \subseteq M$.
By the definition of $\Omega^k_\bbZ(M,M \setminus K)$ in (<ref>),
the cochain $\int_\cdot\, \alpha \in C^k(M,M \setminus K;\bbR)$ induces a relative cohomology class
$\int_\cdot \alpha \in \Hom(\H_k(M,M \setminus K),\bbZ)) \simeq \Hf^k(M,M \setminus K;\bbZ)$.
Passing to the colimit, we interpret $\int_\cdot\, \alpha$ as an element of $\H^k_{{\rm c},\free}(M;\bbZ)$.
Given $\omega \in \Omega^{m-k}_\bbZ(M)$, the cochain $\int_\cdot\, \omega \in C^{m-k}(M;\bbR)$
induces a cohomology class $\int_\cdot \omega \in \Hom(\H_{m-k}(M),\bbZ) \simeq \Hf^{m-k}(M;\bbZ)$.
Similarly to <cit.>,
the cup product for singular cohomology provides an integer-valued pairing
\begin{equation}\label{eqFreeCohoPairing}
\H_\free^{m-k}(M;\bbZ) \times \H^k_{{\rm c},\free}(M;\bbZ) \longrightarrow \bbZ~,
\end{equation}
which gives $(\, \int_\cdot\, \omega \smile \int_\cdot\, \alpha\, ) \mu \in \bbZ$
upon evaluation on $(\, \int_\cdot\, \omega,\int_\cdot\, \alpha\, )$.
Here $\mu \in \H_m(M,M \setminus K)$ denotes the unique relative homology class
which restricts to the orientation of $M$ at each point of $K$.
Since $\wedge$ and $\smile$ are naturally cochain homotopic on differential forms,
we have $\int_M\, \omega \wedge \alpha = (\, \int_\cdot \, \omega
\smile \int_\cdot\, \alpha\, )\mu \in \bbZ$.
$2 \Rightarrow 1$: Let $\alpha \in \Omega^k_{\rm c}(M)$ with
$\int_M\, \omega \wedge \alpha \in \bbZ$, for all $\omega \in \Omega^{m-k}_\bbZ(M)$,
and denote the support of $\alpha$ by $K^\prime:= \supp\, \alpha \subseteq M$.
We have to show that $\int_\cdot\, \alpha \in C^k(M;\bbR)$
induces a $\bbZ$-valued homomorphism on $Z_k(M,M \setminus K)$ for some $K \subseteq M$ compact.
By our assumptions, $\int_M\, \cdot \wedge \alpha$ defines a $\bbZ$-valued homomorphism
on $\Omega^{m-k}_\bbZ(M) / \dd \Omega^{m-k-1}(M) \simeq \Hf^{m-k}(M;\bbZ)$.
Taking into account the pairing (<ref>)
and recalling that $\wedge$ and $\smile$ are naturally cochain homotopic on differential forms,
we obtain $(\, \int_\cdot \, \omega \smile \int_\cdot\, \alpha\, )\mu = \int_M\, \omega \wedge \alpha$,
for all $\omega \in \Omega^{m-k}_\bbZ(M)$.
As a consequence, we have $( \cdot\smile \int_\cdot\,\alpha)\mu \in \Hom(\Hf^{m-k}(M;\bbZ),\bbZ)$.
By <cit.> extended to manifolds of finite-type,
the pairing induced by the cup product is perfect. In particular,
it provides an isomorphism $\H^k_{{\rm c},\free}(M;\bbZ) \simeq \Hom(\Hf^{m-k}(M;\bbZ),\bbZ)$,
hence $\int_\cdot \, \alpha \in \H^k_{{\rm c},\free}(M;\bbZ)$. Recalling the definition
of $\H^k_{{\rm c},\free}(M;\bbZ)$ in terms of a colimit, one finds $K \subseteq M$ compact such that
$\int_\cdot\, \alpha \in \Hf^k(M,M \setminus K;\bbZ) \simeq \Hom(\H_k(M,M \setminus K),\bbZ)$.
The implication then follows from the obvious inclusion
$\Hom(\H_k(M,M \setminus K),\bbZ) \subseteq \Hom(Z_k(M,M \setminus K),\bbZ)$.
Let $M$ be an oriented $m$-dimensional manifold (not necessarily of finite-type).
Let $\omega \in \Omega^k(M)$. Then the following conditions are equivalent:
* $\omega$ has integral periods, i.e. $\omega \in \OmegaZ^k(M)$;
* $\int_M\, \omega \wedge \alpha \in \bbZ$, for each $\alpha \in \Omega^{m-k}_{{\rm c},\bbZ}(M)$.
Let $\omega \in \Omega^k(M)$ satisfy
$\int_M\, \omega \wedge \Omega^{m-k}_{{\rm c},\bbZ}(M) \subseteq \bbZ$.
Since $\dd \Omega_{\rm c}^{m-k-1}(M)$ is a vector space over $\bbR$ and also a subgroup of $\Omega^{m-k}_{{\rm c},\bbZ}(M)$,
we deduce that $\int_M\, \omega \wedge \dd \Omega_{\rm c}^{m-k-1}(M) = \{0\}$,
hence $\dd \omega = 0$ via Stokes' theorem.
This implies that $\int_\cdot\, \omega \in C^k(M;\bbR)$ descends to a homomorphism on $\H_k(M)$.
Recalling Poincaré duality, see e.g. <cit.>,
there is a natural isomorphism $\H_k(M) \simeq \H^{m-k}_{\rm c}(M;\bbZ)$.
Moreover, for each $x \in \H_k(M)$ one has $\int_x\, \omega = (\xi \smile \int_\cdot\, \omega)\mu$,
where $\xi \in \H^{m-k}_{\rm c}(M;\bbZ)$ is the image of $x$ under Poincaré duality. On the right hand side
of this equation we have chosen a representative $\xi \in \H^{m-k}(M,M \setminus K;\bbZ)$ in the colimit,
for $K\subseteq M$ compact, and $\mu \in \H_m(M,M \setminus K)$
is the unique element which agrees with the orientation of $M$ at each point of $K$.
For $\xi \in \H^{m-k}_{{\rm c},\tor}(M;\bbZ)$ we have $(\xi \smile \int_\cdot\, \omega)\mu = 0$
by a similar argument as in <cit.>.
Therefore, showing that $\omega$ has integral periods is equivalent to
checking that the cup product between any $\xi \in \H^{m-k}_{{\rm c},\free}(M;\bbZ)$ and
$\int_\cdot\, \omega \in \H^k(M;\bbR)$ is $\bbZ$-valued.
Taking into account exactness of the right column of the diagram displayed in Theorem <ref>,
and recalling that $\wedge$ and $\smile$ are naturally cochain homotopic on differential forms,
we conclude that $\omega$ has integral periods
if and only if $\int_M\, \omega\wedge \alpha \in \bbZ$, for all $\alpha \in \Omega^{m-k}_{{\rm c},\bbZ}(M)$.
This shows that $1 \Leftrightarrow 2$.
C. Bär and C. Becker,
“Differential characters,”
Lect. Notes Math. 2112, Springer (2014).
C. Becker,
“Cheeger-Chern-Simons theory and differential String classes,”
arXiv:1404.0716 [math.DG].
C. Becker, A. Schenkel and R. J. Szabo,
“Differential cohomology and locally covariant quantum field theory,”
arXiv:1406.1514 [hep-th].
C. Becker, M. Benini, A. Schenkel and R. J. Szabo,
“Abelian duality on globally hyperbolic spacetimes,”
arXiv:1511.00316 [hep-th].
R. Bott and L. W. Tu,
“Differential forms in algebraic topology,”
Springer (1982).
M. Brightwell and P. Turner,
“Relative differential characters,”
Commun. Anal. Geom. 14 (2006) 269
J.-L. Brylinski,
“Loop spaces, characteristic classes and geometric quantization,”
Birkhäuser (2007).
U. Bunke,
“Differential cohomology,”
arXiv:1208.3961 [math.AT].
J. Cheeger and J. Simons,
“Differential characters and geometric invariants,”
Lect. Notes Math. 1167 (1985) 50.
C. Godbillon,
“Eléments de topologie algébrique,”
Hermann (1971).
A. Hatcher,
“Algebraic topology,”
Cambridge University Press (2002).
M. J. Hopkins and I. M. Singer,
“Quadratic functions in geometry, topology, and M-theory,”
J. Diff. Geom. 70 (2005) 329
F. R. Harvey, H. B. Lawson Jr. and J. Zweck,
“The de Rham-Federer theory of differential characters and character duality,"
Amer. J. Math. 125 (2003) 791
M. Kervaire,
“Extension d'un théorème de G. de Rham et expression de l'invariant de Hopf par une intégrale,”
C. R. Acad. Sci. France 237 (1957) 1486.
J. M. Lee,
“Introduction to smooth manifolds,”
Springer (2012).
J. Simons and D. Sullivan,
“Axiomatic characterization of ordinary differential cohomology,"
J. Topol. 1 (2008) 45
C. A. Weibel,
“An introduction to homological algebra,”
Cambridge University Press (1994).
|
1511.00582
|
In <cit.> Paicu and Zarnescu have studied an order tensor system which describes the flow of a liquid crystal. They have proven the existence of weak solutions, the propagation of
higher regularities, namely $H^s$ with $s>1$ and the weak-strong uniqueness in dimension two. This paper is devoted to the propagation of lower regularities,
namely $H^s$ for $0<s\leq 1$ and to prove the uniqueness of the weak solutions. For the completeness of this research, we also propose an alternative approach in order to prove the existence
of weak solutions.
Key words: Q-Tensor, Navier-Stokes Equations, Uniqueness, Regularity, Besov Spaces, Homogeneous Paraproduct
AMS Subject Classification: 76A15, 35Q30, 35Q35, 76D03
§ INTRODUCTION AND MAIN RESULTS
§.§ Liquid Crystal
The Theory of liquid crystal materials has attracted much attention over the recent decades. Generally, the physical state of a material can be determined by the motion degree of
freedom about its molecules. Certainly, the widespread physical states of matter are the solid, the liquid and the gas ones.
If the movement degree of freedom is almost zero, namely the forces which act on the molecules don't allow any kind of movement, forcing the material structure to be confined
in a specific order, then we are classifying a solid material. If such degree still preserves a strong intermolecular force but it is not able to restrict the molecules
to lie on a regular organization, then we are considering a fluid state of matter. Finally in the gas phase the forces and the distance between the molecules are weak and large respectively, so that the material is not confined and it is able to extend its volume.
However, some materials possess some common liquid features as well as some solid properties, namely the liquid crystals. As the name suggests, a liquid crystal is a compound of fluid molecules, which has a state of matter between the ordinary liquid one and the crystal solid one. The molecules have not a positional order but they assume an orientation which can be modified
by the velocity flow. At the same time a variation of the alignment can induce a velocity field as well. In a common liquid (more correctly an isotropic liquid) if we consider the orientation of a single molecule then we should see the random variation of its position. Nevertheless, in a crystal liquid, we see an amount of orientational order.
It is well-documented that liquid crystals have been well-known for more than a century, however they have received a growth in popularity and much study only in recent decades, since they have attracted much attention thanks to their potential applications (see for instance <cit.>).
Commonly, in literature the liquid crystals are categorized by three sub-families, namely the nematics, the cholesterics and the smectics. On a nematic liquid crystal, the molecules have
the same alignment with a preferred direction, however their positions are not correlated.
On a cholesteric liquid crystal we have a foliation of the material where on each plaque the molecules orient themselves with the same direction (which could vary moving on the foliation). As in the nematic case, a cholesteric liquid crystal doesn't require any kind of relation
between the positions of the molecules.
At last, on a smectic liquid crystal we have still a privileged direction for all the molecules, as in the nematic case, however the position of them is bonded by a stratification. In addition to the orientational ordering, the molecules lie in layers.
§.§ The Order Tensor Theory
A first mathematical approach to model the generic liquid crystals has been proposed by Ericksen <cit.> and Leslie <cit.> over the period of 1958 through 1968. Even if they have presented a system which has been extensively studied in literature, for instance in <cit.> and <cit.>, several mathematical challenges and difficulties reside in such
model. Hence, in 1994, Baris and Edwards <cit.> proposed an alternative approach based on the concept of order Q-tensor, that one can find also in physical literature, for example <cit.> and <cit.>. The reader can find an exhaustive introduction to the Q-tensor Theory in a recent paper of Mottram and Newton <cit.>, however we present here some hints in order to introduce the Q-tensor system.
Let us assume that our material lies on a domain $\Omega$ of $\RR^3$. A first natural strategy to model the molecules orientation is to introduce a vector field $n$, the so called director field (see for instance <cit.>, which returns value on $\SSS^2$, the boundary of the unit sphere on $\RR^3$. Here $n(t,x)$ is a specific vector for any fixed time and for any $x\in \Omega$. An alternative approach is not to consider a precise position on $\SSS^2$ but to establish the probability that $n(t,x)$ belongs to some measurable subset $\Aa\subseteq\SSS^2$. Therefore we introduce a continuously distributed measure $\Pp$ on $\SSS^2$, driven by a density $\rho$
\begin{equation*}
\Pp(\Aa) = \int_{\Aa} \rho(P)\dd \sigma(P) = \int_{\Aa}\dd \rho(P).
\end{equation*}
We supposed the molecules to be unpolar, so that there is no difference between the extremities of them, so mathematically the probability $\Pp(\Aa)$ is always equal to $\Pp(-\Aa)$, which yields that the first order momentum vanishes:
\begin{equation*}
\int_{\SSS^2} \rho(P)\dd \sigma = 0.
\end{equation*}
Now considering the second order momentum tensor, given by
\begin{equation*}
M := \int_{\SSS^2} P\otimes P \dd \rho(P) = \left( \int_{\SSS^2} P_iP_j \dd \rho(P) \right)_{i,j=1,2,3} \in \MM_{3}(\RR),
\end{equation*}
where $\MM_{3}(\RR)$ denotes the $3\times 3$ matrices with real coefficients, we observe that $M$ is a symmetric matrix and it has trace $\trc M=1$.
In the presence of an isotropic liquid, the orientation of the molecules is uniform in every direction, hence in this case the probability $\Pp_0$ is given by
\begin{equation*}
\Pp_0(\Aa) = \int_{\Aa} \dd \sigma(P),
\end{equation*}
so that the corresponding second order momentum $M_0$ is exactly $\Id/3$. We denote by $Q$ the difference between a general $M$ and $M_0$ obtaining a tensor which is known as the de Gennes order parameter tensor. Roughly speaking, $Q$ interprets the deviation between a general liquid crystal and an isotropic one. From the definition, it is straightforward that $Q$ is a symmetric tensor and moreover it has null trace. If $Q$ assumes the form $s_+(n\otimes n -\Id/3)$, where $s_+$ is a suitable constant, then the system which models the liquid crystal (and we are going to present) reduces to the general Ericksen-Leslie system (see for instance <cit.>).
§.§ The Q-Tensor System
The present paper is devoted to the global solvability issue for the following system as an evolutionary model for the liquid crystal hydrodynamics:
\begin{equation}\label{main_system}
\tag{$P$}
\begin{cases}
\; \partial_t Q + u\cdot \nabla Q - \Omega Q + Q \Omega = \Gamma H( Q) &\RR_+ \times \RR^2,\\
\; \partial_t u + u \cdot \nabla u-\nu\Delta u +\nabla \Pi= L\Div\,\{\;Q \Delta Q - \Delta Q Q - \nabla Q\odot \nabla Q\;\} &\RR_+ \times \RR^2,\\
\; \Div\,u = 0 &\RR_+ \times \RR^2,\\
\; (u,\,Q)_{t=0} = (u_0,\, Q_0) &\quad\quad\;\;\RR^2,
\end{cases}
\end{equation}
Here $Q=Q(t,x)\in \MM_3(\RR)$ denotes the order tensor, $u=u(t,x)\in \RR^3$ represents the velocity field, $\Pi = \Pi(t,x)\in \RR$ stands for the pressure, everything depending on the time variable $t\in \RR$ and on the space variable $x\in \RR^2$. The symbol $\nabla Q\odot \nabla Q$ denotes the $3\times 3$ matrix whose $(i, j)$-th entry is given by $\trc (\partial_i Q\, \partial_jQ)$, for $i,\,j = 1,\,2,\,3$. Moreover $\Gamma$, $\nu$ and $L$ are three positive constants.
The left hand side of the order tensor equation is composed by a classical transport time derivative while, defining $\Omega$ as the
antisymmetric matrix $\Omega := (\nabla u - \tr \nabla u)1/2$, $Q\Omega- \Omega Q$ is an Oldroyd time derivative and describes how the flow gradient rotates and stretches the order parameter.
On the right-hand-side, $H(Q)$ denotes
\begin{equation*}
H(Q) :=
\underbrace{
-a Q
+b \Big(
Q^2 - \trc (Q^2)\frac{\Id}{3}
\Big)
-c\, \trc(Q^2)Q
+L \Delta Q,
\end{equation*}
and $P$ is the so called Landau-de Gennes thermotropic forces (more precisely it is a truncated taylor expansion about the original one, see for instance <cit.>). Here $a$, $b$ and $c$ are real constant, and from here on we are going to assume $c$ to be positive.
In reality, Systems (<ref>) is a simplification of a more general one. More precisely, fixing a real $\xi\in [0,1]$, we consider
\begin{equation}\label{main_system_xi}
\tag{$P_\xi$}
\begin{cases}
\; \partial_t Q + u\cdot \nabla Q - S(\nabla u, Q) = \Gamma H( Q) &\RR_+ \times \RR^2,\\
\; \partial_t u + u \cdot \nabla u-\nu\Delta u +\nabla \Pi= \Div\,\{\tau + \sigma\} &\RR_+ \times \RR^2,\\
\; \Div\,u = 0 &\RR_+ \times \RR^2,\\
\; (u,\,Q)_{t=0} = (u_0,\, Q_0) &\quad\quad\;\;\RR^2,
\end{cases}
\end{equation}
where $S(\nabla u,\,Q)$ stands for
\begin{equation*}
S(Q, \nabla u) := (\xi\,D + \Omega) \Big( Q + \frac{\Id}{2} \Big) +
\Big( Q + \frac{\Id}{2} \Big) (\xi\,D - \Omega ) -
2\xi\Big( Q + \frac{\Id}{2} \Big) \trc( Q \nabla u ),
\end{equation*}
with $D:= ( \nabla u + \tr \nabla u)1/2$. Moreover $\tau$ and $\sigma$ are the symmetric and antisymmetric part of the the additional stress tensor respectively, namely
\begin{align*}
\tau &:= -\xi (
Q + \frac{\Id}{2}
)H(Q) -
\xi H(Q)(
Q + \frac{\Id}{2}
+2\xi (
Q + \frac{\Id}{2}
\trc \{ Q H(Q) \}
-L \{
\nabla Q\odot \nabla Q + \frac{\Id}{3} | Q |^2
\},\\
\sigma &:= Q H(Q) - H(Q) Q.
\end{align*}
Here $\xi$ is a molecular parameter which describes the rapport between the tumbling and aligning effect that a shear flow exert over the liquid crystal directors. In all this paper we are going to consider the simplest case $\xi=0$, namely system (<ref>), however we predict that all our results are available for the general case and we will prove them in a forthcoming paper.
Before going on, let us recall what we mean by a weak solution of system (<ref>).
Let $Q_0 $ and $u_0$ be a $3\times 3$ matrix a $3$-vector respectively, whose components belong to $L^2( \RR^2 )$. We say that $(u,\,Q)$ is a weak solution for
(<ref>) if $u$ belongs to $L^\infty_{loc}(\RR_+ , L^2_x)\cap L^2_{loc}(\RR_+,\Hh^1)$, $Q$ belongs to $C(\RR_+, H^1)\cap L^2_{loc}(\RR_+,\Hh^2)$ and (<ref>) is
fulfilled in the distributional sense.
§.§ Some Developments in the order tensor Theory
Although the Q-tensor theory has received much attention in several disciplines as Physics <cit.>, numerical analysis <cit.>, mathematical analysis <cit.>, the solvability study of the related system has not received numerous investigations, yet. We recall here
some recent results.
in <cit.>, D. Wang, X. Xu and C. Yu have developed the existence and long time dynamics of globally defined weak solution. In their paper, system (<ref>) has been considered in the compressible and inhomogeneous setting, the fluid density $\rho$ not necessarily constant, described by a transport equation, and moreover a pressure dependent on $\rho$.
In <cit.> J. Fahn and T. Ozawa prove some regularity criteria for a local strong solution of system (<ref>).
In <cit.>, M. Paicu and A. Zarnescu first show the existence of a Lyapunov functional for system (<ref>). Then they prove the existence of a weak solution thanks to a Friedrichs scheme. They also show the propagation of higher regularity, namely $H^{s}(\RR^2)\times H^{1+s}(\RR^2)$ for $(u,\,Q)$, with $s>1$. At last they established an uniqueness result on the condition that one of the two considered solutions is a strong-solution, that is they prove the weak-strong uniqueness.
In <cit.> M. Paicu and A. Zarnescu prove the same results as in <cit.> for system (<ref>) when $\xi$ is a general value of $[0,\xi_0]$ for some $0<\xi_0<1$.
In <cit.> F. G. Guillén-Gonzàlez and L. A. Rodríquez-Bellido show the existence and uniqueness of a local in time weak solution on a bounded domain. They also give a regularity criterion which yields such solutions to be global in time. Moreover they prove the global existence and uniqueness of a strong solution provided a viscosity large enough.
In <cit.> F. G. Guillén-Gonzàlez and L. A. Rodríquez-Bellido prove the existence of global in time weak-solutions, an uniqueness criteria and a mximum principle for $Q$. They also established the traceless and symmetry for $Q$, for any weak solution.
§.§ Main Results
Article <cit.> is probably one of the best-known research interesting the solvability of (<ref>), globally in time and in the whole space. However the author's results present some gaps, therefore this article is mainly devoted to fill them, and complete their paper. Now, let us go into the details.
First Paicu and Zarnescu have proven an uniqueness result on the condition that at least one of the considered solutions is a strong solution. This is due to the necessity to control $(u(t), \,\nabla Q(t))$ in $L^\infty(\RR^2)$, which is strictly correlated to control $(u(t),\,\nabla Q(t))$ in $H^s(\RR^2)$ with $s>1$, thanks to the Sobolev Embedding. However, such necessity turns out from the attempt to estimate the difference between two solutions in the same space the solutions belong to. Here, we are able to overcome the drawbacks thanks to a strategy which is inspired by <cit.> and <cit.>. Indeed, since the difference between two solutions has a null initial datum, then it is possible to estimate such difference in less regular spaces than the ones related to the existence part. Hence the cited difficulties disappear and we are able to prove the uniqueness of the weak solutions. Then, our first result reads as follows:
Let us assume that system (<ref>) admits two weak solutions $(u_i,\,Q_i),\,i=1,\,2$, in the sense of of definition <ref>. Then such solutions are equal,
$(u_1,\,Q_1) \equiv (u_2,\,Q_2)$.
The second (and last) gap concerns the propagation of regularity. Paicu and Zarnescu consider initial data $(u_0,\, Q_0)$ in $H^s(\RR^2)\times H^{1+s}(\RR^2)$, with s greater than $1$. Then, they are able to prove that such high-regularity is preserved by the related solution of (<ref>). Denoting by
\begin{equation*}
f(t) := \|u(t)\|_{\Hh^s}^2 + \|\nabla Q(t)\|_{\Hh^s}^2 ,\quad\quad g(t) := \|\nabla u(t)\|_{\Hh^{s}}^2 + \|\Delta Q(t)\|_{\Hh^s}^2,
\end{equation*}
the major part of their proof releases on the Osgood Theorem, applied on an inequality of the following type:
\begin{equation*}
\frac{\dd}{\dd t}f(t) +g(t) \leq C f(t)\ln\{e+f(t)\},\quad t\in\RR_+,
\end{equation*}
for a suitable positive constant $C$. However such estimate requires again to control $\|(u(t), \,\nabla Q(t))\|_{L^\infty}$ by $\|(u(t), \,\nabla Q(t))\|_{H^s}$, and this is true only if $s$ is greater than $1$. We fix such lack, namely we extend the propagation for $0<s$, passing through an alternative approach. Indeed we control the $L^\infty$-norm by a different method (see Lemma <ref> and (<ref>)). Thus, our second result reads as follows:
Assume that $(u_0,\,Q_0)$ belongs to $H^s(\RR^2)\times H^{1+s}(\RR^2)$, with $0<s$. Then, the solution $(u,\,Q)$ given by Theorem <ref> fulfills
\begin{equation*}
(u,\,\nabla Q)\in L^\infty_{t,loc}\Hh^s(\RR^2)\cap L^2_{t,loc}\Hh^{s+1}(\RR^2) .
\end{equation*}
Now, we have also chosen to perform an existence result, for the completeness of this project, although it has already been proven by Paicu and Zarnescu. Nevertheless, here we use an alternative approach to prove the theorem. Indeed in <cit.>, the authors utilize a Friedrichs scheme, regularizing every equation of (<ref>), while our method is based on a coupled technique between the Friedrichs scheme and the Schaefer's fixed point theorem, so that we have only to regularize the momentum equation of (<ref>). This method is inspired by <cit.>, where F. Lin use a modified Galerkin method coupled with the Schauder fixed point theorem, in the proof of an existence result. Then we are going to prove the following Theorem:
Assume that $(u_0,\,Q_0)$ belongs to $L^2(\RR^2)\times H^1(\RR^2)$, then system (<ref>) admits a global in time weak solution $(u,\,Q)$, in the sense
of definition <ref>.
The structure of this article is over simplistic: in the next section we recall some classical tools which are useful for our proofs, in section three we deal with Theorem <ref>, the existence of weak solutions, in section four and five we establish Theorem <ref>, i.e. such solutions are unique, and finally in section six we determine Theorem (<ref>), proving the propagation of regularities. We put forward in the appendix some technical details, for the simplicity of the reader.
§ PRELIMINARIES AND NOTATIONS
In this section we illustrate some widely recognized mathematical tools and moreover we report some notations which are going to be extensively utilized in this research.
§.§ Sobolev and Besov Spaces
First, let us introduce the spaces we are going to work with (we refer the reader to <cit.> for an exhaustive study and more details) . We recall the well-known definition of Homogeneous Sobolev space $\Hh^s$ and Non-Homogeneous Sobolev Space $H^s$:
Let $s\in\RR$, the Homogeneous Sobolev Space $\Hh^s$ (also denoted $\Hh^s(\RR^2)$) is the space of tempered distribution $u$ over $\RR^2$, the Fourier transform of
which belongs to $L^1_{loc}(\RR^2)$ and it fulfills
\begin{equation*}
\| u \|_{\Hh^s} := \int_{\RR^2} |\xi|^{2s} |\hat{u}(\xi)|^2\dd \xi < \infty.
\end{equation*}
Moreover $u$ belongs to the Non-Homogeneous Sobolev Space $H^s$ (or $H^s(\RR^2)$) if $\hat{u}\in L^2_{loc}(\RR^2)$ and
\begin{equation*}
\| u \|_{\Hh^s} := \int_{\RR^2} (1+|\xi|)^{2s} |\hat{u}(\xi)|^2\dd \xi < \infty.
\end{equation*}
$H^s$ is an Hilbert space for any real $s$, while $\Hh^s$ requires $s<d/2$, otherwise it is Pre-Hilbert. Their inner products are
\begin{equation*}
\langle u,\,v \rangle_{H^s} = \int_{\RR^2} (1+|\xi|)^{2s} \hat{u}(\xi)\overline{\hat{v}(\xi)}\dd \xi
\quad \text{and} \quad
\langle u,\,v \rangle_{\Hh^s} = \int_{\RR^2} |\xi|^{2s} \hat{u}(\xi)\overline{\hat{v}(\xi)}\dd \xi,
\end{equation*}
respectively. Even if such dot-products are the most common ones, from here on we are going to use the ones related to the Besov Spaces (at least for the homogeneous case). Hence, first
we need to define them. In order to do that, it is fundamental to introduce the Dyadic Partition. Let $\chi=\chi(\xi)$ be a smooth function whose support is inside the the ball $|\xi|\leq 1$. Let us assume that $\chi$ is identically equal to $1$ in $|\xi|\leq 3/4$, then, imposing $\varphi_q (\xi ) := \chi(\xi2^{-q-1}) - \chi\big(\xi2^{-q})$ for any $q\in\ZZ$, we define the Homogeneous Litlewood-Paley Block $\Dd_q$ by
\begin{equation*}
\Dd_q f := \Ff^{-1}(\varphi_q \hat{f} ) \in \Ss', \quad \text{for any}\; f\in \Ss'.
\end{equation*}
Moreover we denote by $\Sd_j$ the operator $\sum_{q\leq j-1} \Dd_q$, for any $j\in\ZZ$.
We can now present the definition of Homogeneous Besov Space
For any $s\in \RR$ and $(p,r)\in [1,\infty]^2$, we define $\BB_{p,r}^s$ as the set of tempered
distribution $f$ such that
\begin{equation*}
\|f\|_{\BB_{p,r}^s}:= \|2^{sq}\|\dot{\Delta}_q f\|_{L^p_x}\|_{l^r(\ZZ)}
\end{equation*}
and for all smooth compactly supported function $\theta$ on $\RR^2$ we have
\begin{equation*}
\lim_{\lambda\rightarrow +\infty} \theta (\lambda D)f = 0\quad\text{in}\quad L^\infty(\RR^2).
\end{equation*}
It is straightforward that the space $\BB_{2,2}^s$ and $\Hh^s$ coincides for any real $s$, and their norms are equivalent, so we will use the following abuse of notation from here on:
\begin{equation*}
\langle u,\,v \rangle_{\Hh^s} := \langle u,\,v \rangle_{\BB^s_{2,2}} = \sum_{q\in \ZZ} 2^{2qs} \langle \Dd_q u, \, \Dd_q v\rangle_{L^2_x},
\end{equation*}
where $\langle \cdot, \cdot \rangle_{L^2}$ is the common inner product of $L^2_x:=L^2(\RR^2)$.
A profitable feature of the Homogeneous Besov space with negative index $s$ is the following one (see Proposition $2.33$ of <cit.>)
Let $s<0$ and $1\leq p,r\leq \infty$. Then $u$ belongs to $\BB_{p,r}^s$ if and only if
\begin{equation*}
\big(2^{qs}\|\Sd_q u \|_{L^p_x} \big)_{q\in\ZZ} \in L^r(\ZZ).
\end{equation*}
Moreover there exists two positive constant $c_s$ and $C_S$ such that
\begin{equation*}
c_s\| u \|_{\BB_{p,r}^s} \leq \| \big(2^{qs}\|\Sd_q u \|_{L^p_x} \big)_{q\in\ZZ} \|_{l^r(\ZZ)} \leq C_s \| u \|_{\BB_{p,r}^s}.
\end{equation*}
§.§ Homogeneous Paradifferential Calculus
In this subsection we give some hints about how the product acts between $\Hh^s$ and $\Hh^t$, for some appropriate real $s$ and $t$. We present several tools which will play a major part in all our proofs. First, let us begin with the following Theorem, whose proof is put forward in the appendix:
Let $s$ and $t$ be two real numbers such that $ |s|$ and $|t|$ belong to $[0,1)$. Let us assume that $s+t$ is positive, then for every
$a\in \Hh^s$ and for every $b\in \Hh^t$, the product $ab$ belongs to $\Hh^{s+t-1}$ and there exists a positive
constant (not dependent on $a$ and $b$) such that
\begin{equation*}
\|a b\|_{\Hh^{s+t-1}}\leq C\|a\|_{\Hh^{s}}\|b\|_{\Hh^t}
\end{equation*}
We have already remarked that $\Hh^s$ coincides with $\BB_{2,2}^s$ and this correlation allows us to incorporate in our tools the so-called Bony decomposition:
\begin{equation*}
f g = \Th_fg + \Th_g h +\Rd(f,\,g),\quad
\text{with}\quad
\Th_fg := \sum_{q\in\ZZ} \Sd_{q-1}f \Dd_q g \quad\text{and}\quad
\Rd(f,\,g):= \sum_{q\in\ZZ,\,|l|\leq 1}\Dd_q f \Dd_{q+l}g.
\end{equation*}
However, such decomposition is not going to be useful for every challenging estimation, so that we are going to use the so called symmetric decomposition, in order to overcome the
drawbacks. Here, we present directly the matrix-formulation of such decomposition, since it will be used on such framework. Let $q$ be an integer, and $A$, $B$ be $N\times N$ matrices, whose components are homogeneous temperate distributions. Denoting by
\begin{equation}\label{simmmetric_decomposition}
\begin{array}{ll}
\J_q^1(A,B) :=\sum_{|q-q'|\leq 5 } [\Dd_q, \, \Sd_{q'-1}A] \Dd_{q'} B,
&\J_q^3(A,B) := \Sd_{q -1} A \Dd_{q} B,\\
\\
\J_q^2(A,B) :=\sum_{|q-q'|\leq 5 } ( \Sd_{q'-1}A -\Sd_{q-1}A) \Dd_{q} \Dd_{q'} B,
&\J_q^4(A,B) :=\sum_{ q' \geq q- 5 } \Dd_q(\Dd_{q'}A\, \Sd_{q'+2} B),
\end{array}
\end{equation}
the following classical feature concerning the product $AB$, is satisfied:
\begin{equation}\label{product_decomposition}
\Dd_q(AB) = \J_q^1(A,B)
+ \J_q^2(A,B)
+ \J_q^3(A,B)
+ \J_q^4(A,B),
\end{equation}
for any integer $q$.
§.§ The Frobenius Norm
Before beginning with the proofs of our main results, let us give the following remark:
The most common inner product defined on $\MM_3(\RR)$ (the $3\times3$ real matrices) is determined by:
\begin{equation*}
A\cdot B = \sum_{i,j=1}^3A_{ij}B_{ij} = \trc \{ \tr A B\},\quad\quad\text{for any}\quad A,\,B\in \MM_3(\RR).
\end{equation*}
Hence, if at least one of the two matrices is symmetric, for instance $A$, then we obtain
\begin{equation}\label{Frobenius_inner_product}
A\cdot B = \trc\{AB\},
\end{equation}
which determines the well-known Frobenius norm of a matrix $|A|:= \sqrt{\trc\{A^2\}}$. Since any solution $(u,\,Q)$ for (<ref>) fulfills
\begin{equation*}
Q(t,x)\in S_0 := \big\{ A\in \MM_3(\RR),\,\trc\{ A \} = 0\quad\text{and}\quad \tr A = A \big\},
\end{equation*}
for almost every $(t,x) \in \RR_+\times \RR^2$ (see <cit.> and <cit.>), then from here on we will repeatedly use (<ref>).
Moreover, we will use the symbol $\lesssim$ (instead of $\leq$) which is defined as follows: for any non-negative real numbers $a$ and $b$, $a\lesssim b$ if and only if there exists a positive constant $C$ (not dependent on $a$ and $b$) such that $a \leq C\,b$.
§ WEAK SOLUTIONS
This section deals with the existence of weak solutions for (<ref>) in the sense of definition <ref>. As we have already explained, we are going to proceed with a coupled method between the Friedrichs scheme and the Schaefer's Theorem. Hence, before going on, let us recall the widely recognized Schaefer's fixed point Theorem
Let $\Psi$ be a continuous and compact mapping of a Banach Space $X$ into itself,
such that the set $\{\,x\in X\,:\:x=\lambda\, \Psi x\;\,\text{for some}\;\, 0
\leq \lambda\leq 1\}$ is bounded. Then $T$ has a fixed point.
First, we introduce one of the key ingredient of our proofs, namely the mollifying
operator $J_n$ defined by
\begin{equation*}
\Ff( J_n f)(\xi) = 1_{[\frac{1}{n},\,n]}(\xi) \quad\quad
\text{for } \xi \in \RR^2_\xi,
\end{equation*}
which erases the high and the low frequencies.
We claim the existence and uniqueness of a solution for the following system
\begin{equation}\label{system_friedrichs}
\tag{$P_n$}
\begin{cases}
\; \partial_t Q + ( J_nu\cdot \nabla Q) -
J_n\Omega Q + Q J_n\Omega = \Gamma H( Q) &[0,T) \times \RR^2,\\
\; \partial_t u + J_n \Pp(\, J_nu \cdot
\nabla J_n u\, )
-\nu\Delta u = LJ_n\Pp\Div\,\{\;Q \Delta Q -
\Delta Q Q
- \nabla Q\odot \nabla Q\;\}
&[0,T) \times \RR^2,\\
\; \Div\,u = 0 &[0,T) \times \RR^2,\\
\; (u,\,Q)_{t=0} = (u_0,\, Q_0) &\quad\quad\quad\;\;\RR^2,
\end{cases}
\end{equation}
where $\Pp$ stands for the Leray projector operator, which is determined by
\begin{equation*}
\Ff\{\,\Pp f\,\}(\xi) :=
\hat{f}(\xi) - \frac{\xi}{|\xi|}\,\frac{\xi}{|\xi|}\cdot \hat{f}(\xi),
\quad\quad
\text{for}\quad f \in (L^p_x)^2, \quad\text{with}\quad 1<p<\infty,
\end{equation*}
and $T$ is a positive real number. It is well known that $\Pp$ is a bounded operator of $(L^p_x)^2$ into itself
when $p\in (1,\infty)$.
We say $(u,\,Q)$ is a weak solution of the problem (<ref>),
provided that
\begin{equation*}
u \in C([0,T], L^2_x)\cap L^2(0,T;\, \Hh^1)\,,\quad \quad
Q \in C([0,T];H^1)\cap L^2(0,T; \Hh^2)
\end{equation*}
and (<ref>) is valid in the distributional sense.
The following proposition plays a major part in our main proof, since it allows us
to control the $L^p_x$-norm of $Q$ only by $Q_0$.
Suppose that $u\in C([0,T], L^2_x)\cap L^2(0,T; \Hh^1)$ and moreover that
$Q\in C([0,T],H^1)\cap L^2(0,T; \Hh^2)$ is a
weak solution of
\begin{equation*}
\partial_t Q + u\cdot \nabla Q - \Omega Q + Q \Omega - \Gamma L\Delta Q =
\Gamma P( Q)\quad\text{in}\quad[0,T)\times \RR^2,
\quad\quad\text{and}\quad\quad
Q_{t=0} = Q_0 \in H^1.
\end{equation*}
Then, for every $2\leq q<\infty$, the following estimate is fulfilled
\begin{equation*}
\| Q(t) \|_{L^q_x} \leq \|Q_0\|_{H^1}\exp\{Ct\},
\end{equation*}
for a suitable positive constant $C$ dependent only on $q$, $\Gamma$, $a$, $b$ and $c$.
Fixing $p\in (1,\infty)$, We multiply both left and right-hand side by $2pQ\,\trc\{Q^2\}^{p-1}$, we take the trace and we
integrate in $\RR^2$, obtaining that
\begin{equation*}
\frac{\dd}{\dd t}\|Q(t)\|_{L^{2p}_x}^{2p} -
\Gamma 2Lp \langle Q(t)\trc\{Q(t)^2\}^{p-1},\Delta Q(t)\rangle_{L^2_x} =
2\Gamma p \int_{\RR^2}\trc\{Q(t)^2\}^{p-1} \trc\{\, P(Q(t,x))Q(t,x) \,\}\dd x,
\end{equation*}
for almost every $t\in (0,T)$, where we have used $\Div\, u = 0$ and $\trc\{Q\Omega Q-\Omega Q^2\}=0$ .
First, analyzing the second term on the left-hand side, integrating by parts, we determine the following identity:
\begin{align*}
-\langle 2pQ\trc\{Q^2\}^{p-1},\,&\Delta Q \rangle_{L^2}
\sum_{i=1}^2\Big[
2p \int_{\RR^2}\trc\{Q^2\}^{p-1}\trc\{(\partial_i Q)^2\} +
2p \int_{\RR^N}\partial_i[\trc\{Q^2\}^{p-1}] \trc\{Q\partial_iQ \}\Big]\\
&=2p \int_{\RR^N} \trc\{Q^2\}^{p-1}|\nabla Q|^2 +
4p(p-1) \int_{\RR^N}\trc\{Q^2\}^{p-2} |\nabla [\trc\{Q^2\}] |^2\geq 0,
\end{align*}
which allows us to obtain
\begin{equation*}
\frac{\dd}{\dd t}\|Q(t)\|_{L^{2p}}^{2p}\leq \Gamma \int_{\RR^2}2p\trc\{Q^2\} \trc\{\, P(Q(t,x))Q(t,x) \,\}\dd x.
\end{equation*}
Now, we deal with the right-hand side by a direct computation, observing that
\begin{align*}
\int_{\RR^2}\trc\{Q^2\}^{p-1} \trc\{ P(Q)Q\}\dd x
\Gamma \int_{\RR^2}
\Big[ \,
- a\, \trc\{ Q^2 \}^{p}
+ b\, \trc\{Q^2\}^{p-1}\trc\{ Q^3 \}
- c\, \trc\{ Q^2 \}^{p+1}\,
\Big]\\
\|Q\|_{L^{2p}_x}^{2p} -\frac{c}{2} \|Q\|_{L^{2(p+1)}_x}^{2(p+1)}
\lesssim
\|Q\|_{L^{2p}_x}^{2p},
\end{align*}
where we have used the following feature about a symmetric matrix with null trace:
\begin{equation*}
\big|\int_{\RR^2}\trc\{Q^2\}^{p-1}\trc\{Q^3\} \big|\leq \ee\|Q\|_{L^{2(p+1)}}^{2(p+1)} + \frac{1}{\ee}\|Q\|_{L^{2p}}^{2p},
\end{equation*}
for a positive real $\ee$, small enough.
Indeed, if $Q$ has $\lambda_1$, $\lambda_1$, and $-\lambda_1-\lambda_2$ as eigenvalues, we achieve that
$\trc\{Q^3\}=-3\lambda_1\lambda_2(\lambda_1+\lambda_2)$ and $\trc\{Q^2\} = 2(\lambda_1^2+\lambda_2^2+\lambda_1\lambda_2)$, hence
\begin{equation*}
\lesssim
\ee\lambda_1^2\lambda_2^2 + \frac{1}{\ee}(\lambda_1^2+ \lambda_2^2+2\lambda_1\lambda_2)
\lesssim
\ee(\lambda_1^2+\lambda_2^2+\lambda_1\lambda_2)^2 + \frac{1}{\ee}(\lambda_1^2+ \lambda_2^2+\lambda_1\lambda_2)
\lesssim
\ee\trc\{Q^2\}^2+\frac{1}{\ee}\trc\{Q^2\}.
\end{equation*}
Therefore, we deduce that
\begin{equation}\label{bound_int_Q^3}
\big|\int_{\RR^2}\trc\{Q^2\}^{p-1}\trc\{Q^3\} \big| \lesssim
\ee
\int_{\RR^N}\trc\{Q^2\}^{(p+1)}+
\frac{1}{\ee}
\int_{\RR^N}\trc\{Q^2\}^{p}.
\end{equation}
Summarizing the previous consideration, we get
\begin{equation*}
\frac{1}{2}\frac{\dd}{\dd t}\|Q(t)\|_{L^{2p}_x}^{2p} \lesssim
\|Q(t)\|_{L^{2p}_x}^{2p},
\end{equation*}
so that the statement is proved, thanks to the Gronwall's inequality.
Now, let us focus on one of the main theorems of this section, which reads as follows:
Let $n$ be a positive integer and assume that $(u_0,\,Q_0)$ belongs to $L^2_x$.
Then, system (<ref>) admits a unique local weak solution.
The key method of the proof relies on the Schauder's Theorem. We define the
compact operator $\Psi$ from $C([0,T], L^2_x)^2\cap L^2(0,T;\, \Hh^1)^2$ to itself
as follows: $(\Psi(u),\, Q)=: (\tilde{u},\,Q)$ is the unique weak solution (in
the sense of remark <ref>) of the following Cauchy problem:
\begin{equation*}
\begin{cases}
\; \partial_t Q + (J_n u\cdot \nabla Q)
- J_n\Omega Q + QJ_n\Omega = \Gamma H( Q) &[0,T) \times \RR^2,\\
\; \partial_t \tilde{u} +
J_n \Pp(\, J_n\tilde{u} \cdot
\nabla J_n \tilde{u}\, )
- \nu\Delta \tilde{u} = LJ_n\Pp\Div\,\{\;Q \Delta Q
- \Delta Q Q - \nabla Q\odot \nabla Q\;\} &[0,T) \times \RR^2,\\
\; \Div\,\tilde{u} = 0 &[0,T) \times \RR^2,\\
\; (\tilde{u},\,Q)_{t=0} = (u_0,\, Q_0) &\quad\quad\quad\;\;\RR^2.
\end{cases}
\end{equation*}
We claim that the hypotheses of the Schauder's Theorem are fulfilled, namely
$\Psi$ is a compact mapping of $X:=C([0,T], L^2_x)\cap L^2(0,T;\, \Hh^1)$ into
itself, and the set
$\{\,u=\lambda\,\Psi (u)\;\,\text{for some}\;\,0\leq \lambda\leq 1\}$.
is bounded. First we deal with the compactness of $\Psi$. Considering a bounded
family $\FF$ of $X$, we claim that the closure of $\Psi(\FF)$ is compact in $X$.
If we prove that $\Psi(\FF)$ is an uniformly bounded and equicontinuous
family of $C([0,T]; L^2_x)$ and moreover that
$\{ \,\Psi(u)(t)\;\text{with }t\in [0,T]\;\text{and }u\in \FF\}$ is a compact
set of $L^2_x$, then the result is at least valid as $\Psi$ mapping of $X$ into
$C([0,T], L^2)$, thanks to the Arzelà-Ascoli Theorem. Multiplying the first
equation by $Q-\Delta Q$ and integrating in $\RR^2$, we get
\begin{align*}
\frac{1}{2}
&\frac{\dd}{\dd t}
\Big[
\| \nabla Q \|_{L^2_x}^2 +
\| Q \|_{L^2_x}^2
\Big] +
\Gamma L
\big(
\|\nabla Q\|_{L^2_x}^2+
\|\Delta Q\|_{L^2_x}^2
\big)
\int_{\RR^2}
\Big[
\trc\{(J_n \Omega Q - Q J_n\Omega)\Delta Q\,\} -\\&-
\trc\{(J_n u\cdot \nabla Q ) \Delta Q \,\}
\Big] +
\Gamma L
\int_{\RR^2}
\Big[ \,
a\, \trc\{ Q \Delta Q \} -
b\, \trc\{ Q^2 \Delta Q \} +
c\, \trc\{ Q \Delta Q \}\trc\{ Q^2 \} \,
\Big]+ \\&
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+
\Gamma \int_{\RR^2}
\Big[ \,
a\, \trc\{ Q^2 \} -
b\, \trc\{ Q^3 \} +
c\, \trc\{ Q^2 \}^2 \,
\Big],
\end{align*}
almost everywhere in $(0,T)$, which allows us to achieve
\begin{align*}
\frac{\dd}{\dd t}
\| \nabla Q(t) \|_{L^2_x}^2 +
\| Q(t) \|_{L^2_x}^2
\Big] +
\Gamma L
\| \Delta Q(t) \|_{L^2_x}^2 \leq\\&
\leq
\big( 1 + \| u(t) \|_{L^2_x}^2 \big)
\big( \| Q(t) \|_{L^2}^2 + \| \nabla Q(t) \|_{L^2_x}^2 \big) +
\frac{\Gamma L}{100}
\| \Delta Q(t) \|_{L^2_x},
\end{align*}
where $C_n$ is a positive constant dependent on $n$. Therefore, we realize that
the family composed by $Q=Q(u)$ as $u$ ranges on $\FF$ is a bounded family
in $C([0,T];H^1)\cap L^2(0,T; \Hh^2)$. Now, multiplying the second equation by
$\tilde{u}$ we get the following equality:
\begin{equation*}
\frac{1}{2}
\frac{\dd}{\dd t}
\| \tilde{u}(t) \|_{L^2_x}^2 +
\nu
\| \nabla \tilde{u}(t) \|_{L^2_x}^2
\trc\{\,
\big(
\nabla Q\odot \nabla Q +
Q \Delta Q - \Delta Q\, Q
\big) \nabla \tilde{u} \,\}(t,x)\dd x=: F(t),
\end{equation*}
for almost every $t\in (0,T)$. Thus it turns out that
\begin{equation}\label{Thm_Friedrichs_est_tilde_u}
\frac{\dd}{\dd t}
\| \tilde{u}(t) \|_{L^2_x}^2 +
\nu
\| \nabla \tilde{u}(t) \|_{L^2_x}^2
\leq
\leq
\| (Q(t),\, \nabla Q(t),\, \Delta Q(t)) \|_{L^2_x}^2 +
\| \tilde{u}(t) \|_{L^2_x}^2,
\end{equation}
where $C_n>0$ depends on $n$. Here, we have used the feature $J_n \tilde{u} =
\tilde{u}$, which comes from the uniqueness of the solution for the second equation,
so that $\| \nabla \tilde{u}\|_{L^\infty_x}\leq C_n \|\tilde{u}\|_{L^2_x}$. Summarizing
the previous considerations and thanks to the Gonwall's inequality we discover that
$\Psi(\FF)$ is a bounded family in $X$, so in $C([0,T], L^2)$. Moreover, from
(<ref>) and the previous result,
it turns out that $|F(t)|$ is bounded on $[0,T]$, uniformly in $u\in \FF$.
Hence $\Psi(\FF)$ is an equicontinuous family of $C([0,T]; L^2_x)$. Finally,
because $J_n \tilde{u} = \tilde{u}$, we get that
$\{ \,\Psi(u)(t)\;\text{with }t\in [0,T]\;\text{and }u\in \FF\}$ is a subset of a
bounded $L^2_x$-family composed by functions with Fourier-transform supported in the
anulus $\Cc(1/n, n)$, which is a compact family of $L^2_x$. Summarizing all the
previous consideration, we get that $\Psi(\FF)$ is compact in $C([0,T], L^2_x)$
thanks to the Arzelà-Ascoli Theorem.
It remains to prove that $\Psi(\FF)$ is compact in $L^2(0,T;\Hh^1)$, so
that $\Psi$ is a compact mapping of $X$ into itself. Since $J_n \Psi(u(t)) =
\Psi(u(t))$ for every $u\in \FF$ and $t\in (0,T)$, the precompactness of
$\Psi(\FF)$ in $L^2(0,T;\Hh^1)$ is equivalent to the precompactness of $\Psi(\FF)$
in $L^2( (0,T)\times \RR^2\,)$. Recalling that $\Psi(\FF)$ is precompact in
$C([0,T], L^2_x)$ which is embedded in $L^2( (0,T)\times \RR^2\,)$ (for $T$ finite),
then we determine the result, so that, in conclusion $\Psi$ is a compact operator
from $X$ to itself.
Now, we deal with the Schaefer's Theorem hypotheses, namely the set
$\{\, u=\lambda \Psi(u) \;\text{for some}\;\lambda \in (0,1)\}$
is a bounded family of $X$. First, we point out that if $u=\lambda \psi(u)$, then
the couple $(u,\,Q)$ is a solution for
\begin{equation*}
\begin{cases}
\; \partial_t Q + \lambda J_n u\cdot \nabla Q
- \lambda J_n\Omega Q + \lambda QJ_n\Omega = \Gamma H( Q)
&[0,T) \times \RR^2,\\
\; \partial_t u +
J_n \Pp(\, J_n u \cdot
\nabla J_n u\, )
- \nu\Delta u = LJ_n\Pp\Div\,\{\;Q \Delta Q
- \Delta Q Q - \nabla Q\odot \nabla Q\;\} &[0,T) \times \RR^2,\\
\; \Div\,u = 0 &[0,T) \times \RR^2,\\
\; (u,\,Q)_{t=0} = (u_0,\, Q_0) &\quad\quad\quad\;\;\RR^2.
\end{cases}
\end{equation*}
We multiply the first equation by $Q-\Delta Q$, the second equation by
$u$, we integrate everything in $\RR^N$ and we sum the results, obtaining:
\begin{align*}
\frac{\dd }{\dd t}
\Big[\,
\| Q \|_{L^2}^2 +
\| \nabla Q \|_{L^2}^2 +
\| u \|_{L^2}^2
\Big] +
\Gamma L
\| \nabla Q \|_{L^2}^2 +
\Gamma L
\| \Delta Q \|_{L^2}^2 +
\nu \| \nabla u \|_{L^2}^2 =
\lambda
\langle J_n u\cdot \nabla Q, \, Q - \Delta Q \rangle_{L^2} + \\ +
\lambda
\langle J_n \Omega Q - Q J_n\Omega, \, Q - \Delta Q \rangle_{L^2} +
\Gamma
\langle P(Q), \, Q - \Delta Q \rangle_{L^2} -
\langle J_n u\cdot \nabla J_n u, \,\nabla J_n u \rangle_{L^2} + \\ +
L\langle Q\Delta Q - \Delta Q \,Q, \,\nabla J_n u \rangle_{L^2} +
L\langle \nabla Q\odot\nabla Q, \,\nabla J_n u \rangle_{L^2}.
\end{align*}
According to $\| J_nu \|_{L^\infty} + \|\nabla J_n \|_{L^\infty} \leq C_n \| u \|_{L^2}$,
up to a positive constant $C_n$ dependent on $n$, it is not computationally demanding to
achieve the following estimate:
\begin{align*}
\frac{\dd }{\dd t}
\Big[\,
\| Q \|_{L^2_x}^2 +
\| \nabla Q \|_{L^2_x}^2 +
\| u \|_{L^2_x}^2
\Big] & +
\Gamma L
\| \Delta Q \|_{L^2_x}^2 +
\nu \| \nabla u \|_{L^2_x}^2
\leq \\ &\leq
\tilde{C}_n
\Big[\,
\| Q \|_{L^2_x}^2 +
\| \nabla Q \|_{L^2_x}^2 +
\| u \|_{L^2_x}^2
\Big] +
\frac{\nu}{100}
\| \nabla u \|_{L^2_x}^2+
\frac{\Gamma L}{100}
\| \Delta Q \|_{L^2_x}^2.
\end{align*}
Therefore, thanks to the Gronwall's inequality, we detect the following estimate:
\begin{align*}
\| Q \|_{L^\infty(0,T;L^2_x)}^2 +
\| \nabla Q \|_{L^\infty(0,T;L^2_x)}^2 &+
\| u \|_{L^\infty(0,T;L^2_x)}^2 + \\&+
\| \Delta Q \|_{L^2(0,T;L^2_x)}^2 +
\| \nabla u \|_{L^2(0,T;L^2_x)}^2
\lesssim
\| (u_0,\,Q_0,\, \nabla Q_0)\|_{L^2_x}
e^{C_n T},
\end{align*}
so that, the family $\{\, u=\lambda \Psi(u) \;\text{for some}\;0\leq \lambda \leq 1\}$ is
bounded in $X$. Hence, applying the Schaefer's fixed point Theorem, we conclude that there exists
a fixed point for $\Psi$, namely there exists a weak solution $(u,\,Q)$
(in the sense of remark <ref>) for the system (<ref>).
In the previous proof $T$ has only to be bounded, and it has no correlation with the initial data, so that the solution $(u^n,\,Q^n)$ of system
(<ref>), given by Proposition <ref>, it should be supposed to belong to
\begin{equation*}
C(\RR_+,L^2_x)\cap L^2_{loc}(\RR_+,\Hh^1)\times C(\RR_+,H^1)\cap L^2_{loc}(\RR_+,\Hh^2).
\end{equation*}
We are now able to prove our main existence result, namely Theorem <ref>.
Let us fix a positive real $T$ and let $(u^n,\,Q^n)$ be the solution of (<ref>) given by Proposition <ref>, for any
positive integer $n$. We analyse such solutions in order to develop some $n$-uniform bound for their norms, which will allow us to apply some classical
methods about compactness and weakly convergence.
We multiply the first equation of (<ref>) by $Q^n-L\Delta Q^n$, the second one by $u^n$, we integrate in $\RR^2$ and finally we sum the results,
obtaining the following identity
\begin{equation}\label{neo}
\begin{aligned}
\frac{\dd}{\dd t}\Big[ \| u^n \|_{L^2_x} &+ \| Q^n \|_{L^2_x} + \Gamma L\| \nabla Q^n \|^2_{L^2_x} \Big] +
\nu \| \nabla u^n \|_{L^2_x} +
\Gamma L \| \nabla Q^n \|_{L^2_x} +
\Gamma L^2 \| \Delta Q^n \|_{L^2_x} = \\& =
\underbracket[0.5pt][1pt]{
- \langle u_n \cdot \nabla Q_n,\, Q_n \rangle_{L^2_x}
\underbracket[0.5pt][1pt]{
+ L \langle u_n \cdot \nabla Q_n,\, \Delta Q_n \rangle_{L^2_x}
\underbracket[0.5pt][1pt]{
+ \langle \Omega_n Q_n - Q_n \Omega_n,\, Q_n \rangle_{L^2_x}
}_{=0} -\\&
\underbracket[0.5pt][1pt]{
- L \langle \Omega_n Q_n - Q_n \Omega_n,\, \Delta Q_n \rangle_{L^2_x}
+ \Gamma \langle P(Q_n),\, Q_n \rangle_{L^2_x}
- \Gamma L \langle P(Q_n),\, \Delta Q_n \rangle_{L^2_x} -\\&
\underbracket[0.5pt][1pt]{
- \langle u_n\cdot\nabla u_n,\, u_n \rangle_{L^2_x}
\underbracket[0.5pt][1pt]{
- L \langle Q_n \Delta Q_n - \Delta Q_n Q_n,\, \nabla u_n \rangle_{L^2_x}
}_{\Aa \Aa}
\underbracket[0.5pt][1pt]{
- L \langle\Div\{\nabla Q_n\odot\nabla Q_n\},\, u_n \rangle_{L^2_x}
\end{aligned}
\end{equation}
First, let us observe that $\Aa + \Aa\Aa = 0$ thanks to Lemma <ref>. Moreover $\langle u_n\cdot \nabla Q_n,\,Q_n\rangle_{L^2_x}$ and
$\langle u_n\cdot \nabla u_n, u_n\rangle_{L^2_x}$ are null , because of the divergence-free condition of $u_n$, while
$\langle \Omega_n Q_n - Q_n \Omega_n,\,\Delta Q_n\rangle_{L^2_x}$ is zero since $Q_n$ is symmetric. Furthermore $\Bb+\Bb\Bb = 0$ since the following identity is
\begin{equation*}
\trc\{u_n\cdot\nabla Q_n\,\Delta Q_n\} = \Div \{\nabla Q_n\odot\nabla Q_n\}\cdot u_n - \Div \{ u_n (\,|\partial_1 Q_n|^2 + |\partial_2 Q_n|^2) \}.
\end{equation*}
Recalling (<ref>) with $p=1$, it turns out that
\begin{equation*}
\Gamma \langle P(Q_n),\, Q_n \rangle_{L^2} \lesssim
\| Q_n \|_{L^2_x}^2 -\frac{c}{2}\| Q_n \|_{L^4_x}^4\leq \| Q_n \|_{L^2_x}^2,
\end{equation*}
while, by a direct computation and thanks to Proposition <ref>, we deduce
\begin{align*}
\Gamma L \langle P(Q_n),\, \Delta Q_n \rangle_{L^2_x}
\| \nabla Q_n \|^2_{L^2_x} + \|Q_n\|_{L^6}^3\|\Delta Q_n\|_{L^2_x}
\lesssim
\| \nabla Q_n \|^2_{L^2_x} + \|Q_0\|_{H^1}^6e^{6Ct}+
C_{\Gamma,L}\|\Delta Q_n\|_{L^2_x}^2,
\end{align*}
where $C$ is positive real constant, not dependent on $n$ and $C_{\Gamma,L}>0$ is a suitable small enough constant which will allow to absorb $\|\Delta Q_n\|_{L^2_x}^2$ by the left-hand
side of (<ref>). Thus, summarizing the previous considerations, we get
\begin{align*}
\frac{\dd}{\dd t}\Big[ \| u^n \|_{L^2_x}^2 + \| Q^n \|_{L^2_x}^2 + \Gamma L\| \nabla Q^n \|^2_{L^2_x}\Big] +
\nu \| \nabla u^n \|_{L^2}^2 &+
\Gamma L^2 \| \Delta Q^n \|_{L^2_x}^2\lesssim \\&
\lesssim \|Q^n\|_{L^2_x}^2 + \|\nabla Q^n\|_{L^2_x}^2 + \|Q_0\|_{H^1}^6e^{6Ct},
\end{align*}
which yields
\begin{equation}\label{Thm_fr_estimates}
\begin{aligned}
\| (u^n,\,Q^n,\,\nabla Q^n) \|_{L^\infty(0,T;L^2_x)} &+ \|(\nabla u^n,\,\Delta Q^n) \|_{L^2(0,T;L^2_x)} \lesssim \\&\lesssim
(\| u_0 \|_{L^2_x} +\|Q_0\|_{L^2_x} + \|Q_0\|_{H^1}^6 ) \exp\{\tilde{C}t\},
\end{aligned}
\end{equation}
for a suitable positive constant $\tilde{C}$, independent on $n$.
Thanks to the previous control, we carry out to pass to the limit as $n$ goes to $+\infty$, and we claim to found a weak solution for system (<ref>).
We fix at first a bounded domain $\Omega$ of $\RR^2$, with a smooth enough boundary.
At first we claim that $(Q^n)_\NN$ is a Cauchy sequence in $C([0,T], L^2(\Omega))$, and the major part of the proof releases in the Arzelà-Ascoli Theorem. We have already
proven that $(Q^n)_\NN$ is bounded in such space, moreover, since $Q_n(t)$ belongs to $H^1(\Omega)$ which is compactly embedded in $L^2(\Omega)$, we get that
$\{ Q_n(t)\,:\, n\in \NN\;\text{and}\; t\in [0,T]\}$ is a compact set of $L^2(\Omega)$. Moreover, observing that
\begin{align*}
\| \partial_t &Q^n \|_{L^2(\Omega)}
\leq
\| u^n \|_{L^4_x }
\| \nabla Q^n \|_{L^4_x } +
\| \nabla u^n \|_{L^2_x }
\| Q^n \|_{L^\infty_x} +
\| P( Q^n)\|_{L^2_x }\\
\| u^n \|_{L^2_x }^{\frac{1}{2}}
\| \nabla u^n \|_{L^2_x }^{\frac{1}{2}}
\| \nabla Q^n \|_{L^2_x }^{\frac{1}{2}}
\| \Delta Q^n \|_{L^2_x }^{\frac{1}{2}} +
\| \nabla u^n \|_{L^2_x }
\| Q^n \|_{H^2 } +
\| Q^n \|_{L^2_x } +
\| Q^n \|_{L^4_x }^2 +
\| Q^n \|_{L^6_x }^3.
\end{align*}
Therefore, it turns out that $(\partial_t Q^n)_\NN$ is an uniformly bounded sequence in $L^1(0,T;L^2_x)$ which yields that $(Q^n)_\NN$ is uniformly
equicontinuous in $C([0,T], L^2_x)$, so that, applying the Arzelà-Ascoli Theorem, there exists $Q\in C([0,T], L^2_x)$ such that $Q^n$
strongly converges to $Q$, up to a subsequence. Moreover, thanks to (<ref>), we also obtain that $\nabla Q$ and $\Delta Q$ belong to
$L^\infty(0,T;L^2_x)$ and $L^2(0,T;L^2_x)$ respectively, and we have:
\begin{equation*}
\nabla Q_n \rightharpoonup \nabla Q \quad\quad w - L^2(0,T; L^2_x)\quad\quad\text{and}\quad\quad\quad
\Delta Q^n \rightharpoonup \Delta Q \quad\quad w - L^2(0,T; L^2_x),
\end{equation*}
up to a subsequence. Now, let us fix a bounded smooth domain $\Omega$ of $\RR^2$. Then $\nabla Q^n(t)$ weakly converges to $\nabla Q(t)$ in $H^1(\Omega)$,
for almost every $t\in (0,T)$, up to a subsequence, so that, from the compact embedding $H^1(\Omega)\hookrightarrow\hookrightarrow L^2(\Omega)$, we deduce
that $\nabla Q^n(t)$ strongly converges to $\nabla Q(t)$ in $L^2(\Omega)$, for almost every $t\in (0,T)$. Moreover $\|\nabla Q_n -\nabla Q \|_{L^2(\Omega)}$ belongs to
$L^\infty(0,T)$ and its norm is uniformly bounded in $n$. Hence applying the dominated convergence Theorem, we get
\begin{equation*}
\lim_{n\rightarrow \infty}\int_0^T \|\nabla Q_n(t) -\nabla Q(t) \|_{L^2}^2\dd t =
\int_0^T\lim_{n\rightarrow \infty} \|\nabla Q_n(t) -\nabla Q(t) \|_{L^2}^2\dd t = 0,
\end{equation*}
namely $\nabla Q^n$ strongly converges to $\nabla Q$ in $L^2(0,T;L^2(\Omega))$. Since $\nabla Q^n$ is bounded in $L^2(0,T;L^6_x)$ (from the embedding
$H^1\hookrightarrow L^6_x$ we get also that $\nabla Q^n$ weakly converges to $\nabla Q$ in $w-L^2(0,T;L^6_x)$, so that $\nabla Q^n$ strongly converges to
$\nabla Q$ in $L^2(0,T;L^4(\Omega))$, by interpolation. This range of convergences finally show that $\nabla Q\odot \nabla Q$ and $Q\Delta Q - \Delta Q$ are
the limits of $\nabla Q^n\odot \nabla Q^n$ and $Q^n\Delta Q^n - \Delta Q^n Q^n$, as $n$ goes to infinity, respectively in $L^1(0,T;L^4(\Omega))$ and
$L^1(0,T; L^{4/3}(\Omega))$. the strongly convergence of $P(Q^n)$ to $P(Q)$ in $L^2(0,T;L^2(\Omega))$ is straightforward, while, with a similar strategy, we are
able to prove the existence of $u\in L^\infty(0,T; L^2_x)$ with $\nabla u\in L^2(0,T;L^2_x)$ such that $u^n$ strongly converges to $u$ in $L^2(0,T; L^4(\Omega))$
and $\nabla u^n$ weakly converges to $\nabla u$ in $L^2(0,T; L^{2}(\Omega))$ (everything up to a subsequence). Hence $u^n\cdot\nabla u^n$ and
$\Omega^n Q^n- Q^n\Omega^n$
weakly converges in $L^1(0,T;L^{4/3}(\Omega) )$ to $u\cdot\nabla u$ and $\Omega Q - Q\Omega$ respectively. Finally $u^n\cdot \nabla Q^n$ strongly converges to
$u\cdot \nabla Q$ in $L^1(0,T; L^2(\Omega) )$.
Now, $J_n \phi$ strongly converges to $\phi$ in $L^\infty(0,T; L^p_x)$, for any $\phi\in \DD(\,(0,T)\times \Omega\,)$ and for any $1\leq p<\infty$. Considering all the
previous convergences and since $(u^n,\,Q^n)$ is a weak solution of (<ref>), namely
\begin{align*}
-\int_0^T\int_{\RR^N} \trc\{Q^n \partial_t\Psi\} - \int_{\RR^N}&\trc\{ Q_0 \Psi(0,\cdot) \} + \int_0^T \int_{\RR^N}\trc\{ (u^n\cdot\nabla Q^n)\Psi\} +\\&+
\int_0^T\int_{\RR^N} \trc\{(\Omega^n Q^n - Q^n \Omega^n )\Psi\} =\Gamma \int_0^T \int_{\RR^N} \trc\{H(Q^n)\Psi\},
\end{align*}
for every $N\times N$-matrix $\Psi$ with coefficients in $\DD([0,T)\times \Omega)$ and
\begin{align*}
-\int_0^T\int_{\RR^N} u^n \cdot \partial_t \psi - \int_{\RR^N} u_0 \cdot \psi(0,\cdot) &+ \int_0^T\int_{\RR^N} (u^n\cdot \nabla u^n) \cdot \Pp J_n\psi
-\nu \int_0^T \int_{\RR^N} u^n\cdot \Delta \psi = \\ &=
-L\int_0^T \int_{\RR^N} [ Q^n\Delta Q^n - \Delta Q^nQ^n - \nabla Q^n\odot \nabla Q^n ]\cdot \Pp J_n\nabla \psi,
\end{align*}
for any $N$-vector $\psi$ with coefficients in $\DD([0,T)\times \Omega)$, we pass through the limit as $n$ goes to $\infty$, obtaining
\begin{align*}
-\int_0^T\int_{\RR^N} \trc\{Q \partial_t\Psi\} - \int_{\RR^N}\trc\{ Q_0 \Psi(0,\cdot) \} &+ \int_0^T \int_{\RR^N}\trc\{ (u\cdot\nabla Q)\Psi\} +\\&+
\int_0^T\int_{\RR^N} \trc\{(\Omega Q - Q \Omega )\Psi\} =\Gamma \int_0^T \int_{\RR^N} \trc\{H(Q)\Psi\}
\end{align*}
\begin{align*}
-\int_0^T\int_{\RR^N} u \cdot \partial_t \psi - \int_{\RR^N} u_0 \cdot \psi(0,\cdot) &+ \int_0^T\int_{\RR^N} (u\cdot \nabla u) \cdot \Pp \psi
-\nu \int_0^T \int_{\RR^N} u\cdot \Delta \psi = \\ &=
-L\int_0^T \int_{\RR^N} [ Q\Delta Q - \Delta Q\,Q - \nabla Q\odot \nabla Q ]\cdot \Pp \nabla \psi.
\end{align*}
From the arbitrariness of $T$ and $\Omega$, we finally achieve that $(u,\,Q)$ is a weak solution for (<ref>) in the sense of definition <ref>.
§ THE DIFFERENCE BETWEEN TWO SOLUTIONS
This section is devoted to an important remark which plays a major part in our uniqueness result. We deal with the difference between two weak solutions $(u_i,\,Q_i)$, $i=1,2$, of (<ref>) in the sense of definition <ref>. Denoting by $(\delta u,\,\delta Q)$ the difference between the first and the second one, we claim that such element belongs to a lower regular space than the one the solutions belong to.
For any finite positive $T$, $\delta u$ and $\nabla \delta Q$ belong to $L^\infty(0,T; \Hh^{-1/2})$.
In virtue of Proposition <ref> and since $(\nabla \delta u,\, \Delta \delta Q)$ belongs to $L^2_t L^2_x$ then
\begin{equation*}
(\nabla \delta u,\, \Delta \delta Q) \in L^2(0,T; \Hh^{-1/2} ),
\end{equation*}
for any finite positive $T$, thanks to a classical real interpolation method:
\begin{align*}
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}
\| \nabla \delta u \|_{\Hh^{-\frac{3}{2}} }^{\frac{1}{3}}
\| \nabla \delta u \|_{L^2_x }^{\frac{2}{3}}
\lesssim
\| \delta u \|_{\Hh^{-\frac{1}{2}} }+
\| \nabla \delta u \|_{L^2_x },\\
\| \Delta \delta Q \|_{\Hh^{-\frac{1}{2}}}
\| \Delta \delta Q \|_{\Hh^{-\frac{3}{2}} }^{\frac{1}{3}}
\| \Delta \delta Q \|_{L^2_x }^{\frac{2}{3}}
\lesssim
\| \nabla \delta Q \|_{\Hh^{-\frac{1}{2}} }+
\| \Delta \delta Q \|_{L^2_x }.
\end{align*}
Fixing $T>0$ we are going to prove that $\delta u$ belongs to $L^\infty(0,T; \Hh^{-1/2})$ and $\delta Q$ belongs to $L^\infty(0,T; \Hh^{1/2})$.
We denote by $f_1$ and $f_2$
\begin{align*}
f_1 := - u_1 \cdot \nabla Q_1 + u_2 \cdot \nabla Q_2 & + \Omega_1 Q_1 - \Omega_2 Q_2 - Q_1 \Omega_1 + Q_2 \Omega_2 \,+ \\&+
\Gamma
\Big\{ \;
\frac{b}{3}
\Big(
Q_1^2 - Q_2^2 - \trc\{ Q_1^2 - Q_2^2 \}\frac{\Id}{3}
\Big) -
c\, \trc \{ Q_1^2 \} Q_1 +
c\, \trc \{ Q_2^2 \} Q_2 \;
\Big\} ,
\end{align*}
\begin{align*}
f_2 := \Pp\big[\, - \Div\{ u_1 \otimes u_1 &- u_2 \otimes u_2 \} + L\Div\,\{\; Q_1 \Delta Q_1 - Q_2 \Delta Q_2 -\\
&- \Delta Q_1 Q_1 + \Delta Q_2 Q_2 - \nabla Q_1\odot \nabla Q_1 + \nabla Q_2\odot \nabla Q_2\;\}\big],
\end{align*}
respectively. Then $\delta Q$ and $\delta u$ are weak solutions of the following Cauchy Problems:
\begin{equation*}
\partial_t \delta Q- \Gamma L\Delta \delta Q+\Gamma a \,\delta Q = f_1 \quad \text{and} \quad \partial_t \delta u- \nu \Delta \delta u = f_2
\quad\quad\quad \text{in}\quad [0,T)\times\RR^2,
\end{equation*}
with null initial data. Then, by the classical Theory of Evolutionary Parabolic Equation, it is sufficient to prove that $f_1$ and $f_2$ belong to $L^2(0,T; \Hh^{-1/2})$ and
$L^2(0,T;\Hh^{-3/2})$ respectively in order to obtain
\begin{equation*}
\|(\nabla \delta u,\, \Delta \delta Q) \|_{L^\infty(0,T; \Hh^{-\frac{1}{2}})}\lesssim \| f_1 \|_{L^2(0,T; \Hh^{-\frac{1}{2}})} + \| f_2 \|_{L^2(0,T; \Hh^{-\frac{3}{2}})},
\end{equation*}
and conclude the proof. We start by $f_1$ and Theorem <ref> plays a major part. For any $i=1,\,2$, we get
\begin{equation*}
\begin{alignedat}{5}
\| u_i \cdot \nabla Q_i \|_{ \Hh^{-\frac{1}{2}} }
\| u_i \|_{ \Hh^{ \frac{1}{2}} }
\| \nabla Q_i \|_{ L^2_x }
\| u_i \|_{ L^2_x }^\frac{1}{2}
\| \nabla u_i \|_{ L^2_x }^\frac{1}{2}
\| \nabla Q_i \|_{ L^2_x }
&&&&\in L^4(0, T),\\
\| \Omega_i \, Q_i \|_{ \Hh^{-\frac{1}{2}} }
\| \nabla u_i \|_{ L^2_x }
\| Q_i \|_{ \Hh^{ \frac{1}{2}} }
\| \nabla u_i \|_{ L^2_x }
\| Q_i \|_{ L^2_x }^\frac{1}{2}
\| \nabla Q_i \|_{ L^2_x }^\frac{1}{2}
&&&&\in L^2(0, T),\\
\| Q_i^2 \|_{ \Hh^{-\frac{1}{2}} }
\| Q_i \|_{ \Hh^{ \frac{1}{2}} }
\| Q_i \|_{ L^2_x }
\| Q_i \|_{ L^2_x }
\| \nabla Q_i \|_{ L^2_x }^\frac{1}{2}
\| Q_i \|_{ L^2_x }^\frac{1}{2}
&&&&\in L^\infty(0,T),\\
\| \trc\{ Q_i^2\} Q_i \|_{ \Hh^{-\frac{1}{2}} }
\| Q_i^2 \|_{ L^2_x }
\| Q_i \|_{ \Hh^{ \frac{1}{2}} }
\| Q_i \|_{ L^4_x }^2
\| \nabla Q_i \|_{ L^2_x }^\frac{1}{2}
\| Q_i \|_{ L^2_x }^\frac{1}{2}
\lesssim
\| \nabla Q_i \|_{ L^2_x }^\frac{3}{2}
\| Q_i \|_{ L^2_x }^\frac{3}{2}
&&&&\in L^\infty(0,T).
\end{alignedat}
\end{equation*}
Then, summarizing the previous estimates, we finally deduce that $f_1$ belongs to $L^2(0, T ; \Hh^{-1/2})$. Now, let us handle the terms of $f_2$:
\begin{equation*}
\begin{alignedat}{5}
\| \Div\{ u_i \otimes u_i \} \|_{ \Hh^{-\frac{3}{2}} }
\| u_i \otimes u_i \|_{ \Hh^{-\frac{1}{2}} }
\lesssim
\| u_i \|_{ \Hh^{ \frac{1}{2}} }
\| u_i \|_{ L^2_x }
\| u_i \|_{ L^2_x }^\frac{1}{2}
\| \nabla u_i \|_{ L^2_x }^\frac{1}{2}
\| u_i \|_{ L^2_x }
&&&\in L^4(0,T),\\
\| \Div\{ Q_i \Delta Q_i \} \|_{ \Hh^{-\frac{3}{2}} }
\| Q_i \Delta Q_i \|_{ \Hh^{-\frac{1}{2}} }
\lesssim
\| Q_i \|_{ \Hh^{ \frac{1}{2}} }
\| \Delta Q_i \|_{ L^2_x }
\| Q_i \|_{ L^2_x }^\frac{1}{2}
\| \nabla Q_i \|_{ L^2_x }^\frac{1}{2}
\| \Delta Q_i \|_{ L^2_x }
&&&\in L^\infty(0,T)
\end{alignedat}
\end{equation*}
and moreover
\begin{align*}
\| \Div\{ \nabla Q_i \odot \nabla Q_i \} \|_{ \Hh^{-\frac{3}{2}} }
\lesssim
\| \nabla Q_i \odot \nabla Q_i \|_{ \Hh^{-\frac{1}{2}} }
\| \nabla Q_i \|_{ \Hh^{ \frac{1}{2}} }
\| \nabla Q_i \|_{ L^2_x }\\
\| \nabla Q_i \|_{ L^2_x }^\frac{1}{2}
\| \Delta Q_i \|_{ L^2_x }^\frac{1}{2}
\| \nabla Q_i \|_{ L^2_x }
\in L^4(0,T),
\end{align*}
which finally implies that $f_2$ belongs to $L^2(0,T; \Hh^{-\frac{3}{2}})$. This concludes the proof of Proposition <ref>.
§ UNIQUENESS
In this section we present our first original result. We are going to prove Theorem <ref>, namely the uniqueness of the weak solutions, given by Theorem <ref>. We implement the uniqueness result of Paicu and Zarnescu in <cit.>, concerning the weak-strong uniqueness. Indeed the authors suppose that at least one of the solutions is a classical solution. The leading cause of such restriction relies on the choice to control the difference between two solutions in an $L^2_x$-setting. However, this requires to estimate the $L^\infty_x$-norm of one of the solutions, $\|(u,\,\nabla Q)\|_{L^\infty_x}$, for instance by a Sobolev embedding, therefore the necessity to put $(u(t),\,\nabla Q(t))$ in some $\Hh^s$ with $s>1$, for any real $t$.
In this article we overcome this drawback, performing the weak-weak uniqueness, thanks to an alternative approach which is inspired by <cit.> and <cit.>. The main idea is to evaluate the difference between two weak solutions in a functional space which is less regular than $L^2_x$. Considering two weak solutions $(u_1,\,\nabla Q_1)$ and $(u_2,\,\nabla Q_2)$, we define $(\delta u, \delta Q)$ as the difference between the first one and the second one. It is straightforward that such difference is a weak solution for the following system:
\begin{equation}\label{delta_main_system}
\tag{$\delta P$}
\begin{cases}
\; \partial_t \delta Q + \delta u\cdot \nabla Q_1 + u_2 \cdot \nabla \delta Q
- \delta S(\nabla u,\,Q) -\Gamma L \Delta \delta Q = \Gamma \delta P( Q) &\RR_+ \times \RR^2,\\
\; \partial_t \delta u + \delta u \cdot \nabla u_1 + u_2 \cdot \nabla \delta u -\nu \Delta \delta u +\nabla \delta \Pi=
L\Div\,\big\{\;\delta Q \Delta Q_1 + Q_2 \Delta \delta Q - \\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
- \Delta \delta Q Q_1 - \Delta Q_2 \delta Q -
\nabla \delta Q\odot \nabla Q_1 - \nabla Q_2\odot \nabla \delta Q\;\big\} &\RR_+ \times \RR^2,\\
\; \Div\,\delta u = 0 &\RR_+ \times \RR^2,\\
\; (\delta u,\,\delta Q)_{t=0} = (0,\, 0) &\quad\quad\;\;\RR^2,
\end{cases}
\end{equation}
where we have also defined
\begin{equation*}
\delta \Omega := \Omega_1 - \Omega_2 , \quad
\delta \Pi := \Pi_1 - \Pi_2 , \quad
\delta P(Q) := P(Q_1) - P(Q_2) .
\end{equation*}
and moreover
\begin{equation*}
\delta S(Q,\,\nabla u) := \Omega_1 Q_1 - Q_1 \Omega_1 + \Omega_2 Q_2 - Q_2 \Omega_2
= \delta Q \delta \Omega - \delta \Omega \delta Q + \delta \Omega Q_2 - Q_2 \delta \Omega + \Omega_2 \delta Q-\delta Q \Omega_2.
\end{equation*}
Recalling the previous subsection, we take the $\Hh^{-1/2}$-inner product between the first equation of (<ref>) and $-L\Delta \delta Q$ and moreover we consider the scalar product in $\Hh^{-1/2}$ between the second one and $\delta u$:
\begin{equation}\label{uniqueness_energy_equality}
\begin{split}
\frac{\dd}{\dd t}
\Big[&
\frac{1}{2} \| \delta u \|_{\dot{H}^{-\frac{1}{2}}}^2 +
L \| \nabla \delta Q \|_{\dot{H}^{-\frac{1}{2}}}^2
\Big]+
\nu \| \nabla \delta u \|_{\dot{H}^{-\frac{1}{2}}}^2 +
\Gamma L^2 \| \Delta \delta Q \|_{\dot{H}^{-\frac{1}{2}}}^2
= \\=\;&
L\Gamma \langle \delta P(Q) , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}} -
L \langle \delta u\cdot \nabla Q_1 , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}} +
L \langle u_2 \cdot \nabla \delta Q , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}} + \\&+
L \langle \delta S(Q, \nabla u) , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}} -
L \langle \nabla \delta Q \odot
\nabla Q_1 , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}} +
L \langle \nabla Q_2 \odot
\nabla \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}} - \\&
\quad\quad\quad\quad\quad-
\langle \delta u\cdot\nabla u_1 , \delta u \rangle_{\dot{H}^{-\frac{1}{2}}} -
\langle u_2\cdot\nabla \delta u , \delta u \rangle_{\dot{H}^{-\frac{1}{2}}} +
L \langle \delta Q \Delta \delta Q -
\Delta \delta Q \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}} + \\&
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,+
L \langle Q_2 \Delta \delta Q -
\Delta \delta Q Q_2 , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}+
L \langle \delta Q \Delta Q_2-
\Delta Q_2 \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}.
\end{split}
\end{equation}
Denoting by
\frac{1}{2} \| \delta u(t) \|_{\dot{H}^{-\frac{1}{2}}}^2 +
L \| \nabla \delta Q(t) \|_{\dot{H}^{-\frac{1}{2}}}^2 $
we claim that
\begin{equation*}
\frac{\dd}{\dd t}\Phi(t)\leq \chi(t)\Phi(t),\quad\quad\text{for almost every}\quad t\in \RR_+,
\end{equation*}
where $\chi\geq 0 $ belongs to $L^1_{loc}(\RR_+)$.
Hence, uniqueness holds thanks to the Gronwall Lemma and since $\Phi(0)$ is null. Thus, we need to analyze every terms of the right-hand side of (<ref>). From here on $C_{\Gamma, L}$ and $C_\nu$ are suitable positive constants which will be determined in the end of the proof.
§.§ Simpler Terms
First, we begin evaluating every term which is handleable by Theorem <ref>.
Estimate of $ \Gamma L \langle\delta P(Q) ,
\Delta\delta Q\rangle_{\dot{H}^{-\frac{1}{2}}} $
From the definition of $\delta P(Q)$, and since $\trc\{\Delta Q\} $ is null, we need to control
\begin{equation*}
\begin{split}
\Gamma L \langle \delta P(Q) , \Delta\delta Q &\rangle_{\Hh^{-\frac{1}{2}} }
- \Gamma L a \| \nabla \delta Q \|_{\Hh^{-\frac{1}{2}} }^2
+ \Gamma L b \langle \delta Q\,Q_1 + Q_2 \delta Q , \Delta \delta Q \rangle_{\Hh^{-\frac{1}{2}} } \\&
- \Gamma L c \langle \delta Q\trc\{Q_1^2\} , \Delta \delta Q \rangle_{\Hh^{-\frac{1}{2}} }
- \Gamma L c \langle \trc\{\,\delta Q\,Q_1 + Q_2\delta Q\,\}Q_1 , \Delta \delta Q \rangle_{\Hh^{-\frac{1}{2}} }.
\end{split}
\end{equation*}
We overcome the second term in the right hand-side of the equality as follows:
\begin{equation*}
\begin{split}
\Gamma L b
\langle
\delta Q\,Q_1 + Q_2 \delta Q , \Delta \delta Q
\rangle_{\Hh^{-\frac{1}{2}} }
\| \delta Q \|_{ \Hh^{ \frac{1}{2} } }
\| (Q_1,\,Q_2) \|_{ L^2_x }
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }\\
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\| (Q_1,\,Q_2) \|_{ L^2_x }^2+
C_{\Gamma, L}
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{split}
\end{equation*}
Furthermore, we observe that
\begin{equation*}
\begin{split}
\Gamma& L c
\langle
\delta Q\trc\{Q_1^2\}, \Delta \delta Q
\rangle_{\Hh^{-\frac{1}{2}} }
\lesssim
\| \delta Q \|_{ \Hh^{ \frac{1}{2} } }
\| Q_1^2 \|_{ L^2_x}
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\lesssim
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| Q_1 \|_{ L^4_x }^2
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }\\
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| Q_1 \|_{ L^2_x }
\| \nabla Q_1 \|_{ L^2_x }
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\lesssim
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\| Q_1 \|_{ L^2_x }^2
\| \nabla Q_1 \|_{ L^2_x }^2+
C_{\Gamma, L}
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\end{split}
\end{equation*}
and moreover
\begin{equation*}
\begin{alignedat}{6}
\Gamma &L c
\langle
\trc\{\,\delta Q\,Q_1 + Q_2\delta Q\,\}Q_1 , \Delta \delta Q
\rangle_{\Hh^{-\frac{1}{2}} }
\| \delta Q \|_{ \Hh^{ \frac{1}{2} } }
\Big(
\| |Q_1|^2 \|_{ L^2_x } +
\| |Q_2||Q_1| \|_{ L^2_x }
\Big)
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }\\
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| (Q_1,\,Q_2) \|_{ L^4_x }^2
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| (Q_1,\,Q_2) \|_{ L^2_x }
\| \nabla (Q_1,\,Q_2) \|_{ L^2_x }
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }\\
\lesssim
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2 &&
\| (Q_1,\,Q_2) \|_{ L^2_x }^2
\| \nabla (Q_1,\,Q_2) \|_{ L^2_x }^2+
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{alignedat}
\end{equation*}
Finally, summarizing the previous inequality, we get
\begin{equation*}
\Gamma L \langle \delta P(Q) , \Delta\delta Q \rangle_{\Hh^{-\frac{1}{2}} }
\lesssim
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\| (Q_1,\,Q_2) \|_{ L^2_x }^2
\| \nabla (Q_1,\,Q_2) \|_{ L^2_x }^2+
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\end{equation*}
Estimate of $L \langle \delta u\cdot \nabla Q_1 , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
L \langle &\delta u\cdot \nabla Q_1 , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}
\lesssim
\| \delta u \|_{\Hh^{-\frac{1}{4}} }
\| \nabla Q_1 \|_{\Hh^{ \frac{3}{4}} }
\| \Delta \delta Q \|_{\Hh^{-\frac{1}{2}} }
\lesssim
\| \delta u \|_{\Hh^{-\frac{1}{2}} }^{\frac{3}{4}}
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}} }^{\frac{1}{4}}
\| \nabla Q_1 \|_{ L^2_x }^{\frac{1}{4}}{\scriptstyle \times} \\&{\scriptstyle \times}
\| \Delta Q_1 \|_{ L^2_x }^{\frac{3}{4}}
\| \Delta \delta Q \|_{\Hh^{-\frac{1}{2}} }
\lesssim
\| \Delta \delta Q \|_{\Hh^{-\frac{1}{2}} }^2 +
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}^2 +
\| \nabla Q_1 \|_{ L^2_x }^{\frac{2}{3}}
\| \Delta Q_1 \|_{ L^2_x }^2
\| \delta u \|_{\Hh^{-\frac{1}{2}}}^2.
\end{split}
\end{equation*}
Estimate of $L \langle u_2 \cdot \nabla \delta Q , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
L \langle u_2 \cdot \nabla \delta Q , \Delta \delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}
\|u_2\|_{\Hh^{\frac{3}{4}}}\|\nabla \delta Q\|_{\Hh^{-\frac{1}{4}}}\|\Delta \delta Q\|_{\Hh^{-\frac{1}{2}}}
\lesssim
\|u_2\|_{L^2_x}^\frac{1}{4}\|\nabla u_2\|_{L^2_x}^\frac{3}{4}
\|\nabla \delta Q\|_{\Hh^{-\frac{1}{2}}}^\frac{3}{4}\|\Delta \delta Q\|_{\Hh^{-\frac{1}{2}}}^\frac{5}{4}\\
C_{\Gamma,L}\|\Delta \delta Q\|_{\Hh^{-\frac{1}{2}}}^2 +
\|u_2\|_{L^2_x}^{\frac{2}{3}}\|\nabla u_2\|_{L^2_x}^{2}\|\nabla \delta Q\|_{\Hh^{-\frac{1}{2}}}^{2}.
\end{split}
\end{equation*}
Estimate of $L\langle \delta Q \delta \Omega - \delta \Omega \delta Q ,\Delta\delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{aligned}
\delta Q \delta \Omega - \delta \Omega \delta Q ,\Delta\delta Q
\rangle_{\dot{H}^{-\frac{1}{2}}}
\| \delta Q \|_{ \Hh^{ \frac{1}{2} } }
\| \delta \Omega \|_{ L^2_x }
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }\\
\| \nabla (u_1,\,u_2) \|_{ L^2_x }^2
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{aligned}
\end{equation*}
Estimate of $L\langle \Omega_2 \delta Q-\delta Q \Omega_2 ,\Delta\delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\Omega_2 \delta Q-\delta Q \Omega_2 ,\Delta\delta Q
\rangle_{\dot{H}^{-\frac{1}{2}}}
\lesssim
\| \Omega_2 \|_{ L^2_x }
\| \delta Q \|_{ \Hh^{ \frac{1}{2} } }
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\lesssim
\| \nabla u_2 \|_{ L^2_x }^2
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{equation*}
Estimate of $L \langle \nabla \delta Q \odot \nabla Q_1, \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
L \langle
\nabla \delta Q \odot \nabla Q_1, \nabla \delta u
\rangle_{\dot{H}^{-\frac{1}{2}}}
\| \nabla \delta Q \|_{\Hh^{-\frac{1}{4}}}
\| \nabla Q_1 \|_{\Hh^{ \frac{3}{4}}}
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}\\
\| \nabla \delta Q \|_{\Hh^{-\frac{1}{2}}}^\frac{3}{4}
\| \Delta \delta Q \|_{\Hh^{-\frac{1}{2}}}^\frac{1}{4}
\| \nabla Q_1 \|_{ L^2_x }^\frac{1}{4}
\| \Delta Q_1 \|_{ L^2_x }^\frac{3}{4}
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}\\
C_{\Gamma,L} \|\Delta \delta Q\|_{\Hh^{-\frac{1}{2}}}^2 +
C_{\nu} \|\nabla \delta u\|_{\Hh^{-\frac{1}{2}}}^2 +
\| \nabla Q_1 \|_{ L^2_x }^\frac{2}{3}
\| \Delta Q_1 \|_{ L^2_x }^2
\| \nabla \delta Q \|_{\Hh^{-\frac{1}{2}}}^2
\end{split}
\end{equation*}
Estimate of $L \langle \nabla Q_2\odot\nabla\delta Q, \, \nabla\delta u \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
L \langle
\nabla Q_2\odot\nabla\delta Q,\, \nabla\delta u
\rangle_{\dot{H}^{-\frac{1}{2}}}
\| \nabla \delta Q \|_{\Hh^{-\frac{1}{4}}}
\| \nabla Q_2 \|_{\Hh^{ \frac{3}{4}}}
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}\\
\| \nabla \delta Q \|_{\Hh^{-\frac{1}{2}}}^\frac{3}{4}
\| \Delta \delta Q \|_{\Hh^{-\frac{1}{2}}}^\frac{1}{4}
\| \nabla Q_2 \|_{ L^2_x }^\frac{1}{4}
\| \Delta Q_2 \|_{ L^2_x }^\frac{3}{4}
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}\\
C_{\Gamma,L} \|\Delta \delta Q\|_{\Hh^{-\frac{1}{2}}}^2 +
C_{\nu} \|\nabla \delta u\|_{\Hh^{-\frac{1}{2}}}^2 +
\| \nabla Q_2 \|_{ L^2_x }^\frac{2}{3}
\| \Delta Q_2 \|_{ L^2_x }^2
\| \nabla \delta Q \|_{\Hh^{-\frac{1}{2}}}^2
\end{split}
\end{equation*}
Estimate of $\langle \delta u\cdot\nabla u_1 , \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
\langle
\delta u\cdot\nabla u_1 , \delta u
\rangle_{\dot{H}^{-\frac{1}{2}}}
\lesssim
\| \delta u \|_{\Hh^{ \frac{1}{2}}}
\| \nabla u_1 \|_{ L^2_x }
\| \delta u \|_{\Hh^{-\frac{1}{2}}}
\lesssim
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}^2 +
\| \nabla u_1 \|_{ L^2_x }^2
\| \delta u \|_{\Hh^{-\frac{1}{2}}}^2
\end{split}
\end{equation*}
Estimate of $\langle u_2 \cdot \nabla \delta u , \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
\langle
u_2 \cdot \nabla \delta u , \delta u
\rangle_{\dot{H}^{-\frac{1}{2}}}
\lesssim
\| \delta u \|_{\Hh^{ \frac{1}{2}}}
\| \nabla u_2 \|_{ L^2 }
\| \delta u \|_{\Hh^{-\frac{1}{2}}}
\lesssim
\| \nabla \delta u \|_{\Hh^{-\frac{1}{2}}}^2 +
\| \nabla u_2 \|_{ L^2 }^2
\| \delta u \|_{\Hh^{-\frac{1}{2}}}^2
\end{split}
\end{equation*}
Estimate of $L \langle \delta Q \Delta \delta Q - \Delta \delta Q \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
L\langle \delta Q \Delta \delta Q-\Delta \delta Q \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}
\| \Delta (Q_1,Q_2) \|_{ L^2_x }
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }\\
\| \Delta (Q_1,Q_2) \|_{ L^2_x }^2
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{split}
\end{equation*}
Estimate of $L \langle \delta Q \Delta Q_2 - \Delta Q_2 \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}$
\begin{equation*}
\begin{split}
L\langle \delta Q \Delta Q_2-\Delta Q_2 \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}
\| \Delta Q_2 \|_{ L^2_x }
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }\\
\| \Delta Q_2 \|_{ L^2_x }^2
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{split}
\end{equation*}
§.§ The Residual Terms
Now we deal with the terms in the right-hand side of (<ref>) which we have not evaluated yet, namely
\begin{equation}\label{uniqueness_challenging_term}
L\langle \delta \Omega Q_2 - Q_2 \delta \Omega ,\Delta\delta Q \rangle_{\dot{H}^{-\frac{1}{2}}} +
L \langle Q_2 \Delta \delta Q -
\Delta \delta Q Q_2 , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}.
\end{equation}
Here, the difference between the two solutions appears with the higher derivative-order, more precisely the inner product is driven by $\nabla \delta u$ ( i.e. $\delta \Omega$) and $\Delta \delta Q$. This clearly generates a drawback if we want to analyze every remaining term, proceeding as the previous estimates. Let us remark that if we consider the $L^2_x$-inner product instead of the $\Hh^{-1/2}$-one, then this last sum is null, thanks to Lemma <ref>. However the $\Hh^{-1/2}$-setting force us to analyze such sum, and we overcome the described obstacle, first considering the equivalence between $\Hh^{-1/2}$ and $\BB_{2,2}^{-1/2}$, and moreover thanks to decomposition (<ref>), namely
\begin{equation*}
\begin{array}{ll}
\J_q^1(A,B) :=\sum_{|q-q'|\leq 5 } [\Dd_q, \, \Sd_{q'-1}A] \Dd_{q'} B,
&\J_q^3(A,B) := \Sd_{q -1} A \Dd_{q} B,\\
\\
\J_q^2(A,B) :=\sum_{|q-q'|\leq 5 } ( \Sd_{q'-1}A -\Sd_{q-1}A) \Dd_{q} \Dd_{q'} B,
&\J_q^4(A,B) :=\sum_{ q' \geq q- 5 } \Dd_q(\Dd_{q'}A\, \Sd_{q'+2} B),
\end{array}
\end{equation*}
\begin{equation*}
\Dd_q(AB) = \J_q^1(A,B)
+ \J_q^2(A,B)
+ \J_q^3(A,B)
+ \J_q^4(A,B),
\quad\quad\text{for any integer}\; q.
\end{equation*}
First, let us begin with
\begin{equation*}
L\langle \delta \Omega Q_2 ,\Delta\delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}
\sum_{q\in \ZZ} 2^{-q} L\langle \Dd_q ( \delta \Omega Q_2) , \Dd_q \Delta \delta Q\rangle_{L^2_x}
\sum_{q\in \ZZ}
\sum_{i=1}^4 2^{-q} L\langle \J_q^i ( \delta \Omega,\, Q_2) , \Dd_q \Delta \delta Q\rangle_{L^2_x}.
\end{equation*}
First we separately study the case $i=1,2,4$. The term related to $i=3$ is the challenging one and we are not able to evaluate it. However, we will see how such term is going to be erased.
Let us begin with $i=1$ then
\begin{equation*}
\begin{split}
\I_q^1:=
2^{-q}L\langle \J_q^1 ( \delta \Omega,\, Q_2) , \Dd_q \Delta \delta Q\rangle_{L^2_x}
L 2^{-q}
\sum_{|q-q'|\leq 5 }
\langle
[\Dd_q,\, \Sd_{q'-1}Q_2 ]\Dd_{q'}\delta \Omega, \Dd_q \Delta \delta Q
\rangle_{L^2_x}\\
\sum_{|q-q'|\leq 5 }
\| [\Dd_q,\, \Sd_{q'-1}Q_2 ]\Dd_{q'}\delta \Omega \|_{L^2_x }
\| \Dd_q \Delta \delta Q \|_{L^2_x }.
\end{split}
\end{equation*}
Hence, applying the commutator estimate (see Lemma $2.97$ in <cit.>) we get
\begin{equation*}
\begin{split}
\I_q^1
\lesssim
\sum_{|q-q'|\leq 5 }
\| \Sd_{q'-1}\nabla Q_2 \|_{L^4_x }
\| \Dd_{q'}\delta \Omega \|_{L^4_x }
\| \Dd_q \Delta \delta Q \|_{L^2_x }
\lesssim
\sum_{|q-q'|\leq 5 }
\| \Sd_{q'-1}\nabla Q_2 \|_{L^2_x }^{\frac{1}{2}}
\| \Sd_{q'-1}\Delta Q_2 \|_{L^2_x }^{\frac{1}{2}} {\scriptstyle \times} \\ {\scriptstyle \times}
\| \Dd_{q'}\delta u \|_{L^4_x }
\| \Dd_q \Delta \delta Q \|_{L^2_x }
\lesssim
\sum_{|q-q'|\leq 5 }
\| \nabla Q_2 \|_{L^2_x }^{\frac{1}{2}}
\| \Delta Q_2 \|_{L^2_x }^{\frac{1}{2}}
\| \Dd_{q'}\delta u \|_{L^2_x }
\| \Dd_q \Delta \delta Q \|_{L^2_x },
\end{split}
\end{equation*}
which finally yields
\begin{equation*}
\begin{split}
\I_q^1
\| \nabla Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \Delta Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \delta u \|_{ L^2_x }
\| \Delta \delta Q \|_{ \Hh^{ -\frac{1}{2} } }
\lesssim
\| \nabla Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \Delta Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \delta u \|_{ \Hh^{ -\frac{1}{2} } }^\frac{1}{2}
\| \nabla \delta u \|_{ \Hh^{ -\frac{1}{2} } }^\frac{1}{2}
\| \Delta \delta Q \|_{ \Hh^{ -\frac{1}{2} } },
\end{split}
\end{equation*}
that is
\begin{equation}\label{uniqueness_est_J_1}
\begin{split}
L\sum_{q\in \ZZ}2^{-q}\langle \J_q^1 ( \delta \Omega,\, Q_2) ,& \Dd_q \Delta \delta Q\rangle_{L^2_x}
\lesssim\\
\| \nabla Q_2 \|_{ L^2_x }^2
\| \Delta Q_2 \|_{ L^2_x }^2
\| \delta u \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2+
C_{\Gamma, L}
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{split}
\end{equation}
Now, let us handle the case $i=2$. We argued almost as before:
\begin{equation*}
\begin{split}
\I_q^2:= 2^{-q}L\langle \J_q^2 ( \delta \Omega,\, Q_2) , \Dd_q \Delta \delta Q\rangle_{L^2_x}
L 2^{-q}
\sum_{|q-q'|\leq 5 }
\langle
(\Sd_{q'-1}Q_2 - \Sd_{q-1}Q_2)\Dd_q \Dd_{q'}\delta \Omega, \Dd_q \Delta \delta Q
\rangle_{L^2_x}\\
\| (\Sd_{q'-1}Q_2 - \Sd_{q-1}Q_2) \|_{L^\infty_x}
\| \Dd_q \Dd_{q'}\delta \Omega \|_{L^2_x}
\| \Dd_q \Delta \delta Q \|_{L^2_x},
\end{split}
\end{equation*}
so that, observing that $\Sd_{q'-1}Q_2 - \Sd_{q-1}Q_2$ fulfills
\begin{equation*}
\| \Sd_{q'-1} Q_2 - \Sd_{q-1} Q_2 \|_{L^\infty_x}
\lesssim
\| \Sd_{q'-1} \Delta Q_2 - \Sd_{q-1}\Delta Q_2 \|_{L^\infty_x}
\lesssim
2^{- q}
\| \Sd_{q'-1} \Delta Q_2 - \Sd_{q-1}\Delta Q_2 \|_{L^2_x },
\end{equation*}
then we obtain
\begin{equation*}
\begin{split}
\I_q^2
\lesssim
\sum_{|q-q'|\leq 5 }
\| &(\Sd_{q'-1}\Delta Q_2 - \Sd_{q-1}\Delta Q_2) \|_{L^2_x}
\| \Dd_q \Dd_{q'}\delta \Omega \|_{L^2_x}
\| \Dd_q \Delta \delta Q \|_{L^2_x}
\lesssim
\sum_{|q-q'|\leq 5 }
\| \Delta Q_2 \|_{L^2_x}{\scriptstyle \times} \\ &{\scriptstyle \times}
\| \Dd_{q'}\delta \Omega \|_{L^2_x}
\| \Dd_q \Delta \delta Q \|_{L^2_x}
\lesssim
\sum_{|q-q'|\leq 5 }
\| \Dd_{q'} \delta u \|_{L^2_x}
\| \Dd_q \Delta \delta Q \|_{L^2_x}
\| \Delta Q_2 \|_{L^2_x}.
\end{split}
\end{equation*}
Thus, it turns out that
\begin{equation}\label{uniqueness_est_J_2}
L\sum_{q\in \ZZ} 2^{-q}\langle \J_q^2 ( \delta \Omega,\, Q_2) , \Dd_q \Delta \delta Q\rangle_{L^2_x} \lesssim
\| \Delta Q_2 \|_{ L^2_x }^2
\| \delta u \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\end{equation}
Now, we take into consideration the case $i=4$. Here we will use a convolution method and the Young inequality, since the sum in $q'$ is not finite. Then, let us observe that
\begin{equation*}
\begin{split}
\I_q^4 :=
2^{-q}L\langle \J_q^4 ( \delta \Omega,\, Q_2) , \Dd_q \Delta \delta Q\rangle_{L^2_x}
&= L
\sum_{q-q'\leq 5}
\langle
\Dd_{q'} Q_2 \Sd_{q'+2}\delta \Omega,\, \Dd_q \Delta \delta Q
\rangle_{L^2_x}\\
\sum_{q-q'\leq 5}
\| \Dd_{q'} Q_2 \|_{L^\infty_x }
\| \Sd_{q'+2}\delta \Omega \|_{L^2_x }
\| \Dd_q \Delta \delta Q \|_{L^2_x }.
\end{split}
\end{equation*}
Observing that
$\| \Dd_{q'} Q_2 \|_{L^\infty_x } \lesssim 2^{q'}\| \Dd_{q'} Q_2 \|_{L^2_x } \lesssim 2^{-q'}\| \Dd_{q'}\Delta Q_2 \|_{L^2_x }$ and
$\| \Dd_q \Delta \delta Q \|_{L^2_x }\lesssim 2^q\| \Dd_q \nabla \delta Q \|_{L^2_x }$, it turns out that
\begin{equation*}
\begin{split}
\I_q^4
\sum_{q-q'\leq 5}
\| \Dd_{q'}\Delta Q_2 \|_{L^2_x }
\| \Sd_{q'+2}\delta \Omega \|_{L^2_x }
\| \Dd_q \nabla \delta Q \|_{L^2_x }\\
\sum_{q-q'\leq 5}
\| \Dd_{q'}\Delta Q_2 \|_{ L^2_x }
\| \Sd_{q'+2}\delta \Omega \|_{ L^2_x }
\| \Dd_q \nabla \delta Q \|_{ L^2_x }\\
\| \Delta Q_2 \|_{ L^2_x }
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\sum_{q-q'\leq 5}
\| \Sd_{q'+2}\delta \Omega \|_{ L^2_x }.
\end{split}
\end{equation*}
Then, by convolution, the Young inequality and Proposition <ref>, we finally obtain
\begin{equation}\label{uniqueness_est_J_4}
\begin{aligned}
L\sum_{q\in \ZZ}2^{-q}\langle \J_q^4 ( \delta \Omega,\, Q_2) , \Dd_q \Delta \delta Q\rangle_{L^2_x}
\| \Delta Q_2 \|_{ L^2_x }
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }\\
\| \Delta Q_2 \|_{ L^2_x }^2
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2 +
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2
\end{aligned}
\end{equation}
Summarizing (<ref>), (<ref>) and (<ref>) and recalling the definition of $J^3_q(\delta \Omega, Q_2)$, we finally get
\begin{equation*}
\begin{aligned}
L\langle \delta \Omega Q_2 ,\Delta\delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}
-\sum_{q\in\ZZ}2^{-q}&\langle \Sd_{q-1} \delta \Omega\,\Dd_q Q_2, \Dd_q \Delta \delta Q\rangle_{L^2_x}
\lesssim\\
\tilde{\chi}_1\,\Phi +
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2 +
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2,
\end{aligned}
\end{equation*}
where $\tilde{\chi}_1$ belongs to $L^1_{loc}(\RR_+)$. Hence, we need to analyze
\begin{equation*}
L\sum_{q\in\ZZ}2^{-q}\langle \Sd_{q-1} \delta \Omega\,\Dd_q Q_2, \Dd_q \Delta \delta Q\rangle_{L^2_x}
\end{equation*}
and this term is going to disappear by a simplification.
Now we handle the term $\langle Q_2 \delta\Omega, \Delta \delta Q\rangle_{\Hh^{-\frac{1}{2}}}$ of (<ref>). Observing that it is equal to
$\langle \tr(Q_2 \delta\Omega ), \tr \Delta \delta Q\rangle_{\Hh^{-\frac{1}{2}}}$, that is $-\langle \delta\Omega Q_2, \Delta \delta Q \rangle_{\Hh^{-\frac{1}{2}}}$ then we proceed
exactly as before, obtaining
\begin{equation}\label{uniqueness_first_estimate}
\begin{aligned}
L\langle \delta \Omega Q_2 - Q_2\delta \Omega ,\Delta\delta Q \rangle_{\dot{H}^{-\frac{1}{2}}}
-\sum_{q\in\ZZ}2^{-q}\langle \Sd_{q-1} \delta \Omega\,&\Dd_q Q_2 - \Dd_q Q_2\,\Sd_{q-1} \delta \Omega, \Dd_q \Delta \delta Q\rangle_{L^2_x}
\lesssim\\
\tilde{\chi}\,\Phi +
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2 +
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2,
\end{aligned}
\end{equation}
so that it remains to control
\begin{equation}\label{uniqueness_first_term_to_control}
L\sum_{q\in\ZZ}2^{-q}\langle \Sd_{q-1} \delta \Omega\,\Dd_q Q_2 - \Dd_q Q_2\,\Sd_{q-1} \delta \Omega, \Dd_q \Delta \delta Q\rangle_{L^2_x}.
\end{equation}
Now, we focus on $L\langle Q_2 \Delta \delta Q , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}$ of (<ref>) and we use again decomposition (<ref>) as follows
\begin{equation*}
L\langle Q_2 \Delta \delta Q , \nabla \delta u \rangle_{ \dot{H}^{-\frac{1}{2} } }
\langle \Dd_q( Q_2 \Delta \delta Q ), \Dd_q \nabla \delta u \rangle_{ L^2_x }
\sum_{i=1}^4
\langle \J_q^i( Q_2,\, \Delta \delta Q ), \Dd_q \nabla \delta u \rangle_{ L^2_x }.
\end{equation*}
As before, we estimate the terms related to $i=1,2,4$ while when $i=3$ the associated term is going to be erased. When $i=1$ we get
\begin{equation*}
\begin{split}
L2^{-q}\langle \J_q^1( Q_2,\, \Delta \delta Q ), \Dd_q \nabla &\delta u \rangle_{ L^2_x }
=L\sum_{|q-q'|\leq 5} 2^{-q}
\langle
[\Dd_q,\, \Sd_{q'-1}Q_2 ]\Dd_{q'}\Delta \delta Q, \Dd_q \nabla \delta u
\rangle_{L^2_x}\\
\sum_{|q-q'|\leq 5}
\| [\Dd_q,\, \Sd_{q'-1}Q_2 ]\Dd_{q'}\Delta \delta Q \|_{L^2_x }
\| \Dd_q \nabla \delta u \|_{L^2_x }\\
\sum_{|q-q'|\leq 5}
\| \Sd_{q'-1}\nabla Q_2 \|_{L^4_x }
\| \Dd_{q'}\Delta \delta Q \|_{L^4_x }
\| \Dd_q \nabla \delta u \|_{L^2_x }\\
\sum_{|q-q'|\leq 5}
\| \Sd_{q'-1}\nabla Q_2 \|_{L^2_x }^\frac{1}{2}
\| \Sd_{q'-1}\Delta Q_2 \|_{L^2_x }^\frac{1}{2}
\| \Dd_{q'}\nabla \delta Q \|_{L^4_x }
\| \Dd_q \nabla \delta u \|_{L^2_x }\\
\sum_{|q-q'|\leq 5}
\| \nabla Q_2 \|_{L^2_x }^\frac{1}{2}
\| \Delta Q_2 \|_{L^2_x }^\frac{1}{2}
\| \Dd_{q'}\nabla \delta Q \|_{L^2_x }
\| \Dd_q \nabla \delta u \|_{L^2_x }
\end{split}
\end{equation*}
Hence, taking the sum as $q\in\ZZ$,
\begin{equation*}
\begin{split}
L\sum_{q\in\ZZ}2^{-q}\langle \J_q^1( Q_2,\, \Delta \delta Q ), &\Dd_q \nabla \delta u \rangle_{ L^2_x }
\lesssim
\| \nabla Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \Delta Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \nabla \delta Q \|_{ L^2_x }
\| \nabla \delta u \|_{ \Hh^{ -\frac{1}{2} } }\\
\| \nabla Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \Delta Q_2 \|_{ L^2_x }^\frac{1}{2}
\| \nabla \delta Q \|_{ \Hh^{ -\frac{1}{2} } }^\frac{1}{2}
\| \Delta \delta Q \|_{ \Hh^{ -\frac{1}{2} } }^\frac{1}{2}
\| \nabla \delta u \|_{ \Hh^{ -\frac{1}{2} } }\\
\| \nabla Q_2 \|_{ L^2_x }^2
\| \Delta Q_2 \|_{ L^2_x }^2
\| \nabla \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \nabla \delta u \|_{ \Hh^{ -\frac{1}{2} } }^2+
\| \Delta \delta Q \|_{ \Hh^{ -\frac{1}{2} } }^2.
\end{split}
\end{equation*}
We evaluate the term related to $i=2$ as follows:
\begin{equation*}
\begin{split}
L2^{-q}\langle \J_q^2( Q_2,\, \Delta \delta Q ), &\Dd_q \nabla \delta u \rangle_{ L^2_x }
= L
\langle
(\Sd_{q'-1}Q_2 - \Sd_{q-1}Q_2)\Dd_q \Dd_{q'}\Delta \delta Q, \Dd_q \nabla \delta u
\rangle_{L^2_x}\\
\sum_{|q-q'|\leq 5}
\| (\Sd_{q'-1}Q_2 - \Sd_{q-1}Q_2) \|_{L^\infty_x}
\| \Dd_q \Dd_{q'}\Delta Q \|_{L^2_x}
\| \Dd_q \nabla \delta u \|_{L^2_x},
\end{split}
\end{equation*}
so that
\begin{equation*}
\begin{split}
L2^{-q}\langle \J_q^2( Q_2,\, \Delta \delta Q ), &\Dd_q \nabla \delta u \rangle_{ L^2_x } \\
\sum_{|q-q'|\leq 5}
\| (\Sd_{q'-1}\Delta Q_2 - \Sd_{q-1}\Delta Q_2) \|_{L^2_x}
\| \Dd_q \Dd_{q'}\Delta \delta Q \|_{L^2_x}
\| \Dd_q \nabla \delta u \|_{L^2_x}\\
\sum_{|q-q'|\leq 5}
\| \Sd_{q'-1}\Delta Q_2 \|_{L^2_x}
\| \Dd_{q}\Delta\delta Q \|_{L^2_x}
\| \Dd_q \nabla \delta u \|_{L^2_x}\\
\sum_{|q-q'|\leq 5}
\| \Dd_{q'} \delta u \|_{L^2_x}
\| \Dd_q \Delta \delta Q \|_{L^2_x}
\| \Delta Q_2 \|_{L^2_x}
\end{split}
\end{equation*}
Thus, taking the sum in $q$, it turns out that
\begin{equation*}
L\sum_{q\in\ZZ}2^{-q}\langle \J_q^2( Q_2,\, \Delta \delta Q ), \Dd_q \nabla \delta u \rangle_{ L^2_x }
\lesssim
\| \Delta Q_2 \|_{ L^2_x }^2
\| \delta u \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\end{equation*}
At last, when $i=4$,
\begin{equation*}
\begin{split}
L2^{-q}\langle \J_q^4( Q_2,\, \Delta \delta Q ), &\Dd_q \nabla \delta u \rangle_{ L^2_x }
= L
\sum_{q-q'\leq 5}
\langle
\Dd_{q'} Q_2 \Sd_{q'+2}\Delta \delta Q,\, \Dd_q \nabla \delta u
\rangle_{L^2_x}\\
\sum_{q-q'\leq 5}
\| \Dd_{q'} Q_2 \|_{L^\infty_x }
\| \Sd_{q'+2}\Delta \delta Q \|_{L^2_x }
\| \Dd_q \nabla \delta u \|_{L^2_x }\\
\sum_{q-q'\leq 5}
\| \Dd_{q'}\Delta Q_2 \|_{L^2_x }
\| \Sd_{q'+2}\Delta \delta Q \|_{L^2_x }
\| \Dd_q \delta u \|_{L^2_x }\\
\sum_{q-q'\leq 5}
\| \Dd_{q'}\Delta Q_2 \|_{ L^2_x }
\| \Sd_{q'+2}\Delta \delta Q \|_{ L^2_x }
\| \Dd_q \delta u \|_{ L^2_x }\\
\| \Delta Q_2 \|_{ L^2_x }
\| \delta u \|_{ \Hh^{-\frac{1}{2} } }
\sum_{q-q'\leq 5}
\| \Sd_{q'+2}\Delta \delta Q \|_{ L^2_x }
\end{split}
\end{equation*}
Hence, by convolution, the Young inequalities and Proposition <ref>, we obtain
\begin{align*}
\sum_{q\in \ZZ}2^{-q}\langle \J_q^4( Q_2,\, \Delta \delta Q ), \Dd_q \nabla \delta u \rangle_{ L^2_x }
\| \Delta Q_2 \|_{ L^2_x }
\| \delta u \|_{ \Hh^{-\frac{1}{2} } }
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }\\
\| \Delta Q_2 \|_{ L^2_x }^2
\| \delta u \|_{ \Hh^{-\frac{1}{2} } }^2+
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\end{align*}
$\langle \Delta \delta Q Q_2, \nabla \delta u\rangle_{\Hh^{_\frac{1}{2}}}
= \langle \tr(\Delta \delta Q Q_2), \tr \nabla \delta u\rangle_{\Hh^{_\frac{1}{2}}}
= \langle Q_2 \Delta \delta Q , \tr \nabla \delta u\rangle_{\Hh^{_\frac{1}{2}}}$, then we proceed as for estimate $\langle Q_2\Delta \delta Q, \nabla \delta u\rangle_{\Hh^{_\frac{1}{2}}}$, so that we obtain the following control
\begin{equation}\label{uniqueness_second_estimate}
\begin{aligned}
L \langle Q_2 \Delta \delta Q -
\Delta \delta Q Q_2 , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}} -
L\sum_{q\in\ZZ}2^{-q}\langle \Sd_{q-1} Q_2\,\Dd_q \Delta \delta Q - \Dd_q \Delta \delta Q\,\Sd_{q-1} Q_2 \delta \Omega, \Dd_q \nabla u \rangle_{L^2_x}
\lesssim\\
\lesssim
\tilde{\chi}_2\,\Phi +
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2 +
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\end{aligned}
\end{equation}
where $\chi_2$ belongs to $L^1_{loc}(\RR_+)$. Now, the term we need to erase is
\begin{equation}\label{uniqueness_second_term_to_control}
L\sum_{q\in\ZZ}2^{-q}\langle \Sd_{q-1} Q_2\,\Dd_q \Delta \delta Q - \Dd_q \Delta \delta Q\,\Sd_{q-1} Q_2 \delta \Omega, \Dd_q \nabla u \rangle_{L^2_x}.
\end{equation}
Thus, summing (<ref>) and (<ref>), we obtain
\begin{equation*}
L \sum_{q\in \ZZ}2^{-q}
\Big\{
\langle
\Sd_{q-1} Q_2 \Dd_q \delta \Omega - \Dd_q\delta \Omega \,\Sd_{q-1} Q_2 , \Delta \Dd_q \delta Q
\rangle_{L^2_x}
\langle \Sd_{q-1}Q_2 \Delta \Dd_q \delta Q -
\Delta \Dd_q \delta Q \,\Sd_{q-1} Q_2 , \nabla \delta u
\rangle_{L^2_x}
\Big\},
\end{equation*}
which is a series with every coefficients null, thanks to Lemma <ref>. In virtue of this last result, recalling (<ref>) and
(<ref>), we finally obtain
\begin{equation*}
L\langle \delta \Omega Q_2 - Q_2 \delta \Omega ,\Delta\delta Q \rangle_{\dot{H}^{-\frac{1}{2}}} +
L \langle Q_2 \Delta \delta Q -
\Delta \delta Q Q_2 , \nabla \delta u \rangle_{\dot{H}^{-\frac{1}{2}}}
\lesssim
\tilde{\chi} \Phi +
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2 +
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2.
\end{equation*}
§.§ Conclusion
Recalling (<ref>) and summarizing all the estimate of the previous two sub-sections, we conclude that there exists a function $\chi$ which
belongs to $L^1_{loc}(\RR_+)$ such that
\begin{equation*}
\frac{\dd}{\dd t}\Phi(t) + \nu \| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2 + \Gamma L^2 \| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\lesssim \chi(t)\Phi(t)+ C_{\nu}
\| \nabla \delta u \|_{ \Hh^{-\frac{1}{2} } }^2 +
\| \Delta \delta Q \|_{ \Hh^{-\frac{1}{2} } }^2
\end{equation*}
for almost every $t\in \RR_+$. Thus, choosing $C_{\Gamma,L}$ and $C_{\nu}$ small enough, we absorb the last two terms in the right-hand side by the left-hand side, finally obtaining
\begin{equation*}
\frac{d}{\dd t}\Big[\frac{1}{2} \| \delta u(t) \|_{\dot{H}^{-\frac{1}{2}}}^2 +
L \| \nabla \delta Q(t) \|_{\dot{H}^{-\frac{1}{2}}}^2\Big]
\lesssim
\chi \Big[\frac{1}{2} \| \delta u(t) \|_{\dot{H}^{-\frac{1}{2}}}^2 +
L \| \nabla \delta Q(t) \|_{\dot{H}^{-\frac{1}{2}}}^2\Big].
\end{equation*}
Since the initial datum is null and thanks to the Gronwall inequality, we deduce that $(\delta u, \nabla \delta Q)=0$ which yields
$(\delta u, \delta Q)=0$, since $\delta Q(t)$ decades to $0$ at infinity for almost every $t$. Hence, we have finally achieved the uniqueness of the weak solution for
system (<ref>).
§ REGULARITY PROPAGATION
We now handle the propagation of low regularity, namely we prove Theorem <ref>.
Let us consider the following sequence of system:
\begin{equation}\label{Friedrichs_scheme_complete}
\tag{$\tilde{P}_n$}
\begin{cases}
\; \partial_t Q^n + J_n \Pp \big( J_nu^n \nabla J_n Q^n \big) - J_n \Pp \big( J_n \Omega^n J_n Q^n \big) +\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;
+\,J_n \Pp\big( J_n Q^n J_n \Omega^n \big) - \Gamma L \Delta J_n Q^n =P^n(Q^n) &\RR_+ \times \RR^2,\\
\; \partial_t u^n + J_n\Pp \big( J_n u^n \nabla J_n u^n \big ) - \nu \Delta J_n u^n = \\
\quad\quad\quad\quad
= \Gamma L\Div J_n \Pp \{ J_n Q^n \Delta J_n Q^n - \Delta J_n Q^n J_n Q^n - \nabla J_n Q^n \odot \nabla J_n Q^n\} &\RR_+ \times \RR^2,\\
\; \Div\, u^n = 0 &\RR_+ \times \RR^2,\\
\; (u^n,\,Q^n)_{|t=0} = (u_0,\,Q_0) &\quad\quad\;\;\RR^2,
\end{cases}
\end{equation}
\begin{equation*}
P^n(Q^n):= -a J_n Q^n + b \big[ J_n (J_n Q^n J_n Q^n ) - \trc\{J_n( J_nQ^n J_nQ^n)\}\frac{\Id}{3}\big] - cJ_nQ^n \trc\{J_n( J_nQ^n J_nQ^n)\}.
\end{equation*}
Moreover we recall that $J_n$ is the regularizing operator defined by
\begin{equation*}
\hat{J_n f}(\xi) = 1_{[\frac{1}{n},\,n]}(\xi)\hat{f}(\xi)
\end{equation*}
and $\Pp$ stands for the Leray projector. The Friedrichs scheme related to (<ref>) is not much different to the (<ref>)-one, however here the $Q$-tensor equation has been regularized, as well. System (<ref>) has been utilized in <cit.> and the authors have proven the existence of a strong solution $(u^n,\,Q^n)$ which converges to a weak solution for (<ref>), as $n$ goes to $\infty$ (up to a subsequence). Thanks to our uniqueness result, Theorem <ref>, we deduce that such solution is exactly the one determined by Theorem <ref> and it is unique. Hence, instead of proceeding by a priori estimate (as in <cit.>), we formalize our proof, evaluating directly the (<ref>)-scheme. We will establish some estimates, which are uniformly in $n$, which yields that the weak-solution of (<ref>) fulfills them as well. This is only a strategy in order to formalize the a priori-estimate, while the major part of our proof releases on the inequalities we are going to proof.
Since $(J_nu^n,\,J_nQ^n) = (u^n,\,Q^n)$ (by uniqueness), then $(u^n(t), Q^n(t))$ belongs to $H^{1+s} \times H^{2+s}$ for almost every $t\in \RR_+$ and for every $n\in\NN$. We apply $\Dd_q$ to the first and the second equations of (<ref>), then we apply $\langle\, \cdot\, ,\Dd_q u^n\rangle_{L^2}$ to the first one and
$-L\langle\, \cdot\, ,\Dd_q \Delta Q^n\rangle_{L^2}$ to the second one, obtaining the following identity:
\begin{align*}
&\frac{\dd}{\dd t}\Big[ \|\Dd_q u^n \|_{L^2}^2 + L \| \Dd_q \nabla Q^n \|_{L^2}^2 \Big]
+ \nu \| \Dd_q \nabla u^n \|_{L^2}^2
+ \Gamma L^2 \| \Dd_q \Delta Q^n \|_{L^2}^2
= \\&=
\langle \Dd_q( \Delta Q^n Q^n - Q^n \Delta Q^n ) , \Dd_q \nabla u^n \rangle_{L^2}
- \langle \Dd_q( u^n \cdot \nabla u^n ) , \Dd_q u^n \rangle_{L^2}
+ \langle \Dd_q( \nabla Q^n \odot \nabla Q^n ) , \Dd_q \nabla u^n \rangle_{L^2} +\\&\quad
+L \langle \Dd_q( u^n \cdot \nabla Q^n ) , \Dd_q \Delta Q^n \rangle_{L^2}
+L \langle \Dd_q( \Omega^n Q^n -Q^n \Omega^n ) , \Dd_q \Delta Q^n \rangle_{L^2}
-L \langle \Dd_q P^n(Q^n) , \Dd_q \Delta Q^n \rangle_{L^2}.
\end{align*}
Multiplying both left-hand and the right-hand sides by $2^{2qs}$ and taking the sum as $q\in \ZZ$ we obtain
\begin{equation}\label{sec3_eq_energy_Dq}
\begin{aligned}
&\frac{\dd}{\dd t}\Big[ \|u^n \|_{\Hh^s}^2 + L \|\nabla Q^n \|_{\Hh^s}^2 \Big]
+ \nu \| \nabla u^n \|_{\Hh^s}^2
+ \Gamma L^2 \| \Delta Q^n \|_{\Hh^s}^2
= \\&=
L \langle \Delta Q^n Q^n - Q^n \Delta Q^n , \nabla u^n \rangle_{\Hh^s}
- \langle u^n \cdot \nabla u^n , u^n \rangle_{\Hh^s}
+L \langle \nabla Q^n \odot \nabla Q^n , \nabla u^n \rangle_{\Hh^s} +\\&\quad
+L \langle u^n \cdot \nabla Q^n , \Delta Q^n \rangle_{\Hh^s}
+L \langle \Omega^n Q^n -Q^n \Omega^n , \Delta Q^n \rangle_{\Hh^s}
-L \langle P^n(Q^n) , \Delta Q^n \rangle_{\Hh^s}.
\end{aligned}
\end{equation}
The key part of our proof relies on the Osgood inequality, therefore we need to estimate all the terms of the right-hand side of (<ref>). First, let us proceed estimating the easier terms.
Estimate of $\langle u^n \cdot \nabla u^n , u^n \rangle_{\Hh^s}$
We begin with $\langle \Dd_q( u^n \cdot \nabla u^n ) , \Dd_q u^n \rangle_{L^2}$, with $q\in \ZZ$. Passing through the
Bony decomposition
\begin{align*}
\langle &\Dd_q( u^n \cdot \nabla u^n ,\, \Dd_q u^n \rangle_{L^2}= \\ &=
\underbrace{
\sum_{|q-q'|\leq 5}
\langle \sum_{i=1}^2 \Dd_q T_{u^n_i}\partial_i u^n + \Dd_q T_{\partial_i u^n}u^n_i ,\, \Dd_q u^n \rangle_{L^2}}_{\Aa_q} +
\underbrace{
\sum_{q'\geq q-5}
\langle \sum_{i=1}^2 \Dd_q R( u^n_i,\,\partial_i u^n ) ,\, \Dd_q u^n \rangle_{L^2}}_{\Bb_q}
\end{align*}
We handle the term $\Aa_q$ as follows:
\begin{align*}
\Aa_q &\lesssim
\sum_{|q-q'|\leq 5}
\Big[
\| \Sd_{q'-1} u^n \|_{L^\infty}
\| \Dd_{q' } \nabla u^n \|_{L^2 } +
\| \Sd_{q'-1} \nabla u^n \|_{L^\infty}
\| \Dd_{q' } u^n \|_{L^2 }
\Big]
\| \Dd_q u^n \|_{L^2 }\\
\sum_{|q-q'|\leq 5}
\Big[
\| \Sd_{q'-1} u^n \|_{L^\infty}
\| \Dd_{q' } \nabla u^n \|_{L^2 } +
\| \Sd_{q'-1} u^n \|_{L^\infty}
\| \Dd_{q' } u^n \|_{L^2 }
\Big]
\| \Dd_q u^n \|_{L^2 }\\
\sum_{|q-q'|\leq 5}
\Big[
\| \Sd_{q'-1} u^n \|_{L^\infty}
\| \Dd_{q' } \nabla u^n \|_{L^2 } +
\| \Sd_{q'-1} u^n \|_{L^\infty}
\| \Dd_{q' } \nabla u^n \|_{L^2 }
\Big]
\| \Dd_q u^n \|_{L^2}\\
\| u^n \|_{L^\infty}
\| \Dd_q u^n \|_{L^2 }
\sum_{|q-q'|\leq 5}
\| \Dd_{q' } \nabla u^n \|_{L^2 },
\end{align*}
so that, multiplying by $2^{2sq}$ and taking the sum as $q\in\ZZ$,
\begin{equation}\label{sec3_est1}
\sum_{q\in\ZZ}
2^{2qs} \Aa_q
\lesssim
\| u^n \|_{L^\infty}
\sum_{q\in\ZZ}
\Big\{
2^{2qs} \| \Dd_q u^n \|_{L^2 }
\sum_{|q-q'|\leq 5}
\| \Dd_{q' } \nabla u^n \|_{L^2 }
\Big\}
\lesssim
\| u^n \|_{L^\infty}
\| u^n \|_{\Hh^s }
\| \nabla u^n \|_{\Hh^s }.
\end{equation}
The control of $\Bb_q$ relies on convolution and the Young inequality, indeed
\begin{equation*}
\Bb_q \lesssim
\sum_{\substack{ q'\geq q-5\\|l|\leq 1} }
\| \Dd_{q'+l} u^n \|_{L^\infty}
\| \Dd_{q' } \nabla u^n \|_{L^2}
\| \Dd_{q } u^n \|_{L^2}
\lesssim
\| u^n \|_{L^\infty}
\| \Dd_{q } u^n \|_{L^2 }
\sum_{ q'\geq q-5 }
\| \Dd_{q' } \nabla u^n \|_{L^2 },
\end{equation*}
\begin{align*}
\sum_{q\in\ZZ}
2^{2qs} \Bb_q
\| u^n \|_{L^\infty}
\sum_{q\in\ZZ}
\Big\{
\| \Dd_{q } u^n \|_{L^2 }
\sum_{ q'\geq q-5 }
\| \Dd_{q' } \nabla u^n \|_{L^2 }
\Big\}\\
\| u^n \|_{L^\infty}
\sum_{q\in\ZZ}
\Big\{
\| \Dd_{q } u^n \|_{L^2 }
\sum_{ q'\geq q-5 }
2^{ q' s}
\| \Dd_{q' } \nabla u^n \|_{L^2 }
\Big\}\\
\| u^n \|_{L^\infty}
\| u^n \|_{\Hh^s }
\sum_{q\in\ZZ}
\Big\{
\| \Dd_{q } u^n \|_{L^2 }
\sum_{ q'\in\ZZ }
\Big\},
\end{align*}
where $(b_{q'})_\ZZ$ belongs to $l^2(\ZZ)$. Thus, we obtain
\begin{equation}\label{sec3_est2}
\sum_{q\in\ZZ}
2^{2qs} \Bb_q
\lesssim
\| u^n \|_{L^\infty}
\| u^n \|_{\Hh^s }
\| \nabla u^n \|_{\Hh^s },
\end{equation}
thanks to the Young inequality. Finally, summarizing (<ref>) and (<ref>), we obtain
\begin{equation}\label{sec3_major_est1}
\langle u^n \cdot \nabla u^n , u^n \rangle_{\Hh^s}
\sum_{q\in\ZZ}
\langle
\Dd_q( u^n \cdot \nabla u^n ) ,\, \Dd_q u^n
\rangle_{L^2}
\lesssim
\| u^n \|_{L^\infty}
\| u^n \|_{\Hh^s }
\| \nabla u^n \|_{\Hh^s }.
\end{equation}
Estimate of $\langle u^n\cdot \nabla Q^n , \Delta Q^n \rangle_{\Hh^s}$
Arguing exactly as for proving (<ref>), we obtain
\begin{equation}\label{sec3_major_est3}
\langle u^n\cdot \nabla Q^n , \Delta Q^n \rangle_{\Hh^s} =
\sum_{q\in\ZZ}
\langle \Dd_q( u^n\cdot \nabla Q^n,\,\Dd_q \Delta Q^n\rangle_{L^2}
\lesssim
\| u^n \|_{L^\infty}
\| \nabla Q^n \|_{\Hh^s }
\| \Delta Q^n \|_{\Hh^s }.
\end{equation}
Estimate of $\langle \nabla Q^n \odot \nabla Q^n ,\, \nabla u^n \rangle_{\Hh^s}$
We keep on our control, evaluating the term $\langle \Dd_q (\nabla Q^n \odot \nabla Q^n ),\, \Dd_q \nabla u^n \rangle_{L^2}$, with $q\in\ZZ$. The explicit integral formula of such term is the following one:
\begin{align*}
\int_{\RR^2}
\sum_{i,k=1}^2
\Dd_q (\,\trc\{\partial_i Q \partial_k Q \}\,) \Dd_q \partial_k u^n_i
\int_{\RR^2}
\sum_{i,k=1}^2 \sum_{j,l=1}^3
\Dd_q[\, \partial_i Q_{jl}^n\, \partial_k Q_{lj}^n \,] \Dd_q \partial_k u^n_i \\
\underbrace{
\int_{\RR^2}
\sum_{i,k=1}^2 \sum_{j,l=1}^3
\Dd_q[\, \dot{T}_{\partial_i Q_{jl}^n}\partial_k Q_{lj}^n + \dot{T}_{\partial_k Q_{lj}^n}\partial_i Q_{jl}^n \,] \Dd_q \partial_k u^n_i }_{\Cc_q} +
\underbrace{
\int_{\RR^2} \sum_{i,k,j,l}
\Dd_q \dot{R}(\partial_i Q_{jl}^n,\,\partial_k Q_{lj}^n) \Dd_q \partial_k u^n_i }_{\DD_q},
\end{align*}
where we have used the Bony decomposition again. First, let us observe that
\begin{equation*}
\Cc_q
\lesssim
\sum_{|q-q'|\leq 5}
\| S_{q-1} \nabla Q^n \|_{L^\infty}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }
\| \Dd_q \nabla u^n \|_{L^2 }
\lesssim
\| \nabla Q^n \|_{L^\infty}
\| \Dd_q \nabla u^n \|_{L^2 }
\sum_{|q-q'|\leq 5}
\| \Dd_{q'} \nabla Q^n \|_{L^2 },
\end{equation*}
which yields
\begin{equation}\label{sec3_est3}
\begin{aligned}
\sum_{q\in\ZZ}
2^{2qs} \Cc_q
\| \nabla Q^n \|_{L^\infty}
\sum_{q\in\ZZ}
\Big\{
\| \Dd_q \nabla u^n \|_{L^2 }
\sum_{|q-q'|\leq 5}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }
\Big\}\\
\| \nabla Q^n \|_{L^\infty}
\| \nabla u^n \|_{\Hh^s }
\| \nabla Q^n \|_{\Hh^s }.
\end{aligned}
\end{equation}
Moreover, considering $\DD_q$, we get
\begin{align*}
\DD_q
\sum_{\substack{q'\geq q-5 \\ |l|\leq 5}}
\| \Dd_{q'+l} \nabla Q^n \|_{L^\infty}
\| \Dd_{q' } \nabla Q^n \|_{L^2 }
\| \Dd_{q } \nabla u^n \|_{L^2 }\\
\| \nabla Q^n \|_{L^\infty}
\| \Dd_{q } \nabla u^n \|_{L^2 }
\sum_{ q'\geq q-5 }
\| \Dd_{q' } \nabla Q^n \|_{L^2 },
\end{align*}
so that, proceeding as in the proof of (<ref>),
\begin{equation}\label{sec3_est4}
\begin{aligned}
\sum_{q\in\ZZ}
2^{2qs} \DD_q
\| \nabla Q^n \|_{L^\infty}
\sum_{q\in\ZZ}
\Big\{
\| \Dd_{q } \nabla u^n \|_{L^2 }
\sum_{ q'\in\ZZ }
2^{ q' s}
\| \Dd_{q' } \nabla Q^n \|_{L^2 }
\Big\}\\
\| \nabla Q^n \|_{L^\infty}
\| \nabla u^n \|_{\Hh^s }
\| \nabla Q^n \|_{\Hh^s },
\end{aligned}
\end{equation}
thanks to the Young inequality. Thus, summarizing (<ref>) and (<ref>), we achieve
\begin{equation}\label{sec3_major_est2}
\sum_{q\in\ZZ} 2^{2qs}\langle \Dd_q (\nabla Q^n \odot \nabla Q^n ),\, \Dd_q \nabla u^n \rangle_{L^2}
\lesssim
\| \nabla Q^n \|_{L^\infty}
\| \nabla u^n \|_{\Hh^s }
\| \nabla Q^n \|_{\Hh^s },
\end{equation}
Estimate of $\langle \Delta Q^n Q^n -Q^n \Delta Q^n, \, \nabla u^n\rangle_{\Hh^s}$
Now, we carry out of $\langle \Delta Q^n Q^n -Q^n \Delta Q^n, \, \nabla u^n\rangle_{\Hh^s}$. This is the first non trivial term to evaluate. We choose to use the decomposition (<ref>), presented in the preliminaries, instead of the classical Bony decomposition (which we have used until now). We will remark the presence of a term inside
such decomposition, which is hard to control. However we will see that such drawback is going to be erased. Let us begin controlling
$ \langle \, Q^n \Delta Q^n, \, \nabla u^n \rangle_{\Hh^s}$:
\begin{equation*}
\langle \, Q^n \Delta Q^n, \, \nabla u^n \rangle_{\Hh^s}
\sum_{q\in\ZZ} 2^{2qs}
\langle \Dd_q (Q^n \Delta Q^n), \, \Dd_q \nabla u^n \rangle_{L^2} =
\sum_{q\in\ZZ}\sum_{i=1}^4 2^{2qs}
\langle \J_q^i (Q^n,\,\Delta Q^n), \, \Dd_q \nabla u^n \rangle_{L^2}.
\end{equation*}
where $\J^i_q$ has been defined by (<ref>), for $i=1,\dots, 4$.
When $i=1$, we point out that
\begin{align*}
\langle\, \J_q^1 (Q^n,\,\Delta Q^n) , \,\Dd_q \nabla u^n \rangle_{L^2}
&= \sum_{|q-q'|\leq 5}
\langle [ \Dd_q,\,\Sd_{q'-1}Q^n]\Dd_{q'}\Delta Q^n, \, \Dd_q\nabla u^n \rangle_{L^2} \\
&\lesssim \sum_{|q-q'|\leq 5}
2^{-q'} \| \Sd_{q'-1}\nabla Q^n \|_{L^\infty} \| \Dd_{q'} \Delta Q^n \|_{L^2} \| \Dd_q \nabla u \|_{L^2}\\
&\lesssim \| \nabla Q^n \|_{L^\infty}
\sum_{|q-q'|\leq 5} \| \Dd_{q'} \nabla Q^n \|_{L^2} \| \Dd_q \nabla u \|_{L^2}.
\end{align*}
which yields
\begin{equation}\label{sec3_est5}
\sum_{q\in\ZZ}2^{2qs} \langle \J_q^1 (Q^n,\,\Delta Q^n) , \,\Dd_q \nabla u^n \rangle_{L^2}
\lesssim
\| \nabla Q^n \|_{L^\infty}
\| \nabla Q^n \|_{\Hh^s }
\| \nabla u^n \|_{\Hh^s }.
\end{equation}
On the other hand, for $i=2$, we proceed as follows:
\begin{align*}
\langle \J_q^2 (Q^n,\,\Delta Q^n) , \,\Dd_q \nabla u^n \rangle_{L^2}
&= \sum_{|q-q'|\leq 5}
\langle (\Sd_{q'-1}Q^n-\Sd_{q-1}Q^n)\Dd_q\Dd_{q'}\Delta Q^n, \, \Dd_q\nabla u^n \rangle_{L^2} \\
&\lesssim \sum_{|q-q'|\leq 5}
\| \Sd_{q'-1}Q^n-\Sd_{q-1}Q^n \|_{L^\infty}
\| \Dd_q\Dd_{q'}\Delta Q^n \|_{L^2 }
\| \Dd_q\nabla u^n \|_{L^2 }\\
&\lesssim \sum_{|q-q'|\leq 5}
\| \Sd_{q'-1} \nabla Q^n-\Sd_{q-1} \nabla Q^n \|_{L^\infty}
\| \Dd_q \nabla Q^n \|_{L^2 }
\| \Dd_q \nabla u^n \|_{L^2 }\\
&\lesssim \| \nabla Q^n \|_{L^\infty}
\| \Dd_q \nabla Q^n \|_{L^2 }
\| \Dd_q \nabla u^n \|_{L^2 },
\end{align*}
which yields
\begin{equation}\label{sec3_est6}
\sum_{q\in\ZZ}2^{2qs}\langle \J_q^2 (Q^n,\,\Delta Q^n) , \,\Dd_q \nabla u^n \rangle_{L^2}
\lesssim
\| \nabla Q^n \|_{L^\infty}
\| \nabla Q^n \|_{\Hh^s }
\| \nabla u^n \|_{\Hh^s }.
\end{equation}
The case $i=4$ is handled as follows:
\begin{align*}
\langle \J_q^4 (Q^n,\,\Delta Q^n) ,\, \Dd_q \nabla u^n \rangle_{L^2}
&= \sum_{q'\geq q-5}
\langle \Dd_q[\,\Dd_{q'} Q^n\Sd_{q'+2} \Delta Q^n] ,\, \Dd_q \nabla u^n \rangle_{L^2}\\
\sum_{q'\geq q-5}
\| \Dd_{q'} Q^n \|_{L^2 }
\| \Sd_{q'+2} \Delta Q^n \|_{L^\infty}
\| \Dd_q \nabla u^n \|_{L^2 }\\
\sum_{q'\geq q-5}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }
\| \Sd_{q'+2} \nabla Q^n \|_{L^\infty}
\| \Dd_q \nabla u^n \|_{L^2 }\\
\| \nabla Q^n \|_{L^\infty}
\| \Dd_q \nabla u^n \|_{L^2 }
\sum_{q'\geq q-5}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }.
\end{align*}
Therefore, multiplying by $2^{2qs}$ and taking the sum as $q\in \ZZ$,
\begin{align*}
\sum_{q\in\ZZ}
\langle
& \J_q^4 (Q^n,\,\Delta Q^n),\, \Dd_q \nabla u^n
\rangle_{L^2}
\lesssim\\
\| \nabla Q^n \|_{L^\infty}
\sum_{q\in\ZZ}
\Big(
\| \Dd_q \nabla u^n \|_{L^2 }
\sum_{q'\in\ZZ}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }
\Big)\\
\| \nabla Q^n \|_{L^\infty}
\| \nabla u^n \|_{\Hh^s}
\Big\{
\sum_{q\in\ZZ}
\Big(
\sum_{q'\in\ZZ}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }
\Big)^2
\Big\}^{\frac{1}{2}},
\end{align*}
so that, by convolution and the Young inequality
\begin{equation}\label{sec3_est7}
\sum_{q\in\ZZ}2^{2qs}\langle \J_q^4 (Q^n,\,\Delta Q^n),\, \Dd_q \nabla u^n \rangle_{L^2}
\lesssim
\| \nabla Q^n \|_{L^\infty}
\| \nabla u^n \|_{\Hh^s}
\| \nabla Q^n \|_{\Hh^s}.
\end{equation}
It remains to control the term related to $\J^3_q$, namely
\begin{equation}\label{sec3_est10}
\sum_{q\in\ZZ} 2^{2qs} \langle \J^3(Q^n, \Delta Q^n), \Dd_q \nabla u^n\rangle_{L^2} =
\sum_{q\in\ZZ} 2^{2qs} \langle \Sd_{q-1}Q^n\Dd_q\Delta Q^n, \Dd_q \nabla u^n\rangle_{L^2}
\end{equation}
As already remark in the beginning, such term presents some difficulties. For instance, fixing $q\in\ZZ$ in the sum, the more natural estimate is the following one:
\begin{equation*}
\langle \Sd_{q-1}Q^n\Dd_q\Delta Q^n, \Dd_q \nabla u^n\rangle_{L^2}
\leq
\| \Sd_{q-1} Q^n \|_{L^\infty}
\| \Dd_{q } \Delta Q^n \|_{L^2}
\| \Dd_{q } \nabla u^n \|_{L^2}.
\end{equation*}
The presence of the low frequencies $\Sd_{q-1}$ in the first norm doesn't permit to transport a gradient to $Q^n$, so the best expectation is the following one:
\begin{equation*}
\sum_{q\in\ZZ} 2^{2qs} \langle \Sd_{q-1}Q^n\Dd_q\Delta Q^n, \Dd_q \nabla u^n\rangle_{L^2}
\lesssim
\| Q^n \|_{L^\infty}
\| \Delta Q^n \|_{\Hh^s }
\| \nabla u^n \|_{\Hh^s }.
\end{equation*}
Of course such inequality is not useful for our purpose, i.e. an Osgood type inequality. For example there isn't a term that appears in the time derivative of the left-hand side of (<ref>). Even if there exists a way to overcome such challenging evaluation, we will see that (<ref>) is going to be erased.
Now, let us keep on our control. We have to examine $\langle \Delta Q^n\, Q^n ,\, \nabla u^n\rangle_{\Hh^s}$. Observing that an equivalent formulation is $\langle Q^n \Delta Q^n ,\, \tr\nabla u^n\rangle_{\Hh^s}$ ($Q^n$ and $\Delta Q^n$ are symmetric matrices) we recompute the previous inequality (with $\tr \nabla u$ instead of $\nabla u$), so that
\begin{equation}\label{sec_est11}
\sum_{q\in\ZZ}\sum_{i=1,2,4} 2^{2qs}
\langle \J_q^i (Q^n,\,\Delta Q^n), \, \Dd_q \tr \nabla u^n \rangle_{L^2}
\lesssim
\| \nabla Q^n \|_{L^\infty}
\| \nabla u^n \|_{\Hh^s}
\| \nabla Q^n \|_{\Hh^s}.
\end{equation}
As before, $\J^3_q$ is an inflexible term, so that, recalling (<ref>), we need to erase what follows:
\begin{equation}\label{control1}
\begin{aligned}
\sum_{q\in\ZZ}
\Big\{
\langle \Sd_{q-1}Q^n\Dd_q\Delta Q^n, &\Dd_q \nabla u^n \rangle_{L^2} -
\langle \Sd_{q-1}Q^n\Dd_q\Delta Q^n, \Dd_q \tr \nabla u^n \rangle_{L^2}
\Big\} = \\& =
\sum_{q\in\ZZ}
\langle \Sd_{q-1}Q^n\Dd_q\Delta Q^n -\Dd_q\Delta Q^n\Sd_{q-1}Q^n,\, \Dd_q \nabla u^n \rangle_{L^2}
\end{aligned}
\end{equation}
Estimate of $\langle\, \Omega^n Q^n - Q^n \Omega^n,\,\Delta Q^n \rangle_{\Hh^s}$
Now, let us continue estimating $\langle\, \Omega^n Q^n - Q^n \Omega^n,\,\Delta Q^n \rangle_{\Hh^s}$. The strategy as the same organization of the previous evaluation. We begin analyzing $\langle\, Q^n \Omega^n,\,\Delta Q^n\rangle_{\Hh^s}$
\begin{equation*}
\langle\, Q^n \Omega^n,\,\Delta Q^n\rangle_{\Hh^s} =
\sum_{q\in \ZZ}2^{2qs}\langle \Dd_q (Q^n \Omega^n),\, \Dd_q\Delta Q^n\rangle_{L^2} =
\sum_{q\in \ZZ}\sum_{i=1}^4 2^{2qs}\langle \J^i_q (Q^n,\, \Omega^n),\, \Dd_q\Delta Q^n\rangle_{L^2}
\end{equation*}
First, considering $i=1$ and $q\in\ZZ$, we get
\begin{align*}
\langle \J^1_q (Q^n,\, \Omega^n),\, \Dd_q\Delta Q^n\rangle_{L^2}
&= \sum_{|q-q'|\leq 5}
\langle [ \Dd_q,\,\Sd_{q'-1}Q^n]\Dd_{q'} \Omega^n,\, \Dd_q \Delta Q^n\rangle_{L^2}\\
\sum_{|q-q'|\leq 5}
\| \Sd_{q'-1} \nabla Q^n \|_{L^\infty}
\| \Dd_{q' } \nabla u^n \|_{L^2 }
\| \Dd_{q } \Delta Q^n \|_{L^2 }\\
\| \nabla Q^n \|_{L^\infty}
\| \Dd_{q } \nabla Q^n \|_{L^2 }
\sum_{|q-q'|\leq 5}
\| \Dd_{q' } \nabla u^n \|_{L^2 }.
\end{align*}
therefore, taking the sum as $q\in\ZZ$,
\begin{equation}\label{sec3_est8}
\sum_{q\in\ZZ} 2^{2qs}\langle \J^1_q (Q^n,\, \Omega^n),\, \Dd_q\Delta Q^n\rangle_{L^2}
\lesssim
\| \nabla Q^n \|_{L^\infty}
\| \nabla Q^n \|_{\Hh^s }
\| \nabla u^n \|_{\Hh^s }.
\end{equation}
By a similar method as for proving (<ref>) or (<ref>), the case $i=2$ produces
\begin{equation*}
\sum_{q\in\ZZ}
\langle \J^2_q (Q^n,\, \Omega^n), \, \Dd_q \Delta Q^n \rangle_{L^2}
\lesssim
\| \nabla Q^n \|_{L^\infty}^2
\| \nabla Q^n \|_{\Hh^s }^2+
\frac{\nu}{100}
\| \nabla u^n \|_{\Hh^s }^2,
\end{equation*}
while, for $i=4$, we get
\begin{align*}
\langle \J_q^4 (Q^n,\,\Omega^n), \, \Dd_q \Delta Q^n \rangle_{L^2}
&= \sum_{q'\geq q-5}
\langle \Dd_q[\,\Dd_{q'} Q^n\Sd_{q'+2} \Omega^n], \, \Dd_q \Delta Q^n \rangle_{L^2}\\
\sum_{q'\geq q-5}
\| \Dd_{q' } Q^n \|_{L^2 }
\| \Sd_{q'+2} \Omega^n\|_{L^\infty}
\| \Dd_{q } \Delta Q^n \|_{L^2 }\\
&\lesssim \sum_{q'\geq q-5}
\| \Dd_{q' } \nabla Q^n \|_{L^2 }
\| \Sd_{q'+2} u^n \|_{L^\infty}
\| \Dd_{q } \Delta Q^n \|_{L^2 }\\
\| u^n \|_{L^\infty}
\| \Dd_{q } \Delta Q^n \|_{L^2 }
\sum_{q'\geq q-5}
\| \Dd_{q' } \nabla Q^n \|_{L^2 }.
\end{align*}
Thus, multiplying by $2^{2qs}$ and taking the sum as $q\in \ZZ$, we realize that
\begin{align*}
\sum_{q\in\ZZ}2^{2qs}
\langle &\J_q^4 (Q^n,\,\Omega^n),\, \Dd_q \Delta Q^n \rangle_{L^2}
\lesssim\\
\| u^n \|_{L^\infty}
\sum_{q\in\ZZ}
\Big(
\| \Dd_q \Delta Q^n \|_{L^2 }
\sum_{q'\in\ZZ}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }
\Big)\\
\| u^n \|_{L^\infty}
\| \Delta Q^n \|_{\Hh^s}
\Big[
\sum_{q\in\ZZ}
\Big(
\sum_{q'\in\ZZ}
\| \Dd_{q'} \nabla Q^n \|_{L^2 }
\Big)^2
\Big]^{\frac{1}{2}},
\end{align*}
so that, passing through the Young inequality,
\begin{equation}\label{sec3_est9}
\sum_{q\in\ZZ}2^{2qs}\langle \J_q^4 (Q^n,\,\Omega^n),\, \Dd_q \Delta Q^n \rangle_{L^2}
\lesssim
\| u^n \|_{L^\infty}
\| \Delta Q^n \|_{\Hh^s}
\| \nabla Q^n \|_{\Hh^s}.
\end{equation}
As the reader has already understood, the challenging term is the one related to $\J_q^3$, that is
\begin{equation}\label{sec3_est11}
\sum_{q\in\ZZ}2^{2qs}\langle \J_q^3 (Q^n,\,\Omega^n),\, \Dd_q \Delta Q^n \rangle_{L^2} =
\sum_{q\in\ZZ}2^{2qs}\langle \Sd_{q-1}Q^n,\Dd_q\Omega^n,\, \Dd_q \Delta Q^n \rangle_{L^2}
\end{equation}
As (<ref>), we are not capable to control it, so we claim that such obstacle is going to be simplified.
Going on, we observe that $\langle\, \Omega^n Q^n,\, \Delta Q^n\rangle_{\Hh^s}$ can be reformulated as $\langle\, Q^n \Omega^n ,\, \Delta Q^n\rangle_{\Hh^s}$, which we have just analyzed. Hence we need to control (<ref>) twice, that is
\begin{equation}\label{control2}
\sum_{q\in\ZZ} 2^{2qs} 2\langle S_{q-1} Q^n \Dd_q \Omega_n , \Dd_q \Delta Q^n\rangle_{L^2} =
\sum_{q\in\ZZ} 2^{2qs} \langle S_{q-1} Q^n \Dd_q \Omega_n - \Dd_q \Omega_n S_{q-1} Q^n, \Dd_q \Delta Q^n\rangle_{L^2},
\end{equation}
The Simplification
Recalling (<ref>) and (<ref>), we have not evaluated
\begin{align*}
\sum_{q\in\ZZ} 2^{2qs}
\big\{\,
\langle \Sd_{q-1}Q^n\Dd_q\Delta Q^n -&\Dd_q\Delta Q^n\Sd_{q-1}Q^n,\, \Dd_q \nabla u^n\rangle_{L^2} + \\&+
\langle S_{q-1} Q^n \Dd_q \Omega_n - \Dd_q \Omega_n S_{q-1} Q^n, \Dd_q \Delta Q^n\rangle_{L^2}\,
\big\},
\end{align*}
yet. However, this is a series whose coefficients are null, thanks to Theorem <ref>. Hence, we have overcome all the previous lacks, so that the following inequality is fulfilled:
\begin{equation}\label{sec3_major_est4}
\begin{aligned}
\langle \Delta Q^n Q^n -Q^n \Delta Q^n, \, \nabla u^n\rangle_{\Hh^s} - &\langle\, \Omega^n Q^n - Q^n \Omega^n,\,\Delta Q^n \rangle_{\Hh^s}
\lesssim\\
\| ( u^n,\, \nabla Q^n ) \|_{L^\infty}
\| (\nabla u^n,\, \Delta Q^n ) \|_{\Hh^s}
\| ( u^n,\, \nabla Q^n ) \|_{\Hh^s}.
\end{aligned}
\end{equation}
Estimate of $\langle \Pp(Q^n),\, \Delta Q^n\rangle_{\Hh^s}$
Finally, the last term to estimate is $\langle \Pp(Q^n),\, \Delta Q^n\rangle_{\Hh^s}$. Such evaluation is not a problematic, however it is computationally demanding, therefore we put forward in the appendix the proof of the following inequality:
\begin{equation}\label{estimate_P(Q)_DeltaQ}
\langle\, \Pp(Q^n),\, \Delta Q^n \rangle_{\Hh^s} \lesssim (1+ \|Q\|_{H^2} + \|Q\|_{H^2}^2 )\|\nabla Q\|_{\Hh^s}^2,
\end{equation}
where we remind that $H^2$ is a non-homogeneous Sobolev Space.
The Final Step
Summarizing the equality (<ref>) and the inequalities (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we deduce
\begin{equation}\label{sec3_equ1}
\begin{aligned}
&\frac{\dd}{\dd t}\Big[ \|u^n \|_{\Hh^s}^2 + L \|\nabla Q^n \|_{\Hh^s}^2 \Big]
+ \nu \| \nabla u^n \|_{\Hh^s}^2
+ \Gamma L^2 \| \Delta Q^n \|_{\Hh^s}^2
\lesssim\\
\| (u^n,\,\nabla Q^n) \|_{L^\infty }
\| (u^n,\,\nabla Q^n) \|_{\Hh^{s} }
\| (u^n,\,\nabla Q^n) \|_{\Hh^{1+s} } +
(1+ \|Q^n\|_{H^2} + \|Q^n\|_{H^2}^2 )\|\nabla Q^n\|_{\Hh^s}^2.
\end{aligned}
\end{equation}
For $t\geq 0$, we define the following time-functions
\begin{equation*}
\Phi(t) := \|u^n \|_{\Hh^s}^2 + \|\nabla Q^n \|_{\Hh^s}^2 ,\quad\quad \Psi(t) := \| \nabla u^n \|_{\Hh^s}^2+ \| \Delta Q^n \|_{\Hh^s}^2,
\end{equation*}
so that (<ref>) yields
\begin{align*}
\Phi'(t) + \Psi(t)
\lesssim
\| (u^n(t),\,\nabla Q^n(t)) \|_{L^\infty }
\| (u^n(t),\,\nabla Q^n(t)) \|_{\Hh^{s} }&
\| (u^n(t),\,\nabla Q^n(t)) \|_{\Hh^{1+s} } + \\ &+
(1+ \|Q^n(t)\|_{H^2} + \|Q^n(t)\|_{H^2}^2 )\Phi(t).
\end{align*}
Then, fixing a positive integer $N=N(t)$, we apply Lemma <ref>, obtaining
\begin{equation}\label{sec3_est12}
\begin{aligned}
\Phi' + \Psi
\lesssim
\Big\{
\| (u^n,\,\nabla Q^n) \|_{L^2 } +
& \sqrt{N}
\| (u^n,\,\nabla Q^n) \|_{\Hh^1 } +
\| (u^n,\,\nabla Q^n) \|_{\Hh^{s+1} }
\Big\}
{\scriptstyle \times}\\
& {\scriptstyle \times}
\| (u^n,\,\nabla Q^n) \|_{\Hh^{s} }
\| (u^n,\,\nabla Q^n) \|_{\Hh^{1+s} } +
(1+ \|Q^n\|_{H^2} + \|Q^n\|_{H^2}^2 )\Phi.
\end{aligned}
\end{equation}
For simplicity, let us define
\begin{equation*}
f_1 := \| (u^n,\,\nabla Q^n) \|_{L^2 }^2 + 1+ \|Q^n\|_{H^2} + \|Q^n\|_{H^2}^2 , \quad
f_2 := \| (u^n,\,\nabla Q^n) \|_{\Hh^1}^2,
\end{equation*}
hence (<ref>) implies
\begin{equation}\label{sec3_est13}
\Phi'(t) + \Psi(t) \leq C\big\{ f_1(t)\,\Phi(t) + N f_2(t) \Phi(t) + 2^{-Ns}\| (u^n,\,\nabla Q^n)(t) \|_{\Hh^{s} } \Psi(t) \big\},
\end{equation}
for a positive constant $C$. Now, choosing $N(t)$ to be a positive integer which fulfills
\begin{equation*}
\frac{1}{s}\log_2 \{ 2 + 4C + \Phi(t) \} \leq N(t) \leq \frac{1}{s}\log_2 \{ 2 + 4C + \Phi(t) \}+1
\end{equation*}
it turns out from (<ref>)
\begin{equation*}
\Phi'(t) + \Psi(t) \leq C\big\{ f_1(t)\,\Phi(t) + f_2(t) \Phi(t) (\frac{1}{s}\log_2 \{ 2 + 4C + \Phi(t) \}+1) \big\}+
\frac{1}{2}\Psi(t),
\end{equation*}
so that, finally, increasing the value of $C$, we obtain
\begin{equation}\label{sec3_est14}
\Phi'(t) + \Psi(t)
\leq C\big( f_1(t) + f_2(t)\big) \Phi(t) \log_2 \{ 2 + 4C + \Phi(t) \},
\end{equation}
which yields
\begin{equation*}
\Phi'(t)\leq \frac{C}{\ln 2}\big( f_1(t) + f_2(t)\big) (2+4C +\Phi(t)) \ln \{ 2 + 4C + \Phi(t) \}.
\end{equation*}
By integrating this differential inequality, we obtain
\begin{equation*}
2+4C+\Phi(t)\leq (2+4C+\Phi(0))^{\exp\{ \frac{C}{\ln 2}\int_0^t ( f_1(s) + f_2(s) )\dd s\}}.
\end{equation*}
Recalling the definition of $\Phi$, $f_1$ and $f_2$, we obtain
\begin{equation*}
\|(u^n,\,\nabla Q^n)(t) \|_{\Hh^s}^2 \leq
(2+4C+\|(u_0,\,\nabla Q_0) \|_{\Hh^s}^2)^{ \exp\{ \frac{C}{\ln 2}\int_0^t( \| (u^n(s),\,\nabla Q^n(s)) \|_{L^2 }^2 + 1+ \|Q(s)\|_{H^2} + \|Q(s)\|_{H^2}^2)\dd s\}}
\end{equation*}
Moreover, integrating (<ref>) in time, we get
\begin{equation*}
\int_0^t \Psi(s)\dd s \leq \Phi(0) + C\int_0^t \big( f_1(t) + f_2(t)\big) \Phi(t) \log_2 \{ 2 + 4C + \Phi(t) \}
\end{equation*}
that is
\begin{align*}
\int_0^t \|(u^n,\,\nabla Q^n)(\tau)\|_{\Hh^{s+1}}^2\dd \tau &\leq
\|(u_0,\,\nabla Q_0) \|_{\Hh^s}^2 +
C\int_0^t \big\{ \| (u^n,\,\nabla Q^n) \|_{L^2 }^2 + 1+ \|Q^n\|_{H^2} + \\
&+ \|Q^n\|_{H^2}^2\big\}(\tau)\dd \tau \|(u^n,\,\nabla Q^n)(t) \|_{\Hh^s}^2
\log_2 \{ 2 + 4C + \|(u^n,\,\nabla Q^n)(t) \|_{\Hh^s}^2\},
\end{align*}
Since such estimates are uniform in $n$, we pass to the limit as $n$ goes to $\infty$, obtaining
\begin{equation*}
\|(u,\,\nabla Q)\|_{L^\infty_T \Hh^s} \leq
(2+4C+\|(u_0,\,\nabla Q_0) \|_{\Hh^s}^2)^{ \frac{1}{2}\exp\{ \frac{C}{\ln 2}\int_0^T( \| (u^n(s),\,\nabla Q^n(s)) \|_{L^2 }^2 + 1+ \|Q(s)\|_{H^2} + \|Q(s)\|_{H^2}^2)\dd s\}},
\end{equation*}
\begin{align*}
\int_0^t \|(u,\,\nabla Q)(\tau)\|_{\Hh^{s+1}}^2\dd \tau &\leq
\|(u_0,\,\nabla Q_0) \|_{\Hh^s}^2 +
C\int_0^t \big\{ \| (u,\,\nabla Q) \|_{L^2 }^2 + 1+ \|Q\|_{H^2} + \\
&+ \|Q\|_{H^2}^2\big\}(\tau)\dd \tau \|(u,\,\nabla Q)(t) \|_{\Hh^s}^2
\log_2 \{ 2 + 4C + \|(u,\,\nabla Q)(t) \|_{\Hh^s}^2\},
\end{align*}
where $(u,\,Q)$ is solution of (<ref>) with $(u_0, \,Q_0)$ as initial data. This concludes the proof of Theorem (<ref>).
§.§ Useful tools
Let $Q_1$ and $Q_2$ be two $3\times 3$ symmetric matrices with entries in $H^2(\RR^2)$. Assume that $u$ is a $3$-vector with components in
$H^1(\RR^2)$ and let $\Omega$ be the $3\times 3$ matrix defined by $1/2(\nabla u - \tr \nabla u)$. Then the following identity is satisfied:
\begin{equation*}
\int_{\RR^2}\trc\{(\Omega Q_2 - Q_2 \Omega) \Delta Q_1 \} + \int_{\RR^2} \trc\{ ( \Delta Q_1 Q_2 - Q_1 \Delta Q_2 )\nabla u \} = 0
\end{equation*}
The proof is straightforward, indeed by a direct computation
\begin{align*}
\int_{\RR^2}\trc\{(\Omega Q_2 &-Q_2 \Omega) \Delta Q_1 \}
= \int_{\RR^2}\big[ \trc\{\Omega Q_2 \Delta Q_1 \} - \trc\{ Q_2 \Omega \Delta Q_1 \} \big]
= \int_{\RR^2}\big[ \trc\{\Omega Q_2 \Delta Q_1 \} -\\&- \trc\{ \Delta Q_1 \tr \Omega Q_2 \} \big]
= 2\int_{\RR^2} \trc\{\Omega Q_2 \Delta Q_1 \}
= \int_{\RR^2} \trc\{\nabla u Q_2 \Delta Q_1 - \tr \nabla u Q_2 \Delta Q_1 \}=\\
= \int_{\RR^2} \trc\{ (Q_1 \Delta Q_2 - \Delta Q_1 Q_2 )\nabla u \}.
\end{align*}
Let $f$ be a function in $H^1\cap \Hh^{1+s}$ with $s>0$. Then, there exists $C>0$ such that
\begin{equation*}
\| f \|_{L^\infty} \leq C\big(\,\| f \|_{L^2} + \sqrt{N}\| f \|_{H^1} + 2^{-Ns}\| f \|_{\Hh^{1+s}} \big),
\end{equation*}
for any positive integer $N$.
Let us fix $N>0$. Then $f = \Sd_{N+1}f + (\Id - \Sd_{N+1} )f$ fulfills
\begin{equation*}
\| f \|_{L^\infty} \leq \| \Sd_{N+1}f \|_{L^\infty} + \| \sum_{q\geq N }\Dd_q f \|_{L^\infty} \leq
\underbrace{
\sum_{q < N } \| \Dd_q f\|_{L^\infty}}_{\Aa} +
\underbrace{
\sum_{q \geq N } \| \Dd_q f \|_{L^\infty}}_{\Bb}.
\end{equation*}
First, let us analyze $\Aa$:
\begin{align*}
\sum_{q < N } \| \Dd_q f\|_{L^\infty}
\sum_{ q \leq 0 } \| \Dd_q f\|_{L^\infty} +
\sum_{ q = 1 }^N \| \Dd_q f\|_{L^\infty}
\lesssim
\sum_{ q \leq 0 } 2^{ q } \| \Dd_q f\|_{L^2} +
\sum_{ q = 1 }^N 2^{ q } \| \Dd_q f\|_{L^2}\\
\sum_{ q \leq 0 } \| \Dd_q f\|_{L^2} +
\sqrt{N}
\| f \|_{ \Hh^{1} }
\lesssim
\| f \|_{ L^2 } +
\sqrt{N}
\| f \|_{ \Hh^{1} }.
\end{align*}
Finally, from the definition of $\Bb$
\begin{equation*}
\sum_{q \geq N } \| \Dd_q f \|_{L^\infty}
\sum_{q \geq N} 2^{ q } \| \Dd_q f \|_{L^2}
\sum_{q \geq N} 2^{-sq} 2^{q(1+s)} \| \Dd_q f \|_{L^2}
\lesssim
2^{-Ns} \| f \|_{ \Hh^{1+s} },
\end{equation*}
which concludes the proof of the lemma.
proof of Theorem <ref>
At first we identify the Sobolev Spaces $\Hh^s$ and $\Hh^t$ with the Besov Spaces $\BB_{2,2}^s$ and $\BB_{2,2}^t$ respectively.
We claim that $ab$ belongs to $\BB_{2,2}^{s+t-N/2}$ and
\begin{equation*}
\|a b\|_{\BB_{2,2}^{s+t-N/2}}\leq C\|a\|_{\BB_{2,2}^{s}}\|b\|_{\BB_{2,2}^t},
\end{equation*}
for a suitable positive constant.
We decompose the product $ab$ through the Bony decomposition, namely $ab = \dot{T}_{a}b + \dot{T}_{b}a + R(a,b)$, where
\begin{equation*}
\dot{T}_{a}b := \sum_{q\in\ZZ} \Dd_q a\, \Sd_{q-1} b,\quad\quad
\dot{T}_{b}a := \sum_{q\in\ZZ} \Sd_{q-1} a\, \Dd_q b,\quad\quad
\Rd(a,b) := \sum_{\substack{q\in\ZZ\\|\nu|\leq 1}} \Dd_{q} a\, \Dd_{q+\nu} b.
\end{equation*}
For any $q\in \ZZ$, we have
\begin{align*}
&\| (\Dd_q \dot{T}_ab ,\, \Dd_q \dot{T}_b a) \|_{L^2}
\lesssim\\ &\lesssim
\sum_{|q-q'|\leq 5} 2^{q's} \|\Dd_q a \|_{L^2} 2^{q'(t-\frac{N}{2})} \| \Sd_{q-1} b\|_{L^\infty} +
\sum_{|q-q'|\leq 5} 2^{q'(s-\frac{N}{2})} \|\Sd_{q-1} a \|_{L^\infty} 2^{q't} \| \Dd_q b \|_{L^2},
\end{align*}
so that we determine the following feature
\begin{equation*}
\| (\dot{T}_{a}b,\, \dot{T}_b a) \|_{\BB_{2,2}^{s+t-\frac{N}{2}}}\leq
\| (\dot{T}_{a}b,\, \dot{T}_b a) \|_{\BB_{2,1}^{s+t-\frac{N}{2}}}\lesssim
\| a \|_{\BB_{2, 2}^s} \| b \|_{\BB_{\infty, 2}^{t-\frac{N}{2}}} +
\| a \|_{\BB_{\infty, 2}^{s-\frac{N}{2}}} \| b \|_{\BB_{2, 2}^t}\lesssim
\| a \|_{\BB_{2, 2}^s} \| b \|_{\BB_{2, 2}^t},
\end{equation*}
where we have used the embedding $\BB_{2,2}^{\sigma}\hookrightarrow \BB_{\infty, 2}^{\sigma -N/2}$, for any $\sigma\in\RR$ and Proposition <ref>.
In order to conclude the proof, we have to handle the rest $\Rd(a,b)$. By a direct computation, for any $q\in \ZZ$,
\begin{equation*}
2^{(t+s)q} \| \Dd_q \Rd(a,b) \|_{L^1} \leq
\sum_{\substack{q'\geq q-5\\|\nu|\leq 1}} 2^{(q-q')(s+t)} 2^{q's} \|\Dd_{q'} a\|_{L^2} 2^{(q'+\nu)t}\|\Dd_{q'+\nu} a\|_{L^2},
\end{equation*}
so that, thanks to the Young inequality, we deduce
\begin{equation*}
\| \Rd(a,b) \|_{\BB_{2,2}^{s+t-\frac{N}{2}}} \lesssim
\| \Rd(a,b) \|_{\BB_{1,1}^{s+t}} \lesssim
\| a \|_{\BB_{2,2}^s}
\| b \|_{\BB_{2,2}^t},
\end{equation*}
where we have used the embedding $\BB_{1,1}^{s+t}\hookrightarrow \BB_{2,2}^{s+t-N/2}$ and moreover that $\sum_{q\leq 5} 2^{q(s+t)}$ is
finite, since $s+t$ is positive.
§.§ Proof of estimate (<ref>)
$ $
The purpose of this section is to estimate the following term
\begin{equation*}
\langle \Pp(Q^n),\, \Delta Q^n \rangle_{\Hh^s}.
\end{equation*}
In order to facilitate the reader, we are not going to indicate the index $n$, from here on. We have to examine
\begin{align*}
\langle \Pp(Q),\, \Delta Q \rangle_{\Hh^s} &= \langle -aQ + b [Q^2 - \trc\{Q^2\}\frac{\Id}{3}] - c\trc\{Q^2\}Q,\,\Delta Q\rangle_{\Hh^s}\\
&= \langle -aQ + b Q^2 - c\trc\{Q^2\}Q,\,\Delta Q\rangle_{\Hh^s},
\end{align*}
where $\langle \trc\{Q^2\}\Id, \,\Delta Q\rangle_{\Hh^s}=0$ since $\Delta Q$ has null trace.
It is trivial that
\begin{equation}\label{appx_first_estimate}
-\langle a Q ,\,\Delta Q\rangle_{\Hh^s}\lesssim \|\nabla Q\|_{\Hh^s}.
\end{equation}
Now, let us consider $b\langle Q^2,\, \Delta Q\rangle_{\Hh^s}$. By definition we have
\begin{align*}
b\langle Q^2,\, \Delta Q\rangle_{\Hh^s}
&= b\sum_{q\in\ZZ} 2^{2qs} \langle \Dd_q[ Q^2],\,\Dd_q \Delta Q\rangle_{L^2}\\
b\sum_{q\in\ZZ} 2^{2qs}
\big[
\langle \Dd_q \dot{T}_QQ ,\, \Dd_q \Delta Q\rangle_{L^2}}_{\Aa_q} +
\underbrace{
\langle \Dd_q \dot{R}(Q,\,Q),\, \Dd_q \Delta Q\rangle_{L^2}}_{\Bb_q}
\big]
\end{align*}
We concentrate on $\Aa_q$, getting
\begin{align*}
\Aa_q
&\leq \sum_{|q-q'|\leq 5} \|\Sd_{q'-1} Q \Dd_{q'} Q\|_{L^2} \| \Dd_q \Delta Q\|_{L^2}
\lesssim \|Q\|_{L^\infty}\| \Dd_q \nabla Q\|_{L^2}\sum_{|q-q'|\leq 5}\|\Dd_{q'} \nabla Q\|_{L^2},
\end{align*}
so that
\begin{equation}\label{appx_est_A_q}
b\sum_{q\in\ZZ} 2^{2qs}\Aa_q \lesssim \|Q\|_{L^\infty}\| \nabla Q\|_{\Hh^s}^2.
\end{equation}
Now, analyzing $\Bb_q$, we observe that
\begin{equation*}
\Bb_q
\leq \sum_{\substack{q'\geq q- 5\\|l|\leq 1}} \| \Dd_{q'}Q \Dd_{q'+l} Q \|_{L^2} \|\Dd_q \Delta Q\|_{L^2}
\lesssim \| Q \|_{L^\infty} \|\Dd_q \nabla Q\|_{L^2} \sum_{q'\geq q- 5}2^{q-q'}\| \Dd_{q'}\nabla Q \|_{L^2},
\end{equation*}
so that
\begin{equation*}
b\sum_{q\in\ZZ} 2^{2qs}\Bb_q \lesssim
\| Q \|_{L^\infty}b\sum_{q\in\ZZ} 2^{qs}\|\Dd_q \nabla Q\|_{L^2} \sum_{q'\in\ZZ}2^{(q-q')(s+1)}1_{(-\infty,5)}(q-q')\| \Dd_{q'}\nabla Q \|_{L^2}.
\end{equation*}
Thus, by convolution and young inequality
\begin{equation*}
b\sum_{q\in\ZZ} 2^{2qs}\Bb_q\lesssim \|Q\|_{L^\infty}\| \nabla Q\|_{\Hh^s}^2,
\end{equation*}
and recalling (<ref>), we finally get
\begin{equation}\label{appx_second_estimate}
b\langle Q^2,\, \Delta Q\rangle_{\Hh^s} \lesssim \|Q\|_{L^\infty}\| \nabla Q\|_{\Hh^s}^2.
\end{equation}
Now, it remains to examine $c\langle Q\trc \{Q^2\} ,\, \Delta Q\rangle_{\Hh^s}$. The procedure is quietly similar to the previous one. At first we use the Bony decomposition as follows:
\begin{align*}
\langle Q\trc \{Q^2\} ,\, \Delta Q\rangle_{\Hh^s}
= \sum_{q\in\ZZ} 2^{2qs} \langle \Dd_q( Q &\trc\{Q^2\}) , \, \Dd_q \Delta Q \rangle_{L^2}
= \sum_{q\in\ZZ} 2^{2qs}
\Big[
\underbrace{
\langle \Dd_q \dot{T}_Q (\trc\{Q^2\}\Id), \,\Dd_q \Delta Q \rangle_{L^2}}_{\Aa_q} + \\&+
\underbrace{
\langle \Dd_q \dot{T}_{\trc\{Q^2\}\Id}Q ,\,\Dd_q \Delta Q \rangle_{L^2}}_{\Bb_q} +
\underbrace{
\langle \Dd_q \dot{R}(Q,\trc\{Q^2\}\Id) ,\,\Dd_q \Delta Q \rangle_{L^2}}_{\Cc_q}
\Big]
\end{align*}
First, we concentrate on $\Aa_q$, the more computationally demanding term, obtaining
\begin{align*}
\Aa_q \leq \sum_{[q-q'|\leq 5} \|\Sd_{q'-1}Q \Dd_{q'} (\trc\{Q^2\}\Id)\|_{L^2} \| \Dd_q \Delta Q \|_{L^2} \lesssim
\|Q \|_{L^\infty} \sum_{[q-q'|\leq 5} \| \Dd_{q'} ( Q^2 )\|_{L^2} \| \Dd_q \Delta Q \|_{L^2}\\
\lesssim
\|Q \|_{L^\infty} \sum_{[q-q'|\leq 5}
\Big[
\underbrace{
2 \| \Dd_{q'} \dot{T}_Q Q \|_{L^2}\| \Dd_q \Delta Q \|_{L^2}}_{I_{q,q'}} +
\underbrace{
\| \Dd_{q'} \dot{R}(Q,Q) \|_{L^2}\| \Dd_q \Delta Q \|_{L^2}}_{II_{q,q'}}
\Big]
\end{align*}
The term $I_q$ is the simpler one, indeed
\begin{equation*}
I_{q,q'} \lesssim \sum_{|q'-q''|\leq 5} \| \Sd_{q''-1} Q \Dd_{q''} Q\|_{L^2}\| \Dd_q \Delta Q \|_{L^2}
\lesssim
\| Q \|_{L^\infty}
\sum_{|q'-q''|\leq 5}
\|\Dd_{q''} Q\|_{L^2}\| \Dd_q \Delta Q \|_{L^2},
\end{equation*}
so that
\begin{align*}
\sum_{q\in\ZZ} \|Q\|_{L^\infty}
\sum_{[q-q'|\leq 5}I_{q,q'}
\|Q \|_{L^\infty}^2\sum_{q\in\ZZ} \sum_{[q-q'|\leq 5}
\sum_{|q'-q''|\leq 5}
\|\Dd_{q''} Q\|_{L^2}\| \Dd_q \Delta Q \|_{L^2} \\
\|Q \|_{L^\infty}^2\sum_{q\in\ZZ}
\sum_{|q-q''|\leq 10}
\|\Dd_{q''}\nabla Q\|_{L^2}\| \Dd_q \nabla Q \|_{L^2}\lesssim
\|Q \|_{L^\infty}^2\| \nabla Q\|_{\Hh^s}.
\end{align*}
We overcome the term $II_{q,q'}$ as follows:
\begin{align*}
\|\Dd_q \Delta Q\|_{L^2}\sum_{\substack{q''\geq q'-5\\ |l|\leq 1}} \|\Dd_{q''}Q \|_{L^2}\|\Dd_{q''+l}Q \|_{L^\infty}\lesssim
\| Q \|_{L^\infty}\|\Dd_q \nabla Q\|_{L^2} \sum_{q''\geq q'-5}2^{q-q''} \|\Dd_{q''}\nabla Q \|_{L^2},
\end{align*}
so that
\begin{align*}
\sum_{q\in\ZZ} \|Q\|_{L^\infty}
\sum_{[q-q'|\leq 5}II_{q,q'}
\|Q \|_{L^\infty}^2\sum_{q\in\ZZ}2^{2qs}\|\Dd_q \nabla Q\|_{L^2}\sum_{[q-q'|\leq 5}
\sum_{q''\geq q'-5} 2^{q-q''} \|\Dd_{q''}\nabla Q \|_{L^2}\\
\|Q \|_{L^\infty}^2\sum_{q\in\ZZ}2^{2qs}\|\Dd_q \nabla Q\|_{L^2}
\sum_{q''\geq q-10} 2^{q-q''} \|\Dd_{q''}\nabla Q \|_{L^2}\\
\|Q \|_{L^\infty}^2\sum_{q\in\ZZ}2^{qs}\|\Dd_q \nabla Q\|_{L^2}
\sum_{q''\geq q-10} 2^{(q-q'')(s+1)} 2^{q''s} \|\Dd_{q''}\nabla Q \|_{L^2},
\end{align*}
so that, by convolution and Young inequality
\begin{equation*}
\sum_{q\in\ZZ} \|Q\|_{L^\infty}
\sum_{[q-q'|\leq 5}II_{q,q'} \lesssim \|Q \|_{L^\infty}^2\| \nabla Q\|_{\Hh^s}.
\end{equation*}
Summarizing the previous inequalities, we get
\begin{equation*}
\sum_{q\in\ZZ}2^{2qs}\Aa_q\lesssim \| Q \|_{L^\infty}^2\| \nabla Q\|_{\Hh^s}^2.
\end{equation*}
In order to examine $\Bb_q$ it is sufficient to observe that
\begin{align*}
\sum_{q\in\ZZ}2^{2qs}&\Bb_q \lesssim \sum_{q\in\ZZ}2^{2qs} \sum_{|q-q'|\leq 5}\| \Sd_{q'-1}(\trc\{Q^2\}\Id) \Dd_{q'} Q \|_{L^2}\|\Dd_q \Delta Q\|_{L^2}\\
&\lesssim \|Q^2 \|_{L^\infty}\sum_{q\in\ZZ}2^{2qs} \sum_{|q-q'|\leq 5}\|\Dd_{q'}\nabla Q \|_{L^2}\|\Dd_q \nabla Q\|_{L^2}
\lesssim \|Q \|_{L^\infty}^2\|\nabla Q\|_{\Hh^s}^2.
\end{align*}
It remains indeed $\Cc_q$, which is straightforward, indeed
\begin{align*}
\sum_{q\in\ZZ} 2^{2qs} \Cc_q
\sum_{q\in\ZZ}
\sum_{\substack{q'\geq q-5\\ |l|\leq 1}}
\| \Dd_{q'} Q \Dd_{q+l}(Q^2) \|_{L^2}
\| \Dd_{q} \Delta Q \|_{L^2}\\
\|Q\|_{L^\infty}^2
\sum_{q\in\ZZ} 2^{qs}
\| \Dd_{q} \nabla Q \|_{L^2}
\sum_{q'\geq q-5} 2^{(q-q')(s+1)}
\| \Dd_{q'} \nabla Q \|_{L^2},
\end{align*}
thus, by convolution and the Young inequality,
\begin{equation*}
\sum_{q\in\ZZ} 2^{2qs} \Cc_q \lesssim \|Q\|_{L^\infty}^2 \|\nabla Q\|_{\Hh^s}^2.
\end{equation*}
Summarizing, we finally get
\begin{equation*}
c\langle Q\trc \{Q^2\} ,\, \Delta Q\rangle_{\Hh^s}\lesssim \|Q\|_{L^\infty}^2 \|\nabla Q\|_{\Hh^s}^2
\end{equation*}
and recalling (<ref>)-(<ref>), we finally obtain
\begin{equation*}
\langle \Pp(Q), \Delta Q \rangle_{\Hh^s}
\lesssim
( 1 + \| Q \|_{L^\infty} + \| Q \|_{L^\infty}^2 ) \| \nabla Q \|_{ \Hh^s }^2
\lesssim
( 1 + \| Q \|_{H^2} + \| Q \|_{H^2}^2 ) \| \nabla Q \|_{ \Hh^s }^2,
\end{equation*}
where the last inequality is due to the embedding $H^2(\RR^2)\hookrightarrow L^{\infty}(\RR^2)$. Hence, inequality (<ref>) is proven.
Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier
analysis and nonlinear partial differential equations, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 343, Springer, Heidelberg, 2011. 2768550 (2011m:35004)
Edwards B. J. Beris A. N., Thermodynamics of flowing systems with
internal microstructure, Oxford Engineering Science Series, vol. 36, Oxford
University Press, Oxford, New York, 1994.
de Gennes, P.-G., les cristaux liquides nématiques, J. Phys.
Colloques 32 (1971), C5a–3–C5a–9.
Colin Denniston, Enzo Orlandini, and J. M. Yeomans, Lattice boltzmann
simulations of liquid crystal hydrodynamics, Phys. Rev. E 63
(2001), 056702.
J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational
Mech. Anal. 9 (1962), 371–378. 0137403 (25 #855)
Jishan Fan and Tohru Ozawa, Regularity criteria for a coupled
Navier-Stokes and $Q$-tensor system, Int. J. Anal. (2013), Art. ID
718173, 5. 3081935
Giulia Furioli, Pierre G. Lemarié-Rieusset, and Elide Terraneo,
Unicité dans $L^3(\RR^3)$ et d'autres espaces fonctionnels limites
pour Navier-Stokes, Rev. Mat. Iberoamericana 16 (2000), no. 3,
605–667. 1813331 (2002j:76036)
Francisco Guillén-González and María Ángeles
Rodríguez-Bellido, Weak time regularity and uniqueness for a
$Q$-tensor model, SIAM Journal on Mathematical Analysis 46
(2014), no. 5, 3540–3567.
, Weak solutions for an initial-boundary $Q$-tensor problem
related to liquid crystals, Nonlinear Anal. 112 (2015), 84–104.
F. M. Leslie, Some constitutive equations for liquid crystals, Arch.
Rational Mech. Anal. 28 (1968), no. 4, 265–283. 1553506
Fang-Hua Lin and Chun Liu, Nonparabolic dissipative systems modeling the
flow of liquid crystals, Comm. Pure Appl. Math. 48 (1995), no. 5,
501–537. 1329830 (96a:35154)
, Existence of solutions for the Ericksen-Leslie system,
Arch. Ration. Mech. Anal. 154 (2000), no. 2, 135–156. 1784963
Fanghua Lin and Chun Liu, Static and dynamic theories of liquid
crystals, J. Partial Differential Equations 14 (2001), no. 4,
289–330. 1883167 (2003b:82063)
Craig S. MacDonald, John A. Mackenzie, Alison Ramage, and Christopher J. P.
Newton, Efficient Moving Mesh Methods for $Q$-Tensor Models
of Nematic Liquid Crystals, SIAM J. Sci. Comput. 37 (2015),
no. 2, B215–B238. 3319854
Fabien Marchand and Marius Paicu, Remarques sur l'unicité pour le
système de Navier-Stokes tridimensionnel, C. R. Math. Acad. Sci. Paris
344 (2007), no. 6, 363–366. 2309504 (2007m:35189)
Nigel Mottram and C.J.P. Newton, Introduction to q-tensor theory,
Workingpaper importmodel: Workingpaperimportmodel, 2014.
Marius Paicu and Arghir Zarnescu, Global Existence and Regularity
for the Full Coupled Navier-Stokes and Q-tensor System, SIAM
Journal on Mathematical Analysis 43 (2011), no. 5, 2009–2049.
Marius Paicu and Arghir Zarnescu, Energy dissipation and regularity for a
coupled Navier-Stokes and $Q$-tensor system, Arch. Ration. Mech. Anal.
203 (2012), no. 1, 45–67. 2864407 (2012m:76012)
N. Schopohl and T. J. Sluckin, Defect core structure in nematic liquid
crystals, Phys. Rev. Lett. 59 (1987), 2582–2584.
Géza Tóth, Colin Denniston, and J. M. Yeomans, Hydrodynamics of
domain growth in nematic liquid crystals, Phys. Rev. E 67 (2003),
Dehua Wang, Xiang Xu, and Cheng Yu, Global weak solution for a coupled
compressible Navier-Stokes and $Q$-tensor system, Commun. Math. Sci.
13 (2015), no. 1, 49–82. 3238138
Hao Wu, Xiang Xu, and Chun Liu, On the general Ericksen-Leslie
system: Parodi's relation, well-posedness and stability, Arch. Ration.
Mech. Anal. 208 (2013), no. 1, 59–107. 3021544
|
1511.00098
|
Matching cross-view images is challenging because the appearance and viewpoints are significantly different. While low-level features based on gradient orientations or filter responses can drastically vary with such changes in viewpoint, semantic information of images however shows an invariant characteristic in this respect.
Consequently, semantically labeled regions can be used for performing cross-view matching.
In this paper, we therefore explore this idea and propose an automatic method for detecting and representing the semantic information of an RGB image with the goal of performing cross-view matching with a (non-RGB) geographic information system (GIS).
A segmented image forms the input to our system with segments assigned to semantic concepts such as traffic signs, lakes, roads, foliage, etc.
We design a descriptor to robustly capture both, the presence of semantic concepts and the spatial layout of those segments.
Pairwise distances between the descriptors extracted from the GIS map and the query image are then used to generate a shortlist of
the most promising locations with similar semantic concepts in a consistent spatial layout. An experimental evaluation with challenging query images and a large urban area shows promising results.
§ INTRODUCTION
In this paper, we consider the cross-view and cross-modality matching problem between street-level RGB images and a geographic information system (GIS). Specifically, given an image taken from street-level, the goal is to query a database assembled from a GIS in order to return likely locations of the street-level query image which contain similar semantic concepts in a consistent layout.
Relying only on visual data is important in GPS-denied environments, for images where such tags have been removed on purpose (e.g. for applications in intelligence or forensic sciences), for historical images, or images from the web which are lacking any GPS tags.
Traditionally, such matching problems are solved by establishing pairwise correspondences between interest points using local descriptors such as SIFT <cit.> with a subsequent geometric verification stage. Unfortunately, even if top-down satellite imagery is available, such an approach based on local appearance features is not applicable to the wide-baseline cross-view matching considered in our setting, mainly because of the following two reasons.
Firstly, the extremely wide baseline between top-view GIS imagery and the street-level image leads to a strong perspective distortion, and secondly, there can be drastic changes in appearance, e.g. due to different weather conditions, time of day, camera response function, etc.
In this paper, we present a system to handle those two challenges. We propose to phrase the cross-view matching problem in a semantic way. Our system makes use of two cues: what objects are seen and what their geometric arrangement is. This is very similar to the way we humans try to localize ourselves on a map. For instance, we identify that a house can be seen on the left of a lake and that there are two streets crossing in front of this house. Then we will look for the same semantic concepts in a consistent spatial configuration in the map to find our potential locations.
Typical examples for a query image (left), semantic map from the GIS (middle) and corresponding heat map (right). Our proposed geolocalization scheme captures the semantic layout of the query image and leverages it for matching to a semantic map.
Inspired by this analogy, in our system, instead of matching low-level appearance-based features, we propose to extract segments from the image and label them with a semantic concept employing imperfect classifiers which are trained using images of the same viewpoint and therefore are not invalidated by the viewpoint change.
GIS often already provide highly-accurate semantically annotated top-down views thereby rendering the semantic labeling superfluous for the GIS satellite imagery. Hence, we assume that such a semantic map is provided by the GIS. A typical query image and an excerpt of a semantic map can be seen in fig:exampleimages.
The semantic concepts we focus on (e.g., buildings, lakes, roads, etc) form large (and quite often insignificant in number) segments in the image, and not points. Therefore, we argue that a precise point-based geometric verification, like a RANSAC-search<cit.> with an inlier criterion based on the Euclidean distance between corresponding points, is not applicable.
We address these issues by
a descriptor to robustly capture the spatial layout of those semantic segments. Pairwise asymmetric L2 matching between these descriptors is then used to find likely locations in the GIS map with a spatial layout of semantic segments which is consistent with the one in the query image. We also develop a tree-based search method based on a hierarchical semantic tree to allow fast geo-localization in a geographically broad areas.
Pipeline of our system: We use existing methods to compute the system inputs, namely the vanishing line of the ground plane and a semantic segmentation of the image. The ground plane in the query image is rectified to minimize distortions between a top-down GIS map and the query image. The GIS map is split into multiple overlapping tiles. Descriptors (blue circles) which capture the layout of the semantic regions in the query image and the tiles are extracted. Pairwise distances between those descriptors are then used for a final ranking of promising tiles.
§ RELATED WORK
Cross-view matching in terms of semantic segments between street-level query image and a GIS map joins several previous research directions.
Matching across a wide baseline has traditionally been addressed with local image descriptors for points <cit.>, areas <cit.>, or lines <cit.>.
Registration of street-level images with oblique aerial or satellite imagery is generally based only on geometric reasoning. Previous work, e.g. <cit.>, has reduced the matching problem to a 2D-2D registration problem by projecting ground models along vertical directions and rectifying the ground plane. Unlike our approach, the mentioned work requires a $3$D point cloud at query time, either from a laser scan <cit.> or from multiple views based on structure-from-motion <cit.>.
More recently, <cit.> considered the registration problem of a dense multi-view-stereo reconstruction from street-level images to oblique aerial views.
Building upon accurate 3D city models for assembling a database, contours of skylines in an upward pointing camera can be matched to a city model <cit.> or perspective distortion can be decreased by rectifying regions of the query image according to dominant scene planes <cit.>. <cit.> also relied on rectification of building facades, however, their system relied on the repetitive structure of elements in large facades, enabling a rectification without access to a 3D city model.
Using contours between the sky and landscape has also been shown to provide valuable geometric cues when matching to a digital elevation model <cit.>.
Not using any 3D information, Lin et al. <cit.> proposed a method to localize a street-level image with satellite imagery and a semantic map. Their system relies on an additional, large dataset which contains GPS-annotated street-level images which therefore establish an explicit link between street-level images and corresponding areas in the satellite imagery and semantic map. Similarly to the idea of information transfer in ExemplarSVMs <cit.>, once a short-list of promising images from this additional dataset has been generated by matching appearance-based features, appropriate satellite and semantic map information can be transferred from this short-list to the query image.
Visual location recognition and image retrieval system emphasise the indexing aspect and can handle large image collections: Bag-of-visual-words <cit.>, vocabulary trees <cit.> or global image descriptors such as Fisher vectors <cit.> have been proposed for that purpose, for example.
All those schemes do not account for any higher-level semantic information. More recently, <cit.> has therefore introduced a scheme where pooling regions for local image descriptors are defined in a semantic way: detectors assign each segment a class label and a separate descriptor (e.g. a Fisher Vector) is computed for each such segment.
Those descriptors rely on local appearance features, which fail to handle significant viewpoint changes faced in the cross-view matching problem considered in our paper.
Also, this approach does not encode the spatial layout between semantic segments.
If the descriptors are sufficiently discriminative by themselves, encoding this spatial layout is less important. In our case however, the information available in the query image which is shared with the GIS only captures class labels and a very coarse estimate of the segment shapes.
It is therefore necessary to capture both, the presence of semantic concepts and the spatial layout between those concepts, in a joint representation.
Very recently, Ardeshir et al.'s work <cit.> considered the matching problem between street-level image and a dataset of semantic objects. Specifically, deformable-part-models (DPMs) were trained to detect distinctive objects in urban areas from a single street-level image. The main objective of that paper was improved object detection with a geometric verification stage using a database of objects with known locations and the GPS-tag and viewing direction of the image has been assumed to be known roughly.
They also present an exhaustive search based approach to matching an image against the entire object database. The considered DPMs in <cit.> are well-localized and can be reduced to the centroid of the detection bounding box for a subsequent RANSAC step which searches for the best 2D-affine alignment in image space. Therefore, <cit.> can only handle such "spot" based information and is not designed to handle less accurate and potentially larger semantic segments with uncertain locations such as the ones provided by classifiers for `road' or `lake'.
§ SEMANTIC CROSS-VIEW MATCHING
A graphical illustration of our proposed system is shown in fig:systemoverview. The building blocks proposed by our paper will be described in detail in the next section, here we provide a rough overview of the system and describe the system inputs.
The computation of this input information relies on previous work and is therefore not considered as one of our contributions.
Given a street-level query image, first we split it into superpixel segments and label each segment with a semantic concept by a set of pre-trained classifiers.
We train two different types of classifiers to annotate superpixels with class labels corresponding to a subset of the labels available in the GIS[GIS often provide very fine-grained class labels for regions or areas. For simplicity, we consider a subset of labels for which appearance based classifiers can be trained reliably.].
For semantic concepts with large variation in appearance and shape, we are using the work by Ren et al. <cit.>. However, in street level images, it is also quite common to spot highly informative objects with small with-in class variation. Typical examples are traffic signs or lamp posts. For each of those classes, we therefore employ a deformable-part-model (DPM), similar to the ones of <cit.>.
The second piece of input information is an estimate of the vertical vanishing point or the horizon line, e.g. <cit.> describe how those entities can be estimated from a single image.
The perspective distortion of the ground plane can then be undone by warping the query image with a suitable rectifying homography, which is fully determined by the horizon line, assuming a rough guess of the camera intrinsic matrix is available.
We have opted to estimate the two inputs for our system, namely the ground plane location and the superpixel segmentation, in two entirely independent steps. We note however, that at the expense of higher computational cost this could be estimated jointly <cit.>.
The semantic map from the GIS and the warped and labelled query image now share the same modality and suffer from less perspective distortion. The cross-view matching problem between query and semantic GIS map is then cast as a search for consistent spatial layouts of those semantically labeled regions.
However, special care must be taken to model the propagation of errors due to potentially inaccurately detected segments in the original image
and due to segments which are not contained in the ground plane.
In order to do so,
we design a Semantic Segment Layout (SSL) descriptor which captures the spatial and semantic arrangement of those segments within a local support area located at a point in the query image or the semantic map.
Upon extracting such descriptors from the rectified image and semantic map, the problem can be reduced to a well-understood matching problem.
§ SEMANTIC SEGMENT LAYOUT (SSL) DESCRIPTOR
The goal of the SSL descriptor is to capture the presence of semantic concepts and at the same time encode the rough geometric layout of those concepts.
Similar to several previous descriptors <cit.>, the neighborhood around the descriptor centre is captured with pooling regions arranged in an annular pattern where the size of the regions increases with increasing distance to the descriptor centre.
We instantiate separate pooling regions for each semantic concept and the overall descriptor is the concatenation of the per-concept descriptors.
§.§ Placement of Descriptors
Pipeline for the SSL descriptor extraction in a query image: Due to the challenges in the contact region detection, in practice a simplified version of the illustration shown here is used, see text for more details.
(a) The pooling regions of the SSL descriptor are a set of Gaussians. The descriptor visualized here contains 2 rings of six pooling regions each. (b) Contact regions of vertical objects are detected in the rectified image by shooting rays from the camera position projected onto the ground plane (visualized as the lower vertex of the blue triangle and computed as $\mymatrix{H} \myvector{v}_{|}$, where $\mymatrix{H}$ is the rectifying homography and $\myvector{v}_{|}$ is the vertical vanishing point) and recording the closest intersection point with the segment. (c) Semantic segments are approximated with a Gaussian (for clarity, visualization is only for a single segment). (d) Uncertainty in contact regions are handled by convolving the Gaussians of vertical segments with another Gaussian whose covariance is elongated along the line between the camera position and the segment centroid. (e) A descriptor is extracted at each segment centroid.
Similar to appearance based local descriptors, we have to choose the locations where to extract descriptors and its orientation.
We are not aware of a good way to find reliable interest points in arrangements of semantic regions.
We initially experimented with placing a separate descriptor at the center of each semantic segment. Unfortunately, this choice turns out to be very sensitive to the location of the segment projected onto the ground plane. This projection depends on an accurate estimate of the contact region of that segment with the ground plane. Based on our experiments, it is challenging to get a sufficiently accurate estimate without manual user intervention.
In some preliminary experiments, we therefore also tried to factor in the contact region uncertainty by 'blurring' the contributions of a neighbouring segment along the line of sight between the camera and that segment, with an increasing amount of blur the further away the segment is from the camera center. We think this is an elegant and theoretically sound way to account for those uncertainties
and we refer to fig:descrextraction for a graphical illustration of the descriptor and of the subsequent processing steps of our preliminary pipeline.
We plan to explore this pipeline in future work in more detail.
In this work however, we settled for a simpler choice: a single descriptor is placed either in the center of the rectified image (denoted CI in the experiments) or at the center of the camera (CC in the experiments). Also, despite the SSL descriptor being more general, we only choose one annular pooling region thereby putting more emphasis on capturing the direction of semantic segments rather than the direction and distance. This choice is again motivated by the difficulty of accurately estimating contact regions.
We suggest using descriptors which are not rotation invariant and an orientation therefore needs to be assigned to each descriptor as well.
The reason for this choice is that the alternative of defining a rotation-invariant descriptor leads to a considerably less discriminative descriptor and pairwise matching score.
It is straightforward to define a canonical orientation in the semantic GIS map. For example, the first pooling region can be chosen to point to geographic North. However, unless compass direction is available, it is not easily possible to define a canonical orientation for the rectified query image.
Hence, we choose an arbitrary
direction for
the descriptors extracted from the rectified query image, and cope with not knowing the rotation parameters by employing a rotation invariant distance metric at the query time (see section sec:matching).
§.§ Aggregation over Pooling Region
There are several ways to capture the 'overlap' between a pooling region and a semantic segment. An intuitive approach is to compute the area of intersection between the pooling region and the segment.
This can be fairly slow for irregularly shaped segments. Moreover, the shape of the segments in the query image are quite imprecise, so an accurate computation of the intersection area might be unnecessary or even harmful.
Here, we propose a scheme motivated by a probabilistic point of view. The segments can be considered as a spatial probability distribution, that a point sampled in or close to this segment takes a certain label. Similarly, the pooling regions are interpreted as probability distributions of sampling a point at a certain location. We then have to compute a statistical measure for the 'similarity' between the two distributions. For discrete distributions, the mutual information is a good candidate. In our setting however, we have to handle continuous distributions defined over the ground plane. The Bhattacharyya distance <cit.> is a good way to measure the overlap between two continuous distributions and can be efficiently computed when we deal with Gaussian distributions.
Hence, in practice, in order to keep the computational requirements low, we will use a two-dimensional Gaussian to define a pooling region and each segment will be approximated by a Gaussian, as well.
In detail, let ${\cal G}^s(x;\mu^s,\Sigma^s)$ and ${\cal G}^p(x;\mu^p,\Sigma^p)$ denote the Gaussian for the segment and the pooling region, respectively. The Bhattacharyya distance is then given by
\begin{align}
d_B({\cal G}^s,{\cal G}^p) = \frac{1}{8}(\mu^s-\mu^p)^T\Sigma^{-1}(\mu^s-\mu^p)+ \nonumber \\
\frac{1}{2}\ln\frac{\det{\Sigma}}{\sqrt{\det{\Sigma^s}\det{\Sigma^p}}},
\end{align}
where $\Sigma=\frac{\Sigma^s+\Sigma^p}{2}$.
If multiple segments that are labeled with the same semantic concept are present, they can be treated as a Gaussian Mixture Model (GMM). The Bhattacharyya distance between two GMMs can be approximated <cit.> by
d_B({\cal M}^s,{\cal M}^p)=\sum_{i=1}^{N^s}\sum_{j=1}^{N^p}\alpha_i\beta_j d_B({\cal G}^s_i,{\cal G}^p_j)
where ${\cal M}^s=\sum_{i=1}^{N^s}\alpha_i{\cal G}^s_i(x;\mu^s_i,\Sigma^s_i)$ and ${\cal M}^p=\sum_{j=1}^{N^p}\beta_j{\cal G}^p_j(x;\mu^p_j,\Sigma^p_j)$ are a GMM for the segment and the pooling region, respectively. In our case, the pooling region is always represent by a single Gaussian, so $N^p = 1$ and $\beta_1 = 1$.
The Bhattacharyya distance is then converted to the Hellinger distance[The Hellinger distance satisfies the triangle inequality whereas the Bhattacharyya does not.]
\begin{align}
d_H({\cal M}^s,{\cal G}^p) &= \sqrt{1 - BC({\cal M}^s,{\cal G}^p)}
\text{,}
\end{align}
$ BC({\cal M}^s,{\cal G}^p) = \exp\left(-d_B({\cal M}^s,{\cal G}^p) \right) $
is the Bhattacharyya coefficient.
Our descriptor is the concatenation of all those Hellinger distances
$ \myvector{d} = \left( d_H({\cal M}^1, {\cal G}^1), d_H({\cal M}^1, {\cal G}^2), \ldots,
d_H({\cal M}^{N_s}, {\cal G}^{N_p}) \right) \in \Re^{N_s N_p}$.
Each block of this descriptor corresponding to a semantic concept is then L2-normalized independently of the other blocks.
If a concept is not present, the Hellinger distance for all pooling regions of that concept are set to zero.
Special care must be taken during the extraction of descriptors from the rectified query image, due to geometric imprecision in the localization of semantic segments. We also refer to fig:descrextraction for a graphical illustration of the necessary processing steps.
Superpixels which are not contained in the ground plane will be mapped to a wrong location by the rectifying homography. Furthermore, those superpixels are likely not visible from the top-down view in the GIS database. However, what is often visible from the top-down view is the projection of those superpixels onto the ground plane, which defines the silhouette of the object as viewed from the top.
We therefore estimate the contact region of a vertical object with the ground plane and only aggregate this contact region in our descriptor, see fig:descrextraction(b) for an illustrative explanation.
Unfortunately, accurate detection of the contact regions is a challenging task. We therefore need a procedure to handle inaccurate estimates.
An important observation is that the location uncertainty of the contact regions is highly anisotropic. For example, typical street-level images have the y-axis roughly aligned with the vertical vanishing point. In that case, the contact region of a tree or traffic sign is fairly well localized side-wise, however the depth uncertainty w.r.t. the camera can be large. Inaccuracies in contact point estimation therefore mostly lead to displacements in the rectified image along the line between the true contact point and the camera center projected onto the ground plane, i.e. $\mymatrix{H} \myvector{v}_{|}$, where $\mymatrix{H}$ is the rectifying homography and $\myvector{v}_{|}$ is the vertical vanishing point.
We design our descriptor to account for these inaccuracies by blurring the contribution of each semantic segment along this line with a spatially varying anisotropic blur kernel. The covariance matrix of the blur kernel is largest along the line of sight and increases with distance to the camera, see also fig:descrextraction(d).
§.§ Descriptor Computation in Semantic Map
The descriptor extraction from the semantic map is considerably simpler than from the query image: no rectification needs to performed as the overhead view can be readily rendered and the segment boundaries are exact.
The semantic map is divided into overlapping tiles for the subsequent matching stage.
A single SSL descriptor is extracted at the center of each tile (CI) or at the center of the camera (CC).
§ MATCHING
Given a street-level query image, our goal is to generate a `heat-map' of likely locations where this image has been taken. The semantic GIS map is therefore split into fix sized overlapping tiles, based on parameters such as average query image field-of-view and height of camera above ground.
As described previously, the rotation alignment between the descriptor used in the semantic map and the query image is unknown. In order to cope with that, we propose to rotate one of the descriptors in discrete rotation steps and compute the L2 distance for each step.
This boils down to the computation of a circular correlation (or circular convolution) between blocks of the two descriptors corresponding to the same semantic concept. This can be implemented efficiently with a circulant matrix multiplication or even with a FFT using the circular convolution theorem <cit.>.
In our implementation, we are currently using the L2 distance between two descriptors. However, especially for the descriptor placed at the camera centre, several pooling regions will not be contained in the field of view. We therefore employ an asymmetric L2 distance where the distance contribution of pooling regions in the query descriptor which are not within the field of view of the camera is set to zero. The field of view of the camera can easily be estimated given the image resolution and focal length.
§ HIERARCHICAL SEMANTIC TREE REPRESENTATION OF A GIS MAP
Two top-most layers of the semantic tree. The tiles of each layer have been split into $L = 3$ groups with the hierarchical spectral clustering, as described in the paper. We can see how this clustering produces semantically similar clusters. Note that the 'white' areas in the top-most layer denote tiles which are mostly empty, i.e. had no semantic entries in the GIS map.
The reference area covered by GIS maps is often very broad. This leads to a large number of reference tiles which the query descriptor should be compared against. Several techniques, such as k-means or kd-tree, have developed for fast nearest neighbor search <cit.> which we can employ to speed up this process.
We describe a procedure inspired by k-means trees <cit.> suitable for pre-computing a hierarchical semantic tree which arranges the tiles of the map in a semantically and spatially meaningful way. This speeds up the matching and also enables fast semantic queries in a GIS system.
Our tree construction is based on hierarchical spectral clustering with $L$ branches on each level. The similarity matrix required for spectral clustering is composed of the previously described asymmetric distance between pairs of tiles of the GIS map. This provides us with a $N \times N$ similarity matrix, where $N$ denotes the number of tiles. We use spectral clustering, rather than k-means as used in k-means trees or similar methods, since we are employing our own customized distance function (asymmetric L2) while those methods often assume a standard distance.
fig:tree shows the two top layers of the hierarchical tree ($L=3$) obtained by applying the procedure just defined on a large GIS image. What we observe is that the area has been partitioned into three well-defined semantic concepts: water (in blue), area scarcely populated (yellow) and densely populated area (grey) in the first layer. In the second layer, each of those three areas are decomposed into three other areas, and construction of the tree continues as such. The tree being semantically meaningful is a byproduct of the fact that our descriptor is targeted towards capturing semantic information.
At query time, we start traversing the tree at the root. The query image is matched against a random subset of tiles contained in each cluster of the current level. The most promising child node out of the $L$ children at each level, which represents the cluster, is found in this way. We use this technique since the cluster center is not straightforward to define for our customized distance functions. The tree is traversed all the way down to the leaf nodes to find the final match[Spilling could be introduced, as well, where more than just the most promising child is explored.]. The tree depth is $\log_L(N)$, which leads to a speed-up of roughly $\frac{N}{M \log_L(N)}$ where $M$ denotes the cardinality of the random subset of tiles explored at each level.
§ EXPERIMENTS AND RESULTS
The area on which the cross-view matching has been tested is located in the District of Columbia, US. On the bottom we report the color coding for the semantic classes used in the experiments.
Our framework has been tested for the geolocalization of generic outdoor images taken in the entire ($\sim159 km^2$) of the District of Columbia, US, as extensive GIS databases of this area are made available to public <cit.>. The area is depicted in Figure <ref> and includes a variety of different regions (water, suburban, urban). We have gathered a set of $50$ geo-tagged images from Google Maps and Panoramio taken at different locations, which serves as a benchmark for our system.
We have $N_c=7$ semantic classes, that are $C=${Road, Tree, Building, Water, Lamp Post, Traffic Signal, Traffic Sign}, and in the following experiments we are going to use different sets of classes. The size of a tile in the GIS is around $\sim 30m^2$. We set the number of rings in our descriptors to $1$, and the number of pooling regions to $8$.
We assume that the focal length is approximately known[A good estimate of the focal length is often available in the EXIF-header. If not, then the focal length can also be estimated from three orthogonal vanishing points, for example.]. The homography which rectifies the ground plane is then determined by the vertical vanishing point or the horizon line.
We assume that the y-axis of the image is roughly aligned with the vertical direction and the vertical vanishing point is then given by the MLE including all the lines segments $\pm 20$ degrees w.r.t. the y-axis, see also <cit.>. As ground plane estimation is not the topic of our paper, we manually check and correct highly inaccurate estimates of the vanishing point in our benchmark images in order not to bias our evaluation toward mistakes in the ground plane estimation (this is done for all baselines).
Since the semantic GIS map uses metric units and the rectified image is in pixels, the size or scale of the pooling regions need to be converted between those two units. A reasonable assumption for street-level images is that the camera is roughly at $d = 1.7m$ above ground. The conversion factor between metric units and pixels is then given by $[\text{pixels}] = \frac{f}{d} [\text{meters}]$. The scale of the pooling regions is chosen such that a SSL descriptor captures significant contributions from segments within roughly $30$m.
Qualitative results: Each row shows a separate query, the query image is shown in the first column.
The second column shows the segmented and rectified query image in the top left and the top-15 tiles returned by our system (from left to right and top to bottom).
The third column shows the ground truth location (blue circle) on top of a heat map visualization of the matching scores between the query image and the GIS map tiles (color-coded in log-scale). It can be seen that semantically similar tiles have a higher score and are therefore ranked higher. This ranking is visualized in the fourth column: each plot shows the empirical cumulative-distribution-function (CDF) of the (normalized) score between the query and all the tiles. The red star denotes the bin which contains the ground truth tile.
Quantitative results: Analogously to the visualization in <cit.>, this figure plots the percentage of queries which contained the ground truth tile in a short-list of the best scoring tiles against the normalized size of that short-list (i.e. the x-axis denotes the fraction of tiles contained in the short-list). The red curve shows how the combination of SSL descriptor in the center of camera and Presence term outperforms the other methods.
fig:quali1 reports detailed qualitative results of our proposed cross-view matching scheme for several sample queries. Several interesting observations can be made in this figure. First, as the covered area is very large, even such generic semantic cues can narrow down the search space to often $<5\%$. Second, the semantic and geometric similarity among the top 15 matching GIS tiles shows the proposed method is successfully capturing such properties and is yet forgiving of the modeled uncertainties by not being overly discriminative; note that the ground truth location (marked in the CDF) is often among the top few percents. Third, the heat map correlates well with the semantic content of the image.
As for quantitative results, we compare different Nearest Neighbor (NN) classifiers with the following feature vectors: SSL descriptor in the center of image (per query image/GIS tile), SSL descriptor in the center of the camera, binary indicator vector encoding the presence or absence of semantic concepts (Presence term), SSL descriptor in the center of the camera plus the Presence term. The last method is random matching. The superior results of the SSL plus Presence matching reported in fig:quant1 confirm the necessity of jointly using semantic and coarse geometry for a successful localization task.
We also suspect that placing the SSL descriptor in the center of camera results in better performance than placing the descriptor in the center of the image because the former approach is less sensible to tiling quantization artifacts: e.g. objects on the left side of the field of view will generally remain on the left side even when the viewpoint is moved to the closest tile location.
We used the subset $C^s=${Building, Lamp Post, Traffic Signal, Traffic Sign} of the $N_c$ semantic classes since this subset yielded the best overall geo-localization result.
For further evaluation, fig:excl depicts the results of SSL plus Presence matching over four different sets of semantic classes, which shows the contribution of each semantic class in the overall geo-localization results.
Interestingly, the curves also show how certain sets of semantic classes may even mislead the geo-localization process. We believe this is due to un-informativeness (e.g. being too common) of some classes as well as several sources of noise whose magnitude can vary between different classes (e.g. inaccurate entries in GIS map, semantic segmentation misclassifications, etc.).
If enough training data were available, appropriate weights could be learned for each class.
Quantification of the impact of different semantic configurations (in the legend R is road, T is tree, B is building, W is water, L is lamp post, T is traffic signal, S is traffic sign).
§ SUMMARY
This paper proposed an approach for cross-view matching between a street-level image and a GIS map. This problem was addressed in a semantic way. A fast Semantic Segment Layout descriptor has been proposed which jointly captures the presence of segments with a certain semantic concept and the spatial layout of those segments. As our experimental evaluation showed, this enabled matching a street-level image to a large reference map based on purely semantic cues of the scene and their coarse spatial layout. The results confirm that the semantic and topological cues captured by our method significantly narrow down the search area. This can be used as an effective pre-processing for other less efficient but more accurate localization techniques, such as street view based methods.
This work has been supported by the Max Planck Center for Visual Computing and Communication. We also acknowledge the support of NSF CAREER grant (N1054127).
Washington DC Open GIS Data.
K. T. Abou–Moustafa and F. P. Ferrie.
A note on metric properties of some divergence measures: The gaussian
In ACML, JMLR W&CP, 2012.
S. Ardeshir, A. R. Zamir, A. Torroella, and M. Shah.
Gis-assisted object detection and geospatial localization.
In ECCV, pages 602–617. Springer, 2014.
G. Baatz, K. Köser, D. Chen, R. Grzeszczuk, and M. Pollefeys.
Leveraging 3d city models for rotation invariant place-of-interest
IJCV, 96(3):315–334, 2012.
G. Baatz, O. Saurer, K. Köser, and M. Pollefeys.
Large scale visual geo-localization of images in mountainous terrain.
In ECCV, pages 517–530. Springer, 2012.
M. Bansal, K. Daniilidis, and H. Sawhney.
Ultra-wide baseline facade matching for geo-localization.
In Computer Vision–ECCV 2012. Workshops and Demonstrations,
pages 175–186. Springer, 2012.
H. Bay, V. Ferrari, and L. Van Gool.
Wide-baseline stereo matching with line segments.
In CVPR, pages 329–336. IEEE, 2005.
J.-C. Bazin, Y. Seo, C. Demonceaux, P. Vasseur, K. Ikeuchi, I. Kweon, and
M. Pollefeys.
Globally optimal line clustering and vanishing point estimation in
manhattan world.
In CVPR. IEEE, 2012.
S. Belongie, J. Malik, and J. Puzicha.
Shape context: A new descriptor for shape matching and object
In NIPS, volume 2, page 3, 2000.
P. Denis, J. H. Elder, and F. J. Estrada.
Efficient edge-based methods for estimating manhattan frames in urban
In ECCV, pages 197–210. Springer, 2008.
T. Durand, D. Picard, N. Thome, and M. Cord.
Semantic pooling for image categorizatin using multiple kernel
In ICIP, 2014.
M. A. Fischler and R. C. Bolles.
Random sample consensus: a paradigm for model fitting with
applications to image analysis and automated cartography.
Communications of the ACM, 24(6):381–395, 1981.
C. Frueh and A. Zakhor.
Constructing 3d city models by merging ground-based and airborne
In CVPR, volume 2, pages 554–562. IEEE, 2003.
R. Hartley and A. Zisserman.
Multiple view geometry in computer vision.
Cambridge University Press, 2nd edition, 2004.
D. Hoiem, A. A. Efros, and M. Hebert.
Putting objects in perspective.
IJCV, 80(1):3–15, 2008.
H. Jégou, F. Perronnin, M. Douze, J. Sanchez, P. Perez, and C. Schmid.
Aggregating Local Image Descriptors into Compact Codes.
PAMI, 34(9), 2012.
R. S. Kaminsky, N. Snavely, S. M. Seitz, and R. Szeliski.
Alignment of 3d point clouds to overhead images.
In CVPR Workshop, pages 63–70. IEEE, 2009.
T.-Y. Lin, S. Belongie, and J. Hays.
Cross-view image geolocalization.
In CVPR, pages 891–898. IEEE, 2013.
D. Lowe.
Distinctive image features from scale-invariant keypoints.
IJCV, 60(2), 2004.
T. Malisiewicz, A. Gupta, and A. A. Efros.
Ensemble of exemplar-svms for object detection and beyond.
In ICCV. IEEE, 2011.
J. Matas, O. Chum, M. Urban, and T. Pajdla.
Robust wide-baseline stereo from maximally stable extremal regions.
Image and vision comp., 22(10):761–767, 2004.
D. Nister and H. Stewenius.
Scalable recognition with a vocabulary tree.
In CVPR, 2006.
A. V. Oppenheim and A. S. Willsky.
Signals and systems.
Prentice-Hall, 1997.
S. Ramalingam, S. Bouaziz, P. F. Sturm, and M. Brand.
Skyline2gps: Localization in urban canyons using omni-skylines.
In IROS, pages 3816–3823, 2010.
X. Ren, L. Bo, and D. Fox.
Rgb-(d) scene labeling: Features and algorithms.
In CVPR. IEEE, 2012.
A. Saxena, M. Sun, and A. Y. Ng.
Learning 3-d scene structure from a single still image.
In ICCV. IEEE, 2007.
G. Sfikas, C. Constantinopoulos, A. Likas, and N. P. Galatsanos.
An analytic distance metric for gaussian mixture models with
application in image retrieval.
In ANN, pages 835–840. Springer, 2005.
Q. Shan, C. Wu, B. Curless, Y. Furukawa, C. Hernandez, and S. M. Seitz.
Accurate geo-registration by ground-to-aerial image matching.
In 3DV. IEEE, 2014.
J. Sivic and A. Zisserman.
Video google: A text retrieval approach to object matching in videos.
In ICCV, 2003.
E. Tola, V. Lepetit, and P. Fua.
A fast local descriptor for dense matching.
In CVPR, pages 1–8. IEEE, 2008.
B. Verhagen, R. Timofte, and L. Van Gool.
Scale-invariant line descriptors for wide baseline matching.
In WACV, pages 493–500. IEEE, 2014.
|
1511.00145
|
On the optimal control of opinion dynamics on evolving networks
G. Albi, L. Pareschi, M. Zanella
TU München, Faculty of Mathematics,
Boltzmanstraße 3, D-85748, Garching (München), Germany.
Department of Mathematics and Computer Science,
Via N. Machiavelli 35, 44121, University of Ferrara, Italy.
In this work we are interested in the modelling and control of opinion dynamics spreading on a time evolving network with scale-free asymptotic degree distribution. The mathematical model is formulated as a coupling of an opinion alignment system with a probabilistic description of the network. The optimal control problem aims at forcing consensus over the network, to this goal a control strategy based on the degree of connection of each agent has been designed. A numerical method based on a model predictive strategy is then developed and different numerical tests are reported. The results show that in this way it is possible to drive the overall opinion toward a desired state even if we control only a suitable fraction of the nodes.
§ INTRODUCTION
Graph theory has emerged in recent years as one of the most active fields of research <cit.>. In fact, the study of technological and communication networks earned a special attention thanks to a huge amount of data coming from empirical observations and more recently from online platforms like Facebook, Twitter, Instagram and many others. This fact offered a real laboratory for testing on a large-scale the collective behavior of large populations of agents <cit.> and new challenges for the scientific research has emerged. In particular, the necessity to handle millions, and often billions, of vertices implied a substantial shift to large-scale statistical properties of graphs giving rise to the study of the so-called scale-free networks <cit.>.
In this work, we will focus our attention on the modelling and control of opinion dynamics on a time evolving network.
We consider a system of agents, each one belonging to a node of the network, interacting only if they are connected through the network. Each agent modifies his/her opinion through a compromise function which depends both on opinions and the network <cit.>. At the same time new connections are created and removed from the network following a preferential attachment process. For simplicity here we restrict to non-growing network, that is a graph where the total number of nodes and the total number of edges are conserved in time. An optimal control problem is then introduced in order to drive the agents toward a desired opinion.
The rest of the paper is organized as follows. In Section <ref> we describe the alignment model for opinions spreading on a non-growing network. In order to control the trajectories of the model we introduce in Section <ref> a general setting for a control technique weighted by a function on the number of connections. A numerical method based on model predictive control is then developed. Finally in Section <ref> we perform numerical experiments showing the effectiveness of the present approach. Some conclusion are then reported in the last Section.
§ MODELLING OPINION DYNAMICS ON NETWORKS
In the model each agent $i=1,\dots,N$ is characterized by two quantities $(w_i,c_i), i=1,\dots,N$, representing the opinion and the number of connections of the agent $i$th respectively. This latter term is strictly related to the architecture of the social graph where each agent shares its opinion and influences the interaction between individuals. Each agent is seen here as a node of a time evolving graph $\mathcal{G}^N=\mathcal{G}^N(t), t\in[t_0,t_f]$ whose nodes are connected through a given set of edges. In the following we will indicates the density of connectivity the constant $\gamma\ge 0$.
§.§ Network evolution without nodes' growth
In the sequel we will consider a graph with both a fixed number of nodes $N$ and a fixed number of edges $E$. In order to describe the network's evolution we take into account a preferential attachment probabilistic process. This mechanism, known also as Yule process or Matthew effect, has been used in the modeling of several phenomena in biology, economics and sociology, and it is strictly connected to the generation of power law distributions <cit.>. The initial state of the network, $\mathcal{G}^N(0)$, is chosen randomly and, at each time step an edge is randomly selected and removed from the network. At the same time, a node is selected with probability
\begin{equation}\label{eq:probability_pi}
\Pi_{\alpha}(c_i) = \dfrac{c_i+\alpha}{\sum_{j=0}^N (c_j+\alpha)}=\dfrac{c_i+\alpha}{2E+N\alpha},\qquad i=1,\dots,N,
\end{equation}
among all possible nodes of $\mathcal{G}^N$, with $\alpha>0$ an attraction coefficient. Based on the probability (<ref>) another node is chosen at time $t$ and connected with the formerly selected one. The described process is repeated at each time step. In this way both the number of nodes and the total number of edges remains constant in the reference time interval. Let $p(c,t)$ indicates the probability that a node is endowed of degree $c$ at time $t$, we have
\begin{equation}\begin{split}
\sum_c p(c,t)=1, \qquad
\sum_c c~p(c,t)=\gamma.
\end{split}\end{equation}
Then we have that the described process is in agreement with the following master equation
\begin{equation}\begin{split}\label{eq:master}
\dfrac{d}{dt} p(c,t)=&\dfrac{D}{E}\left[(c+1)p(c+1,t)-cp(c,t)\right]\\
\end{split}\end{equation}
where $D>0$ characterizes the relaxation velocity of the network toward an asymptotic degree distribution $p_{\infty}(c)$, the righthand side consists of four terms, the first and the third terms account the rate of gaining a node of degree $c$ and respectively the second and fourth terms the rate of losing a node of degree $c$. The equation (<ref>) holds in the interval $c\le E$, whereas for each $c>E$ we set $p(c,t)=0$. While most the random graphs models with fixed number of nodes and vertices produces unrealistic degree distributions as the Watts and Strogatz generation model, called small-world model <cit.>, the main advantage of the graph generated through the described process is the possibility to recover the scale-free properties. Indeed we can easily show that if $\gamma=2E/N \ge 1$ with attraction coefficient $\alpha\ll 1$ then the stationary degree distribution $p_{\infty}(c)$ obeys a power-law of the following form
\begin{equation}
\end{equation}
When $\alpha\gg 1$ we loose the features of the preferential attachment mechanism, in fact high degree nodes are selected approximately with the same probability of the nodes with low degree of connection. Then the selection occurs in a non preferential way and the asymptotic degree distribution obeys the Poisson distribution
\begin{equation}
\end{equation}
A simple graph is sketched in Figure <ref> where we can observe how the initial degree of the nodes influences the evolution of the connections. In order to correctly observe the creation of the new links, that preferentially connect nodes with the highest connection degree, we marked each node with a number $i=1,\dots,20$ and the nodes' diameters are proportional with their number of connections.
Left: initial configuration of the sample network $\mathcal{G}^{20}$ with density of connectivity $\gamma=5$. Right: a simulation of the network $\mathcal{G}^{20}$ after $10$ time steps of the preferential attachment process. The diameter of each node is proportional to its degree of connection.
§.§ The opinion alignment dynamics
The opinion of the $i$th agent ranges in the closed set $I=[-1,1]$, that is $w_i=w_i(t)\in I$ for each $t\in [t_0,t_f]$, and its opinion changes over time according to the following differential system
\begin{equation}\label{eq:dynamics}
\dot{w}_i = \frac{1}{|S_i|} \sum_{j\in S_i} P_{ij} (w_j-w_i), \qquad i=1,\dots,N
\end{equation}
where $S_i$ indicates the set of vertex connected with the $i$th agent and reflects the architecture of the chosen network, whereas $c_i=|S_i| < N$ stands for the cardinality of the set $S_i$, also known as degree of vertex $i$. Note that the number of connections $c_i$ evolves in time accordingly to the process described in Section <ref>. Furthermore we introduced the interaction function $P_{ij}\in[0,1]$, depending on the opinions of the agents and the graph $\mathcal{G}^N$ which can be written as follows
\begin{equation}
P_{ij}=P(w_i,w_j; \mathcal{G}^N).
\end{equation}
A possible choice for the interaction function is the following
\begin{equation}\label{eq:interaction_fun}
P(w_i,w_j; \mathcal{G}^N)=H(w_i,w_j)K(\mathcal{G}^N),
\end{equation}
where $H(\cdot,\cdot)$ represents the positive compromise propensity,
and $K$ a general function taking into account statistical properties of the graph $\mathcal{G}$. In what follows we will consider $K=K(c_i,c_j)$, a function depending on the verticesÕ connections.
§ OPTIMAL CONTROL PROBLEM OF THE ALIGNMENT MODEL
In this section we introduce a control strategy which characterizes the action of an external agent with the aim of driving opinions toward a given target $w_d$. To this goal, we consider the evolution of the network $\mathcal{G}^N(t)$ and the opinion dynamics in the interval $[t_0,t_f]$. Therefore we introduce the following optimal control problem
\begin{equation}\label{eq:functional}
\min_{u\in\mathcal{U}}J(\textbf{w},u):= \frac{1}{2}\int_{t_0}^{t_f} \Big\{ \frac{1}{N}\sum_{j=1}^N (w_j(s)-w_d)^2 +\nu u(s)^2 \Big\} ds,
\end{equation}
subject to
\begin{equation}\begin{split}\label{eq:constrain}
\dot w_i &= \dfrac{1}{|S_i|}\sum_{j\in S_i}P_{ij}(w_j-w_i)+u\chi(c_i\ge c^*),\quad
\end{split}\end{equation}
where we indicated with $\mathcal{U}$ the set of admissible controls, with $\nu>0$ a regularization parameter which expresses the strength of the control in the overall dynamics and $w_d\in[-1,1]$ the target opinion.
Note that the action of the control $u$ is weighted by an indicator function $\chi(\cdot)$, which is active only for the nodes with degree $c_i\ge c^*$.
In general this selective control approach models an a-priori strategy of a policy maker, possibly acting under limited resources or unable to influence the whole ensemble of agents.
The solution of this kind of control problems is in general a difficult task, given that their direct solution is prohibitively expensive for a large number of agents. Different strategies have been developed for alignment modeling in order to obtain feedback controls or more general numerical control techniques <cit.>.
To tackle numerically the described problem a standard strategy makes use of a model predictive control (MPC) approach, also referred as receding horizon strategy.
In general MPC strategies solves a finite horizon open-loop optimal control problem predicting the dynamic behavior over a predict horizon $t_p\leq t_f$, with initial state sampled at time $t$ (initially $t = t_0$), and computing the control on a control horizon $t_c\leq t_p$. The optimization is computed introducing
a new integral functional $J_p(\cdot,\cdot)$, which is an approximation of (<ref>) on the time interval $[t,t+t_p]$, namely
\begin{equation}\label{eq:sub_functional}
J_p(\textbf{w},\bar{u}):= \frac{1}{2}\int_t^{t+t_p} \Big\{ \frac{1}{N}\sum_{j=1}^N (w_j(s)-w_d)^2 +\nu_p \bar{u}(s)^2 \Big\} ds
\end{equation}
where the control, $\bar{u}: [t,t+t_p]\to \mathcal{U}$, is supposed to be an admissible control in the set of admissible control $\mathcal{U}$, subset of $\mathbb{R}$, and $\nu_p$ a possibly different penalization parameter with respect to the full optimal control problem.
Thus the computed optimal open-loop control $\bar{u}(\cdot)$ is applied feedback to the system dynamic until the next sampling time $t+t_s$ is evaluated, with $t_s\leq t_c$, thereafter the procedure is repeated taking as initial state of the dynamic at time $t+t_s$ and shifting forward the prediction and control horizons, until the final time $t_f$ is reached. This process generates a sub-optimal solution with respect to the solution of the full optimal control problem (<ref>)-(<ref>).
Let us consider now the full discretize problem, defining the time sequence $[t_0,t_1,\ldots,t_M]$, where $t_{n}-t_{n-1}=t_s=\Delta t>0$ and $t_M:=M\Delta t = t_f$, for all $n=1,\ldots, M $, assuming furthermore that $t_c = t_p = p\Delta t$, with $p>0$. Hence the linear MPC method look for a piecewise control on the time frame $[t_0,t_M]$, defined as follows
\begin{align}
\bar{u}(t) = \sum_{n=0}^{M-1} \bar{u}^n\chi_{[t_n,t_{n+1}]}(t).
\end{align}
In order to discretize the evolution dynamic we consider a Runge-Kutta scheme, the full discretized optimal control problem on the time frame $[t_n,t_n+p\Delta t]$ reads
\begin{equation}\label{eq:disc_functional}
\min_{\bar{u}\in\mathcal{U}}J_p(\textbf{w},\bar{u}):= \frac{1}{2}\int_{t_n}^{t_n+p\Delta t} \Big\{ \frac{1}{N}\sum_{j=1}^N (w_j(s)-w_d)^2 +\nu_p \bar{u}^2 \Big\} ds
\end{equation}
subject to
\begin{equation}
\begin{subequations}
\begin{aligned}%\begin{split}
W^{(n)}_{i,l} &= w_{i}^n + \Delta t\sum_{k=1}^s a_{l,k} \left(F(t+\theta_k\Delta t,W^{(n)}_{i,k})+\bar{U}^{(n)}_kQ_i(t+\theta_k\Delta t)\right),\\
w_i^{n+1} &= w_{i}^n + \Delta t\sum_{l=1}^s b_{l} \left(F(t+\theta_l\Delta t,W^{(n)}_{i,l})+\bar{U}^{(n)}_lQ_i(t+\theta_l\Delta t)\right),\\
w_i^n &= w_i(t_n),
%\bar{u}(t)&\in \mathcal{U},\, \forall t\in[t_n,t_n+t_c], \,\bar{u}(t) = \bar{u}(t+t_c),\, \forall t\in[t_n+t_c,t_n+t_p]
\end{aligned}
\end{subequations}
\end{equation}
for all $n=1,\ldots, p-1$; $l=1,\ldots,s$; $i,\ldots, N$ and having defined the following functions
\begin{align*}
F(t,w_i)=\dfrac{1}{|S_i(t)|}\sum_{j\in S_i(t)}P_{ij}(w_j-w_i),\quad
Q_i(t) = \chi(c_i(t)\ge c^*).
\end{align*}
The coefficients $(a_{l,k})_{l,k}$, $(b_l)_l$ and $(\theta_{l})_l$, with $l,k=1,\ldots,s$, define the Runge-Kutta method and $(\bar{U}^{(n)})_l, (W^{(n)}_{i,l})_l$ are the internal stages associated to $\bar{u}(t), w_i(t)$ on time frame $[t_n,t_{n+1}]$.
(Instantaneous control).
Let us restrict to the case of a single prediction horizon, $p = 1$, where we discretize the dynamic with an explicit Euler scheme ( $a_{1,1}=\theta_1=0$ and $ b_1 = 1$). Notice that since the control $\bar{u}$ is a constant value and assuming that the network, $\mathcal{G}^N$ remains fixed over the time interval $[t_n,t_n+\Delta t]$ the discrete optimal control problem (<ref>) reduces to
\begin{equation}
\min_{\bar{u}\in\mathcal{U}}J_p(\textbf{w},\bar{u}^n):= \Delta t \Big\{ \frac{1}{N}\sum_{j=1}^N (w_j^{n+1}(\bar{u}^n)-w_d)^2 +\nu_p (\bar{u}^n)^2 \Big\}
\end{equation}
\begin{align}\label{eq:EE}
w_i^{n+1} &= w_{i}^n + \Delta t\left(F(t_n,w^n_i)+\bar{u}^nQ_i^n\right),\quad w_i^n= w_i(t_n).
\end{align}
In order to find the minima of (<ref>) is sufficient to find the value $\bar{u}$ satisfying $ \partial_{\bar{u}} J_p(\textbf{w},\bar{u})=0$, which can be computed by a straightforward calculation
\begin{align}\label{eq:IC}
\bar{u}^n = -\frac{1}{N\nu+ \Delta t\sum_{j=1}^N(Q^n_j)^2}\left( \sum_{j=1}^NQ^n_j\left(w^n_j-w_d\right)+\Delta t\sum_{j=1}^NQ^n_jF(t_n,w^n_i)\right).
\end{align}
where we scaled the penalization parameter with $\nu_p= \Delta t \nu$.
§ NUMERICAL RESULTS
In this section we present some numerical results in order to show the main features of the control introduced in the previous paragraphs. We considered a population of $N=100$ agents, each of them representing a node of an undirected graph with density of connectivity $\gamma=30$. The network $\mathcal{G}^{100}$ evolves in the time interval $[0,50]$ with attraction coefficient $\alpha=0.01$ and represents a single sample of the evolution of the master equation (<ref>) with $D=20$. The control problem is solved by the instantaneous control method described in Remark <ref> with $\Delta t=5~10^{-2}$.
In Figure <ref> we present the evolution over the reference time interval of the constrained opinion dynamics. The interaction terms have been chosen as follows
\begin{equation}\label{eq:KH}
K(c_i,c_j)=e^{-\lambda c_i}\left(1-e^{-\beta c_j}\right), \qquad H(w_i,w_j)=\chi(|w_i-w_j|\le \Delta),
\end{equation}
where the function $H(\cdot,\cdot)$ is a bounded confidence function with $\Delta=0.4$, while $K(\cdot,\cdot)$ defines the interactions between the agents $i$ and $j$ taking into account that agents with a large number of connections are more difficult to influence and at the same time they have more influence over other agents. The action of the control is characterized by a parameter $\kappa=0.1$ and target opinion $w_d=0.8$. We present the resulting opinion dynamics for a choice of constants $\lambda=1/100,\beta=1$ in Figure <ref>. We report the evolution of the network and of the opinion in Figure <ref>, here the diameter of each node is proportional with its degree of connection whereas the color indicates its opinion.
As a measure of consensus over the agents we introduce the quantity
\begin{equation}
V_{w_d}=\dfrac{1}{N-1}\sum_{i=1}^N (w_i(t_f)-w_d)^2,
\end{equation}
where $w_i(t_f)$ is the opinion of the $i$th agent at the final time $t_f$. In Figure <ref> we compare different values of $V_{w_d}$ as a function of $c^*$. Here we calculated the size of the controlled agents and the values of $V_{w_d}$ both, starting from a given uniform initial opinion and the same graph with initial uniform degree distribution.
It can be observed how the control is capable to drive the overall dynamics toward the desired state acting only on a portion of the nodes.
Evolution of the constrained opinion dynamics with uniform initial distribution of opinions over the time interval $[0, 50]$ for different values of $c^*=10,15,30$ with target opinion $w_d=0.8$, control parameter $\kappa=0.1$, $\Delta t=10^{-3}$ and confidence bound $\Delta=0.4$.
Evolution of opinion and connection degree of each node of the previously evolved graph $\mathcal{G}^{100}$. From left to right: graph at times $t=0,25,50$. From the top: opinion dynamics for threshold values $c^*=10,20,30$. The target opinion is set $w_d=0.8$ and the control parameter $\kappa=0.1$.
Left: the red squared plot indicates the size of the set of controlled agent at the final time $t_f$ in dependence on $c^*$ whereas the blue line indicates the mean square displacement $V_{w_d}$. Right: values of the control $u$ at each time step for $c^*=10,20,30$. In the numerical test we assumed $\Delta=0.4, \Delta t=5~10^{-3}, \kappa=0.1$.
§ CONCLUSIONS AND PERSPECTIVES
In this short note we focus our attention on a control problem for the dynamic of opinion over a time evolving network. We show that the introduction of a suitable selective control depending on the connection degree of the agent's node is capable to drive the overall opinion toward consensus. In a related work we will consider this problem in a mean-field setting where the number of agents, and therefore nodes, is very large <cit.>.
§ ACKNOWLEDGMENTS
GA acknowledges the support of the ERC-Starting Grant project High-Dimensional Sparse Optimal Control (HDSPCONTR).
AB Albert, R., Barabˆsi, A.-L.: Statistical mechanics of complex networks. Reviews of modern physics, 74(1), 2002.
ABCK Albi, G., Bongini, M., Cristiani, E., Kalise, D.: Invisible control of self-organizing agents leaving unknown environments. preprint 2015.
AHP Albi, G., Herty, M., Pareschi, L.: Kinetic description of optimal control problems and applications to opinion consensus. Communications in Mathematical Sciences, 13(6): 1407–1429, 2015.
APZa Albi, G., Pareschi, L., Zanella, M.: Boltzmann-type control of opinion consensus through leaders. Philosophical Transactions of the Royal Society A, 372(2028): 20140138, 2014.
APZb Albi, G., Pareschi, L., Zanella, M.: Uncertainty quantification in control problems for flocking models. Mathematical Problems in Engineering, 2015, 14 pp., 2015.
APZc Albi, G., Pareschi, L., Zanella, M.: Modelling and control of opinion dynamics over large evolving networks. Work in progress.
ASBS Amaral, L. A. N., Scala A., BarthŽlemy, M., Stanley, H. E.: Classes of small-world networks. Proceedings of the National Academy of Sciences of the United States of America, 97(21): 11149–11152, 2000.
BAJ Barabˆsi, A.-L., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Physica A: Statistical Meachanics and its Applications, 272(1): 173–187, 1999.
BBV Barrat, A., BarthŽlemy, M., Vespignani, A.: Dynamical Processes on Complex Networks, Cambridge University Press, 2008.
BBSZ Benczik, I. J., Benczick, S. Z., Schmittmann, B., Zia, R. K.: Opinion dynamics on an adaptive random network. Physical Review E, 79(4): 046104, 2009.
BF Bongini, M., Fornasier, M., Fröhlich, F., Haghverdi, L.: Sparse stabilization of dynamical systems driven by attraction and avoidance forces. Networks and Heterogeneous Media, 9(1): 1–31, 2014.
WCB Wongkaew, S., Caponigro, M., Borzì, A.: On the control through leadership of the Hegselmann-Krause opinion formation model. Mathematical Models and Methods in Applied Sciences, 25(2): 255–282, 2015.
C Chi, L.: Binary opinion dynamics with noise on random networks. Chinese Science Bulletin, 56(34): 3630–3632, 2011.
DGM Das, A., Gollapudi, S., Munagala, K.: Modeling opinion dynamics in social networks. Proceedings of the 7th ACM International conference on Web search and data mining, ACM, 2014.
JGN Jin, E. M., Girvan, M., Newman, M. E. J.: Structure of growing social networks. Physical Review E, 64(4): 046132, 2001.
KGH Kramer, A. D. I., Guillory, J. E., Hancock, J. T.: Experimental evidence of massive scale emotional contagion through social networks. Proceedings of the National Academy of Sciences, 111(24): 8788–8789, 2014.
N Newman, M. E. J.: The structure and function on complex networks. SIAM Review, 45(2): 167–256, 2003.
PT Pareschi, L., Toscani, G.: Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods. Oxford University Press, 2013.
S Strogatz, S. H.: Exploring complex networks. Nature, 410(6825): 268-276, 2001.
SWS Sznajd-Weron, K., Sznajd, J.: Opinion evolution in closed community. International Journal of Modern Physics C, 11(6): 1197-1165, 2000.
WSWatts, D.J., Strogatz, S.H.: Collective dynamics of 'small-world' networks. Nature, 393: 440-442, 1998.
WWeisbuch, G.: Bounded confidence and social networks. The European Physical Journal B-Condensed Matter and Complex Systems, 38(2): 339-343, 2004.
XZWXie, Y.-B., Zhou, T., Wang, B.-H.: Scale-free networks without growth. Physica A, 387: 1683-1688, 2008.
|
1511.00396
|
\begin{equation}#1\end{equation}
\begin{align}#1\end{align}
.05em ind
SF ^ind
mod-$\K Q$
mod-$\K Q/I$
nil-$\K Q$
nil-$\K Q/I$
#1|#1 |
Let $\CF(\fObj_\A)$ denote the vector space of $\Q$-valued constructible functions on a given stack $\fObj_\A$ for an exact category $\A$. By using the Ringel–Hall algebra approach, Joyce proved that $\CF(\fObj_\A)$ is an associative $\mathbb{Q}$-algebra via the convolution multiplication and the subspace $\CFi(\fObj_\A)$ of constructible functions supported on indecomposables is a Lie subalgebra of $\CF(\fObj_\A)$ in <cit.>. In this paper, we show that there is a subalgebra $\CF^{\text{KS}}(\fObj_\A)$ of $\CF(\fObj_\A)$ isomorphic to the universal enveloping algebra of $\CFi(\fObj_\A)$. Moreover we construct a comultiplication on $\CF^{\text{KS}}(\fObj_\A)$ and a degenerate form of Green's theorem. This generalizes Joyce's work, as well as results of <cit.>.
§ INTRODUCTION
Let $\Lambda$ be a finite dimensional $\mathbb{C}$-algebra such that it is a representative-finite algebra, i.e., there are finitely many finite dimensional indecomposable $\Lambda$-modules up to isomorphism. Let $\mathcal{I}(\Lambda)=\{X_1,\ldots,X_n\}$ be a set of representatives. Let $\mathcal{P}(\Lambda)$ be a set of representatives for the all isomorphism classes of $\Lambda$-modules. There is a free $\mathbb{Z}$-module $R(\Lambda)$ with a basis $\{u_X~|~X\in\mathcal{P}(\Lambda)\}$. Using the Euler characteristic, $\mathcal{P}(\Lambda)$ can be endowed with a multiplicative structure (see <cit.> and <cit.>). The multiplication is defined by
u_X\cdot u_Y=\sum\limits_{A\in\mathcal{P}(\Lambda)}\chi(V(X,Y;A))u_A,
where $V(X,Y;A)=\{0\subseteq A_1\subseteq A~|~A_1\cong X, A/A_1\cong Y\}$ and $\chi(V(X,Y;A))$ is the Euler characteristic of $V(X,Y;A)$. Thus $(R(\Lambda),+,\cdot)$ is a $\mathbb{Z}$-algebra with identity $u_0$. Let $L(\Lambda)$ be a submodule of $R(\Lambda)$ which is spanned by $\{u_X~|~X\in\mathcal{I}(\Lambda)\}$. It follows that $L(\Lambda)$ is a Lie subalgebra of $R(\Lambda)$ with the Lie bracket $[u_X,u_Y]=u_X\cdot u_Y-u_Y\cdot u_X$. Riedtmann studied the universal enveloping algebra of $L(\Lambda)$. Let $R(\Lambda)^{\prime}$ be the subalgebra of $R(\Lambda)$ generated by $\{u_X~|~X\in\mathcal{I}(\Lambda)\}$. Riedtmann showed that $R(\Lambda)^{\prime}$ is isomorphic to the universal enveloping algebra of $L(\Lambda)$. These results have been generalized by two ways.
Joyce generalized Riedtmann's work in the context of constructible functions (also stack functions) over moduli stacks. In <cit.>, Joyce defined the Euler characteristics of constructible sets in $\mathbb{K}$-stacks, pushforwards and pullbacks for constructible functions, where $\mathbb{K}$ is an algebraically closed field. Let $\A$ be an abelian category and $\CF(\fObj_\A)$ the vector space of $\mathbb{Q}$-valued constructible functions on $\fObj_\A(\mathbb{K})$, where $\fObj_\A$ is the moduli stack of objects in $\A$ and $\fObj_\A(\mathbb{K})$ the collection of isomorphism classes of objects in $\A$. Joyce proved that $\CF(\fObj_\A)$ is an associative $\mathbb{Q}$-algebra. The algebra $\CF(\fObj_\A)$ can be viewed as a variant of the Ringel-Hall algebra. Let $\CFi(\fObj_\A)$ be a subspace of $\CF(\fObj_\A)$ satisfying the condition that $f([X])\neq0$ implies $X$ is an indecomposable object in $\A$ for every $f\in\CFi(\fObj_\A)$. Then $\CFi(\fObj_\A)$ is shown to be a Lie subalgebra of $\CF(\fObj_\A)$ (<cit.>). Let $\CF_{\rm fin}(\fObj_\A)$ be the subspace of $\CF(\fObj_\A)$ such that
\text{supp}(f)=\big\{[X]\in\fObj_\A(\mathbb{K})~|~f([X])\neq0\big\}
is a finite set for every $f\in\CF_{\rm fin}(\fObj_\A)$. Let
\CFi_{\rm fin}(\fObj_\A)=\CF_{\rm fin}(\fObj_\A)\cap\CFi(\fObj_\A).
Assume that a conflation $X\rightarrow Y\rightarrow Z$ in $\A$ implies that the number of isomorphism classes of $Y$ is finite for all $X,Z\in\Obj(A)$. With the assumption, Joyce proved that $\CF_{\rm fin}(\fObj_\A)$ is an associative algebra and $\CFi_{\rm fin}(\fObj_\A)$ a Lie subalgebra of $\CF_{\rm fin}(\fObj_\A)$. It follows that $\CF_{\rm fin}(\fObj_\A)$ is isomorphic to the universal enveloping algebra of $\CFi_{\rm fin}(\fObj_\A)$. Joyce defined a comultiplication on $\CF_{\rm fin}(\fObj_\A)$ and proved that $\CF_{\rm fin}(\fObj_\A)$ is a bialgebra.
In <cit.>, the authors extended Riedtmann's results to algebras of representation-infinite type, i.e., the cardinality of isomorphism classes of indecomposable finite dimensional $\Lambda$-modules is infinite. Let $R(\Lambda)$ be the $\mathbb{Z}$-module spanned by $1_{\mathcal{O}}$, where $1_{\mathcal{O}}$ is the characteristic function over a constructible set of stratified Krull-Schmidt $\mathcal{O}$ (see <cit.>). The subspace $L(\Lambda)$ of $R(\Lambda)$ is spanned by $1_{\mathcal{O}}$, where $\mathcal{O}$ are indecomposable constructible sets. The multiplication is defined by
where $X$ is a $\Lambda$-module. Then $R(\Lambda)$ is an associative algebra with identity $1_0$ and $L(\Lambda)$ a Lie subalgebra of $R(\Lambda)$ with Lie bracket. The algebra $R(\Lambda)\otimes\mathbb{Q}$ is the universal enveloping algebra of $L(\Lambda)\otimes\mathbb{Q}$. The authors gave the degenerate form of Green's formula and established the comultiplication of $R(\Lambda)$ in <cit.>.
The goal of this paper is to explicitly construct the enveloping algebra of $\CFi(\fObj_\A)$ by the methods in <cit.>. Let $\A$ be an exact category satisfying some properties. Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a conflation in $\A$ and $\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$ the automorphism group of $X\xrightarrow{f}Y\xrightarrow{g}Z$. The key idea in <cit.> is that $V(X,Y;L)$ has the same Euler characteristic as its fixed point set under the action of $\mathbb{C}^*$. In this paper, we consider exact categories instead of categories of modules. Then as a substitute of the action of $\mathbb{C}^*$, we analyze the action of a maximal torus of $\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$ on $X\xrightarrow{f}Y\xrightarrow{g}Z$. The universal enveloping algebra of $\CFi(\fObj_\A)$ can be endowed with a comultiplication structure (Definition <ref>). It is compatible with multiplication (Theorem <ref>). The compatibility can be viewed as the degenerate form of Green's theorem on Ringel-Hall algebras (see <cit.> or <cit.>).
The paper is organized as follows. In Section 2 we recall the basic concepts about stacks, constructible sets and constructible functions. In Section 3 we define the constructible sets of stratified Krull-Schmidt. We study the the subspace $\CF^{\text{KS}}(\fObj_\A)$ of $\CF(\fObj_\A)$ generated by characteristic functions supported on constructible sets of stratified Krull-Schmidt. Then $\CF^{\text{KS}}(\fObj_\A)$ provides a realization of the universal enveloping algebra of $\CFi(\fObj_\A)$. In Section $4$ we give the comultiplication $\Delta$ in $\CF^{\rm KS}(\fObj_\A)$ and prove that $\Delta$ is an algebra homomorphism.
§ PRELIMINARIES
§.§ Constructible sets and constructible functions
From now on, let $\mathbb{K}$ be an algebraically closed field with characteristic zero. We recall the definitions of constructible sets and constructible functions on $\mathbb{K}$-stacks. These definitions are taken from Joyce <cit.>.
Let $\mathcal{F}$ be a $\mathbb{K}$-stack. Let $\mathcal{F}(\mathbb{K})$ denote the set of $2$-isomorphism classes $[x]$ where $x:\Spec\mathbb{K}\rightarrow\mathcal{F}$ are $1$-morphisms. Every element of $\mathcal{F}(\mathbb{K})$ is called a geometric point (or $\mathbb{K}$-point) of $\mathcal{F}$. For $\mathbb{K}$-stacks $\mathcal{F}$ and $\mathcal{G}$, let $\phi: \mathcal{F}\rightarrow\mathcal{G}$ be a 1-morphism of $\mathbb{K}$-stacks. Then $\phi$ induces a map $\phi_{*}:\mathcal{F}(\mathbb{K})\rightarrow\mathcal{G}(\mathbb{K})$ by $[x]\mapsto [\phi\circ x]$.
For any $[x]\in\mathcal{F}(\mathbb{K})$, let $\Iso_{\mathbb{K}}(x)$ denote the group of 2-isomorphisms $x\rightarrow x$ which is called a stabilizer group. For ease of notations, $\Iso_{\mathbb{K}}(x)$ is used to denote the group instead of $\Iso_{\mathbb{K}}([x])$. If $\Iso_{\mathbb{K}}(x)$ is an affine algebraic $\mathbb{K}$-group for each $[x]\in\mathcal{F}(\mathbb{K})$, then we say $\mathcal{F}$ with affine geometric stabilizers. A morphism of algebraic $\mathbb{K}$-groups $\phi_x:\Iso_{\mathbb{K}}(x)\rightarrow \Iso_{\mathbb{K}}(\phi_{*}(x))$ is induced by $\phi: \mathcal{F}\rightarrow\mathcal{G}$ for each $[x]\in\mathcal{F}(\mathbb{K})$.
A subset $\mathcal{O}\subseteq\mathcal{F}(\mathbb{K})$ is called a constructible set if $\mathcal{O}=\amalg_{i=1}^{n} \mathcal{F}_{i}(\mathbb{K})$ for some $n\in\mathbb{N}^+$, where every $\mathcal{F}_{i}$ is a finite type algebraic $\mathbb{K}$-substack of $\mathcal{F}$. A subset $S\subseteq\mathcal{F}(\mathbb{K})$ is called a locally constructible set if $S\cap\mathcal{O}$ are constructible for all constructible subsets $\mathcal{O}\subseteq\mathcal{F}(\mathbb{K})$. If $\mathcal{O}_1$ and $\mathcal{O}_2$ are constructible sets, then $\mathcal{O}_1\cup\mathcal{O}_2$, $\mathcal{O}_1\cap\mathcal{O}_2$ and $\mathcal{O}_1\setminus\mathcal{O}_2$ are constructible sets by <cit.>.
Let $\Phi:\mathcal{F}(\mathbb{K})\rightarrow\mathcal{G}(\mathbb{K})$ be a map. The set $\Gamma_{\Phi}=\{(x,\Phi(x))~|~x\in\mathcal{F}(\mathbb{K})\}$ is called the graph of $\Phi$. Recall that $\Phi$ is a pseudomorphism if $\Gamma_{\Phi}\bigcap(\mathcal{O}\times\mathcal{G}(\mathbb{K}))$ are constructible for all constructible subsets $\mathcal{O}\subseteq\mathcal{F}(\mathbb{K})$. By <cit.>, if $\phi:\mathcal{F}\rightarrow\mathcal{G}$ is a 1-morphism then $\phi_*$ is a pseudomorphism, $\Phi(\mathcal{O})$ and $\Phi^{-1}(y)\cap\mathcal{O}$ are constructible sets for all constructible subset $\mathcal{O}\subseteq\mathcal{F}(\mathbb{K})$ and $y\in\mathcal{G}(\mathbb{K})$. If $\Phi$ is a bijection and $\Phi^{-1}$ is also a pseudomorphism, we call $\Phi$ a pseudoisomorphism.
Then we will recall the definition of the naïve Euler characteristic of a constructible subset of $\mathcal{F}(\mathbb{K})$ in <cit.>.
This is a useful result due to Rosenlicht <cit.>.
Let $G$ be an algebraic $\mathbb{K}$-group acting on a $\mathbb{K}$-variety $X$. There exist an open dense $G$-invariant subset $X_{1}\subseteq X$ and a $\mathbb{K}$-variety $Y$ such that there is a morphism of varieties $\phi:X_{1}\rightarrow Y$ which induces a bijection form $X_{1}(\mathbb{K})/G$ to $Y(\mathbb{K})$.
Let $X$ be a separated $\mathbb{K}$-scheme of finite type, the Euler characteristic $\chi(X)$ of $X$ is defined by
\chi(X)=\sum\limits_{i=0}^{2\dim X}(-1)^{i}\dim_{\mathbb{Q}_{p}}H_{\text{cs}}^{i}(X,\mathbb{Q}_{p}),
where $p$ is a prime number, $\mathbb{Z}_p=\lim\limits_{\longleftarrow}$ $\mathbb{Z}/p^r\mathbb{Z}$ is the ring of $p$-adic integers, $\mathbb{Q}_p$ is its field of fractions and $H_{\text{cs}}^{i}(X,\mathbb{Q}_{p})$ are the compactly-supported $p$-adic cohomology groups of $X$ for $i\geq0$.
The following properties of Euler characteristic follow <cit.> and <cit.>.
Let $X$, $Y$ be separated, finite type $\mathbb{K}$-schemes and $\varphi:X\rightarrow Y$ a morphism of schemes. Then:
(1) If $Z$ is a closed subscheme of $X$, then $\chi(X)=\chi(X\setminus Z)+\chi(Z)$.
(2) $\chi(X\times Y)=\chi(X)\times\chi(Y)$.
(3) Let $X$ be a disjoint union of finitely many subschemes $X_{1},\ldots,X_{n}$, we have
\chi(X)=\sum\limits_{i=1}^{n}\chi(X_{i}).
(4) If $\varphi$ is a locally trivial fibration with fibre $F$, then $\chi(X)=\chi(F)\cdot\chi(Y)$.
(5) $\chi(\mathbb{K}^{n})=1$, $\chi(\mathbb{K}\mathbb{P}^{n})=n+1$ for all $n\geq0$.
An algebraic $\mathbb{K}$-stack $\mathcal{F}$ is said to be stratified by global quotient stacks if $\mathcal{F}(\mathbb{K})=\amalg_{i=1}^s\mathcal{F}_{i}(\mathbb{K})$ for finitely many locally closed substacks $\mathcal{F}_{i}$ where each $\mathcal{F}_{i}$ is 1-isomorphic to a quotient stack $[X_i/G_i]$, where $X_i$ is an algebraic $\mathbb{K}$-variety and $G_i$ a smooth connected linear algebraic $\mathbb{K}$-group acting on $X_i$. By <cit.>, if $\mathcal{F}$ is a finite type algebraic $\mathbb{K}$-stack with affine geometric stabilizers, then $\mathcal{F}$ is stratified by global quotient stacks.
Let $\mathcal{F}=\amalg_{i=1}^s\mathcal{F}_{i}(\mathbb{K})$ where each $\mathcal{F}_{i}\cong[X_i/G_i]$ as above. By Theorem <ref>, there exists an open dense $G_i$-invariant subvariety $X_{i1}$ of $X_i$ for each $i$ such that there exists a morphism of varieties $\phi_{i1}:X_{i1}\rightarrow Y_{i1}$, which induces a bijection between $X_{i1}(\mathbb{K})/G_i$ and $Y_{i1}(\mathbb{K})$. Then $\phi_{i1}$ induces a 1-morphism $\theta_{i1}:\mathcal{G}_{i1}\rightarrow Y_{i1}$, where $\mathcal{G}_{i1}$ is 1-isomorphic to $[X_{i1}/G_i]$. Note that
$$\text{dim}(X_{i(j-1)}\setminus X_{ij})<\dim X_{i(j-1)}$$
for $j=1,\ldots,k_i$. Using Theorem <ref> again, we get a stratification
\mathcal{F}(\mathbb{K})=\amalg_{i=1}^{s}\amalg_{j=1}^{k_i}\mathcal{G}_{ij}(\mathbb{K})
for $s,k_i\in\mathbb{N}^+$, where $\mathcal{G}_{ij}\cong[X_{ij}/G_i]$ such that $\phi_{ij}:X_{ij}\rightarrow Y_{ij}$ is a morphism of $\mathbb{K}$-varieties and $\theta_{ij}:\mathcal{G}_{ij}\rightarrow Y_{ij}$ a 1-morphism induced by $\phi_{ij}$. Let
\Theta=\amalg_{i=1}^{s}\amalg_{j=1}^{k_i}(\theta_{ij})_*:\mathcal{F}(\mathbb{K})\rightarrow Y(\mathbb{K}).
Then $Y$ is a a separated $\mathbb{K}$-scheme of finite type and $\Theta$ a pseudoisomorphism (see <cit.>).
Let $\mathcal{F}$ be an algebraic $\mathbb{K}$-stack with affine geometric stabilizers and $\mathcal{C}\subseteq\mathcal{F}(\mathbb{K})$ a constructible set. Then $\mathcal{C}$ is pseudoËËisomorphic to $Y(\mathbb{K})$, where $Y$ is a separated $\mathbb{K}$-scheme of finite type by <cit.>. The naïve Euler characteristic of $\mathcal{C}$ is defined by $\chi^{\na}(\mathcal{C})=\chi(Y)$.
The following lemma is a generalization of Proposition <ref> (4).
Let $\mathcal{F}$ and $\mathcal{G}$ be algebraic $\mathbb{K}$-stacks with affine geometric stabilizers. If $\mathcal{C}\subseteq\mathcal{F}(\mathbb{K})$, $\mathcal{D}\subseteq\mathcal{G}(\mathbb{K})$ are constructible sets, and $\Phi:\mathcal{C}\rightarrow\mathcal{D}$ is a surjective pseudomorphism such that all fibers have the same naïve Euler characteristic $\chi$, then $\chi^{\na}(\mathcal{C})=\chi\cdot\chi^{\na}(\mathcal{D})$.
Because $\mathcal{C}$, $\mathcal{D}$ are constructible sets, there exist separated finite type $\mathbb{K}$-schemes $X$, $Y$ such that $\mathcal{C}$, $\mathcal{D}$ are pseudoisomorphic to $X(\mathbb{K})$, $Y(\mathbb{K})$ respectively. Therefore $\chi^{\na}(\mathcal{C})=\chi(X)$, $\chi^{\na}(\mathcal{D})=\chi(Y)$. Then $\Phi$ induces a surjective pseudomorphism between $X(\mathbb{K})$ and $Y(\mathbb{K})$, say $\phi:X(\mathbb{K})\rightarrow Y(\mathbb{K})$. There exist two projective morphisms
$\pi_{1}:\Gamma_{\phi}\rightarrow X(\mathbb{K})$ and $\pi_{2}:\Gamma_{\phi}\rightarrow Y(\mathbb{K})$. Note that $\pi_{1}$ is also a pseudoisomorphism, that is $\chi^{\na}(\Gamma_{\phi})=\chi(X)$, and all fibres of $\pi_{2}$ have the same naïve Euler characteristic $\chi$. Then $\chi^{\na}(\Gamma_{\phi})=\chi\cdot\chi(Y)$. Hence $\chi(X)=\chi\cdot\chi(Y)$. We finish the proof.
A function $f:\mathcal{F}(\mathbb{K})\rightarrow\mathbb{Q}$ is called a constructible function on $\mathcal{F}(\mathbb{K})$ if the codomain of $f$ is a finite set and $f^{-1}(a)$ is a constructible subset of $\mathcal{F}(\mathbb{K})$ for each $a\in f(\mathcal{F}(\mathbb{K}))\setminus\{0\}$. Let $\CF(\mathcal{F})$ denote the $\mathbb{Q}$-vector space of all $\mathbb{Q}$-valued constructible functions on $\mathcal{F}(\mathbb{K})$.
Let $S\subseteq\mathcal{F}(\mathbb{K})$ be a locally constructible set. The integral of $f$ on $S$ is
\int_{x\in S}f(x)=\sum\limits_{a\in f(S)\setminus\{0\}}a\chi^{\na}(f^{-1}(a)\cap S)
for each $f\in\CF(\mathcal{F})$.
We recall the pushforwards and pullbacks of constructible functions due to Joyce <cit.>.
Let $\mathcal{F}$ and $\mathcal{G}$ be algebraic $\mathbb{K}$-stacks with affine geometric stabilizers and $\phi: \mathcal{F}\rightarrow\mathcal{G}$ a 1-morphism. For each $f\in\CF(\mathcal{F})$, the naïve pushforward $\phi^{\na}_!(f):\mathcal{F}(\mathbb{K})\rightarrow\mathbb{Q}$ of $f$ is
\phi^{\na}_!(f)(t)=\sum\limits_{a\in f(\phi^{-1}_{*}(t))\setminus\{0\}}a\chi^{\na}(f^{-1}(a)\cap\phi^{-1}_{*}(t))
for each $t\in\mathcal{G}(\mathbb{K})$. Then $\phi^{\na}_!(f)$ is a constructible function for each $f\in\CF(\mathcal{F})$ by <cit.>.
Similarly, if $\Phi:\mathcal{F}(\mathbb{K})\rightarrow\mathcal{G}(\mathbb{K})$ is a pseudomorphism, the naïve pushforward $\Phi^{\na}_!(f):\mathcal{F}(\mathbb{K})\rightarrow\mathbb{Q}$ of $f\in\CF(\mathcal{F})$ is defined by
\Phi^{\na}_!(f)(t)=\sum\limits_{a\in f(\Phi^{-1}(t))\setminus\{0\}}a\chi^{\na}(f^{-1}(a)\cap\Phi^{-1}(t))
for $t\in\mathcal{G}(\mathbb{K})$. Recall that $\Phi^{\na}_!(f)\in\CF(\mathcal{G})$ by <cit.>.
If $\phi: \mathcal{F}\rightarrow\mathcal{G}$ is a 1-morphism such that $\chi(\text{Ker}(\phi_x))=1$ for all $x\in \mathcal{F}(\mathbb{K})$, we can define a function $m_{\phi}:\mathcal{F}(\mathbb{K})\rightarrow\mathbb{Q}$ by
for each $x\in\mathcal{F}(\mathbb{K})$. For each $f\in\CF(\mathcal{F})$, the pushforward $\phi_!(f):\mathcal{G}(\mathbb{K})\rightarrow\mathbb{Q}$ of $f$ is defined by
\phi_!(f)=\phi^{na}_!(f\cdot m_{\phi}),
where $(f\cdot m_{\phi})(x)=f(x)m_{\phi}(x)$ for $x\in\mathcal{F}(\mathbb{K})$. Note that $\phi_!(f)\in\CF(\mathcal{G})$ (see <cit.>).
If $\phi$ is a 1-morphism of finite type, then $\phi_{*}^{-1}(\mathcal{D})\subset\mathcal{F}(\mathbb{K})$ is a constructible set for each constructible subset $\mathcal{D}$ of $\mathcal{G}(\mathbb{K})$. Then $g\circ\phi_{*}\in\CF(\mathcal{F})$ for $g\in\CF(\mathcal{G})$. Recall that the pullback $\phi^{*}:\CF(\mathcal{G})\rightarrow\CF(\mathcal{F})$ of $\phi$ is defined by $\phi^{*}(g)=g\circ\phi_{*}$ and it is linear.
§.§ Stacks of objects and conflations in $\A$
From now on, let $(\A,\mathcal{S})$ be a Krull-Schmidt exact $\mathbb{K}$-category with idempotent complete (see <ref> and <ref>). For simplicity, we write $\A$ instead of $(\A,\mathcal{S})$.
The isomorphism classes of $X\in\Obj(\A)$ and conflations $X\xrightarrow{i}Y\xrightarrow{d}Z$ in $\A$ are denoted by $[X]$ and $[X\xrightarrow{i}Y\xrightarrow{d}Z]$ (or $[(X,Y,Z,i,d)]$), respectively. Two conflations $X\xrightarrow{i}Y\xrightarrow{d}Z$ and $A\xrightarrow{f}B\xrightarrow{g}C$ are isomorphic if there exist isomorphisms $a:X\rightarrow A$, $b:Y\rightarrow B$ and $c:Z\rightarrow C$ in $\A$ such that the following diagram is communicative
\begin{equation}
\xymatrix{
X \ar[d]_{a} \ar[r]^{i} & Y \ar[d]_{b} \ar[r]^{d} & Z \ar[d]^{c} \\
A \ar[r]^{f} & B \ar[r]^{g} & C }
\end{equation}
The morphism $(a,b,c)$ is called an isomorphism of conflations in $\A$.
Assume that $\text{dim}_{\mathbb{K}}\Hom_{\A}(X,Y)$ and $\text{dim}_{\mathbb{K}}\Ext^1_{\A}(X,Y)$ are finite for all $X,Y\in\text{Obj}(\A)$. Let $K(\A)$ denote the quotient group of the Grothendieck group $K_0(\A)$ such that $\tilde{[X]}=0$ in $K(\A)$ implies that $X$ is a zero object in $\A$, where $\tilde{[X]}$ denotes the image of $X$ in $K(\A)$.
The following $2$-categories are defined in <cit.>.
Let $\Sch_{\mathbb{K}}$ be a $2$-category of $\mathbb{K}$-schemes such that objects are $\mathbb{K}$-schemes, $1$-morphisms morphisms of schemes and $2$-morphisms only the natural transformations $\id_{f}$ for all $1$-morphisms $f$. Let $\text{(exactcat)}$ denote the $2$-category of all exact categories with $1$-morphisms exact functors of exact categories and $2$-morphisms natural transformations between the exact functors. If all morphisms of a category are isomorphisms, then the category is called a groupoid. Let (groupoids) be the $2$-category with objects groupoids, $1$-morphisms functors of groupoids and $2$-morphisms natural transformations (also see <cit.>).
In <cit.>, Joyce defined a stack $\mathcal{F}_{\A}:\Sch_{\mathbb{K}}:\rightarrow\text{(exactcat)}$ associated to the exact category $\A$ (the original definition is for abelian category, it can be extended to exact categories directly), where $\mathcal{F}_{\A}$ is a contravariant 2-functor and satisfies the condition $\mathcal{F}_{\A}(\Spec(\mathbb{K}))=\A$. Applying $\mathcal{F}_{\A}$, he defined two moduli stacks
\fObj_\A,\fExact_\A:\Sch_{\mathbb{K}}\rightarrow\text{(groupoids)}
which are contravatiant $2$-functors (<cit.>). The $2$-functor
where $F:\text{(exactcat)}\rightarrow\text{(groupoids)}$ is a forgetful $2$-functor as follows. For an exact category $G$, $F(G)$ is a groupoid such that $\text{Obj}(F(G))=\text{Obj}(G)$ and morphisms are isomorphisms in $G$. For $U\in\Sch_{\mathbb{K}}$, a category $\fExact_\A(U)$ is a groupoid whose objects are conflations in $\mathcal{F}_\A(U)$ and morphisms isomorphisms of conflations in $\mathcal{F}_\A(U)$.
Let $\eta:U\rightarrow V$ and $\theta:V\rightarrow W$ be morphisms of schemes in $\Sch_{\mathbb{K}}$. Obviously, the functors $\fObj_\A(\eta):\fObj_\A(V) \rightarrow\fObj_\A(U)$ and $\fExact_\A(\eta):\fExact_\A(V)\rightarrow\fExact_\A(U)$ are induced by $\mathcal{F}_\A(\eta):\mathcal{F}_\A(V)\rightarrow\mathcal{F}_\A(U)$. The natural transformations $\epsilon_{\theta,\eta}:\fObj_{\A}(\eta)\circ\fObj_{\A}(\theta)\rightarrow \fObj_{\A}(\theta\circ\eta)$ and $\epsilon_{\theta,\eta}:\fExact_{\A}(\eta)\circ\fExact_{\A}(\theta)\rightarrow \fExact_{\A}(\theta\circ\eta)$ are also induced by $\epsilon_{\theta,\eta}:\mathcal{F}_{\A}(\eta)\circ\mathcal{F}_{\A}(\theta)\rightarrow \mathcal{F}_{\A}(\theta\circ\eta)$.
K^{\prime}(\A)=\{\tilde{[X]}\in K(\A)~|~X\in\text{Obj}(\A)\}\subset K(\A).
For each $\alpha\in K^{\prime}(\A)$, Joyce defined $\fObj_{\A}^{\alpha}:\Sch_{\mathbb{K}}\rightarrow\text{(groupoids)}$ which is a substack of $\fObj_\A$ in <cit.>. For each $U\in\Sch_{\mathbb{K}}$, $\fObj_{\A}^{\alpha}(U)$ is a full subcategory of $\fObj_{\A}(U)$. For each object $X$ in $\fObj_{\A}^{\alpha}(U)$, the image of $\fObj_{\A}(f)(X)$ in $K(\A)$ is $\alpha$ for each morphism $f:\Spec(\mathbb{K})\rightarrow U$.
Let $\eta:U\rightarrow V$ and $\theta:V\rightarrow W$ be morphisms in $\Sch_{\mathbb{K}}$. The functor $\fObj_{\A}^{\alpha}(\eta): \fObj_{\A}^{\alpha}(V)\rightarrow\fObj_{\A}^{\alpha}(U)$ is defined by restriction from $\fObj_{\A}(\eta): \fObj_{\A}(V)\rightarrow\fObj_{\A}(U)$. The natural transformation $\epsilon_{\theta,\eta}:\fObj_{\A}^{\alpha}(\eta)\circ\fObj_{\A}^{\alpha}(\theta)\rightarrow \fObj_{\A}^{\alpha}(\theta\circ\eta)$ is restricted from $\epsilon_{\theta,\eta}:\fObj_{\A}(\eta)\circ\fObj_{\A}(\theta)\rightarrow \fObj_{\A}(\theta\circ\eta)$.
For $\alpha,\beta,\gamma\in K^{\prime}(\A)$ and $\beta=\alpha+\gamma$, $\fExact_{\A}^{\alpha,\beta,\gamma}:\Sch_{\mathbb{K}}\rightarrow \text{(groupoids)}$ is defined as follows. For $U\in\Sch_{\mathbb{K}}$, $\fExact_{\A}^{\alpha,\beta,\gamma}(U)$ is a full subcategory of $\fExact_{\A}(U)$. The objects of $\fExact_{\A}^{\alpha,\beta,\gamma}(U)$ are conflations
X\xrightarrow{i}Y\xrightarrow{d}Z \in\text{Obj}(\fExact_{\A}(U)),
where $X\in\text{Obj}(\fObj_{\A}^{\alpha}(U))$, $Y\in\text{Obj}(\fObj_{\A}^{\beta}(U))$ and $Z\in\text{Obj}(\fObj_{\A}^{\gamma}(U))$. Similarly, the morphism $\fExact_{\A}^{\alpha,\beta,\gamma}(\eta)$ and natural transformation $\epsilon_{\theta,\eta}$ are defined by restriction.
The following theorem is taking from <cit.>.
The $2$-functors $\fObj_\A$, $\fExact_\A$ are $\mathbb{K}$-stacks, and $\fObj_{\A}^{\alpha}$, $\fExact_{\A}^{\alpha,\beta,\gamma}$ are open and closed $\mathbb{K}$-substacks of them respectively. There are disjoint unions
$$\fObj_\A=\amalg_{\alpha\in K^{\prime}(\A)}\fObj_\A^{\alpha},\fExact_{\A}=\amalg_{\alpha,\beta,\gamma\in K^{\prime}(\A)\atop \beta=\alpha+\gamma}\fExact_{\A}^{\alpha,\beta,\gamma}.$$
Assume that $\fObj_\A$ and $\fExact_\A$ are locally of finite type algebraic $\mathbb{K}$-stacks with affine algebraic stabilizers. Recall that $\fObj_\A(\mathbb{K})$ and $\fExact_\A(\mathbb{K})$ are the collection of isomorphism classes of objects in $\A$ and the collection of isomorphism classes of conflations in $\A$, respectively. For each $\alpha\in K^{\prime}(\A)$, $\fObj_\A^{\alpha}(\mathbb{K})$ is the collection of isomorphism classes of $X\in\text{Obj}(\A)$ such that $\tilde{[X]}=\alpha$ (see <cit.>).
Let $Q=(Q_0,Q_1,s,t)$ be a finite connected quiver, where $Q_0=\{1,\ldots,n\}$ is the set of vertices, $Q_1$ is the set of arrows and $s:Q_1\rightarrow Q_0$ (resp. $t:Q_1\rightarrow Q_0$) is a map such that $s(\rho)$ (resp. $t(\rho)$) is the source (resp. target) of $\rho$ for $\rho\in Q_1$. Let $A=\mathbb{C}Q$ be the path algebra of $Q$ and mod-$A$ denote the category of all finite dimensional right $A$-modules.
Let $\underline{d}=(d_j)_{j\in Q_0}$ for all $d_j\in\mathbb{N}$. There is an affine variety
\text{Rep}(Q,\underline{d})=\bigoplus\limits_{\rho\in Q_1}\Hom(\mathbb{C}^{d_{s(\rho)}},\mathbb{C}^{d_{t(\rho)}}).
For each $x=(x_\rho)_{\rho\in Q_1}\in\text{Rep}(Q,\underline{d})$, there is a $\mathbb{C}$-linear representation $M(x)=(\mathbb{C}^{d_j},x_\rho)_{j\in Q_0,\rho\in Q_1}$ of $Q$. Let $\text{rep}(Q)$ denote the category of finite dimensional $\mathbb{C}$-linear representations of $Q$. Recall that $\text{rep}(Q)\cong \text{mod-}\A$. We identify $\text{rep}(Q)$ with mod-$\A$. The linear algebraic group
\text{GL}(\underline{d})=\prod\limits_{j\in Q_0}\text{GL}(d_j,\mathbb{C})
acts on $\text{Rep}(Q,\underline{d})$ by $g.x=(g_{t(\rho)}x_\rho g^{-1}_{s(\rho)})_{\rho\in Q_1}$ for $g=(g_j)_{j\in Q_0}\in\text{GL}(\underline{d})$.
A complex $M^\bullet=(M^{(i)},\partial^{i})$, where $M^{(i)}\in\text{Obj}(\text{mod-}\A)$ and $\partial^{i+1}\partial^{i}=0$, is bounded if there exist some positive integers $n_0$ and $n_1$ such that $M^{(i)}=0$ for $i\leq -n_0$ or $i\geq n_1$. Let $\underline{\dim}M^{(i)}=\underline{d}^{(i)}$ be the dimension vector of $M^{(i)}$ for each $i\in\mathbb{Z}$. The vector sequence $(\underline{d}^{(i)})_{i\in\mathbb{Z}}$ of $M^{\bullet}$ is denoted by $\underline{\textbf{ds}}(M^{\bullet})$.
Let $\mathcal{C}(Q,\mathbf{\underline{d}})$ denote the affine variety consisting of all complexes $M^\bullet$ with $\underline{\textbf{ds}}(M^{\bullet})= \mathbf{\underline{d}}$. The group $G(\mathbf{\underline{d}})= \prod\limits_{i\in\mathbb{Z}}\text{GL}(\underline{d}^{(i)})$ is a linear algebraic group acting on $\mathcal{C}^{b} (Q,\mathbf{\underline{d}})$. The action is induced by the actions of $\text{GL}(\underline{d}^{(i)})$ on $\text{Rep}(Q,\underline{d}^{(i)})$ for all $i\in\mathbb{Z}$, that is
Let $\{P_1,\ldots,P_n\}$ be a set of representatives for all isomorphism classes of finite dimensional indecomposable projective $A$-modules. A complex $P^\bullet=$
\ldots\rightarrow P^{(i-1)}\xrightarrow{\partial^{i-1}}P^{(i)}\xrightarrow{\partial^{i}}P^{(i+1)}\rightarrow\ldots
is projective if $P^{(i)}\cong\bigoplus\limits_{j=1}^n m^{(i)}_{j}P_{j}$ for $m^{(i)}_{j}\in\mathbb{N}$ and $i\in\mathbb{Z}$. Let
\underline{e}(P^{(i)})=\underline{m}^{(i)}=(m_{1}^{(i)}, \ldots,m_{n}^{(i)})
be a vector corresponding to $P^{(i)}$. By the Krull-Schmidt Theorem, $\underline{e}(P^{(i)})$ is unique. The dimension vector of $P^\bullet$ can be defined by
\mathbf{\underline{dim}}(P^\bullet)=(\ldots,\underline{m}^{(i-1)},\underline{m}^{(i)}, \underline{m}^{(i+1)},\ldots).
A dimension vector $\mathbf{\underline{dim}}(P^\bullet)$ is bounded if $P^\bullet$ is bounded.
let $\mathbf{\underline{m}}=(\underline{m}^{(i)})_{i\in\mathbb{Z}}$ be a bounded dimension vector and $\mathbf{\underline{d}}(\mathbf{\underline{m}})=(\underline{d}^{(i)})_{i\in\mathbb{Z}}$ be the vector sequence of a complex whose dimension vector is $\mathbf{\underline{m}}$. Let $\mathcal{P}^{b}(Q,\mathbf{\underline{m}})$ be the set of all bounded project complexes $P^\bullet$ with $\mathbf{\underline{dim}}(P^\bullet) =\mathbf{\underline{m}}$ and $\underline{\textbf{ds}}(P^{\bullet})= \mathbf{\underline{d}}(\mathbf{\underline{m}})$. Note that $\mathcal{P}^{b}(Q,\mathbf{\underline{m}})$ is a locally closed subset of $\mathcal{C}^{b}(Q,\mathbf{\underline{d}}(\mathbf{\underline{m}}))$. An action of $G(\mathbf{\underline{d}}(\mathbf{\underline{m}}))$ on the variety $\mathcal{P}^{b}(Q,\mathbf{\underline{m}})$ is induced by the action of $G(\mathbf{\underline{d}}(\mathbf{\underline{m}}))$ on $\mathcal{C}^{b}(Q,\mathbf{\underline{d}}(\mathbf{\underline{m}}))$.
Let $\mathcal{P}^b(Q)$ denote the exact category with objects bounded project complexes and morphisms $\phi:P^\bullet\rightarrow Q^\bullet$ morphisms between bounded projective complexes. The Grothendieck group
where $\mathbb{Z}^{n}_{(i)}=\mathbb{Z}^{n}$. Note that $K(\mathcal{P}^b(Q))=K_0(\mathcal{P}^b(Q))$ and
where $\mathbb{N}^{n}_{(i)}=\mathbb{N}^{n}$.
Joyce defined $\mathcal{F}_{\text{mod}-\mathbb{K}Q}$ in <cit.>. Similarly, for each $U\in\Sch_{\mathbb{K}}$, we define $\mathcal{F}_{\mathcal{P}^b(Q)}(U)$ to be the category as follows.
The objects of $\mathcal{F}_{\mathcal{P}^b(Q)}(U)$ are complexes of sheaves $P^{\bullet}=(P^{(i)},\partial^i)_{i\in\mathbb{Z}}$, where $P^{(i)}=(\bigoplus_{j\in Q_0}X^{(i)}_{j},x^i)$ and $\partial^{i+1}\partial^{i}=0$. The data $X^{(i)}_{j}$ are locally free sheaves of finite rank on $U$ and $x^i=(x_{\rho}^i)_{\rho\in Q_1}$, where $x_{\rho}^i:X^{(i)}_{s(\rho)}\rightarrow X^{(i)}_{t(\rho)}$ are morphisms of sheaves, such that $P^{(i)}=(\bigoplus_{j\in Q_0}X^{(i)}_{j},x^i)$ are projective $\mathbb{C}Q$-modules for all $i\in\mathbb{Z}$. The morphisms of $\mathcal{F}_{\mathcal{P}^b(Q)}(U)$ are morphisms of complexes $\phi^{\bullet}:(P^{(i)},\partial^i)\rightarrow (Q^{(i)},d^i)$, where $Q^{(i)}=(\bigoplus_{j\in Q_0}Y^{(i)}_{j},y^i)$ and $\phi^{\bullet}$ is a sequence of morphisms
$$(\phi^{i}:P^{(i)} \rightarrow Q^{(i)})_{i\in\mathbb{Z}}$$
with $\phi^i=(\phi^{i}_{j}:X^{(i)}_{j} \rightarrow Y^{(i)}_{j})_{j\in Q_0}$ such that $\phi^{i+1}\partial^i=d^{i}\phi^i$ and $\phi^{i}_{t(\rho)}x_{\rho}^i=y_{\rho}^i\phi^{i}_{s(\rho)}$ for all $i\in\mathbb{Z}$ and $\rho\in Q_1$. It is easy to see that $\mathcal{F}_{\mathcal{P}^b(Q)}(U)$ is an exact category.
Let $\eta:U\rightarrow V$ be a morphism in $\Sch_{\mathbb{K}}$. A functor
\mathcal{F}_{\mathcal{P}^b(Q)}(\eta):\mathcal{F}_{\mathcal{P}^b(Q)}(V)\rightarrow \mathcal{F}_{\mathcal{P}^b(Q)}(U)
is defined as follows. If $(P^{(i)},\partial^i)_{i\in\mathbb{Z}}\in\text{Obj}(\mathcal{F}_{\mathcal{P}^b(Q)}(V))$,
\mathcal{F}_{\mathcal{P}^b(Q)}(\eta)(P^{(i)},\partial^i)_{i\in\mathbb{Z}}=(\eta^*(P^{(i)}),\eta^*(\partial^i))_{i\in\mathbb{Z}}
for $\eta^*(P^{(i)})=\big(\bigoplus_{j\in Q_0}\eta^{*}(X^{(i)}_{j}),(\eta^{*}(x_{\rho}^i))_{\rho\in Q_1}\big)$, where $\eta^{*}(X^{(i)}_{j})$ are the inverse images of $X^{(i)}_{j}$ by the morphism $\eta$, $\eta^*(\partial^i):\eta^*(P^{(i)}) \rightarrow\eta^*(P^{(i+1)})$ with $\eta^*(\partial^{i+1})\eta^*(\partial^i)=0$ for $i\in\mathbb{Z}$ and
\eta^{*}(x_{\rho}^i):\eta^{*}(X^{(i)}_{s(\rho)})\rightarrow \eta^{*}(X^{(i)}_{t(\rho)})
for $\rho\in Q_1$ are pullbacks of morphisms between inverse images. For a morphism $\phi^{\bullet}:(P^{(i)},\partial^i)\rightarrow (Q^{(i)},d^i)$ in $\mathcal{F}_{\mathcal{P}^b(Q)}(V)$, the morphism
\mathcal{F}_{\mathcal{P}^b(Q)}(\eta)(\phi^{\bullet}):\big(\eta^{*}(P^\bullet), \eta^{*}(\partial^i)\big) \rightarrow \big(\eta^{*}(Q^\bullet),\eta^{*}(d^i)\big)
is a sequence of morphisms
\Big(\eta^{*}(\phi^{i}):\big(\bigoplus_{j\in Q_0}\eta^{*}(X^{(i)}_{j}),(\eta^{*}(x_{\rho}^i))_{\rho}\big)\rightarrow \big(\bigoplus_{j\in Q_0}\eta^{*}(Y^{(i)}_{j}),(\eta^{*}(y_{\rho}^i))_{\rho}\big)\Big)_{i\in\mathbb{Z}},
with $\eta^{*}(\phi^{i+1})\eta^{*}(\partial^i)=\eta^{*} (d^i)\eta^{*}(\phi^i)$, where $\eta^{*}(d^i)$ are pullbacks of morphisms between inverse images which satisfy $\eta^{*}(d^{i+1})\eta^{*}(d^i)=0$, and
\eta^{*}(Q^\bullet)=\Big(\bigoplus_{j\in Q_0}\eta^{*}(Y^{(i)}_{j}),(\eta^{*}(y_{\rho}^i))_{\rho\in Q_1}\Big)_{i\in\mathbb{Z}}
such that the pullbacks
\eta^*(\phi^{i}_{j}):\eta^{*}(X^{(i)}_{j})\rightarrow\eta^{*}(Y^{(i)}_{j})
satisfy $\eta^*(\phi^{i}_{t(\rho)})\eta^*(x_{\rho}^i)=\eta^*(y_{\rho}^i)\eta^*(\phi^{i}_{s(\rho)})$. Because locally free sheaves are flat, $\mathcal{F}_{\mathcal{P}^b(Q)}(\eta)(\phi^{\bullet})$ is an exact functor.
Let $\eta:U\rightarrow V$ and $\theta:V\rightarrow W$ be morphisms in $\Sch_{\mathbb{K}}$. As in <cit.>, for each $P^\bullet\in\text{Obj}(\mathcal{F}_{\mathcal{P}^b(Q)}(W))$, there is a canonical isomorphism $\epsilon_{\theta,\eta}(P^\bullet):\mathcal{F}_{\mathcal{P}^b(Q)}(\eta)\circ\mathcal{F}_{\mathcal{P}^b(Q)}(\theta)(P^\bullet)\rightarrow \mathcal{F}_{\mathcal{P}^b(Q)}(\theta\circ\eta)(P^\bullet)$. We get a 2-isomorphism of functors
\epsilon_{\theta,\eta}:\mathcal{F}_{\mathcal{P}^b(Q)}(\eta)\circ\mathcal{F}_{\mathcal{P}^b(Q)}(\theta)\rightarrow \mathcal{F}_{\mathcal{P}^b(Q)}(\theta\circ\eta)
by the canonical isomorphisms. Thus we have the $2$-functor $\mathcal{F}_{\mathcal{P}^b(Q)}$.
The set $\fObj_{\mathcal{P}^{b}(Q)}(\mathbb{C})$ consists of all isomorphism classes of complexes in $\mathcal{P}^{b}(Q)$.
As in <cit.> and <cit.>, we have the following $1$-morphisms
\mathbf{\pi}_l: \fExact_\A\rightarrow\fObj_\A
which induces a map $(\mathbf{\pi}_l)_*: \fExact_\A(\mathbb{K})\rightarrow\fObj_\A(\mathbb{K})$ defined by $[X\xrightarrow{i}Y\xrightarrow{d}Z]\mapsto[X]$;
\pi_m: \fExact_\A\rightarrow\fObj_\A
such that the induced map $(\pi_m)_*: \fExact_\A(\mathbb{K})\rightarrow\fObj_\A(\mathbb{K})$ maps $[X\xrightarrow{i}Y\xrightarrow{d}Z]$ to $[Y]$;
\pi_r: \fExact_\A\rightarrow\fObj_\A
inducing the map $(\pi_r)_*: \fExact_\A(\mathbb{K})\rightarrow\fObj_\A(\mathbb{K})$ by $[X\xrightarrow{i}Y\xrightarrow{d}Z]\mapsto[Z]$.
The map $\mathbf{\pi}_{l*}\times\mathbf{\pi}_{r*}:\fExact_\A(\mathbb{K})\rightarrow\fObj_\A(\mathbb{K})\times\fObj_\A(\mathbb{K})$ is defined by $(\mathbf{\pi}_{l*}\times\mathbf{\pi}_{r*})([X\xrightarrow{i}Y\xrightarrow{d}Z])=([X],[Z])$. Note that $(\pi_l\times\pi_r)_*=\mathbf{\pi}_{l*}\times\mathbf{\pi}_{r*}$.
§ HALL ALGEBRAS
§.§ Constructible sets of stratified Krull-Schmidt
These definitions are related to <cit.>.
Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two constructible subsets of $\fObj_\A(\mathbb{K})$, the direct sum of $\mathcal{O}_1$ and $\mathcal{O}_2$ is
\mathcal{O}_1\oplus\mathcal{O}_2=\big\{[X_1\oplus X_2]~|~[X_1]\in \mathcal{O}_1,[X_2]\in \mathcal{O}_2 ~\text{and}~X_1, X_2\in \Obj(\mathcal{A})\big\}.
Let $n\mathcal{O}$ denote the direct sum of $n$ copies of $\mathcal{O}$ for $n\in\mathbb{N}^+$ and $0\mathcal{O}=\{[0]\}$. Similarly, let $nX$ denote the direct sum of $n$ copies of $X\in\text{Obj}(\A)$. A constructible subset $\mathcal{O}$ of $\fObj_\A(\K)$ is called indecomposable if $X\in\Obj(\mathcal{A})$ is indecomposable and $X\ncong0$ for every $[X]\in \mathcal{O}$.
A constructible set $\mathcal{O}$ is called to be of Krull-Schmidt if
\mathcal{O}= n_1\mathcal{O}_1\oplus n_2\mathcal{O}_2\oplus\ldots\oplus n_k\mathcal{O}_k,
where $\mathcal{O}_i$ are indecomposable constructible sets and $n_i\in\mathbb{N}$ for $i=1,\ldots,k$. If a constructible set $\mathcal{Q}=\amalg_{i=1}^{n}\mathcal{Q}_i$, where $\mathcal{Q}_i$ are constructible sets of Krull-Schmidt for $1\leq i\leq n$, namely $\mathcal{Q}$ is a disjoint union of finitely many constructible sets of Krull-Schmidt, then $\mathcal{Q}$ is said to be a constructible set of stratified Krull-Schmidt.
Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two indecomposable constructible sets. If $\mathcal{O}_1\cap\mathcal{O}_2\neq\emptyset$ and $\mathcal{O}_1\neq\mathcal{O}_2$, we have
\mathcal{O}_1\oplus\mathcal{O}_2=2(\mathcal{O}_1 \cap \mathcal{O}_2) \amalg \Big(\big(\mathcal{O}_1 \setminus (\mathcal{O}_1 \cap \mathcal{O}_2)\big)\oplus \big(\mathcal{O}_2 \setminus (\mathcal{O}_1 \cap \mathcal{O}_2)\big)\Big)
\amalg \Big((\mathcal{O}_1 \cap \mathcal{O}_2) \oplus \big(\mathcal{O}_2 \setminus (\mathcal{O}_1 \cap \mathcal{O}_2)\big)\Big) \amalg \big((\mathcal{O}_1 \setminus (\mathcal{O}_1 \cap \mathcal{O}_2)) \oplus (\mathcal{O}_1 \cap \mathcal{O}_2)\big).
If $\mathcal{Q}=m_{1}\mathcal{O}_{1}\oplus\ldots\oplus m_{l}\mathcal{O}_{l}$ is a constructible set of Krull-Schmidt, we can write $\mathcal{Q}=\amalg_{i=1}^{n}\mathcal{Q}_i$ as a constructible set of stratified Krull-Schmidt, where
\mathcal{Q}_{i}=n_{i1}\mathcal{O}_{i1}\oplus n_{i2}\mathcal{O}_{i2}\oplus\ldots\oplus n_{ik_{i}}\mathcal{O}_{ik_{i}}
for indecomposable constructible sets $\mathcal{O}_{ij}$ which are disjoint each other. Hence we can assume that $\mathcal{O}_{1}, \ldots,\mathcal{O}_{l}$ are disjoint each other.
Let $\CF^{\text{KS}}(\fObj_\A)$ be the subspace of $\CF(\fObj_\A)$ which is spanned by characteristic functions $1_{\mathcal{O}}$ for constructible sets of stratified Krull-Schmidt $\mathcal{O}$, where each $1_{\mathcal{O}}$ satisfies that $1_{\mathcal{O}}([X])=1$ for $[X]\in\mathcal{O}$, and $1_{\mathcal{O}}([X])=0$ otherwise.
Let $\mathbb{P}^{1}$ be the projective line over $\mathbb{K}$ and $\text{coh}(\mathbb{P}^{1})$ denote the category of coherent sheaves on $\mathbb{P}^1$.
Let $O(n)$ denote an indecomposable locally free coherent sheaf whose rank and degree are equal to $1$ and $n$ respectively. Let $S_{x}^{[r]}$ be an indecomposable torsion sheaf such that $\rk(S_{x}^{[r]})=0$, $\text{deg}(S_{x}^{[r]})=r$ and the support of $S_{x}^{[r]}$ is $\{x\}$ for $x\in\mathbb{P}^1$. The Grothendieck group $K_0(\text{coh}(\mathbb{P}^{1}))\cong\mathbb{Z}^{2}$. The data $K(\text{coh}(\mathbb{P}^{1}))$ and $\mathcal{F}_{\text{coh}(\mathbb{P}^{1})}$ are defined in <cit.>. The set of isomorphism classes of indecomposable objects in $\text{coh}(\mathbb{P}^{1})$ is
\{[S_{x}^{[d]}]~|~x\in\mathbb{P}^1,d\in\mathbb{N}\}\cup\{[O(n)]~|~n\in\mathbb{Z}\}.
Recall that a non-trivial subset $U\subset\mathbb{P}^1$ is closed (resp. open) if $U$ is a finite (resp. cofinite) set. Let $\mathcal{O}_d$ be a finite or cofinite subset of $\{[S_{x}^{[d]}]~|~x\in\mathbb{P}\}$ for each $d\in\mathbb{Z}^{+}$ and $\mathcal{O}_{0}$ a finite subset of $\{[O(n)]~|~n\in\mathbb{Z}\}$. Then $\mathcal{O}_d$ and $\mathcal{O}_{0}$ are indecomposable constructible subsets of $\fObj_{\text{coh}(\mathbb{P}^{1})}(\mathbb{K})$. Note that every indecomposable constructible subset of $\fObj_{\text{coh}(\mathbb{P}^{1})}(\mathbb{K})$ is of the form
\mathcal{O}_{0}\amalg\mathcal{O}_{i_1}\amalg\ldots\amalg\mathcal{O}_{i_n}
for $1\leq i_1<\ldots<i_n$. Then the finite direct sum $\oplus(\mathcal{O}_{0}\amalg\mathcal{O}_{i_1}\amalg\ldots\amalg \mathcal{O}_{i_n})$ is a constructible set of Krull-Schmidt. Every constructible set of Krull-Schmidt in $\fObj_{\text{coh}(\mathbb{P}^{1})}(\mathbb{K})$ is of the form. A constructible set of stratified Krull-Schmidt is a disjoint union of finitely many constructible sets of Krull-Schmidt.
In Example <ref>, $\fObj_{\mathcal{P}^{b}(Q)}^{\mathbf{\underline{m}}}(\mathbb{C})$ is the set of all isomorphism classes of project complexes in $\mathcal{P}^{b}(Q,\mathbf{\underline{m}})$. Note that
\fObj_{\mathcal{P}^{b}(Q)}(\mathbb{C})=\amalg_{\mathbf{\underline{m}}\in K^{\prime}(\mathcal{P}^b(Q))} \fObj_{\mathcal{P}^{b}(Q)}^{\mathbf{\underline{m}}}(\mathbb{C}).
There is a canonical map
p_{\mathbf{\underline{m}}}:\mathcal{P}^{b}(Q,\mathbf{\underline{m}})\rightarrow \fObj_{\mathcal{P}^{b}(Q)}^{\mathbf{\underline{m}}}(\mathbb{C})
which maps $P^\bullet$ to $[P^\bullet]$. A subset $U\subseteq\fObj_{\mathcal{P}^{b}(Q)}^{\mathbf{\underline{m}}}(\mathbb{C})$ is closed (resp. open) if $p_{\mathbf{\underline{m}}}^{-1}(U)$ is closed (resp. open) in $\mathcal{P}^{b}(Q,\mathbf{\underline{m}})$. A subset $V_{\mathbf{\underline{m}}} \subseteq \fObj_{\mathcal{P}^{b}(Q)}^{\mathbf{\underline{m}}}(\mathbb{C})$ is locally closed if it is an intersection of a closed subset and an open subset of $\fObj_{\mathcal{P}^{b}(Q)}^{\mathbf{\underline{m}}}(\mathbb{C})$. A subset $\mathcal{O}\subseteq \fObj_{\mathcal{P}^{b}(Q)}(\mathbb{C})$ is constructible if it is a finite disjoint union of locally closed sets $V_{\mathbf{\underline{m}}}$. Every indecomposable constructible set $\mathcal{O}$ is of the form $\coprod_{\mathbf{\underline{m}}\in S}V_{\mathbf{\underline{m}}}$, where $S$ is a finite set and each complex in $p_{\mathbf{\underline{m}}}^{-1} (V_{\mathbf{\underline{m}}})$ is an indecomposable complex.
§.§ Automorphism groups of conflations
For each $X\in\text{Obj}(\A)$, suppose that $X=n_1X_1\oplus n_2X_2\oplus\ldots\oplus n_tX_t$, where $X_i$ are indecomposable for $i=1,\ldots,t$ and $X_i\ncong X_j$ for $i\neq j$. Then we have
\Aut(X)\cong(1+rad\End(A))\rtimes\sum\limits_{i=1}^{t}\GL(n_{i},\mathbb{K}).
The rank of maximal torus of $\Aut(X)$ is denoted by $\rk\Aut(X)$. Let $n=n_{1}+n_{2}+\ldots+n_{t}$. Thus the number of indecomposable direct summands of $X$ is $n$, which is denoted by $\gamma(X)$. Note that $\gamma(X)=\rk\Aut(X)$. Let
\gamma(\mathcal{O})=\max\{\gamma(X)~|~[X]\in\mathcal{O}\}
for each constructible set $\mathcal{O}$ in $\fObj_\A(\mathbb{K})$.
Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a conflation in $\A$ and $\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$ denote the group of $(a_1,a_2,a_3)$ for $a_1\in\Aut(X)$, $a_2\in\Aut(Y)$ and $a_3\in\Aut(Z)$ such that the following diagram is commutative
\begin{equation*}
\xymatrix{
X \ar[d]_{a_1} \ar[r]^{f} & Y \ar[d]_{a_2} \ar[r]^{g} & Z \ar[d]^{a_3} \\
X \ar[r]^{f} & Y \ar[r]^{g} & Z }
\end{equation*}
The homomorphism
is defined by $(a_1,a_2,a_3)\mapsto a_2$. If $p_1((a_1,a_2,a_3))=p_1((a^{\prime}_1,a_2,a^{\prime}_3))$ then $f(a_1-a^{\prime}_1)=0$ and ($a_3-a^{\prime}_3)g=0$. We have $a_1=a^{\prime}_1$ and $a_3=a^{\prime}_3$ since $f$ is an inflation and $g$ a deflation. Hence $p_1$ is an injective homomorphism of affine algebraic $\mathbb{K}$-groups and
\begin{equation}\label{equ3}
\rk(\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z))=\rk~\text{Im}p_1\leq\rk\Aut(Y)
\end{equation}
be a homomorphism given by $(a_1,a_2,a_3)\mapsto (a_1,a_3)$. If $p_2((a_1,a_2,a_3))=p_2((a_1,a_2^{\prime},a_3))$, then $(a_2-a^{\prime}_2)f=0$ and $g(a_2-a^{\prime}_2)$=0, we have
Observe that $\text{Ker}p_2$ is a linear space. It follows that $\chi(\text{Ker}p_2)=1$ and
\begin{equation}\label{equ4}
\rk~\text{Im}(p_2)\leq\rk\Aut(X)+\rk\Aut(Z).
\end{equation}
Two conflations $X\xrightarrow{i}Y\xrightarrow{d}Z$ and $X^{\prime}\xrightarrow{i^{\prime}}Y\xrightarrow{d^{\prime}}Z^{\prime}$ in $\A$ are said to be equivalent if there exists a commutative diagram
\begin{equation*}
\xymatrix{
X \ar[d]_{f} \ar[r]^{i} & Y \ar[d]_{1_{Y}} \ar[r]^{d} & Z \ar[d]^{g} \\
X^{\prime} \ar[r]^{i^{\prime}} & Y \ar[r]^{d^{\prime}} & Z^{\prime} }
\end{equation*}
where both $f$ and $g$ are isomorphisms. If the two conflations are equivalent, we write $X\xrightarrow{i}Y\xrightarrow{d}Z\sim X^{\prime}\xrightarrow{i^{\prime}}Y\xrightarrow{d^{\prime}}Z^{\prime}$. The equivalence class of $X\xrightarrow{i}Y\xrightarrow{d}Z$ is denoted by $\langle X\xrightarrow{i}Y\xrightarrow{d}Z\rangle$. Define
V(\mathcal{O}_{1},\mathcal{O}_{2};Y)=\big\{\langle X\xrightarrow{i} Y\xrightarrow{d}Z\rangle~|~X\xrightarrow{i} Y\xrightarrow{d}Z\in\mathcal{S}, [X]\in\mathcal{O}_{1},[Z]\in\mathcal{O}_{2}\big\},
where $\mathcal{S}$ is the collection of all conflations of $\A$.
§.§ Associative algebras and Lie algebras
For $f,g\in\CF(\fObj_\A)$, define $f\cdot g$ by $(f\cdot g)([X],[Y])=f([X])g([Y])$ for $([X],[Y])\in\fObj_\A(\mathbb{K})\times \fObj_\A(\mathbb{K})$. Thus $f\cdot g\in \CF(\fObj_\A\times\fObj_\A)$. The pushforward of $\pi_m$ is well-defined since $p_1$ is injective. The following definition of multiplication is taken from <cit.>.
Using the following diagram
\begin{equation*}
\fObj_\A\times\fObj_\A\xleftarrow{\pi_l\times\pi_r}\fExact_\A\xrightarrow{\pi_m}\fObj_\A,
\end{equation*}
we can define the convolution multiplication
\begin{equation*}
\begin{gathered}
\CF(\fObj_\A\times\fObj_\A)\xrightarrow{(\pi_l\times\pi_r)^*}
\CF(\fExact_\A)\xrightarrow{(\pi_m)_!}\CF(\fObj_\A).
\end{gathered}
\end{equation*}
The multiplication $*:\CF(\fObj_\A)\times\CF(\fObj_\A)\rightarrow\CF(\fObj_\A)$ is a bilinear map defined by
\begin{equation*}
f*g=(\pi_m)_![(\pi_l\times\pi_r)^*(f\cdot g)]=(\pi_m)_![\pi_{l}^*(f)\cdot\pi_{r}^*(g)].
\end{equation*}
Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be constructible subsets of $\fObj_\A(\mathbb{K})$, the meaning of $1_{\mathcal{O}_1}\ast1_{\mathcal{O}_2}$ can be understood as follows. The function $m_{\mathbf{\pi}_m}:\fExact_\A(\mathbb{K})\rightarrow\mathbb{Q}$, which is defined by
m_{\mathbf{\pi}_m}([X\xrightarrow{f}Y\xrightarrow{g}Z])= \chi\big[\Aut(Y)/p_1\big(\Aut(X\xrightarrow{f} Y\xrightarrow{g}Z)\big)\big],
is a locally constructible function on $\fExact_\A(\mathbb{K})$ by <cit.>, namely $m_{\mathbf{\pi}_m}|_{\mathcal{O}}$ is a constructible function on $\mathcal{O}$ for every constructible subset $\mathcal{O}\subseteq\fExact_\A(\mathbb{K})$.
For each $[Y]\in\fObj_\A(\mathbb{K})$,
1_{\mathcal{O}_1} \ast 1_{\mathcal{O}_2}([Y])=\sum\limits_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y}c\chi^{na}(Q_{c}),
\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y=\{c=m_{\pi_m}([A\xrightarrow{f}Y\xrightarrow{g}B])~|~[A]\in\mathcal{O}_1, [B]\in\mathcal{O}_2\}\setminus\{0\}
is a finite set, and
Q_{c}=\{[ A\xrightarrow{f}Y\xrightarrow{g}B]~|~[A]\in\mathcal{O}_{1},[B]\in\mathcal{O}_{2},m_{\mathbf{\pi}_m}([ A\xrightarrow{f}Y\xrightarrow{g}B])=c\}
are constructible sets for $c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y$. In fact, the 1-morphism $\pi_l\times\pi_r$ is of finite type by <cit.>. Hence $(\pi_{l*}\times\pi_{r*})^{-1}(\mathcal{O}_1\times\mathcal{O}_2)$ is a constructible subset of $\fExact_\A$. Then
\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y=m_{\mathbf{\pi}_m}\big[\big((\pi_{l*}\times\pi_{r*})^{-1}(\mathcal{O}_1\times\mathcal{O}_2)\big) \cap \big((\pi_{m*})^{-1}([Y])\big)\big]\setminus\{0\}
is a finite set by <cit.>. Therefore
are constructible for all $c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y$.
For each $([X],[Z])\in\mathcal{O}_{1}\times\mathcal{O}_{2}$, let
\Lambda_{XZ}^Y=\big\{c=m_{\pi_m}([X\xrightarrow{f}Y\xrightarrow{g}Z])~|~[X\xrightarrow{f}Y\xrightarrow{g}Z] \in\fExact_{\A}(\mathbb{K})\big\}
where $\Lambda_{XZ}^Y$ is a finite set and $Q_{c}^{X,Z}$ are constructible sets for all $c\in\Lambda_{XZ}^Y$. Then
$$(1_{[X]} \ast 1_{[Z]})([Y])=\sum\limits_{c\in\Lambda_{XZ}^Y}c\chi^{na}(Q_{c}^{X,Z}).$$
\pi_{1}:V(\mathcal{O}_{1},\mathcal{O}_{2};Y)\rightarrow\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y}Q_{c}
be a morphism given by $\langle X\xrightarrow{f} Y\xrightarrow{g}Z\rangle\mapsto([X\xrightarrow{f} Y\xrightarrow{g}Z])$. For each fibre of $\pi_{1}$, $\chi^{\na}(\pi_{1}^{-1}([X\xrightarrow{f} Y\xrightarrow{g}Z]))=\chi\Big(\Aut(Y)/p_1\big(\Aut(X\xrightarrow{f} Y\xrightarrow{g}Z)\big)\Big)$. The set
\Big\{\chi\Big(\Aut(Y)/p_1\big(\Aut(X\xrightarrow{f} Y\xrightarrow{g}Z)\big)\Big)~|~[X\xrightarrow{f} Y\xrightarrow{g}Z]\in\bigcup_{c \in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y} Q_{c}\Big\}
is finite since $\chi(\Aut(Y)/\text{Im}p_1)=m_{\mathbf{\pi}_m}([X\xrightarrow{f} Y\xrightarrow{g}Z])$.
If $U\subseteq V(\mathcal{O}_{1},\mathcal{O}_{2};Y)$ is a constructible set, then
\begin{equation}\label{formula1}
\chi^{\na}(U)=\sum_{c}c\chi^{na}(P_c),
\end{equation}
where $P_c=\big\{[X\xrightarrow{f} Y\xrightarrow{g}Z]~|~\langle X\xrightarrow{f} Y\xrightarrow{g}Z\rangle\in U, m_{\pi_m}([X\xrightarrow{f} Y\xrightarrow{g}Z])=c\big\}$.
Consequently, we have the naïve Euler characteristics of $V([X],[Z];Y)$ and $V(\mathcal{O}_{1},\mathcal{O}_{2};Y)$.
Let $X,Y,Z\in\rm{Obj}(\A)$ and $\mathcal{O}_{1},\mathcal{O}_{2}$ be constructible sets. Then
\chi^{\na}(V([X],[Z];Y))=\sum\limits_{c\in\Lambda_{XZ}^Y}c\chi^{na}(Q_{c}^{X,Z})=1_{[X]}*1_{[Z]}([Y]),
\chi^{\na}(V(\mathcal{O}_{1},\mathcal{O}_{2};Y))=\sum\limits_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y}c\chi^{\na}(Q_c) =1_{\mathcal{O}_1} \ast 1_{\mathcal{O}_2}([Y]).
The following result is due to <cit.> and <cit.>.
The $\mathbb{Q}$-space $\CF(\fObj_\A)$ is an associative $\mathbb{Q}$-algebra, with convolution multiplication $*$ and identity $1_{[0]}$, where $1_{[0]}$ is the characteristic function of $[0]\in\fObj_\A(\mathbb{K})$.
The proof of the theorem is quite similar to that in <cit.> and so is omitted.
Joyce defined $\CFi(\fObj_\A)$ to be the subspace of $\CF(\fObj_\A)$ such that if $f([X])\neq0$ then $X$ is an indecomposable object in $\A$ for every $f\in\CFi(\fObj_\A)$. There is a result of <cit.> and <cit.>.
The $\mathbb{Q}$-space $\CFi(\fObj_\A)$ is a Lie algebra under the Lie bracket $[f,g]=f*g-g*f$ for $f,g\in\CFi(\fObj_\A)$.
The proof is the same as the one used in <cit.>.
§.§ The algebra $\CF^{\text{KS}}(\fObj_\A)$
Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two constructible subsets of $\fObj_\A(\K)$. For any $Y\in \Obj(\A)$, if $1_{\mathcal{O}_1} \ast 1_{\mathcal{O}_2} ([Y])\neq 0$, then there exist $X,Z \in \Obj(\A)$ such that $[X]\in \mathcal{O}_1$, $[Z]\in \mathcal{O}_2$ and $1_{[X]}*1_{[Z]}([Y])\neq0$.
Let $Q_{c}$ and $\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y$ be as in Section <ref>. Let
\mathbf{\pi}_{2}:\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y}Q_{c}\rightarrow(\mathbf{\pi}_{l*} \times\mathbf{\pi}_{r*})\Big(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y}Q_c\Big)
be a map which maps $[X\xrightarrow{i} Y\xrightarrow{d}Z]$ to $([X],[Z])$ and
It is easy to see that $m_{m}$ is a constructible function over $\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^Y}Q_{c}$.
Because $\pi_l\times\pi_r$ is a $1$-morphism, $\pi_2$ is a pseudomorphism by <cit.>. Thus $\pi_2(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}}Q_c)$ is constructible and the naïve pushforward $(\pi_{2})_!^{\na}(m_{m})$ of $m_{m}$ to $\pi_2(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}}Q_c)$ exists. Note that $(\pi_{2})_!^{\na}(m_{m})$ is a constructible function on $\pi_2(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}}Q_c)$. In fact
for all $([X],[Z])\in\pi_2(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}}Q_c)$. Therefore
\big\{1_{[X]}*1_{[Z]}([Y])~|~([X],[Z])\in\pi_{2}(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}}Q_c)\big\}
is a finite set.
Let $\{([X_1],[Z_1]),\ldots,([X_n],[Z_n])\}$ be a complete set of representatives for $([X],[Z])\in\pi_{2}(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}} Q_c)$ such that
$$1_{[X_i]}*1_{[Z_i]}([Y])\neq 1_{[X_j]}*1_{[Z_j]}([Y])$$
for $i\neq j$. Set
\mathcal{P}^{X_k,Z_k}=\Big\{([A],[B])\in\pi_{2}\Big(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}} Q_c\Big)~|~1_{[A]}*1_{[B]}([Y])=1_{[X_k]}*1_{[Z_k]}([Y])\Big\}
for all $1\leq k\leq n$.
Then $\mathcal{P}^{X_k,Z_k}$ are constructible sets for all $1\leq k\leq n$ since
\mathcal{P}^{X_k,Z_k}=\big((\pi_{2})_!^{\na}(m_{m})\big)^{-1}(c_k),
where $c_k=(\pi_{2})_!^{\na}(m_{m})([X_k],[Z_k])$. Let $\pi=\pi_2\circ\pi_1$ which maps $\langle X\xrightarrow{i} Y\xrightarrow{d}Z\rangle$ to $([X],[Z])$. For each $([X],[Z])\in\pi_{2}(\bigcup_{c\in\Lambda_{\mathcal{O}_1\mathcal{O}_2}^{Y}}Q_c)$,
\chi^{\na}(\pi^{-1}([X],[Z]))=\chi^{\na}(V([X_k],[Z_k];Y))
for some $k$. According to Lemma <ref>, we have
1_{\mathcal{O}_1} \ast 1_{\mathcal{O}_2}([Y])=\chi^{\na}(V(\mathcal{O}_{1},\mathcal{O}_{2};Y))=\sum\limits_{k=1}^{n}\chi^{\na}(V([X_k],[Z_k];Y))\cdot\chi^{\na} (\mathcal{P}^{X_k,Z_k})
There exists $([X_k],[Z_k])$ for some $k\in\{1,\ldots,n\}$ such that $1_{[X_k]}*1_{[Z_k]}([Y])\neq0$ since $1_{\mathcal{O}_1} \ast 1_{\mathcal{O}_2}([Y])\neq0$.
Let $\textbf{D}_n(\mathbb{K})$ denote the group of invertible diagonal matrices in $\textbf{GL}(n,\mathbb{K})$.
The following lemma is related to Riedtmann<cit.>.
Let $X,Y,Z\in\Obj(\A)$ and $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a conflation in $\A$. If $m_{\pi_m}([X\xrightarrow{f}Y\xrightarrow{g}Z])\neq0$, then $\gamma(Y)\leq\gamma(X)+\gamma(Z)$. In particular, $\gamma(Y)=\gamma(X)+\gamma(Z)$ if and only if $Y\cong X\oplus Z$.
Recall that $m_{\pi_m}([X\xrightarrow{f}Y\xrightarrow{g}Z])=\chi(\Aut Y/\text{Im}(p_1))$.
If $\rk\Aut(Y)>\rk~\text{Im}(p_1)$, then the fibre of the action of a maximal torus of $\Aut(Y)$ on $\Aut Y/\text{Im}(p_1)$ is $(\mathbb{K}^*)^k$ for some $k\geq1$, it forces $\chi(\Aut Y/\text{Im}(p_1))=0$. Hence we have $\rk\Aut(Y)=\rk~\text{Im}(p_1)\leq\rk\Aut(X)+\rk\Aut(Z)$.
We prove the second assertion by induction. First of all, suppose that $X\ncong0$ and $Z\ncong0$. If $\rk\Aut(Y)=2$ and $Y=Y_1\oplus Y_2$, then $\rk\Aut(X)=\rk\Aut(Z)=1$ since $X$ and $Z$ are not isomorphic to $0$. For $t\in\mathbb{K}^*\setminus\{1\}$, $\left(
\begin{array}{cc}
\end{array}
\right)\in\Aut(Y)$ and it is an element of a maximal torus $\textbf{D}_2(\mathbb{K})$ of $\Aut(Y)$. A maximal torus of $\text{Im}(p_1)$ is also a maximal torus of $\Aut(Y)$ since $\rk\Aut(Y)=\rk~\text{Im}(p_1)$. Because two maximal tori of a connected linear algebraic group are conjugate, there exists $\alpha\in\Aut(Y)$ such that $\alpha\left(
\begin{array}{cc}
\end{array}
\right)\alpha^{-1}$ lies in a maximal torus of $\text{Im}(p_1)$. Hence there exist $a\in\Aut(X)$ and $b\in\Aut(Z)$ satisfying $(a,\alpha\left(
\begin{array}{cc}
\end{array}
\right)\alpha^{-1},b)\in\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$, namely
\begin{array}{cc}
\end{array}
\right),b)\in\Aut(X\xrightarrow{\alpha^{-1}f}Y\xrightarrow{g\alpha}Z).
Let $f^{\prime}=\alpha^{-1}f$ and $g^{\prime}=g\alpha$. Observe $(t,\left(
\begin{array}{cc}
\end{array}
\right),t)\in\Aut(X\xrightarrow{f^{\prime}}Y\xrightarrow{g^{\prime}}Z)$. Hence $f^{\prime}(a-t)=\left(
\begin{array}{cc}
\end{array}
\right)f^{\prime}$. Let $s=\frac{1}{t^2-t}(a-t)\in\End(X)$ ($t\neq0,1$). Then $f^{\prime}s=\left(
\begin{array}{cc}
\end{array}
\right)f^{\prime}$. Because $f^{\prime}$ is an inflation and
\left(
\begin{array}{cc}
\end{array}
\right)f^{\prime}s=\left(
\begin{array}{cc}
\end{array}
\right)f^{\prime}=f^{\prime}s,
$s^2=s$. The category $\A$ is idempotent completion, consequently $s$ has a kernel and an image such that $X=\text{Ker}s\oplus\text{Im}s$. But $X$ is indecomposable, without loss of generality we can assume $X=\text{Ker}s$. Then $s=0$. Let $f^{\prime}= {f_{1}\choose f_{2}}$ and $g^{\prime}=(g_1,g_2)$. It follows that
\left(
\begin{array}{c}
\end{array}
\right)=f^{\prime}s=\left(
\begin{array}{cc}
\end{array}
\right)\left(
\begin{array}{c}
\end{array}
\right)=\left(
\begin{array}{c}
\end{array}
\right).
We have $f_2=0$ and $f^{\prime}=\left(
\begin{array}{c}
\end{array}
\right)$. The morphism $Y_1\oplus Y_2\xrightarrow{(0,1)}Y_2$ is a deflation by <cit.>. Because $(0,1){f_1\choose0}=0$, there exits $h\in\Hom(Z,Y_1)$ such that $(0,1)=h(g_1,g_2)$. We have $hg_1=0$ and $hg_2=1_{Y_2}$. Observe $g_2h\in\End(Z)$ and $(g_2h)(g_2h)=g_2h$, so $g_2h$ has a kernel $k:K\rightarrow Z$ and an image $i:I\rightarrow Z$. Moreover $Z\cong K\oplus I$. It follows that $Z\cong K$ or $Z\cong I$ since $Z$ is indecomposable. If $Z\cong K$ then $g_2h=0$. But $hg_2h=h$, $K=0$. Thus $h$ is an isomorphism and $g_1=0$. We have $Z\cong Y_2$. Similarly $X\cong Y_1$. Hence $X\oplus Z\cong Y_1\oplus Y_2$.
Assume that the assertion is true for $\rk\Aut(Y)=n<N$. When $n=N$, we can assume $\rk\Aut(X)=n_1$ where $0<n_1<N$, then $\rk\Aut(Z)=N-n_1=n_2$. Let $Y=Y^{\prime}\oplus Y_N$ and $Y^{\prime}=Y_1\oplus\ldots\oplus Y_{N-1}$, where $Y_i$ are indecomposable. Observe that $\left(
\begin{array}{cc}
\end{array}
\right)$ lies in a maximal torus of $\Aut(Y)$ for $t\in\mathbb{K}^*\setminus\{1\}$. There exists $(a,c,b)\in\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$ such that $c$ and $\left(
\begin{array}{cc}
\end{array}
\right)$ are conjugate in $\Aut(Y)$. For simplicity we assume $c=
\left(
\begin{array}{cc}
\end{array}
\right)$. So we have the following commutative diagram
\begin{equation*}
\xymatrix{
X \ar[d]_{a} \ar[r]^{f} & Y^{\prime}\oplus Y_N \ar[d]_{c} \ar[r]^{g} & Z \ar[d]^{b} \\
X \ar[r]^{f} & Y^{\prime}\oplus Y_N \ar[r]^{g} & Z }
\end{equation*}
where $f=(f_1,f_2,\ldots,f_N)^t$ and $g=(g_1,g_2,\ldots,g_N)$.
There is another commutative diagram
\begin{equation*}
\xymatrix{
X \ar[d]_{tI_{n_1}} \ar[r]^{(f^*,f_N)^t~} & Y^{\prime}\oplus Y_N \ar[d]_{tI_N} \ar[r]^{~~(g^*,g_N)} & Z \ar[d]^{tI_{n_2}} \\
X \ar[r]^{(f^*,f_N)^t~} & Y^{\prime}\oplus Y_N \ar[r]^{~~(g^*,g_N)} & Z }
\end{equation*}
where $f^*=(f_1,f_2,\ldots,f_{N-1})^T$ and $g^*=(g_1,g_2,\ldots,g_{N-1})$. Then $f=(f^*,f_N)^t$, $g=(g^*,g_N)$ and $f(a-tI_{n_1}) =\left(
\begin{array}{cc}
\end{array}
\right)f$.
Then $fs_{N}=\text{diag}\{0,\ldots,0,1\}f$. It follows $f^*s_N=0$, $f_Ns_N=f_N$ and $g_Nf_N=g\left(
\begin{array}{cc}
\end{array}
\right)f=gfs_N=0$. Moreover $s_{N}$ is an idempotent, we know that $X=\text{Ker}s_N\oplus\text{Im}s_N$. If $f_N\neq0$ then $\text{Im}s_N$ is not isomorphic to $0$. Similarly we can define $s_1$, $s_2$, $\ldots$, $s_{N-1}\in\End(X)$ with the property that $fs_i=\text{diag}\{0,\ldots,0,1,0,\ldots,0\}f=(0,\ldots,0,f_i,0,\ldots,0)^t$. Hence $s_i$ is idempotent and if $f_i\neq0$ then $\text{Im}s_i$ is not isomorphic to $0$ for each $i$. Note that $s_1+s_2+\ldots+s_N=1_X\in\Aut(X)$, it follows
Hence $f_i=0$ for some $i$ since $\rk\Aut(X)<N$. Without loss of generality, we assume $f_N=0$. Let $(0,\ldots,0,1):Y_1\oplus\ldots\oplus Y_N\rightarrow Y_N$, then
$$(0,\ldots,0,1)(f_1,\ldots,f_N)^t =0$$
Hence there exists $h\in\Hom(Z,Y_N)$ such that $h(g_1,\ldots,g_N)=(0,\ldots,0,1)$, namely $hg_1=0,\ldots,hg_{N-1}=0$ and $hg_N=1$. Therefore $Y_N$ is isomorphic to a direct summand of $Z$. Assume that $Z=Z^{\prime}\oplus Y_N$ where $\gamma(Z^{\prime})=\gamma(Z)-1$. The morphism $(1,0):Z^{\prime}\oplus Y_N\rightarrow Z^{\prime}$ is a deflation, so $g^\prime=g^*(1,0):Y^\prime\rightarrow Z^\prime$ is a deflation by Definition <ref>. Obviously, $(f_1,\ldots,f_{N-1})^t:X\rightarrow Y_1\oplus\ldots\oplus Y_{N-1}$ is a kernel of $g^\prime$. Thus
X\xrightarrow{(f_1,\ldots,f_{N-1})^t}Y_1\oplus\ldots\oplus Y_{N-1}\xrightarrow{g^\prime}Z^{\prime}
is a conflation. By hypothesis, $Y_1\oplus\ldots\oplus Y_{N-1}\cong X\oplus Z^{\prime}$. Hence $Y=Y_1\oplus\ldots\oplus Y_{N}\cong X\oplus Z$. The proof is completed.
If $1_{[X]}*1_{[Z]}([Y])\neq0$, then $\gamma(Y)\leq\gamma(X)+\gamma(Z)$, where the equality holds if and only if $Y\cong X\oplus Z$.
Let $X,Y,Z\in\Obj(\A)$ and $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a conflation in $\A$. If $m_{\pi_m}([X\xrightarrow{f}Y\xrightarrow{g}Z])\neq0$, $\gamma(Y)<\gamma(X)+\gamma(Z)$ and $Y=Y_1\oplus Y_2$, then there exist two conflations $X_1\xrightarrow{f_1}Y_1\xrightarrow{g_1}Z_1$ and $X_2\xrightarrow{f_2}Y_2\xrightarrow{g_2}Z_2$ in $\A$ such that $X\cong X_1\oplus X_2$, $Z\cong Z_1\oplus Z_2$ and $f=\text{diag}\{f_1,f_2\},g=\text{diag}\{g_1,g_2\}$.
Suppose that $\rk\Aut(X)=n_1$, $\rk\Aut(X)=N$ and $\rk\Aut(Z)=n_2$. Then $N<n_1+n_2$. For simplicity, we use the notation as above. Let $Y=Y_1\oplus\ldots\oplus Y_{N}$, $f=(f_1,f_2,\ldots,f_N)^t$, $g=(g_1,g_2,\ldots,g_N)$ and the isomorphisms $(a,c,b),(tI_{n_1},tI_{N},tI_{n_2})\in\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$, where $c=\left(
\begin{array}{cc}
\end{array}
\right)$. Recall that
is an idempotent such that
and $X=\text{Ker}s_N\oplus\text{Im}s_N$. Similarly, there exists an idempotent
in $\End(Z)$ such that $r_{N}g=(0,\ldots,0,g_N)$ and $Z=\text{Ker}r_N\oplus\text{Im}r_N$. Without loss of generality, we assume that $f_N\neq0$ and $g_N\neq0$. Because $f_Ns_N=f_N$ and $r_Ng_N=g_N$,
It is clear that $i:\text{Ker}s_N\hookrightarrow X$ is a kernel of $f_N:X\rightarrow Y_N$. There exists a morphism $f_{N}^\prime:\text{Im}s_N\rightarrow Y_N$ which is an image of $f_N$ since $X=\text{Ker}s_N\oplus\text{Im}s_N$. Similarly we can find a morphism $g_{N}^\prime:Y_N\rightarrow\text{Im}r_N$ which is a coimage of $g_N$ such that $g_N=jg_{N}^\prime$, where $j:\text{Im}(r_N)\hookrightarrow Z$ is an image of $g_N$. It is easy to check that $f_{N}^\prime$ is an inflation, $g_{N}^\prime$ a deflation and $g_{N}^\prime f_{N}^\prime=0$. Let $h:Y_N\rightarrow A$ be a morphism in $\A$ such that $hf_{N}^\prime=0$. The morphism
(0,\ldots,0,h):Y_1\oplus\ldots\oplus Y_N\rightarrow A
satisfies $(0,\ldots,0,h)f=0$. There exists $k\in\Hom_{\A}(Z,A)$ such that
since $g$ is a cokernel of $f$. It follows that $h=kg_N=kjg_{N}^\prime$. Hence $g_{N}^\prime$ is a cokernel of $f_{N}^\prime$. Therefore $\text{Im}s_N\xrightarrow{f_{N}^\prime}Y_N\xrightarrow{g_{N}^\prime}\text{Im}r_N$ is a conflation. By induction, every indecomposable direct summand of $Y$ is extended by the direct summands of $X$ and $Z$. The proof is finished.
Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two indecomposable constructible subsets of $\fObj_\A(\K)$. Let $A\in\Obj(\A)$ and $\gamma(A)\geq2$. If $[A]\notin\mathcal{O}_1\oplus \mathcal{O}_2$, then $1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([A])=0$.
If $1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([A])\neq0$, then there exist $X$, $Y\in \Obj(\mathcal{A})$ such that $[X]\in\mathcal{O}_{1}$, $[Y]\in\mathcal{O}_{2}$ and $1_{[X]}*1_{[Y]}(A)\neq0$ by Lemma <ref>. It follows that $\gamma(A)=2$ and $A\cong X\oplus Y$ by Lemma <ref> (also see <cit.>). This leads to a contradiction.
Let $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ be indecomposable subsets of $\fObj_{\A}(\mathbb{K})$. If $\mathcal{O}_{1}\cap\mathcal{O}_{2}=\emptyset$, then
where $\mathcal{P}_{i}$ are indecomposable constructible subsets and $a_i=1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([X])$ for $[X]\in\mathcal{P}_{i}$.
Let $[M]\in\mathcal{O}_{1}$ and $[N]\in\mathcal{O}_{2}$. Then $M$ is not isomorphic to $N$ since $\mathcal{O}_{1}\cap\mathcal{O}_{2}= \emptyset$. Using the fact that
$m_{\pi_{m}}([M\xrightarrow{(1,0)^t}M\oplus N\xrightarrow{(0,1)}N])=1$, we obtain
1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([M\oplus N])
=m_{\pi_{m}}([M\xrightarrow{(1,0)^t}M\oplus N\xrightarrow{(0,1)}N])\cdot \chi^{\na}([M\xrightarrow{(1,0)^t}M\oplus N\xrightarrow{(0,1)}N])=1.
By Lemma <ref>, we know that if $1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([X]) \neq0$ and $[X]\notin\mathcal{O}_{1}\oplus\mathcal{O}_{2}$, then $X$ is an indecomposable object. Note that
\big(1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}(\fObj_{\A}(\mathbb{K})\setminus\mathcal{O}_{1}\oplus\mathcal{O}_{2})\big) \setminus\{0\}=\{a_1,a_2,\ldots,a_m\}.
Then $\mathcal{P}_{i}= (1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}})^{-1}(a_i)\setminus\mathcal{O}_{1}\oplus\mathcal{O}_{2}$ for $1\leq i\leq m$. We complete the proof.
Using Lemma <ref> and Lemma <ref>, it is easy to see the following corollary:
Let $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ be two constructible sets. There exist finitely many constructible sets $\mathcal{Q}_1,\mathcal{Q}_2,\ldots,\mathcal{Q}_n$ such that
where $\gamma(\mathcal{Q}_{i})\leq \gamma(\mathcal{O}_{1})+ \gamma(\mathcal{O}_{2})$ and $a_i=(1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}})([X])$ for any $[X]\in\mathcal{Q}_{i}$.
For indecomposable constructible sets $\mathcal{O}_{1},\ldots ,\mathcal{O}_{k}$ and $X\in \Obj(\mathcal{A})$, $1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}*\ldots*1_{\mathcal{O}_{k}}([X])\neq0$ implies that $\gamma(X)\leq k$. In particular, $\gamma(X)=k$ means $X=X_{1}\oplus\ldots\oplus X_{k}$ with $[X_{i}]\in \mathcal{O}_{i}$ for $1\leq i\leq k$.
Let $X_{1},\ldots,X_{m}\in \Obj(\A)$ and there be $r$ isomorphic classes, we can assume that $X_{1},\ldots,X_{m_{1}}$ are isomorphic, $X_{m_{1}+1},\ldots,X_{m_{2}}$ are isomorphic, $\ldots$, and $X_{m_{r-1}+1},\ldots,X_{m_{r}}$ are isomorphic, where $m_{1}+\ldots+m_{r}=m$. By <cit.>, we have
\begin{equation}\label{equ1}
\Aut(X_{1}\oplus\ldots\oplus X_{m})/\Aut(X_{1})\times\ldots\times\Aut(X_{m})
\cong\mathbb{K}^{l}\times\prod\limits_{i=1}^{r}(\GL(m_{i},\mathbb{K})/(\mathbb{K}^{*})^{m_{i}}),
\end{equation}
\begin{equation}\label{equ2}
\chi(\Aut(X_{1}\oplus X_{2}\oplus\ldots\oplus X_{m})/\Aut(X_{1})\times\ldots\times\Aut(X_{m}))=\prod\limits_{i=1}^{r}m_{i}!.
\end{equation}
Let $\mathcal{O}$ be an indecomposable constructible set. Then
where $\gamma(\mathcal{P}_{i})<k$ for each $i$ and $m_{i}=1^{*k}_{\mathcal{O}}([X])$ for $[X]\in\mathcal{P}_{i}$.
We prove the proposition by induction. When $k=1$, it is easy to see that the formula is true. If $k=2$, then
1^{*2}_{\mathcal{O}}([X\oplus X])=1_{\mathcal{O}}([X])\cdot1_{\mathcal{O}}([X])\cdot\chi(\Aut(X\oplus X)/\Aut(X)\times\Aut(X))=2
for $[X]\in\mathcal{O}$ and
1^{*2}_{\mathcal{O}}([X\oplus Y])=
\big(1_{\mathcal{O}}([X])\cdot1_{\mathcal{O}}([Y])+1_{\mathcal{O}}([Y])\cdot1_{\mathcal{O}}([X])\big)\cdot\chi\big(\Aut(X\oplus Y)/\Aut(X)\times\Aut(Y)\big)=2,
where $[X],[Y]\in \mathcal{O}$ and $X\ncong Y$. If $[X]\notin\mathcal{O}\oplus\mathcal{O}$ and $\gamma(X)\geq2$ then $1^{*2}_{\mathcal{O}}([X])=0$ by Lemma <ref>. Hence $1^{*2}_{\mathcal{O}}=2\cdot1_{\mathcal{O}\oplus\mathcal{O}}+\sum\limits_{i}m_{i}\mathcal{P}_{i}$ where $\mathcal{P}_{i}$ are indecomposable constructible sets by Corollary <ref>.
Now we suppose that the formula is true for $k\leq n$. When $k=n+1$, we have
1^{*(n+1)}_{\mathcal{O}}=1^{*(n)}_{\mathcal{O}}*1_{\mathcal{O}}=(n!1_{n\mathcal{O}}+\sum c_{\mathcal{P}^{\prime}}1_{\mathcal{P}^{\prime}})*1_{\mathcal{O}},
where $\mathcal{P}^{\prime}$ are constructible sets with $\gamma(\mathcal{P}^{\prime})<n$. If the formula is true for $k=n+1$, then
n!1_{n\mathcal{O}}*1_{\mathcal{O}}= (n+1)!1_{(n+1)\mathcal{O}}+\sum c_{\mathcal{Q}}1_{\mathcal{Q}},
where $\mathcal{Q}$ are constructible sets with $\gamma(\mathcal{Q})<n+1$.
Hence it suffices to show that the initial term of $1_{n\mathcal{O}}*1_{\mathcal{O}}$ is $(n+1)1_{(n+1)\mathcal{O}}$, namely $(1_{n\mathcal{O}}*1_{\mathcal{O}})([X])=n+1$ for all $[X]\in(n+1)\mathcal{O}$.
Assume that $X=m_1X_1\oplus m_2X_2\oplus\ldots\oplus m_rX_r$, where $X_1,\ldots,X_r\in\text{Obj}(\A)$ which are not isomorphic to each other, $[X_i]\in\mathcal{O}$ for $1\leq i\leq r$, $m_1,\ldots,m_r$ are positive integers and $m_1+m_2+\ldots+m_r=n+1$.
(1_{n\mathcal{O}}*1_{\mathcal{O}})([X])=(1_{[(m_1-1)X_1\oplus m_2X_2\oplus\ldots\oplus m_rX_r]}*1_{[X_1]})([X])
+(1_{[m_1X_1\oplus (m_2-1)X_2\oplus\ldots\oplus m_rX_r]}*1_{[X_2]})([X])
+(1_{[m_1X_1\oplus\ldots\oplus m_{r-1}X_{r-1}\oplus (m_r-1)X_r]}*1_{[X_r]})([X])
Using Equation (<ref>), it follows that
1_{[X_1]}^{*(m_1-1)}*1_{[X_2]}^{*m_2}*\ldots*1_{[X_r]}^{*m_r}=(m_1-1)!m_2!\ldots m_r!1_{[(m_1-1)X_1\oplus m_2X_2\oplus\ldots\oplus m_rX_r]}+\ldots,
1_{[X_1]}^{*(m_1-1)}*1_{[X_2]}^{*m_2}*\ldots*1_{[X_r]}^{*m_r}*1_{[X_1]}=(\prod\limits_{i=1}^{r}m_i!)1_{[m_1X_1\oplus m_2X_2\oplus\ldots\oplus m_rX_r]}+\ldots
Compare the initial monomials of the two equations, it follows that
1_{[(m_1-1)X_1\oplus m_2X_2\oplus\ldots\oplus m_rX_r]}*1_{[X_1]}=m_11_{[m_1X_1\oplus m_2X_2\oplus\ldots\oplus m_rX_r]}+\ldots
Thus $1_{[(m_1-1)X_1\oplus m_2X_2\oplus\ldots\oplus m_rX_r]}*1_{[X_1]}([X])=m_1$.
Similarly, we have $1_{[m_1X_1\oplus \ldots\oplus(m_i-1)X_i\oplus\ldots\oplus m_rX_r]}*1_{[X_i]}([X])=m_i$ for $i=2,\ldots,r$. Hence $(1_{n\mathcal{O}}*1_{\mathcal{O}})([X])=\sum\limits_{i=1}^r m_i=n+1$
which completes the proof.
By induction, we have the following corollary.
Let $\mathcal{O}_{1},\mathcal{O}_{2},\ldots,\mathcal{O}_{k}$ be indecomposable constructible sets which are pairwise disjoint. Then we have the following equations
1^{*n_{1}}_{\mathcal{O}_{1}}*1^{*n_{2}}_{\mathcal{O}_{2}}\ldots*1^{*n_{k}}_{\mathcal{O}_{k}}= n_{1}!n_{2}!\ldots n_{k}!1_{n_{1}\mathcal{O}_{1}\oplus\ldots\oplus n_{k}\mathcal{O}_{k}}+\ldots,
1_{m_{1}\mathcal{O}_{1}\oplus\ldots\oplus m_{k}\mathcal{O}_{k}}*1_{n_{1}\mathcal{O}_{1}\oplus\ldots\oplus n_{k}\mathcal{O}_{k}}= \prod\limits_{i=1}^{k}\frac{(m_{i}+n_{i})!}{m_{i}!n_{i}!}1_{(m_{1}+n_{1})\mathcal{O}_{1}\oplus\ldots\oplus (m_{k}+n_{k})\mathcal{O}_{k}}+\ldots,
where $k$ is a positive integer and $m_1,\ldots,m_k,n_1,\ldots,n_k\in\mathbb{N}$.
Let $\text{Ind}(\alpha)$ be the subset of $\fObj_\A^{\alpha}(\mathbb{K})$ such that X are indecomposable for all $[X]\in\text{Ind}(\alpha)$.
For each $\alpha\in K^{\prime}(\A)$, $\rm Ind(\alpha)$ is a locally constructible set.
Assume $\alpha,\beta,\gamma\in K^{\prime}(\A)\setminus\{0\}$. The map
is defined by $([B],[C])\mapsto[B\oplus C]$. It is clear that $f$ is a pseudomorphism. Every $\fObj_\A^{\beta}(\mathbb{K})\times\fObj_\A^{\gamma}(\mathbb{K})$ is a locally constructible set. For any constructible set $\mathcal{C}\subseteq\fObj_\A(\mathbb{K})\times\fObj_\A(\mathbb{K})$, there are finitely many $\fObj_\A^{\beta}(\mathbb{K})\times\fObj_\A^{\gamma}(\mathbb{K})$ such that $\mathcal{C}\cap (\fObj_\A^{\beta}(\mathbb{K})\times\fObj_\A^{\gamma} (\mathbb{K}))\neq\emptyset$. Hence $\amalg_{\beta,\gamma;\beta+\gamma=\alpha}\fObj_\A^{\beta}(\mathbb{K})\times\fObj_\A^{\gamma}(\mathbb{K})$ is locally constructible. Then $\text{Im}f$ is a locally constructible set. It follows that $\text{Ind}(\alpha)=\fObj_\A^{\alpha}(\mathbb{K})\setminus\text{Im}f$ is locally constructible.
The following proposition is due to <cit.>.
Let $\mathcal{O}_{1},\mathcal{O}_{2}$ be two constructible sets of Krull-Schmidt. It follows that
1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}=\sum\limits_{i=1}^{c}a_{i} 1_{\mathcal{Q}_{i}}
for some $c\in\mathbb{N}^+$, where $a_{i}=1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([X])$ for each $[X]\in\mathcal{Q}_{i}$ and $\mathcal{Q}_{i}$ are constructible sets of stratified Krull-Schmidt such that $\gamma(\mathcal{Q}_i)\leq \gamma(\mathcal{O}_1)+\gamma(\mathcal{O}_2)$.
Because $\mathcal{O}_{1},\mathcal{O}_{2}$ are constructible sets, the equation holds for some constructible sets $\mathcal{Q}_{i}$ with $\gamma(\mathcal{Q}_i)\leq \gamma(\mathcal{O}_1)+\gamma(\mathcal{O}_2)$ by Corollary <ref>.
For every $[Y_{i}]\in\mathcal{Q}_{i}$, $1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([Y_i])\neq0$. By Lemma <ref>, there exist $X_{i},Z_{i}\in\Obj(\mathcal{A})$ such that $[X_{i}]\in \mathcal{O}_{1}$, $[Z_{i}]\in \mathcal{O}_{2}$ and $1_{[X_{i}]}*1_{[Z_{i}]}([Y_{i}])\neq0$ since $1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}([Y_{i}])\neq0$. Thanks to Lemma <ref>, we have that $\gamma(Y_i)\leq\gamma(X_i)+\gamma(Z_i)$. According to Lemma <ref>, all indecomposable direct summands of $Y_{i}$ are extended by the direct summands of $X_{i}$ and $Z_{i}$ since $1_{[X_{i}]}*1_{[Z_{i}]}([Y_{i}])\neq0$.
By the discussion in Section <ref>, we can suppose that $\mathcal{O}_{1}=\bigoplus\limits_{i=1}^{t}a_{i}\mathcal{C}_{i}$ and $\mathcal{O}_{2}=\bigoplus\limits_{j=1}^{t}b_{j}\mathcal{C}_{j}$, where $a_{i},b_{j}\in\{0,1\}$ for all $i,j$ and $\mathcal{C}_{i}$ are indecomposable constructible sets such that $\mathcal{C}_{i}\cap\mathcal{C}_{j}=\emptyset$ or $\mathcal{C}_{i}=\mathcal{C}_{j}$ for all $i\neq j$. Let $1\leq r\leq t$, the set
\{A_{1},A_{2},\ldots,A_{r}~|~\emptyset\neq A_{i}\subseteq \{1,\ldots,n\}~\text{for}~i=1,\ldots,r\}
is called an $r$-partition of $\{1,2,\ldots,t\}$ if $A_{1}\cup A_{2}\cup\ldots\cup A_{r}=\{1,2,\ldots,t\}$ and $A_{i}\cap A_{j}=\emptyset$ for all $i\neq j$. Obviously, the cardinal number of all partitions of $\{1,2,\ldots,t\}$ is finite. Let $\{A_{1},A_{2},\ldots,A_{r}\}$, $\{B_{1},B_{2},\ldots,B_{r}\}$ be two $r$-partitions of $\{1,2,\ldots,t\}$ and $c_{k}\in\mathbb{Q}\setminus\{0\}$ for $k=1,2,\ldots,r$. Set $\mathcal{O}_{A_{k}}=\bigoplus\limits_{i\in A_{k}}a_{i}\mathcal{C}_{i}$ and $\mathcal{O}_{B_{k}}=\bigoplus\limits_{j\in B_{k}}b_{j}\mathcal{C}_{j}$ for $1\leq k\leq r$. Then we have
\mathcal{R}_{A_{k},B_{k},c_{k}}=\{[X]\in\mathcal{O}_{A_{k}}\oplus \mathcal{O}_{B_{k}}~|~1_{\mathcal{O}_{A_{k}}}*1_{\mathcal{O}_{B_{k}}}([X])=c_{k}\},
\mathcal{I}_{A_{k},B_{k},c_{k}}=\{[X]~|~X~\text{indecomposable},1_{\mathcal{O}_{A_{k}}}*1_{\mathcal{O}_{B_{k}}}([X])=c_{k}\}.
This means that for each $[X]\in\mathcal{R}_{A_{k},B_{k},c_{k}}$, there exist $[A]\in\mathcal{O}_{A_{k}}$ and $[B]\in\mathcal{O}_{B_{k}}$ such that $X\cong A\oplus B$. For each $[Y]\in\mathcal{I}_{A_{k},B_{k},c_{k}}$, there exist $[C]\in\mathcal{O}_{A_{k}}$ and $[D]\in\mathcal{O}_{B_{k}}$ such that $C\rightarrow Y\rightarrow D$ is a non-split conflation in $\A$. Note that
\mathcal{R}_{A_{k},B_{k},c_{k}}=((1_{\mathcal{O}_{A_{k}}}*1_{\mathcal{O}_{B_{k}}})^{-1}(c_{k}))\cap(\mathcal{O}_{A_{k}}\oplus \mathcal{O}_{B_{k}}).
By Corollary <ref>, $\mathcal{R}_{A_{k},B_{k},c_{k}}=\emptyset$ or $\mathcal{O}_{A_{k}}\oplus \mathcal{O}_{B_{k}}$. Hence $\mathcal{R}_{A_{k},B_{k},c_{k}}$ is a constructible set of Krull-Schmidt. There exist $\alpha_1,\ldots,\alpha_s\in K^{\prime}(\A)$ such that $\mathcal{I}_{A_{k},B_{k},c_{k}}=(\amalg_{i=1}^s\text{Ind}(\alpha_i))\cap ((1_{\mathcal{O}_{A_{k}}}*1_{\mathcal{O}_{B_{k}}})^{-1}(c_{k}))$. By Lemma <ref>, $\mathcal{I}_{A_{k},B_{k},c_{k}}$ is an indecomposable constructible set.
Finally, $1_{\mathcal{O}_{1}}*1_{\mathcal{O}_{2}}$ is a $\mathbb{Q}$-linear combination of finitely many $1_{\oplus_{k=1}^{r}\mathcal{O}_{A_{k},B_{k},c_{k}}}$, where $\mathcal{O}_{A_{k},B_{k},c_{k}}$ run through $\mathcal{R}_{A_{k},B_{k},c_{k}}$ and $\mathcal{I}_{A_{k},B_{k},c_{k}}$ for all $r$-partitions and $r=1,2,\ldots,t$. We finish the proof.
Thus we summarize what we have proved as the following theorem which is due to <cit.>.
The $\mathbb{Q}$-space $\CF^{\rm{KS}}(\fObj_\A)$ is an associative $\mathbb{Q}$-algebra with convolution multiplication $*$ and identity $1_{[0]}$.
§.§ The universal enveloping algebra of $\CFi(\fObj_\A)$
Let $U(\CFi(\fObj_\A))$ denote the universal enveloping algebra of $\CFi(\fObj_\A)$ over $\mathbb{Q}$. The multiplication in $U(\CFi(\fObj_\A))$ will be written as $(x,y)\mapsto xy$. There is a $\mathbb{Q}$-algebra homomorphism
\Phi :U(\CFi(\fObj_\A))\rightarrow\CF^{\text{KS}}(\fObj_\A)
defined by $\Phi(1)=1_{[0]}$ and $\Phi(f_1f_2\ldots f_n)=f_1*f_2*\ldots*f_n$, where $f_1,f_2,\ldots,f_n$ belong to $\CFi(\fObj_\A)$.
The following theorem is related to <cit.>.
$\Phi :U(\CFi(\fObj_\A))\rightarrow\CF^{\rm KS}(\fObj_\A)$ is an isomorphism.
For simplicity of presentation, let
Suppose that $\mathcal{O}_{1},\mathcal{O}_{2},\ldots,\mathcal{O}_{k}$ are indecomposable constructible subsets of $\fObj_\A(\mathbb{K})$ which are pairwise disjoint. It follows that $1_{\mathcal{O}_{1}},1_{\mathcal{O}_{2}},\ldots, 1_{\mathcal{O}_{k}}$ are linearly independent in $\CFi(\fObj_\A)$.
Let $U_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}$ denote the subspace of $U$ which is spanned by all $1_{\mathcal{O}_{1}}^{n_1}1_{\mathcal{O}_{2}}^{n_2}\ldots1_{\mathcal{O}_{k}}^{n_k}$ for $n_i\in\mathbb{N}$ and $i=1,\ldots,k$.
Define $\CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{n}}$ to be the subalgebra of $\CF$ which is generated by the elements $1_{n_1\mathcal{O}_{1}\oplus n_2\mathcal{O}_{2}\oplus\ldots\oplus n_k\mathcal{O}_{k}}$ of $\CF$, where $n_i\in\mathbb{N}$ for $i=1,2,\ldots,k$.
The homomorphism $\Phi$ induces a homomorphism
\Phi_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}:U_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}\rightarrow \CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}
which maps $1_{\mathcal{O}_{1}}^{n_1} 1_{\mathcal{O}_{2}}^{n_2}\ldots1_{\mathcal{O}_{k}}^{n_k}$ to $1_{\mathcal{O}_{1}}^{*n_1}*1_{\mathcal{O}_{2}}^{*n_2}*\ldots*1_{\mathcal{O}_{k}}^{*n_k}$.
First of all, we want to show that $\Phi_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}$ is injective.
For $m\in\mathbb{N}$, let $U_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}$ be the subspace of $U$ which is spanned by
\big\{1_{\mathcal{O}_{1}}^{n_1}1_{\mathcal{O}_{2}}^{n_2}\ldots1_{\mathcal{O}_{k}}^{n_k}~|~\sum\limits_{i=1}^{k}n_i\leq m, n_i\geq0 ~\text{for}~i=1,\ldots,k\big\}
Using the PBW Theorem, we obtain that
\big\{1_{\mathcal{O}_{1}}^{n_1}1_{\mathcal{O}_{2}}^{n_2}\ldots1_{\mathcal{O}_{k}}^{n_k}~|~\sum\limits_{i=1}^{k}n_i= m, n_i\geq0 ~\text{for}~i=1,\ldots,k\big\}
is a basis of the $\mathbb{Q}$-vector space $U_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}/ U_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m-1)}$ for $m\geq1$.
Similarly, we define $\CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}$ to be a subspace of $\CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}$ such that each $f\in\CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}$ is of the form $\sum\limits_{i=1}^l c_{i}1_{\mathcal{C}_i}$, where $l\in\mathbb{N}^+$, $c_{i}\in\mathbb{Q}$, $1_{\mathcal{C}_i}\in \CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}$ and $\mathcal{C}_i$ are constructible sets of Krull-Schmidt such that $\gamma(\mathcal{C}_i)\leq m$.
In $\CF^{(m)}/\CF^{(m-1)}$, the set
\{1_{n_1\mathcal{O}_{1}\oplus n_2\mathcal{O}_{2}\oplus\ldots\oplus n_k\mathcal{O}_{k}}~|~\sum\limits_{i=1}^{k}n_i= m, n_i\geq0 ~\text{for}~i=1,\ldots,k\}
is linearly independent by the Krull-Schmidt Theorem.
For each $m\geq1$, $\Phi_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}$ induce a map
\Phi_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}:U_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}/ U_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m-1)}\rightarrow\CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}/ \CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m-1)}
which maps $1_{\mathcal{O}_{1}}^{n_1}1_{\mathcal{O}_{2}}^{n_2}\ldots1_{\mathcal{O}_{k}}^{n_k}$ to $n_1!n_2!\ldots n_k! 1_{n_1\mathcal{O}_{1}\oplus n_2\mathcal{O}_{2}\oplus\ldots\oplus n_k\mathcal{O}_{k}}$ (also see Corollary <ref>), where $\sum\limits_{i=1}^{k}n_i= m$ and $m_i\geq0$. From this we know that $\Phi_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}^{(m)}$ is injective for all $m\in\mathbb{N}$.
Obviously, both $U_{\mathcal{O}_{1}\mathcal{O}_{2}\ldots\mathcal{O}_{n}}$ and $\CF_{\mathcal{O}_{1}\ldots\mathcal{O}_{n}}$ are filtered. From the properties of filtered algebra, we know that $\Phi_{\mathcal{O}_{1}\ldots\mathcal{O}_{k}}$ is injective. Hence $\Phi :U\rightarrow\CF$ is injective.
Finally, we show that $\Phi$ is surjective by induction. When $m=1$, the statement is trivial. Then we assume that every constructible function $f=\sum\limits_{i=1}^t a_{i}1_{\mathcal{Q}_i}$ lies in $\Im(\Phi)$, where $a_{i}\in\mathbb{Q}$ and $\mathcal{Q}_i$ are constructible sets of stratified Krull-Schmidt with $\gamma(\mathcal{Q}_i)<m$.
Let $n_1+n_2+\ldots+n_k=m$ and $n_i\in\mathbb{N}$ for $1\leq i\leq k$. Then
\Phi(1_{\mathcal{O}_{1}}^{n_1}1_{\mathcal{O}_{2}}^{n_2}\ldots1_{\mathcal{O}_{k}}^{n_k})= 1_{\mathcal{O}_{1}}^{*n_1}*1_{\mathcal{O}_{2}}^{*n_2}*\ldots*1_{\mathcal{O}_{k}}^{*n_k}
=n_{1}!n_{2}!\ldots n_{n}!1_{n_{1}\mathcal{O}_{1}\oplus n_{2}\mathcal{O}_{2}\oplus\ldots\oplus n_{k}\mathcal{O}_{k}}+ \sum\limits_{j=1}^s b_{j}1_{\mathcal{P}_{j}},
where $b_j\in\mathbb{Q}$ and $\mathcal{P}_{j}$ are constructible sets of stratified Krull-Schmidt with $\gamma(\mathcal{P}_{j})<m$. By the hypothesis, $\sum\limits_{j=1}^{s}b_{j}1_{\mathcal{P}_{j}}\in\Im(\Phi)$. Hence $1_{n_{1}\mathcal{O}_{1}\oplus n_{2}\mathcal{O}_{2}\oplus\ldots\oplus n_{k}\mathcal{O}_{k}}$ lies in $\Im(\Phi)$. The algebra $\CF$ is generated by all $1_{n_{1}\mathcal{O}_{1}\oplus\ldots\oplus n_{k}\mathcal{O}_{k}}$, which proves that $\Phi$ is surjective, the proof is finished.
§ COMULTIPLICATION AND GREEN'S FORMULA
§.§ Comultiplication
We now turn to define a comultiplication on the algebra $\CF^{\text{KS}}(\fObj_\A)$. For $f,g\in\CF(\fObj_\A)$, $f\otimes g$ is define by $f\otimes g([X],[Y])=f([X])g([Y])$ for $([X],[Y])\in (\fObj_\A\times\fObj_\A)(\mathbb{K})= \fObj_\A(\mathbb{K})\times\fObj_\A(\mathbb{K})$ (see <cit.>). Let $X\xrightarrow{f}Y\xrightarrow{g} Z$ be a conflation in $\A$. Recall that the map $p_2:\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)\rightarrow\Aut(X)\times\Aut(Z)$ is defined by $(a_1,a_2,a_3)\mapsto (a_1,a_3)$ and $\text{Ker}p_2=1$. Therefore the pushforward of $\mathbf{\pi}_l\times\mathbf{\pi}_r$ is well-defined.
The following definitions are related to <cit.> and <cit.>.
From now on, assume that $\mathbf{\pi}_m:\fExact_\A\rightarrow\fObj_\A$ is of finite type. Then we have the following diagram
\begin{equation*}
\begin{gathered}
\CF^{\text{KS}}(\fObj_\A\times\fObj_\A)\xleftarrow{(\mathbf{\pi}_l\times\mathbf{\pi}_r)_!}
\CF^{\text{KS}}(\fExact_\A)\xleftarrow{(\mathbf{\pi}_m)^{*}}\CF^{\text{KS}}(\fObj_\A).
\end{gathered}
\end{equation*}
The comultiplication
\begin{equation*}
\Delta:\CF^{\text{KS}}(\fObj_\A)\rightarrow\CF^{\text{KS}}(\fObj_\A\times\fObj_\A)
\end{equation*}
is defined by $\Delta=(\mathbf{\pi}_l\times\mathbf{\pi}_r)_!\circ(\mathbf{\pi}_m)^*$, where $\CF^{\text{KS}}(\fObj_\A\times\fObj_\A)$ is regarded as a topological completion of $\CF^{\text{KS}}(\fObj_\A)\otimes\CF^{\rm KS}(\fObj_\A)$.
The counit $\varepsilon:\CF^{\text{KS}}(\fObj_\A)\rightarrow\mathbb{Q}$ maps $f$ to $f([0])$.
Note that $\Delta$ is a $\mathbb{Q}$-linear map since $(\mathbf{\pi}_l\times\mathbf{\pi}_r)_!$ and $(\mathbf{\pi}_m)^*$ are $\mathbb{Q}$-linear map.
Let $\alpha=[A],\beta=[B]\in\fObj_\A(\mathbb{K})$ and $\mathcal{O}\subseteq\fObj_\A(\mathbb{K})$ be a constructible set of stratified Krull-Schmidt, define
Let $\mathcal{O}_1$ and $\mathcal{O}_2\subseteq\fObj_\A(\mathbb{K})$ be constructible sets, define
Because $\Delta(1_{\mathcal{O}})$ is a constructible function, $\Delta(1_{\mathcal{O}})= \sum\limits_{i=1}^{n}h^{\beta_i\alpha_i}_{\mathcal{O}} 1_{\mathcal{O}_i}$ for some $\alpha_i,\beta_i\in\fObj_\A(\mathbb{K})$ and $n\in \mathbb{N}$, where $\mathcal{O}_i$ are constructible subsets of $\fObj_\A(\mathbb{K})\times\fObj_\A(\mathbb{K})$.
Let $X$, $Y$, $Z\in\Obj(\A)$. If $X\oplus Z$ is not isomorphic to $Y$, then $\Delta(1_{[Y]})([X],[Z])=0$.
If $\Delta(1_{[Y]})([X],[Z])\neq0$, there exists a conflation $X\xrightarrow{f}Y\xrightarrow{g}Z$ in $\A$ such that $m_{\pi_l\times\pi_r}([X\xrightarrow{f}Y\xrightarrow{g}Z])\neq0$. Recall that
If $\rk~\text{Im}p_{2}<\rk\big(\Aut(X)\times\Aut(Z)\big)$, the fibre of the action of a maximal torus of $\Aut(X)\times\Aut(Z)$ on $(\Aut(X)\times\Aut(Z))/\text{Im}p_2$ is $(\mathbb{K}^*)^{l}$ for $l>0$. Then $\chi\big((\Aut(X)\times\Aut(Z))/\text{Im}p_2\big)=0$, which is a contradiction. Hence $\rk\big(\Aut(X)\times\Aut(Z)\big)=\rk~\text{Im}p_{2}$.
Assume that $\rk\Aut(X)=n_1$, $\rk\Aut(Z)=n_2$ and $\rk\Aut(Y)=n$ for some positive integers $n_1$, $n_2$ and $n$. Note that $\textbf{D}_{n_1}\times\textbf{D}_{n_2}$ is a maximal torus of $\Aut(X) \times\Aut(Z)$. Because $\rk\big(\Aut(X)\times\Aut(Z)\big)=\rk~\text{Im}(p_{2})$, each maximal torus of $\text{Im}p_2$ is also a maximal torus of $\Aut(X) \times\Aut(Z)$. Therefore every maximal torus of $\text{Im}p_2$ and $\textbf{D}_{n_1}\times\textbf{D}_{n_2}$ are conjugate. For simplicity, we can assume that $\textbf{D}_{n_1}\times\textbf{D}_{n_2}$ is a maximal torus of $\text{Im}p_{2}$. For $(t_1I_{n_1},t_2I_{n_2})\in\textbf{D}_{n_1}\times\textbf{D}_{n_2}$, where $t_1\neq t_2$, there exists $\tau\in\Aut(Y)$ such that $(t_1I_{n_1},\tau,t_2I_{n_2})\in\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$. Then we have the commutative diagram
\begin{equation*}
\xymatrix{
X \ar[d]_{t_1I_{n_1}} \ar[r]^{f} & Y \ar[d]_{\tau} \ar[r]^{g} & Z \ar[d]^{t_2I_{n_2}} \\
X \ar[r]^{f} & Y \ar[r]^{g} & Z }
\end{equation*}
The morphism $(t_2I_{n_1},t_2I_{n},t_2I_{n_2})$ is also in $\Aut(X\xrightarrow{f}Y\xrightarrow{g}Z)$. The following diagram is commutative
\begin{equation*}
\xymatrix{
X \ar[d]_{t_2I_{n_1}} \ar[r]^{f} & Y \ar[d]_{t_2I_{n}} \ar[r]^{g} & Z \ar[d]^{t_2I_{n_2}} \\
X \ar[r]^{f} & Y \ar[r]^{g} & Z }
\end{equation*}
Consequently $g(\tau-t_2I_{n})=0$. Because $f$ is a kernel of $g$, there exists $h\in\Hom(Y,X)$ such that $\tau-t_2I_{n}=fh$. Then $\tau=fh+t_2I_{n}$. We have
f(t_1I_{n_1})=\tau f=(fh+t_2I_{n}))f,
it follows that
Then $hf=(t_1-t_2)I_{n_1}$ since $f$ is an inflation. Let $f^\prime=\frac{1}{t_1-t_2}h$, then $f^\prime f=1_X$. Hence $X$ is isomorphic to a direct summand of $Y$. The proof is completed.
For an indecomposable object $X\in\Obj(\A)$, direct summands of $X$ are only $X$ and $0$. Thus $\Delta(1_{[X]})=1_{[X]}\otimes1_{[0]}+ 1_{[0]}\otimes1_{[X]}$. It follows that $\Delta(f)=f\otimes1_{[0]}+1_{[0]}\otimes f$ for $f\in\CFi(\fObj_\A)$.
By Lemma <ref>, $h^{\beta\alpha}_{\mathcal{O}}=1$ if $\alpha\oplus\beta\in\mathcal{O}$, and $h^{\beta\alpha}_{\mathcal{O}}=0$ otherwise. Let $\mathcal{O}=n_1\mathcal{O}_1\oplus\ldots\oplus n_m\mathcal{O}_m$ be a constructible set of stratified Krull-Schmidt, where $\mathcal{O}_i$ are indecomposable constructible sets for all $1\leq i\leq m$. By Lemma <ref>, the formula $\Delta(1_{\mathcal{O}})= \sum\limits_{i=1}^{n}h^{\beta_i\alpha_i}_{\mathcal{O}} 1_{\mathcal{O}_i}$ can be written as
\Delta(1_{\mathcal{O}})=\sum\limits_{1\leq i\leq m;0\leq k_i\leq n_i}1_{k_1\mathcal{O}_1\oplus\ldots\oplus k_m\mathcal{O}_m}\otimes 1_{(n_1-k_1)\mathcal{O}_1\oplus\ldots\oplus (n_m-k_m)\mathcal{O}_m}.
Hence we have the following proposition.
Let $\mathcal{O}$ be a constructible set of stratified Krull-Schmidt, then $\Delta(1_{\mathcal{O}})\in\CF^{\rm KS}(\fObj_\A)\otimes\CF^{\rm KS}(\fObj_\A)$, i.e., the map
\Delta:\CF^{\rm KS}(\fObj_\A)\rightarrow\CF^{\rm KS}(\fObj_\A)\otimes\CF^{\rm KS}(\fObj_\A))
is well-defined.
§.§ Green's formula on stacks
Recall that
\int_{x\in S}f(x)=\sum\limits_{a\in f(S)\setminus\{0\}}a\chi^{\na}(f^{-1}(a)\cap S),
where $f$ is a constructible function and $S$ a locally constructible set.
Let $\mathcal{O}_1,\mathcal{O}_2,\mathcal{O}_\rho,\mathcal{O}_\sigma,\mathcal{O}_\epsilon,\mathcal{O}_\tau,\mathcal{O}_{\lambda}$ be constructible sets and $\alpha\in\mathcal{O}_1,\beta\in\mathcal{O}_2,\rho\in\mathcal{O}_\rho, \sigma\in\mathcal{O}_\sigma,\epsilon\in\mathcal{O}_\epsilon, \tau\in\mathcal{O}_\tau, \lambda\in\mathcal{O}_{\lambda}$ such that $\mathcal{O}_\rho\oplus\mathcal{O}_\sigma=\mathcal{O}_1$ and $\mathcal{O}_\epsilon\oplus\mathcal{O}_\tau=\mathcal{O}_2$.
The following theorem is the degenerate form of Green's formula which is related to <cit.>.
Let $\mathcal{O}_1,\mathcal{O}_2$ be constructible subsets of $\fObj_\A(\mathbb{K})$ and $\alpha^\prime,\beta^\prime\in\fObj_\A(\mathbb{K})$, then we have
g^{\alpha^\prime\oplus\beta^\prime}_{\mathcal{O}_2\mathcal{O}_1}=\int_{\rho,\sigma,\epsilon,\tau\in\fObj_\A(\mathbb{K}); \rho\oplus\sigma\in\mathcal{O}_1,\epsilon\oplus\tau\in\mathcal{O}_2} g^{\alpha^\prime}_{\epsilon\rho}g^{\beta^{\prime}}_{\tau\sigma}.
By the proof of Lemma <ref>, $g^{\alpha^\prime\oplus\beta^\prime}_{\mathcal{O}_2\mathcal{O}_1}=\int_{\alpha\in\mathcal{O}_1,\beta\in\mathcal{O}_2} g^{\alpha^\prime\oplus\beta^\prime}_{\beta\alpha}$. It suffices to prove the following formula
g^{\alpha^\prime\oplus\beta^\prime}_{\beta\alpha}=\int_{\rho,\sigma,\epsilon,\tau\in\fObj_\A(\mathbb{K}); \rho\oplus\sigma=\alpha,\epsilon\oplus\tau=\beta}g^{\alpha^\prime}_{\epsilon\rho}g^{\beta^{\prime}}_{\tau\sigma}.
Suppose that $[A]=\alpha,[B]=\beta,[A^{\prime}]=\alpha^{\prime},[B^{\prime}]=\beta^{\prime},[C]=\rho,[D]=\sigma, [E]=\epsilon$ and $[F]=\tau$ for $A,B,C,D,E,F\in\text{Obj}(\A)$. There are finitely many $(\rho,\sigma)$ and $(\epsilon,\tau)$ such that $\rho\oplus\sigma=\alpha$ and $\epsilon\oplus\tau=\beta$. The morphism
i:\bigcup\limits_{[C],[D],[E],[F];\atop[C\oplus D]=[A],[E\oplus F]=[B]}V([C],[E];A^\prime)\times V([D],[F];B^{\prime})\rightarrow V([A],[B];A^\prime\oplus B^\prime)
is defined by
(\langle C\xrightarrow{f_1}A^\prime\xrightarrow{g_1}E\rangle, \langle D\xrightarrow{f_2}B^\prime\xrightarrow{g_2}F\rangle)\mapsto \langle C\oplus D\xrightarrow{f}A^\prime\oplus B^\prime\xrightarrow{g}E\oplus F\rangle,
where $f=\left(
\begin{array}{cc}
\end{array}
\right)$ and $g=\left(
\begin{array}{cc}
\end{array}
\right)$. Because both $C\xrightarrow{f_1}A^\prime\xrightarrow{g_1}E$ and $D\xrightarrow{f_2}B^\prime\xrightarrow{g_2}F$ are conflations, $C\oplus D\xrightarrow{f} A^\prime\oplus B^\prime\xrightarrow{g}E\oplus F$ is a conflation by <cit.>. Hence the morphism is well-defined. Note that $i$ is injective and $g^{\alpha^\prime}_{\epsilon\rho}g^{\beta^{\prime}}_{\tau\sigma}=\chi^{\na}(V([C],[E];A^\prime)\times V([D],[F];B^{\prime}))$.
By <cit.>, we have
\chi^{\na}(V([A],[B];A^\prime\oplus B^\prime))=\chi^{\na}(\text{Im}i)+\chi^{\na}\big(V([A],[B];A^\prime\oplus B^\prime) \setminus\text{Im}i\big).
According to Lemma <ref>, if $m_{\pi_m}([A\xrightarrow{f}A^\prime\oplus B^\prime\xrightarrow{g}B])\neq0$, then there exist two conflations $C\xrightarrow{f_1}A^\prime\xrightarrow{g_1}E$ and $D\xrightarrow{f_2}B^\prime\xrightarrow{g_2}F$ in $\A$ such that $A\cong C\oplus D$, $B\cong E\oplus F$, $f=\left(
\begin{array}{cc}
\end{array}
\right)$ and $g=\left(
\begin{array}{cc}
\end{array}
\right)$. Thus
m_{\pi_{m}}([A\xrightarrow{f}A^\prime\oplus B^\prime\xrightarrow{g}B])=0
for any $\langle A\xrightarrow{f}A^\prime\oplus B^\prime\xrightarrow{g}B\rangle \in V([A],[B];A^\prime\oplus B^\prime)\setminus \text{Im}i$. Using (<ref>), it follows that $\chi^{\na}(V([A],[B];A^\prime\oplus B^\prime)\setminus\text{Im}i)=0$. Hence
\begin{eqnarray*}
g^{\alpha^\prime\oplus\beta^\prime}_{\beta\alpha}=\chi^{\na}(V([A],[B];A^\prime\oplus B^\prime))=\chi^{\na}(\text{Im}i)\\
=\int_{\rho,\sigma,\epsilon,\tau\in\fObj_\A(\mathbb{K}); \rho\oplus\sigma=\alpha,\epsilon\oplus\tau=\beta}g^{\alpha^\prime}_{\epsilon\rho}g^{\beta^{\prime}}_{\tau\sigma}.
\end{eqnarray*}
This completes the proof.
For all $f_1,f_2,g_1,g_2\in\CF^{\rm KS}(\fObj_\A)$, define $(f_1\otimes g_1)*(f_2\otimes g_2)=(f_1*f_2)\otimes(g_1*g_2)$. Using Green's formula, we have the following theorem due to <cit.>.
The map $\Delta:\CF^{\rm KS}(\fObj_\A)\rightarrow\CF^{\rm KS}(\fObj_\A)\otimes\CF^{\rm KS}(\fObj_\A)$ is an algebra homomorphism.
The proof is similar to the one in <cit.>. Let $\mathcal{O}_1, \mathcal{O}_2\in\fObj_\A(\mathbb{K})$ be constructible sets of stratified Krull-Schmidt. Then
\begin{eqnarray*}
\Delta(1_{\mathcal{O}_1}*1_{\mathcal{O}_2})=\Delta(\sum\limits_{\lambda}g_{\mathcal{O}_2\mathcal{O}_1}^{\lambda}1_{\mathcal{O}_\lambda}) =\sum\limits_{\lambda}g_{\mathcal{O}_2\mathcal{O}_1}^{\lambda}\Delta(1_{\mathcal{O}_\lambda})\\
=\sum\limits_{\lambda} g_{\mathcal{O}_2\mathcal{O}_1}^{\lambda}(\sum\limits_{\alpha^{\prime},\beta^{\prime}} h_{\mathcal{O}_{\lambda}}^{\beta^{\prime}\alpha^{\prime}}1_{\mathcal{O}_{\alpha^{\prime}}}\otimes1_{\mathcal{O}_{\beta^{\prime}}}) =\sum\limits_{\alpha^{\prime},\beta^{\prime}}g_{\mathcal{O}_2\mathcal{O}_1}^{\alpha^{\prime}\oplus\beta^{\prime}} 1_{\mathcal{O}_{\alpha^{\prime}}}\otimes1_{\mathcal{O}_{\beta^{\prime}}},
\end{eqnarray*}
\begin{eqnarray*}
\Delta(1_{\mathcal{O}_1})*\Delta(1_{\mathcal{O}_2})=(\sum\limits_{\rho,\sigma}h_{\mathcal{O}_1}^{\sigma\rho} 1_{\mathcal{O}_{\rho}}\otimes1_{\mathcal{O}_{\sigma}})*(\sum\limits_{\epsilon,\tau}h_{\mathcal{O}_2}^{\tau\epsilon} 1_{\mathcal{O}_{\epsilon}}\otimes1_{\mathcal{O}_{\tau}})\\
=\sum\limits_{\rho,\sigma,\epsilon,\tau}h_{\mathcal{O}_1}^{\sigma\rho}h_{\mathcal{O}_2}^{\tau\epsilon} (1_{\mathcal{O}_{\rho}}*1_{\mathcal{O}_{\epsilon}})\otimes(1_{\mathcal{O}_{\sigma}}*1_{\mathcal{O}_{\tau}})\\
= \sum\limits_{\rho,\sigma,\epsilon,\tau}h_{\mathcal{O}_1}^{\sigma\rho}h_{\mathcal{O}_2}^{\tau\epsilon} (\sum\limits_{\alpha^{\prime},\beta^{\prime}}g_{\mathcal{O}_{\epsilon}\mathcal{O}_{\rho}}^{\alpha^{\prime}} g_{\mathcal{O}_{\tau}\mathcal{O}_{\sigma}}^{\beta^{\prime}}1_{\mathcal{O}_{\alpha^{\prime}}}\otimes 1_{\mathcal{O}_{\beta^{\prime}}})\\
=\sum\limits_{\alpha^{\prime},\beta^{\prime}}(\sum\limits_{\rho,\sigma,\epsilon,\tau}h_{\mathcal{O}_1}^{\sigma\rho} h_{\mathcal{O}_2}^{\tau\epsilon}g_{\mathcal{O}_{\epsilon}\mathcal{O}_{\rho}}^{\alpha^{\prime}} g_{\mathcal{O}_{\tau}\mathcal{O}_{\sigma}}^{\beta^{\prime}}1_{\mathcal{O}_{\alpha^{\prime}}}\otimes 1_{\mathcal{O}_{\beta^{\prime}}}).
\end{eqnarray*}
According to Theorem <ref>, it follows that
\sum\limits_{\rho,\sigma,\epsilon,\tau}h_{\mathcal{O}_1}^{\sigma\rho} h_{\mathcal{O}_2}^{\tau\epsilon}g_{\mathcal{O}_{\epsilon}\mathcal{O}_{\rho}}^{\alpha^{\prime}} g_{\mathcal{O}_{\tau}\mathcal{O}_{\sigma}}^{\beta^{\prime}}=g_{\mathcal{O}_2\mathcal{O}_1}^{\alpha^{\prime}\oplus\beta^{\prime}}.
Therefore $\Delta(1_{\mathcal{O}_1}*1_{\mathcal{O}_2})=\Delta(1_{\mathcal{O}_1})*\Delta(1_{\mathcal{O}_2})$. We have thus proved the theorem.
§ EXACT CATEGORIES
We recall the definition of an exact category (see <cit.>).
Let $\mathcal{A}$ be an additive category. A sequence
in $\mathcal{A}$ is called exact if $f$ is a kernel of $g$ and $g$ is a cokernel of $f$. The morphisms $f$ and $g$ are called inflation and deflation respectively. The short exact sequence is called a conflation. Let $\mathcal{S}$ be the collection of conflations closed under isomorphism and satisfying the following axioms
A0 $1_{0}:0\rightarrow0$ is a deflation.
A1 The composition of two deflations is a deflation.
A2 For every $h\in\Hom(X,X^{\prime})$ and every inflation $f\in\Hom(X,Y)$ in $\mathcal{A}$, there exists a pushout
\begin{equation*}
\xymatrix{
X \ar[r]^f \ar[d]_h & Y \ar[d]^{h^{\prime}} \\
X^{\prime} \ar[r]^{f^{\prime}} & Y^{\prime}
\end{equation*}
where $f^{\prime}\in\Hom(X^{\prime},Y^{\prime})$ is an inflation.
A3 For every $l\in\Hom(Z^{\prime},Z)$ and every deflation $g\in\Hom(Y,Z)$ in $\mathcal{A}$, there exists a pullback
\begin{equation*}
\xymatrix{
Y^{\prime} \ar[r]^{g^{\prime}} \ar[d]_{l^{\prime}} & Z^{\prime} \ar[d]^{l} \\
Y \ar[r]^{g} & Z }
\end{equation*}
where $g^{\prime}\in\Hom(Y^{\prime},Z^{\prime})$ is an deflation.
Then $(\A,\mathcal{S})$ is called an exact category.
The definition of idempotent complete is taken from<cit.>.
Let $\A$ be an additive category. The category $\A$ is idempotent complete if for every idempotent morphism $s:A\rightarrow A$ in $\A$, $s$ has a kernel $k:K\rightarrow A$ and a image $i:I\rightarrow A$ (a kernel of a cokernel of $s$) such that $A\cong K\oplus I$. We write $A\cong\text{Ker}s \oplus\text{Im}s$, for simplicity.
ass06I. Assem, D. Simson, A. Skowronski.
Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory.
London Mathematical Society Student Texts, 65. Cambridge University Press, Cambridge, 2006.
buhler10T. Bühler.
Exact categories.
Expo. Math. 28 (2010), no. 1, 1-69.
dxx10M. Ding, J, Xiao, F. Xu.
Realizing enveloping algebras via varieties of modules.
Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 1, 29-48.
greenJ. A. Green.
Hall algebras, hereditary algebras and quantum groups.
Invent. Math. 120 (1995), no. 2, 361-377.
gomezT. L. Gómez.
Algebraic stacks.
Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 1, 1-31.
hartshorneag77R. Hartshorne.
Algebraic geometry.
Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
jsz2005B. T. Jensen, X. Su, A. Zimmermann.
Degenerations for derived categories.
J. Pure Appl. Algebra 198 (2005), no. 1-3, 281-295.
joyceA1D. Joyce.
Configurations in abelian categories. I. Basic properties and moduli stacks.
Adv. Math. 203 (2006), no. 1, 194-255.
joyceJLMS06D. Joyce.
Constructible functions on Artin stacks.
J. London Math. Soc. (2) 74 (2006), no. 3, 583-606.
joyceA2D. Joyce.
Configurations in abelian categories. II. Ringel-Hall algebras.
Adv. Math. 210 (2007), no. 2, 635-706.
keller90B. Keller.
Chain complexes and stable categories.
Manuscripta Math. 67 (1990), no. 4, 379-417.
kresch99A. Kresch.
Cycle groups for Artin stacks.
Invent. Math. 138 (1999), no. 3, 495-536.
lusztigkyoto90G. Lusztig.
Intersection cohomology methods in representation theory.
Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 155-174, Math. Soc. Japan, Tokyo, 1991.
mozgovoyS. Mozgovoy.
Motivic Donaldson-Thomas invariants and McKay correspondence.
arXiv:1107.6044v1 [math.AG].
mumford76D. Mumford.
Algebraic geometry. I. Complex projective varieties.
Reprint of the 1976 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.
px00L. Peng, J. Xiao.
Triangulated categories and Kac-Moody algebras.
Invent. Math. 140 (2000), no. 3, 563-603.
quillen73D. Quillen.
Higher algebraic K-theory. I.
Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147. Lecture Notes in Math., Vol. 341, Springer, Berlin 1973.
rie94C. Riedtmann.
Lie algebras generated by indecomposables.
J. Algebra 170 (1994), no. 2, 526-546.
ringel90C. M. Ringel.
Hall algebras and quantum groups.
Invent. Math. 101 (1990), no. 3, 583-591.
ringel96C. M. Ringel.
Green's theorem on Hall algebras.
Representation theory of algebras and related topics (Mexico City, 1994), 185-245, CMS Conf. Proc., 19, Amer. Math. Soc., Providence, RI, 1996.
rosenlicht63M. Rosenlicht.
A remark on quotient spaces.
An. Acad. Brasil. Ci. 35 1963 487-489.
schofieldA. Schofield.
Notes on constructing Lie algebras from finite-dimensional algebras.
xxz06 J. Xiao, F. Xu, G. Zhang.
Derived categories and Lie algebras.
arXiv:0604564v2 [math.QA].
|
1511.00104
|
Today, much of our sensitive information is stored inside mobile applications (apps), such as the browsing histories and chatting logs. To safeguard these privacy files, modern mobile systems, notably Android and iOS, use sandboxes to isolate apps' file zones from one another. However, we show in this paper that these private files can still be leaked by indirectly exploiting components that are trusted by the victim apps. In particular, we devise new indirect file leak () attacks that exploit browser interfaces, command interpreters, and embedded app servers to leak data from very popular apps, such as Evernote and QQ. Unlike the previous attacks, we demonstrate that these IFLs can affect both Android and iOS. Moreover, our IFL methods allow an adversary to launch the attacks remotely, without implanting malicious apps in victim's smartphones. We finally compare the impacts of four different types of IFL attacks on Android and iOS, and propose several mitigation methods.
§ INTRODUCTION
Mobile applications (apps) are gaining significant popularity in today's mobile cloud computing era <cit.>.
Much sensitive user information is now stored inside mobile apps (on mobile devices), such as Facebook authentication tokens, Chrome browsing histories, and Whatsapp chatting logs.
To safeguard these privacy files, modern mobile systems, notably Android and iOS, use sandboxes to isolate apps' file zones from one another.
However, it is still possible for an adversary to steal private app files in an indirect manner by exploiting components that are trusted by the victim apps.
We refer to this class of attacks as indirect file leaks (IFLs).
<ref> illustrates a high-level model.
Initially, an adversary cannot directly access a private file, formulated as $a \nLeftarrow s$.
If the adversary can send crafted inputs to a deputy[We borrow this term from the classic confused deputy problem <cit.> to represent a trusted component in victim apps.] inside the victim app ($a \rightarrow d$) and these inputs can trigger the deputy to send the private file to the adversary ($d \rightarrow s \Rightarrow a$), then the adversary can indirectly steal the private file.
The whole process, $a \rightarrow d \rightarrow s \Rightarrow a$, achieves the goal of $s \Rightarrow a$, causing an .
A high-level model.
In this paper, we devise new attacks that exploit browser interfaces, command interpreters, and embedded app servers to leak files from very popular apps, such as Evernote and Tencent QQ.
Unlike prior works <cit.> that only show local attacks on Android, we demonstrate that three out of our four IFL attacks affect both Android and iOS.
We summarize these attacks below.
* attacks bypass the same-origin policy (SOP), which is enforced to protect resources originating from different origins (i.e., the combination of scheme, domain, and port), to steal private files via browsing interface deputies.
Although our prior work <cit.> has demonstrated such attacks on Android by exploiting numerous Android browsers' SOP flaws on the scheme, we are extending it in this paper to a number of vulnerable iOS apps, such as the very popular Evernote, Tencent QQ, and Mail.Ru.
We also confirm that the latest iOS 8 fails to enforce the appropriate SOP on .
In our analysis, the root case of this attack is that the legacy web SOP is found to be inadequate for the local schemes, such as . Eradicating the problem may call for an enhanced SOP.
* attacks also leverage browsing interfaces as deputies, but they do not need to violate SOP.
It can do so by injecting and executing unauthorized JavaScripts directly on target files, instead of requiring a malicious file to bypass SOP to access target files as in the attacks.
Popular Android browsers, such as Baidu, Yandex, and 360 Browser, can be easily compromised in this way, allowing their private files (e.g., cookie and browsing history) to be stolen.
The high-profile 360 Mobile Safe and Baidu Search are also exploitable.
Besides these Android apps, we further uncover a vulnerable iOS app, myVault.
* attacks exploit command interpreters as deputies inside victim apps to execute unauthorized commands for file leaks.
We demonstrate that the top command apps on Android, Terminal Emulator and SSHDroid, can be stealthily exploited to execute arbitrary commands, possibly with the root privilege.
This will jeopardize their own files (e.g., command histories and private configuration files), sensitive user photos stored on SD cards, and even other apps' private files.
* attacks send unauthorized file extraction requests to embedded app server deputies inside victim apps to obtain private files.
All of the tested popular server apps, such as WiFi File Transfer (Android) and Simple Transfer (iOS), are vulnerable to these attacks.
It is worth noting that both the and attacks use previously unexplored deputies—command interpreters and embedded app servers—to launch attacks for the first time in mobile platforms.
Besides the cross-platform vulnerability, our methods also allow an adversary to launch the attacks remotely, without implanting malicious apps in victim's phones as in prior works <cit.>.
These attacks can be launched both locally (in the same phone) and remotely (in an Intranet/Internet).
Table <ref> highlights the identified vulnerabilities and their major attack channels.
Besides local attacks, we show that browsers, such as Baidu and Yandex Browser, can be remotely exploited by enticing the victim to access a web page.
Email apps (e.g., Mail.Ru) and social apps (e.g., QQ) can be similarly compromised if the victim opens a malicious attachment or file transmitted by a remote adversary. In other remote attacks, the adversary can scan the whole Intranet, locate open ports, and exploit vulnerable server apps installed on the victim phone.
Furthermore, we analyze the differences between Android and iOS in terms of the impact of the attacks.
These differences are caused by different system architectures and app design practices between Android and iOS.
Our analysis shows that a common iOS app practice could lead to more powerful and pervasive attacks on iOS than Android. On the other hand, three iOS system characteristics help lessen the impacts on iOS for the other three attacks. These findings can help developers and OS providers build more secure apps and mobile systems.
Ethical considerations.
All of our vulnerability testing is conducted using our own mobile devices and test accounts.
The tests never affect the data security of real-world users.
As the attacks in mobile apps are client-side vulnerabilities, they cannot affect the server-side integrity.
Real-world impacts.
We have reported most of the identified vulnerabilities to their vendors in a responsible way and are in the process of reporting the remaining vulnerabilities.
All of the contacted vendors have acknowledged our reports. For example, Evernote has listed us in its security hall of fame.
Baidu has ranked one of our reports as the most valuable vulnerability report of the second quarter of 2014,
and Qihoo 360 has issued us the highest award in its mobile bug bounty program history.
We have also offered our suggestions to the vendors to fix the identified vulnerabilities.
Four vulnerabilities and their attack channels.
2c|Attack Channels 2-3
Remote Local
1|c|2* Evernote, Tencent QQ UC & QQ browsers 1|c| Mail.Ru 360 & Mail.Ru Cloud
1|c|2* 2c|Baidu, Yandex, and 360 browsers 1|c| 2c|360 Mobile Safe, Baidu Search, myVault
1|c| SSHDroid Terminal Emulator
1|c| 2c|WiFi File Transfer, Simple Transfer
To summarize, we make the following three contributions:
* We devise four new attacks that, for the first time, can affect both Android and iOS and are exploitable not only locally but also remotely. (Section <ref> & <ref>)
* We identify a number of zero-day vulnerabilities in popular Android and iOS apps and uncover a serious SOP issue in the latest iOS 8 system. (Section <ref>)
* We analyze the differences between Android and iOS in terms of the attacks' impacts and propose several methods to mitigate the attacks. (Section <ref> & <ref>)
§ BACKGROUND
§.§ Sandbox-based App Isolation
Both Android and iOS use sandbox-based app isolation to build a trustworthy mobile environment.
Each app resides in its own sandbox, with its code and data isolated from other apps.
This isolation is usually enforced at the kernel level.
For example, Android uses UNIX UID-based protection to isolate each app, in which each app is treated as an independent user and runs in a separate process with a unique .
Each app's sensitive files are stored in their own system-provided isolated (or private) file zone.
Unless an app actively leaks a file (i.e., a direct file leak), other apps have no access to the protected files.
The widely deployed SEAndroid MAC system <cit.> further thwarts the risks incurred by direct file leaks.
However, IFLs can still occur in the presence of both sandbox-based isolation and MAC. Although the actual executer of the file access is the legal victim app, the file request is actually initiated and crafted by an adversary.
Encryption-based defenses, such as encrypting all private app files, face similar limitations as the MAC. To sum up,
remains a serious, and yet unsolved, threat, which motivates our study.
§.§ Terminology
In Table <ref>, we summarize the terms used throughout this paper.
Terms and their descriptions.
Term Description
2*Private files The files stored in apps' isolated file zones. In nonlocal attacks, they also include files on a SD card, e.g., user photos. Target file A private file the adversary wants to steal in an attack. 2*Permission A form of privilege representation. For example, the permission on Android and the permission on iOS. 2*Root A superuser privilege. For example, a rooted phone is a phone that enables superuser privilege for apps. Browsing Or browsing component. A component with browsing capabili- interface ty, usually built with Android's WebView or iOS's UIWebView. Command 2*An app component that can interpret and execute commands. interpreter App server A server component embedded in an app.
§ THE ATTACKS
In this section, we first describe the adversary model and then detail the four types of attacks introduced in Section <ref>.
§.§ Adversary Model
We consider the following three types of adversaries in our attacks. A local adversary can launch only local attacks, whereas the Intranet and Internet adversaries remote attacks.
* A local adversary is an attack app installed on the same smartphone as the victim app.
It requires few or no permissions and does not exhibit any typical malicious app behavior <cit.>.
The root privilege is never used by this attack app.
We also do not consider screenshot attacks <cit.> that require strong assumptions.
* An Intranet adversary resides in the same Intranet as the victim's mobile device.
It can send network requests to any other node within the Intranet.
It can sniff the nearby wireless traffic and retrieve unencrypted content.
We do not assume that it can launch effective ARP spoofing attacks, as network administrators can detect such anomalous events.
* An Internet adversary can be located in any host in the Internet.
It remotely compromises a victim by (i) enticing a victim to browse a web page under the adversary's control, and/or (ii) sending the victim a malicious file via email, chatting app, social network, and other means.
§.§ Bypassing SOP on Browsing Interfaces
The attacks bypass SOP to steal private files via browsing interfaces.
The deputy in this attack is the browsing interface or rendering engine, whose SOP enforcement is flawed and cannot prevent malicious JavaScript codes from accessing a private file.
All apps that contain browsing components are potential victims.
The scheme is an ideal medium to launch the attacks.
Two parts of SOP enforcement on can be exploited to steal local private files.
The adversary can cross the origin from a web domain to access local file content, if the cross-scheme SOP enforcement for to (labelled as ) is broken.
Alternatively, the adversary can leverage a local malicious HTML file in one path to steal a target file in another path.
In this case, the SOP enforcement between the two file origins (labelled as ) must be bypassed.
Since failure of enforcing is rare in modern rendering engines, we focus on the attacks that will bypass the .
Our recent study <cit.> shows that Android does not effectively enforce .
Android prior to 4.1 does not enforce this policy at all.
Although the succeeding Android versions fix this logical flaw at the engine level, a new application program interface (API) <cit.> is available for developers to loosen the corresponding SOP restriction.
An app compiled with a vulnerable software development kit (i.e., before 4.1) is still exploitable on more recent Android platforms, including Android 4.4, which uses the Chrome Blink engine as its default engine.
Our recent study <cit.> shows that Android does not effectively enforce .
However, little is known about iOS.
This is where our contribution for the attacks lies.
Contrary to our expectation, these attacks can have higher impact on the iOS ecosystem than the Android's.
Our testing using iPhone 6 reveals that even the latest iOS 8 does not properly enforce .
Indeed, iOS never guarantees this policy (Section <ref>).
In Section <ref>, we identify a common practice among the iOS apps that could lead to more pervasive and powerful attacks on iOS than Android.
We believe that the root problem is that the legacy SOP cannot adequately cover the local schemes, such as .
The typical web SOP principle (i.e., the legacy SOP) allows file A (at file:///dir1/a.html) to access file B (at file:///dir2/b.txt) because the two origins share the same scheme, domain (i.e., 127.0.0.1 or localhost), and port.
In practice, this legal behavior fails to meet the security requirements for , especially in the mobile environment.
Therefore, an enhanced SOP for local schemes, such as adding the “path” element to the current three-element SOP tuples, is needed for eradicating this vulnerability.
We reported our iOS findings to Apple on 19 January 2015 and suggested them to use an enhanced SOP at the system or engine level.
§.§ Unauthorized JavaScript Execution on Target Files
The attacks could be regarded as an advanced variant of the attacks.
Both attacks use the browsing interface as the deputy, but the attack does not violate SOP.
It can do so by injecting and executing unauthorized JavaScripts directly on target files, instead of requiring a malicious file to bypass SOP to access target files as in the attacks.
The attacks usually consist of two steps.
Unauthorized JavaScript codes are first injected into the target file.
This can be achieved in different ways, depending on the type of the target files.
For example, if the target is a cookie file, the JavaScript codes can be injected via a website's cookie field.
The target files are then loaded and rendered.
The previously injected JavaScript is executed to steal the current file content via an HTML document object model variable, such as .
As JavaScript only accesses the current document, this attack does not violate SOP, thus setting this attack apart from the attack.
It is worth noting that the two steps can also be performed simultaneously without the user's knowledge.
A victim user's private files will be stolen when he browses a web page under the attacker's control.
In this paper, we focus on designing remote attacks, although they can also be conducted locally.
We identify two types of remote attacks illustrated in <ref>.
In the first type, a web page tries to load a target file through local schemes like and .
The file can be loaded by a file link or an HTML .
The link-based loading requires an extra user clicking, whereas the iframe-based loading is automatic and does not require any user action.
Before loading the target cookie file, the web page injects malicious JavaScript codes (e.g., <script>alert(document.body.innerHTML)</script>) via the web cookie field.
Once the cookie file is successfully loaded, the JavaScript inside it can steal the cookie content.
As will be shown in Section <ref>, the popular 360 Mobile Safe, Baidu, and Yandex browsers are all vulnerable to this type of attack.
[Attacks that actively load a target file.]
[Attacks that exploit victim apps' file loading features.]
Remote attacks.
The second type of the attacks does not actively load and render target files.
Instead, it exploits victim apps' ability to load the content of target files into a renderable user interface (UI), such as those containing WebView widgets.
For example, some browser apps load browsing histories from the history file into a renderable UI.
Using this app-loading feature, an adversary simply injects the JavaScript into the target file.
For example, as illustrated in <ref>, the adversary injects an unauthorized JavaScript into the history file through the title or URL field of a web page.
When the user switches to the renderable UI, a new history log containing the malicious title or URL is displayed.
The embedded JavaScript is then executed to steal the history file.
These passively loaded files are usually rendered in the local domain (e.g., under ).
The adversary can also combine an attack with a attack to steal other private files that are not loaded by the victim apps.
We will detail the affected apps in Section <ref>.
§.§ Unauthorized Command Execution on Command Interpreters
The attacks exploit command interpreters (as deputies) inside victim apps to execute file operation related commands for attacks.
We consider explicitly embedded command interpreters, such as those in command terminal apps.
In other words, (remote) code execution vulnerabilities contained in host apps are out of the scope.
An app is vulnerable to a attack only when it can be injected with unauthorized commands.
To leak private files, the injected commands can
(i) set a world-readable file permission by invoking the command,
(ii) export a file to a public SD card via the command,
or (iii) send a file to a remote server through commands like .
If the victim app has root permissions, all of these commands can be used to steal private files in other, possibly more sensitive, apps.
To discover and exploit a vulnerability, an adversary needs to identify channels that can be used to confuse command interpreters to accept unauthorized commands.
The cross-app component communication channel <cit.> on Android can be used to launch local attacks, if the command interpreter components are exposed.
An adversary can also exploit the URL scheme <cit.> to achieve the same, if not better, attack impacts.
Likewise, the URL scheme <cit.> can achieve the same, if not better, attack effects.
Moreover, general remote attacks are also possible, if the command interpreter accepts remote command requests. In addition, victim apps' configurations stored in public storage can be changed to indirectly inject commands. Accessibility services can also be misused to mimic user commands <cit.>.
A attack could also be launched through a GUI to attack file manager apps.
An adversary can force these apps to perform unauthorized UI-based file operations to leak file contents.
However, this is not a real threat, because such file operations have to be conducted by victim users and thus are easily noticed.
A smarter adversary can wait for users to open a sensitive file and launch screenshot attacks <cit.> to sniff the content.
We exclude this scenario also from our treat model, because this requires the adversary to closely monitor the victim's activity.
§.§ Unauthorized File Extraction via Embedded App Servers
The confused deputies here are the app servers embedded in victim apps.
An adversary sends unauthorized file extraction requests to exploit vulnerable embedded app servers for obtaining private files.
The affected app servers are mainly file servers (e.g., those that support and requests) that provide users with file transmission service between phones and desktops.
The command servers mentioned in the last section are another type of candidate servers.
Some apps that support multi-function servers are also affected, such as the very popular AirDroid app, which has over 10 million installs.
A attack can be conducted in three ways.
A attack can be conducted in three ways, as shown in <ref>.
First, a local attack can be launched from another app installed on the same smartphone as the victim app.
It scans the local hosts' ports and sends packets to the port listened to by the victim app.
Second, adversarial nodes (e.g., phones or laptops) in the same Intranet can attack the victim app in another device.
Within the same Intranet, it is also easy for an adversary to identify a port opened by the victim app.
Third, we show that remote attacks are also possible.
Attack vectors are delivered through the victim's desktop browsers when they browse a malicious web page.
Three kinds of attacks.
A successful attack may need to bypass authentication set up to protect the victim app, such as an authentication code (e.g., user password) or a confirmation action (e.g., clicking a confirmation button).
Our evaluation in Section <ref>, however, shows that the current authentication in server apps is nearly broken.
Some apps use no or weak authentication (e.g., using only four random numbers).
Almost all channels for transmitting authentication codes and subsequent session tokens are unencrypted.
An Intranet adversary can easily sniff these secrets to bypass the authentication step.
§ UNCOVERING VULNERABILITIES IN ANDROID AND IOS APPS
§.§ SopIFL Vulnerabilities
We extend the prior work <cit.> by demonstrating that the vulnerabilities also exist in non-browser Android apps.
Specifically, we discover two vulnerable Android apps, 360 Cloud and On The Road (a Chinese travel app).
Theses apps are not browsers, but they contain exposed browsing interfaces that can be exploited by local attacks.
We uncover a number of vulnerable iOS apps summarized in Table <ref>.
Our main result is uncovering a number of vulnerable iOS apps.
We evaluate them using an iPhone 6 (with the latest iOS 8) and an iPad 3 (with iOS 7).
Our evaluation shows that both iOS 7 and 8—which are used in around 95 percent of all iOS devices <cit.>—do not enforce at the engine level. Since all iOS apps can use only the default web engine, the current solution for mitigating the vulnerabilities can only be done on the application level.
iOS apps vulnerable to the attacks.
Category Vulnerable Apps Attack Channel 2*Browser UC, Mercury 2*Local Baidu, Sogou, QQ browsers 2*Cloud Drive Mail.Ru Cloud 2*Local & Web Baidu Cloud, 360 Cloud Note/Read Evernote, QQ Reader Local & Web Email Mail.Ru Remote Social Tencent QQ Remote Utility Foxit Reader, OliveOffice Local
To launch a attack on the iOS platform, a “malicious” HTML file must be delivered to the victim app.
We have identified three such channels.
Local The adversary can design stealthy iOS apps (e.g., the Jekyll app <cit.>) to launch local attacks, because some iOS apps accept external HTML files from other apps through the “open with” feature[Two “open with” demos implemented in Dropbox and WeChat are available at <http://goo.gl/H7KXeM>.].
Similarly, Android browsers often use exposed browsing interfaces <cit.>.
We notice that local attacks can also be conducted “remotely” by leveraging other apps' remote channels.
For example, an adversary first sends an HTML file to the popular WeChat app installed on a victim device.
Since WeChat will not open this type of file, it will ask the user to open the file using another app.
We find that browser and cloud drive apps are likely to be affected by these local attacks.
Web An adversary can deliver attack vectors through the web service interfaces of mobile apps.
For example, cloud drive and note apps support file sharing on the web.
An adversary can share an HTML file with a victim via web interfaces.
Remote Remote attacks are possible for some iOS apps.
The attachment mechanism in email apps is an ideal channel to launch targeted remote attacks.
For example, once a Mail.Ru iOS user opens an attached HTML file from an adversary email, the adversary can steal the victim's private Mail.Ru files remotely.
Similarly, the file sending mechanism in social apps, such as the popular Tencent QQ, can also be exploited.
We have reported these vulnerabilities to their vendors (mostly in early June 2014).
The vendors have acknowledged our reporting and some of them have patched the issues[Currently developers have to implement application-level defenses (e.g., refusing external files from other apps and disabling JavaScript in the domain) to patch their apps, as the underlying iOS engine is still vulnerable.].
In particular, Mail.Ru and 360 have awarded us with several bug bounties.
For example, Mail.Ru has issued a total of $1,000 for its two vulnerabilities.
Other vendors, such as Evernote, Baidu, and Tencent, have either listed us in their security halls of fame or given us bug bounty gifts.
Besides HTML files, we anticipate other types of files for launching attacks.
For example, the commonly used PDF and flash file formats can execute embedded JavaScript codes in a desktop environment <cit.>.
The current mobile systems have limited support for flash and only run basic JavaScript in PDF files.
The current mobile systems have limited support for flash <cit.> and only run basic JavaScript in PDF files <cit.>.
Once these systems are improved, we expect that these new attack vectors will bypass the existing protection.
For example, WeChat is immune to the current attacks by disabling the opening of HTML files. However, it allows opening PDF files to be opened, which could be exploited for future attacks.
§.§ AimIFL Vulnerabilities
We summarize the vulnerabilities in Table <ref>.
As discussed in Section <ref>, these vulnerabilities can be classified into two types: and .
aimIFL-1 attacks.
We attempt an attack via on two Android apps, Baidu Browser and On The Road.
We find it difficult to directly load a content (e.g., via an HTML iframe) from a web page on Android.
We thus use an alternative method that asks users to click a link embedded in a web page.
This method is able to exploit On The Road, as the app renders the link clickable.
This method is able to exploit On The Road (see <ref> in Appendix <ref>), as the app renders the link clickable.
However, similar to desktop browsers, links in Baidu Browser are not clickable.
We find that an adversary can entice a victim to long-press the link, allowing Baidu Browser to pop up a dialog that the user can click.
The target file is then rendered and its contents are stolen automatically.
The attack procedure is illustrated in <ref>.
Apps vulnerable to the attacks.
Attack Name Vulnerable Apps
via Baidu Browser, On The Road via 360 Mobile Safe 2*via Yandex and 360 browsers Baidu Search, Baidu Browser
3*on Android Internet Browser, Smart Browser Shady Browser, Zirco Browser on iOS myVault
An attack exploiting Baidu Browser.
We find the scheme on Android can also be exploited by attacks.
This scheme is used to retrieve content or data from the corresponding content provider components.
Surprisingly, we find that a web page can load a local file via the scheme if the associated content provider implements the API.
Launching attacks via is therefore even easier than .
We have successfully launched this attack to remotely exploit 360 Mobile Safe.
We can use this attack to remotely exploit 360 Mobile Safe (see <ref> in Appendix <ref>).
Because of the seriousness of this exploit, 360 has issued us the highest award in its mobile bug bounty program history.
Terada <cit.> points out that the scheme can be used to remotely attack local Android components.
Following this idea, we independently identify several popular browsers and the Baidu Search app that can be exploited by attacks via .
These victim apps satisfy the following three conditions, which make them exploitable.
* They contain to intercept an URI and generate an structure.
This can invoke any component of the victim, even a private component that has not been exposed to other apps.
An adversary can thus design a crafted URI to deliver attack vectors to a target component.
* They include a component that imports external parameters to .
An adversary can thus control this component to render an arbitrary URI.
* The victim component in the last step allows access and its JavaScript execution.
The victim can therefore render a target file via and execute its embedded JavaScript codes.
All the attacks discussed above affect only Android apps.
We will explain why it is hard to launch attacks on iOS in Section <ref>.
aimIFL-2 attacks.
Launching the attacks successfully requires two conditions.
The victim app must be able to load the content of a target file into a WebView-based UI gadget, and the adversary must be able to inject an unauthorized JavaScript into the target file.
We use the first condition to facilitate the search for vulnerabilities.
We focus on Android browsers and iOS WebView-based apps in <cit.> and inspect their screenshots to choose which apps to install and test.
The WebView.loadDataWithBaseURL API is useful for locating vulnerable Android browsers, because developers often invoke this API to load the local content and their associated assets.
We use this API and its first parameter value (e.g., starting with “file://”) to search Android browser codes and identify a vulnerable open source Zirco browser.
Interestingly, this vulnerable Zirco design is also used in several other browsers, including “Browser for Android” () that affects around one million users.
One iOS app, myVault, is exploitable by the attack.
This app allows users to store their private photos, bookmarks, and passwords.
Its bookmark store page is an entry point where a “malicious” bookmark can be injected to steal the victim's bookmarks.
Even worse, as iOS does not enforce SOP well on , an adversary can therefore steal other sensitive content through a crafted bookmark.
§.§ CmdIFL Vulnerabilities
Table <ref> lists the identified vulnerabilities.
We select command terminal and server apps in Google Play, because these apps are more likely to contain command interpreters than normal apps.
More specifically, we evaluate the top apps, Terminal Emulator and SSHDroid, as they are the most likely to be installed by users with these requirements.
For example, Terminal Emulator for Android has over 10 million installs, whereas the top two terminal apps have less than 0.5 million installs.
2*Apps 2*Vulnerability Cause Attack # of Channel Installs
Terminal 2*The command component is exposed. 2*Local 2*10M+ Emulator 1-4
2*SSHDroid The command server is Local & 2*500K+ weakly protected. Intranet
Manifest excerpt of Terminal Emulator.
security weaknesses in the top 10 server apps.
2*App 2*App 2*Transmission Immune to Effective 3*# of Installs* Platform 2*Id 2*Name Protocol Port 2*Encryption Authentication File Upload Connection CSRF Alert
5*Android 1 AirDroid http 8888 (setting) (user confirm) 10M - 50M 2 WiFi File Transfer http 1234 (setting) (setting) 5M - 10M 3 Xender http 6789 (four numbers) 1M - 5M 4 WiFi File Explorer http 8000 (setting) 1M - 5M 5 ftp 2121 100K - 500K
5*iOS 6 Simple Transfer http 80 (setting) 1,504 Ratings 7 Photo Transfer WiFi http 8080 (six bytes) 865 Ratings 8 WiFi Photo Transfer http 15555 (setting) 661 Ratings 9 USB & Wi-Fi Flash Drive http 8080 462 Ratings 10 Air Transfer http 8080 (setting) 138 Ratings
* The app install numbers were counted on November 1, 2014. We use rating numbers to estimate the popularity of the iOS apps.
Both tested command apps are found to be vulnerable to the attacks.
A local attack app can execute arbitrary commands using the identity of Terminal Emulator, such as exporting its private files (e.g., command histories and private configuration files) to a public SD card.
The root cause of this vulnerability is that the exposed terminal command component can be invoked by other local apps with arbitrary command parameters.
Interestingly, we find that Terminal Emulator tries to protect its command component with a -level permission, as shown in part (1) of <ref>.
The rationale behind this design is that all command invocations (via the action) now have the appropriate authorization through the permission mechanism.
Unfortunately, the adversary does not need to touch the protected command proxy component (“RunScript”).
Instead, it directly invokes the underlying command component (“RemoteInterface” in part (2) of <ref>), which is by default exposed by Android's intent filter mechanism <cit.>.
Consequently, a crafted input can force the “RemoteInterface” component to execute commands.
We reported this issue and its demo attack code to Terminal Emulator's github page, and helped the open-source community patch the app <cit.>.
The top command server app, SSHDroid, is vulnerable to the local and Intranet attacks.
This app works as an SSH server listening to the default port of 22, but it cannot prevent unauthorized connectors.
An adversary does not need to fingerprint SSHDroid, because SSHDroid gives this information to any connector.
SSHDroid only has two user name choices, the “user” in the normal case or the “root” on a rooted phone.
It also uses the default “admin” password if the user does not change it in the settings.
Hence, an adversary has enough pre-knowledge to exploit this app and steal its private files.
More alarmingly, SSHDroid always tries to work as the “root”.
If it is exploited on a rooted phone, an adversary can execute root commands and steal the private files of all the installed apps.
§.§ ServerIFL Vulnerabilities
We test the top ten server apps from Google Play and the Apple App Store to evaluate the vulnerabilities.
Table <ref> summarizes the statistics of their security metrics.
All of the tested apps have at least one security weakness that can be exploited to launch the attacks.
Surprisingly, none of these apps provide encrypted transmission between file requesters (e.g., users' desktop browsers) and file servers (i.e., the tested apps).
Eighty percent of the apps do not implement this important security guarantee at all.
This can cause serious consequences in the wireless setting, which is assumed in these apps' user models.
Two apps also provide this functionality which, however, is not enabled by default.
We find that the apps' SSL encryption (when manually enabled in the setting) accepts only self-signed certificates, which causes security warnings in client-side browsers, thus hurting the user experience.
The authentication used in these apps is very weak.
Seven of the ten tested apps do not enforce authentication, including the most popular iOS server app, Simple Transfer.
An Intranet adversary can thus easily send unauthorized file extraction requests.
The apps that conduct authentication still do not have guaranteed security due to the aforementioned lack of encrypted transmission.
An adversary can sniff wireless traffic to obtain the secret information used for authentication or the cookies used for post-authentication transmission.
The secret information used for authentication is generally not strong, such as the four-number verification code used in app #3 and the six-character password in app #7.
Brute-force attacks are therefore practical.
For example, to crack Xender's authentication, an adversary only needs to try 10,000 times at most.
Remote attacks are also possible.
We propose an improved file upload CSRF (cross-site request forgery) attack <cit.> for this purpose.
An adversary uploads an HTML or PDF file with malicious JavaScript codes through the file upload CSRF.
After the victim opens the uploaded file in his desktop browser, the embedded JavaScript runs in the same domain as other target files and can steal arbitrary file content.
We only describe the test results here:
* Apps #2 and #9 suffer from file upload CSRF, making remote attacks possible.
* Apps #4, #5, and #8 do not support file upload functionalities, making them immune to file upload CSRF. However, the paid version of app #4 supports file uploading.
* Apps #6 and #7 allow only uploading photo files, making them non-vulnerable.
* Apps #3 and #10 do not support viewing uploaded files. Therefore, the attack vector (i.e., the embedded JavaScript) cannot be executed in the victim's desktop browser.
* One app, AirDroid, embeds a secret token into each GET/POST request URL. As CSRF cannot obtain this token, the app is safe.
Launching stealthy attacks usually requires that victim apps do not have an effective mechanism to detect illegal connections.
Our evaluation reveals that only two apps have such detection capabilities.
AirDroid alerts users to confirm or reject each new incoming connection and breaks the last connection.
This mechanism is effective in preventing stealthy connections, because it is hard for a local attack app to disable or envelop AirDroid alerts.
Simple Transfer displays a “connected” UI when it accepts its first connection.
However, it does not further implement multiple-connection detections or warnings.
This weakness allows an adversary to stealthily connect to a victim app after the victim has established its initial connection with a legal user browser.
To launch effective attacks, an attacker could fingerprint common server apps in advance.
As shown in Table <ref>, the protocols used in the tested apps are quite indistinguishable (basically HTTP). However, the opened ports have sufficient variability for identifying the apps.
Moreover, for the apps with the same port numbers, an adversary can leverage different HTTP responses to distinguish them.
Once the adversary has constructed a database of fingerprints, it can launch targeted attacks on the apps.
§ ANDROID VS IOS
Our evaluation reveals four major differences between Android and iOS in terms of the impact of the attacks.
We discuss their implications below.
Implication 1:
The common practice in iOS apps to open (untrusted) files in their own app domain could lead to more pervasive and powerful attacks on iOS than Android.
Table <ref> lists a detailed comparison of the file-opening behavior in the iOS and Android versions of our tested apps in different categories.
We choose two file types, HTML and PDF, for their ability to carry attack vectors.
Popular apps that do not support sending HTML or PDF files, such as Facebook and Whatsapp, are not listed.
We use “in” to show that the tested app opens files within its own app and “out” to show that it opens files outside the original app (i.e., in other apps).
A comparison of the file-opening behavior in the iOS and Android versions of representative apps.
3*Category 3*Apps 4c|File-opening Behavior 3-6
2c|HTML file 2c|PDF file 3-6
iOS* Andr iOS Andr
5*Email Gmail in (web) out in out 2-6
Yahoo Mail in (noJS) out in out 2-6
Mail.Ru in (vuln) out in out 2-6
QQ Mail in (noJS) out in out 2-6
Netease Mail in (noJS) out in out
Dropbox in (dbcache) out in out 2-6
Google Drive out out in out 2-6
Cloud Mail.Ru Cloud in (vuln) out in out 2-6
Drive Baidu Cloud in (vuln) out in out 2-6
360 Cloud in (vuln) out in out 2-6
Tencent Cloud in (text) out in out
2*Social WeChat out out in out 2-6
Tencent QQ in (vuln) out in out
* Further description of the file-opening behavior is given in each parenthesis. They include opening the file using a web link (web), without JavaScript support (noJS), leading to a vulnerability (vuln), under a custom scheme (dbcache), and as a text file (text).
We compare the file-opening behavior in the iOS and Android versions of representative apps in different categories.
We choose two file types, HTML and PDF, for their ability to carry attack vectors.
We find that most of the tested iOS apps open the HTML files by themselves.
According to Table <ref>, most of the tested iOS apps open HTML files by themselves.
In contrast, the corresponding Android versions choose to let dedicated apps (e.g., browsers) handle the HTML files.
The PDF files have similar results.
Opening untrusted files within the app's own domain is thus a common practice in iOS apps, whereas Android apps generally ask dedicated apps to open files.
However, Google Drive and WeChat for iOS also require explicit user actions to open HTML files outside the app. But similar to other iOS apps, they open PDF files internally.
This common practice produces more attack surfaces for iOS apps than their Android versions.
Asking dedicated apps to handle untrusted files is a more secure design, because potential attack vectors are then kept away from the user's private files.
The tested Android apps generally use this practice, which makes them immune to attacks.
Hence, attacks on Android are nearly local attacks that force files opened in exposed browsing interfaces, which only affects browser apps and careless apps.
In contrast, iOS cases are more pervasive and span multiple app categories.
In contrast, iOS cases are more pervasive and span multiple app categories (see Table <ref>).
They are also more powerful and can be local, web, or remote attacks, as they do not necessarily require locally exposed components.
There are many possible reasons for iOS's this design.
We believe that the lack of flexible data sharing on iOS is an important reason responsible for the apps to open files by themselves.
Indeed, the iOS data sharing involves a non-lightweight process of “exporting and importing,” possibly due to the lack of public SD cards and a content URI mechanism, both of which are supported on Android.
Implication 2:
The randomized app data directory on iOS makes it difficult to conduct the attacks on iOS.
The attacks usually require the knowledge of a full file path.
However, iOS assigns a random directory for each app's data zone, which makes it difficult for a remote attacker to construct the full path of a target file.
Moreover, this iOS randomness is performed at every installation. Therefore, the directory of an app reinstalled on the same phone will be different after each new installation.
An example of a randomized app directory is 3570E343-2A5A-484E-BC86-7B3CC611D634, with the unified path prefix /private/var/mobile/Applications/ (on iOS 7) and /private/var/mobile/Containers/Bundle/Application/ (on iOS 8).
In comparison, Android names app data directories according to the app package name.
An adversary can easily construct the app directory using the pattern /data/data/packagename/.
As apps generally do not use their own randomness within this directory, it is straightforward to obtain the full file path.
We only find one exception: Firefox uses a random path strategy in its Android app design.
An example of its full file path is /data/data/org.mozilla.firefox/files/mozilla/62x7scuo.default with the randomized directory underlined.
As randomizing the app directory is useful for thwarting the attacks on iOS, we recommend the Android developers to use this practice in their own app design.
Implication 3:
Apple's strict app review prevents iOS apps from executing bash commands. An adversary therefore cannot find targets to launch the attacks on iOS.
As stated in <cit.>, Apple has strict regulations for reviewing iOS apps submitted to the App Store.
Apple's app review guidelines <cit.> briefly describe many scenarios that can lead to an app rejection.
Among them, rule 2.8 states:
Apps that install or launch other executable code will be rejected.
This rule implies that running interpreted codes (e.g., bash scripts) is forbidden by Apple.
This rule implies that running interpreted codes (e.g., bash scripts) is forbidden by Apple, as discussed by <cit.>.
We thus cannot locate any iOS apps that contain command interpreters, which is a necessary condition for launching the attacks.
Although a few iOS apps (e.g., ipash ME) claim that they provide command execution for iOS, they actually only mimic the output and do not run the native commands.
They therefore receive customer reviews saying “It's a fake command line.”
To sum up, this iOS restriction makes it nearly impossible to launch the attacks on iOS, because there are no suitable app targets in the App Store.
Implication 4:
iOS generally does not allow background server behavior, which reduces the chance of the attacks on iOS.
The success of launching the attacks depends on whether the adversary can attack victim apps when the phone screen is off or locked.
If victim apps do not support background servers, then the attack timing window is shortened, thus reducing the chance of a successful attack.
The evaluation in Section <ref> indicates that iOS server apps usually only work in the foreground.
Of the top five iOS server apps, only Air Transfer can be attacked when the screen is off.
In contrast, all of the top five Android apps support background server behavior and are thus exploitable in the same phone setting.
We find that it is not easy for iOS developers to implement background server behavior.
They require advanced tricks <cit.> and have to worry about violating Apple app review policies.
Thus, developing an app server that can run in the background is uncommon on iOS.
§ MITIGATION METHODS
Application-specific defenses are required to mitigate existing risks.
Developers can refer to Section <ref> for avoiding the same flaws shown in the existing vulnerable apps.
System flaws in Android and iOS, such as the SOP flaw mentioned in Section <ref>, should be also timely fixed.
Four implications in Section <ref> will be useful to improve both app and system security at different levels.
For example, it is prudent for iOS apps not to open untrusted files in their own app domain and instead to ask dedicated apps to handle them.
We now offer two more suggestions, NoJS and AuthAccess, to further mitigate attacks.
* NoJS: disabling JavaScript execution in local schemes to safeguard against the and attacks. The attacks based on file upload CSRF can be similarly stopped by opening uploaded files as plain texts.
* AuthAccess: restricting commands and network requests to access apps' private file zone. Each access should be explicitly authorized by users. By doing so, the and attacks can be mitigated.
Beyond vendors' own ad hoc fixes, a central mitigation solution is desirable.
A possible way is to extend the existing SEAndroid MAC system <cit.> by leveraging the fine-grained context information to differentiate attack requests from normal requests.
Prior works <cit.> have shown how to collect and enforce process-related context for tackling local permission leak attacks.
The local attacks can be handled in a similar way.
However, it is challenging for context-based enforcement to mitigate the remote attacks because remote entities are usually not under defender's control and thus their context cannot be easily obtained.
To address this problem, recently proposed user-driven and content-based access control <cit.> may be useful.
We will investigate how to leverage them to develop an enhanced context-based MAC system for the attack mitigation in our future work.
§ RELATED WORK
File leaks in mobile apps.
Compared with the IFLs studied in this paper, direct file leaks are a more straightforward type of file leak.
Many direct leaks in mobile apps have been reported <cit.>.
Many of these leaks were caused by the setting an insecure (e.g., world-readable) permission for its private files in the apps.
For example, Opera <cit.> and Lookout <cit.> have made this error.
For example, Skype <cit.>, Opera <cit.>, Lookout <cit.>, and Firefox <cit.> have made this mistake.
On the other hand, the victim app writes sensitive files to public storage (e.g., SD cards and system debug logs).
Outlook <cit.> and Evernote <cit.> put their users at risk in this way.
The recent SEAndroid MAC system defends against these direct leaks, whereas our is still an unsolved threat.
Moreover, these direct cases are just local leaks on Android, whereas we propose multiple forms of remote file leaks across both Android and iOS.
Some attack instances have been studied before, but they focus only on their specific problems.
For example, Zhou et al. <cit.> study an attack with exposed Android content providers as confused deputies.
By issuing unauthorized database queries to these components, an adversary can steal victim apps' database files.
This attack is one kind of local attacks and only exists on Android.
Another example is our previous FileCross attacks <cit.>, which belong to the attacks discussed in Section <ref>.
However, it <cit.> only shows local attacks on Android, whereas we demonstrate that issues also exist in a number of iOS apps and can be remotely exploited.
It is worth noting that a blog post <cit.> reported two local issues in Dropbox and Google Drive iOS apps over two years ago, but did not mention how to deliver the attack vectors (as we do in Section <ref>).
This blog post used the old iOS systems before the recent iOS 7 and 8 to test the problem and did not show that this issue is widespread in the current iOS ecosystem.
We are also the first to identify its fundamental cause: the legacy web SOP does not adequately cover the local schemes.
Compared with all of these isolated studies, we are the first to systematically study both local and remote attacks across Android and iOS.
Confused deputy problems on mobile.
The attack is a class of the general confused deputy problem <cit.>.
A number of previous works have studied the permission-related confused deputy problem on Android, called permission leak or privilege escalation <cit.>.
They have proposed detection systems based on control- and data-flow analysis, including ComDroid <cit.>, Woodpecker <cit.>, CHEX <cit.>, ECVDetector <cit.>, Epicc <cit.>, and SEFA <cit.>.
Some Android app analysis frameworks, such as FlowDroid <cit.> and Amandroid <cit.>, can be extended to detect this problem.
However, it is difficult for these static tools to analyze the vulnerabilities, because most of the attacks do not have explicit vulnerable code patterns.
Using dynamic analysis tools (e.g., <cit.>) to construct automatic detectors is therefore desirable.
We leave this for our future work.
Furthermore, none of the aforementioned studies identify confused deputy problems on iOS.
But as shown in our attacks, it is equally, if not more, important to develop detection tools for iOS.
To mitigate the privilege escalation, researchers have devoted efforts to design several solutions <cit.>.
Access control frameworks, such as FlaskDroid <cit.> and ASM <cit.>, could be also plugged with corresponding defense policies.
They address the problem from different aspects, but all require extracting context information—inter-component communication caller-callee relationship—to defeat untrusted requests attacking confused deputies.
Such context-based approaches is mainly effective to thwart local confused deputy attacks.
It is hard for them to stop remote attacks (e.g., our and attacks), because remote entities are usually not under defender's control (i.e., no suitable context could be extracted from them).
To bypass this fundamental limitation, the recent user-driven and content-based access control <cit.> might be new directions.
We will investigate how to leverage them to mitigate our attacks in the further research.
Mobile browser security.
Mobile browser security and SOP.
Our and attacks are related to mobile browser security.
Related works have studied the threats to Android WebView <cit.>, the security risks in HTML5-based mobile libraries <cit.> and apps <cit.>, and unauthorized origin crossing in several popular Android and iOS apps <cit.>.
Differently, our focus is file leaks via vulnerable browser components.
Only the aforementioned work <cit.> shares the same goal as our study.
In addition, a prior work <cit.> injects unauthorized JavaScripts into HTML5-based apps, and their technique is similar to our attacks.
However, as their goal is to compromise the victim website's online credentials (instead of our goal of stealing local files), they do not need to overcome our challenge of launching local file-stolen attacks from a web origin due to the SOP restriction.
Moreover, our attacks apply to all mobile apps that contain browser components, rather than just the HTML5-based apps shown in <cit.>.
In Section <ref>, we call for the standardization of enhanced SOP (that considers the “path” ingredient into the current tuple of scheme, domain, and port) for local schemes.
We find some modern browsers have adopted this principle, e.g., the full support in Chrome and partial support in Firefox <cit.>.
Unfortunately, no formal standard has been drafted or even no previous efforts have been made to call for such standardization, to the best of our knowledge.
The legacy SOP was invented years ago to protect web resources at that time, so it is hard to thwart the attacks under new threat models.
For example, two enhanced SOPs were proposed to thwart dynamic pharming attacks <cit.> and DNS rebinding attacks <cit.>, respectively.
We believe the enhance SOP for local schemes is also necessary.
§ CONCLUSION
In this paper, we systematically studied indirect file leaks (IFLs) in mobile applications.
In particular, we devised four new attacks that exploit browser interfaces, command interpreters, and embedded app servers to leak private files from popular apps.
Unlike the previous attacks that work only on Android, we demonstrated that these IFLs (three of them) can affect both Android and iOS.
The vulnerable apps include Evernote, QQ, and Mail.Ru on iOS, and Baidu Browser, 360 Mobile Safe, and Terminal Emulator on Android.
Moreover, we showed that our four attacks can be launched remotely, without implanting malicious apps in victim's smartphones.
This remote attack capability significantly increases the impact of the attacks.
Finally, we analyzed the differences between Android and iOS in terms of the attacks' impacts and proposed several methods to mitigate the attacks.
We thank all three anonymous reviewers for their helpful comments.
This work was partially supported by a grant (ref. no. ITS/073/12) from the Innovation Technology Fund in Hong Kong.
Additional materials.
We will provide supplementary materials, such as detailed vulnerability reports, at this link (<https://daoyuan14.github.io/pp/most15.html>).
|
1511.00468
|
Real Options and Threshold Strategies
Real Options and Threshold Strategies
Central Economics and Mathematics Institute, Russian Academy of Sciences,
Nakhimovskii pr. 47, 117418 Moscow, Russia
Lecture Notes in Computer Science
Authors' Instructions
The paper considers an investment timing problem appearing in real options theory. Present values from an investment project are modeled by general diffusion process. We prove necessary and sufficient conditions under which an optimal investment time is induced by threshold strategy. We study also the conditions of optimality of threshold strategy (over all threshold strategies) and discuss the connection between solutions to investment timing problem and to free-boundary problem.
§ INTRODUCTION
One of the fundamental problems in real options theory concerns the determination of optimal time for investment into a given project (see, e.g., classical monograph <cit.>).
Let us consider an investment project, for example, a creating of new firm in the real sector of economy. This project is characterized by a
pair $(X_t ,t \ge 0,\,\,I)$, where $X_t$ is a present value of the
firm created at time $t$, and $I$ is a cost of investment required to implement the project (for example, to create the firm). Prices on input and output production are assumed to be stochastic, so $X_t$ is considered as a stochastic process,
defined at a probability space with filtration $(\Omega, \F,
\{\F_t, t \ge 0\},{\bf P})$. This model supposes that:
- at any moment, a decision-maker (investor) can either
accept the project and proceed with the investment or
delay the decision until he obtains new information;
investment are considered to be instantaneous and irreversible so
that they cannot be withdrawn from the project any more and used
for other purposes.
The investor's problem is to evaluate the project and to determine an
appropriate time for the investment (investment timing problem). In real option theory investment times are considered as stopping times (regarding to flow of $\s$-algebras $\{\F_t, t \ge 0\}$).
In real options theory there are two different approaches to solving investment timing problem (see <cit.>).
The value of project under the first approach is the maximum of
net present value (NPV) from the implemented project over all stopping times (investment rules):
F = max_τ(X_τ-
where $\r$ is the given discount rate. An optimal stopping
time $^*$ in (\ref{1}) is viewed as optimal investment time
(investment rule).
Within the second approach an opportunity to invest is considered
as an American call option -- the right but not obligation to buy the asset on
predetermined price. At that an exercise time is viewed as
investment time, and value of option is accepted as a value of
investment project. In these framework a project is spanned with
some traded asset, which price is completely correlated
with present value of the project $X_t$.
In order to evaluate a (rational) value of this real option one can use methods of financial options pricing theory, especially, contingent claims analysis (see, e.g., \cite{DP}).
In this paper we follow the approach that optimal investment timing decision can be mathematically determined as a solution of an optimal stopping problem (\ref{1}). Such an approach started from the well-known McDonald--Siegel model (see \cite{MS}, \cite{DP}), in which the underlying present value's dynamics is modeled by a geometric Brownian motion. The majority of results on this problem (optimal investment strategy) has a threshold structure: to invest when present value from the project exceeds the certain level (threshold).
In the heuristic level this is so for the cases of geometric Brownian motion, arithmetic Brownian motion, mean-reverting process and some other (see \cite{DP}). And the general question arises: For what underlying processes an optimal decision to an investment timing problem will have a threshold structure?
Some sufficient conditions in this direction was obtained in \cite{Al}. In this paper we focus on necessary and sufficient conditions for optimality of threshold strategies in investment timing problem. Since this problem is a special case of optimal stopping problem, the similar question may be addressed to a general optimal stopping problem: Under what conditions (on both process and payoff function) an optimal stopping time will have a threshold structure? Some results in this direction (in the form of necessary and sufficient conditions) were obtained in \cite{AS}, \cite{CM}, \cite{A} under some additional assumptions on underlying process and/or payoffs.
The paper is organized as follows. After a formal description of investment timing problem and assumptions on underlying process (Section 2.1) we go to study of threshold strategies for this problem. Since an investment timing problem in threshold strategies is reduced to one-dimensional maximization, a related problem is to find an optimal threshold. In Section 2.2 we give necessary and sufficient conditions for optimal threshold (over all thresholds).
Solving a free-boundary problem (based on smooth-pasting principle) is the most accepted method (but not the only method, see, e.g., \cite{PS}) that allows to find a solution to optimal stopping problem. In Section 2.3 we discuss the connection between solutions to investment timing problem and to free-boundary problem. Finally, in Section 2.4 we prove the main result on necessary and sufficient conditions under which an optimal investment time is generated by threshold strategy.
\section{Investment Timing Problem}
Let $I$ be a cost of investment required for implementing a project, and $X_t$ is the present value from the project started at time $t$. As usual investment supposed to be instantaneous and irreversible, and the project --- infinitely-lived.
At any time a decision-maker (investor) can either {\it accept} the project and proceed with the investment or
{\it delay} the decision until he/she obtains new information
regarding its environment (prices of the product and resources,
demand etc.). The goal of a decision-maker in this
situation is to find, using the available information, an
optimal time for investing the project (investment timing
problem), which maximizes the net present value from the project:
\beq
\label{optinv0}
{\Ex}\left(X_\t - I\right) e^{-\r \t}\II_{\{\t{<}\infty\}} \to \max_{\t\in\M},
\eeq
where $$ means the expectation for the process $X_t$ starting from the
initial state $x$, $_A$ is indicator function of the set $A$,
and the maximum is considered over stopping times $$ from a certain class $$ of stopping times\footnote{In this paper we consider stopping times which can take infinite values (with positive probability)}.
We consider the case $I<r$ else the optimal time in (\ref{optinv0}) will be $+∞$.
\subsection{Mathematical Assumptions}
Let $X_t$ be a diffusion process with values in the interval $D⊆^1$ with boundary points $l$ and $r$, where $ -∞≤l< r≤+∞$, open or closed (i.e. it may be $(l,r)$,\, $[l,r)$,\, $(l,r]$,\, or $[l,r]$), which is a solution to stochastic differential equation:
\beq \label{infop1}
dX_t=a(X_t)dt + \s(X_t) dw_t,\quad X_0=x,
\eeq
where $w_t$ is a standard Wiener process, $a: D →^1$ and $: D →^1_+$ are the drift and diffusion functions, respectively. Denote $=int (D)=(l,r)$.
The process $X_t$ is assumed to be regular; this means
that, starting from an arbitrary point $x∈$, this process
reaches any point $y∈$ in finite time with positive
It is known that the following local integrability condition:
\beq \label{regular}
\int_{x-\varepsilon}^{x+\varepsilon}\frac{1+|a(y)|}{\s^2(y)} dy <\infty \quad \mbox{for some }\varepsilon>0,
\eeq
at any $x∈$ guarantees the existence of weak solution of equation (\ref{infop1}) and its regularity (see, e.g. \cite{KS}).
\smallskip
The process $X_t$ is associated with infinitesimal operator
\beq\label{operator}
\L f(x)= a(x)f'(x) +\frac12\s^2(x) f''(x).
\eeq
Under the condition (\ref{regular})
there exist (unique up to constant positive multipliers) increasing and decreasing functions $ψ(x)$ and $φ(x)$ with absolutely continuous derivatives, which are the fundamental solutions to the ODE
\beq\label{diffur}
\L f(p)=\r f(p)
\eeq
almost sure (in Lebesque measure) on the interval $$ (see, e.g. \cite[Chapter 5, Lemma 5.26]{KS}). Moreover, $0<ψ(p), φ(p)<∞$ for $p∈$.
Note, if functions $a(x), (x)$ are continuous, then \ $ψ, φ∈C^2()$.
\subsection{Optimality of Threshold Strategies}
Let us define $=(x)=inf{t≥ 0: X_t≥p}$ --- the first time when the process $X_t$, starting from $x$, \emph{exceeds} level $p$. We will call $$ as threshold stopping time generated by the threshold strategy --- to stop when the process exceeds threshold $p$. Let $_th={, p∈}$ be a class of all such threshold stopping times.
For the class $_th$ of threshold stopping times the investment timing problem (\ref{optinv0}) can be written as follows:
\beq
\label{optinv1}
\left(p - I\right) {\Ex}e^{-\r \tp} \to \max_{p\in(l,r)}.
\eeq
Such a problem appeared in \cite{DPS} as the heuristic method for solving a general investment timing problem (\ref{optinv0}) over class of all stopping times.
We say that threshold $p^*$ is optimal for the investment timing problem (\ref{optinv1}) if threshold stopping time $_p^*$ is optimal in (\ref{optinv1}).
The following result gives necessary and sufficient conditions for optimal threshold.
\begin{theorem}
Threshold $p^*\in\I$ is optimal in the problem \emph{(\ref{optinv1})} for all $x\in \I$, if and only if the following conditions hold:
\begin{eqnarray}
&& \frac{p-I}{\psi(p)}\le \frac{p^*-I}{\psi(p^*)}\quad \mbox{\rm whenever } p<p^*; \label{criteria0}\\
&& \frac{p-I}{\psi(p)}\quad \mbox{\rm does not increase for } p\ge p^* \label{criteria00},
\end{eqnarray}
where $\psi(p)$ is an increasing solution to ODE \emph{(\ref{diffur})}.
\end{theorem}
\begin{proof} Let us denote the left-hand side in (\ref{optinv1}) as $V(p;x)$.
Obviously, $V(p;x)=x-I$ for $x\ge p$.
Along with the above stopping time let us define the first hitting time to threshold: $T_p= \inf\{t{\ge} 0: X_t= p\},\ p\in (l,r)$.
For $x<p$, obviously, $\tp=T_p$ and using known formula $\ds \Ex e^{-\r T_p}= \psi(x)/\psi(p)$ (see, e.g., \cite{IM}, \cite{BS})we have:
\beq\label{Vpx}
V(p;x)=(p-I)\Ex e^{-\r\tp} \II_{\{\tp{<}\infty\}}=(p-I)\Ex e^{-\r T_p} =\frac{p-I}{\psi(p)}\psi(x).
\eeq
Denote $h(x)=(x-I)/\psi(x)$.
i) Let $p^*\in\I$ be an optimal threshold in the problem (\ref{optinv1}) for all $x\in \I$. Then for $p<p^*$ we have
V(p;p)=p-I\le V(p^*;p)=\frac{p^*-I}{\psi(p^*)} \psi(p),
i.e. (\ref{criteria0}) holds. If $p^*\le p_1 < p_2$, then
V(p_2;p_1)=h(p_2) \psi(p_1)\le V(p^*;p_1)= p_1-I=
h(p_1) \psi(p_1),
and it follows (\ref{criteria00}).
ii) Now, let (\ref{criteria0})--(\ref{criteria00}) hold.
Let $p<p^*$.
If $x \ge p^*$, then $V(p;x)= x-I=
If $p\le x<{p^*}$, then, due to (\ref{criteria0}), $V_p(x)=x-I=h(x)\psi(x) \le h(p^*)\psi(x)= V(p^*;x)$.
Finally, if $x<p$, then, using (\ref{criteria0}) and (\ref{Vpx}), we have: $V(p;x)= h(p)\psi(x)\le h(p^*)\psi(x)= V(p^*;x)$.
Consider the case $p>p^*$. If $x\ge p$, then $V(p;x)= x-I=
Whenever $p^*\le x < p$, then, due to (\ref{criteria00}),
$V(p;x)= h(p)\psi(x)\le h(x)\psi(x)= x-I=V(p^*;x)$.
When $x < p^*$, then $V(p;x)= h(p)\psi(x)\le h(p^*)\psi(x)= V(p^*;x)$, since $h(p)\le h(p^*)$.
Theorem is completely proved.
\end{proof}
\begin{remark} The condition (\ref{criteria00}) is equivalent to the inequality
(p -I) \psi'(p) \ge \psi(p) \quad {\rm for } \ p\ge p^*.
This relation implies, in particular, that optimal threshold $p^*$ must be strictly greater than investment cost $I$ (because $\psi(p^*),\ \psi'(p^*)$ are positive values).
\end{remark}
\begin{remark} Assume that $\log \psi(x)$ is a convex function, i.e. $\psi'(x)/\psi(x)$ increases. For this case there exists a unique point $p^*$ which satisfies the equation
\beq\label{smooth}
(p^* -I) \psi'(p^*) = \psi(p^*)
\eeq
and constitutes the optimal threshold in the problem (\ref{optinv1}) for all $x\in \I$. Indeed, the sign of derivative of the function $(p-I)/\psi(p)$ coincides with the sign of $\psi(p)-(p -I) \psi'(p)$. Therefore, in the considered case the conditions (\ref{criteria0})--(\ref{criteria00}) in Theorem 1 are true automatically.
\end{remark}
We can give a number of cases of diffusion processes which are more or less realistic for modeling a process $X_t$ of present values from a project. Some of them are presented below.
\smallskip
1) \emph{Geometric Brownian motion (GBM)}:
\beq \label{GBM}
dX_t=X_t(\a dt + \s dw_t).
\eeq
For this case $ψ(x)=x^$̱, where $\b$ is the positive root of the equation\\ $1/2^2(̱-̱1)+-̱=̊ 0$.
2) \emph{Arithmetic Brownian motion (ABM)}:
\beq \label{ABM}
dX_t=x+ \a dt + \s \,dw_t.
\eeq
For this case $ψ(x)=e^x̱$, where $$̱ is the positive root of the equation
$\ds \textstyle\frac12\s^2\b^2+\a\b-\r = 0$.
3) Mean-reverting process (or geometric Ornstein–Uhlenbeck process):
dX_t=(x̅-X_t)X_tdt + X_t dw_t.
For this case $\ds \psi(x)=x^\b {}_1\! F_1\left(\b,2\b+\frac{2\a \bar x}{\s^2}; \frac{2\a}{\s^2}x\right) $, where $\b$ is the positive root of equation $1/2^2(̱-̱1)+x̅-̱=̊ 0$, and $_1 F_1(p,q;x)$ is confluent hypergeometric function satisfying Kummer's equation $xf”(x)+(q-x)f'(x)-pf(x)=0$.
4) \emph{Square-root mean-reverting process \ (or Cox--Ingersoll--Ross process)}:
\beq \label{SRMRP}
dX_t=\a (\bar x-X_t)dt + \s \sqrt {X_t}\, dw_t.
\eeq
For this case $ψ(x)= _1F_1(/,2x̅/^2; 2/^2x) $.
\smallskip
The above processes are well studied in the literature (in connection with real options and optimal stopping problems see, for example, \cite{DP}, \cite{JZ}).
For the first two processes (\ref{GBM}) and (\ref{ABM}) the conditions of Theorem 1 give explicit formulas for optimal threshold in investment timing problem:
p^*=\frac{\b}{\b-1}I \ \ {\rm for \ GBM}, \quad {\rm and } \ \ p^*=I+\frac1{\b} \ \ {\rm for \ ABM}.
On the contrary, for mean-reverting processes (\ref{MRP}) and (\ref{SRMRP}) the function $ψ(x)$ is represented as infinite series, and optimal threshold can be find only numerically.
\smallskip
So, Theorem 1 states that optimal threshold $p^*$ is a point of maximum for the function $h(x)=(x-I)/ψ(x)$. This implies the first-order optimality condition $h'(p^*)=0$, i.e. the equality (\ref{smooth}), and smooth-pasting principle:
\ds V'_x(p^*;x)\bigr|_{x=p^*}=1.
In the next section we discuss smooth-pasting principle and appropriate free-boundary problem more closely.
\subsection{Threshold Strategies and Free-Boundary Problem}
It is almost common opinion (especially among engineers and economists) that solution to free-boundary problem always gives a solution to optimal stopping problem.
A free-boundary problem for the case of threshold strategies in investment timing problem can be written as follows:
to find threshold $p^*∈(l,r)$ \ and twice differentiable function $H(x), l<x<p^*$, such that
\begin{eqnarray}
&&\L H(x)=\r H(x), \quad l<x<p^*;\label{fbp1} \\
&& H(p^*{-}0)= p^* -I, \quad H'(p^*{-}0)=1 \label{fbp2}.
\end{eqnarray}
If $ψ(x)$ is twice differentiable, then solution to the problem (\ref{fbp1})--(\ref{fbp2}) has the type
\beq\label{fbp0}
H(x)= \frac{p^*-I}{\psi(p^*)}\psi(x),\quad l<x<p^*,
\eeq
where $ψ(x)$ is an increasing solution to ODE (\ref{diffur}) and $p^*$ satisfies the smooth-pasting condition (\ref{smooth}). In further, we will call such $p^*$ the solution to free-boundary problem.
According to Theorem 1 the optimal threshold in problem (\ref{optinv1}) must be a point of maximum of the function $h(x)=(x-I)/ψ(x)$, but smooth-pasting condition (\ref{smooth}) provides only a stationary point for $h(x)$. Thus, we can apply standard second-order optimality conditions to derive relations between solutions to investment timing problem and to free-boundary problem.
Let $p^*$ be a solution to free-boundary problem (\ref{fbp1})--(\ref{fbp2}). If $p^*$ is also an optimal threshold in investment timing problem (\ref{optinv1}), then, of course, $h”(p^*)≤0$. It means that
\psi''(p^*)=-\frac{h''(p^*)\psi(p^*)+ 2h'(p^*)\psi'(p^*) }{h(p^*)}= -\frac{h''(p^*)\psi(p^*)}{h(p^*)} \ge 0.
Thus, the inequality $ψ”(p^*)≥0$ may be viewed as necessary condition for a solution of free-boundary problem to be optimal in investment timing problem.
The inverse relation between solutions can be state as follows.
\medskip
\noindent {\bf Statement 1.} {\it If \ $p^*$ is the unique solution to free-boundary problem \emph{(\ref{fbp1})--(\ref{fbp2})}, and $\psi''(p^*)> 0$, then $p^*$ is optimal threshold in the problem \emph{(\ref{optinv1})} for all $x\in \I$.
\begin{proof} \ Since $h'(p^*)=0$ and $\psi''(p^*)> 0$ then $h''(p^*)=-h(p^*)\psi''(p^*)/\psi(p^*){<}0$. Therefore, $h'(p)$ strictly decreases at some neighborhood of $p^*$.
Then, it is easy to see that $h'(p)>0$ for $p<p^*$ and $h'(p)<0$ for $p>p^*$, else $h'(q)=0$ for some $q\neq p^*$, that contradicts to the uniqueness of solution to free-boundary problem (\ref{fbp1})--(\ref{fbp2}). So, conditions (\ref{criteria0})--(\ref{criteria00}) hold and Theorem 1 gives the optimality of threshold ${p^*}$.
\end{proof}
The following result concerns the general case when free-boundary problem has several solutions.\medskip
\noindent{\bf Statement 2.} {\it Let \ $p^*$ and \ $\tilde p$ \ are two solutions to free-boundary problem \emph{(\ref{fbp1})--(\ref{fbp2})}, such that \ $\psi''(p^*)> 0$ and $(x-I)/\psi(x)\le (p^*-I)/\psi(p^*)$ for $l{<}x{<} p^*$. If $\tilde p >p^*$, and
$\psi^{(k)}(\tilde p)=0$ \ $(k=2,...,n-1)$, $\psi^{(n)}(\tilde p)> 0$ for some $n>2$,
then ${p^*}$ is optimal threshold in the problem \emph{(\ref{optinv1})} for all $x\in \I$.
\medskip
\begin{proof} \
Let us prove that $h'(p)\le 0$ for all $p>p^*$. Inequality $\psi''(p^*)> 0$ implies (as above) that $h''(p^*)< 0$, and, therefore, $h'(p)< 0$ for all $p^*<p<p_1$ with some $p_1$. If we suppose that $h'(p_2)>0$ for some $p_2 >p^*$, then there exists $p_0\in (p_1,p_2)$ such that $h'(p_0)=0$ and $h'(p)>0$ for all $p_0<p<p_2$. Therefore, $p_0$ is another solution to free-boundary problem (\ref{fbp1})--(\ref{fbp2}), and due to conditions of the Statement $h^{(k)}(p_0)=0$ \ $(k=2,...,n-1)$, $h^{(n)}(p_0)< 0$ for some $n>2$, that contradicts to positivity of $h'(p)$ for $p_0<p<p_2$.
Hence, $h'(p)\le 0$ for all $p>p^*$ and conditions (\ref{criteria0})--(\ref{criteria00}) hold. Thus, according to Theorem 1, ${p^*}$ is optimal threshold in the problem (\ref{optinv1}).
\end{proof}
\medskip
\subsection{Optimal Strategies in Investment Timing Problem}
Now, return to `general' investment timing problem (\ref{optinv0}) over the class $$ of \emph{all stopping times}.
The sufficient conditions under which an optimal investment time in (\ref{optinv0}) will be a threshold stopping time were derived in \cite{Al}. In this section we give necessary and sufficient conditions (criterion) for optimality of threshold stopping time in investment timing problem (\ref{optinv0}).
To reduce some technical difficulties we assume below that drift $a(x)$ and diffusion $(x)$ of the underlying process $X_t$ are continuous functions.
\begin{theorem} \
Threshold stopping time $\t_{p^*}$, $p^*{\in} (l,r)$, is optimal in the investment timing problem \emph{(\ref{optinv0})} for all $x{\in}\, \I$ if and only if the following conditions hold:
\begin{eqnarray}
(p-I){\psi(p^*)}\le (p^*-I){\psi(p)}\quad \mbox{\rm for } p<p^*;\label{criteria110}\\[3pt]
&& \psi(p^*)=(p^*-I)\psi'(p^*);
\label{criteria11}\\
&& a(p)\le \r (p- I) \quad \mbox{\rm for } p>p^*, \label{criteria21}
\end{eqnarray}
where $\psi(x)$ is an increasing solution to ODE \emph{(\ref{diffur})} and $a(p)$ is the drift function of the process $X_t$.
\end{theorem}
\begin{proof}
Define the value function for the problem (\ref{optinv0}) over the class $\M$ of all stopping times
V(x)=\sup_{\t\in\M}{\Ex}\left(X_\t - I\right) e^{-\r \t}\II_{\{\t{<}\infty\}}.
i) Let conditions (\ref{criteria110})--(\ref{criteria21}) hold.
Take the function
\Phi(x)=V({p^*};x)=\left\{
\begin{array}{ll}\ds\frac{p^*-I}{\psi(p^*)}
\psi(x), & \ {\rm for }\ x<p^*,\\[8pt]
x-I, & \ {\rm for }\ x\ge p^*.
\end{array}
\right.
Obviously, $\Phi(x)>0$ (due to condition (\ref{criteria11})) and $ V(x)\ge \Phi(x)$.
On the other hand, (\ref{criteria110}) implies
\frac{p^*-I}{\psi(p^*)}\psi(x)\ge \frac{x-I}{\psi(x)}\psi(x)=x-I,
therefore $\Phi(x)\ge x-I$ for all $x\in(l,r)$, i.e. $\Phi(x)$ is a majorant of payoff function $x-I$.
For any stopping time $\t\in \M$ and $N>0$ put $\tilde\t=\t{\wedge}N$. From It\^{o}--Tanaka--Meyer formula (see, e.g. \cite{KS}) we have:
\begin{eqnarray}
\Ex \Phi(X_{\tilde\t}) e^{-\r\tilde\t}&=& \Phi (x)+\Ex \int_0^{\tilde\t}(\L\Phi -\r\Phi)(X_t)e^{-\r t}dt\nonumber\\
&+&\frac12 \s^2(p^*)[\Phi'(p^*{+}0)-\Phi'(p^*{-}0)]\Ex \int_0^{\tilde\t} e^{-\r t}dL_t(p^*),
\end{eqnarray}
where $L_t(p^*)$ is the local time of the process $X_t$ at the point $p^*$.
By definition we have
\Phi'(p^*{+}0)-\Phi'(p^*{-}0)= 1-
\frac{p^*-I}{\psi(p^*)} \psi'(p^*)= 0
due to (\ref{criteria11}).
Take $T_1=\{0\le t\le {\tilde\t}:\, X_t < p^*\}$, \ $T_2=\{0\le t\le {\tilde\t}:\, X_t > p^*\}$. We have:
\begin{eqnarray*}
&&\L\Phi(X_t) -\r\Phi(X_t) =\frac{p^*-I}{\psi(p^*)} \Bigl(\L\psi(X_t)-\r\psi(X_t)\Bigr)=0 \quad {\rm for } \ t\in T_1,\\
&&\L\Phi(X_t) -\r\Phi(X_t) =a(X_t) -\r (X_t-I)\le 0 \quad {\rm for } \ t\in T_2
\end{eqnarray*}
by definition of the function $\psi(x)$ and (\ref{criteria21})).
\begin{eqnarray*}
\Ex \Phi(X_{\tilde\t}) e^{-\r\tilde\t}&\le &\Phi (x)+\Ex \left( \int\limits_{T_1}(\L\Phi {-}\r\Phi)(X_t)e^{-\r t}dt \right.
+ \left.\int\limits_{T_2}(\L\Phi {-}\r\Phi)(X_t)e^{-\r t}dt\right)\\
&\le &\Phi (x).
\end{eqnarray*}
Since $\Phi(X_{\tilde\t}) e^{-\r \tilde\t}\stackrel{\mbox{\scriptsize a.s.}}{\longrightarrow} \Phi(X_{\t}) e^{-\r\t}\II_{\{\t{<}\infty\}}$ when $N\to \infty$, then due to Fatou's Lemma : $\ds \Ex \Phi(X_{\t}) e^{-\r\t}\II_{\{\t{<}\infty\}}\le \Phi (x)$ for all $\t\in \M$ and $x\in \I$. Therefore, $\Phi(x)$ is $\r$-excessive function, which majorates payoff function $x-I$. Since, by Dynkin's characterization, value function $V(x)$ is the least $\r$-excessive majorant, then $ V(x)\le \Phi(x)$.
Therefore, $ V(x)= \Phi(x)=V({p^*};x)$, i.e. $\t_{p^*}$ is the optimal stopping time in problem (\ref{optinv0}) for all $x$.
ii) Now, let $\t_{p^*}$ be optimal stopping time in the problem (\ref{optinv0}). Note, that $\t_{p^*}$ will be an optimal stopping time in the problem (\ref{optinv1}) also. Therefore, Theorem 1 implies (\ref{criteria110}) and (\ref{criteria11}), since $p^*$ is point of maximum for the function $(x-I)/\psi(x)$.
Further, assume that inequality (\ref{criteria21}) is not true at some point $p_0 > p^*$, i.e. $a(p)> \r (p-I)$ in some interval $J\subset (p^*,r)$ (by virtue of continuity). For some $\tilde x\in J$ define $ \t=\inf\{t\ge 0:\, X_t\notin J\}$, where process $X_t$ starts from the point $\tilde x$. Then for any $N>0$ from Dynkin's formula
\E^{\tilde x} (X_{\t{\wedge}N}-I) e^{-\r (\t{\wedge}N)}= \tilde x-I+\E^{\tilde x} \int_0^{\t{\wedge}N}[a(X_t) -\r (X_t-I)]e^{-\r t}dt >\tilde x-I.
Therefore, $V(\tilde x) > \tilde x-I$ that contradicts to $V(\tilde x)=V({p^*};\tilde x)=g(\tilde x)$, since $\tilde x>p^*$.
\end{proof}
\begin{example}
Let $X_t$ be the process of geometric Brownian motion (\ref{GBM}). Then Theorem 2 implies that threshold stopping time $\t_{p^*}$ will be optimal in the investment timing problem (\ref{optinv0}) over all investment times if and only if
where $\b$ is the positive root of the equation $\ds \textstyle\frac12\s^2\b(\b-1)+\a\b-\r = 0$.
\end{example}
\subsubsection*{Acknowledgments.} The work was supported by Russian Foundation for Basic Researches (project 15-06-03723) and Russian Foundation for Humanities (project 14-02-00036).
\begin{thebibliography}{4}
\bibitem{Al} Alvarez, L.H.R.: Reward functionals, salvage values, and optimal stopping. Math. Methods Oper. Res. 54, 315--337 (2001)
\bibitem{A} Arkin, V.I.: Threshold Strategies in Optimal Stopping Problem for One-Dimensional Diffusion Processes. Theory Probab. Appl. 59, 311--319 (2015)
\bibitem{AS} Arkin, V.I., Slastnikov A.D.: Threshold stopping rules for diffusion processes and Stefan's problem. Dokl. Math. 86, 626-–629 (2012)
\bibitem{BS} Borodin, A.N., Salminen, P.: Handbook of Brownian Motion -- Facts and Formulae. Birkhauser-Verlag (2002)
\bibitem{CM} Crocce, F., Mordecki, E.: Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions. Stochastics, 86, 491-–509 (2014)
\bibitem{DP} Dixit, A., Pindyck, R.S.: Investment under Uncertainty. Princeton University Press, Princeton (1994)
\bibitem{DPS} Dixit, A., Pindyck, R.S., S{\o}dal, S.: A Markup Interpretation of Optimal Investment Rules. The Economic Journal. 109, 179--189 (1999)
\bibitem{JZ} Johnson, T.C., Zervos, M.: A discretionary stopping problem with applications to the optimal timing of investment decisions. Preprint, \url{https://vm171.newton.cam.ac.uk/files/preprints/ni05045.pdf} (2005)
\bibitem{IM} Ito, K., McKean, H.:
Diffusion Processes and Their Sample Paths. Springer-Verlag, Berlin (1974)
\bibitem{KS} Karatzas, I., Shreve, S.~E.: Brownian Motion and Stochastic Calculus. Springer-Verlag, Berlin (1991)
\bibitem{MS} McDonald, R., Siegel, D.: The value of waiting to invest. The Quarterly Journal of Economics, 101, 707--728 (1986)
\bibitem{PS} Peskir, G., Shiryaev, A.: Optimal Stopping and Free-Boundary Problems. Birkhauser (2006)
\end{thebibliography}
\end{document}
|
1511.00010
|
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
[email protected], [email protected]
September 12, 2025
Recent hydrodynamic (HD) simulations have shown that galactic disks
evolve to reach well-defined statistical equilibrium states.
The star formation rate (SFR) self-regulates until
energy injection by star formation feedback balances
dissipation and cooling in the interstellar medium (ISM),
and provides vertical pressure support to balance gravity.
In this paper, we extend our previous models to allow for a range of initial magnetic field strengths and configurations,
utilizing three-dimensional, magnetohydrodynamic (MHD) simulations.
We show that a quasi-steady equilibrium state is established
as rapidly for MHD as for HD models unless the initial
magnetic field is very strong or very weak, which requires
more time to reach saturation.
Remarkably, models with initial magnetic
energy varying by two orders of magnitude approach the same
asymptotic state.
In the fully saturated state of the fiducial model,
the integrated energy proportions
$\Ekin:\Eth:\Emagt:\Emago$ are
while the proportions of midplane support
$\Pturb:\Pth:\Pmagt:\Pmago$ are
Vertical profiles of total effective pressure
satisfy vertical dynamical equilibrium with the
total gas weight at all heights.
We measure the “feedback yields” $\eta_c\equiv P_c/\SigSFR$
(in suitable units) for
each pressure component, finding that $\etaturb\sim 4$ and
$\etath\sim 1$ are the same for MHD as in previous HD simulations,
and $\etamagt \sim 1$.
These yields can be used to predict the equilibrium SFR for a
local region in a galaxy based on its observed gas and stellar
surface densities and velocity dispersions.
As the ISM weight (or dynamical equilibrium pressure) is fixed, an
increase in $\eta$ from
turbulent magnetic fields reduces the predicted $\SigSFR$ by
$\sim 25\%$ relative to the HD case.
§ INTRODUCTION
The diffuse atomic interstellar medium (ISM), the mass reservoir in galaxies
from which molecular clouds and stars are born, is highly turbulent
<cit.> and strongly magnetized
<cit.>. The roles of turbulence and magnetic
fields in the context of star formation have been emphasized many times
Additionally, the importance of turbulence and magnetic fields on large scales
is clear from comparison of each component's pressure to the vertical weight
of the ISM disk <cit.>. In
the Solar neighborhood, representing fairly typical conditions among star
forming galactic disks, turbulent and magnetic pressures are
roughly comparable to each
other <cit.>, and about three times larger than
the thermal pressure in diffuse gas <cit.>.
Therefore, turbulence and magnetic fields provide most of the vertical support
to the gas layer, and are essential to include in models of the vertical
structure in galactic disks. Moreover, the magnetic field has a significant
random component <cit.>, which
presumably is related to observed turbulent velocities. This implies an
intriguing additional dimension to models in which the star formation rate
(SFR) is connected to the vertical dynamical equilibrium of the ISM disk via
turbulence driven by star formation feedback
Another important characteristic of the diffuse atomic ISM is its multiphase
nature, which emerges naturally as a consequence of thermal instability with a
realistic treatment of cooling and heating <cit.>. The
temperature and density of the cold and warm medium differ by about two orders
of magnitude
<cit.>. The
inhomogeneous, multiphase structure of the diffuse atomic ISM is observed to be
In addition to the warm and cold phases, the hot phase that is created by
supernovae (SNe) is an essential ISM component
<cit.>. This
phase contains very little mass, but is of great dynamical importance because
the expansion of high-pressure SN remnants and superbubbles is the main driver
of turbulence in the surrounding denser phases of the ISM (and also drives
galactic winds)
Owing to the huge difference between cooling and dynamical times and
correspondingly large range of spatial scales, directly modeling the multiphase
ISM is numerically quite challenging. However, the combination of increasing
computational resources, with robust algorithms for evolving gas dynamics and
thermodynamics simultaneously, have enabled development of highly sophisticated
ISM models in the last decade. Direct numerical simulations of vertically
stratified galactic disks with SN driven turbulence and stiff source terms to
follow heating and cooling include those of
Among those, only <cit.> and
<cit.> have a time-dependent SFR and SN rate
self-consistently set by the self-gravitating localized collapse of gas.
The time-dependent response of the SFR to large-scale ISM dynamics and
thermodynamics is key to self-consistent regulation of turbulent and thermal
pressures, since both the turbulent driving rate (from SNe) and thermal heating
rate (from far-UV-emitting massive stars) are proportional to the SFR. If
heating rates and turbulent driving rates drop due to a low SFR, the increased
mass of cold, dense gas subsequently leads to a rebound in the SFR. The
opposite is also true, with an excessively high SFR checked by the ensuing
reduction in the mass of gas eligible to collapse. Within less than an orbital
time, the SFR self-regulates such that the ISM achieves a
statistical equilibrium state.
In this state, there are balances between (1) vertical gravity and the combined
pressure forces, (2) turbulent driving and dissipation, and (3) cooling and
heating (see
analytic theory, and
for numerical
confirmations). Paper I successfully reproduces important aspects of the
diffuse atomic ISM, demonstrating self-regulated SFR and vertical equilibrium
following the analytic theory for a range of galactic conditions including
those of the Solar neighborhood as well as regions with higher and lower total
ISM surface density. In addition, the warm/cold ratios and properties of 21 cm
emission/absorption for H1 (including column density and spin
temperature distributions) are in agreement with observations
Despite of the success of hydrodynamic (HD) models in Paper I, the dynamical
role of magnetic fields cannot be ignored. For example,
<cit.> have run magnetized ISM disk models with
self-gravity and SN feedback, showing a factor of two decrement in the SFR
compared to the HD case. However, a systematic exploration of the effect of
magnetic fields is still lacking. In this paper, a set of three-dimensional,
magnetohydrodynamic (MHD) simulations is carried out, to test the effects of
different initial magnetic field strengths and configuration.
The remainder of the paper is organized as follows. In
Section <ref>, we begin by reviewing the analytic theory and
previous simulation results to identify and explain the properties and
parameters we will measure here. Section <ref> summarizes our
numerical methods and model parameters. Our basis is the fiducial Solar
neighborhood model from Paper I with gas surface density of 10$\Surf$, but here
we slightly alter the initial and boundary conditions to minimize early
vertical oscillations that were exaggerated in Paper I.
Section <ref> contains our main new results. Time evolution of
disk diagnostics in Section <ref> shows that a quasi-steady state is
achieved. In Section <ref>, we analyze temporally and horizontally
averaged vertical profiles to confirm vertical dynamical equilibrium. Finally,
Section <ref> presents the measurements of “feedback yields,”
exploring the mutual relationship between measured SFR surface density and the
various midplane pressure supports. We compare the numerical results with
analytic expectations including magnetic terms. In Section <ref>,
we summarize and discuss the model disk properties at saturation in comparison
to observations and previous simulations, especially focusing on the level of
mean and turbulent magnetic fields.
§ THE EQUILIBRIUM THEORY
Before describing the results of our new simulations, it is informative to
summarize the analytic theory for mutual equilibrium of the ISM and star
formation in disk systems, as developed in our previous work
<cit.>. This
includes a list of the parameters to be measured for detailed investigation.
The equilibrium theory assumes a quasi-steady state that satisfies force
balance between vertical gravity and pressure support, and energy balance
between gains from star formation feedback and losses in the dissipative ISM.
We shall test the validity of these assumptions and quantify the relative
contributions to the vertical support from turbulent, thermal, and magnetic
terms. We shall also measure the efficiency of feedback (feedback yield) for
each support term.
Using the temporally- and horizontally- averaged momentum equation, it is
straightforward to show that vertical dynamical equilibrium requires a balance
between the total momentum flux difference between surfaces at $z=0$ and
$\zmax$, and the weight of the gas: $\Delta \Ptot=\mathcal{W}$. In magnetized,
turbulent galactic disks, the vertical momentum flux consists of thermal
($\Pth\equiv\rho \vth^2$), turbulent ($\Pturb\equiv\rho v_z^2$), and magnetic
($\Pmag\equiv|\mathbf{B}|^2/8\pi-B_z^2/4\pi$) terms
More generally, radiation and cosmic ray pressure terms could also contribute
<cit.>, but are not included in the present
numerical simulations]. Note that the magnetic term includes both pressure and
tension, and can be rewritten as $\Pmag=\rho(v_A^2/2-v_{A,z}^2)$,
where $v_A\equiv \left(|\mathbf{B}|^2/4\pi\rho\right)^{1/2}$ is the Alfvén
velocity from the total magentic field and $v_{A,z}=|B_z|/(4\pi \rho)^{1/2}$ is
from its $z$ component. We hereafter refer to $\Pmag$ as the magnetic
“support” to distinguish from the usual magnetic “pressure” ($P_{\rm
mag}\equiv|\mathbf{B}|^2/8\pi$), while the thermal and turbulent “supports”
are equivalent to the thermal and turbulent “pressures,” respectively. If we
choose $\zmax$ where the gas density is sufficiently small, $\Pth(\zmax)$,
$\Pturb(\zmax)\rightarrow 0$ by definition, but $\Pmag(\zmax)$ can in general
be nonzero and significant. We thus have
\begin{equation}\label{eq:Ptot}
\Delta \Ptot= {\Pth}_{,0}+{\Pturb}_{,0}+\Delta\Pmag
\equiv \rho_0 \sigma_z^2(1+\mathcal{R}),
\end{equation}
where $\sigma_z^2\equiv \vth^2+v_z^2$ is the sum of thermal and turbulent
velocity dispersions, and
\begin{equation}\label{eq:R}
\mathcal{R}\equiv \frac{\Delta\Pmag}{\rho_0 \sigma_z^2}
\end{equation}
is the relative contribution to the vertical support from magnetic to kinetic
(thermal plus turbulent) terms. If $\mathbf{B}(\zmax)\rightarrow 0$, $\Delta
\Pmag \rightarrow (|\mathbf{B}_0|^2/2 -B_{z,0}^2)/4\pi$. Here and hereafter,
we use the subscript `0' to indicate quantities evaluated at the midplane
The weight of the gas under self- and external gravity is respectively defined
\begin{equation}\label{eq:wsg}
\Wsg = \int_0^{\zmax} \rho \frac{d\Phi_{\rm sg}}{dz} dz = \frac{\pi G\Sigma^2}{2}
\end{equation}
\begin{equation}\label{eq:wext}
\Wext = \int_0^{\zmax} \rho \gext(z) dz \equiv \zeta_d\gext(H)\Sigma
\end{equation}
Here, $\Sigma\equiv \int_{-\infty}^\infty \rho dz$ is the gaseous surface
density, $H\equiv \Sigma/(2\rho_0)$ is the gaseous disk's effective scale
height, and $\zeta_d\equiv (1/2)\int_{0}^{\zmax} \rho
|\gext(z)/\gext(H)|dz/\int_{0}^{\zmax} \rho dz$ is a dimensionless parameter
that characterizes the vertical distribution of gas and the shape of external
gravity profile. For example, with linear external gravity profile
($\gext\propto z$), density following exponential, Gaussian, and sech$^2$
distributions gives $\zeta_d=1/2$, $1/\pi$, and $(\ln2)/2$, respectively.
We now specialize to the case of a galactic disk with a midplane density of
stars + dark matter equal to $\rhosd$, for which the external gravity near the
midplane is $g_\mathrm{ext}=4 \pi G \rhosd z$. Vertical dynamical equilibrium
can be expressed as
\begin{equation}\label{eq:de}
\rho_0 \sigma_z^2(1+\mathcal{R})=\Wsg(1+\chi),
\end{equation}
\begin{equation}\label{eq:chidef}
\chi\equiv \frac{\Wext}{\Wsg} =\frac{4\zeta_d\rhosd}{\rho_0}
\end{equation}
is the ratio of external to self gravity. By substituting from Equation
(<ref>) for $\Wsg$ and from Equation (<ref>) for $\chi$ in
Equation (<ref>), we can solve to obtain
\begin{equation}\label{eq:H}
H = H_{\rm sg}\frac{1+\mathcal{R}}{1+\chi}
= H_{\rm ext}\rbrackets{\frac{1+\mathcal{R}}{1+1/\chi}}^{1/2},
\end{equation}
where $H_{\rm sg}\equiv\sigma_z^2/\pi G\Sigma$ and $H_{\rm ext}\equiv
\sigma_z/(8\pi G\zeta_d\rhosd)^{1/2}$ are the scale heights for self- and
external- gravity dominated cases, respectively. By defining
\begin{equation}\label{eq:Cdef}
C\equiv \frac{H_{\rm sg}^2}{H_{\rm ext}^2}=\frac{8\zeta_d \rhosd \sigma_z^2}{\pi G\Sigma^2},
\end{equation}
Equation (<ref>) can also be solved for $\chi$ to obtain
\begin{equation}\label{eq:chi}
\chi=\frac{2C(1+\mathcal{R})}{1+\sqrt{1+4C(1+\mathcal{R})}}.
\end{equation}
If we neglect dark matter and consider a stellar disk with surface density
$\Sigma_*$ and vertical stellar velocity dispersion $\sigma_*$ such that
$\rhosd \rightarrow \pi G \Sigma_*^2/(2\sigma_*^2)$, we have
\begin{equation}\label{eq:C}
C(1+\mathcal{R})=4\zeta_d \rbrackets{\frac{\sigmaeff\Sigma_*}{\sigma_*\Sigma}}^2.
\end{equation}
\begin{equation}\label{eq:sigmaeff}
\sigmaeff\equiv\left(\vth^2+v_z^2+\frac{v_A^2}{2}-v_{A,z}^2\right)^{1/2}=\sigma_z(1+ \mathcal{R})^{1/2}
\end{equation}
is the effective vertical velocity dispersion including magnetic terms for
gaseous support, and we have assumed that $\mathbf{B}(\zmax)\rightarrow 0$.
We shall show from results of our simulations in Section <ref> that
$\sigmaeff\sim 5-6\kms$ and $\zeta_d\sim 0.4-0.5$ are quite insensitive to the
initial magnetization (see Tables <ref> and <ref>). In
Paper I, we also found relatively little variation in $\sigma_z$, even with a
factor 10 variation in $\Sigma$ and two orders of magnitude variation of
$\rhosd/\Sigma^2$, the input parameter that controls the ratio of external- to
self-gravity. In this paper, we fix $\Sigma$ and $\rhosd$ to isolate the
effect of magnetic fields. Further investigations would therefore be needed to
investigate constancy or variation of $\sigmaeff$ and $\zeta_d$ with $\Sigma$
and $\rhosd$, which we defer to future work.
In simulations, $\Sigma$ and $\rhosd$ (as well as a seed magnetic field) are
input parameters, and $\sigma_z$, $\mathcal R$, and $\zeta_d$ are measured
outputs. From the right-hand side of Equation (<ref>) combined with
Equations (<ref>) and (<ref>), the predicted dynamical
equilibrium midplane pressure can be obtained. In observed (relatively
face-on) galactic disks, the gas and stellar surface densities $\Sigma$ and
$\Sigma_*$ together with $\sigma_z$ and $\sigma_*$ may be considered basic
observables <cit.>. The magnetic term $v_A^2/2- v_{A,z}^2$
in $\sigmaeff^2$ is more difficult to measure directly. We shall show,
however, that if the model disk is fully saturated, $\mathcal{R}=(v_A^2/2-
v_{A,z}^2)/\sigma_z^2$ appears to be insensitive to the initial magnetic
geometry or strength. This implies that the predicted equilibrium midplane
pressure support may similarly be obtained from observables using Equations
(<ref>), (<ref>), (<ref>), (<ref>) if estimates
of $\mathcal{R}$ and $\zeta_d$ from simulations are adopted.
To test the validity of vertical dynamical equilibrium, for our simulations we
will compare full vertical profiles of the total (turbulent, thermal, and
magnetic) support with the vertical profiles of the weight of the gas. These
profiles are based on horizontal and temporal averages of the simulation
outputs at all values of $z$, including the midplane.
In equilibrium, the weight and total support profiles must match each other.
Due to the highly dissipative nature of the ISM, continuous energy injection is
necessary to maintain thermal as well as turbulent kinetic and magnetic
components of the vertical support at a given level. Massive young stars
inject prodigious energy, providing the feedback that is key to self-regulation
of the SFR. When the system is out of equilibrium, with not enough massive
young stars, lack of energy injection leads the entire ISM to become
dynamically and thermally cold. A cold disk is highly susceptible to
gravitational collapse; it forms new stars that supply the “missing" feedback
and restore equilibrium. In the opposite case, with too many massive stars,
the ISM becomes dynamically and thermally hot, quenching further star formation
by suppressing gravitational instability. Simulations of local model disks
show that the SFR converges to the quasi-steady value predicted by theory, in
which the vertical support produced by feedback matches the requirements set by
vertical dynamical equilibrium. The ISM state and SFR predicted by the
feedback-regulated theory are expected to hold in real galaxies provided all
equilibria can be established in less than the disk's secular evolution
In Paper I <cit.>, we
showed that the balance between energy gains and losses is established within
one vertical crossing time, the turbulence dissipation timescale. We then
quantified the efficiency of energy conversion to each support component for a
given SFR by measuring the “feedback yield.”
Using a suitable normalization[
Dimensionally, $\eta$ has units of velocity. To obtain $\eta$ in
$\kms$, the values reported in this paper should be multiplied by 209.],
we define yield parameters $\eta_c$ as
\begin{equation}\label{eq:eta}
\eta_c\equiv\frac{P_{c,3}}{\SigSFRnorm}
\end{equation}
where $P_{c,3}=P_{c,0}/10^3\kbol\Punit$ and
$\SigSFRnorm=\SigSFR/10^{-3}\sfrunit$. The subscript `$c$' denotes “turb,”
“th," or “mag" for the respective component of vertical support ($\Pturb$,
$\Pth$, or $\Pmag$). For magnetic pressure, the support is further divided
into turbulent and mean components, $\Pmagt$ and $\Pmago$, respectively (see
Section <ref> for definitions). <cit.> and
<cit.> respectively showed that $\etath$ and $\etaturb$
are expected to be nearly independent of $\SigSFR$. In Paper I <cit.>, which omitted magnetic fields but covered a wide
range of disk conditions such that $0.1<\SigSFRnorm<10$, we obtained
$\etaturb=4.3\SigSFRnorm^{-0.11}$ and $\etath=1.3\SigSFRnorm^{-0.14}$. The
values (and weak $\SigSFR$ dependence) of these numerically calibrated yield
parameters are in very good agreement with analytic expectations.
The MHD simulations of this paper allow us to study the evolution and
saturated-state properties of magnetic fields in star-forming, turbulent,
differentially-rotating galactic disks with vertical stratification. ISM
turbulence driven by star formation feedback can generate and deform magnetic
fields. The small scale turbulent dynamo, combined with buoyancy and sheared
rotation, creates turbulent magnetic fields and modifies mean magnetic fields,
both of which can provide vertical support to the ISM. We shall consider three
different initial magnetic field strengths (as well as two initial vertical
profiles), and directly measure the saturated-state magnetic field strengths
and feedback yields, while also testing how magnetization affects the thermal
and turbulent feedback yield components.
§ METHODS AND MODELS
In this paper, we extend our previous three dimensional HD simulations from
Paper I to include magnetic fields. We solve ideal MHD equations in a local,
shearing box with self- and external gravity, thermal conduction,
and optically thin cooling.
To solve the MHD equations, we utilize Athena
<cit.> with the van Leer integrator
<cit.>, HLLD solver, and second-order spatial
We also consider feedback from massive young stars
using time-varying heating rate and momentum feedback from SNe.
The probability of massive star formation is calculated
based on a predicted local SFR ($\dot{M}_*$)
in cells with density exceeding a threshold, assuming an efficiency per
free-fall time $\epsilon_{\rm ff}=1\%$ <cit.>,
and adopting a total mass in new stars per massive star of $m_*=100\Msun$.
When a massive star forms (i.e., when a uniform random number in (0,1)
is less than the probability $\dot{M}_*/m_*\Delta t$),
we immediately inject total radial momentum $p_*=3\times10^5 \Msun\kms$ in a
surrounding $10\pc$ sphere.
The global SFR surface density is then calculated by
\begin{equation}\label{eq:sfr}
\SigSFR = \frac{\mathcal{N}_{*} m_*}{L_xL_y t_{\rm bin}},
\end{equation}
where $\mathcal{N}_{*}$ denotes the total number of massive stars in the time
interval $(t-t_{\rm bin},t)$, and $t_{\rm bin}=10\Myr$ is UV-weighted lifetime
of OB stars <cit.>.
The heating rate is proportional to the mean FUV intensity,
which is assumed to be linearly proportional to the SFR surface density,
$\Gamma\propto J_{\rm FUV}\propto \SigSFR$.
See Paper I for more details and other source terms.
The models in this paper are magnetically-modified versions of the fiducial
simulation QA10 from Paper I, with conditions similar to the Solar
neighborhood. Our simulation domain is a local Cartesian grid far from the
galactic center, with center rotating at angular velocity of
$\Omega=28\kms\kpc^{-1}$ (the corresponding orbital time is
$\torb=2\pi/\Omega=219\Myr$) and shear parameter $q\equiv-d\ln\Omega/d\ln R=1$
for a flat rotation curve. In the $\hat z$ direction, we adopt outflow
boundary conditions to allow magnetic flux loss at the vertical boundaries;
this is in contrast to Paper I, where we adopted periodic vertical boundary
conditions (except for the gravitational potential) to prevent any mass loss.
In order to minimize mass loss at the vertical boundaries, we double the
vertical domain size to $L_z=1024\pc$ compared to Paper I. Shearing-periodic
boundary conditions are employed in the horizontal directions, with
$L_x=L_y=512\pc$ the same as in Paper I. The spatial resolution is set to
$2\pc$ as in Paper I. We adopt a linear external gravity profile
$\mathbf{g}_{\rm ext} = - 4\pi G\rhosd z \zhat$, where $\rhosd$ is the midplane
volume density of stellar disk plus dark matter.
In order to explore the independent effect of the magnetic fields, we fix the
background gravity and gas surface density parameters to $\rhosd=0.05\rhounit$
and $\Sigma=10\Surf$ for all models. Initial profiles of the gas density are
set by an exponential function as $\rho=\rho_0\exp(-|z|/H)$, with midplane
density $\rho_0=\Sigma/2H$ and scale height set using $\sigma_z=4\kms$ in
Equations (<ref>) and (<ref>), also using $\zeta_d=1/2$ and
$\mathcal{R}$ as described below. The initial thermal pressure has the same
profile as the density with midplane thermal pressure of $P_{\rm
th,0}/\kbol=3000\Punit$. We vary the plasma beta at the midplane
$\beta_0\equiv P_{\rm th,0}/P_{\rm mag,0} =8\pi P_{\rm th,0}/B_0^2$ with two
different vertical field distributions; uniform plasma beta with
$\mathbf{B}=B_0\exp(-|z|/2H)\yhat$ (Models MA), and uniform magnetic fields
with $\mathbf{B}=B_0\yhat$ (Models MB). The suffixes of magnetized models
denote the values of $\beta_0=1$, 10, and 100. The initial magnetic energy is
then largest for Model MB1 and smallest for Model MA100. For comparison, we
also present the results from an unmagnetized counterpart (Model HL) as well as
the previous Model QA10 from Paper I (here, renamed HS). These hydrodynamic
models differ in the size of the vertical domain (“S”=small, $L_z=512$ pc;
“L”=large, $L_z=1024$ pc).
For our initial conditions, we have $\zeta_d=1/2$ and $C=2.37$ for all models.
With vertically stratified magnetic fields (Models MA;
$\mathcal{R}=1/\beta_0$), $\chi=1.12$, 1.13, 1.19, and 1.73 for
$\beta_0=\infty$, 100, 10, and 1, respectively, while $\chi=1.12$ for all MB
models ($\mathcal{R}=0$). The scale height is $H=77
[(1+\mathcal{R})(1+1/\chi)]^{1/2}\pc$ and
$\rhomid=\rho_0/1.4\mh=1.9[(1+\mathcal{R})(1+1/\chi)]^{-1/2}\pcc$. The midplane
magnetic field strength is
\begin{equation}
\rbrackets{\frac{P_{\rm th,0}/\kbol}{3000\Punit}}^{1/2}\mu G .
\end{equation}
In contrast to Paper I, we drive turbulence for $\torb$ in order to provide
turbulent support at early stages before SN feedback generates sufficient
turbulence. We utilize divergence-free turbulent velocity fields following a
Gaussian random distribution with a power spectrum of the form $|\delta
\vel_k^2| \propto k^6 \exp(-8k/k_{\rm pk})$ where the peak driving is at
$k_{\rm pk}L_x/2\pi=4$ and $1\ge kL_x/2\pi\ge128$. A new turbulent velocity
perturbation field is generated every $10\Myr$, with total energy injection
rate of $\dot{E}_{\rm turb} = 500L_\odot$ to the turbulence. This perturbation
corresponds to the saturation level of one-dimensional velocity dispersion
$\sim 4\kms$. Turbulence is driven at full strength up to $\torb/2$, and then
slowly turned off from $\torb/2$ to $\torb$. The spatially uniform
photoelectric heating rate is set to constant for the first $\torb/2$
($\Gamma=0.8\Gamma_0$, where $\Gamma_0=2\times10^{-26}\ergs$ from
). From $\torb/2$ to $\torb$ this imposed heating
is slowly turned off and replaced by the self-consistent heating rate that is
proportional to the SFR surface density ($\Gamma=0.4\Gamma_0\SigSFRnorm$).
By initially driving turbulence, and allowing smooth changes from early to
saturated stages in the thermal and turbulent supports, the current models
minimize abrupt early collapse that was seen in the models of Paper I that were
initialized without turbulence (as reproduced in model HS here). There, the
initial vertical collapse also triggered a strong burst of star formation,
leading to exaggerated vertical oscillations. However, we shall show that
these and other differences in numerical treatment (including initial
turbulence driving, vertical boundary conditions, and domain size) make little
or no differences in the physical quantities (see Section <ref>) and
averaged vertical distributions (see Section <ref>). Since some
magnetized models saturate slowly, we run magnetized models longer to achieve a
quasi-steady state for the turbulent magnetic fields. For the purpose of
computing time-averaged quantities, the saturated stages are considered from
$t_1=3\torb$ to $t_2=4\torb$ for magnetized models and from $t_1=1.5\torb$ to
$t_2=2\torb$ for unmagnetized models. Note that 4 orbits is still not long
enough for complete saturation of the mean magnetic field in the cases of
initially strongest and weakest magnetization.
§ SIMULATION RESULTS
§.§ Time Evolution and Saturated State
In this section, we shall show that model disks achieve a quasi-steady state,
using evolution of diagnostics that describe the average disk properties at a
given time. Volume and mass-weighted means for quantities $q_{ijk}(t)$ are
respectively calculated by
\begin{equation}\label{eq:mwavg}
\abrackets{q}_V(t)\equiv \frac{\sum q \Delta V}{L_x L_y L_z},
\quad
\abrackets{q}_M(t)\equiv \frac{\sum \rho q \Delta V}{\sum\rho \Delta V},
\end{equation}
where the summation is over all grid zones (indices $ijk$), and the volume
element is $\Delta V=(2\pc)^3$. We also calculate the horizontally-averaged
vertical profile as
\begin{equation}\label{eq:havg}
\overline{q}(z;t) \equiv \frac{\sum_{i,j} q \Delta x \Delta y}{L_xL_y},
\end{equation}
where the summation is only over horizontal planes (indices $ij$) at each
vertical coordinate $z$ (index $k$).
We use a horizontal average to obtain the mean magnetic field,
$\overline{\mathbf{B}}(z)$. The turbulent magnetic field is then defined by
$\delta \mathbf{B} = \mathbf{B}-\overline{\mathbf{B}}$. Note that in our
simulations the mean field is dominated by $\yhat$-component with small
$\xhat$-component and negligible $\zhat$-component. The turbulent magnetic
pressure is given by $\delta P_\mathrm{mag} \equiv |\delta \mathbf{B}|^2/8\pi$
and the mean magnetic pressure is given by $\bar P_\mathrm{mag} \equiv
Time evolution of
(a) SFR surface density $\SigSFR$, (b) midplane number density $\rhomid$, and
(c) mass-weighted mean thickness $\abrackets{|z|}_M$. Physical model
parameters differ only in the initial magnetic energy, increasing from zero in
HS and HL to a maximum in MB1 (see Section <ref>).
Time evolution of mass-weighted RMS (a) turbulent kinetic, (b)
thermal, (c) turbulent Alfvén and (d) mean Alfvén velocity.
Mean Disk Properties at Saturation
$\alpha_{\rm ss,R}$
$\alpha_{\rm ss,M}$
HS $ 1.59 \pm 0.38$ $ 1.96 \pm 0.56$ $61.5 \pm 10.4$ $73.8 \pm 21.1$ $ 0.42 \pm 0.14$ $ 1.23 \pm 0.54$ $ 0.22 \pm 0.07$
HL $ 1.46 \pm 0.28$ $ 2.15 \pm 0.41$ $60.6 \pm 6.3$ $67.3 \pm 12.9$ $ 0.45 \pm 0.10$ $ 1.21 \pm 0.35$ $ 0.25 \pm 0.10$
MA100 $ 0.91 \pm 0.16$ $ 2.87 \pm 0.45$ $47.1 \pm 3.1$ $50.3 \pm 7.9$ $ 0.47 \pm 0.08$ $ 0.94 \pm 0.22$ $ 0.22 \pm 0.13$ $ 0.23 \pm 0.07$
MA10 $ 0.81 \pm 0.16$ $ 2.87 \pm 0.35$ $47.4 \pm 4.2$ $50.4 \pm 6.2$ $ 0.47 \pm 0.07$ $ 0.95 \pm 0.18$ $ 0.22 \pm 0.09$ $ 0.27 \pm 0.07$
MB10 $ 0.74 \pm 0.15$ $ 2.85 \pm 0.31$ $47.6 \pm 3.6$ $50.7 \pm 5.5$ $ 0.47 \pm 0.06$ $ 0.95 \pm 0.16$ $ 0.21 \pm 0.09$ $ 0.32 \pm 0.08$
MA1 $ 0.75 \pm 0.16$ $ 3.00 \pm 0.37$ $48.2 \pm 4.2$ $48.2 \pm 6.0$ $ 0.50 \pm 0.08$ $ 0.96 \pm 0.19$ $ 0.23 \pm 0.10$ $ 0.22 \pm 0.10$
MB1 $ 0.56 \pm 0.14$ $ 2.97 \pm 0.43$ $55.1 \pm 4.3$ $48.7 \pm 7.0$ $ 0.57 \pm 0.09$ $ 1.10 \pm 0.24$ $ 0.17 \pm 0.09$ $ 0.27 \pm 0.10$
The mean and standard deviation are taken over
$t/\torb=1.5-2$ for HD models and $t/\torb=3-4$ for MHD models.
See Sections <ref> and <ref> for definition of each quantity.
$\SigSFR$ is in units of $10^{-3}\sfrunit$,
$\rhomid$ is in units of $\rm cm^{-3}$,
$\abrackets{|z|}_M$ and $H$ are in units of pc.
Saturated State Velocities
$\vrms{\delta v}$
$\vrms{\delta v_A}$
$\sigma_{\rm z}$
HS $ 7.31 \pm 0.65$ $ 3.89 \pm 0.20$ $ 5.96 \pm 0.36$ $ 5.96 \pm 0.36$
HL $ 6.95 \pm 0.53$ $ 3.89 \pm 0.16$ $ 5.67 \pm 0.27$ $ 5.67 \pm 0.27$
MA100 $ 5.22 \pm 0.55$ $ 3.16 \pm 0.13$ $ 3.07 \pm 0.13$ $ 1.93 \pm 0.17$ $ 4.36 \pm 0.28$ $ 4.78 \pm 0.27$
MA10 $ 5.10 \pm 0.55$ $ 3.07 \pm 0.16$ $ 3.18 \pm 0.10$ $ 2.47 \pm 0.17$ $ 4.27 \pm 0.29$ $ 4.83 \pm 0.25$
MB10 $ 4.89 \pm 0.55$ $ 3.01 \pm 0.15$ $ 3.19 \pm 0.12$ $ 2.74 \pm 0.13$ $ 4.17 \pm 0.30$ $ 4.81 \pm 0.27$
MA1 $ 4.94 \pm 0.52$ $ 3.05 \pm 0.17$ $ 3.15 \pm 0.14$ $ 3.16 \pm 0.15$ $ 4.21 \pm 0.28$ $ 4.97 \pm 0.24$
MB1 $ 4.46 \pm 0.45$ $ 3.09 \pm 0.17$ $ 3.19 \pm 0.16$ $ 5.39 \pm 0.22$ $ 4.11 \pm 0.27$ $ 5.78 \pm 0.22$
The mean and standard deviation are taken over
$t/\torb=1.5-2$ for HD models and $t/\torb=3-4$ for MHD models.
See Sections <ref> and <ref> for definition of each quantity.
All velocities are in units of km/s.
Figure <ref> for detailed statistical information about energies related to each velocity component.
Figures <ref> and <ref> plot time evolution of selected
diagnostics that describe overall disk properties and energetics. In
Figure <ref>, we plot (a) SFR surface density $\SigSFR$, (b)
midplane number density $\rhomid=\overline{\rho}_0/(1.4\mh)$ and (c)
mass-weighted mean height $\abrackets{|z|}_M$. Figure <ref> plots
mass weighted means of (a) three-dimensional turbulent velocity $\vrms{\delta
v}$, (b) sound speed $\vrms{\vth}$, and (c) turbulent $\vrms{\delta {v}_A}$ and
(d) mean $\vrms{\overline{v}_A}$ Alfvén velocities. Here, we subtract the
azimuthal velocity arising from the background shear (not the horizontal
average) for turbulent velocity, $\delta \mathbf{v}\equiv \mathbf{v}+q\Omega
x\yhat$. The mean and standard deviations of values from
Figure <ref> are listed in Table <ref>. Time averages are
taken over the “saturated state" time interval of $(t_1, t_2)$. We also list
in Table <ref> the mean and standard deviation of $H=\Sigma/2\rho_0$,
$\zeta_d=\abrackets{|z|}_M/2H$, and $\chi=4\zeta_d\rhosd/\rho_0$. In
Table <ref>, we report the mean and standard deviations of the
velocities shown in Figure <ref>, as well as
$\sigma_z=[\vrms[]{\vth}+\vrms[]{v_z}]^{1/2}$, and
Comparing Models HS (yellow) and HL (black), Figure <ref> shows
distinct differences at early stages ($t<\torb$), but no systematic differences
in mean values of physical quantities at later stages. In Model HS, initial
vertical collapse at $t\sim20\Myr$ triggers an abrupt increase of $\rhomid$ and
hence $\SigSFR$, which then produces feedback that causes a strong reduction in
$\rhomid$ and $\SigSFR$, leading to further bounces. These exaggerated
vertical oscillations are a direct consequence of the lack of turbulence in the
initial conditions and early evolution, before feedback has developed. In
contrast to Model HS, Model HL (and all magnetized models) shows no strong
oscillation at early stages, although more limited vertical oscillation emerges
and persists at later times. The early driven turbulence and constant heating
in Model HL (and magnetized models) prevent strong, global vertical
oscillations, keeping the midplane density and the thickness of the disk more
or less constant. Without initial vertical collapse, there is no bursting star
formation; rather, the SFR in Model HL remains moderate. After one orbit time,
when the turbulence driving and heating are fully self-consistent with feedback
from the star formation, all diagnostics in the unmagnetized models quickly
saturate. The convergence of Models HS and HL to the same saturated state,
despite their completely different early evolution, confirms the robustness of
our previous work.
The magnetized models also achieve a quasi-steady state, but not so rapidly as
the unmagnetized models. Even after saturation of thermal and turbulent
velocities (both $\delta v$ and $\delta v_A$) as well as the midplane density
and scale height at $\sim 2 \torb$, clear secular evolution continues for the
mean Alfvén velocity (or mean magnetic fields; see
Figure <ref>(d)). The models with initially strongest magnetic
fields (MA1 and MB1) slowly lose energy from the mean magnetic field as buoyant
magnetic fields escape through the vertical boundaries. Conversely, the mean
magnetic energy of models with initially weakest magnetic fields (MA100 and
MA10) slowly grows. We note, however, that since our horizontal dimension is
only 512 pc and assumed to be periodic, magnetic energy loss might be somewhat
overestimated in our simulations compared to the case in which azimuthal fields
are anchored at larger scales. In principle, if numerical reconnection in our
models is faster than realistic small-scale reconnection (which is uncertain)
should be, we might also overestimate the growth of mean magnetic fields in
weak-field models.
Modulo these potential numerical effects, the interesting tendency seen in
Figure <ref>(d) is that $\vrms{\overline{v}_A}$ converges toward
similar values for cases with widely varying initial magnetic fields.
The magnetized models show distinguishably different final saturated states
compared to the unmagnetized models. The SFR surface density and turbulent and
thermal velocity dispersions are lower in the magnetized models, while
variations among the set of magnetized models is small. This is completely
consistent with expectations from the equilibrium theory, in which (1) the sum
of all pressures must offset a given ISM weight, so the addition of magnetic
pressure reduces the need for turbulent and thermal pressure, and (2) an
increase in “feedback yields” due to magnetic fields implies that a lower
star formation rate is needed for equilibrium. We examine this issue in detail
in Section <ref>.
Similarities and differences among models are clearest in the energetics. In
Figure <ref>, we clearly see the saturation of turbulent and thermal
velocity dispersions for all models immediately after $\torb$. The turbulent
Alfvén velocity also converges rapidly except in Model MA100, which converges
after $3\torb$. Model MB10 achieves a quasi-steady state earliest among the
magnetized models since it has initial magnetic energy comparable to that of
the final state. We thus consider Model MB10 as the fiducial run. All other
magnetized models converge toward the same saturated state as Model MB10. As
seen in Figure <ref> and Table <ref>, the saturated-state
values of $\vrms{\delta {v}}$, $\vrms{\vth}$, and $\vrms{\delta {v}_A}$, are
essentially indistinguishable for all magnetized models, whereas
$\vrms{\overline{v}_A}$ values show some variations as they have not yet
reached asymptotic values.
Box and whisker plot of the energy ratios (a) turbulent (kinetic)
$\Ekin/E_{\rm tot}$, (b) thermal $\Eth/E_{\rm tot}$, (c) turbulent magnetic
$\Emagt/E_{\rm tot}$, and (d) mean magnetic $\Emago/E_{\rm tot}$, averaged over
$(t_1, t_2)$. The bottom and top of the rectangular box denote the first
($Q_1$) and third ($Q_3$) quartiles, respectively, while the red line within
the box is the median (second quartile; $Q_2$) of the data. The lower/upper
whisker reaches to $1.5\times(Q_3-Q_1)$ or the minimum/maximum of the data,
whichever is larger/smaller.
In Figure <ref>, we summarize the saturated-state energy ratios
with “box and whisker" plots omitting outliers, and with the mean values (red
squares). From left to right in each panel, the initial degree of
magnetization increases. Panels show the fraction of total energy in each
component: turbulent kinetic, thermal, turbulent magnetic, and mean magnetic.
The turbulent kinetic energy is defined by $\Ekin\equiv\sum \Delta
V\rho|\delta\mathbf{v}|^2/2 $, and the thermal energy is $\Eth = \sum \Delta
V\Pth/(\gamma-1) $, where $\gamma=5/3$ in our simulations. The mean and
turbulent magnetic energies are given by $\Emago\equiv \sum \Delta V
|\overline{\mathbf{B}}|^2/8\pi $ and $\Emagt\equiv \sum \Delta
V|\delta\mathbf{B}|^2/8\pi $. Since $\overline{\delta \mathbf{B}}=0$ by
definition, $E_{\rm mag}=\Emago+\Emagt$. Note that these energies are
integrated over the whole volume, which compared to the midplane (see
Section <ref> and Table <ref>) increases the relative
proportion of thermal to turbulent energy.
For the unmagnetized models, the turbulent kinetic and thermal energies are
nearly in equipartition. For Model MB10, which is the most saturated
magnetized model, the total energy is portioned into kinetic (35%), thermal
(39%), turbulent magnetic (15%), and mean magnetic (11%) terms. Thus,
kinetic, thermal, and magnetic components are each close to 1/3 of the total
energy. The ratio between turbulent kinetic and turbulent magnetic energies is
about $\Ekin: \Emagt=7:3$, similar to what has been found for saturation of
small scale dynamo simulations at large Reynolds number <cit.>. Although the mean
magnetic energy is not yet fully saturated (see Figure <ref>(d)) for
all models, there is a clear trend of convergence toward the saturated state of
Model MB10. Figure <ref> and Table <ref> show that
component energies in the magnetized models are nearly the same except the mean
magnetic term, which still reflects initial mean field strengths.
It is of interest to characterize the angular momentum transport by both
turbulence and magnetic fields in our models. We measure the $R$-$\phi$
component of Reynolds and Maxwell stresses as a function of $z$,
$R_{xy}\equiv\overline{\rho v_x \delta v_y}$ and $M_{xy}\equiv\overline{B_x
B_y/4\pi}$, respectively.[ We confirm that the other off-diagonal
terms of stress tensors are one or two orders of magnitude smaller.] Since the
stresses are non-negligible only within one gas scale height, it is most
informative to calculate the mass-weighted mean values of the stresses
normalized by the mean midplane thermal pressure, the
“$\alpha_\mathrm{ss}$-parameters” of <cit.>:
$\alpha_\mathrm{ss,R}=\abrackets{R_{xy}}_M/\overline{P}_{\rm th,0}$ and
$\alpha_\mathrm{ss,M}=\abrackets{M_{xy}}_M/\overline{P}_{\rm th,0}$ (see
Table <ref> for $\overline{P}_{\rm th,0}$). As listed in
Table <ref>, the $\alpha_\mathrm{ss}$-parameters are comparable to
each other and $\sim 0.2-0.3$. Although the ratio between Reynolds and Maxwell
stresses is completely different compared to the the case of turbulence driven
by magnetorotational instability (where the Maxwell stress dominates), the
total stress $\sim 0.4-0.5$ is similar
Note that gravitational stress in our simulations is generally lower, with
$\alpha_\mathrm{ss}\sim 0.05$. This is also true for other simulations in
which turbulence is driven by gravitational instability combined with sheared
rotation, which give $\alpha_\mathrm{ss}$ less than $0.1$
<cit.>. The total stress gives
$\alpha_\mathrm{ss}\sim0.4-0.5$, implying that the gas accretion time $t_{\rm
acc}\sim R^2\Omega/(\alpha_{\rm ss} \vth^2)=390\Gyr$ for $\vth=3\kms$ using
$R=8$ kpc.
Gas and magnetic structure in Models (a) MA100, (b) MB10, and (c) MB1
at $t/\torb=3$. Top: Surface density in units of $\Surf$. Bottom: 3D
visualization of magnetic field lines, with vertical slices of hydrogen number
density shown in color scale on box boundaries. Model MB1 shows field lines
preferentially along $\yhat$ within $|z|<300\pc$ since its mean magnetic fields
are still dominant. From right to left, randomness of the magnetic field
structure increases as the mean field strength decreases, since the turbulent
field strengths are all similar at this time. Differences in the magnetic
field produce no clear signature in surface density maps, however, with similar
cloudy structure in all cases.
To illustrate the overall saturated-state disk structure, in
Figure <ref> we display surface density (top) and magnetic field lines
(bottom) for Models (a) MA100, (b) MB10, and (c) MB1 at $t/\torb=3$. As shown
in Figures <ref> and <ref>, the strength of the mean
magnetic field increases from MA100 to MB10 to MB1 (from left to right in
Figure <ref>). Averaged over the saturation period ($t/\torb=3-4$),
the mean and turbulent magnetic field values at the midplane are
and $\delta\mathbf{B}_{\rm rms}=(1.3,1.6,1.1)\muG$ for Model MA100,
and $\delta\mathbf{B}_{\rm rms}=(1.4,1.7,1.2)\muG$ for Model MB10, and
and $\delta\mathbf{B}_{\rm rms}=(1.4,1.8,1.2)\muG$ for Model MB1.
For all models, the azimuthal ($\hat y$) component is the largest of
$\overline{\mathbf{B}}$. However, this component exceeds the turbulent
components only for model MB1; for model MB10 it is comparable to the largest
turbulent component, and for model MB100 it is smaller than the largest
turbulent component. As a result, field lines are more complex and random in
Model MA100 ($\Emagt>\Emago$) and more aligned in a preferential direction
(along $\yhat$) in Model MB1 ($\Emagt<\Emago$). The dominance of the mean
magnetic fields at all heights in Model MB1 is also evident.
Despite of the strong distinctions in the structure of field lines, the surface
density maps look quite similar for all models. In particular, there is no
visually prominent evidence of alignment of dense filaments either
perpendicular or parallel to the mean magnetic field direction. However,
traces of the shear are evident in overall pattern of striations (consistent
with trailing wavelets), particularly for Model MB1. More quantitative
analysis of the morphology of filaments and magnetic field lines may be
obtained using maps of synthetic 21 cm emission, dust emission, and
polarization <cit.>. We
defer this interesting study to future work.
§.§ Vertical Dynamical Equilibrium
In this subsection, we investigate the vertical dynamical equilibrium of model
disks using horizontally and temporally averaged profiles of $\Pturb$, $\Pth$,
$\Pmagt$, and $\Pmago$, in comparison to profiles of the ISM weight
$\mathcal{W}$. We also compare midplane values.
Time evolution of horizontally-averaged components of support at the
midplane: (a) turbulent kinetic $\Pturb$, (b) thermal $\Pth$, (c) turbulent
magnetic $\Pmagt$, and (d) mean magnetic $\Pmago$.
Vertical Support and Feedback Yield at Saturation
HS $ 5.24 \pm 3.05$ $ 1.78 \pm 0.40$ $ 3.30 \pm 2.08$ $ 1.12 \pm 0.37$
HL $ 5.15 \pm 2.61$ $ 1.61 \pm 0.29$ $ 3.53 \pm 1.91$ $ 1.10 \pm 0.29$
MA100 $ 3.40 \pm 2.42$ $ 1.24 \pm 0.18$ $ 0.94 \pm 0.21$ $ 0.46 \pm 0.15$ $ 3.72 \pm 2.73$ $ 1.35 \pm 0.31$ $ 1.03 \pm 0.29$
MA10 $ 3.38 \pm 2.66$ $ 1.14 \pm 0.20$ $ 1.04 \pm 0.18$ $ 0.73 \pm 0.18$ $ 4.17 \pm 3.38$ $ 1.41 \pm 0.36$ $ 1.29 \pm 0.33$
MB10 $ 2.85 \pm 1.46$ $ 1.04 \pm 0.18$ $ 1.03 \pm 0.17$ $ 0.87 \pm 0.23$ $ 3.83 \pm 2.12$ $ 1.40 \pm 0.38$ $ 1.38 \pm 0.37$
MA1 $ 3.02 \pm 2.11$ $ 1.05 \pm 0.16$ $ 1.10 \pm 0.22$ $ 0.91 \pm 0.29$ $ 4.00 \pm 2.92$ $ 1.39 \pm 0.37$ $ 1.46 \pm 0.43$
MB1 $ 2.18 \pm 1.12$ $ 0.83 \pm 0.16$ $ 1.16 \pm 0.18$ $ 1.83 \pm 0.45$ $ 3.92 \pm 2.23$ $ 1.49 \pm 0.47$ $ 2.08 \pm 0.61$
The mean and standard deviation are taken over
$t/\torb=1.5-2$ for HD models and $t/\torb=3-4$ for MHD models.
The vertical support terms at the midplane ($\Pturb$, $\Pth$, $\Pmagt$, and $\Pmago$)
are given in units of $10^3\kbol\Punit$.
See Equation (<ref>) for $\eta$ definition and
Figure <ref> for detailed statistical information about feedback yields.
Figure <ref> plots time evolution of horizontally-averaged midplane
support terms: (a) $\Pturb$, (b) $\Pth$, (c) $\Pmagt$, and (d) $\Pmago$. The
mean and standard deviation values over $(t_1, t_2)$ are summarized in
Table <ref>. Table <ref> also lists the feedback yields for
each support component in the units of Equation (<ref>). Similar to
Figures <ref> and <ref>, convergence of each support term
except the mean magnetic field is evident after $t/\torb>3$, regardless of
initial magnetic field strength. The mean magnetic support $\Pmago$ in
Figure <ref>(d) more gradually converges toward the value of the
fiducial run, Model MB10.
Temporally and horizontally averaged vertical profiles of (a)
$\Pturb$, (b) $\Pth$, (c) $\Pmagt$, (d) $\Pmago$, (e)
$\alpha\equiv(\Pturb+\Pth)/\Pth$, and (f) $\mathcal{R}\equiv
Figure <ref>(a)-(d) plots vertical profiles of the four individual
support terms averaged over $(t_1, t_2)$ and the horizontal direction. Panel
(e) shows the ratio of the kinetic (turbulent + thermal) to the thermal term,
$\alpha\equiv(\Pturb+\Pth)/\Pth$; and panel (f) shows the ratio of total
magnetic to kinetic term, $\mathcal{R}\equiv (\Pmagt+\Pmago)/(\Pturb+\Pth)$.
First of all, Figure <ref> confirms that Models HS and HL agree very
well not only for the mean and midplane values, but also for the overall
profiles. The periodic vertical boundary conditions in Model HS introduce a
small anomaly near the vertical boundaries.
In the presence of magnetic support, the turbulent and thermal terms are both
reduced. However, the relative contribution between turbulent and thermal terms
remains similar (Figure <ref>(e)): $\alpha\sim 4$ within one scale
height, decreasing to $\alpha\sim 1$ at high-$|z|$. This is an important
consequence of self-regulation by star formation feedback, explained in the
equilibrium theory (see Section <ref>). The turbulent magnetic support
in all magnetized models converges to very similar profiles
(Figure <ref>(c)), while the mean magnetic support still shows
differences (Figure <ref>(d)), especially for two extreme cases
(Models MA100 and MB1). For all models except MB1, Figure <ref>(f)
shows that the midplane value $\mathcal{R}_0\sim$0.2-0.5, while $\mathcal{R}$
becomes very small at high-$|z|$; Model MB1 has $\mathcal{R}\sim 0.3-1$.
Vertical saturated-state density profiles of cold (blue; $T<184\Kel$),
intermediate-temperature (green; $184\Kel<T<5050\Kel$), and warm (red;
$T>5050\Kel$) gas phases. The profiles of total density (black) and combined
non-cold phases (yellow; $T>184\Kel$) are also shown.
Figure <ref> presents vertical profiles of the
horizontally-averaged density from different thermal components of gas, for
each model. We separately plot the profiles of cold ($n_c$; $T<184\Kel$),
intermediate-temperature ($n_i$; $184\Kel<T<5050\Kel$), and warm ($n_w$;
$T>5050\Kel$) phases, as well as the whole medium ($n_H=n_c+n_i+n_w$) and
combined non-cold ($n_{i+w}$; $T>184\Kel$) components. The distribution of
cold medium is mostly limited to one gas scale height, while the warm medium
extends to high $|z|$ with a dip near the midplane. The intermediate
temperature gas is not as concentrated toward the midplane as the cold medium,
but does not extend to high $|z|$. Decomposition into cold ($n_c$) and
non-cold ($n_{i+w}$) components gives two smooth profiles that resemble the
results for two-component fits to observed H1 gas from 21 cm emission.
Because most of cold medium resides within one gas scale height (and we do not
consider runaway O stars), most of SN explosions occur there. The $\Pturb$ and
$\alpha$ profiles in Figure <ref> are peaked near the midplane, with
scale heights similar to the cold medium scale height. The value of $\alpha$
is close to unity at $|z| > 100$pc, implying that much of the turbulent energy
dissipates very efficiently within the driving layer without propagating to
high $|z|$. Thermal pressure is flat in the central layer, implying that the
two-phase medium is well-mixed and in pressure equilibrium. Beyond the
turbulent, two-phase layer, the ISM is mostly warm gas and is supported mainly
by thermal pressure. Note that the shape of vertical profiles of the Reynolds
and Maxwell stress are respectively similar to those of $\Pturb$ and $\Pmagt$
(see Figure <ref>(a) and (c)).
In order to check vertical dynamical equilibrium quantitatively, for each model
we calculate the weight of the gas using the horizontally and temporally
averaged density and gravitational potentials, with $\Wsg(z)$ and $\Wext(z)$ as
in Equations (<ref>) and (<ref>) except integrated between $z$
and $\zmax$. Figure <ref> plots the vertical profiles for the
total support $\Delta\Ptot(z) = {\Ptot}(z) - {\Ptot}(\zmax)$ (blue) and the
weight $\mathcal{W}(z)=\Wsg(z)+\Wext(z)$ of the ISM (red). We also plot each
component of the vertical support shown in Figure <ref> along with
the individual weights, to show the relative importance of the contributing
terms in each model.
Profiles of saturated-state vertical support (blue) and gas weight
(red). Total support and total weight are shown in thick lines. Thin blue
solid, dotted, dashed, dot-dashed lines denote turbulent kinetic, thermal,
turbulent magnetic, and mean magnetic support, respectively. Thin red dotted
and dashed lines denote the weight of the ISM under self- and external gravity,
Figure <ref> shows that vertical equilibrium is remarkably well
satisfied (thick blue and red lines overlap almost completely). In the present
simulations, the magnetic scale height is not very different from the gas scale
height, especially for the turbulent component. Thus, similarly to the gas
pressure terms we can simply replace $\Delta\Pmag\rightarrow {\Pmag}_{,0}$ in
Equations (<ref>) and (<ref>).[Note, however, that in
observations the mean magnetic field has a scale height much larger than that
of the warm/cold ISM. This may be due to effects (not included in the present
models) that help drive magnetic flux out of galaxies, including a hot ISM and
cosmic rays. For this reason, Equations (<ref>) and (<ref>) are
most generally written in terms of differences in the magnetic support, and
also allow for differences in radiation and cosmic ray pressure across the
warm/cold ISM layer <cit.>.] Since the weight of the gas
(RHS of Eq. (<ref>)) is more or less the same for all of the present
models ($\chi\sim1$ within 25% from Table <ref>), the additional
support from both the mean and turbulent magnetic fields necessarily implies a
reduction in the turbulent and thermal pressures compared to unmagnetized
models. Although external gravity dominates the weight at high-$|z|$, self-
and external gravity contributions are almost the same close to the midplane.
Within the turbulent driving layer, turbulent pressure dominates other support
terms for all models, while the thermal pressure dominates at high-$|z|$. The
turbulent and mean magnetic support terms are as important as the thermal
pressure at the midplane. Only for Model MB1, the mean magnetic support is
substantial at all $z$. Table <ref> includes the midplane values of
the contribution to vertical support from each component.
§.§ Regulation of Star Formation Rates
Feedback yields, defined in Equation (<ref>): (a) $\etaturb$,
(b) $\etath$, (c) $\etamagt$, and (d) $\etamago$. Box and whisker plot is as in
Figure <ref>. Note that the mean (square symbol) and median (red
line) of $\etaturb$ are different since the spikes in time evolution of
$\Pturb$ (see Figure <ref>(a)) caused by SN feedback affect the mean
more than the median.
We compute the feedback yield (ratio of midplane support to surface density of
star formation rate) for each component at saturation, and report values in
Table <ref> in the units of Equation (<ref>). These yields
are a quantitative measure of the efficiency of feedback for controlling star
formation. Figure <ref> presents a box and whisker plot for each
component feedback yield. As shown in Figure <ref>, the individual
feedback yields except $\etamago$ are quite similar for all models. Notably,
the magnetized models and unmagnetized models have comparable value for both
turbulent and thermal feedback yields, $\etaturb\sim 3.5-4$ and
$\etath\sim1.1-1.4$. The agreement of $\etaturb$ between HD and MHD models
stems from the similarity in dissipation timescales for HD and MHD turbulence
Except for models MA100 and MB1, the magnetic field and especially the
turbulent magnetic energy become fairly well saturated, due to efficient
generation from the small-scale turbulent dynamo. The resulting turbulent
magnetic feedback yield for these models is $\etamagt\sim1.3-1.5$, providing a
ratio $\etaturb/\etamagt\sim 3-4$. This is equivalent to the energy ratios of
$\Ekin/\Emagt\sim (1.5-2)$ found in driven turbulence MHD simulations
Table <ref> includes $\etamago$ for reference. However, it should be
kept in mind that the mean magnetic field support is less directly related to
the SFR than the other terms. Mean magnetic fields arise from the mean-field
dynamo, which depends on the turbulent magnetic field and therefore indirectly
on the SFR, but also depends on other physical effects in a complex manner that
is not well understood. From the present simulations, we simply note that
$\etamago$ is comparable to $\etamagt$ for the saturated models.
The theoretical idea of near-linear relationships between SFRs and turbulent
and thermal pressures for self-regulated equilibrium ISM states was introduced
and initially quantified by
<cit.>. In the
present simulations, our momentum feedback prescription for SNe injects radial
momentum of
$p_* = 3\times 10^5 \Msun\kms$
within a $10\pc$ sphere (see
Paper I for details); other work confirms this value for the momentum from a
single SN of energy $10^{51} \erg$, insensitive to the mean value of the
ambient density or cloudy structure in the ambient ISM <cit.>. Our adopted value for the total mass
of new stars formed per SN is $m_*=100\Msun$
<cit.>. For these feedback parameters, the
momentum flux/area in the vertical direction is then
this is the
effective turbulent driving rate in the vertical direction. If dissipation of
turbulence occurs in approximately a vertical crossing time over $H$ (the main
energy-containing scale), the expected saturation level for turbulent pressure
is ${\Pturb}_{,0}\approx\Pdriv$, giving $\etaturb=3.6$. Direct calibration in
Paper I gives ${\Pturb}_{,0}/\Pdriv=1.20\SigSFRnorm^{-0.11}$ <cit.>. Including results from both previous and current
simulations, we obtain ${\Pturb}_{,0}/\Pdriv=0.9$-$1.1$ from all HD and MHD
models. Thus, the turbulent pressure is consistent with theoretical
predictions, insensitive to magnetization.
As the SFR varies in our simulations, the time-varying heating rates move the
thermal equilibrium curves up and down in the density-pressure phase plane. To
maintain a two-phase medium within the midplane layer, the actual thermal
pressure of gas must change to be self-consistent with the changing heating
rate. Specifically, for a two-phase medium the range of the midplane thermal
pressure is between the minimum and maximum pressures of the cold and warm
medium, $\Pmin$ and $\Pmax$, respectively. Our adopted cooling and heating
formalism gives ${\Pmin}_{,3}=0.7\SigSFRnorm$ and ${\Pmax}_{,3}=2.2\SigSFRnorm$
<cit.>. The geometric mean of
these two pressures is representative of the expected thermal pressure in a
two-phase medium <cit.>,
yielding ${\Ptwo}_{,3}\equiv({\Pmin}_{,3}{\Pmax}_{,3})^{1/2}=1.2\SigSFRnorm$.
argued that the self-consistent expected midplane pressure for a
star-forming disk in equilibrium is ${\Pth}_{,0}\sim\Ptwo$, corresponding to
$\etath=1.2$ for the thermal feedback yield. Here, we obtain
${\Pth}_{,0}/\Ptwo=1.0$-$1.3$ for all HD and MHD models, consistent with the
theory in <cit.>,
and with the numerical results in Paper I <cit.>, ${\Pth}_{,0}/\Ptwo=1.09\SigSFRnorm^{-0.14}$.
In addition to the turbulent and thermal pressures, the turbulent magnetic
support is also directly related to the SFR. As demonstrated in Figures
<ref> and <ref>, the small-scale turbulent dynamo generates
turbulent magnetic fields efficiently. The turbulent magnetic energy is expect
to saturate at roughly equipartition level with turbulent kinetic energy. In
our simulations, the saturation level of turbulent magnetic energy is $\sim
40\%$ or slightly smaller compared to the turbulent kinetic energy (see
Figure <ref>). If turbulent kinetic and magnetic components are all
isotropic so that $\Pturb=(1/3)\rho |\delta \mathbf{v}|^2=(2/3)\Ekin$ and
$\Pmagt=(1/3)|\delta \mathbf{B}|^2/8\pi=(1/3)\Emagt$, then $\Ekin\sim 2\Emagt$
(Figure <ref>) would give $\Pturb\sim 4\Pmagt$. We find
${\Pturb}_{,0}\sim \Pdriv \sim 3 {\Pmagt}_{,0}$, except for Model MB1, see
Table <ref>. In idealized, driven MHD turbulence simulations,
$\Ekin/\Emagt\sim 1-2.5$ for various initial magnetic fields, Mach number, and
compressibility of gas
Our simulations are generally consistent with saturation energy levels from
idealized driven turbulence experiments in a periodic box, keeping in mind that
identical results are not expected considering differences in the setup. (That
is, rather than an idealized periodic box with spectral driving of turbulence,
our simulations incorporate complex physics to model realistic galactic disks
including vertical stratification, compressibility, spatially localized
turbulent driving, self-consistent evolution of mean fields, self-gravity,
presence of cooling and heating, and so on.)
Our simulations are suggestive, but not conclusive, with respect to the
asymptotic equilibrium state of the mean magnetic support and its connection to
SFRs. The mean fields for all models appear to be converging to the same level
of support as the turbulent fields (e.g., as in Model MB10; see Figure
<ref>), which as shown above have ${\Pmagt}\propto\SigSFR$. Note
that the mean magnetic field is anisotropic and dominated by the azimuthal
component, so $\Pmago = \Emago$ in contrast to $\Pmagt = (1/3) \Emagt$ for the
isotropic case. However, longer-term simulations would be needed to confirm
convergence of $\Pmago$, because the evolution time scale for the mean field is
much longer than for the turbulent field (see Figure <ref>). In
addition, the mean field level may in principle be affected by numerical
parameters including the horizontal box size and effective numerical
resistivity. Thus, it remains uncertain whether the mean magnetic support
should be considered as directly related to the SFR or not. Models MA10, MB10,
and MA1 appear closest to saturation in their mean magnetic field, and have
${\Pmago}_{,0}/\Pdriv\sim 0.3$, which would correspond to $\etamago\sim1$. For
reference, Models MA100 and MB1 respectively have ${\Pmago}_{,0}/\Pdriv\sim
0.14$ and 0.9, although with the caveat that these models are not
asymptotically converged.
The SFR surface density as a function of midplane values of the
effective pressure, (a) total pressure $\Ptot\equiv\Pturb+\Pth+\Pmagt+\Pmago$,
(b) feedback pressure $\Pfb\equiv\Pturb+\Pth+\Pmagt$, and (c) kinetic pressure
$\Pturb+\Pth$. The solid line denotes the fit from a wide parameter space of
HD models in Paper I (see text). The vertical dotted line in (a) is the
midplane pressure required for the vertical dynamical equilibrium, $\PDE=\pi G
\Sigma^2(1+\chi)/2=6.6\times10^3\kbol\Punit$ for $\chi=1$. The dashed line in
(b) is the relationship between $\SigSFR$ and $\Pfb$ using
Equation (<ref>) with total feedback yield for the fiducial run, MB10,
Figure <ref> plots $\SigSFR$ as functions of three midplane (effective)
pressures, (a) total pressure
${\Ptot}\equiv{\Pturb}_{}+{\Pth}_{}+{\Pmagt}_{}+{\Pmago}_{}$, (b) total
“feedback” pressure ${\Pfb}\equiv{\Pturb}_{}+{\Pth}_{}+{\Pmagt}_{}$, and (c)
kinetic pressure ${\Pturb}_{}+{\Pth}_{}$. We also plot as a solid line the
fitting result from HD models in Paper I for a wide range of $\Sigma$ and
$\rhosd$, $\SigSFRnorm=2.1[{(\Pturb+\Pth)}/10^4\kbol\Punit)]^{1.18}$. The
vertical dotted line in panel (a) denotes the total ISM weight when $\chi=1$
(as in Table <ref>), $\PDE=\mathcal{W}_{0}=\pi G \Sigma^2(1+\chi)/2
\rightarrow 6.6\times10^3\kbol\Punit$. As shown in Figure <ref>,
vertical dynamical equilibrium indeed holds so that the total midplane pressure
adjusts to match this weight. We note that the total midplane pressure and
weight are very similar for all models ($\chi\sim1$ agrees within 25% for all
cases), since we fix $\Sigma$ and $\rhosd$, and $\sigmaeff$ and $\zeta_d$ are
insensitive to the model parameters. The vertically aligned symbols in
Figure <ref>(a) show graphically the consistency of ${\Ptot}_{,0}$ and
$\mathcal W_{0}$ in all models, and also make clear that the inclusion of
magnetic terms reduces $\SigSFR$ for the same ${\Ptot}_{,0}$.
Since the mean magnetic field can supply vertical support without star
formation (at least on timescales comparable to the disk's vertical crossing
time), the total support requirement from star formation feedback for the
vertical equilibrium can be reduced to ${\Pfb}=\PDE-{\Delta\Pmago}$ (where we
assume $\Pmagt(\zmax) \rightarrow 0$, as turbulent fields are driven only
within the star-forming layer, but we keep $\Pmago(\zmax)$ for generality,
allowing for the possibility that the mean fields are carried into the halo by
winds). Thus, it is interesting to plot $\SigSFR$ as a function of midplane
values of ${\Pfb}$ instead of ${\Ptot}$, which we show in
Figure <ref>(b). In the presence of magnetic fields, even with very
weak mean fields, star formation feedback can generate turbulent magnetic
fields at a significant level in a very short time. The additional support
from turbulent magnetic fields enhances overall feedback yield with
$\etamagt\sim1-2$ on top of $\etaturb+\etath\sim 4-5$. Thus, the feedback
pressure is larger by 20%-40% at a given SFR for a magnetized medium compared
to an unmagnetized medium. The dashed line in panel (b) represents
$\SigSFRnorm={\Pfb}_{,3}/(\etaturb+\etath+\etamagt)$ with
$\etaturb+\etath+\etamagt=6.6$, the feedback yield for Model MB10. As
expected, the symbols and the dashed line in Figure <ref>(b) for
magnetized models fall systematically below (or to the right) of the solid
line, which was based on simulation results from HD models with lower total
$\eta$ than MHD models.
Figure <ref>(c) shows another important property of self-regulated
star-forming ISM disks. Since $\etaturb$ and $\etath$ are nearly unchanged
with and without magnetic fields, the same relationship between kinetic
midplane pressure ${\Pturb+\Pth}$ and SFR holds as we found in Paper I (solid
line). However, the present work considers only a single value of $\Sigma$
and $\rhosd$, and simulations for a wider range of these parameters, for
magnetized models, would be needed to confirm the robustness of this
In short, the presence of the magnetic fields will reduce the SFR by increasing
the total efficiency of feedback ($\etaturb+\etath+\etamagt$) and reducing the
required dynamical equilibrium pressure from feedback ($\PDE-\Delta{\Pmago}$).
The general relationship between the SFR surface density and the midplane
pressure including magnetic terms can then be written as
\begin{equation}\label{eq:sfrP}
\SigSFR =10^{-3}\sfrunit\rbrackets{\frac{\etaturb+\etath+\etamagt}{5}}^{-1}
\rbrackets{\frac{\PDE-{\Delta\Pmago}}{5\times10^3\kbol\Punit}}.
\end{equation}
If $\Delta\Pmago$ is asymptotically proportional to $\SigSFR$ (which would be
true if the mean-field dynamo arises from feedback-driven fluctuations in the
magnetic and velocity fields, similar to the small-scale turbulent dynamo), a
relationship of the form $\SigSFR \propto \PDE$ with larger total efficiency of
feedback would also hold. Alternatively, $\Delta \Pmago$ may be negligible
compared to $\PDE$ if the scale height of the mean magnetic field is large
compared to that of the neutral ISM.
§ SUMMARY AND DISCUSSION
In Paper I, we developed realistic galactic disk models with cooling and
heating, self-gravity, sheared galactic rotation, and self-consistent star
formation feedback using direct momentum injection by SNe and time-varying
heating rates. Here, we extend the fiducial Solar-neighborhood-like model of
Paper I to include magnetic fields with varying initial field strengths and
distributions. We also alter the initial and boundary conditions from the
original model to minimize artificial initial vertical oscillations. We show
in Section <ref> that the overall evolution remains the same in HD
models irrespective of differences in initial and boundary conditions,
confirming the robustness of results in Paper I. In both HD and MHD models, the
time evolution of ISM properties reaches a saturated equilibrium state within
$\sim \torb$, except for the mean magnetic field, which continues to secularly
evolve over several $\torb$ (see Figures <ref> and <ref>).
In what follows, we summarize the main results derived from analyzing the time
evolution of mean ISM properties and final saturated-state vertical profiles,
with careful consideration of the slow variation of mean magnetic fields.
* Generation and Saturation of Magnetic Fields – Turbulent motions
in our simulations quickly develop and saturate. The same is true for
turbulent magnetic fields (see Figure <ref>(b)). Beyond $\sim
\torb$, saturated levels of turbulent kinetic and magnetic energies are similar
for all MHD models except Model MA100, which has a very weak initial field.
Model MA100 converges to the same levels of $\Ekin$ and $\Emagt$ as other
models by $\sim 3 \torb$. The final saturated state has mass-weighted mean
$\vrms{\delta {v}}\approx 5\kms $ and $\vrms{\delta {v}_A}\approx 3\kms$ for
all MHD models, corresponding to $\Ekin/\Emagt\sim 2.5$.
The growth of magnetic fields in turbulent flows is believed to be a
consequence of turbulent dynamo action, which generally refers to mechanisms of
energy conversion from kinetic to magnetic. Dynamos are classified into
large-scale (mean field) dynamos and small-scale (fluctuation or turbulent)
dynamos. The former and latter respectively generate magnetic fields with
scales larger and smaller than the turbulence injection scale. In many spiral
galaxies, there are significant large-scale, regular azimuthal magnetic fields
<cit.>, which may result from the so-called
$\alpha-\Omega$ dynamo driven by turbulence and differential rotation
<cit.>. Our initial conditions with mean fields
in the $\yhat$ direction is motivated by observed preferentially azimuthal (or
spiral-arm aligned) fields. Self-consistent growth of mean fields from the
very weak primordial magnetic fields is an important and active research area
<cit.>. Although growth of mean magnetic
fields is not the main focus of this paper, some hints can be found in the
evolution of mean fields in cases with initially weak fields (Models MA100 and
MA10 in Figure <ref>(d)).
More straightforward results from our simulations are the saturation properties
of turbulent magnetic fields, which can be understood in the context of
small-scale dynamos. It is broadly known that turbulent conductive flows can
amplify their own magnetic fields through random field line stretching,
twisting, and folding <cit.>.
Recently, <cit.> have quantified three stages of evolution
using a comprehensive set of nonlinear simulations for incompressible MHD
turbulence <cit.>. The
magnetic energy grows exponentially until it reaches equipartition at the
kinetic energy dissipation scale. Then, growth becomes linear until the entire
energy spectrum reaches equipartition up to the energy injection scale. The
saturation amplitude of turbulent magnetic energy for small-scale
incompressible dynamos approaches $\sim 40\%$ of the total energy at large
Reynolds number <cit.>. <cit.>
obtained $\sim 30\%$ of the total energy from their compressible simulations.
<cit.> obtained similar results for various Mach numbers
from isothermal, compressible MHD simulations <cit.>. Nearly irrespective of the Mach number, the
ratios of magnetic energy to total (kinetic+magnetic) energy in fluctuations
are $30-40\%$.[ <cit.> investigated the growth of
magnetic energy in forced turbulence from very weak initial fields with a
variety of Mach numbers and two different forcing schemes (solenoidal and
compressive). Their ratios of magnetic to kinetic energy at saturation are
0.4-0.6 for subsonic, solenoidally driven turbulence but only 1% or less for
supersonic turbulence. However, their saturated magnetic energy is not
measured at final saturation but the end of the exponential growth stage.
Based on our results (slow convergence of turbulent magnetic energy in Model
MA100), the low saturation level in the supersonic (or transonic) cases they
report may also owe to their extremely weak initial mean fields.] With stronger
initial field, saturation is achieved more rapidly.
The small-scale dynamo provides a fast and universal mechanism for magnetic
field amplification <cit.>. Thus, we
expect that qualitatively similar processes will occur in our simulations in
spite of their greater physical complexity, including compressibility, vertical
stratification, SN driven turbulence, self-gravity, and multiphase gas with
cooling and heating. In early phases $t<\torb$, rapid growth and saturation of
the turbulent magnetic energy are induced by driven turbulence (see
Figure <ref>). After $\torb$, turbulence is driven entirely by SN
feedback, with a rate that depends on the SFR. Vertical dynamical equilibrium
(see below) dictates the turbulent pressure level, and hence SFR, that is
self-consistently reached in these disk systems. The final saturated-state
turbulent magnetic energy level is about half of turbulent kinetic energy, as
expected in small-scale dynamo simulations.
* Vertical Structure of the Diffuse ISM –
The vertical distribution and gravitational support of the diffuse ISM in the
Milky Way has been investigated many times under the assumption of vertical
dynamical equilibrium[We prefer to use the term “vertical dynamical
equilibrium” instead of “hydrostatic equilibrium” since the gas is not
“static” – turbulent pressure arising from large-scale gas motions is
crucial to the vertical force balance.]
<cit.>. In
Section <ref>, we validate that vertical dynamical equilibrium is
satisfied in turbulent, star-forming, magnetized galactic disks, extending
results from previous multiphase ISM simulations with various physical
At the midplane, vertical support is dominated by the turbulent (kinetic)
pressure except in model (MB1) with very strong mean magnetic fields (see
Figure <ref> and Table <ref>). Thermal and turbulent
magnetic supports are comparable to each other and about 2-3 times smaller than
the turbulent pressure. Compared to HD models, MHD disk models can maintain
smaller turbulent and thermal pressures owing to additional vertical support
from the turbulent and mean magnetic fields. Despite of different
magnetization, the total effective velocity dispersion $\sigmaeff$ varies by
only $\sim 25\%$ over all models (see Table <ref>), resulting in
similar variation in $\chi$ and hence dynamical equilibrium midplane pressure
Our models demonstrate that vertical dynamical equilibrium is indeed a good
assumption even though disks are magnetized and highly dynamic. However,
practical application of vertical equilibrium in observations is not simple.
Even in the Solar neighborhood, uncertainty is substantial. For example,
<cit.> calculated the total equilibrium pressure based on
observations of the vertical gas distribution and gravity
<cit.>, and taking the
difference with observed thermal+turbulent pressure inferred significant
contributions from non-thermal (cosmic-ray and magnetic) pressures. However,
<cit.> have argued that local (high-latitude) H1
21 cm emission data can be fitted with three-component Gaussians, and that
non-thermal vertical support is not necessary to explain the H1
distribution. Considering the large observed scale height of non-thermal
pressure terms, the vertical support of H1 within $|z|<1\kpc$ (which
depends on the pressure gradient, not pressure itself), can be explained solely
by thermal + turbulent terms. Later, <cit.>
simultaneously analysed the distribution of emission at 21 cm from warm/cold
H1 with H$\alpha$ from diffuse ionized gas, soft X-rays for the hot
medium, and synchrotron radiation combined with $\gamma$-ray emission probing
magnetic fields and cosmic rays. They concluded that within $400\pc$ of the
midplane, cosmic ray support is not required, and turbulent magnetic fields
contribute with $\mathcal{R}\sim 1/3$ ($\alpha=1/3$ in their notation). The
contribution of turbulent magnetic support compared to thermal+turbulent
kinetic pressure in our simulations is $\sim 25\%$ from Table <ref>,
in good agreement with this result. In our simulations, the mean magnetic
field has a relatively small scale height (Figure <ref>) and vertical
support slightly less than that of the turbulent magnetic field. In contrast,
observations indicate that magnetic fields extend into the halo in equipartion
with the pressures of cosmic rays and hot gas. The decline in mean magnetic
fields with height in our present simulations may be due to the absence of a
hot medium and cosmic rays; this issue will be addressed in future work.
Adoption of vertical equilibrium is useful to estimate the midplane pressure in
external galaxies. The pressure cannot be measured directly except for nearby
edge-on disks, and even in that case requires deprojection and an assumption
for the turbulent velocity dispersion
<cit.>. As outlined in
Section <ref>, to get the total dynamical equilibrium pressure (or
the total weight of the gas; see Equation (<ref>)), one may need to
determine $\Sigma$, $\Sigma_*$, $\sigma_*$, and $\sigmaeff$ (or $\sigma_z$ and
$\mathcal{R}$), assuming the parameter $\zeta_d\sim 0.4-0.5$ is nearly
constant. In several systematic studies of nearby galaxies
$\sigmaeff$ and/or $H_*\propto \sigma_*^2/\Sigma_*$ are assumed to be constant
to determine the total midplane pressure. Observational measurements have
suggested that H1 velocity dispersions are nearly constant with
$\sigma_z\sim 5-10\kms$, especially for the atomic gas dominated regime, using
single Gaussian fitting <cit.> and intensity-weighted
second velocity moments <cit.>. Recent studies of global
H1 kinematics analysis to obtain “superprofiles” (averaged H1
profiles for the entire galaxy) in nearby spiral and dwarf galaxies have
arrived at similar conclusions <cit.>.
In this paper, we show that the contribution from turbulent magnetic field to
$\mathcal{R}$ (or, equivalently, $[\delta {v}_A^2/2 - \delta
v_{A,z}^2]/\sigma_z^2)$ is $\approx 0.3$ insensitive to the mean magnetic field
strength for saturated models (except Model MB1). This suggests that if the
total velocity dispersion $\sigma_z$ can be measured accurately, an estimate of
$\sigmaeff$ needed to compute $C$ and $\chi$ and therefore the total midplane
pressure may be obtained using a typical value of $\mathcal{R}\sim0.3$ even if
the observed magnetic field strength is not measured.
The density structure in our simulations is well characterized by two
components, a cold and non-cold medium (see Figure <ref>). For the
present models, we find scale heights of cold and non-cold components are
$H\sim30\pc$ and $110\pc$, respectively. <cit.> have used
three-component Gaussians to fit H1 emission line observations near the
Sun, obtaining vertical scale heights equivalent to $H\sim80-130\pc$, $\sim
150-300\pc$, and $\sim 600-750\kpc$ <cit.>. A
recent Herschel galactic survey of [C2], along with ancillary
H1, ${}^{12}$CO, ${}^{13}$CO, and C${}^{18}$O data, shows that the
equivalent vertical scale heights of [C2] sources with CO and without
CO which trace colder/denser vs. warmer/more diffuse gas, respectively, are
$\sim70\pc$ and $170\pc$ <cit.>. Our
simulations agree with both of these observational studies in the sense that
the warmer component has $\sim 2-3$ times the scale height of the colder
component. However, our measured $H$ values are somewhat lower than observed
Solar-neighborhood values, as our measured values of $\sigma_z$ and $\SigSFR$.
Simulations currently underway that include a hot ISM suggest that $H$,
$\sigma_z$, and $\SigSFR$ may increase.
* Feedback Efficiencies and Star Formation Laws –
One of the important conclusions of this study is that the feedback yields for
turbulent and thermal pressure are unchanged by the presence of magnetic
fields. This is mainly because the HD and MHD turbulent energy dissipation
rates are similar. The dissipation time scale is always of order the crossing
time at the driving scale (or main energy-containing scale) both for HD
turbulence <cit.> and MHD turbulence
and both compressible and incompressible flows. Turbulent magnetic fields
saturate at a similar timescale to turbulent velocities. Balancing turbulent
driving with turbulent dissipation therefore leads to direct proportionality
between $\SigSFR$ and the turbulent kinetic and magnetic pressures. Similarly,
balancing heating and cooling leads to a direct proportionality between
$\SigSFR$ and the thermal pressure. turbulent kinetic and magnetic pressures.
Defining “yield” as the pressure-to-$\SigSFR$ ratio (in convenient units; see
Equation <ref>), we obtain turbulent, thermal, and turbulent magnetic
feedback yields as $\etaturb\sim 3.5-4$, $\etath\sim 1.1-1.4$, and
$\etamagt\sim 1.3-1.5$, respectively. Since the ISM weight and therefore total
pressure is nearly the same irrespective of magnetization, the addition of
magnetic terms to the total $\eta$ reduces $\SigSFR$ for MHD compared to HD
simulations at a given $\Sigma$ and $\rhosd$.
Correlations between the estimated ISM equilibrum pressure $\PDE$ and the
molecular content and star formation rate have been identified empirically
In particular, the $P-\SigSFR$ relation has much less scatter than the
classical Kennicutt-Schmidt relationship <cit.>
between gas surface density $\Sigma$ and $\SigSFR$ for the atomic-dominated
regime <cit.>. As argued in
our previous work
<cit.>, this is
because the total midplane pressure is directly related to the SFR, whereas ISM
properties (and the SFR) can vary considerably at a given value of $\Sigma$
depending on the gravitational potential confining the disk. As shown in
<cit.>, in the
starburst regime where self-gravity dominates the potential, both pressure and
surface density correlate well with SFR surface density, giving
$\SigSFR\propto\PDE\propto\Sigma^2$ (shallower reported slopes are arguably due
to too-high assumed CO-to-H$_2$ ratios at high $\Sigma$; see
). In the atomic-dominated regime, as modelled in
this paper, the dynamical equilibrium pressure depends on both
$\mathcal{W}_{\rm sg}\propto\Sigma^2$ and $\mathcal{W}_{\rm ext}\propto
\Sigma\sqrt{\rhosd}$ (see Equations (<ref>) and (<ref>)). In
outer disks where $\rhosd$ varies more than $\Sigma_{\mathrm{atomic}}\sim
6-10\Msun \pc^{-2}$, simple Kennicutt-Schmidt laws fail.
As demonstrated in this paper, there can also be scatter introduced in
$\SigSFR$ vs. $\PDE$ if support from the mean magnetic field contributes
substantially without itself having $\Delta\Pmago \propto \SigSFR$. Even if a
mean-field dynamo secularly leads to a well-define asymptotic ratio between
$\Delta\Pmago$ and $\SigSFR$, the long timescale to reach this may mean that
$\Delta\Pmago$ is not well correlated with the recent SFR. If the vertical
scale height of the mean field is larger than that of the star-forming gas,
however, the support from $\Delta\Pmago$ would be small even if the mean field
pressure is non-negligible, which would reduce scatter in $\SigSFR$ vs. $\PDE$.
This work was supported by grants AST-1312006 from the National Science
Foundation and NNX14AB49G from NASA. We thank E. Blackman and J.-M. Shi for
discussions of dynamos, and the referee for helpful comments on the
§ NUMERICAL CONVERGENCE
We adopt a standard spatial resolution of $\Delta=$2pc throughout this paper.
Here, we present a convergence study for physical properties at the saturated
state. Since a simulation with $\Delta=$1pc for the same box size would be
too computationally expensive for our current facilities, we instead have rerun
Model MB10 with $\Delta=$1pc, 2pc, and 4pc using a smaller horizontal box
($L_x=L_y=256\pc$) and a halved the turbulence driving period ($t_{\rm
drive}=0.5\torb$) and final time ($t_{\rm end}=2\torb$).
Figure <ref> summarizes the midplane support components and the
energy ratios for $(\torb, 2\torb)$ with box-and-whisker plots. We plot the
quantities by taking the logarithms for clearer presentation of temporal
fluctuations. We label the original MB10 model (full box with $\Delta=2pc$)
as `MB10' and the smaller box counterparts at three different resolutions
as `1pc', `2pc', and `4pc'. Although the smaller box increases temporal
fluctuations as expected, all components except the mean magnetic component
show statistically converged results.
The mean magnetic component in the 1pc simulation is systematically reduced,
and has the largest temporal fluctuation, also causing a slightly smaller
saturated value of the turbulent magnetic component. Considering that the mean
magnetic component always shows the slowest convergence, it is possible that
with a longer integration time, the mean magnetic field in the 1pc model would
approach that of the other models. However, it is also possible that it would
remain at this reduced level. Since the evolution of mean magnetic fields is
not the main focus of this paper, we defer further study of the mean magnetic
component to future work, which will include more comprehensive investigation
of mean field dynamo.
Box and whisker plots of the logarithms of the midplane support components
(top; (a)-(d)) and the energy ratios (bottom; (e)-(h)) over the
interval $(\torb,2\torb)$. From left to right columns, we show the
turbulent kinetic, thermal, turbulent magnetic, and mean magnetic components.
We label the original MB10 model as `MB10', while the smaller box counterparts
at three different resolutions are labeled `1pc', `2pc', and `4pc'.
|
1511.00267
|
Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, California 91125, USA
QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, Netherlands
Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Center for Computation and Technology, Louisiana State University, Baton
Rouge, Louisiana 70803, USA
The entropic uncertainty relation with quantum side information (EUR-QSI) from
[Berta et al., Nat. Phys. 6, 659 (2010)] is a unifying
principle relating two distinctive features of quantum mechanics: quantum
uncertainty due to measurement incompatibility, and entanglement. In these
relations, quantum uncertainty takes the form of preparation uncertainty where
one of two incompatible measurements is applied. In particular, the
“uncertainty witness” lower bound in the
EUR-QSI is not a function of a post-measurement state. An insightful proof of
the EUR-QSI from [Coles et al., Phys. Rev. Lett. 108, 210405
(2012)] makes use of a fundamental mathematical consequence of the postulates
of quantum mechanics known as the non-increase of quantum relative entropy
under quantum channels. Here, we exploit this perspective to establish a
tightening of the EUR-QSI which adds a new state-dependent term in the lower
bound, related to how well one can reverse the action of a quantum
measurement. As such, this new term is a direct function of the
post-measurement state and can be thought of as quantifying how much
disturbance a given measurement causes. Our result thus quantitatively unifies
this feature of quantum mechanics with the others mentioned above. We have
experimentally tested our theoretical predictions on the IBM Quantum
Experience and find reasonable agreement between our predictions and
experimental outcomes.
§ INTRODUCTION
The uncertainty principle is one of the cornerstones of modern physics,
providing a striking separation between classical and quantum mechanics
<cit.>. It is routinely used to reason about the behavior of quantum
systems, and in recent years, an information-theoretic refinement of it that
incorporates quantum side information has been helpful for witnessing
entanglement and in establishing the security of quantum key distribution
<cit.>. This latter refinement, known as the entropic uncertainty
relation with quantum side information (EUR-QSI), is the culmination of a
sequence of works spanning many decades
and is the one on which we focus here.
Tripartite uncertainty relations. There are two variations of the
EUR-QSI <cit.>, one for tripartite and one for bipartite scenarios.
For the first, let $\rho_{ABE}$ denote a tripartite quantum state shared
between Alice, Bob, and Eve, and let $\mathbb{X}\equiv\{P_{A}^{x}\}$ and
$\mathbb{Z}=\{Q_{A}^{z}\}$ be projection-valued measures (PVMs) that can be
performed on Alice's system (note that considering PVMs implies statements for
the more general positive operator-valued measures, by invoking the Naimark
extension theorem <cit.>). If Alice chooses to measure $\mathbb{X}$,
then the post-measurement state is as follows:
\begin{align}
\sigma_{XBE} & \equiv\sum_{x}|x\rangle\langle x|_{X}\otimes\sigma_{BE}%
\sigma_{BE}^{x} & \equiv\operatorname{Tr}_{A}\{(P_{A}^{x}\otimes I_{BE}%
)\rho_{ABE}\}. \label{eq:sig_X_state}%
\end{align}
Similarly, if Alice instead chooses to measure $\mathbb{Z}$, then the
post-measurement state is
\begin{align}
\omega_{ZBE} & \equiv\sum_{z}|z\rangle\langle z|_{Z}\otimes\omega_{BE}%
\omega_{BE}^{z} & \equiv\operatorname{Tr}_{A}\{(Q_{A}^{z}\otimes I_{BE}%
)\rho_{ABE}\}. \label{eq:om_Z_state}%
\end{align}
In the above, $\{|x\rangle_{X}\}_{x}$ and $\{|z\rangle_{Z}\}_{z}$ are
orthonormal bases that encode the classical outcome of the respective
measurements. The following tripartite EUR-QSI in (<ref>)
quantifies the trade-off between Bob's ability to predict the outcome of the
$\mathbb{X}$ measurement with the help of his quantum system $B$ and Eve's
ability to predict the outcome of the $\mathbb{Z}$ measurement with the help
of her system $E$:
\begin{equation}
H(X|B)_{\sigma}+H(Z|E)_{\omega}\geq-\log c, \label{eq:EUR-QSI-1}%
\end{equation}
where here and throughout we take the logarithm to have base two. In the
\begin{equation}
H(F|G)_{\tau}\equiv H(FG)_{\tau}-H(G)_{\tau}=H(\tau_{FG})-H(\tau_{G})
\end{equation}
denotes the conditional von Neumann entropy of a state $\tau_{FG}$, with
$H(\tau)\equiv-\operatorname{Tr}\{\tau\log\tau\}$, and the parameter $c$
captures the incompatibility of the $\mathbb{X}$ and $\mathbb{Z}$
\begin{equation}
c\equiv\max_{x,z}\Vert P_{A}^{x}Q_{A}^{z}\Vert_{\infty}^{2}\in\lbrack0,1].
\label{eq:c-param}%
\end{equation}
The conditional entropy $H(F|G)_{\tau}$ is a measure of the uncertainty about
system $F$ from the perspective of someone who possesses system $G$, given
that the state of both systems is $\tau_{FG}$. The uncertainty relation in
(<ref>) thus says that if Bob can easily predict $X$ given $B$
(i.e., $H(X|B)$ is small) and the measurements are incompatible, then it is
difficult for Eve to predict $Z$ given $E$ (i.e., $H(Z|E)$ is large). As such,
(<ref>) at the same time captures measurement incompatibility and
the monogamy of entanglement <cit.>. A variant of
(<ref>) in terms of the conditional min-entropy <cit.> can
be used to establish the security of quantum key distribution under particular
assumptions <cit.>.
The EUR-QSI in (<ref>) can be summarized informally as a game
involving a few steps. To begin with, Alice, Bob, and Eve are given a state
$\rho_{ABE}$. Alice then flips a coin to decide whether to measure
$\mathbb{X}$ or $\mathbb{Z}$. If she gets heads, she measures $\mathbb{X}$ and
tells Bob that she did so. Bob then has to predict the outcome of her
$\mathbb{X}$ measurement and can use his quantum system $B$ to help do so. If
Alice gets tails, she instead measures $\mathbb{Z}$ and tells Eve that she did
so. In this case, Eve has to predict the outcome of Alice's $\mathbb{Z}$
measurement and can use her quantum system $E$ as an aid. There is a trade-off
between their ability to predict correctly, which is captured by (<ref>).
Bipartite uncertainty relations. We now recall the second variant of
the EUR-QSI from <cit.>. Here we have a bipartite state $\rho_{AB}%
$ shared between Alice and Bob and again the measurements $\mathbb{X}$ and
$\mathbb{Z}$ mentioned above. Alice chooses to measure either $\mathbb{X}$ or
$\mathbb{Z}$, leading to the respective post-measurement states $\sigma_{XB}$
and $\omega_{ZB}$ defined from (<ref>) and
(<ref>) after taking a partial trace over the $E$ system. The
following EUR-QSI in (<ref>) quantifies the trade-off between
Bob's ability to predict the outcome of the $\mathbb{X}$ or $\mathbb{Z}$
\begin{equation}
H( Z|B) _{\omega}+H( X|B) _{\sigma}\geq-\log c+H( A|B) _{\rho},
\label{eq:EUR-QSI-2}%
\end{equation}
where the incompatibility parameter $c$ is defined in (<ref>) and
the conditional entropy $H( A|B) _{\rho}$ is a signature of both the mixedness
and entanglement of the state $\rho_{AB}$. For (<ref>) to hold,
we require the technical condition that the $\mathbb{Z}$ measurement be a
rank-one measurement <cit.> (however see also <cit.> for a
lifting of this condition). The EUR-QSI in (<ref>) finds
application in witnessing entanglement, as discussed in <cit.>.
The uncertainty relation in (<ref>) can also be summarized
informally as a game, similar to the one discussed above. Here, we have Alice
choose whether to measure $\mathbb{X}$ or $\mathbb{Z}$. If she measures
$\mathbb{X}$, she informs Bob that she did so, and it is his task to predict
the outcome of the $\mathbb{X}$ measurement. If she instead measures
$\mathbb{Z}$, she tells Bob, and he should predict the outcome of the
$\mathbb{Z}$ measurement. In both cases, Bob is allowed to use his quantum
system $B$ to help in predicting the outcome of Alice's measurement. Again
there is generally a trade-off between how well Bob can predict the outcome of
the $\mathbb{X}$ or $\mathbb{Z}$ measurement, which is quantified by
(<ref>). The better that Bob can predict the outcome of either
measurement, the more entangled the state $\rho_{AB}$ is.
§ MAIN RESULT
The main contribution of the present paper is to refine and tighten both of the
uncertainty relations in (<ref>) and (<ref>) by
employing a recent result from <cit.> (see also <cit.>).
This refinement adds a term involving measurement reversibility, next to the
original trade-offs in terms of measurement incompatibility and entanglement.
An insightful proof of the EUR-QSIs above makes use of an entropy inequality
known as the non-increase of quantum relative entropy <cit.>.
This entropy inequality is fundamental in quantum physics, providing
limitations on communication protocols <cit.> and thermodynamic processes
<cit.>. The main result of <cit.> offers a
strengthening of the non-increase of quantum relative entropy,
quantifying how well one can recover from the deleterious effects of a noisy
quantum channel. Here we apply the particular result from <cit.> to
establish a tightening of both uncertainty relations in (<ref>)
and (<ref>) with a term related to how well one can
“reverse” an additional $\mathbb{X}$
measurement performed on Alice's system at the end of the uncertainty game, if
the outcome of the $\mathbb{X}$ measurement and the $B$ system are available.
The upshot is an entropic uncertainty relation which incorporates measurement
reversibility in addition to quantum uncertainty due to measurement
incompatibility, and entanglement, thus unifying several genuinely quantum
features into a single uncertainty relation.
In particular, we establish the following refinements of (<ref>)
and (<ref>):
\begin{align}
H(Z|E)_{\omega}+H(X|B)_{\sigma} & \geq-\log c-\log
H(Z|B)_{\omega}+H(X|B)_{\sigma} & \geq-\log c-\log f+H(A|B)_{\rho},
\label{eq:EUR-QSI-2-new}%
\end{align}
where $c$ is defined in (<ref>),
\begin{equation}
f\equiv F(\rho_{AB},\mathcal{R}_{XB\rightarrow AB}(\sigma_{XB})),
\end{equation}
and in (<ref>) we need the projective $\mathbb{Z}$
measurement to be a rank-one measurement (i.e., $Q_{A}^{z}=|z\rangle\langle
z|$). In addition to the measurement incompatibility $c$, the term $f$
quantifies the disturbance caused by one of the measurements, in particular,
how reversible such a measurement is. $F(\rho_{1},\rho_{2})\equiv\Vert
\sqrt{\rho_{1}}\sqrt{\rho_{2}}\Vert_{1}^{2}$ denotes the quantum fidelity
between two density operators $\rho_{1}$ and $\rho_{2}$ <cit.>, and
$\mathcal{R}_{XB\rightarrow AB}$ is a recovery quantum channel with
input systems $XB$ and output systems $AB$. Appendix <ref> details a
proof for (<ref>) and (<ref>). In Section <ref>, we discuss several simple exemplary states and measurements to which (<ref>) applies, and in Section <ref>, we detail the results of several experimental tests of the theoretical predictions, finding reasonable agreement between the experimental results and our predictions.
In the case that the $\mathbb{Z}$ measurement has the form $\{Q_{A}%
^{z}=|z\rangle\langle z|_{A}\}_{z}$ for an orthonormal basis $\{|z\rangle
_{A}\}_{z}$, the action of the recovery quantum channel $\mathcal{R}%
_{XB\rightarrow AB}$ on an arbitrary state $\xi_{XB}$ is explicitly given as
follows (see Appendix <ref> for details):
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}(\xi_{XB})=\sum_{z,x,z^{\prime}}|z\rangle\langle
z|_{A}P_{A}^{x}|z^{\prime}\rangle\langle z^{\prime}|_{A}\otimes\mathcal{R}%
_{XB\rightarrow B}^{x,z,z^{\prime}}(\xi_{XB}), \label{eq:main:recovery1}%
\end{equation}
\begin{equation}
\mathcal{R}_{XB\rightarrow B}^{x,z,z^{\prime}}(\xi_{XB})\equiv\int_{-\infty
}^{\infty}dt\ p(t)\left( \omega_{B}^{z}\right) ^{\frac{1-it}{2}}\left(
\theta_{B}^{x}\right) ^{\frac{-1+it}{2}}\operatorname{Tr}_{X}\{|x\rangle
\langle x|_{X}(\xi_{XB})\}\left( \theta_{B}^{x}\right) ^{\frac{-1-it}{2}%
\end{equation}
with the probability density $p(t)\equiv\frac{\pi}{2}(\cosh(\pi t)+1)^{-1}$.
(Note that $\mathcal{R}_{XB\rightarrow B}^{x,z,z^{\prime}}$ is not a
channel—we are merely using this notation as a shorthand.) In the above,
$\theta_{XB}$ is the state resulting from Alice performing the $\mathbb{Z}%
$ measurement, following with the $\mathbb{X}$ measurement, and then
discarding the outcome of the $\mathbb{Z}$ measurement:
\begin{align}
\theta_{XB} & \equiv\sum_{x}|x\rangle\langle x|_{X}\otimes\theta_{B}%
\theta_{B}^{x} & \equiv\sum_{z}\langle z|P_{A}^{x}|z\rangle\ \omega^{z}_{B}.
\label{eq:theta-state}%
\end{align}
For this case, $\omega^{z}_{B}$ from (<ref>) reduces to
$\omega^{z}_{B} = (\langle z|_{A}\otimes I_{B})\rho_{AB}(|z\rangle_{A}\otimes
I_{B})$. As one can readily check by plugging into (<ref>),
the recovery channel $\mathcal{R}$ has the property that it perfectly reverses
an $\mathbb{X}$ measurement if it is performed after a $\mathbb{Z}%
$ measurement:
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}(\theta_{XB})=\sum_{z}|z\rangle\langle
z|_{A}\otimes\omega^{z}_{B}. \label{eq:Z-X-recover-perfect}%
\end{equation}
The fidelity $F(\rho_{AB},\mathcal{R}_{XB\rightarrow AB}(\sigma_{XB}))$ thus
quantifies how much disturbance the $\mathbb{X}$ measurement causes to the
original state $\rho_{AB}$ in terms of how well the recovery channel
$\mathcal{R}$ can reverse the process. We note that there is a trade-off
between reversing the $\mathbb{X}$ measurement whenever it is greatly
disturbing $\rho_{AB}$ and meeting the constraint in
(<ref>). Since the quantum fidelity always takes a
value between zero and one, it is clear that (<ref>) and
(<ref>) represent a state-dependent tightening of
(<ref>) and (<ref>), respectively.
§ INTERPRETATION
It is interesting to note that just as the original relation in
(<ref>) could be used to witness entanglement, the new relation
can be used to witness both entanglement and recovery from measurement, as
will be illustrated using the examples below. That is, having low conditional
entropy for both measurement outcomes constitutes a recoverability witness,
when given information about the entanglement.
We recalled above the established “uncertainty
games” in order to build an intuition for
(<ref>) and (<ref>). In order to further understand
the refinements in (<ref>) and (<ref>), we
could imagine that after either game is completed, we involve another player
Charlie. Regardless of which measurement Alice performed in the original game,
she then performs an additional $\mathbb{X}$ measurement. Bob sends his
quantum system $B$ to Charlie, and Alice sends the classical outcome of the
final $\mathbb{X}$ measurement to Charlie. It is then Charlie's goal to
“reverse” the $\mathbb{X}$ measurement in
either of the scenarios above, and his ability to do so is limited by the
uncertainty relations in (<ref>) and
(<ref>). Figure <ref> depicts this
game. In the case that (a) Alice performed an $\mathbb{X}$ measurement in the
original game, the state that Charlie has is $\sigma_{XB}$. In the case that
(b) Alice performed a $\mathbb{Z}$ measurement in the original game, then the
state that Charlie has is $\theta_{XB}$. Not knowing which state he has
received, Charlie can perform the recovery channel $\mathcal{R}$ and be
guaranteed to restore the state to
\begin{equation}
\sum_{z}|z\rangle\langle z|\otimes\big(\langle z|\otimes I_{B}\big)\rho
_{AB}\big(|z\rangle\otimes I_{B}\big)
\end{equation}
in the case that (b) occurred, while having a performance limited by
(<ref>) or (<ref>) in the case that (a) occurred.
Measurement reversibility game. How well can Charlie reverse
the action of the $\mathbb{X}$ measurement in either scenario (a) or (b)? The
quantities in (<ref>) and (<ref>) other than
$f$ constitute a “recoverability witness,” quantifying Charlie's ability to
do so.
§ EXAMPLES
It is helpful to examine some examples in order to build an intuition for our
refinements of the EUR-QSIs. Here we focus on the bipartite EUR-QSI
in (<ref>) and begin by evaluating it for some “minimum
uncertainty states” <cit.> (see also <cit.>). These are states
for which the original uncertainty relation in (<ref>) is already
tight, i.e., an equality. Later, we will consider the case of a representative
“maximum uncertainty state,” that is, a state for which the original
uncertainty relation (<ref>) is maximally non-tight. This last
example distinguishes our new contribution in (<ref>) from
the previously established bound in (<ref>).
For all of the forthcoming examples, we take the $\mathbb{X}$ measurement to
be Pauli $\sigma_{X}$ and the $\mathbb{Z}$ measurement to be Pauli $\sigma
_{Z}$, which implies that $-\log c=1$. We define the “BB84” states $|0\rangle$, $|1\rangle$, $|+\rangle$, and
$|-\rangle$ from the following relations:
\begin{equation}
\sigma_{Z}|0\rangle=|0\rangle,\qquad\sigma_{Z}|1\rangle=(-1)|1\rangle
\end{equation}
So this means that the $\mathbb{X}$ and $\mathbb{Z}$ measurements have the
following respective implementations as quantum channels acting on an
input $\xi$:
\begin{align}
\xi & \rightarrow\langle+|_{A}\xi|+\rangle_{A}|0\rangle\langle0|_{X}%
\xi & \rightarrow\langle0|_{A}\xi|0\rangle_{A}|0\rangle\langle0|_{Z}%
\end{align}
§.§ Minimum uncertainty states
§.§.§ $X$ eigenstate on system $A$
First suppose that $\rho_{AB}=|+\rangle\langle+|_{A}\otimes\pi_{B}$, where
$\pi$ is the maximally mixed state. In this case, Bob's system $B$ is of no
use to help predict the outcome of a measurement on the $A$ system because the
systems are in a product state. Here we find by direct calculation that
$H(A|B)_{\rho}=0$, $H(X|B)_{\sigma}=0$, and $H(Z|B)_{\omega}=1$. By
(<ref>), this then implies that there exists a recovery
channel $\mathcal{R}^{(1)}$ such that (<ref>) is
satisfied and, given that $\sigma_{XB}=|0\rangle\langle0|_{X}\otimes\pi_{B}$,
we also have the perfect recovery
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(1)}\big(|0\rangle\langle0|_{X}\otimes\pi
\end{equation}
To determine the recovery channel $\mathcal{R}^{(1)}$, consider that
\begin{equation}
\sum_{z}|z\rangle\langle z|_{Z}\otimes\omega_{B}^{z}=\pi_{Z}\otimes\pi
_{B},\qquad\sum_{x}|x\rangle\langle x|_{X}\otimes\theta_{B}^{x}=\pi_{X}%
\otimes\pi_{B},
\end{equation}
with the states on the left in each case defined in (<ref>) and
(<ref>), respectively. Plugging into (<ref>),
we find that the recovery channel in this case is given explicitly by
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(1)}(\xi_{XB})=|+\rangle\langle+|_{A}%
\otimes\operatorname{Tr}_{X}\{|0\rangle\langle0|_{X}\xi_{XB}\}+|-\rangle
\langle-|_{A}\otimes\operatorname{Tr}_{X}\{|1\rangle\langle1|_{X}\xi
\end{equation}
so that we also see that
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(1)}(\pi_{X}\otimes\pi_{B})=\pi_{A}\otimes
\pi_{B}.\label{eq:r1-recovery-theta-omega}%
\end{equation}
§.§.§ $Z$ eigenstate on system $A$
The situation in which $\rho_{AB}=|0\rangle\langle0|_{A}\otimes\pi_{B}$ is
similar in some regards, but the recovery channel is different—i.e., we have by direct calculation that
$H(A|B)_{\rho}=0$, $H(X|B)_{\sigma}=1$, and $H(Z|B)_{\omega}=0$, which implies
the existence of a different recovery channel $\mathcal{R}^{(2)}$ such that
(<ref>) is satisfied, and given that $\sigma_{XB}%
=\pi_{X}\otimes\pi_{B}$, we also have the perfect recovery
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(2)}\big(\pi_{X}\otimes\pi_{B}\big)=|0\rangle
\langle0|_{A}\otimes\pi_{B}.
\end{equation}
To determine the recovery channel $\mathcal{R}^{(2)}$, consider that
\begin{equation}
\sum_{z}|z\rangle\langle z|_{Z}\otimes\omega_{B}^{z}=|0\rangle\langle
0|_{Z}\otimes\pi_{B},\qquad\sum_{x}|x\rangle\langle x|_{X}\otimes\theta
\end{equation}
with the states on the left in each case defined in (<ref>) and
(<ref>), respectively. Plugging into (<ref>),
we find that the recovery channel in this case is given explicitly by
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(2)}(\xi_{XB})=|0\rangle\langle0|_{A}%
\otimes\operatorname{Tr}_{X}\{\xi_{XB}\}.
\end{equation}
§.§.§ Maximally entangled state on systems $A$ and $B$
Now suppose that $\rho_{AB}=|\Phi
\rangle\langle\Phi|_{AB}$ is the maximally entangled state, where
$|\Phi\rangle_{AB}\equiv(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}$. In this
case, we have that both $H(X|B)_{\sigma}=0$ and $H(Z|B)_{\omega}=0$, but the
conditional entropy is negative: $H(A|B)_{\rho}=-1$. So here again we find
the existence of a recovery channel $\mathcal{R}^{(3)}$ such that
(<ref>) is satisfied, and given that $\sigma
_{XB}=(|0+\rangle\langle0+|_{XB}+|1-\rangle\langle1-|_{XB})/2$, we also have
the perfect recovery
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(3)}\Big(\big(|0+\rangle\langle0+|_{XB}%
\end{equation}
To determine the recovery channel $\mathcal{R}^{(3)}$, consider that
\begin{align}
\sum_{z}|z\rangle\langle z|_{Z}\otimes\omega_{B}^{z} & =\frac{1}{2}\left(
1|_{Z}\otimes|1\rangle\langle1|_{B}\right) ,\\
\sum_{x}|x\rangle\langle x|_{X}\otimes\theta_{B}^{x} & =\pi_{X}\otimes\pi
\end{align}
with the states on the left in each case defined in (<ref>) and
(<ref>), respectively. Plugging into (<ref>),
we find that the recovery channel in this case is given explicitly by
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(3)}(\xi_{XB})=\sum_{z,z^{\prime},x\in\left\{
0,1\right\} }\left( -1\right) ^{x(z+z^{\prime})}|z\rangle\langle z^{\prime
}|_{A}\otimes|z\rangle\langle z^{\prime}|_{B}\operatorname{Tr}\{|x\rangle
\langle x|_{X}\otimes|z^{\prime}\rangle\langle z|_{B}\xi_{XB}%
\},\label{eq:max-ent-state-recovery-op}%
\end{equation}
i.e., with the following Kraus operators:
\begin{equation}
\left\{ \sum_{z}\left( -1\right) ^{xz}\left( |z\rangle_{A}\otimes
|z\rangle_{B}\right) \left( \langle x|_{X}\otimes\langle z|_{B}\right)
\right\} _{x}.
\end{equation}
These Kraus operators give the recovery map $\mathcal{R}_{XB\rightarrow
AB}^{(3)}$ the interpretation of 1) measuring the $X$ register and
2) coherently copying the contents of the $B$ register to the $A$ register
along with an appropriate relative phase. It can be implemented by performing
a controlled-NOT gate from $B$ to $A$, followed by a controlled-phase gate on
$X$ and $B$ and a partial trace over system $X$.
All of the examples mentioned above involve a perfect recovery or a perfect
reversal of the $\mathbb{X}$ measurement. This is due to the fact that the
bound in (<ref>) is saturated for these examples. However, the
refined inequality in (<ref>) allows to generalize these
situations to the approximate case, in which $\rho_{AB}$ is nearly
indistinguishable from the states given above. It is then the case that the
equalities in (<ref>)–(<ref>) become
approximate equalities, with a precise characterization given by (<ref>).
§.§ Maximum uncertainty states
We now investigate the extreme opposite situation, when the bound in
(<ref>) is far from being saturated but its refinement in
(<ref>) is saturated. Let $\rho_{AB}=|+_{Y}\rangle
\langle+_{Y}|_{A}\otimes\pi_{B}$, where $|+_{Y}\rangle$ is defined from the
relation $\sigma_{Y}|+_{Y}\rangle=|+_{Y}\rangle$. In this case, we find that
both $H(X|B)_{\sigma}=1$ and $H(Z|B)_{\omega}=1$. Thus, we could say that
$\rho_{AB}$ is a “maximum uncertainty state”
because the sum $H(X|B)_{\sigma}+H(Z|B)_{\omega}$ is equal to two bits and
cannot be any larger than this amount. We also find that $H(A|B)_{\rho}=0$,
implying that (<ref>) is one bit away from being saturated. Now
consider that $\sigma_{XB}=\theta_{XB}=\pi_{X}\otimes\pi_{B}$ and $\omega
_{ZB}=\pi_{Z}\otimes\pi_{B}$, and thus one can explicitly calculate the
recovery channel $\mathcal{R}^{(4)}$ from (<ref>) to take
the form:
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}^{(4)}(\xi_{XB})\equiv|+\rangle\langle
\}+|-\rangle\langle-|_{A}\otimes\operatorname{Tr}_{X}\{|1\rangle\langle
\end{equation}
Note that the recovery channel $\mathcal{R}_{XB\rightarrow AB}^{(4)}$ is the
same as $\mathcal{R}_{XB\rightarrow AB}^{(1)}$ in (<ref>).
This implies that
\begin{align}
\mathcal{R}_{XB\rightarrow AB}^{(4)}(\sigma_{XB}) & =\pi_{A}\otimes\pi
\mathcal{R}_{XB\rightarrow AB}^{(4)}(\theta_{XB}) & =\pi_{A}\otimes\pi
\end{align}
and in turn that
\begin{equation}
-\log F(\rho_{AB},\mathcal{R}_{XB\rightarrow AB}^{(4)}(\theta_{XB}))=1.
\end{equation}
Thus the inequality in (<ref>) is saturated for this example.
The key element is that there is one bit of uncertainty when measuring a $Y$
eigenstate with respect to either the $X$ or $Z$ basis. At the same time, the
$Y$ eigenstate is pure, so that its entropy is zero. This leaves a bit of
uncertainty available and for which (<ref>) does not account, but
which we have now interpreted in terms of how well one can reverse the
$\mathbb{X}$ measurement, using the refined bound in (<ref>).
One could imagine generalizing the idea of this example to higher dimensions
in order to find more maximum uncertainty examples of this sort.
§ EXPERIMENTS
Circuits for experimental testing of our entropic uncertainty
relation in (<ref>). (a) Four different experimental
tests in which one can prepare system $A$ as either $|+\rangle_{A}$ or
$|+_{Y}\rangle_{A}$ and then perform either a $\sigma_Z$ Pauli measurement or not.
Afterward, a $\sigma_X$ measurement is performed followed by the recovery operation
from (<ref>), whose aim it is to undo the effect of the
$\sigma_X$ measurement. If $|+\rangle_{A}$ is prepared and $\sigma_Z$ is not measured, then
it is possible to undo the effect of the $\sigma_X$ measurement and recover the qubit
$|+\rangle$ perfectly in system $A^{\prime}$. If $|+\rangle_{A}$ is prepared
and $\sigma_Z$ is then measured, it is possible to undo the effect of the $\sigma_X$
measurement with the same recovery operation. The same results hold if
$|+_{Y}\rangle_{A}$ is prepared in system $A$ (i.e., the recovery operation
undoes the effect of the $\sigma_X$ measurement). (b) Two different
experimental tests in which one can prepare a maximally entangled Bell state
$|\Phi\rangle_{AB}$ in systems $A$ and $B$, perform a Pauli $\sigma_Z$ measurement or
not, perform a $\sigma_X$ measurement, followed by a recovery operation whose aim it
is to undo the effect of the $\sigma_X$ measurement. In the case that $\sigma_Z$ is not
measured, the recovery operation perfectly restores the maximally entangled
state in systems $A^{\prime}$ and $B$. In the case that $\sigma_Z$ is measured, the
recovery operation undoes the effect of the $\sigma_X$ measurement by restoring the
maximally correlated state $(|00\rangle\langle00|_{A^{\prime}B}+|11\rangle
\langle11|_{A^{\prime}B})/2$ in systems $A^{\prime}$ and $B$.
We have experimentally tested three of the examples from the previous section,
namely, the $X$ eigenstate, the maximally entangled state, and the $Y$
eigenstate examples. We did so using the recently available IBM Quantum
Experience (QE) <cit.>. Three experiments have already appeared on the arXiv, conducted remotely by theoretical groups testing out experiments which had never been performed previously <cit.>. The QE architecture consists of five
fixed-frequency superconducting transmon qubits, laid out in a
“star geometry” (four “corner” qubits and one in the center). It is possible to
perform single-qubit gates $X$, $Y$, $Z$, $H$, $T$, $S$, and $S^{\dag}$, a
Pauli measurement $Z$, and Bloch sphere tomography on any single qubit.
However, two-qubit operations are limited to controlled-NOT gates with any one
of the corner qubits acting as the source and the center qubit as the target.
Thus, one must “recompile” quantum circuits
in order to meet these constraints. More information about the architecture is
available at the user guide at <cit.>.
Our experiments realize
and test three of the examples from the previous section and, in particular,
are as follows:
* Prepare system $A$ in the state $|+\rangle$. Measure Pauli $\sigma_{X}$
on qubit $A$ and place the outcome in register $X$. Perform the recovery channel
given in (<ref>), with output system $A^{\prime}$.
Finally, perform Bloch sphere tomography on system $A^{\prime}$.
* Prepare system $A$ in the state $|+\rangle$. Measure Pauli $\sigma_{Z}$
on qubit $A$ and place the outcome in register $Z$. Measure Pauli $\sigma_{X}$ on
qubit $A$ and place the outcome in register $X$. Perform the recovery channel
given in (<ref>), with output system $A^{\prime}$.
Finally, perform Bloch sphere tomography on system $A^{\prime}$.
* Same as Experiment 1 but begin by preparing system $A$ in the state $|+_{Y}%
\rangle_{A}$.
* Same as Experiment 2 but begin by preparing system $A$ in the state $|+_{Y}%
\rangle_{A}$.
* Prepare systems $A$ and $B$ in the maximally entangled Bell state
$|\Phi\rangle_{AB}$. Measure Pauli $\sigma_{X}$ on qubit $A$ and place the
outcome in register $X$. Perform the recovery channel given in
(<ref>), with output systems $A^{\prime}$ and
$B$. Finally, perform measurements of $\sigma_{X}$ on system $A^{\prime}$ and
$\sigma_{X}$ on system $B$, or $\sigma_{Y}$ on system $A^{\prime}$ and
$\sigma_{Y}^{\ast}$ on system $B$, or $\sigma_{Z}$ on system $A^{\prime}$ and
$\sigma_{Z}$ on system $B$.
* Prepare systems $A$ and $B$ in the maximally entangled Bell state
$|\Phi\rangle_{AB}$. Measure Pauli $\sigma_{Z}$ on qubit $A$ and place the
outcome in register $Z$. Measure Pauli $\sigma_{X}$ on qubit $A$ and place the
outcome in register $X$. Perform the recovery channel given in
(<ref>), with output systems $A^{\prime}$ and
$B$. Finally, perform measurements of $\sigma_{X}$ on system $A^{\prime}$ and
$\sigma_{X}$ on system $B$, or $\sigma_{Y}$ on system $A^{\prime}$ and
$\sigma_{Y}^{\ast}$ on system $B$, or $\sigma_{Z}$ on system $A^{\prime}$ and
$\sigma_{Z}$ on system $B$.
A quantum circuit that can realize Experiments 1–4 is given in
Figure <ref>(a), and a quantum circuit that can
realize Experiments 5–6 is given in Figure <ref>
(b). These circuits make use of standard quantum computing gates, detailed in
<cit.>, and one can readily verify that they ideally have
the correct behavior, consistent with that discussed for the examples in the
previous section. As stated above, it is necessary to recompile these circuits
into a form which meets the constraints of the QE architecture.
Results of experimental tests on the IBM QE quantum computer.
Subfigures (a)–(f) correspond to Experiments 1–6 outlined in the main text,
respectively. (a) The ideal state of the recovered qubit is
$|+\rangle$, as predicted by (<ref>)
and depicted on the Bloch sphere as a blue dot. The figure plots the result of Bloch sphere tomography from the experimental tests (as a red dot). (b)–(d) The
ideal state of the recovered qubit in each case is $\pi$ (the maximally mixed
state), as predicted by (<ref>),
(<ref>), and (<ref>),
respectively (again depicted as blue dots).
The figure again plots the result of Bloch sphere tomography as red dots.
(e) The ideal state of the recovered qubits is
$|\Phi\rangle$ as predicted by (<ref>). In such a case,
measurement of the Pauli observables $\sigma_{i}$ on $A^{\prime}$ and
$\sigma_{i}^{\ast}$ on $B$ for $i\in\{X,Y,Z\}$ should return 00 and 11 with
probability 0.5 and 01 and 10 with probability zero. The plots reveal
significant noise in the experiments, especially from the $\sigma_X$ and $\sigma_Y$ measurements. (f) The ideal state of the
recovered qubits is the maximally correlated state $(|00\rangle\langle
00|+|11\rangle\langle11|)/2$ as predicted in
Section <ref>. In such a case, measurement of the
Pauli observables $\sigma_{Z}$ on $A$ and $\sigma_{Z}$ on $B$ should return 00
and 11 with probability 0.5 and 01 and 10 with probability zero. Measurement
of the Pauli observables $\sigma_{i}$ on $A$ and $\sigma_{i}^{\ast}$ on $B$
for $i\in\{X,Y\}$ should return all outcomes with equal probabilities. Again,
the plots reveal significant noise in the experiments.
Figure <ref> plots the results of Experiments 1–6. Each
experiment consists of three measurements, with Experiments 1–4 having
measurements of each of the Pauli operators, and Experiments 5–6 having three
different measurements each as outlined above. Each of these is repeated 8192
times, for a total of $6\times3\times8192=147,456$ experiments. The standard
error for each kind of experiment is thus $\sqrt{p_{c}(1-p_{c})/8192}$, where $p_{c}$
is the estimate of the probability of a given measurement outcome in a given experiment. The caption of
Figure <ref> features discussions of and comparisons between
the predictions of the previous section and the experimental outcomes. While
it is clear that the QE chip is subject to significant noise, there is still
reasonable agreement with the theoretical predictions of the previous section. One observation we make regarding Figure <ref>(e) is that the frequencies for the outcomes of the $\sigma_Z$ and $\sigma_Z$ measurements are much closer to the theoretically predicted values than are the other measurement outcomes.
§ CONCLUSION
The entropic uncertainty relation with quantum side information is a unifying
principle relating quantum uncertainty due to measurement incompatibility and
entanglement. Here we refine and tighten this inequality with a
state-dependent term related to how well one can reverse the action of a
measurement. The tightening of the inequality is most pronounced when the
measurements and state are all chosen from mutually unbiased bases, i.e., in
our “maximum uncertainty” example with the
measurements being $\sigma_{X}$ and $\sigma_{Z}$ and the initial state being a
$\sigma_{Y}$ eigenstate. We have experimentally tested our theoretical
predictions on the IBM Quantum Experience and find reasonable
agreement between our predictions and experimental outcomes.
We note that in terms of the conditional min-entropy, other refinements
of (<ref>) are known <cit.> that look at the measurement
channel and its own inverse channel, and it would be interesting to understand
their relation. Going forward, it would furthermore be interesting to
generalize the results established here to infinite-dimensional and multiple
measurement scenarios.
Acknowledgments—The authors acknowledge discussions with Siddhartha
Das, Michael Walter, and Andreas Winter.
We are grateful to the team at IBM and the IBM Quantum Experience project. This work does not reflect the views or opinions of IBM or any of its employees.
MB acknowledges funding provided by
the Institute for Quantum Information and Matter, an NSF Physics Frontiers
Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore
Foundation (GBMF-12500028). Additional funding support was provided by the ARO
grant for Research on Quantum Algorithms at the IQIM (W911NF-12-1-0521). SW
acknowledges support from STW, Netherlands and an NWO VIDI Grant. MMW is
grateful to SW and her group for hospitality during a research visit to QuTech
in May 2015 and acknowledges support from startup funds from the Department of
Physics and Astronomy at LSU, the NSF under Award No. CCF-1350397, and the
DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019.
§ PROOF OF (<REF>) AND (<REF>)
The main idea of the proof of (<ref>) follows the approach
first put forward in <cit.> (see also <cit.>), for which the core
argument is the non-increase of quantum relative entropy. Here we instead
apply a refinement of this entropy inequality from <cit.> (see also
<cit.>). In order to prove (<ref>), we start by
noting that it suffices to prove it when $\rho_{ABE}=|\psi\rangle\langle
\psi|_{ABE}$ (i.e., the shared state is pure). This is because the conditional
entropy only increases under the discarding of one part of the conditioning
system. We consider the following isometric extensions of the measurement
channels <cit.>, which produce the measurement outcomes and
post-measurement states:
\begin{align}
U_{A\rightarrow XX^{\prime}A} & \equiv\sum_{x}|x\rangle_{X}\otimes
|x\rangle_{X^{\prime}}\otimes P_{A}^{x},\\
V_{A\rightarrow ZZ^{\prime}A} & \equiv\sum_{z}|z\rangle_{Z}\otimes
|z\rangle_{Z^{\prime}}\otimes Q_{A}^{z}.
\end{align}
We also define the following pure states, which represent purifications of the
states $\sigma_{XBE}$ and $\omega_{ZBE}$ defined in (<ref>)
and (<ref>), respectively:
\begin{align}
|\sigma\rangle_{XX^{\prime}ABE} & \equiv U_{A\rightarrow XX^{\prime}A}%
|\omega\rangle_{ZZ^{\prime}ABE} & \equiv V_{A\rightarrow ZZ^{\prime}A}%
\end{align}
Consider from duality of conditional entropy for pure states (see, e.g.,
<cit.>) that
\begin{equation}
\end{equation}
where $D(\rho\Vert\sigma)\equiv\operatorname{Tr}\{\rho\lbrack\log\rho
-\log\sigma]\}$ is the quantum relative entropy <cit.>, defined as such
when $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$ and as
$+\infty$ otherwise. Now consider the following quantum channel
\begin{equation}
\mathcal{P}_{ZZ^{\prime}A}(\cdot)\rightarrow\Pi(\cdot)\Pi+(I-\Pi)(\cdot
\end{equation}
where $\Pi\equiv VV^{\dag}$. From the monotonicity of quantum relative entropy
with respect to quantum channels <cit.>, we find that
\begin{equation}
D(\omega_{ZZ^{\prime}AB}\Vert I_{Z}\otimes\omega_{Z^{\prime}AB})\geq
\end{equation}
Consider that $\mathcal{P}_{ZZ^{\prime}A}(\omega_{ZZ^{\prime}AB}%
)=\omega_{ZZ^{\prime}AB}$. Due to the fact that
\begin{equation}
\end{equation}
and from the direct sum property of the quantum relative entropy (see, e.g.,
<cit.>), we have that
\begin{equation}
\end{equation}
Consider that
\begin{equation}
\Pi(I_{Z}\otimes\omega_{Z^{\prime}AB})\Pi=VV^{\dag}(I_{Z}\otimes
\omega_{Z^{\prime}AB})VV^{\dag}=V\!\left( \sum_{z}Q_{A}^{z}\rho_{AB}Q_{A}%
^{z}\right) V^{\dag}.
\end{equation}
This, combined with $\omega_{ZZ^{\prime}AB}=V\rho_{AB}V^{\dag}$, then implies
\begin{align}
D(\omega_{ZZ^{\prime}AB}\Vert\Pi(I_{Z}\otimes\omega_{Z^{\prime}AB})\Pi) &
=D\!\left( V\rho_{AB}V^{\dag}\middle\Vert V\!\left( \sum_{z}Q_{A}^{z}%
\rho_{AB}Q_{A}^{z}\right) V^{\dag}\right) \\
& =D\!\left( \rho_{AB}\middle\Vert\sum_{z}Q_{A}^{z}\rho_{AB}Q_{A}%
^{z}\right) ,
\end{align}
where the last equality follows from the invariance of quantum relative
entropy with respect to isometries. Now consider the following quantum
\begin{equation}
\mathcal{M}_{A\rightarrow X}\equiv\operatorname{Tr}_{X^{\prime}A}%
\circ\ \mathcal{U}_{A\rightarrow XX^{\prime}A},
\end{equation}
where $\mathcal{U}_{A\rightarrow XX^{\prime}A}(\cdot)\equiv U(\cdot)U^{\dag}$.
Consider that $\mathcal{M}_{A\rightarrow X}(\rho_{AB})=\sigma_{XB}$. Also, we
can calculate
\begin{equation}
\mathcal{M}_{A\rightarrow X}\!\left( \sum_{z}Q_{A}^{z}\rho_{AB}Q_{A}%
\end{equation}
as follows:
\begin{equation}
(\operatorname{Tr}_{X^{\prime}A}\circ\mathcal{U}_{A\rightarrow XX^{\prime}%
A})\left( \sum_{z}Q_{A}^{z}\rho_{AB}Q_{A}^{z}\right) =\theta_{XB}.
\end{equation}
From <cit.>, we have the following inequality holding for a density
operator $\rho$, a positive semi-definite operator $\sigma$, and a quantum
channel $\mathcal{N}$:
\begin{equation}
F(\rho,\mathcal{R}(\mathcal{N}(\rho))), \label{eq:mono-RE-refine}%
\end{equation}
where $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$ and
$\mathcal{R}$ is a recovery channel with the property that $\mathcal{R}%
(\mathcal{N}(\sigma))=\sigma$. Specifically, $\mathcal{R}$ is what is known as
a variant of the Petz recovery channel, having the form
\begin{align}
\mathcal{R}(\cdot)\equiv & \int\mathrm{d}t\,p(t)\,\sigma^{-it/2}%
\mathcal{R}_{\sigma,\mathcal{N}}(\mathcal{N}(\sigma)^{it/2}(\cdot
& \mathrm{with}\quad p(t)\equiv\frac{\pi}{2}(\cosh(\pi t)+1)^{-1},
\label{eq:petz-recovery-rotated}%
\end{align}
where $\mathcal{R}_{\sigma,\mathcal{N}}$ is the Petz recovery channel
<cit.> defined as
\begin{equation}
\mathcal{R}_{\sigma,\mathcal{N}}(\cdot)\equiv\sigma^{1/2}\mathcal{N}^{\dag
}\!\left( \mathcal{N}(\sigma)^{-1/2}(\cdot)\mathcal{N}(\sigma)^{-1/2}\right)
\sigma^{1/2},
\end{equation}
with $\mathcal{N}^{\dag}$ the adjoint of $\mathcal{N}$ (with respect to the
Hilbert–Schmidt inner product). Applying this to our case, we find that
\begin{multline}
D\!\left( \rho_{AB}\middle\Vert\sum_{z}Q_{A}^{z}\rho_{AB}Q_{A}^{z}\right)
\geq D\!\left( \mathcal{M}_{A\rightarrow X}(\rho_{AB})\middle\Vert
\mathcal{M}_{A\rightarrow X}\!\left( \sum_{z}Q_{A}^{z}\rho_{AB}Q_{A}%
^{z}\right) \right) \label{eq:EUR-QSI-mono-RE-refine}\\
-\log F(\rho_{AB},\mathcal{R}_{XB\to AB}(\mathcal{M}_{A\rightarrow X}%
\end{multline}
where the recovery channel is such that
\begin{equation}
\mathcal{R}_{XB\to AB}\!\left( \mathcal{M}_{A\rightarrow X}\!\left( \sum
_{z}Q_{A}^{z}\rho_{AB}Q_{A}^{z}\right) \right) =\sum_{z}Q_{A}^{z}\rho
_{AB}Q_{A}^{z}. \label{eq:Z-X-recover-perfect-general}%
\end{equation}
Consider from our development above that
\begin{align}
D\!\left( \mathcal{M}_{A\rightarrow X}(\rho_{AB})\middle\Vert\mathcal{M}%
_{A\rightarrow X}\!\left( \sum_{z}Q_{A}^{z}\rho_{AB}Q_{A}^{z}\right)
\right) & =D(\sigma_{XB}\Vert\theta_{XB})\\
& \geq D(\sigma_{XB}\Vert I_{X}\otimes\sigma_{B})-\log c,
\end{align}
where we have used $\sigma\leq\sigma^{\prime}\Rightarrow D(\rho\Vert
\sigma^{\prime})\leq D(\rho\Vert\sigma)$ (see, e.g., <cit.>), applied
to $Q_{A}^{z}P_{A}^{x}Q_{A}^{z}=|Q_{A}^{z}P_{A}^{x}|^{2}\leq c\cdot I_{A}$,
with $c$ defined in (<ref>). Putting everything together, we
conclude that
\begin{equation}
D(\omega_{ZZ^{\prime}AB}\Vert I_{Z}\otimes\omega_{Z^{\prime}AB})\geq
D(\sigma_{XB}\Vert I_{X}\otimes\sigma_{B})-\log c-\log F(\rho_{AB}%
,\mathcal{R}_{XB\to AB}(\sigma_{XB})),
\end{equation}
which, after a rewriting, is equivalent to (<ref>) coupled
with the constraint in (<ref>).
The inequality in (<ref>) follows from
(<ref>) by letting $|\psi\rangle_{ABE}$ be a purification of
$\rho_{AB}$ and observing that
\begin{equation}
\end{equation}
whenever $\rho_{ABE}$ is a pure state and $Q_{A}^{z}=|z\rangle\langle z|_{A}$
for some orthonormal basis $\{|z\rangle_{A}\}_{z}$.
§ EXPLICIT FORM OF RECOVERY MAP
Here we establish the explicit form given in
(<ref>) for the recovery map, in the case that $\{Q_{A}%
^{z}=|z\rangle\langle z|_{A}\}$ for some orthonormal basis $\{|z\rangle
_{A}\}_{z}$. The main idea is to determine what $\mathcal{R}_{XB\to AB}$ in
(<ref>) should be by inspecting
(<ref>) and (<ref>). For our setup,
we are considering a bipartite state $\rho_{AB}$, a set $\{Q_{A}^{z}\}$ of
measurement operators, and the measurement channel
\begin{equation}
\mathcal{M}_{A\rightarrow X}(\zeta_{A})\equiv\sum_{x}\operatorname{Tr}%
\{P_{A}^{x}\zeta_{A}\}|x\rangle\langle x|_{X},
\end{equation}
where $\{P_{A}^{x}\}_{x}$ is a set of projective measurement operators. The
entropy inequality in (<ref>) reduces to
\begin{multline}
D\!\left( \rho_{AB}\middle\Vert\sum_{z}|z\rangle\langle z|_{A}\otimes
\omega_{B}^{z}\right) -D\!\left( \mathcal{M}_{A\rightarrow X}(\rho
_{AB})\middle\Vert\sum_{x}|x\rangle\langle x|_{X}\otimes\theta_{B}^{x}\right)
\\
\geq-\log F(\rho_{AB},\mathcal{R}_{XB\rightarrow AB}(\mathcal{M}_{A\rightarrow
\end{multline}
\begin{equation}
\omega_{B}^{z}\equiv(\langle z|_{A} \otimes I_{B})\rho_{AB}(|z\rangle_{A}
\otimes I_{B}),\qquad\theta_{B}^{x}\equiv\sum_{z}\langle z|_{A}P_{A}%
\end{equation}
Observe that
\begin{equation}
\sum_{x}|x\rangle\langle x|_{X}\otimes\theta_{B}^{x}=\mathcal{M}_{A\rightarrow
X}\!\left( \sum_{z}|z\rangle\langle z|_{A}\otimes\omega_{B}^{z}\right) .
\end{equation}
Writing the measurement channel as
\begin{align}
\mathcal{M}_{A\rightarrow X}(\zeta_{A}) & \equiv\sum_{x}\operatorname{Tr}%
\{P_{A}^{x}\zeta_{A}P_{A}^{x}\}|x\rangle\langle x|_{X}=\sum_{x,j}\langle
j|_{A}P_{A}^{x}\zeta_{A}P_{A}^{x}|j\rangle_{A}|x\rangle\langle x|_{X}\\
& =\sum_{x,j}|x\rangle_{X}\langle j|_{A}P_{A}^{x}\zeta_{A}P_{A}^{x}%
|j\rangle_{A}\langle x|_{X},
\end{align}
we can see that a set of Kraus operators for it is $\{|x\rangle_{X}\langle
j|_{A}P_{A}^{x}\}_{x,j}$. So its adjoint is as follows:
\begin{align}
\left( \mathcal{M}_{A\rightarrow X}\right) ^{\dag}(\kappa_{X}) &
=\sum_{x,j}P_{A}^{x}|j\rangle_{A}\langle x|_{X}\kappa_{X}|x\rangle_{X}\langle
j|_{A}P_{A}^{x}=\sum_{x,j}\langle x|_{X}\kappa_{X}|x\rangle_{X}P_{A}%
^{x}|j\rangle_{A}\langle j|_{A}P_{A}^{x}\\
& =\sum_{x}\langle x|_{X}\kappa_{X}|x\rangle_{X}P_{A}^{x}.
\end{align}
So by inspecting (<ref>) and
(<ref>), we see that the recovery map has the
following form:
\begin{align}
& \mathcal{R}_{XB\rightarrow AB}(\xi_{XB})\nonumber\\
& =\int dt\ p(t)\ \left( \sum_{z}|z\rangle\langle z|_{A}\otimes\omega
_{B}^{z}\right) ^{\frac{1-it}{2}}\sum_{x}P_{A}^{x}\left( \langle
x|_{X}\otimes I_{B}\right) \left( \sum_{x^{\prime}}|x^{\prime}\rangle\langle
x^{\prime}|_{X}\otimes\theta_{B}^{x^{\prime}}\right) ^{\frac{-1+it}{2}}%
& \qquad\qquad\left( \sum_{x^{\prime\prime}}|x^{\prime\prime}\rangle\langle
^{\frac{-1-it}{2}}\left( |x\rangle_{X}\otimes I_{B}\right) \left(
\sum_{z^{\prime}}|z^{\prime}\rangle\langle z^{\prime}|_{A}\otimes\omega
_{B}^{z^{\prime}}\right) ^{\frac{1+it}{2}}\\
& =\int dt\ p(t)\ \left( \sum_{z}|z\rangle\langle z|_{A}\otimes\left(
\omega_{B}^{z}\right) ^{\frac{1-it}{2}}\right) \sum_{x}P_{A}^{x}\left(
\langle x|_{X}\otimes I_{B}\right) \nonumber\\
& \qquad\qquad\left[ \left( \sum_{x^{\prime}}|x^{\prime}\rangle\langle
x^{\prime}|_{X}\otimes\left( \theta_{B}^{x^{\prime}}\right) ^{\frac
{-1+it}{2}}\right) (\xi_{XB})\left( \sum_{x^{\prime\prime}}|x^{\prime\prime
}\rangle\langle x^{\prime\prime}|_{X}\otimes\left( \theta_{B}^{x^{\prime
\prime}}\right) ^{\frac{-1-it}{2}}\right) \right] \nonumber\\
& \qquad\qquad\left( |x\rangle_{X}\otimes I_{B}\right) \left(
\sum_{z^{\prime}}|z^{\prime}\rangle\langle z^{\prime}|_{A}\otimes\left(
\omega_{B}^{z^{\prime}}\right) ^{\frac{1+it}{2}}\right)
\end{align}
\begin{align}
& =\int dt\ p(t)\ \left( \sum_{z}|z\rangle\langle z|_{A}\otimes\left(
\omega_{B}^{z}\right) ^{\frac{1-it}{2}}\right) \sum_{x,x^{\prime}%
,x^{\prime\prime}}P_{A}^{x}\langle x|_{X}|x^{\prime}\rangle\langle x^{\prime
}|_{X}\otimes\left( \theta_{B}^{x^{\prime}}\right) ^{\frac{-1+it}{2}}%
& \qquad\qquad|x^{\prime\prime}\rangle\langle x^{\prime\prime}|_{X}%
|x\rangle_{X}\otimes\left( \theta_{B}^{x^{\prime\prime}}\right)
^{\frac{-1-it}{2}}\left( \sum_{z^{\prime}}|z^{\prime}\rangle\langle
z^{\prime}|_{A}\otimes\left( \omega_{B}^{z^{\prime}}\right) ^{\frac{1+it}%
{2}}\right) \\
& =\int dt\ p(t)\ \left( \sum_{z}|z\rangle\langle z|_{A}\otimes\left(
\omega_{B}^{z}\right) ^{\frac{1-it}{2}}\right) \sum_{x}P_{A}^{x}\langle
x|_{X}\otimes\left( \theta_{B}^{x}\right) ^{\frac{-1+it}{2}}(\xi
_{XB})|x\rangle_{X}\otimes\left( \theta_{B}^{x}\right) ^{\frac{-1-it}{2}%
& \qquad\qquad\left( \sum_{z^{\prime}}|z^{\prime}\rangle\langle z^{\prime
}|_{A}\otimes\left( \omega_{B}^{z^{\prime}}\right) ^{\frac{1+it}{2}}\right)
\\
& =\int dt\ p(t)\ \sum_{z,x,z^{\prime}}|z\rangle\langle z|_{A}P_{A}%
^{x}|z^{\prime}\rangle\langle z^{\prime}|_{A}\otimes\left( \omega_{B}%
^{z}\right) ^{\frac{1-it}{2}}\left( \theta_{B}^{x}\right) ^{\frac{-1+it}%
{2}}\operatorname{Tr}_{X}\{|x\rangle\langle x|_{X}(\xi_{XB})\}\left(
\theta_{B}^{x}\right) ^{\frac{-1-it}{2}}\left( \omega_{B}^{z^{\prime}%
}\right) ^{\frac{1+it}{2}}.
\end{align}
We can thus abbreviate its action as
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}(\xi_{XB})=\ \sum_{z,x,z^{\prime}}%
|z\rangle\langle z|_{A}P_{A}^{x}|z^{\prime}\rangle\langle z^{\prime}%
|_{A}\otimes\mathcal{R}_{XB\rightarrow B}^{x,z,z^{\prime}}(\xi_{XB}),
\end{equation}
\begin{equation}
\mathcal{R}_{XB\rightarrow B}^{x,z,z^{\prime}}(\xi_{XB})\equiv\int
dt\ p(t)\left( \omega_{B}^{z}\right) ^{\frac{1-it}{2}}\left( \theta_{B}%
^{x}\right) ^{\frac{-1+it}{2}}\operatorname{Tr}_{X}\{|x\rangle\langle
x|_{X}(\xi_{XB})\}\left( \theta_{B}^{x}\right) ^{\frac{-1-it}{2}}\left(
\omega_{B}^{z^{\prime}}\right) ^{\frac{1+it}{2}}.
\end{equation}
(Note that $\mathcal{R}_{XB\rightarrow B}^{x,z,z^{\prime}}$ is not a channel.)
So then the action on the classical–quantum state $\sigma_{XB}$, defined as
\begin{equation}
\sigma_{XB}\equiv\sum_{x}|x\rangle\langle x|_{X}\otimes\sigma_{B}^{x},
\end{equation}
with $\sigma_{B}^{x}\equiv\operatorname{Tr}_{A}\{P_{A}^{x}\rho_{AB}\}$, is as
\begin{equation}
\mathcal{R}_{XB\rightarrow AB}(\sigma_{XB})=\sum_{z,x,z^{\prime}}%
|z\rangle\langle z|_{A}P_{A}^{x}|z^{\prime}\rangle\langle z^{\prime}%
|_{A}\otimes\int dt\ p(t)\left( \omega_{B}^{z}\right) ^{\frac{1-it}{2}%
}\left( \theta_{B}^{x}\right) ^{\frac{-1+it}{2}}\sigma_{B}^{x}\left(
\theta_{B}^{x}\right) ^{\frac{-1-it}{2}}(\omega_{B}^{z^{\prime}}%
\end{equation}
|
1511.00470
|
J. V. Smoker
European Southern Observatory, Alonso de Cordova 3107,
Vitacura, Santiago, Chile
Astrophysics Research Centre, School of Mathematics and Physics,
Queen's University Belfast, Belfast, BT7 1NN, U.K.
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, U.S.A.
By utilising spectra of early-type stellar probes of known distances in the same region of the sky, the large and small-scale (pc) structure of the
Galactic interstellar medium can be investigated. This paper
determines the variation in line strength of Ca ii at 3933.661Å as a function of probe separation for a large sample of stars,
including a number of sightlines in the Magellanic Clouds.
FLAMES-GIRAFFE data taken with the Very Large Telescope towards early-type stars in 3 Galactic and 4 Magellanic open
clusters in Ca ii are used to obtain the velocity, equivalent width,
column density and line width of interstellar Galactic calcium for a total of 657 stars, of which
443 are
Magellanic Cloud sightlines. In each cluster there are between 43-110 stars observed. Additionally, FEROS and UVES Ca ii K
and Na i D spectra of 21 Galactic
and 154 Magellanic early-type stars are presented and combined with data from the literature
to study the calcium column density – parallax relationship.
For the four Magellanic clusters studied with FLAMES, the strength of the Galactic interstellar Ca ii K equivalent width over transverse
scales from $\sim$0.05–9 pc is found to vary by factors of
$\sim$1.8–3.0, corresponding to column density variations of $\sim$0.3–0.5 dex in
the optically-thin approximation.
Using FLAMES, FEROS and UVES archive spectra, the minimum and maximum reduced equivalent width for
Milky Way gas is found to lie in the range $\sim$35–125 mÅ and $\sim$30–160 mÅ for Ca ii K and Na i D, respectively.
The range is consistent with a simple model of the ISM published by van Loon et al. (2009)
consisting of spherical cloudlets of filling factor $\sim$0.3, although other geometries are not ruled out.
Finally, the derived functional form for parallax ($\pi$) and Ca ii column density ($N_{\rm CaII}$)
is found to be $\pi$(mas)=1/(2.39$\times$10$^{-13}\times N_{\rm CaII}$(cm$^{-2}$) + 0.11). Our derived parallax is $\sim$25 per cent lower than
predicted by Megier et al. (2009) at a distance of $\sim$100 pc and $\sim$15 percent lower at a distance of $\sim$200 pc,
reflecting inhomogeneity in the Ca ii distribution in the different sightlines studied.
The full version including online material is available via the Astronomy and Astrophysics website http://www.aanda.org/articles/aa/olm/2015/10/aa25190-14/aa25190-14.html
Ca ii K interstellar observations of 7 Galactic and MC open clusters
§ INTRODUCTION
FLAMES-GIRAFFE sample: Galactic coordinates of the sightlines for which low-velocity interstellar Ca ii K was detected.
Since its discovery in the interstellar medium more than 100 years ago (Hartmann 1904), the Ca ii K line has been
extensively studied. At 3933Å it lies in a region of the spectrum free of strong telluric features and
where CCD detectors are sensitive. Perhaps most importantly the transition itself is strong and hence
easily detected in most high-resolution stellar spectra of medium signal to noise (S/N) ratio. Recent studies of Ca ii include
those by Albert et al (1993), Sembach et al. (1993), Welty et al. (1996; at very high
spectral resolution to obtain detailed information on cloud velocity components), Wakker & Mathis (2000;
to determine the spread in the Ca/H ratio as a function of H i column density), and Smoker et al (2003; to determine
the Galactic scale-height of the transition). More recently, Megier et al. (2005, 2009) compared the strength of the
Ca ii K interstellar absorption with distances from the Hipparcos catalogue (ESA 1997) and open cluster
distances to determine whether the line strength can be used to estimate distances to faraway objects. They
found several cases of significant column density difference in the interstellar component of stars
in the same cluster or association, but were unable to determine whether this was caused by a local
contribution to the derived profile, or confusion caused by background or foreground stars being mis-classified
as cluster objects. Similar results were obtained by Smoker et al. (2011) who observed three open
clusters with UVES and found, within the same cluster, variations in interstellar column density in Ca ii and
Na i D of up to $\sim$0.5 and 1 dex, respectively. Authors such as Bowen et al. (1991), Kennedy et al. (1998),
Meyer & Lauroesch (1999), Andrews et al. (2001), Smoker et al. (2003, 2011), van Loon et al. (2009, 2013), Welsh et al. (2009)
and Nasoudi-Shoar et al. (2010) attempted to eliminate the confusion issue by observing the centre of Globular clusters, QSOs or stars at high Galactic latitude
in order to obtain the reduced Ca ii or Na i equivalent width. The current paper builds on some previous work by observing a total
of 609 stars within 7 open clusters, of which 403 objects lie in the Large and Small Magellanic Clouds. By comparison with many previous studies,
smaller scales ($\sim$12 arcsec to 27 arcmin corresponding to $\sim$0.05 to 15 pc) are studied, corresponding to
the fieldsize of FLAMES[FLAMES (Pasquini et al. 2002) is a multi-object, intermediate and high
resolution spectrograph, mounted at the VLT/Unit Telescope 2 (Kueyen)
at Cerro Paranal, Chile, operated by ESO.] which is $\sim$30 arcmin, but still at a reasonable spectral resolution ($\sim$16 km s$^{-1}$).
Additionally to the small-scale FLAMES observations, the large-scale structure of the Milky Way is studied using
FEROS[FEROS (Kaufer et al. 1999) is a high-resolution echelle spectrograph,
mounted at the 2.2 m Telescope at La Silla, Chile, operated by ESO.] and UVES[UVES (Dekker et al. 2000;
Smoker et al. 2009) is a a high-resolution echelle spectrograph, mounted at the 8.2-m Unit Telescope 2 at Paranal, Chile]
via the observation in Ca ii K and Na i D of 165 stars in the Magellanic Clouds and 29 within the Galaxy at a
resolution of $\sim$4–8 km s$^{-1}$. Our aim is to determine if current models of the ISM match with the observations.
Section <ref> describes the sample of open clusters and field stars, plus the data reduction and analysis performed to estimate
the equivalent width and column density in the Galactic component in Ca ii and/or Na i D of the
sightlines studied. In Sect. <ref> we give the main results, including figures showing the interstellar profiles of the
sightlines studied as well as tables of the component fit parameters. Section <ref> contains the discussion,
in particular focusing on the large- and small-scale variation observed in the Ca ii K and Na i D profiles towards the target stars and
how variation in the former impacts on the use of the former species as a distance indicator. Using the FEROS and UVES Galactic sightlines plus
archive data we determine the parallax – column density relationship for Ca ii and other lines and compare with the distance
derived using spectroscopic parallax. Finally, Sect. <ref> contains the summary and suggestions for future work. In
what follows we define Low Velocity gas as having absolute values of velocity less than $\sim$35 km s$^{-1}$, intermediate velocity
gas with $\sim$35$<v<\sim$90 km s$^{-1}$, and high velocity gas with absolute velocities greater than $\sim$90 km s$^{-1}$.
§ THE SAMPLE, DATA REDUCTION AND ANALYSIS
The data on which the current paper is based were extracted from the ESO archive and are FLAMES-GIRAFFE observations
towards three open clusters located in the Milky Way, and two in each of the Large and Small Magellanic Clouds, plus FEROS
and UVES observations towards stars located in the Magellanic system and Milky Way.
§.§ FLAMES-GIRAFFE archive sample towards open clusters in the MW and Magellanic System
The FLAMES data for the seven open clusters were taken from the ESO archive. Data reduction was performed using the
ESO FLAMES pipeline within gasgano (Izzo et al. 2004) using calibrations taken on the day following the observation. For some observations
the simultaneous calibration fibre was used. The HR2 grating was
employed and measurements from a few spectra with the simultaneous calibration fibre enabled shows that the wavelength
range was from $\sim$3850 to 4045Å with a spectral resolution around Ca ii K of $R$=$\lambda/\Delta\lambda\sim$18,500
or $\sim$16 km s$^{-1}$. The re-binned data from the pipeline were imported into iraf[
iraf is distributed by the National Optical Astronomy Observatories, U.S.A.] where they were co-added using
median addition within scombine and then read into dipso (Howarth et al. 1996). Subsequently, the spectra were
normalised by fitting the continuum in regions bereft of interstellar features, and the signal-to-noise (S/N) ratio measured.
At this stage a small number of late-type stars were excluded from the sample as in these cases distinguishing stellar from
interstellar lines was problematical. For the early-type stars the stellar features were normally broad (c.f. Mooney et al. 2002, 2004)
and hence easily removed in the normalisation process. Table <ref> shows basic data on the open clusters for which FLAMES data were analysed.
Columns 1–5 give the cluster name, alternative name, location (Galactic or Magellanic), Galactic coordinates of the plate centre and
distance in kpc. Columns 6–8 give the total exposure time in seconds, median S/N ratio per pixel around Ca ii K
and the number of stars used in the analysis. This number only includes objects where the
Galactic interstellar component was useful and varies from 43 usable spectra for NGC 6611 to 110 for NGC 330, or a total
of 609 sightlines with Ca ii interstellar measurements. Finally, columns 9–10 give the range of Galactic scales probed
with the current measurements, being the minimum or maximum star-star separation at the distance of the cluster. For the
Magellanic objects we assumed the scale height of the Galaxy in Ca ii K is $\sim$800 pc (Smoker et al. 2003).
Fig. <ref> shows the ($l,b$) coordinates of the stars used for each of the
clusters. Tables <ref> to <ref> (available online) show the coordinates and magnitudes of
the individual stars on which the FLAMES-MEDUSA fibres were placed.
Open cluster basic data observed with FLAMES sorted in terms of increasing NGC number. Distances to Milky Way clusters
are from the WEBDA database with the distances to the LMC and SMC being taken from Keller & Wood (2006). The scales
probed column corresponds to the minimum and maximum transverse star-star separation at the distance of the cluster.
For the Galactic scales probed by the Magellanic objects we have assumed that the scaleheight of the Galaxy is
$\sim$800 pc (Smoker et al. 2003). Note that the Galactic scales
are upper limits, as the ISM absorption can arise anywhere in the
line-of-sight between the Earth and the cluster in question.
Cluster Alternative Location ($l,b$) Dist Exp time Median S/N Stars Scales Galactic scales
name (deg.) (kpc) (s) at Ca ii K used probed (arcmin) probed (pc)
NGC 330 Kron 35 SMC 302.42, –44.66 61 13650 30 111 0.2 – 27.4 0.07 – 9.1
NGC 346 Kron 39 SMC 302.14, –44.94 61 6825 60 110 0.3 – 20.7 0.12 – 8.9
NGC 1761 LH 09 LMC 277.23, –36.07 51 13650 135 111 0.2 – 22.3 0.09 – 8.8
NGC 2004 KMHK 991 LMC 277.45, –32.63 51 13650 95 111 0.2 – 20.0 0.05 – 4.6
NGC 3293 Collinder 224 Galaxy 285.85, +0.07 2.327 795 70 90 0.2 – 22.3 0.14 – 15.1
NGC 4755 The Jewel Box Galaxy 303.21, +2.50 1.976 795 80 81 0.2 – 21.8 0.11 – 12.5
NGC 6611 M 16 Galaxy 16.98, +0.80 1.749 750 50 43 0.2 – 23.5 0.10 – 12.0
§.§ FEROS and UVES archive data towards the Magellanic system
FEROS and UVES spectra of stars within the Large and Small Magellanic Clouds from several observing runs were also
extracted from the ESO archive.
They are typically observations towards bright early-type O- and B- type stars. The Na i D observations
are sometimes affected by Na in emission, although this always appears slightly offset from the Galactic
absorption and was not fitted when performing profile fits.
Telluric correction around the Na i D lines was performed by removing a scaled model of the sky smoothed to the spectral
resolution of the observations using skycalc[skycalc is available at https://www.eso.org/observing/etc/skycalc].
The aim of extracting these spectra was to determine the variation in the Ca ii and Na i D Galactic column density over large scales.
Authors such as Megier et al. (2009) found large variations in the Ca ii K line strength in Galactic open clusters,
although could not rule out non-cluster contamination by background and foreground stars. The current observations of Magellanic
targets eliminate the distance uncertainty and have median S/N ratios of $\sim$35 and $\sim$80 per pixel in Ca ii and Na i, respectively.
Table <ref> (available online) lists details of the 154 Magellanic Cloud targets observed. They are plotted in Fig. <ref>
and probe structures of size $\approx$5 degrees, and hence act as a useful comparison to FLAMES observations of LMC and SMC targets in Sect. <ref>
that probe scales of less than $\sim$30 arcminutes.
[]Positions on the sky of stars taken from the FEROS and UVES archives used to investigate the large-scale structure
variation of interstellar Galactic Ca ii K and Na i D. (a) LMC sample (b) SMC sample.
§.§ FEROS archive data towards Galactic early-type stars
Data towards Milky Way early-type stars from two observing runs were extracted from the FEROS archive.
They are typically observations towards bright early-type O- and B- type stars.
Authors such as Struve (1928), Megier et al. (2005, 2009)
and references therein) have postulated the use of the Ca ii K line strength as a distance indicator for objects close to the Galactic plane.
Table <ref> (available online) lists details of the Milky Way targets observed, with Fig. <ref>
showing their sky positions. The median S/N ratio was $\sim$200 and $\sim$230 in Ca ii and Na i D, respectively.
In Sect. <ref> these observations are used with archive data to investigate the parallax – column density relationship
for Ca ii K.
[]Positions on the sky of data taken from the FEROS archive used to investigate the large-scale structure
variation of Galactic Ca ii K and Na i D and the relationship between parallax and Ca ii column density.
§.§ Data analysis; component fitting
For the Galactic absorption, interstellar components were fitted using both Gaussian fitting using the elf
routine within the dipso software package, and also full profile Voigt-profile fitting using vapid (Howarth et al. 2002).
Simultaneous fitting was used for the Ca ii H and K line profiles within the whole sample and for
Na i D$_1$ and D$_2$ in the FEROS and UVES datasets. The Ca ii H line lies in the wing of H$\epsilon$, and was hence normalised to provide a
profile which could be fitted simultaneously with Ca ii K. Examples of Ca ii H
and K spectra are shown in online Fig. <ref>.
ELF gives velocity centroids, full width half maxima and equivalent widths of the interstellar components. In addition, vapid
yields estimates of the $b$-values and column densities of the profiles.
The wavelength for Ca ii of 3933.661Å and f-value of 0.627 were taken from Morton et al. (2003, 2004); conversion from
Topocentric to the Local Standard of Rest (LSR) reference frame was performed using rv (Wallace & Clayton 1996). We note that due to the
spectral resolution of the FLAMES-GIRAFFE data ($\sim$16 km s$^{-1}$) it is likely that the interstellar profiles observed are
in fact a superposition of many different components. For example, observations of Galactic gas in Ca ii K by Welty
et al. (1996) show that the vast majority of components in their sample have $b$-values of between 0.5–3.0 km s$^{-1}$,
components that would be unresolved in the current dataset. In total Galactic interstellar profiles towards 609 stars were fitted. Smoker et al. (2015) separately describe the
analysis of intermediate and high velocity clouds observed in the spectra towards the Magellanic Cloud targets.
Errors in the interstellar components were estimated using procedures outlined in Hunter et al. (2006). Briefly,
these involve changing the column density and $b$-value of each component in turn until the residual in the model-data exceeds
1$\sigma$ in 3 adjacent velocity bins. For the cases where no residual was above the limit even when the change
equaled the measurement value, the error was set to the value of the measurement. This most frequently happened in
the components with small $b$-values.
Two of the FEROS sample stars (HD 53244 and HD 76728) have very strong stellar lines around Ca ii and Na i hence
profile fitting was not performed for these objects.
§ RESULTS
In this section we present the reduced spectra and model fits for the FLAMES-GIRAFFE and FEROS/UVES sample.
§.§ FLAMES-GIRAFFE spectra
Figures <ref> to <ref> (available online) show the
Ca ii K spectra towards each of the stars in the sample as well as the model fit obtained using Gaussian (ELF) fitting and the
(data-model) residual fit. In order to assess the variations in the profiles, Figs. <ref> to
<ref> (also available online) show the 16 star-star pairs in each cluster with the largest
differences in equivalent width, with no star being plotted more than once.
Tables <ref> to <ref> (available on-line) show the corresponding
Voigt profile fit results for each of the seven clusters studied.
§.§ FEROS and UVES spectra
Figure <ref> (available online) shows the FEROS and UVES Ca ii K, Na i D and
corresponding GASS and LABS Survey 21-cm H i (Kalberla et al. 2005; McClure-Griffiths et al. 2009)
spectra towards the 165 Magellanic Cloud and 29 Milky Way stars. The latter
data have velocity resolution of $\sim$1 km s$^{-1}$, brightness temperature sensitivity of 0.06–0.07 K and
spatial resolution of $\sim$0.5$^{\circ}$ (LAB) and 16 arcmin (GASS). Tables A16 and A17
show the corresponding profile fits and total column densities derived from the optical data, plus the total derived H i Galactic column density
derived from the equation; $N_{\rm HI}$=1.823$\times$10$^{18}\times \int T_{\rm B}$ $dv$, where $T_{\rm B}$
is the detected brightness temperature and $dv$ is in km s$^{-1}$.
Figures <ref> and <ref> show the 16 Magellanic
sightlines pairs in Ca ii K and Na i D for which there is the greatest difference
in column density. Each sightline is only plotted once.
§ DISCUSSION
In this section we discuss the composite Ca ii K spectra and their comparison with single-dish H i 21-cm observations,
the reduced equivalent widths and column densities for the local Ca ii gas as a function of sky position, the variation
in the equivalent width with sky position and the velocity structure, the Ca/H i and Na/H i ratios as a function
of $N$(H i), the Ca ii/Na i ratio and finally
the parallax – Ca ii column density relationship.
§.§ Composite GIRAFFE Ca ii K spectra and comparison with 21-cm H i observations
Figure <ref> shows the composite Ca ii K spectra towards each of the seven clusters, formed by median-combining the
individual normalised spectra, weighting by the square of the S/N ratio, and boxcar smoothing using a box of 3 pixels. We note that the FWHM of the arc lines
is 4 pixels, so no degradation in resolution occurs due to the smoothing. The composite spectra have S/N ratios ranging from $\sim$500-1200, and display between 1-2 main
components with velocities from –35 to +35 km s$^{-1}$. Also shown in Fig. <ref> are 21-cm H i data taken from
the LABS and GASS surveys. Tables <ref> and <ref> show the results of component fitting
to the composite Ca ii K and single-dish H i spectra, with values of the abundance $A$=log($N$(Ca)/$N$(H i)) given.
Although the ATCA-Parkes H i survey (Kim et al. 2003) of the Magellanic Clouds covers the LMC,
no velocity information is available for the Galactic component.
For the three Galactic clusters the H i in emission has significantly more velocity structure than the Ca ii in absorption, which is
simply a reflection of the difference in path lengths studied, with the clusters being at distances of $\sim$2-kpc compared with the extent of the
Milky Way disc in H i that extends beyond a radius of 40-kpc (e.g. Kalberla et al. 2007 and references therein). For three of the four
Magellanic clusters, the main low-velocity peaks observed in the H i spectra are also visible in the Ca ii data. The
exception is NGC 2004 for which there are two bright H i components separated by $\sim$8 km s$^{-1}$ plus an IV component at +40 km s$^{-1}$
but for which only one Ca ii component is detected at a spectral resolution of $\sim$16 km s$^{-1}$. This Ca ii feature has a
FWHM (corrected for instrumental broadening) of $\sim$27 km s$^{-1}$, indicative of two or more components (otherwise the kinetic temperature
of the gas would be extremely high).
We comment finally on the NGC 1761 interstellar spectra in H i and Ca ii K. In H i there is a weak emission
feature at $\sim$–20 km s$^{-1}$ and another much stronger one at $\sim$3 km s$^{-1}$. The component at –20 km s$^{-1}$ has a
[Ca ii/H i] equivalent width/column density ratio approximately 4 times larger than that at +3 km s$^{-1}$. This
increasing ratio with velocity is frequently seen and is generally thought to be caused by calcium being
liberated from dust into the gas phase in intermediate- and high-velocity gas (e.g. Wakker & Matthis 2000).
Ca ii ELF fit results for the composite FLAMES-GIRAFFE spectra. Velocities are in the LSR. FWHM velocities are observed and
do not take into account instrumental broadening of $\sim$16 km s$^{-1}$. Equivalent widths of the Ca ii K lines are
in mÅ. See Sect. <ref> for details.
Cluster $v$ FWHM $EW$
(km s$^{-1}$) (km s$^{-1}$) (mÅ)
NGC 330 -0.3 24.9 93.6
" 63.0 35.8 36.4
" 118.8 23.1 97.9
" 133.4 57.4 126.1
" 202.3 14.9 2.2
" 352.9 20.1 5.0
NGC 346 0.0 24.6 95.8
" 84.7 46.8 58.2
" 117.3 26.5 75.4
" 140.5 24.0 90.8
" 160.2 34.5 63.2
" 202.2 18.5 3.9
NGC 1761 -41.9 12.5 2.2
" -19.1 20.1 45.1
" 1.6 18.1 56.9
" 46.6 16.3 3.3
" 76.1 28.6 17.0
" 142.1 26.7 12.8
" 205.0 62.1 12.1
" 252.8 47.6 66.7
" 263.4 24.4 69.9
NGC 2004 -18.8 31.4 1.4
" 0.5 19.9 72.0
" 46.5 22.6 23.0
" 102.1 25.5 19.7
" 239.8 33.8 46.8
" 283.2 38.7 24.1
" 334.4 35.0 4.1
NGC 3293 -86.8 77.4 9.2
" -48.6 19.7 5.5
" -9.6 30.9 276.1
" 9.9 46.6 18.5
NGC 4755 -44.3 23.6 39.9
" -21.3 26.0 215.9
" -1.6 18.5 96.8
NGC 6611 -48.5 74.2 12.6
" -28.6 14.2 3.9
" 2.4 14.1 24.9
" 15.8 41.2 478.3
" 52.5 36.6 31.5
H i fit results for the LABS spectrum at the coordinate of the centre of the FLAMES-GIRAFFE pointing for
the Magellanic objects.
See Sect. <ref> for details.
Cluster $v$ FWHM $T_{\rm peak}$ $N_{\rm HI}$
(km s$^{-1}$) (km s$^{-1}$) (K) (cm$^{-2}$)
NGC 330 -0.5 5.4 10.5 20.0
" 1.7 16.7 6.1 20.3
" 114.8 30.3 22.3 21.1
" 123.3 19.6 34.8 21.1
" 159.5 30.6 49.2 21.5
NGC 346 -0.4 6.1 7.5 19.9
" 0.0 2.5 5.2 19.4
" 0.3 20.0 4.2 20.2
" 99.1 15.3 1.4 19.6
" 122.9 20.8 18.8 20.9
" 158.5 23.4 47.1 21.3
" 173.3 24.1 24.1 21.1
NGC 1761 -19.9 11.3 1.8 19.6
" 3.0 18.4 7.5 20.4
" 5.0 6.2 3.7 19.7
" 256.8 12.0 1.5 19.6
" 278.7 44.5 10.1 20.9
" 280.1 21.3 36.5 21.2
NGC 2004 -3.1 12.9 11.3 20.5
" 5.9 6.1 7.0 19.9
" 7.4 14.8 3.4 20.0
" 269.2 43.5 3.5 20.5
" 297.7 22.6 8.3 20.6
" 287.2 16.5 6.8 20.3
" 331.6 23.4 0.7 19.5
Top panels: Composite Ca ii K spectra towards each of the 7 clusters, formed by median combining the individual fibres after
continuum normalisation. Bottom panels: 21-cm H i data taken from the GASS (full lines) and LABS survey (dotted lines) towards the FLAMES plate centres.
(a) NGC 330 (S/N$\sim$500), (b) NGC 346 (S/N$\sim$900), (c) NGC 1761 (S/N$\sim$1200), (d) NGC 2004 (S/N$\sim$1000),
(e) NGC 3293 (S/N$\sim$800),
(f) NGC 4755 (S/N$\sim$800), (g) NGC 6611 (S/N$\sim$500). Absorption-line features at $\sim$50 km s$^{-1}$ are caused by IV gas.
§.§ Reduced equivalent widths and column densities for Ca ii K and Na i D and comparison with previous work
Figure <ref> (available online)
shows the point-to-point variation in total column density and
percentage difference in equivalent width as a function of transverse separation for each of the 7 clusters observed with FLAMES. The
column densities and equivalent widths were integrated between the velocity limits shown on the figures in order to
exclude intermediate-, high- and Magellanic Cloud velocity components. We again note that due to the relatively low spectral resolution
of the FLAMES-GIRAFFE dataset, unresolved components are likely to be present that make the column densities very
Bowen et al. (1991) and Smoker et al. (2003) find reduced equivalent width (REW) values in the Ca ii K line for objects at
infinity of $\sim$110 mÅ (with 95 percent of lines lying between 60-310 mÅ) and 113 mÅ, respectively, where the REW
is defined as EW$\times$sin($b$). In the current FLAMES-GIRAFFE dataset for the 4 Magellanic clusters in Ca ii K we find REW values for Galactic
gas ranging from $\sim$35–125 mÅ (see Table <ref>), with median values of $\sim$40–70 mÅ on scales of $\sim$0.05–6 pc.
For the individual stars observed by FEROS and UVES, the corresponding range is $\sim$30–125 mÅ in Ca ii K and $\sim$50–155 mÅ
in Na i D, with median values of 45 and 100 mÅ, respectively. Figure <ref>
shows histogrammes of the REW for low-velocity gas observed in absorption towards four Magellanic clusters, with
Fig. <ref> showing the corresponding histogrammes for the Galactic clusters. For the MC sightlines the
reduced equivalent widths are approximately half the values of those observed in previous work (Bowen et al. 1991), and again indicate large-scale variations
in the EW of optical absorption lines.
For the FLAMES-GIRAFFE sightlines, the maximum variation in REW in low-velocity gas over the $\sim$5 pc field of view is a
factor 3.0 for NGC 330 (which has lower S/N ratio than the other sightlines), 1.8 for NGC 346, 1.8 for
NGC 1761 and 1.6 for NGC 2004. These variations are somewhat smaller than observed in the intermediate velocity and high velocity gas towards
the same sightlines, where the Ca ii K REW for example towards NGC 2004
varies by factors exceeding 10 (Smoker et al. 2015). Of course, the IV gas is likely to be at larger distances than the LV gas, hence the transverse
scales sampled are bigger. Previous studies of small-scale ($\sim$0.03 pc) structure using binaries or the cores of Globular
Clusters (e.g. Meyer & Blades 1996, Lauroesch & Meyer 1999, Lauroesch 2007 and references therein) have found
strong variations in Na i D profiles on small scales, but much smaller changes in Ca ii K equivalent widths or
column densities. The current observations confirm that such equivalent width variations in Ca ii also exist on
scales of $\sim$0.05–6 pc, with variation of $\sim$0.3–0.5 dex in the optically thin approximation.
Figure <ref> shows histograms of the reduced equivalent width and reduced
column density for the two FEROS/UVES-observed species.
For the LMC-only
sightlines, 68 percent of the Ca ii reduced column densities lie within $\pm$0.16 dex of log[$N$(Ca ii cm$^{-2}$)]=11.85.
For the Na i data, 68 percent of the reduced column densities lie within $\pm$0.32 dex of log[$N$(Na i cm$^{-2}$)]=11.93,
reflecting the generally higher clumpiness of this neutral species compared with Ca ii.
Finally, Fig. <ref> shows the sightlines that display the biggest and smallest reduced
equivalent width values for the low velocity gas in Ca ii K and Na i D for the Magellanic Cloud targets.
For over fifty years astronomers have thought that hierarchical structures and turbulence exist in the ISM (von Weizsacker 1951;
von Hoerner 1951; see reviews by Elmegreen & Scalo 2004, Dickey 2007, Hennebelle & Falgarone 2012 and Falceta-Goncalves et al. 2014).
The power spectrum of the ISM has often been used to provide coarse-scale information on
structures present in the ISM and indicate how much material is present at each scale. However, it provides
little information about the
shape of the structures themselves, i.e. very different structures can produce similar power spectra (Chappell & Scalo 2001). In any
case, with the incompletely-sampled FLAMES and FEROS data we cannot obtain a reliable power spectrum of the column density variations. Hence we
restrict ourselves to a comparison between our work and the similar observational and theoretical study of Van Loon et al. (2009).
Their observations towards $\omega$ Cen found that the real fluctuations in the column density maps over scales
of half a degree were 7 per cent in Ca ii (1 standard deviation). The fluctuations detected
in Ca ii K are shown in Table <ref> for our seven clusters. In the case of
the Magellanic clusters,
the 1$\sigma$ variation in equivalent width for the FLAMES-observed field-of-view ranges from 9–15 per cent (for NGC 330 and
NGC 346, respectively) for Ca ii in the gas phase with velocities between -35 and +35 km s$^{-1}$. For the three Galactic
clusters the variation is 63 percent (NGC 3293), 10 per cent (NGC 4755) and 15 per cent for NGC 6611. These are upper
limits, not taking into account the errors on the measurements. For the FEROS/UVES spectra which span tens of degrees on the
plane of the sky, the variation in column density is unsurprisingly much larger, being $\sim$51 per cent in Ca ii.
Van Loon et al. (2009) presented a simple model of the ISM as a collection of spherical cloudlets with filling factor 0.3 and sizes
between 1 AU and 10 pc (their Appendix B). The model predicts observed fluctuations in the column density of the ISM on
scales of 0.5 degrees of 0.1–0.2,
consistent with our observed values. However, other physical
forms of the ISM such as sheets or filaments may also be consistent with the observed variations (e.g. Heiles 1997,
Gómez & Vázquez-Semadeni 2014 and refs. therein). Indeed, Herschel observations of
molecular clouds have detected
a wealth of filaments towards the Gould Belt, with typical widths of around 0.1 pc (André et al. 2010), although to our knowledge
there are no existing optical absorption line data that show filaments of such size in the warm ISM.
[]Reduced equivalent width values for Galactic (low-velocity) gas using stellar probes. Minimum, maximum,
median and 1-sigma variations values are shown for each dataset. The cluster objects are FLAMES-GIRAFFE observations and
the field objects FEROS and UVES observations.
Cluster Species REW$_{\rm Min}$ REW$_{\rm Max}$ REW$_{\rm Med}$ $\sigma$
(mÅ) (mÅ) (mÅ) (mÅ)
NGC 330 Ca ii 36.9 111.3 64.5 10.0
NGC 346 Ca ii 46.7 87.8 68.0 6.3
NGC 1761 Ca ii 46.1 81.6 60.2 7.6
NGC 2004 Ca ii 28.3 49.1 39.9 7.9
NGC 3293 Ca ii 0.03 1.54 0.41 0.26
NGC 4755 Ca ii 12.0 26.2 15.3 1.5
NGC 6611 Ca ii 4.8 11.8 7.8 1.2
Mag. field Ca ii 27.9 138.8 56.8 29.3
Panels (a)-(d) Histogrammes of reduced equivalent width in the Ca ii K interstellar line for Magellanic open
clusters NGC 330, NGC 346, NGC 1761 and NGC 2004. The integration limits are the same as in Fig. <ref>.
Panels (a)-(c) Histogrammes of reduced equivalent width in the Ca ii K interstellar line for Milky Way
clusters NGC 3293, NGC 4755, NGC 6611. The integration limits are the same as in Fig. <ref>
[]Histogrammes showing the reduced equivalent width (REW) (panels (a) and (c)), and reduced column density
(panels (b) and (d)) for Ca ii K and Na i D data from the Magellanic FEROS and UVES sample for gas with LSR velocities
between –35 and +35 km s$^{-1}$. Note that a number of targets do not have REW measurements due to the presence of
stellar lines. See Table 9 for details.
[]FEROS and UVES spectra. Top panel: Ca ii spectra of Magellanic targets for the sightlines that have largest
and smallest reduced equivalent widths between –35 and +35 km s$^{-1}$. One star is from the SMC and one from the LMC
although we are probing Galactic gas. Bottom panel: Corresponding data
for the same two stars for Na i D. Filled lines: LHA 115-S 23 (SMC). Dotted lines: SK -67 256 (LMC).
§.§ Variation in the Galactic velocity centroid and component structure
Figures <ref> to <ref> (available online) show the velocity centroid of the
main Galactic Ca ii K component for each of the clusters observed with FLAMES-GIRAFFE. There are hints of gradients in the velocity
centroid for this low-velocity component in the GIRAFFE data only towards NGC 1761 (north to south in Galactic coordinates with a magnitude
of a few km s$^{-1}$) and towards NGC 2004 (north-west to south-east of a few km s$^{-1}$ over a 0.5 degree field). These probe Galactic gas with maximum transverse
scales of $\sim$5 pc. Figures <ref> and <ref> show the corresponding plots for the
LV gas for the Magellanic stars observed with FEROS.
The SMC clusters NGC 330 and NGC 346
are generally well fitted by only one component at low velocities, although in particular for NGC 330 there are sometimes indications
of two-component structure that would need a better S/N ratio and/or spectral resolution to resolve. Considering the LMC clusters,
for NGC 1761 there are generally two strong low-velocity components, and one for NGC 2004 (although both frequently show
intermediate and high-velocity gas, discussed in Smoker et al. 2015). For the Galactic clusters, NGC 3293 can often be
fit with a single component (e.g. Star 2372 in Fig. <ref>), although the residuals in other
sightlines (e.g. Star 2341) imply another component may be needed, and in yet other sightlines (e.g. Star 2303) there is
clearly more than one component. Nevertheless, the overall shape of the Ca ii K profile is similar in all sightlines. More
variation in profile shape is apparent towards Galactic cluster NGC 4755, with all sightlines needing two or three components
to be well-fitted. Finally, towards NGC 6611 there is also a large variation in the two to four interstellar components present.
§.§ Ca ii/H i and Na i/H i ratios in the Galactic ISM from observations of stars in the Magellanic Clouds
Values of the estimated Galactic Ca ii and Na i abundances $A$ were derived from log($N_{\rm opt}$)/log($N_{\rm HI})$
(where N$_{\rm opt}$ is the column density of either Ca ii or Na i).
Figures <ref>(a) and (b) show the corresponding fits of $A$ against $N_{\rm HI}$ for Ca ii
and Na i respectively, with the best-fitting lines from Wakker & Mathis (2000)
overlaid. We note that Wakker & Mathis only plotted data up to log[$N$(H i cm$^{-2}$)]$\sim$21.4, and that
for the current dataset ionisation effects have not been taken into account. Neither has H$_{2}$ nor
the large difference in resolution between the optical and H i observations. For both Ca ii and Na i,
at column densities smaller than log[$N$(H i)]=21.4, the values of $A$ lie within the 1$\sigma$ scatter
of 0.42 and 0.52 dex, respectively, given in Wakker & Mathis (2000).
However, at higher H i column densities the extrapolation of the best fit of Wakker & Mathis (2000) lies $\sim$1 dex above
the observed $A$ values for Ca ii. Also, the scatter in $A$ for Na i markedly increases at high values of H i. This
may be caused in part by saturation effects, and ideally the current sightlines should be re-observed in Na i at 3303Å to
eliminate this possibility.
[]$A$ vs $N$(H i) for FEROS and UVES Magellanic sightlines for gas with LSR velocities between –35
and +35 km s$^{-1}$. The full lines are from Wakker & Mathis (2000) (a) Ca ii K. (b) Na i D.
§.§ The Ca ii/Na i ratio in the FEROS/UVES Magellanic sightlines
For LV gas the [Ca ii/Na i] ratio ranges from $\sim$–0.9 to +0.6 dex which is within the range of –0.1 to 100
derived for example by Siluk & Silk (1974) and Vallerga et al. (1993). The Ca ii/Na i column density ratio
is a common diagnostic of the ISM (Hobbs 1975; Welty et al. 1999; van Loon et al. 2009; Welsh et al. 2009 amongst others), due to the fact
that Ca shows a large range in depletion, depending on the temperature and presence of dust (e.g. Bertin et al. 1993).
Welty et al. (1999) note that if Ca ii is the dominant species, then the ratio [Ca ii/Na i]
depends primarily on the Ca depletion and the temperature. In warm gas (T$\sim$3000 K), Ca i is enhanced and Ca iii
can also be a major contributor to the total Ca column density (Sembach et al. 2000).
Figure <ref> shows this
ratio plotted against Ca ii K column density for the FEROS and UVES Magellanic sightlines only. A weak trend in increasing [Ca ii/Na i]
with Ca ii K column density is present, although this could be in part explained by saturation issues. A future paper will look at these
and other data in more detail to investigate the [Ca ii/Na i] ratio as a function of velocity (the Routly-Spitzer effect;
Routly & Spitzer 1952, Vallerga et al. 1993).
Figure <ref> shows the FEROS and UVES-observed [Ca ii/Na i] ratio for low-velocity gas, derived using
Magellanic objects, as a function of H i column density for LV gas as obtained from the LABS survey. There is an anti-correlation
between the two quantities, explained by the fact that the neutral species Na i and H i both probe cooler parts of the ISM than
Ca ii which tends to be depleted onto dust as the H i column density increases.
[][Ca ii K/Na i D] ratio plotted against log[Ca ii K] for low velocity gas for the FEROS and UVES Magellanic sightlines only.
[][Ca ii K/Na i D] ratio plotted against log[H i] for low velocity gas for the FEROS and UVES Magellanic sightlines only.
§.§ The parallax – column density correlation for Ca ii and other species
Authors including Beals & Oke (1953), Megier et al. (2005, 2009) and Welsh et al. (2010) find a slowly-increasing Ca ii equivalent width
with increasing distance, although with a large scatter. At distances greatly exceeding 100 pc, Welsh et al. (1997)
note that the increase in column density is more associated with the number of clouds sampled along a particular
sight-line as opposed to the actual distance. Figure <ref>(a) shows the Hipparcos parallax plotted against
the log of the Ca ii K column density for all the objects in the current paper for which both quantities are
available. Additionally, we have used results from Sembach et al. (1993), Hunter et al. (2006) and Smoker et al. (2011) to
produce a sample of 125 sightlines with parallaxes ranging from $\sim$11 mas down to zero. Superimposed on the plot is the best-fit
line from Megier et al. (2009) which has the form $\pi$=1/(2.29$\times$10$^{-13}\times N_{\rm CaII}$ + 0.77) (n=262),
where $N_{\rm CaII}$ is in cm$^{-2}$. At small distances (large values of parallax), the equation from Megier et al. predicts
larger parallaxes at a given log($N_{\rm CaII}$) than observed in our present dataset, ranging from a $\sim$25 per cent difference
at 100 pc distance, to $\sim$15 per cent at 200 pc. This is likely just a reflection of the differing sightlines used in the two datasets, as previously
observed when results from Megier et al. (2005) and Megier et al. (2009) are compared. We note that the Local Bubble has dimensions of around $\sim$100 pc
(e.g. Breitschwerdt et al. 1998, Welsh & Shelton (2009) and references therein), and hence the column densities in this region are
frequently very low (e.g. Frisch & York 1983). Therefore, any analytical formula for the parallax/column density is
likely to fail in this regime. The best fit that we obtain is;
\begin{equation}
\pi(mas)=1/(2.39 \times 10^{-13} \times N_{\rm CaII} (cm^{-2}) + 0.11),
\end{equation}
which is also displayed on the figure. To further evaluate the relationship, we have plotted in Fig. <ref>(b)-(h)
the parallaxes against Ca ii column densities taken from the compilation of Gudennavar et al. (2012),
the most extensive
dataset of such measurements available in the literature. In particular, Fig. <ref>(b) shows 419 data points with absolute values of Galactic
latitude less than 10.0$^{\circ}$, with the best-fit lines from Megier et al. (2009) and the current result superimposed.
Figures EA29 to <ref> (available online) shows the corresponding plot of parallax against column density
for the 38 species from Gudennavar et al. (2012). Clear correlations (although with large scatters) are visible in
Al i (few data points), Ca ii, D i (few data points), Mn ii, Na i, O vi, P ii
and Ti ii. Of these, only Mn ii and P ii are typically the dominant ionisation stage in the warm ISM
(Sembach et al. 2000). Hence it is likely that other factors, such as dust depletion and inherant clumpiness of the
element, causes much of the observed scatter. In particular, S ii is both the dominant ionisation stage and typically
the element is thought to be little depleted onto dust grains in the diffuse ISM. However, the parallax/column density
relationship for this line shows no reduced scatter compared with other elements.
Finally, in Fig. <ref> we plot the Hipparcos parallax against the Ca ii column density,
the Hipparcos parallax against the spectroscopic parallax and the spectroscopic parallax against the Ca ii
column density. Data are taken from Gudennavar et al. (2012) and we only include stars of spectral type V in the
comparision. The spectroscopic parallaxes used absolute magnitudes and colours from Schmidt-Kaler (1982) and Wegner (1994),
from which we estimated the reddening and hence the distance to the stars in question. Straight line
fits between 100 pc and 1000 pc for CaII/Hipparcos, Spectroscopic/Hipparcos and CaII/Spectroscopic result
in scatters in the ordinate of 0.67, 0.35 and 0.60, respectively. Hence between these distance limits the
scatter in the parallax vs spectroscopic parallax fit is 0.25 dex or a factor 1.8 smaller than the
parallax vs Ca ii column density fit. We have attempted to improve the correlation by only including sightlines
where log(Ca ii/Na i) exceeds 0.5, to exclude instances
with cooler gas present in which
the Ca ii is locked up in dust grains. However, the correlation between parallax and Ca ii is not
significantly improved when using this subsample.
[](a) Hipparcos parallax against log$N$(Ca ii K) for stars in the current Galactic FEROS sample plus objects taken from the
literature. The red line is the best-fit from Megier et al. (2009). The green line is the best fit derived from the current data
to the function $\pi$=1/(A$\times N$+b) where A=2.39$\times$10$^{-13}$ and b=0.11. (b)-(h) Corresponding plots with data taken from
the compilation of Gudennavar et al. (2012) and a range of Galactic latitude and longitudes.
[](a) Log(Ca ii cm$^{-2}$) against Log(paralactic distance pc).
(b) Log(Spectroscopic parallax distance pc) against Log(paralactic distance pc)
(c) Log(Ca ii cm$^{-2}$) against Log(Spectroscopic parallax distance pc).
Parallaxes and spectral types are from simbad. Only stars of spectral type
V are included in the plot. Ca ii column densities are taken from
Gudennavar et al. (2012).
§ SUMMARY AND SUGGESTIONS FOR FUTURE WORK
We have described the use of FLAMES, FEROS and UVES archive data towards field stars and open clusters in the Milky Way and
Magellanic Clouds to obtain information on the variability of Ca ii in the Galactic interstellar medium
and its possible use as a distance indicator. We find that towards 4 Magellanic open clusters
the maximum variation observed is between a factor of $\sim$1.8 and 3 in equivalent width or $\sim$0.3–0.5 dex
in column density in the optically thin approximation over fields of size $\sim$0.05–6 pc. These observations can
be explained by a simple model of the ISM presented in van Loon et al. (2009) although likely other functional
forms of the ISM would also match the observations. Using archive observations and results from the literature we
derive a parallax – column density relationship for Milky Way gas in Ca ii of
$\pi$(mas)=1/(2.39$\times$10$^{-13}\times N_{\rm CaII}$(cm$^{-2}$ + 0.11) that predicts parallaxes to within 15 percent
of the Megier et al. (2009) values for distances of 200 pc.
A future paper will use the FLAMES HR4 grating setting to study the variation in the Ca i and CH$^{+}$ lines at 4226Å
and 4232Å, respectively, for the sample of clusters discussed here, to determine the variation in neutral and
molecular column density. New observations using the Na i D line could probe the variation in the Routley-Spitzer effect
on small scale and how it changes with reddening.
This paper makes use of data taken from the Archive of the European Southern Observatory. This research has made use of the simbad database,
operated at CDS, Strasbourg, France. JVS thanks Queen's University Belfast for financial support under the
visiting scientist scheme and to ESO for financial support
under the Director General's Discressionary fund. We would like to thank an anonymous referee for their careful reading of the
original manuscript.
[1993]alb03 Albert C. E., Blades J. C., Morton D. C., Lockan F. J., Proulx M., Ferrarese L., 1993,
ApJS, 88, 81
[2010]and10 André Ph., et al., 2010, A&A. 518, 102
[2001]and01 Andrews S. M., Meyer D. M., Lauroesch J. T., 2001,
ApJ, 552, 73
[1953]bea53 Beals C. S., Oke J. B. 1953, MNRAS, 113, 530
[1993]ber93 Bertin P., Lallement R., Ferlet R., Vidal-Madjar A., 1993, A&A, 278, 549
[1991]bow91 Bowen D. V., 1991,
MNRAS, 251, 649
[1998]bre98 Breitschwerdt D., Freyberg M., Truemper J.,1998, In: IAU Coloq 166, "The local bubble
and beyond". LNP, vol 506, Springer, Berlin
[2001]cha01 Chappell D., Scalo J., 2001, MNRAS, 325, 1
[2000]dek00 Dekker H., D'Odorico S., Kaufer A., Delabre B., Kotzlowski H., 2000, SPIE, 4008, 534
[2007]dic07 Dickey J. M., 2007, IAUS, 237, 1
[2004]elm04 Elmegreen B. G., Scalo J., ARA&A, 2004, 42, 211
[1997]esa07 ESA, 1997, The Hipparcos and Tycho Catalogues (ESA SP-1200) (Noordwijk: ESA)
[2013]fal13 Falceta-Goncalves D., Kowal G., Falgarone E., Chian A. C.-L., 2014,
Nonlin. Processes Geophys., 21, 587
[1983]fri83 Frisch P. C., York D. G., 1983,
ApJ, 271, 59
[2014]gom98 Gómez Gilberto C., Vázquez-Semadeni E., 2014, ApJ, 791, 124
[2012]gud12 Gudennavar S.B., Bubbly S.G., Preethi K., Murthy J., 2012,
ApJS, 199, 8
[1904]har04 Hartmann J., 1904, ApJ, 19, 268
[1997]hei97 Heiles C., 1997, ApJ, 481, 193
[2012]hen12 Hennebelle P., Falgarone E., 2012, A&ARv, 20, 55
[1975]hob75 Hobbs L., 1975, ApJ, 202, 628
[2002]how02 Howarth I. D., Price R. J., Crawford I. A., Hawkins I., 2002,
335, 267
[2003]how03 Howarth I. D., Murray J., Mills D., Berry D. S., 1996,
Starlink User Note SUN 50, Rutherford Appleton Laboratory/CCLRC
[2006]hun06 Hunter I., Smoker J. V., Keenan F. P., et al, 2006, MNRAS, 367, 1478
[2004]izz04 Izzo C., Kornweibel N., McKay D., Palsa R., Peron M., Taylor M., 2004,
The Messenger, 117, 33
[2005]kal05 Kalberla P. M. W., Burton W. B., Hartmann D., Arnal E. M., Bajaja E., Morras R.,
Pöppel, W. G. L. 2005,
A&A, 440, 775
[2007]kal07 Kalberla P. M., W., Dedes L., Kerp J., Haud U., 2007,
A&A, 469, 511
[1999]kau99 Kaufer, A., Stahl, O., Tubbesing, S. Norregaard P., Avila G., Francois P., Pasquini L., Pizzella A.,. 1999,
The Messenger 95, 8
[1998]ken98 Kennedy D. C., Bates B., Kemp S. N., 1998,
A&A, 336, 315
[2006]kel06 Keller S. C., Wood P. R., 2006, ApJ, 642, 834
[2003]kim03 Kim S., Staveley-Smith L., Dopita M. A., Sault R. J., Freeman K. C., Lee Y., Chu Y., 2003,
ApJS, 148, 473
[1999]lau99 Lauroesch J. T., Meyer D. M., 1999,
Apj, 519, 181
[2007]lau07 Lauroesch J. T., 2007, ASP Conference Series, SINS – Ionized
and Neutral Structures in the diffuse interstellar medium, Vol 365, 40
[2009]mcl09 McClure-Griffiths N. M., Pisano, D. J., Calabretta, M. R. et al., 2009, ApJS, 181, 398
[2005]meg05 Megier A., Strobel A., Bondar A., Musaev F. A., Han I., Krelowski J., Galazutdinov G. A.,
2005, ApJ, 634, 451
[2008]meg08 Megier A., Strobel A., Galazutdinov G. A., Krelowski J., 2009, A&A, 507, 833
[1996]mey96 Meyer D. M., Blades J. C., 1996,
ApJ, 464, 179
[1999]mey99 Meyer D., Lauroesch J. T., 1999,
ApJ, 520, 103
[2002]moo02 Mooney C. J., Rolleston W. R. J., Keenan F. P., Dufton P. L.,
Smoker J. V., Ryans R. S. I., Aller L. H., 2002, MNRAS, 337, 851
[2004]moo04 Mooney C. J., Rolleston W. R. J., Keenan F. P., Dufton P. L.,
Smoker J. V., Ryans R. S. I., Aller L. H., Trundle C., 2004, A&A, 419, 1123
[2003]mor03 Morton D. C., 2003, ApJS, 149, 205
[2004]mor04 Morton D. C., 2004, ApJS, 151, 403
[2010]nas10 Nasoudi-Shoar S., Richter P., de Boer K. S., Wakker B. P., 2010,
A&A, 520, 26
[2002]pas02 Pasquini, L., Avila, G., Blecha, A., et al. 2002, The Messenger 110, 1
[1952]rs52 Routly P. M., Spitzer L., 1952,
ApJ, 115, 227
[1982]sch82 Schmidt-Kaler Th., 1982, in Schaifers K., Voigt H. H.,
eds, Landolt-Boernstein, group VI, subvol. b, Stars and Star Clusters, Vol. 2
Springer, Berlin, p. 17
[1993]sem93 Sembach K. R., Danks A. C., Savage B. D., 1993, A&AS, 100, 107
[2000]sem00 Sembach K. R., Howk J. C., Ryans R. S. I., Keenan F. P., 2000,
ApJ, 528, 310
[1974]sil74 Siluk R. S., Silk J., 1974,
ApJ, 192, 51
[2003]smo93 Smoker J. V., et al. 2003, MNRAS, 346, 119
[2009]smo09 Smoker J. V., et al., 2009, Msngr, 138, 8
[2011]smo11 Smoker J. V., Fox F. P., Keenan F. P., 2011, MNRAS, 415, 1105
[2015]smo15 Smoker J. V., Fox A. J., Keenan F. P., 2015, MNRAS, 451, 4346
[2011]smo11 Smoker J. V., Bagnulo S., Cabanac R., et al., 2011,
MNRAS, 414, 59
[1928]str28 Struve O., 1928, ApJ, 108, 242
[1993]val93 Vallerga J. V., Vedder P. W., Craig N., Welsh B. Y., 1993,
ApJ, 411, 729
[1951]van51b von Hoerner S., 1951, Astrophysik, 30, 17
[1999]van99 van Loon J. T., Smith K. T., McDonald I., Sarre P. J., Fossey S. J., Sharp R. G., 2009,
MNRAS, 399, 195
[2013]van13 van Loon J. T., et al., 2013, A&A, 550, 108
[1951]van51a von Weizsacker C. F., 1951, ApJ, 114, 165
[2000]wak00 Wakker B. P., Mathis J. S., 2000,
ApJ, 544, 107
[1996]wal96 Wallace P., Clayton C., 1996, rv, Starlink User Note
SUN 78, Rutherford Appleton Laboratory/CCLRC
[1994]weg94 Wegner, W., 1994, MNRAS, 270, 229
[1997]wel97 Welsh B. Y., Sasseen T., Craig N., et al. 1997,
ApJS, 112, 507
[2009]wel09a Welsh B. Y., Shelton R. L., 2009,
ApSS, 323, 1
[2009]wel09b Welsh B. Y., Wheatley J., Lallement R., 2009,
PASP, 121, 606
[2010]wel10 Welsh B. Y., Lallement R., Vergely J. -L., Raimond S., 2010,
A&A, 510, 54
[1996]wel96 Welty D. E., Morton D. C., Hobbs L. M., 1996,
ApJS, 106, 533
[1999]wel99 Welty D. E., Frisch P. C., Sonnerborn G., York D. G., 1999,
ApJ, 512, 636
|
1511.00215
|
Bidirectional Long Short-Term Memory Recurrent Neural Network (BLSTM-RNN) has been shown to be very effective for modeling and predicting sequential data, e.g. speech utterances or handwritten documents.
In this study, we propose to use BLSTM-RNN for a unified tagging solution that can be applied to various tagging tasks including part-of-speech tagging, chunking and named entity recognition.
Instead of exploiting specific features carefully optimized for each task, our solution only uses one set of task-independent features and internal representations learnt from unlabeled text for all tasks.
Requiring no task specific knowledge or sophisticated feature engineering, our approach gets nearly state-of-the-art performance in all these three tagging tasks.
§ INTRODUCTION
Long short-term memory (LSTM) <cit.> is a type of promising recurrent architecture, able to bridge long time lags between relevant input and target output, and thereby incorporate long range context.
This type of structure is theoretically well suited and has been proven a powerful model for tagging tasks.
For applications in natural language processing (NLP), LSTM has proved advantageous in language modeling <cit.>, language understanding <cit.>, and machine translation <cit.>.
A bidirectional LSTM (BLSTM) <cit.>, furthermore, introduces two independent layers to accumulate contextual information from the past and future histories.
It seems natural to expect BLSTM to be an effective model for tagging tasks in NLP while to our best knowledge no successful case of this application has been reported.
In this work, we apply BLSTM-RNN to three typical tagging tasks: part-of-speech (POS) tagging, chunking and named entity recognition (NER).
As a neural network model, BLSTM-RNN is awkward for using discrete features.
Since these features have to be represented as one-hot vector usually with very large size, using this type of features would lead to too large input layer to operate.
Therefore, we only use word form and simple capital features, disregarding of all the other discrete conventional NLP features, such as morphological features.
Using such simple task independent features assures our model quite unified so that it can be directly applied to various tagging tasks.
To further improve the performance of our approach without disrupting the universality, we introduce word embedding, which is a real-valued vector associated with each word.
It is an internal representation that is considered containing syntactic and semantic information and has shown a very attractive feature for various NLP tasks <cit.>.
Word embedding can be obtained by training a neural network language model <cit.>, shallow neural network <cit.>, or a recurrent neural network <cit.>.
In this work, we also propose a novel method to train word embedding on unlabeled data with BLSTM-RNN.
The main contributions of this work include:
First, it shows an effective way to use BLSTM-RNN for dealing with various NLP tagging tasks.
Second, it proposes a unified tagging system that can get competitive tagging accuracy without using any task specific features, which makes this approach more practical for tagging tasks that lack of prior knowledge.
The remainder of this paper is organized as follows.
Section 2 gives a brief introduction of BLSTM architecture.
Section 3 describes the BLSTM-RNN based tagging approach and Section 4 introduces the training and usage of word embeddings.
Section 5 presents experimental results.
Section 6 discusses related works and concluding remarks are given in Section 7.
§ BIDIRECTIONAL LSTM ARCHITECTURE
Recurrent neural network (RNN) is a kind of artificial neural network that contains cyclic connections, which can model contextual information dynamically.
Given an input sequence $x_1,x_2,...,x_n$, a standard RNN computes the output vector $y_t$ of each word $x_t$ by iterating the following equations from $t$ = 1 to $n$:
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
where $h_t$ is the vector of hidden states, $W$ denotes weight matrix connecting two layers (e.g. $W_{xh}$ is the weights between input and hidden layer), $b$ denotes bias vector (e.g. $b_h$ is the bias vector of hidden layer) and $H$ is the activation function of hidden layer.
Note that $h_t$ persists information from previous step's hidden state $h_{t-1}$, and thus theoretically RNN can make use of all input history.
However, in practice, the range of input history that can be accessed is limited, since the influence
An LSTM network. The network has five input units, a hidden layer composed of two LSTM memory blocks and three output units. Each memory block has four inputs but only one output.
All connections of the left block are drawn and other connections are skipped for simplicity.
of a given input would decay or blow up exponentially as it circulates around the hidden states, which is known as vanishing gradient problem <cit.>.
The most effective solution of this problem so far is the long short-term memory (LSTM) architecture <cit.>.
An LSTM network is formed like the standard RNN except that the self-connected hidden units are replaced by special designed units called memory blocks as illustrated in Figure <ref>.
The output of LSTM hidden layer $h_t$ given input $x_t$ is computed as following composite function <cit.>:
\begin{equation*}
\begin{split}
i_t &= \sigma (W_{xi}x_t+W_{hi}h_{t-1}+W_{ci}c_{t-1}+b_i) \\
f_t &= \sigma (W_{xf}x_t+W_{hf}h_{t-1}+W_{cf}c_{t-1}+b_f) \\
c_t &= f_{t}c_{t-1}+i_{t}\tanh(W_{xc}x_t+W_{hc}h_{t-1}+b_c) \\
o_t &= \sigma (W_{xo}x_{t}+W_{ho}h_{t-1}+W_{co}c_t+b_o) \\
h_t &= o_t \tanh(c_t)
\end{split}
\end{equation*}
where $\sigma$ is the logistic sigmoid function, and $i$, $f$, $o$ and $c$ are respectively the input gate, forget gate, output gate and cell activation vectors.
Weights matrices are represented as arrows in Figure <ref>.
These multiple gates allow the cell in LSTM memory block to store information over long periods of time, thereby avoiding the vanishing gradient problem.
More interpretation about this architecture can be found in <cit.>.
Another shortcoming of conventional RNN is that only historic context can be exploited.
In a typical tagging task where the whole sentence is given, it is helpful to exploit future context as well.
text centered,
inner sep=0.6em,
text width=12em,
inner sep=0.6em,
minimum width=3em,
text centered,
inner sep=0.3em,
[x=1em, y=1em, >=stealth]
[bigcircle](bht1) at (-5.5,0)$\overleftarrow{h}_{t-1}$;
[bigcircle](bht2) at (0,0)$\overleftarrow{h}_{t}$;
[bigcircle](bht3) at (5.5,0)$\overleftarrow{h}_{t+1}$;
[bigcircle](fht1) at (-5.5,-4)$\overrightarrow{h}_{t-1}$;
[bigcircle](fht2) at (0,-4)$\overrightarrow{h}_{t}$;
[bigcircle](fht3) at (5.5,-4)$\overrightarrow{h}_{t+1}$;
[label](x0) at (-9,-8)$\dots$;
[label](x1) at (-5.5,-8)$x_{t-1}$;
[label](x2) at (0,-8)$x_{t}$;
[label](x3) at (5.5,-8)$x_{t+1}$;
[label](x4) at (9,-8)$\dots$;
[label](y0) at (-9,4)$\dots$;
[label](y1) at (-5.5,4)$y_{t-1}$;
[label](y2) at (0,4)$y_{t}$;
[label](y3) at (5.5,4)$y_{t+1}$;
[label](y4) at (9,4)$\dots$;
[namelabel](textoutputs) at (-12,4)Outputs;
[namelabel](textbklayer) at (-12,0)Backward Layer;
[namelabel](textflayer) at (-12,-4)Forward Layer;
[namelabel](textinputs) at (-12,-8)Inputs;
[->](x1) to[bend left=50] (bht1);
[->](x2) to[bend left=50] (bht2);
[->](x3) to[bend left=50] (bht3);
[->](fht1) to[bend right=50] (y1);
[->](fht2) to[bend right=50] (y2);
[->](fht3) to[bend right=50] (y3);
[->] (-10,-4) – (fht1);
[->](fht1) – (fht2);
[->](fht2) – (fht3);
[->](fht3) – (10,-4);
[->] (10,0) – (bht3);
[->] (bht3) – (bht2);
[->] (bht2) – (bht1);
[->] (bht1) – (-10,0);
[->] (x1) – (fht1);
[->] (x2) – (fht2);
[->] (x3) – (fht3);
[->] (bht1) – (y1);
[->] (bht2) – (y2);
[->] (bht3) – (y3);
Bidirectional RNN
Bidirectional RNN (BRNN) <cit.> offers an effective solution that can access both the preceding and succeeding contexts by involving two separate hidden layers.
As illustrated in Figure <ref>, BRNN first computes the forward hidden sequence $\overrightarrow{h}$ and the backward hidden sequence $\overleftarrow{h}$ respectively, and then combines $\overrightarrow{h_t}$ and $\overleftarrow{h_t}$ to generate output $y_t$.
The process can be expressed as:
\begin{equation*}
\begin{split}
\overrightarrow{h_t} &= H(W_{x\overrightarrow{h}}x_t+W_{\overrightarrow{h}\overrightarrow{h}}\overrightarrow{h}_{t-1}+b_{\overrightarrow{h}}) \\
\overleftarrow{h_t} &= H(W_{x\overleftarrow{h}}x_t+W_{\overleftarrow{h}\overleftarrow{h}}\overleftarrow{h}_{t+1}+b_{\overleftarrow{h}}) \\
y_t &= W_{\overrightarrow{h}y}\overrightarrow{h}_t+W_{\overleftarrow{h}y}\overleftarrow{h}_t+b_y
\end{split}
\end{equation*}
Replacing the hidden states in BRNN with LSTM memory blocks gives bidirectional LSTM (BLSTM) <cit.>, i.e., the main architecture used in this paper, which can incorporate long periods of contextual information from both directions.
Besides, as feed-forward layers stacked in deep neural networks, the BLSTM layer can also be stacked on the top of the others to form a deep BLSTM architecture.
As a type of RNN, deep BLSTM can be trained via various gradient-based algorithms designed for general RNN, for example, real-time recurrent learning (RTRL) and back-propagation through time (BPTT).
In this work, we use the BPTT algorithm as described in <cit.> since it is conceptually simple and efficient for computation.
§ TAGGING SYSTEM
The schematic diagram of BLSTM-RNN based tagging system is illustrated in Figure <ref>.
text centered,
inner sep=0em,
[x=1em, y=1em, >=stealth]
[label](input) at (0,0) sentence;
[label](inputsample) at (11,0) $\dots w_{n-1}, w_{n}, w_{n+1} \dots$;
[rectangle,draw, minimum width=8em, thick](BLSTM) at (0,-2.5)BLSTM-RNN;
[label, text width=8em](ioutput) at (0,-6) tag probability distribution;
[label](ioutputsample) at (11,-6) $\dots o(w_{n-1}), o(w_{n}), o(w_{n+1}) \dots$;
[rectangle,draw, minimum width=8em, thick](Decorder) at (0,-9)Decorder;
[label, text width=8em](output) at (0,-11.5) tags;
[label](inputsample) at (11,-11.5) $\dots y'_{n-1}, y'_{n}, y'_{n+1} \dots$;
[->](input) – (BLSTM);
[->](BLSTM) – (ioutput);
[->](ioutput) – (Decorder);
[->](Decorder) – (output);
BLSTM-RNN based tagging system
Given a sentence $w_1,w_2,...,w_n$ with tags $y_1,y_2,...,y_n$, BLSTM-RNN is first used to predict the tag probability distribution $o(w_i)$ of each word,
then a decoding algorithm is proposed to generate the final predicted tags $y'_1,y'_2,...,y'_n$.
§.§ BLSTM-RNN for tagging
The usage of BLSTM RNN is illustrated in Figure <ref>.
text width=12em,
text centered,
minimum height=1.2em,
inner sep=0em
text width=6em,
text centered,
minimum height=1.2em,
inner sep=0em
text width=6em,
text centered,
minimum height=1.2em,
inner sep=0em
text width=6em,
text centered,
minimum height=1.2em,
inner sep=0em
text width=8em,
text centered,
minimum height=1.2em,
text width=8em,
text centered,
minimum height=1em,
inner sep=0em,
text width=2em,
text centered,
minimum height=1em,
inner sep=0em,
[x=1em, y=1em, >=stealth]
[wordvector] (w1) at (0,0) $\underline{w_{i}}$;
[input] (w2) at (10,0)$f(w_{i})$;
[wordembedding] (we1) at (4,3) $I_i$;
[block_noborder](tinput) at (11.5,3) input layer;
[NN] (NN) at (4,6) BLSTM;
[block_noborder](tinput) at (11.5,6) hidden layer;
[output] (w3) at (4,9)$o(w_i)$;
[block_noborder](tinput) at (11.5,9) output layer;
[-,dotted,thick] (-6,0.6) – (1,2.4);
[-,dotted,thick] (6,0.6) – (7,2.4);
[weightsymbol](w1) at (1,1.3) $W_1$;
[-,dotted,thick] (7,0.6) – (1,2.4);
[-,dotted,thick] (13,0.6) – (7,2.4);
[weightsymbol](w2) at (8,1.3) $W_2$;
[-,dotted,thick] (1,3.6) – (-0.4,5.4);
[-,dotted,thick] (7,3.6) – (8.4,5.4);
[-,dotted,thick] (-0.4,6.6) – (1,8.4);
[-,dotted,thick] (8.4,6.6) – (7,8.4);
Usage of BLSTM-RNN for tagging
Here $\underline{w_i}$ is the one hot representation of the current word which is a binary vector with dimension $|V|$ where $V$ is the vocabulary.
To reduce $|V|$, each letter of input word is transferred to its lowercase.
The upper case information is kept by introducing a three-dimensional binary vector $f(w_i)$ to indicate if $w_i$ is full lowercase, full uppercase or leading with a capital letter.
The input vector $I_i$ of the network is computed as:
\begin{equation*}
\end{equation*}
where $W_1$ and $W_2$ are weight matrixes connecting two layers.
$W_1\underline{w_i}$ is also known as the word embedding of $w_i$ which is a real-valued vector with a much smaller dimension than $\underline{w_i}$.
In practice, to reduce the computational cost, $W_1$ is implemented as a lookup table, $W_1\underline{w_i}$ is returned by referring to $w_i$'s word embedding stored in this table.
The output layer is a softmax layer whose dimension is the number of tag types.
It outputs the tag probability distribution of word $w_i$.
§.§ Decoding
According to BLSTM-RNN, the obtained probability distribution of each step is supposed independent with each other.
However, in some tasks such as NER and chunking, tags are highly related with each other and a few of types of tags can only follow specific types of tags.
To make use of this kind of labeling constraints, we introduce a transition matrix $A$ between each step's output as illustrated in Figure <ref>.
minimum size=0.3cm
text height=0.1cm,
execute at begin node=$\vdots$
text height=0.1cm,
execute at begin node=$\dots$
[x=1cm, y=0.8cm, >=stealth]
[align=center] (output0_1) at (-1,-0.8) $t_1$;
[align=center] (output0_2) at (-1,-1.6) $t_2$;
[align=center] (output0_n) at (-1,-3.2) $t_m$;
[align=center] at (0,0) $o(w_1)$;
[circlenode,red,thick] (output1_1) at (0,-0.8) ;
[circlenode] (output1_2) at (0,-1.6) ;
[circlemissing] (output1_m) at (0,-2.6) ;
[circlenode] (output1_n) at (0,-3.2) ;
[align=center] at (1.2,0) $o(w_2)$;
[circlenode] (output2_1) at (1.2,-0.8) ;
[circlenode,red,thick] (output2_2) at (1.2,-1.6) ;
[circlemissing] (output2_m) at (1.2,-2.6) ;
[circlenode] (output2_n) at (1.2,-3.2) ;
[align=center] at (2.4,0) $o(w_2)$;
[circlenode] (output3_1) at (2.4,-0.8) ;
[circlenode] (output3_2) at (2.4,-1.6) ;
[circlemissing] (output3_m) at (2.4,-2.6) ;
[circlenode,red,thick] (output3_n) at (2.4,-3.2) ;
[circlehmissing] at (3.6,0) ;
[circlehmissing] at (3.6,-1.6) ;
[align=center] at (4.8,0) $o(w_n)$;
[circlenode] (output4_1) at (4.8,-0.8) ;
[circlenode,red,thick] (output4_2) at (4.8,-1.6) ;
[circlemissing] (output4_m) at (4.8,-2.6) ;
[circlenode] (output4_n) at (4.8,-3.2) ;
[->, blue] (output1_1) – (output2_1);
[->, red, thick] (output1_1) – (output2_2);
[->, blue] (output1_1) – (output2_n);
[->, blue] (output1_2) – (output2_1);
[->, blue] (output1_2) – (output2_2);
[->, blue] (output1_2) – (output2_n);
[->, blue] (output1_n) – (output2_1);
[->, blue] (output1_n) – (output2_2);
[->, blue] (output1_n) – (output2_n);
[->, blue] (output2_1) – (output3_1);
[->, blue] (output2_1) – (output3_2);
[->, blue] (output2_1) – (output3_n);
[->, blue] (output2_2) – (output3_1);
[->, blue] (output2_2) – (output3_2);
[->, red,thick] (output2_2) – (output3_n);
[->, blue] (output2_n) – (output3_1);
[->, blue] (output2_n) – (output3_2);
[->, blue] (output2_n) – (output3_n);
[align=center, blue] at (0.5,-3.6) $A$;
[align=center, blue] at (1.8,-3.6) $A$;
Illustration of decoding. $o(w_i)$ is the output of BLSTM-RNN, a probability distribution of $m$ tag types.
Each circle represents a tag probability predicted by BLSTM-RNN, $A_{ij}$ stores the score of transition from tag $t_i$ to $t_j$.
The score is determined in a very simple way that if $t_j$ appears immediately behind $t_i$ in training corpus, $A_{ij}$ is 1, otherwise 0.
It implies that tag bigrams that do not appear in training corpus are supposed invalid and would not appear in test case, no matter whether they are actually valid.
The score of a sentence $w_1,w_2,...,w_n$ ($[w]_1^n$ for short) along a path of tags $y_1,y_2,...,y_n$ ($[y]_1^n$ for short) is then given by the product of transition score and BLSTM-RNN output probability:
\begin{equation*}
s([w]_1^n, [y]_1^n)=\prod_{i=1}^n (A_{y_{i-1}y_{i}}\times o(w_i)_{y{i}})
\end{equation*}
The goal of decoding is to find the path which gives the highest sentence score:
\begin{equation*}
[y']_1^n=\argmax \limits_{[y]_1^n} s([w]_1^n, [y]_1^n)
\end{equation*}
This is a typical dynamic programming problem and can be solved with Viterbi algorithm <cit.>.
§ WORD EMBEDDING
As a neural network, BLSTM-RNN can easily adopt already trained word embedding by initializing $W_1$ (illustrated in Figure <ref>) with those external embeddings.
Currently, many word embeddings trained on very large corpora are available on line.
However, these embeddings are trained by neural networks that are very different from BLSTM-RNN.
This inconsistency is supposed as an shortcoming to make the most of these trained word embeddings.
To conquer this shortcoming, we also propose a novel method to train word embedding on unlabeled data with BLSTM-RNN.
In this method, BLSTM-RNN is applied to perform a tagging task with only two types of tags to predict: incorrect/correct.
The input is a sequence of words which is a normal sentence with some words replaced by words randomly chosen from vocabulary.
The words to be replaced are chosen randomly from the sentence.
For those replaced words, their tags are 0 (incorrect) and for those that are not replaced, their tags are 1 (correct).
A simple sample is shown in Figure <ref>.
text centered,
inner sep=0em,
text width=3em,
inner sep=0em,
[x=1em, y=1em, >=stealth]
[label,text width=10em, align=left](t1) at (0,0) original sentence:;
[labelword](w1) at (-3.5,-1) They;
[labelword](w2) at (-3.5+3,-1) seem;
[labelword](w3) at (-3.5+3+3,-1) to;
[labelword](w4) at (-3.5+3+3+2,-1) be;
[labelword](w5) at (-3.5+3+3+2+2,-1) prepared;
[labelword](w6) at (-3.5+3+3+2+2+4.5,-1) to;
[labelword](w7) at (-3.5+3+3+2+2+4.5+2,-1) make;
[labelword](w7) at (-3.5+3+3+2+2+4.5+2+3,-1) $\dots$;
[label,text width=10em, align=left](t2) at (0,-3) input sentence:;
[labelword](w1) at (-3.5,-4) They;
[labelword](w2) at (-3.5+3,-4) beast;
[labelword](w3) at (-3.5+3+3,-4) to;
[labelword](w4) at (-3.5+3+3+2,-4) be;
[labelword](w5) at (-3.5+3+3+2+2,-4) austere;
[labelword](w6) at (-3.5+3+3+2+2+4.5,-4) to;
[labelword](w7) at (-3.5+3+3+2+2+4.5+2,-4) make;
[labelword](w7) at (-3.5+3+3+2+2+4.5+2+3,-4) $\dots$;
[label,text width=10em, align=left](t3) at (0,-6) tags:;
[labelword](w1) at (-3.5,-7) 1;
[labelword](w2) at (-3.5+3,-7) 0;
[labelword](w3) at (-3.5+3+3,-7) 1;
[labelword](w4) at (-3.5+3+3+2,-7) 1;
[labelword](w5) at (-3.5+3+3+2+2,-7) 0;
[labelword](w6) at (-3.5+3+3+2+2+4.5,-7) 1;
[labelword](w7) at (-3.5+3+3+2+2+4.5+2,-7) 1;
[labelword](w7) at (-3.5+3+3+2+2+4.5+2+3,-7) $\dots$;
Sample of constructed corpus for training word embedding.
Although it is possible that some replaced words are also reasonable in the sentence, they are still considered “incorrect”.
Then BLSTM-RNN is trained to minimize the binary classification error on the training corpus.
The neural network structure is the same as that in Figure <ref>.
When the neural network is trained, $W_1$ contains all trained word embeddings.
§ EXPERIMENTS
All of our approaches are implemented based on CURRENT <cit.>, an open source GPU-based toolkit of BLSTM-RNN.
For constructing and training the neural network, we follow the default setup of CURRENT:
The activation functions of input layer and hidden layers are logistic function, while the output layer uses softmax function for multiclassification.
Neural network is trained using statistical gradient descent algorithm with constant learning rate.
In all experiments, consecutive digits occurring within a word are relpaced with the symbol “#” .
For example, both words “Tel192” and “Tel6” are converted into “Tel#”.
The vocabulary we used is the most common 100,000 words in North American news corpus <cit.>, plus one single “UNK” symbol for replacing all out of vocabulary words.
§.§ Tasks
In this section, we briefly introduce three typical tagging tasks and their experimental setup on which we evaluate the performance of the proposed approach: part-of-speech tagging (POS), chunking (CHUNK) and named entity recognition (NER).
POS is the task of labeling each word with its part of speech, e.g. noun, verb, adjective, etc.
Our POS tagging experiment is conducted on the Wall Street Journal data from Penn Treebank III <cit.>.
Training, development and test sets are split following setup in <cit.>.
Table <ref> lists statistical information of the three data sets.
Data Set WSJ sec. IDs Sentences# Tokens#
Training 0-18 38,219 912,344
Develop 19-21 5,527 131,768
Test 22-24 5,462 129,654
3|c|# of tag types 1c|45
POS tagging corpus (WSJ in PTB III)
Performance is evaluated by the accuracy of predicted tags on test set.
CHUNK, also known as shallow parsing, divides a sentence into phrases that each phrase contains syntactically related words, such as noun phrase (NP), verb phrase (VP), etc.
To identify the phrase boundaries, we use a commonly used IOBES tagging scheme that further fractionizes each tag type into four subtypes to indicate whether the word is inside (I), outside (O), begin (B), end(E) a multiple words chunk or a single word chunk (S).
We conduct our experiment on a standard experimental setup of CHUNK according to the CoNLL-2000 shared task <cit.>.
Basic information about this setup is listed in Table <ref>.
Data Set WSJ sec. IDs Sentences# Tokens#
Training 15-18 8,936 211,727
Develop N/A N/A N/A
Test 20 2,012 47,377
3|c|# of tag types (IOBES scheme) 1c|42
CHUNK corpus (WSJ in PTB III: CoNLL-2000)
Performance is assessed by the F1 score computed by the evaluation script released by the CoNLL-2000 shared task[<http://www.cnts.ua.ac.be/conll2000/chunking>].
NER recognizes phrases of named entities such as names of persons, organizations and locations.
IOBES tagging scheme is also applied in this task.
Our experimental setup follows the CoNLL-2003 shared task <cit.>.
Table <ref> shows its basic information.
Data Set Sentences# Tokens#
Training 14,987 203,621
Develop 3,466 51,362
Test 3,684 46,435
2|c|# of tag types (IOBES scheme) 1c|17
NER corpus (CoNLL-2003)
Performance is measured by the F1 score calculated by the evaluation script of the CoNLL-2003 shared task [<http://www.cnts.ua.ac.be/conll2003/ner/>].
§.§ Network Structure
In all experiments, without specific description, the input layer size is fixed to 100 and output layer size is set as the number of tag types according to the specific tagging task.
In this experiment, we evaluate different sizes of hidden layer in BLSTM-RNN to pick up the best size for later experiments.
Performances on three tasks are shown in Figure <ref>.
Performance of BLSTM-RNN with different hidden layer sizes. Horizontal axis is the hidden layer size, vertical axis is respectively accuracy for POS, or F1 score for CHUNK and NER.
It shows that hidden layer size has a limited impact on performance when it becomes large enough.
To keep a good trade-off of accuracy, model size and training time, we choose 100 which is the smallest layer size to get a “reasonable” performance as the hidden layer size in all the following experiments.
Besides, we also evaluate deep structure which uses multiple BLSTM layers.
This deep BLSTM has been reported achieving significantly better performance than single layer BLSTM in various applications such as speech synthesis <cit.>, speech recognition <cit.> and handwriting recognition <cit.>.
Table <ref> compares the performance of BLSTM-RNNs with one (B) and two (BB) hidden layers.
Size of all hidden layers is set 100.
Sys POS(Acc) CHUNK(F1) NER(F1)
B 96.60 91.91 82.52
BB 96.63 91.76 82.66
Comparison of systems with one and two BLSTM hidden layers.
Using more layers brings a slightly improvement for NER task, while does not show much help for POS and slightly decreases the performance of CHUNK.
A possible explanation is that one BLSTM layer is adequate to learn an effective model for tasks like POS and CHUNK which are relatively simple compared with NER or speech tasks, thus in these cases involving more layers would not provide a further help.
Meanwhile additional more parameters makes the network harder to converge to a locally optimal model which leads to a worse performance.
Based on this observation, in following experiments, BLSTM-RNN is set only one hidden layer.
§.§ Decoder
In this experiment, we test the effect of decoder.
WE Dim Vocab# Train Corpus (Toks #) POS (Acc) CHUNK (F1) NER
<cit.> 80 82K Broadcast news (400M) 96.97 92.53 84.69
<cit.> 50 130K RCV1+Wiki (221M+631M) 97.02 93.76 89.34
<cit.> 300 3M Google news (10B) 96.85 92.45 85.80
<cit.> 100 1193K Twitter (27B) 97.02 93.01 87.33
BLSTMWE(10m) 100 100K US news (10M) 96.61 91.91 84.66
BLSTMWE(100m) 100 100K US news (100M) 97.10 93.86 86.47
BLSTMWE(all) 100 100K US news (536M) 97.26 94.44 88.38
BLSTMWE(all) + <cit.> 100 113K US news (536M) 97.26 94.59 89.64
RANDOM 100 100K N/A 96.61 91.71 82.52
Comparison different word embeddings.
Performance of BLSTM tagging systems without and with decoder are listed in Table <ref>.
Sys POS CHUNK NER
BLSTM 96.60 90.60 80.85
BLSTM+Decoder 96.60 91.71 82.52
Comparison of systems without and with decoder.
Without using decoder, predicted tag $y'$ is determined by directly selecting the tag with the highest probability among network output $o(w_i)$.
The results show that decoder significantly improves the performance of CHUNK and NER tasks, though shows no help for POS.
For this difference of improvment, we provide a possible explanation.
In tasks like CHUNK and NER which uses IOBES tagging scheme, tags are highly dependent with their previous tags.
For example, I-X can only appear behind B-X.
CHUNK task has 42 tag types that can combine $42 \times 42 = 1764$ tag bigrams, but only 252 (14.3%) of them actually appear in training corpus.
In NER task, 78 (27.0%) of total 289 tag bigrams have occurred more than once.
Decoding in this case filters considerable invalid paths including more than half of candidates and thus improves the performance.
As a contrast, in POS task, tags are directly predicted on each word without using any tagging schemes and thus do not have such strong dependence among tags.
In POS training corpus, 1439 (71.0%) of total 2025 tag bigrams have occurred more than once.
In this case, the dependences of tags are mainly learnt by BLSTM layer and the help from the decoder is very limited.
The improvement provided by decoder shows that although BLSTM is considered can adopt contextual information automatically, the model is still far from ideal and a decoder with prior knowledge of tagging scheme is essential for achieving a good performance.
§.§ Word Embedding
In this experiment, we evaluate BLSTM-RNN tagging approach using various of word embeddings including those trained by the proposed approach in Section <ref> as well as four types of published word embeddings.
To train word embeddings with our approach, we use North American news <cit.> as the unlabeled data.
To construct corpus for training embeddings, the North American news data is first tokenized with the Penn Treebank tokenizer script [<https://www.cis.upenn.edu/ treebank/tokenization.html>].
Then about 20% words in normal sentences of the corpus are replaced with randomly selected word.
BLSTM-RNN is trained to judge which word has been replaced as described in Section <ref>.
To use word embedding, we just initialize the word embedding lookup table ($W_1$) with these already trained embeddings.
For words without corresponding external embeddings, their word embeddings are initialized with uniformly distributed random values, ranging from -0.1 to 0.1.
Table <ref> lists the basic information of involved word embeddings and performances of BLSTM-RNN tagging approaches using these embeddings,
where RCV1 represents the Reuters Corpus Volume 1 news set.
RANDOM is the word embedding set composed of random values which is the baseline.
BSLTMWE(10m), BSLTMWE(100m) and BSLTMWE(all) are word embeddings respectively trained by BLSTM-RNN on the first 10 million words, first 100 million words and all 536 million words of North American news corpus.
While BSLTMWE(10m) does not bring about obvious improvement, BSLTMWE(100m) and BSLTMWE(all) significantly improve the performance.
It shows that BLSTM-RNN can benefit from word embeddings trained by our approach and larger training corpus indeed leads to a better performance.
This suggests that the result may be further improved by using even bigger unlabeled dataset.
In our experiment, BSLTMWE(all) can be trained in about one day (23 hrs) on a NVIDIA Tesla M2090 GPU.
The training time increases linearly with the training corpus size.
System Acc
<cit.> 97.35
<cit.> 97.24
BLSTM 97.26
System F1
<cit.> 94.34
<cit.> 93.48
BLSTM 94.59
System F1
<cit.> 89.31
<cit.> 88.76
BLSTM 89.64
Top systems on three tagging tasks. State-of-the-art system is marked with bold type.
When uses published word embeddings, the input layer size of BLSTM-RNN is set to the dimension of utiembeddings.
All of the four published word embeddings significantly enhance BLSTM-RNN.
It proves that word embedding is a useful feature and is an effective way to make use of big unlabeled data.
Among all mentioned embeddings, BLSTMWE(all) achieves the best performance on POS and CHUNK tasks while slightly falls behind <cit.> in NER task.
One possible explanation is that <cit.> is trained on Wikipedia data which contains more named entities, thereby their word embedding contains more useful information for NER task.
BLSTMWE(all) is trained on a news corpus which is written with more formal grammar, thus it learns better representations for tagging syntactic role.
Based on this conjecture, it is natural to expect the combination of <cit.> and BLSTMWE(all) to bring a further improvement.
This idea yields the BLSTMWE(all) + <cit.>.
This word embeddings are first initialized with <cit.> and then trained by BLSTM-RNN on North American corpus as BLSTMWE(all).
This embeddings help BLSTM-RNN obtain the best performance in all three tasks.
§.§ Comparison with Previous Systems
In this section, we compare our approach with previous state-of-the-art systems on POS, CHUNK and NER tasks.
Table <ref> lists the related works of these three tasks.
BLSTM represents the BLSTM-RNN tagging approach using BLSTMWE(all) + <cit.> word embeddings.
<cit.> reports the highest accuracy on WSJ test set (97.35%).
Besides, <cit.> (97.34%) and <cit.> (97.33%) also reach accuracy above 97.3%.
These three systems are considered as state-of-the-art systems in POS tagging.
<cit.> is one of the most commonly used approaches which is also known as Stanford tagger.
All of these methods utilize rich morphological features proposed in <cit.> which involves $n$-gram prefix and suffix ($n$ = 1 to 4).
Moreover, <cit.> also involves prefix and suffix of length from 5 to 9.
<cit.> adds extra elaborately designed features, including flags indicating if word ends with $-ed$ or $-ing$, etc.
<cit.> (93.48%) is the system ranked first in CoNLL-2000 shared task challenge.
Later, <cit.> reports the state-of-the-art F1 score (94.34%) on CoNLL-2000 task.
Besides, <cit.> (94.29%) and <cit.> (94.29%) both report F1 score around 94.3%.
These systems use features composing of words and POS tags.
<cit.> (88.76%) is the top system in CoNLL-2003 shared task challenge.
<cit.> (89.31%) reports a better F1 score with a semi-supervised approach.
The unlabeled corpus they used is 27M words from Reuters.
Features used in these works include words, POS tags, CHUNK tags, prefixes, suffixes, and a large gazetteer (geographical dictionary).
All of these top systems use rich features and feature sets in different tasks are quite different.
In contrast, our system only uses one set of task-independent features in all tasks and do not require any feature engineering to achieve the state-of-the-art performance in CHUNK and NER tasks.
In POS, our approach also get a competitive performance that is comparable with Stanford POS tagger.
Our results show that feature engineering in conventional methods can be effectively replaced by built-in modeling of BLSTM-RNN.
§ RELATED WORKS
<cit.> is the most similar work as ours.
It is a unified tagging solution based on neural network which also uses simple task-independent features and word embeddings learnt from unlabeled text.
The main difference is that <cit.> uses feedforward neural network instead of BLSTM-RNN.
A comparison of <cit.> and our approach is listed in Table <ref>,
Sys POS CHUNK NER
NN 96.37 90.33 81.47
NN+WE 97.20 93.63 88.67
BLSTM 96.61 91.71 82.52
BLSTM+BLSTMWE 97.26 94.59 89.64
Comparison of <cit.> and our approach.
where NN is the unified tagging system of <cit.> and NN+WE is that system using word embedding trained by their approach.
Without using word embedding, BLSTM outperforms NN in all three tasks.
It is consistent with the observations in previous works that BLSTM-RNN is a more powerful model for sequential labeling than feedforward network.
With word embeddings, our approach also significantly surpasses NN+WE in all three tasks.
<cit.> proposes a semi-supervised approach which incorporates one billion words of unlabeled data during training.
They claim that their model can be applied to various tasks and report the state-of-the-art performance on all these three tasks (97.40 for POS, 95.15 for CHUNK, 89.92 for NER).
However, they also utilize rich feature templates for each task (47 feature templates for POS, 39 templates for CHUNK, 79 templates for NER) which makes their system not so unified.
Their method still requires prior knowledge to design these feature templates which limits its application to tasks that lack of knowledge or tools to extract necessary features.
Besides, they have to train model with quite big unlabeled data for each task which would lead to a time consuming training process.
In contrast, our approach separates the lengthy training for word embedding from the relatively fast training for the supervised tagging task.
Once our word embeddings are trained, they can be regarded as a kind of linguistic resources like semantic dictionary and be directly used.
§ CONCLUSIONS
In this paper, we propose a unified tagging solution based on BLSTM-RNN.
This system avoids involving task-specific features, instead it utilizes word embeddings learnt automatically from unlabeled text.
Without reliance on feature engineering or prior knowledge, this approach can be easily applied to various tagging tasks.
Experiments are conducted on three typical tagging tasks: POS tagging, chunking and named entity recognition.
Using simple task-independent input features, our approach gets nearly state-of-the-art results on all these three tasks.
Our results suggest that BLSTM-RNN with word embedding is an effective unified tagging solution and worth further exploration.
|
1511.00115
|
[email protected] St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia.
[email protected] St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia.
An asymptotic investigation of monochromatic electromagnetic fields in a layered periodic
medium is carried out in the assumption that the wave frequency is close to the frequency of a stationary
point of a dispersion surface.
We find solutions of Maxwell equations by the method of two-scale asymptotic expansions.
We establish that the principal order of the expansion of a solution dependent on three spatial
coordinates is the sum of two differently polarized Floquet-Bloch solutions, each of which is
multiplied by a slowly varying envelope function. We derive that the envelope functions satisfy a
system of differential equations with constant coefficients. In new variables, it is reduced to a system of two independent equations, both of them are
either hyperbolic or elliptic, depending on the type of the stationary point. The envelope functions are
independent only in the planar case. Some consequences are discussed.
§ INTRODUCTION
The propagation of electromagnetic waves in media with periodic changes of dielectric permittivity
and magnetic permeability, in the so-called photonic crystals, is a subject of many investigations
(see, for example, <cit.>). The popularity of such studies is caused, on the one hand,
by unusual properties of such media, which are yet not completely known and which promise new
On the other hand, there is a technological progress in the creation of artificial structures with
prescribed properties.
The problem of wave propagation in a quickly oscillating medium can be usually reduced to a
problem of wave propagation in an effective homogeneous medium. The mathematic methods applied in
finding the effective medium are methods of homogenization <cit.>, <cit.>,
<cit.>, i.e., asymptotic methods in the long wavelength approximation: $kb \to
\infty$, where $k$ is the wavenumber, $b$ is the period of oscillations of the medium. One of the methods
of the derivation of asymptotic formulas is the method of two-scale expansions. The solution is
assumed to depend on fast and slow distance variables. It turns out that the principal term is a
function of only a slow variable and satisfies homogenized equations, which may be
interpreted as equations in an effective medium. Another approach is based on the spectral point
of view: the field can be represented as a superposition of Floquet-Bloch solutions
corresponding to the lower part of the spectrum.
A lot of physical works use the concept of effective medium. Peculiarities of a
dispersive surface in an effective medium are responsible for unusual phenomena of wave propagation in
such structures, which are often not observed in nature. For example, if one of the principal
components of the effective electric or magnetic tensor has an opposite sign with respect to other two
principal components, the dispersive surface in such a medium has a hyperbolic point. Such media are
named hyperbolic ones and are studied intensively; see, for example, <cit.>. Another
example is a composite material consisting of layers of metals and dielectrics. The
isofrequency dispersive surface for such a structure may have a plane part, i.e., one of the
components of the wavevector may be almost independent of the other two. The waves may propagate
without distortion in such a medium, see <cit.>.
The present work was motivated by papers of Longhi
<cit.>, <cit.>, in which the possibilities of the existence of localized waves in
2D and 3D periodic structures were studied. The idea was to take the frequency of a monochromatic
electromagnetic field equal to the frequency of the stationary point of the dispersive (band)
surface in the periodic structure. The field in 2D and 3D photonic crystals was represented as a
superposition of Wannier functions with an envelope, which satisfies the equation with
constant coefficients. For hyperbolic (saddle) point the envelope satisfies the wave equation,
where one of the spatial coordinates stands for time. By choosing for the envelope one of the
known solutions of the wave equation with finite energy, see, for example, <cit.>, one may
obtain localized waves in the periodic structure. The papers <cit.>, <cit.>
were mainly concentrated on physical aspects of the problem. We are more interested in careful
mathematical analysis and obtain new results in the 3D case.
We noted in our paper <cit.> that the local hyperbolic behavior of a
dispersive surface occurs even for the simplest dielectric layered periodic structures, for
example, for a structure with alternating layers of dielectric. In this case we studied a
field dependent only on two spatial coordinates in <cit.> and found and
numerically confirmed the following physical phenomenon: undistorted beams can propagate in such a structure and there are
only two permitted directions of a beam in the medium, which differs by a sign from the angle with
the normal to layers. A similar phenomenon was found before by numerical simulation in 2D crystals in
<cit.>; but it was not investigated analytically.
In the present paper, we study monochromatic electromagnetic fields with a frequency close to that of a
stationary point of the dispersive surface, which may be a hyperbolic or an elliptic one. The
field is studied in a layered structure with periodically varying dielectric permittivity and the
magnetic permeability. We deal with fields dependent on all the three spatial coordinates. Our
aim is to elaborate a mathematical asymptotic approach for the description of solutions
of Maxwell equations.
The two-scale method in the homogenization theory is applied usually to equations written in
divergent form. Our asymptotic scheme is applied to Maxwell equations in matrix form, which
was first suggested in the book by Felsen and Marcuvitz <cit.>. This form was also used in
the turning points problem for the Maxwell equations in <cit.>, for other
particular equations in <cit.>, <cit.>,
<cit.>, and for the operators in the general form <cit.>. Therefore we believe that results may have generalizations to other
problems. The matrix form of the Maxwell equations is given in Section <ref>.
In Section <ref> we discuss difficulties in studying electromagnetic fields dependent
on three spatial variables caused by the fact that the stationary points are common for waves of
both the TM and the TE polarizations, and that the concepts of TM and TE polarizations depend on
the direction of propagation and at the stationary point itself they loose their meaning. In Section
<ref>, we obtain some integral formulas for second derivatives of the dispersion functions,
which are applied in the next section. In Section <ref>, we develop the two-scale
expansions method for our problem. Apart from the period $b$ in $z$ direction we introduce the second
scale of length, which is a scale of field variation in the $(x,y)$ plane denoted by $L$. We regard
the ratio between $b$ and $L$ as a small parameter $\chi$. We consider the entire formal
asymptotic series and show that the recurrent system for subsequent approximations can be solved
step by step. In Section <ref>, we found that the principal term of the two-scaled field
is obtained as a sum of two differently polarized Floquet-Bloch solutions at the stationary point
and each of these solutions has a slowly varying envelope function. The envelope functions satisfy
a system of two partial differential equations with constant coefficients.
In Section
<ref>, we show that these equations are independent only if the envelopes depend on two spatial coordinates. A qualitative consequence of equations is that undistorted beams can propagate in the medium.
In the general 3D case,
each envelope function can be expressed in terms of two functions. These functions are solutions of
two equations, which are of the Helmholtz type or the Klein-Gordon-Fock type for elliptic
or hyperbolic stationary points, respectively. The fact that the wave field is described by a system of two equations was not predicted in the papers <cit.>, <cit.>.
Our paper is concluded with two Appendices. Appendix 1 contains known results important for the present paper from the Floquet-Bloch theory, which are given in our notation. Appendix 2 contains proofs of two lemmas.
Our case may be regarded as homogenization near a stationary point of the dispersive surface in
contrast to a more usual situation, where the frequency corresponds to the lower part of the spectrum
and obtained in the assumption that $kb \to 0$. Our case demands an assumption $k b \sim
1/\sqrt{\varepsilon_{av} \mu_{av}}$, where $\varepsilon_{av}$, $\mu_{av}$ are typical permittivity
and permeability.
The equations for envelopes are analogous to equations in an effective medium and can be used for
a qualitative description of the wavefield.
§ STATEMENT OF THE PROBLEM
A monochromatic electromagnetic field satisfies the Maxwell
\begin{equation}\label{eq}
\begin{array}{ccc}
{\rm rot}\mathbb{E} = i k \mu \mathbb{H}, \\
{\rm rot}\mathbb{H} = -i k \varepsilon \mathbb{E},
\end{array}
\end{equation}
where $\varepsilon(z+b) = \varepsilon(z)$, $\mu(z+b) = \mu(z)$; $\varepsilon$ and $\mu$ are
piecewise continuous.
The medium with alternating dielectric layers is a practically important special case.
We seek solutions under two assumptions. The first assumption is about the parameters of the
problem. The Maxwell equations contain two parameters of length dimension: the wavelength $\lambda
= 2 \pi/k$ and the period of the medium $b$. We introduce the third parameter of length dimension
$L$, which is the scale of variation of the field in the $(x,y)$ plane. We assume that the
parameter $\chi \equiv b/L$ is small:
\begin{equation}
\chi = b/L \ll 1.
\end{equation}
The second assumption is related to the frequency of the monochromatic field $\omega$. To state
this assumption, we need some concepts: the quasimomentum $\vp$, the dispersive surface
$\omega=\omega(\vp)$, and Floquet-Bloch solutions. We determine and discuss in detail all these
concepts in Appendix 1. This assumption means that the frequency $\omega$ is close to that of the stationary
point of the dispersive surface $\omega_*$:
\begin{equation}
\omega = \omega_* +\chi^2 \delta\omega, \qquad {\delta \omega \sim 1,}
\end{equation}
where $\omega_*$ satisfies the condition
\begin{equation}\label{stat-point}
\left. \nabla \omega \right|_{\vp_*} =0, \quad
\omega_*=\omega(\vp_*).
\end{equation}
We shall see that these stationary points are minima and saddle
points. There is also an additional condition, which we shall
introduce later.
§ MATRIX FORM OF MAXWELL EQUATIONS
We represent the Maxwell equations in matrix form for the sake of
brevity and generality of subsequent asymptotic considerations:
\begin{align}
& k P \bPs = - i \widehat{ \G}\cdot\nabla \bPs, \quad \bPs = \left(
\begin{array}{c} \mathbb{E} \\ \mathbb{H} \end{array} \right), \label{Maxwell} \\
& \hG\cdot\nabla \equiv\G_1 \frac{\partial}{\partial x} + \G_2
\frac{\partial}{\partial y} + \G_3 \frac{\partial}{\partial z},
\qquad \label{scal-def}
\end{align}
\begin{equation}\label{P-G1-G2}
\begin{aligned}
\P = \left( \begin{array}{cc} \varepsilon I & 0 \\ 0 & \mu I \\
\end{array} \right), \,\, \G_1 = \left( \begin{array}{cc}
0 & \gamma_1 \\ -\gamma_1 & 0 \\
\end{array} \right), \,\, \G_2 = -\left( \begin{array}{cc}
0 & \gamma_2 \\ -\gamma_2 & 0 \\
\end{array} \right), \,\, \G_3 = \left( \begin{array}{cc}
0 & \gamma_3 \\ -\gamma_3 & 0 \\
\end{array} \right),
\nonumber
\end{aligned}
\end{equation}
\begin{equation}\label{g2-g3}
\begin{aligned}
\gamma_1 = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1
& 0 \\ \end{array} \right), \quad \gamma_2 =
\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \\
\end{array} \right), \quad \gamma_3 =
\left(\begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \\
\end{array} \right),
\end{aligned}
\end{equation}
and $k/\sqrt{\varepsilon_{av} \mu_{av}} = \omega/c$, where $c$ is
the speed of light in vacuum.
We introduce two types of inner products. Let ${\bf v}(z)$ and ${\bf w}(z)$ be 6-component
complex-valued vector functions; then
\begin{equation}\label{scal-angle}
<{\bf v}, {\bf w}> = \sum\limits_{j=1}^6 \overline{v^j} w^j, \qquad
j=1,\ldots 6,
\end{equation}
where the bar over the symbol stands for complex conjugation. If
${\bf v}$ and ${\bf w}$ are periodic with period $b$ and piecewise
continuous, we define $({\bf v},{\bf w})$ as follows:
\begin{equation}\label{scalar-int}
({\bf v},{\bf w}) = \int\limits_0^b <{\bf v}(z), {\bf w}(z)> dz.
\end{equation}
The Umov-Poynting vector, averaged over time, for a monochromatic
field of frequency $\omega$, i.e., the energy flux density of this
field, averaged over $T=2\pi/\omega$, is determined as
\begin{equation}\label{def-Sk}
\vec{s} = \frac{1}{2} \Re{\, \overline{\mathbb{E}}\times
\end{equation}
It is easy to check that
\begin{equation}\label{Point}
<{\bPs},\G_j {\bPs}> = 4s_j, \quad j=1,2,3; \quad ({\bPs},\G_j {\bPs}) = {2} \int\limits_0^b
\Re{\, \left[\overline{\mathbb{E}} \times {\mathbb{H}}\right]}_j dz.
\end{equation}
The density of electromagnetic energy, averaged over time, in the
case of real $\varepsilon$ and $\mu$ reads
\begin{equation}\label{def-u}
u = \frac{\varepsilon}{4} |\mathbb{E}|^2 + \frac{\mu}{4}
\end{equation}
We note that
\begin{equation}\label{def-en}
<{\bPs},\P {\bPs}> = 4 u, \qquad ({\bPs},\P {\bPs}) =
\int\limits_0^b ( \varepsilon |\mathbb{E}|^2 + \mu |\mathbb{H}|^2)
\end{equation}
§ THE FLOQUET-BLOCH SOLUTIONS AND THE DISPERSION RELATION
Since the properties of the medium do not depend on $x,y$, we shall seek particular
solutions in the form
\begin{equation}\label{plane_wave}
\bPs_B(x, y, z;\vp) = e^{i(p_x x + p_y y)} \bPh(z;\vec{p}), \quad
\bPh = \left( \begin{array}{c} {\mathbf E} \\ {\mathbf{H}}
\end{array} \right),
\end{equation}
where the parameters $p_x$ and $p_y$ are lateral components of the wave vector. The components of
the vector-valued function $\bPh(z;\vec{p})$ satisfy a system of ordinary differential equations
with periodic coefficients:
\begin{equation}\label{Maxwell-pxpy}
k\mathbf{P} \bPh + i\G_3\frac{\partial \bPh}{\partial z} = p_x\G_1 \bPh + p_y\G_2 \bPh.
\end{equation}
We are going to obtain Floquet-Bloch solutions of this system. However there are difficulties owing
to the vector nature of the problem. It is well known (see, for example, <cit.>) that an
appropriate choice of the coordinate system allows one to split the system
(<ref>) into two independent subsystems. These subsystems describe waves of two
polarizations: the transverse electric wave and the transverse magnetic waves, which are named the
TE and TM waves, respectively. We call the coordinates, in which such a splitting occurs, the natural
§.§ Two types of Floquet-Bloch solutions in the natural coordinates
To determine the type of a solution, we should clarify, which component of the wave is
transverse to the propagation plane, the electric or the magnetic one. The propagation plane passes
through the lateral wave vector $(p_x,p_y)$ and the $z$ axis. To find the TM and TE modes, it is
convenient to rotate the axes in the $(x,y)$ plane through an angle $\gamma$, so that in the new
coordinates, $\widetilde{p_y} = 0$, $\widetilde{p_x} \equiv p_{\parallel} = \sqrt{p_x^2 + p_y^2}$.
We denote fields in the rotated coordinate system by a tilde. The waves of TE and TM types are as follows:
\begin{equation}\label{TE-TM}
{\widetilde {\bPh}}^{E} = \left( \begin{array}{cccccc} 0,
E_{\perp}, 0, H_{\parallel}, 0, \frac{p_{\parallel}}{k\mu} E_{\perp}
\end{array} \right)^t,\quad {\widetilde {\bPh}}^{H} =
\left(\begin{array}{cccccc} E_{\parallel}, 0,
-\frac{p_{\parallel}}{k\varepsilon} H_{\perp}, 0, H_{\perp}, 0
\end{array} \right)^t.
\end{equation}
The system of equations (<ref>) in such coordinates splits into two subsystems:
\begin{equation}\label{Maxwell-TM}
\left\{\begin{array}{rcl} \displaystyle{ i\frac{\partial
{H}_{\perp}}{\partial z}} & = & - k\varepsilon {E}_{\parallel}, \\
\\ \displaystyle{ i\frac{\partial {E}_{\parallel}}{\partial z}} & = &
-\left(\frac{k^2\varepsilon\mu -
p_{\parallel}^2}{k\varepsilon}\right) {H}_{\perp}.
\end{array} \right., \quad \left\{\begin{array}{rcl} \displaystyle{
i\frac{\partial {H}_{\parallel}}{\partial z}} & = &
\left(\frac{k^2\varepsilon\mu - p^2_{\parallel}}{k\mu}\right)
{E}_{\perp}, \\ \\ \displaystyle{ i\frac{\partial
{E}_{\perp}}{\partial z}} & = & k\mu {H}_{\parallel}.
\end{array} \right.
\end{equation}
In the special case $p_{\parallel}=0$, the splitting into TM and TE waves has no meaning:
both systems can be reduced to the following one:
\begin{equation}\label{E0-H0}
i \frac{dE_0}{dz} = - k \mu H_0;\quad i \frac{d H_0}{dz} = - k
\varepsilon E_0,
\end{equation}
\begin{equation}\label{limEH-perp-par}
{E}_{\parallel}|_{p_{\parallel}=0} = E_0, \quad {H}_{\perp}|_{p_{\parallel}=0} = H_0, \quad {E}_{\perp}|_{p_{\parallel}=0} = -E_0, \quad {H}_{\parallel}|_{p_{\parallel}=0} = H_0.
\end{equation}
We introduce the new notation for vector-functions obtained by means of the passage to the limit $p_{\parallel}\to 0$
in ${\widetilde \bPh}^{E}$,${\widetilde \bPh}^{H}$:
\begin{equation}
{\widetilde {\bPh}}^{E} |_{p_{\parallel} \to 0} \to \quad \bPh^X, \quad {\widetilde {\bPh}}^{H}
|_{p_{\parallel} \to 0} \to \quad \bPh^Y,
\end{equation}
\begin{equation}\label{Bas-XY}
\bPh^X=(E_0,0,0,0,H_0,0)^t, \quad \bPh^Y=(0,-E_0,0,H_0,0,0)^t.
\end{equation}
The superscript $X$ (or $Y$) indicates that the vector-function has the nonzero first (or second) component.
The obtained systems of ordinary linear differential equations (<ref>), (<ref>)
for ${E}_{\parallel}$, ${H}_{\perp}$ and ${E}_{\perp}$,${H}_{\parallel}$ and for $E_0$ and $H_0$,
respectively, are systems with piecewise continuous periodic coefficients, because $\varepsilon$ and
$\mu$ are piecewise continuous. Each system has two Floquet-Bloch solutions; for details, see
Appendix 1. We denote the components of the second solution by the subscript $2$. For
example, the two solutions of the first subsystem of (<ref>) for TM waves read
\begin{equation}\label{Floquet-TE}
{\left(\begin{array}{c} E_{\parallel}\\
H_{\perp} \end{array}\right) = e^{ i p_z z} {\bf
U}^H_{+}(z;p_z,p_{\parallel}^2,\omega), \quad \left(\begin{array}{c} E_{\parallel\, 2}\\
H_{\perp \,2} \end{array}\right) = e^{-i p_z z} {\bf U}^H_{-}(z;p_z,p_{\parallel}^2,\omega).}
\end{equation}
The solutions depend on the parameters of the equations $p_{\parallel}^2$ and $\omega$, and
also on the real-valued parameter $p_z$, which is called the quasimomentum and which is related to
$p_{\parallel}^2$ and $\omega$ by the formula
\begin{equation}\label{disp-p-o}
p_z = p_z(p_{\parallel}^2,\omega ).
\end{equation}
This relation ensures that the functions ${\bf U}^H_{\pm}$ are periodic,
\begin{equation}
{\bf U}^H_{\pm}(z+b;p_z,p_{\parallel}^2,\omega)={\bf U}^H_{\pm}(z;p_z,p_{\parallel}^2,\omega),
\end{equation}
and continuous as functions of the $z$ variable. We call these functions Floquet-Bloch
amplitudes. The relation (<ref>) can be reduced to the following one:
\begin{equation}\label{disp-relH}
\omega=\omega^H(\vp).
\end{equation}
We call this equation the dispersion relation. The function $\omega^H(\vp)$ is called the dispersion function. It is a multisheeted function on $[-\pi/b,\pi/b)\times \mathbb{R}^2_+$, and
its derivation is discussed in the Appendix 1. Substituting (<ref>) into (<ref>), we obtain $E_{\parallel}$ and $H_{\perp}$ as functions of $z$ and $\vp$.
The Floquet-Bloch solutions and the dispersion
function for the waves of TE type $({E}_{\perp}$, ${H}_{\parallel})$ and for the solutions $E_0$ and
$H_0$ of the system (<ref>) are obtained analogously. The solutions (<ref>) are
linearly independent if $p_z \ne 0, \pm \pi/b$; see Appendix 1. However, this particular case $p_z = 0, \pm \pi/b$ is important in the present paper.
We are interested here in effects that arise if the frequency of the problem is
close to the frequency of one of the stationary points of some sheet of the dispersive surface. In
Appendix 1, we show that the multisheeted dispersive surfaces $\omega = \omega^H(\vp)$
and $\omega = \omega^E(\vp)$ have stationary points on each sheet, at the points $\vp_*$,
$p_{\parallel*} = 0$, $p_{z*} = 0, \pm\pi/b$, where $\nabla\omega^E_{\vp*}=0$. In these
points the sheets of the dispersive surfaces for TM and TE polarizations touch each other, and
therefore $\nabla\omega^H_{\vp_*}=\nabla\omega^E_{\vp_*}=0$ and $\omega^E(\vp_*)=\omega^H(\vp_*) \equiv
\omega_*$. At the point $\vp_*$, formulas (<ref>) may give the same solution, which is
bounded at infinity and which is periodic for $p_{z*} = 0$ and anti-periodic for $p_{z*} =
\pm\pi/b.$ Then the second solution, which is linearly independent with the first one, grows at infinity
( for details, see Appendix 1). It is this case that is of interest to us in the present paper. The solutions read
\begin{equation}\label{two_sol2}
\begin{aligned}
\left.\left(\begin{array}{c} E_{\parallel}\\
H_{\perp} \end{array}\right)\right|_{\vp=\vp_*} = \left.\left(\begin{array}{c} E_{0}\\
H_{0} \end{array}\right)\right|_{p_{z}=p_{z*}}\, & = &
e^{ip_{z*}z} \vec{U}^H_+(z;p_{z*},0,\omega_*), \qquad \qquad \qquad \qquad\\
\left.\left(\begin{array}{c} E_{\parallel\, 2}\\
H_{\perp \,2} \end{array}\right)\right|_{\vp=\vp_*} = \left.\left(\begin{array}{c} E_{02}\\
H_{02} \end{array}\right)\right|_{p_{z}=p_{z*}} & = & e^{ip_{z*}z} \left[e^{-ip_{z*}b}
\frac{z}{b} \vec{U}^H_+(z;p_{z_*},0,
\omega_*) + \vec{Q}^H(z;p_{z_*}, \omega_*) \right],
\end{aligned}
\end{equation}
where the vector-valued function ${\bf Q}^H$ is a periodic and continuous function of the
variable $z$. The function ${\bf Q}^H$ does not depend on the argument $p_{\parallel*}^2,$ because
$p_{\parallel*}^2=0,$ and we are not going to use the second solution (<ref>) for $p_{\parallel}\ne0$.
The solutions of the second subsystem (<ref>) are expressed in the following
\begin{equation}
\left.\left(\begin{array}{c} E_{\perp}\\
H_{\parallel} \end{array}\right)\right|_{\vp=\vp_*}\, = \left.\left(\begin{array}{c} -E_{0}\\
\quad H_{0} \end{array}\right)\right|_{p_{z}=p_{z*}}, \quad \left.\left(\begin{array}{c} E_{\perp \, 2}\\
H_{\parallel \,2} \end{array}\right)\right|_{\vp=\vp_*} = \left.\left(\begin{array}{c} - E_{02}\\
\quad H_{02} \end{array}\right)\right|_{p_{z}=p_{z*}}.
\end{equation}
§.§ Floquet-Bloch solutions in an arbitrary coordinate system
We have found solutions $E_{\parallel},
H_{\perp}$ and $E_{\perp}, H_{\parallel}$ of the systems (<ref>). These solutions were found in the coordinate system
rotated by an angle $\gamma$, which characterizes the direction of wave propagation. In the initial
coordinate system, these solutions read:
\begin{equation}\label{Phi_E_Phi_H}
{{\bPh}}^H = \left(\begin{array}{c} E_{\parallel} \cos \gamma \\
E_{\parallel} \sin\gamma \\ -\frac{1}{k \varepsilon}p_{\parallel}
H_{\perp} \\ -H_{\perp} \sin\gamma
\\ H_{\perp} \cos\gamma \\ 0
\end{array} \right), \quad
{{\bPh}}^E = \left(\begin{array}{c} -E_{\perp} \sin\gamma \\
E_{\perp} \cos \gamma \\ 0 \\ H_{\parallel} \cos\gamma
\\ H_{\parallel} \sin\gamma \\ \frac{1}{k \mu}p_{\parallel} E_{\perp}
\end{array} \right).
\end{equation}
The solutions (<ref>) do not have a limit for $p_{\parallel} \to 0$ as a function of two
variables $p_x,p_y$. This limit exists in any direction determined by $\gamma$
and depends on the direction $\gamma$:
\begin{equation}\label{lim-old}
{\bPh}^H |_{{p_{\parallel}}\to 0} \to \cos{\gamma} \bPh^X -
\sin{\gamma} \bPh^Y,\quad {\bPh}^E|_{{p_{\parallel}}\to 0} \to
\sin{\gamma} \bPh^X + \cos{\gamma} \bPh^Y.
\end{equation}
Here we take into account (<ref>) and the definitions of $\bPh^X$ and $\bPh^Y$ from (<ref>).
Below we will use the derivatives $\partial {{\bPh}}^H / \partial p_j$, $\partial
{{\bPh}}^E/\partial p_j,$ $j=x,y,z$, which are linear combinations of the derivatives of the
functions $E_{\parallel}, H_{\perp}$ and $E_{\perp}, H_{\parallel}$. These functions are
analytic functions of the parameter $p_{\parallel}^2$, since the coefficients of the system
(<ref>) are analytic functions of $p_{\parallel}^2$. It is easy to obtain
\begin{equation}
\frac{\partial E_{\parallel}}{\partial p_x} = 2 p_{\parallel} \frac{\partial
E_{\parallel}}{\partial p_{\parallel}^2} \cos\gamma.
\end{equation}
This yields $\partial E_{\parallel}/\partial p_x |_{p_{\parallel}=0}=0.$ Similarly, we
derive that the derivatives of $H_{\perp}$, $E_{\perp}$ and $H_{\parallel}$ with respect to the
variables $p_x$ and $p_y$ are zero for $p_{\parallel}=0$. Finally, we get
\begin{eqnarray}
&& \frac{\partial {\bPh}^H}{\partial p_x}\left|_{p_x=p_y=0}\right. = \frac{\partial
{\bPh}^H}{\partial p_y}\left|_{p_x=p_y=0}\right.
= \left(0,0,-\frac{H_0}{k\varepsilon},0,0,0\right)^t , \label{der-p1}\\
&& \frac{\partial {\bPh}^E}{\partial p_x}\left|_{p_x=p_y=0}\right. = \frac{\partial
{\bPh}^E}{\partial p_y}\left|_{p_x=p_y=0}\right. = \left(0,0,0,0,0,-\frac{E_0}{k\mu}\right)^t.
\label{der-p2}
\end{eqnarray}
These derivatives do not depend on the direction $\gamma$.
We introduce the unified notation $\bPh^f$ for the Floquet-Bloch solutions of different types:
\begin{equation}\label{Bl-sol-amp}
\bPh^f{(z;\vp)} = e^{ip_zz}\bvph^f(z;\vp), \quad f=H, E, X, \, {\rm or} \, Y,
\end{equation}
where $\bvph^f(z+b;\vp)=\bvph^f(z;\vp)$.
The six-component vector-functions $\bvph^f(z,\vp)$ are
called the Floquet-Bloch amplitudes.
Let us discuss formula (<ref>) in more detail.
The Floquet-Bloch solutions $\bPh^f{(z;\vp)}$ for
$f=H,E$ are determined by formula (<ref>), where $E_{\parallel}, H_{\perp}$ and
$E_{\perp}, H_{\parallel}$ are the first solutions (<ref>) of the systems
(<ref>). We assume that the frequency $\omega$ in the formulas (<ref>) is expressed in terms of $\vp$ on one of the sheets of the multisheeted function $\omega=\omega^f(\vp)$. We also take into account the fact that $p_{\parallel}^2 = p_x^2 + p_y^2$, $\cos{\gamma}=p_x/p_{\parallel},$ $\sin{\gamma} = p_y/p_{\parallel}$. The number of the sheet is omitted for brevity. The Floquet-Bloch solutions $\bPh^f(z;\vp)$, $f=X,Y$,
are defined only for $p_{\parallel}=0$, i.e., for $\vp = \vp_0 \equiv (0,0,p_z)$; their
polarization is not defined.
The limit of the Floquet-Bloch amplitudes in the direction determined by $\gamma$ follows from
\begin{equation}\label{lim}
{\bvph}^H |_{{p_{\parallel}}\to 0} \to \cos{\gamma} \bvph^X - \sin{\gamma} \bvph^Y,\quad
{\bvph}^E|_{{p_{\parallel}}\to 0} \to \sin{\gamma} \bvph^X + \cos{\gamma} \bvph^Y.
\end{equation}
In particular, if $\gamma=0$, then $p_{\parallel}=p_x$. If $\gamma=\pi/2$, then
$p_{\parallel}=p_y$. From (<ref>) we obtain
\begin{eqnarray}
&&{\bvph}^H{(z;\vp)} |_{p_x\to 0,p_y=0} \to \quad \bvph^X{(z;\vp_0)},\quad
{\bvph}^E{(z;\vp)}|_{p_x\to 0, p_y=0} \to \bvph^Y{(z;\vp_0)}, \label{lim-p1}\\
&&{\bvph}^H{(z;\vp)}|_{p_x=0, p_y\to 0} \to - \bvph^Y{(z;\vp_0)},\quad {\bvph}^E{(z;\vp)}|_{p_x=0,
p_y\to 0} \to \bvph^X{(z;\vp_0)}.\label{lim-p2}
\end{eqnarray}
Next, we need to obtain the derivatives of the solutions $\bvph^H$ and $\bvph^E$ with respect to
the parameters $p_x$ and $p_y$ at the point $ p_{\parallel}= 0$. Owing to (<ref>)
and(<ref>) and the definition (<ref>), we get
\begin{eqnarray}
&& \frac{\partial {\bvph}^H}{\partial p_x}\left|_{\vp_*}\right. = \frac{\partial
{\bvph}^H}{\partial p_y}\left|_{\vp_*}\right.
= \left.\left(0,0,-\frac{H_0}{k\varepsilon},0,0,0\right)^t\right|_{p_{z*}} e^{-ip_{z*}z} , \label{der-p1v}\\
&& \frac{\partial {\bvph}^E}{\partial p_x}\left|_{\vp_*}\right. = \frac{\partial
{\bvph}^E}{\partial p_y}\left|_{\vp_*}\right. =
\left.\left(0,0,0,0,0,-\frac{E_0}{k\mu}\right)^t\right|_{p_{z*}} e^{-ip_{z*}z}. \label{der-p2v}
\end{eqnarray}
These derivatives do not depend on the direction $\gamma$. These
functions are periodic in $z$, because ${\bvph}^H$, ${\bvph}^E$ are
periodic for all $\vp$.
§ SOME AUXILIARY RELATIONS
Our aim is to obtain relations for the derivatives of the dispersion functions by means of
the Floquet-Bloch amplitudes. To do this, we give equations for these
amplitudes and their derivatives with respect to the parameters $p_j$, $j=x,y,z$.
§.§ Equations for the Floquet-Bloch amplitudes and their derivatives
We deal with six-component solutions
of the Maxwell equations:
\begin{equation}\label{sol-Bl-am}
\bPs^f_B {(x,y,z;\vp)} = e^{i(p_xx+p_yy)}\bPh^f {(z;\vp)}, \quad f=E, H,
\end{equation}
where $\bPh^f$ is of the form (<ref>).
Here the superscript $f$ stands for the type of the field: $f=E$ corresponds to the TE polarization
and $f=H$ corresponds to the TM one. Inserting (<ref>) into Maxwell equations, we rewrite
them in the form
\begin{equation}\label{Af}
\A^f(\vp)\bvph^f {(z;\vp)} =0,
\end{equation}
\begin{equation}
\A^f(\vp) \equiv
\frac{\omega^f(\vp)}{c}
\P + i \G_3\frac{\partial}{\partial z} - \vp \cdot \hG, \qquad
\vp \cdot \hG \equiv p_x\G_1 + p_y \G_2 + p_z \G_3. \label{Bl-amp2}
\end{equation}
We note that the equations for both types differ only by the
dispersion function $\omega^f(\vp)$. These equations are valid for
any $\vec{p}$, and thus we can take the derivatives of these
equations with respect to $\vec{p}$.
By taking the derivatives with respect to parameters $p_j, j=x,y$, we obtain
\begin{eqnarray}
&&\A^f(\vp) \frac{\partial \bvph^f}{\partial p_j} =
- \frac{\partial \A^f}{\partial p_j} \bvph^f, \label{eq-der}\\
&&\A^f(\vp) \frac{\partial^2 \bvph^f}{\partial p_j^2} = -
\frac{1}{c} \frac{\partial^2 \omega^f}{\partial p_j^2} \P
\bvph^f - 2 \frac{\partial \A^f(\vp)}{\partial p_j}
\frac{\partial \bvph^f}{\partial p_j}, \label{eq-der-2}
\end{eqnarray}
\begin{equation}
\frac{\partial \A^f}{\partial p_j} = \frac{1}{c}
\frac{\partial \omega^f}{\partial p_j} \P - \G_j. \label{der-A}
\end{equation}
The derivatives with respect to $p_z$ are discussed below at the end of the Section. Now we proceed with specifying
formulas (<ref>), (<ref>) at the stationary points $\vp_*$. Such points for each
branch of the multisheeted dispersion functions $\omega^f(\vp)$, $f=E,H$, differ only by $p_{z*}$, and all of them
have $p_{\parallel*}=0$. Solutions $\bvph^f(z;\vp)$ are not continuous
as functions of two variables $p_x$ and $p_y$ near the point $p_x=p_y=0$. However, for any fixed
angle $\gamma$, they can be calculated by means of the passage to the limit, see (<ref>). The derivatives at $p_{\parallel}=0$ do not depend on $\gamma$, see (<ref>), (<ref>). We
indicate these derivatives at $\vp=\vp_*$ by the asterisk subscript. Upon substitution $\vp=\vp_*$, the operator $\A$
is denoted as
\begin{equation}\label{A-star}
\A^f(\vp_*)\equiv \A_*(\vp_*), \quad f = E, H,
\end{equation}
and no more depends on the wave type TM or TE.
Let $\mathcal{M}$ be a class of six-component vector-valued functions of $z$, which are periodic with a period $b$,
piecewise smooth on the period and their components with numbers $1,2,4,5$ are continuous; see details in Appendix 2.
Lemma 1 (see Appendix 2) shows that the
operator $\A_*(\vp_*)$ is symmetric on the functions from $\mathcal{M}$.
Finally, passing to the limit in (<ref>), (<ref>), in view of (<ref>),
(<ref>) we obtain
\begin{eqnarray}
&&\A_* \frac{\partial \bvph^H_*}{\partial p_x} = \G_1 \bvph^X_*,
\qquad \A_* \frac{\partial \bvph^E_*}{\partial p_x} =
\G_1 \bvph^Y_*, \label{der-1-0}\\
&&\A_* \frac{\partial {\bvph}^H_*}{\partial p_y} = - \G_2
{\bvph}^Y_*, \qquad \, \A_* \frac{\partial {\bvph}^E_*}{\partial
p_y} = \G_2 {\bvph}^X_*. \label{der-2-0}
\end{eqnarray}
Here we have introduced the notation
\begin{equation}\label{vph-XY-def}
\bvph^X(z;\vp_*)=\bvph^X_*, \quad \bvph^Y(z;\vp_*)=\bvph^Y_*.
\end{equation}
We note that the terms containing the derivative $\partial \omega^f/\partial p_j$ vanish at the
stationary point, since this derivative vanishes.
For the second derivatives of the Floquet-Bloch amplitudes, in view of (<ref>),
(<ref>), and (<ref>), (<ref>) we derive
\begin{eqnarray}
&&\A_* \frac{\partial^2 \bvph^H_*}{\partial p_x^2} = - \frac{1}{c} \frac{\partial^2
\omega^H_*}{\partial p_x^2}
\P \bvph^X_* + 2 \G_1 \frac{\partial \bvph^H_*}{\partial p_x}, \label{der-2H1}\\
&&\A_* \frac{\partial^2 \bvph^E_*}{\partial p_x^2} = - \frac{1}{c} \frac{\partial^2
\omega^E_*}{\partial p_x^2}
\P \bvph^Y_* + 2 \G_1 \frac{\partial \bvph^E_*}{\partial p_x},\label{der-2E1}\\
&&\A_* \frac{\partial^2 \bvph^H_*}{\partial p_y^2} = \frac{1}{c} \frac{\partial^2
\omega^H_*}{\partial p_y^2}
\P \bvph^Y_* + 2 \G_2 \frac{\partial \bvph^H_*}{\partial p_y}, \label{der-2H2} \\
&&\A_* \frac{\partial^2 \bvph^E_*}{\partial p_y^2} = - \frac{1}{c} \frac{\partial^2
\omega^E_*}{\partial p_y^2} \P \bvph^X_* + 2 \G_2 \frac{\partial \bvph^E_*}{\partial
\end{eqnarray}
We emphasize that the directional limit of ${\bvph}^H$ and ${\bvph}^E$ at the point $\vp_*$ can be
expressed by ${\bvph}^X_*$, as well as by ${\bvph}^Y_*$ depending on the direction $\gamma$.
Now we proceed to the calculation of the derivatives with respect to $p_z$ at the point $\vp_*$, i.e., $p_{\parallel*}=0$, $p_z=p_{z*}$.
If $p_{\parallel}= p_{\parallel*}=0$, i.e., $\vp = \vp_0 \equiv
(0,0,p_z)$, we have
\begin{equation}\label{om0}
\omega^H(\vp_0)=\omega^E(\vp_0)\equiv \omega^0(\vp_0)
\end{equation}
and $\A^H(\vp_0)=\A^E(\vp_0)\equiv \A^0(\vec{p}_0)$. The Floquet-Bloch amplitudes
$\bvph^X$, $\bvph^Y$ (see (<ref>), (<ref>) )
satisfy the same equation
\begin{equation}\label{A0}
\A^0(\vp_0) \bvph^f(z;\vp_0) = 0, \qquad f = X, \,\, Y.
\end{equation}
By differentiating (<ref>) with respect to $p_z$ and then by passing to the limit $p_z \to
p_{z*}$, we get
\begin{eqnarray}
&&\A_* \frac{\partial {\bvph}^X_*}{\partial p_z} = \G_3 {\bvph}^X_*,
\quad \A_* \frac{\partial {\bvph}^Y_*}{\partial p_z} = \G_3
{\bvph}^Y_*. \label{der-3-0}
\end{eqnarray}
By taking the second derivatives of (<ref>) and then by passing to the limit $p_z \to p_{z*}$, we
\begin{eqnarray}
&&\A_* \frac{\partial^2 \bvph^X_*}{\partial p_z^2} = - \frac{1}{c}
\frac{\partial^2 \omega^0_*}{\partial p_z^2} \P {\bvph}^X_* +
2 \G_3\frac{\partial \bvph^X_*}{\partial p_z},\label{der-2X3} \\
&&\A_* \frac{\partial^2 \bvph^Y_*}{\partial p_z^2} = - \frac{1}{c}
\frac{\partial^2 \omega^0_*}{\partial p_z^2} \P {\bvph}_*^Y +
2 \G_3\frac{\partial \bvph^Y_*}{\partial p_z}.\label{der-2Y3}
\end{eqnarray}
§.§ Derivatives of dispersion functions
Now we get integral relations containing derivatives of the Floquet-Bloch amplitudes and
derivatives of dispersion functions. We take the inner product of (<ref>), (<ref>),
and (<ref>) with $\bvph^X_*$ or $\bvph^Y_*$ and, taking into account the fact that
\begin{equation}
\A_* \bvph^X_* = 0, \quad \A_* \bvph^Y_* = 0,
\end{equation}
by Lemma 1 (see Appendix 2) we find
\begin{equation}\label{matr-el1}
\left(\bvph^{f_2}_*,\G_j \bvph^{f_1}_*\right)= 0 \quad {\rm
for\quad any}\quad j=1,2,3; \quad f_1=X,Y; \quad f_2=X,Y.
\end{equation}
Henceforth, we use the fact that $\bvph^{f_2}$ and its derivatives with respect to parameters $p_j$, $j=x,y,z$ belong to $\mathcal{M}$.
We are going to apply the same operations to formulas (<ref>–<ref>) and
(<ref>–<ref>). We introduce the notation
\begin{equation}\label{u_def}
u_*^{f_1f_2}\equiv\left(\bvph_*^{f_1},\P \bvph_*^{f_2}\right),
\quad f_1=X \,\, \mathrm{or} \,\, Y, \quad f_2=X \,\, \mathrm{or} \,\,Y,
\end{equation}
where $u^{ff}$ has the meaning of the density of energy averaged over time (see Section <ref>), and
\begin{equation}
\ddot{\omega}^f_{11*} \equiv \left. \frac{\partial^2 \omega^f}{\partial p_x^2}
\right|_{\vp=\vp*}, \quad \ddot{\omega}^f_{22*} \equiv \left. \frac{\partial^2
\omega^f}{\partial p_y^2} \right|_{\vp=\vp*}, \quad f=H \,\, \mathrm{or} \,\,
E; \quad \ddot{\omega}^0_{33*} \equiv \left. \frac{\partial^2 \omega^0}{\partial
p_z^2} \right|_{\vp=\vp*},\label{omega_dot_dot}
\end{equation}
where $\ddot{\omega}^0_{33*}=\ddot{\omega}^H_{33*}=\ddot{\omega}^E_{33*}$, because (<ref>) is
valid for any $p_z$. We arrive at the relations
\begin{eqnarray}
&&\frac{\ddot{\omega}^H_{11*}}{c} u_*^{XX} = 2\left(\bvph_*^X,\G_1
\frac{\partial \bvph^H_*}{\partial p_x}\right), \quad
\frac{\ddot{\omega}^E_{11*}}{c} u_*^{YY} = 2\left(\bvph_*^Y,\G_1
\frac{\partial \bvph^E_*}{\partial p_x}\right), \label{om2-1}\\
&&\frac{\ddot{\omega}^H_{22*}}{c} u_*^{YY} = -2\left(\bvph_*^Y,\G_2
\frac{\partial \bvph^H_*}{\partial p_y}\right), \quad
\frac{\ddot{\omega}^E_{22*}}{c} u_*^{XX} = 2\left(\bvph_*^X,\G_2
\frac{\partial \bvph^E_*}{\partial p_y}\right), \label{om2-2}\\
&&\frac{\ddot{\omega}^0_{33*}}{c} u_*^{XX} =2\left(\bvph_*^X,\G_3
\frac{\partial \bvph^X_*}{\partial p_z}\right), \quad
\frac{\ddot{\omega}^0_{33*}}{c} u_*^{YY} = 2\left(\bvph_*^Y,\G_3
\frac{\partial \bvph^Y_*}{\partial p_z}\right). \label{om2-3}
\end{eqnarray}
If $f_1 \neq f_2$, then $u^{f_1f_2}\equiv 0$. Moreover, $u^{XX} = u^{YY}$ by the definitions of $\bvph_*^X$, $\bvph_*^Y$, and $\P$.
§.§ Additional relations
Multiplying equations (<ref>), (<ref>) and
(<ref>) by $\bvph_*^X$ and $\bvph_*^Y$ in such a way that on
the left-hand side we obtain $(\bvph_*^X,\P\bvph_*^Y)$, which
vanishes, we find the following relations:
\begin{eqnarray}
&&\left(\bvph_*^Y,\G_1 \frac{\partial \bvph^H_*}{\partial
p_x}\right)= \left(\bvph_*^X,\G_1 \frac{\partial \bvph^E_*}{\partial
\left(\bvph_*^Y,\G_2 \frac{\partial \bvph^H_*}{\partial p_y}\right)=0, \label{help1}\\
&&\left(\bvph_*^X,\G_2 \frac{\partial \bvph^E_*}{\partial
p_y}\right)=\left(\bvph_*^Y,\G_3 \frac{\partial \bvph^X_*}{\partial
p_z}\right)=\left(\bvph_*^X,\G_3 \frac{\partial \bvph^Y_*}{\partial
p_z}\right)=0. \label{help2}
\end{eqnarray}
Now we mention some other useful relations with the derivatives of
$\bvph^f$. The derivatives of $\bvph^f$ with respect to $p_x$ and
$p_y$ coincide; see (<ref>) and (<ref>). This fact
\begin{equation}\label{eq-G-bv}
\G_1 \bvph^X_* = - \G_2 {\bvph}^Y_*, \quad \G_1 \bvph^Y_* = \G_2
\end{equation}
The same relations can be obtained by direct computations by (<ref>), (<ref>) not
only at the point $\vp_*$ but at the point $\vp=(0,0,p_z)$ for any $p_z$. The direct calculations
with the help of (<ref>), (<ref>) and (<ref>) show that
\begin{equation}\label{g3d}
\G_3 \frac{\partial {\bvph}^H_*}{\partial p_j}=0, \quad \G_3
\frac{\partial {\bvph}^E_*}{\partial p_j}=0, \quad j=x,y.
\end{equation}
§ THE TWO-SCALED ASYMPTOTIC DECOMPOSITION
We give an asymptotic representation of some special solutions of Maxwell equations in the entire
space under several assumptions:
* the vertical period $b$ of the medium is small as compared with the horizontal scale
of the field, and the relation between the scales is characterized by the small parameter $\chi$,
* the frequency $\omega$ is close to the frequency $\omega_*$ of the stationary point $\vp_*$ of one
of the sheets of the dispersion function $\omega=\omega^f(\vp),$ $f=H,E$, i.e., the frequency
$\omega_*$ is determined by the relations
\begin{equation}
\omega_*=\omega^E(\vp_*)=\omega^H(\vp_*),\quad \nabla \omega^f(\vp_*)=0,\quad f=H,E.
\end{equation}
We assume that
\begin{equation}
\omega = \omega_* + \chi^2 \delta\omega, \quad \delta\omega \sim 1.
\end{equation}
* We assume that there is one bounded and one unbounded Floquet-Bloch solution of the
periodic problem (<ref>) at the point $\vp_*$.
Our aim is to find the asymptotics of solutions of the Maxwell equations in the following
\begin{equation}\label{Psi-form}
\bPs = \bPs(z, \brho), \quad \xi\equiv\chi x, \quad \eta \equiv \chi y, \quad \zeta \equiv \chi z,
\quad \brho=(\xi,\eta,\zeta)
\end{equation}
where $\chi \ll 1.$ In the direction transverse to the layers, the field has two scales, one of them is
determined by the slow variable $\zeta= \chi z$, and the other is given by the variable $z$. In the plane of the layers in the directions $x$ and $y$, the field depends only on the slow variables $\xi=\chi x$, $\eta
= \chi y$.
We seek a solution in the form of a two-scaled asymptotic series
\begin{eqnarray} \label{anz}
&&\bPs(z,\brho) = \bPh(z,\brho) e^{i (p_{x*}\xi +
p_{y*}\eta)/{\chi}}, \quad
\bPh(z,\brho) = \bphi(z,\brho) e^{i p_{z*} z},\\
&&\bphi(z,\brho) = \sum\limits_{n \geq 0} \chi^n
\bphi^{(n)}(z,\brho), \quad \bphi^{(n)}(z + b,\brho) =
\bphi^{(n)}(z,\brho), \quad \brho = (\xi,\eta,\zeta).
\end{eqnarray}
In the case under
consideration, the stationary point is $p_{x*}=p_{y*}=0$.
We assume that $\bphi^{(n)} \in \mathcal{M}$ for every $n$ as functions of $z$, see the definition after formula (<ref>). Also these functions are infinitely
differentiable with respect to slow variables.
The Maxwell equations in new variables read
\begin{equation}\label{Maxw}
k_* \P \bPs + i\G_3\frac{\partial\Psi}{\partial z} = -i\chi
\widehat{\G}\cdot \nabla_{\brho} \bPs - \chi^2 \frac{\delta
\omega}{c} \P \bPs, \quad k_* = \frac{\omega_*}{c},
\end{equation}
\begin{equation}
\widehat{\G}\cdot \nabla_{\brho} \equiv \G_1 \frac{\partial}{\partial
\xi} + \G_2 \frac{\partial}{\partial \eta} + \G_3
\frac{\partial}{\partial \zeta}. \nonumber
\end{equation}
Substituting the asymptotic series (<ref>) in (<ref>), we
obtain a set of equations
\begin{equation}
\A_* \bphi^{(0)} = 0, \qquad \A_* \bphi^{(n)} = \vec{F}^{(n)}, \label{Maxwell-series}
\end{equation}
\begin{eqnarray}
&&\A_*\bPs = k_* \P \bPs + i\G_3\frac{\partial\bPs}{\partial z} - p_{3*} \G_3 \bPs,\label{def-A*}\\
&&\vec{F}^{(1)} = -i \widehat{\G}\cdot \nabla_{\rho}\bphi^{(0)},\\
&&\vec{F}^{(n)} = -i \widehat{\G}\cdot \nabla_{\rho}\bphi^{(n-1)} -
\frac{\delta \omega}{c} \P \bphi^{(n-2)},\quad n \ge2. \label{Fn}
\end{eqnarray}
Prior to solving the set of equations, we are going to find the relations between the
parameters of the problem, which ensures that all nonzero terms on the right-hand side of
(<ref>) are of the same order. We assume that the variables $\mathbb{E}$ and $\mathbb{H}$
and the parameters $\varepsilon$ and $\mu$ in formulas (<ref>) are already normalized as follows
\begin{equation}\nonumber
\mathbb{E} = \sqrt{\frac{\varepsilon_{av}}{\mu_{av}}}\widetilde{\mathbb{E}}, \quad \mathbb{H} =
\sqrt{\frac{\varepsilon_{av}}{\mu_{av}}}\widetilde{\mathbb{H}}, \quad {\rm where} \quad
\varepsilon= \frac{\widetilde{\varepsilon}}{\varepsilon_{av}}, \quad \mu =
\frac{\widetilde{\mu}}{\mu_{av}}, \quad k_* = \sqrt{\varepsilon_{av}\mu_{av}} \widetilde{k_*},
\end{equation}
where $\varepsilon_{av}$, $\mu_{av}$ are typical dielectric permittivity and magnetic permeability,
these parameters may be large, $\varepsilon$ and $\mu$ are of order unity, and
$\widetilde{\varepsilon}, \widetilde{\mu}$ are the original parameters of the equation. The
variables ${\widetilde{k}}_*$ and $k_*$ mean the wave number in vacuum and in the medium with
parameters $\varepsilon_{av}$ and $\mu_{av}$, respectively, $k_* = \sqrt{\varepsilon_{av}
\mu_{av}}\omega/c $, where $c$ is the speed of light in vacuum. The second and the third (if
nonzero) terms in the right-hand side of (<ref>) are of order of $1/b$. The
first and the second terms are of the same order if
\begin{equation}
\sqrt{\varepsilon_{av} \mu_{av}}\omega/c \sim 1/b.
\end{equation}
This means that the case under consideration differs from the well-known case $\omega b/c \to
§.§ The principal order
The equation of principal order term is the equation for the Floquet-Bloch amplitudes at
the stationary point $\vp_*$. It does not contain the derivatives with respect to slow
variables $\brho=(\xi, \eta, \zeta)$ and its coefficients do not depend on $\brho$. Its solutions
may depend on $\brho$ as on parameters. We seek the principal term in the form of
\begin{equation}\label{princip}
\bphi^{(0)}(z, \brho) = \alpha_{1}(\brho)\bvph^{X}_*(z) +
\alpha_{2}(\brho) \bvph^{Y}_*(z), \quad \bphi^{(0)} \in \mathcal{M},
\end{equation}
\begin{equation}
\bvph^f_*(z) = \bvph^f(z;\vp_*) = e^{-ip_{z*}z} \bPh^f{(z;\vp_*)},
\end{equation}
$f = X,\,\,Y$, and $\bPh^f{(z;\vp)}$ for $p_{\parallel}=0$ are defined in (<ref>) by means of the functions $E_0$ and $H_0$, which satisfy a system (<ref>). The
functions $\alpha_{1}, \alpha_{2}$ are arbitrary scalar functions of slow variables $\brho$.
Additional restrictions on these arbitrary functions will arise later.
§.§ First-order approximation
The equation for the first-order term $\bphi^{(1)}$ of the expansion has the form
\begin{equation}
\A_* \bphi^{(1)} = -i\hG\cdot\nabla_{\brho} \bphi^{(0)}
\label{Maxwell-1-order}, \quad \bphi^{(1)} \in \mathcal{M}.
\end{equation}
In order to get the solution of the system belonging to the class $\mathcal{M}$, we must impose additional conditions.
Lemma 2. A solution from the class $\mathcal{M}$ of the equation $\A_* \bphi = \vec{F}$ exists if and only if the following conditions are satisfied:
\begin{equation}\label{lemma2}
\left(\bvph^X_*,\vec{F}\right)=0,\quad
\left(\bvph^Y_*,\vec{F}\right)=0.
\end{equation}
The proof of the Lemma 2 is given in the Appendix 2.
Now we check the solvability conditions for the first-order
approximation (<ref>), i.e., we must check that
\begin{equation}
\left(\bvph^X_*,\hG\cdot\nabla_{\brho} \bphi^{(0)}\right)=0,\quad
\left(\bvph^Y_*,\hG\cdot\nabla_{\brho} \bphi^{(0)}\right)=0,
\end{equation}
which are reduced to the following conditions
\begin{eqnarray}
&&\left(\bvph^X_*,\hG\bvph^X_*\right)\cdot \nabla_{\brho}\alpha_1 +
\left(\bvph^X_*,\hG\bvph^Y_*\right)\cdot \nabla_{\brho}\alpha_2 =0,\\
&&\left(\bvph^Y_*,\hG\bvph^X_*\right)\cdot \nabla_{\brho}\alpha_1 +
\left(\bvph^Y_*,\hG\bvph^Y_*\right)\cdot \nabla_{\brho}\alpha_2 =0,
\end{eqnarray}
where, for example,
\begin{equation}
\left(\bvph^X_*,\hG\bvph^X_*\right)\cdot \nabla_{\brho}\alpha_1 \equiv \left(\bvph^X_*,
\G_1\bvph^X_*\right) \frac{\partial \alpha_1}{\partial \xi} + \left(\bvph^X_*, \G_2\bvph^X_*\right)
\frac{\partial \alpha_1}{\partial \eta} + \left(\bvph^X_*, \G_3\bvph^X_*\right) \frac{\partial
\alpha_1}{\partial \zeta}.
\end{equation}
These conditions are satisfied at the stationary point owing to
Now let us find the exact formula for the solution $\bphi^{(1)}$. The right-hand side of the
equation (<ref>) can be written as follows:
\begin{equation}\label{ord1}
\vec{F}^{(1)} = -i \hG\cdot (\nabla_{\brho}\alpha_1) \bvph^X_* -i
\hG\cdot (\nabla_{\brho}\alpha_2 )\bvph^Y_*.
\end{equation}
Taking into account (<ref>), we replace the terms containing $\G_2$ by the terms containing
$\G_1$. Collecting the resulting terms, we obtain
\begin{equation}
\vec{F}^{(1)} = -i \left( \frac{\partial \alpha_1}{\partial \xi} -
\frac{\partial \alpha_2}{\partial \eta}\right) \G_1 \bvph^X_* -i
\left( \frac{\partial \alpha_1}{\partial \eta} + \frac{\partial
\alpha_2}{\partial \xi}\right) \G_1 \bvph^Y_* -i \frac{\partial
\alpha_1}{\partial \zeta} \G_3 \bvph^X_* - i\frac{\partial
\alpha_2}{\partial \zeta} \G_3 \bvph^Y_*.
\end{equation}
Now the right-hand side contains four terms. Instead of solving (<ref>), we solve
four independent vector equations with right-hand sides containing each of the terms and then take
the sum of their solutions. We add also a solution of the homogeneous equation. The particular
solutions of four equations coincide with solutions of (<ref>), (<ref>), and
(<ref>). We find the following solution:
\begin{eqnarray}
\label{Phi_1_solution} &&\bphi^{(1)} = -i \left( \frac{\partial
\alpha_1}{\partial \xi} - \frac{\partial \alpha_2}{\partial
\eta}\right) \frac{\partial \bvph^H_*}{\partial p_x} -i \left(
\frac{\partial \alpha_1}{\partial \eta} + \frac{\partial
\alpha_2}{\partial \xi}\right) \frac{\partial \bvph^E_*}{\partial
\nonumber\\
&& -i \frac{\partial \alpha_1}{\partial \zeta} \frac{\partial
\bvph^X_*}{\partial p_z} - i\frac{\partial \alpha_2}{\partial \zeta}
\frac{\partial \bvph^Y_*}{\partial p_z}
+ \alpha_{1}^{(1)}\bvph^{X}_*(z) + \alpha_{2}^{(1)}
\bvph^{Y}_*(z),
\end{eqnarray}
where $\alpha_{1,2}^{(1)}$ are new arbitrary functions of the slow variables
$\brho=(\xi,\eta,\zeta)$. The subscript $*$ stands to show that all the derivatives with the
respect to $p_j, j=x,y,z$ are taken at $\vp=\vp_*$.
§.§ The second-order approximation
Now let us consider the equation on the second-order approximation
\begin{equation}
\A_* \bphi^{(2)} = -i\hG \cdot\nabla_{\brho}\bphi^{(1)}, \quad
\bphi^{(2)} \in \mathcal{M}. \label{Maxwell-2-order}
\end{equation}
The solution $\bphi^{(1)}$ depends on four unknown functions
$\alpha_{j}, j=1,2$ and $\alpha^{(1)}_{j}, j=1,2.$ The solvability
conditions of (<ref>) yield equations for two of
\begin{eqnarray}
&&i \left(\bvph^X_*,\hG\cdot\nabla_{\brho} \bphi^{(1)}\right) +
\left(\bvph^X_*,\frac{\delta \omega}{c} \P \bphi^{(0)}\right)=0, \label{cond-2X}\\
&& i \left(\bvph^Y_*,\hG\cdot\nabla_{\brho} \bphi^{(1)}\right) +
\left(\bvph^Y_*,\frac{\delta \omega}{c} \P \bphi^{(0)}\right)=0.
\label{cond-2Y}
\end{eqnarray}
For brevity, we introduce the notation
\begin{equation}\label{tau-def}
\tau_1 \equiv \left( \frac{\partial \alpha_1}{\partial \xi} -
\frac{\partial \alpha_2}{\partial \eta}\right), \quad \tau_2 \equiv
\left( \frac{\partial \alpha_1}{\partial \eta} + \frac{\partial
\alpha_2}{\partial \xi}\right).
\end{equation}
Let us calculate every term separately
\begin{eqnarray}\label{cond2X-0}
i\left(\bvph^X_*,\hG\cdot\nabla_{\brho} \bphi^{(1)}\right) =
\left(\bvph^X_*,\hG \frac{\partial
\bvph^H_*}{\partial p_x}\right) \cdot \nabla_{\brho}\tau_1 + \left(\bvph^X,\hG
\frac{\partial \bvph^E_*}{\partial p_x}\right) \cdot \nabla_{\brho}\tau_2
\nonumber \\
+ \left(\bvph^X_*,\hG \frac{\partial \bvph^X_*}{\partial p_z}\right)\cdot \nabla_{\brho}
\frac{\partial \alpha_1}{\partial \zeta} + \left(\bvph^X_*,\hG \frac{\partial \bvph^Y_*}{\partial
p_z}\right) \cdot \nabla_{\brho} \frac{\partial \alpha_2}{\partial \zeta}.
\end{eqnarray}
Here we take into account that the coefficients of $\alpha_{2}^{(1)}$ and $\alpha_{1}^{(1)}$
vanish owing to (<ref>).
For example, the first term in the right-hand side of (<ref>) reads
\begin{eqnarray}
\left(\bvph^X_*,\hG \frac{\partial
\bvph^H_*}{\partial p_x}\right) &\cdot& \nabla_{\brho}\tau_1 \equiv \nonumber\\
& &\left(\bvph^X_*,\G_1 \frac{\partial \bvph^H_*}{\partial
p_x}\right) \frac{\partial \tau_1}{\partial
\xi} + \left(\bvph^X_*,\G_2 \frac{\partial \bvph^H_*}{\partial
p_x}\right)\frac{\partial \tau_1}{\partial
\eta} + \left(\bvph^X_*,\G_3 \frac{\partial \bvph^H_*}{\partial
p_x}\right) \frac{\partial \tau_1}{\partial
\zeta}.\label{cond2X-01}
\end{eqnarray}
According to (<ref>), the first term is proportional to $\ddot{\omega}_{11*}^H$. The second
term vanishes owing to (<ref>) and the last relation of (<ref>), the third term
vanishes by (<ref>). Analogously,
\begin{eqnarray}
\left(\bvph^X_*,\hG \frac{\partial
\bvph^E_*}{\partial p_x}\right) &\cdot& \nabla_{\brho}\tau_2 \equiv \nonumber\\
& & \left(\bvph^X_*,\G_1
\frac{\partial \bvph^E_*}{\partial p_x}\right)\frac{\partial \tau_2}{\partial
\xi} + \left(\bvph^X_*,\G_2 \frac{\partial \bvph^E_*}{\partial
p_x}\right)\frac{\partial \tau_2}{\partial
\eta} + \left(\bvph^X_*,\G_3 \frac{\partial \bvph^E_*}{\partial
p_x}\right) \frac{\partial \tau_2}{\partial
\zeta}.
\label{cond2X-02}
\end{eqnarray}
According to (<ref>), the second term is proportional to $\ddot{\omega}_{22*}^E$ and two
other terms vanish: the first one owing to (<ref>), and the third one because of (<ref>).
Now we proceed to the last two terms in (<ref>). By (<ref>)-(<ref>), we obtain
\begin{eqnarray}
&& \left(\bvph^X_*,\widehat{\G} \frac{\partial \bvph^X_*}{\partial
p_z}\right) \cdot \nabla_{\brho}\frac{\partial \alpha_1}{\partial \zeta} = \left(\bvph^X_*,\G_3
\frac{\partial \bvph^X_*}{\partial p_z}\right)\frac{\partial^2 \alpha_1}{\partial \zeta^2} =
\frac{ \ddot{\omega}^0_{33*}}{2c} u_*^{XX} \frac{\partial^2 \alpha_1}{\partial \zeta^2}, \label{cond2X-03}\\
&& \left(\bvph^X_*,\widehat{\G} \frac{\partial \bvph^Y_*}{\partial
p_z}\right) \cdot \nabla_{\brho}\frac{\partial \alpha_2}{\partial \zeta} = \left(\bvph^X_*,
\G_3 \frac{\partial \bvph^Y_*}{\partial p_z}\right) \frac{\partial^2 \alpha_2}{\partial \zeta^2}= 0. \label{cond2X-04}
\end{eqnarray}
We note that since $u_*^{XY}=0$, we get, with account of (<ref>) and (<ref>),
\begin{equation}
\left(\bvph^X_*,\frac{\delta \omega}{c} \P \bphi^{(0)}\right)= \frac{\delta \omega}{c}u_*^{XX}
\alpha_1.
\end{equation}
Finally, the condition (<ref>) yields
\begin{equation}\label{eq0-1}
\frac{\partial \tau_1}{\partial \xi} \ddot{\omega}_{11*}^H +
\frac{\partial \tau_2}{\partial \eta} \ddot{\omega}_{22*}^E +
\frac{\partial^2 \alpha_1}{\partial \zeta^2}\ddot{\omega}_{33*}^0 +
2\frac{\delta \omega}{c}\alpha_1=0.
\end{equation}
We omit here the nonzero common factor $u_*^{XX}/2=u_*^{YY}/2$.
Now we obtain the second equation by considering the condition
(<ref>). We use a similar line of argument. First, we take
the relation that differs from (<ref>) only by the first
factor in inner products
\begin{eqnarray}
i\left(\bvph^Y_*,\hG\cdot\nabla_{\brho} \bphi^{(1)}\right) =
\left(\bvph^Y_*,\hG \frac{\partial
\bvph^H_*}{\partial p_x}\right) \cdot \nabla_{\brho}\tau_1 + \left(\bvph^Y,\hG
\frac{\partial \bvph^E_*}{\partial p_x}\right) \cdot \nabla_{\brho}\tau_2 \nonumber\\
+ \left(\bvph^Y_*,\hG \frac{\partial \bvph^X_*}{\partial
p_z}\right) \cdot \nabla_{\brho} \frac{\partial \alpha_1}{\partial \zeta} +
\left(\bvph^Y_*,\hG \frac{\partial \bvph^Y_*}{\partial p_z}\right) \cdot
\nabla_{\brho} \frac{\partial \alpha_2}{\partial \zeta}.
\label{cond2Y-0}
\end{eqnarray}
Here, for example,
\begin{eqnarray}
\left(\bvph^Y_*,\hG \frac{\partial
\bvph^H_*}{\partial p_x}\right) &\cdot& \nabla_{\brho} \tau_1 \equiv \nonumber \\
& & \left(\bvph^Y_*,
\G_1 \frac{\partial \bvph^H_*}{\partial p_x}\right)\frac{\partial \tau_1}{\partial
\xi} + \left(\bvph^Y_*,\G_2 \frac{\partial \bvph^H_*}{\partial
p_x}\right) \frac{\partial \tau_1}{\partial
\eta} + \left(\bvph^Y_*,\G_3 \frac{\partial \bvph^H_*}{\partial
p_x}\right) \frac{\partial \tau_1}{\partial
\zeta}.
\end{eqnarray}
The second term here is proportional to $\ddot{\omega}^H_{22_*}$ according to (<ref>). The
first and the third terms vanish by (<ref>) and (<ref>), respectively. The term
$\left(\bvph^Y_*,\widehat{\G} \left.\partial \bvph^E_*\right/\partial
p_x\right)\cdot\nabla_{\brho}\tau_2$ is treated analogously to (<ref>) by using the
second relation of (<ref>), (<ref>) with (<ref>), and (<ref>). Analogously
to (<ref>) and (<ref>), we obtain
\begin{equation}
\left(\bvph^Y_*,\widehat{\G} \frac{\partial \bvph^X_*}{\partial p_z}\right)\cdot \nabla_{\brho}
\frac{\partial \alpha_1}{\partial \zeta} = 0, \quad \left(\bvph^Y_*,\widehat{\G} \frac{\partial
\bvph^Y_*}{\partial p_z}\right) \cdot\nabla_{\brho}\frac{\partial \alpha_2}{\partial \zeta} =
\frac{\ddot{\omega}^0_{33*}}{2c}u_*^{XX} \frac{\partial^2 \alpha_2}{\partial \zeta^2} .
\end{equation}
\begin{equation}
\left(\bvph^Y_*,\hG \frac{\partial \bvph^H_*}{\partial p_x}\right) \cdot\nabla_{\brho}
\tau_1=-\frac{\ddot{\omega}^H_{22_*}} {2c} u_*^{XX}\frac{\partial \tau_1}{\partial
\eta}.\label{cond2Y-01}
\end{equation}
Again, by (<ref>) and (<ref>), we get
\begin{equation}\label{eq0-2}
- \frac{\partial \tau_1}{\partial \eta} \ddot{\omega}_{22*}^H + \frac{\partial \tau_2}{\partial
\xi} \ddot{\omega}_{11*}^E + \frac{\partial^2 \alpha_2}{\partial \zeta^2}\ddot{\omega}_{33*}^0 +
2\frac{\delta \omega}{c}\alpha_2=0.
\end{equation}
Taking into account the definition of $\tau_1$ and $\tau_2$ (<ref>) and the fact that
$\ddot{\omega}_{11*}^H=\ddot{\omega}_{22*}^H$ and $\ddot{\omega}_{11*}^E=\ddot{\omega}_{22*}^E$, we
rewrite the equations for $\alpha_1$ and $\alpha_2$ as follows:
\begin{equation}\label{eq0-sys}
\begin{aligned}
\frac{\partial^2 \alpha_1}{\partial \xi^2} \ddot{\omega}_{11*}^H +
\frac{\partial^2 \alpha_1}{\partial \eta^2} \ddot{\omega}_{11*}^E +
\frac{\partial^2 \alpha_1}{\partial \zeta^2}\ddot{\omega}_{33*}^0 +
2\frac{\delta \omega}{c}\alpha_1- \frac{\partial^2
\alpha_2}{\partial \xi \partial \eta}
(\ddot{\omega}_{11}^H-\ddot{\omega}_{11}^E)=0, \\
\frac{\partial^2 \alpha_2}{\partial \xi^2} \ddot{\omega}_{11*}^E +
\frac{\partial^2 \alpha_2}{\partial \eta^2} \ddot{\omega}_{11*}^H +
\frac{\partial^2 \alpha_2}{\partial \zeta^2}\ddot{\omega}_{33*}^0 +
2\frac{\delta \omega}{c}\alpha_2-\frac{\partial^2 \alpha_1}{\partial
\xi \partial \eta}(\ddot{\omega}_{11}^H-\ddot{\omega}_{11}^E)=0.
\end{aligned}
\end{equation}
§.§ Higher-order approximations
Now we turn to the set of equations
(<ref>-<ref>). By considering several recurrent equations, we conclude that the approximation of $n$th order has the form
\begin{eqnarray}\label{sol-phi-n}
&&\bphi^{(n)} =\alpha_{1}^{(n)}\bvph^{X}_*(z) + \alpha_{2}^{(n)}\bvph^{Y}_*(z) +
{G^{(n)}\left(\alpha_1^{(n-1)}, \alpha_2^{(n-1)}, \ldots \alpha_1, \alpha_2 \right)},
\end{eqnarray}
where $G^{(n)}$ is the linear combination of the derivatives of the
functions $\alpha_j^{(k)}, j = 1,2, \, k = 1 \ldots n-1$, with respect to the variables
$\xi,\eta,\zeta$ with known coefficients. For example, in the principal order, owing to the formula
(<ref>), $G^{(0)} \equiv 0$. In the first order approximation, by
(<ref>), $G^{(1)}$ contains the derivatives of $\alpha_j^{(0)} \equiv \alpha_j, j =
1,2,$ up to the first order:
\begin{equation}
G^{(1)} \equiv -i \left( \frac{\partial \alpha_1}{\partial \xi} - \frac{\partial \alpha_2}{\partial
\eta}\right) \frac{\partial \bvph^H_*}{\partial p_x} -i \left( \frac{\partial \alpha_1}{\partial
\eta} + \frac{\partial \alpha_2}{\partial \xi}\right) \frac{\partial \bvph^E_*}{\partial p_x}-
i\frac{\partial \alpha_1}{\partial \zeta} \frac{\partial \bvph^X_*}{\partial p_z} - i\frac{\partial
\alpha_2}{\partial \zeta} \frac{\partial \bvph^Y_*}{\partial p_z}.
\end{equation}
To find the approximation $\bphi^{(2)}$ we consider the following inhomogeneous equations:
\begin{eqnarray}
\A_* \Upsilon^{2H}_{j} = \G_j \frac{\partial \bvph^H_*}{\partial p_x},&& \quad \A_*
\Upsilon^{2E}_{j} = \G_j\frac{\partial
\bvph^E_*}{\partial p_x},\label{eq-upsilon1} \\
\A_* \Upsilon^{2X}_{j} = \G_j \frac{\partial \bvph^X_*}{\partial p_z},&& \quad \A_*
\Upsilon^{2Y}_{j} = \G_j\frac{\partial \bvph^Y_*}{\partial p_z}, \quad j=1,2,3.\label{eq-upsilon2}
\end{eqnarray}
Each of these equations is a system of the form (<ref>) with nonzero right-hand sides. It is
necessary to check the solvability of these equations in the class $\mathcal{M}$. To do this, we check the conditions imposed in Lemma 2. This means that the conditions (<ref>) must be satisfied:
\begin{equation}
\left(\bvph^X_*,\vec{F}\right)=0,\quad \left(\bvph^Y_*,\vec{F}\right)=0,
\end{equation}
where $\vec{F}$ stands for each expression on the right-hand sides of the equations
(<ref>), (<ref>). By taking all the possible combinations of the form
\begin{equation}
\begin{aligned}
& \left(\bvph^f_*, \G_j \frac{\partial \bvph^H_*}{\partial p_x} \right)=0, \quad \left(\bvph^f_*,
\G_j\frac{\partial
\bvph^E_*}{\partial p_x} \right)=0, \\
&\left(\bvph^f_*, \G_j \frac{\partial \bvph^X_*}{\partial p_z} \right)=0, \quad \left(\bvph^f_*,
\G_j \frac{\partial \bvph^Y_*}{\partial p_z} \right)=0, \,\, j = 1,2,3, \quad f = X,Y,
\end{aligned}
\end{equation}
we obtain 24 conditions, 18 of them are satisfied (which follows from (<ref>), (<ref>),
(<ref>)), and 6 are not satisfied (which follows from (<ref>), (<ref>) and
(<ref>)). Nonhomogeneous equations from the set (<ref>), (<ref>),
for which the solvability conditions are satisfied, have solutions in the class $\mathcal{M}$.
The other equations also have solutions, which do not belong to the class $\mathcal{M}$ and are
Next we take the sum of the equations for $\Upsilon^{2H}_{j}$, $\Upsilon^{2E}_{j}$,
$\Upsilon^{2X}_{j}$, $\Upsilon^{2Y}_{j}$, $j=1,2,3,$ with coefficients such that the sum of the
right-hand side expressions coincides with $F^{(2)} \equiv -i\hG \cdot\nabla_{\brho}\bphi^{(1)}$.
The solution of the obtained sum of equations
belongs to the class $\mathcal{M}$, because the conditions of solvability, which are reduced to (<ref>), are satisfied. Thus, we get
\begin{equation}\label{Phi_2_G}
\begin{aligned}
G^{(2)} \equiv &-i \left( \frac{\partial \alpha_1^{(1)}}{\partial \xi} - \frac{\partial
\alpha_2^{(1)}}{\partial \eta}\right) \frac{\partial \bvph^H_*}{\partial p_x} -i \left(
\frac{\partial \alpha_1^{(1)}}{\partial \eta} + \frac{\partial \alpha_2^{(1)}}{\partial \xi}\right)
\frac{\partial
\bvph^E_*}{\partial p_x}- \\
& -i\frac{\partial \alpha_1^{(1)}}{\partial \zeta} \frac{\partial \bvph^X_*}{\partial p_z} -
i\frac{\partial \alpha_2^{(1)}}{\partial \zeta} \frac{\partial \bvph^Y_*}{\partial p_z}-
\\
& -\sum\limits_{j=1}^3\nabla_j \G_j \left[ \left( \frac{\partial \alpha_1}{\partial \xi} -
\frac{\partial \alpha_2}{\partial \eta}\right) \Upsilon^{2H}_{j}(z) + \left( \frac{\partial
\alpha_1}{\partial
\eta} + \frac{\partial \alpha_2}{\partial \xi}\right) \Upsilon^{2H}_{j}(z)\right.+ \\
& \left. + \frac{\partial \alpha_1}{\partial \zeta}\Upsilon^{2X}_{j} + \frac{\partial
\alpha_2}{\partial \zeta}\Upsilon^{2Y}_{j}\right],\quad \nabla_1 = \frac{\partial}{\partial \xi},
\nabla_2 = \frac{\partial}{\partial \eta}, \nabla_3 = \frac{\partial}{\partial \zeta}.
\end{aligned}
\end{equation}
In an analogous manner, we can derive that the term $G^{(3)}$ contains the first derivatives of $\alpha_1^{(2)}$ and $\alpha_2^{(2)}$, the
second derivatives of $\alpha_1^{(1)}$ and $\alpha_2^{(1)}$, and the third derivatives of
$\alpha_1$ and $\alpha_2$. The approximation $\bphi^{(k)}$ contains yet unknown functions
$\alpha^{(k)}_j, \, j=1,2$, and $G^{(k)}$, which depends on the first derivatives
of $\alpha_1^{(k-1)}$ and $\alpha_2^{(k-1)}$, the second derivatives of $\alpha_1^{(k-2)}$ and
$\alpha_2^{(k-2)}$, and the derivatives of the $k$th order of $\alpha_1$ and $\alpha_2$. The
solvability conditions of $\bphi^{(k+1)}$,
\begin{equation}\label{solv-n}
\left(\bvph^X_*,\hG\cdot\nabla_{\brho} \bphi^{(k)}\right)=0,\quad
\left(\bvph^Y_*,\hG\cdot\nabla_{\brho} \bphi^{(k)}\right)=0,
\end{equation}
provide equations for $\alpha_1^{(k-1)}$ and $\alpha_2^{(k-1)}$. By analogy with the section
<ref>, we obtain a system of partial differential equations:
\begin{eqnarray}
&& \frac{\partial^2 \alpha^{(k-1)}_1}{\partial \xi^2} \ddot{\omega}_{11*}^H + \frac{\partial^2
\alpha^{(k-1)}_1}{\partial \eta^2} \ddot{\omega}_{11*}^E + \frac{\partial^2
\alpha^{(k-1)}_1}{\partial \zeta^2}\ddot{\omega}_{33*}^0 + 2\frac{\delta
\omega}{c}\alpha^{(k-1)}_1- \nonumber
\\ && - \frac{\partial^2 \alpha^{(k-1)}_2}{\partial \xi \partial \eta}
(\ddot{\omega}_{11{*}}^H-\ddot{\omega}_{11{*}}^E)= A_1^{(k-1)}\left(\alpha_1^{(k-2)},
\alpha_2^{(k-2)}, \ldots, \alpha_1, \alpha_2 \right),
\end{eqnarray}
\begin{eqnarray}
&& \frac{\partial^2 \alpha^{(k-1)}_2}{\partial \xi^2} \ddot{\omega}_{11*}^E + \frac{\partial^2
\alpha^{(k-1)}_2}{\partial \eta^2} \ddot{\omega}_{11*}^H + \frac{\partial^2
\alpha^{(k-1)}_2}{\partial \zeta^2}\ddot{\omega}_{33*}^0 + 2\frac{\delta
\omega}{c}\alpha^{(k-1)}_2- \nonumber
\\&& - \frac{\partial^2 \alpha^{(k-1)}_1}{\partial \xi \partial
\eta}(\ddot{\omega}_{11{*}}^H-\ddot{\omega}_{11{*}}^E)=A_2^{(k-1)}\left(\alpha_1^{(k-2)},
\alpha_2^{(k-2)}, \ldots, \alpha_1, \alpha_2 \right), \label{al-higher}
\end{eqnarray}
where $A_1^{(0)}=A_2^{(0)}=0$, $A_1^{(k-1)}, A_2^{(k-1)}$ for $k>1$ are linear combinations of
derivatives of the already known functions $\alpha_j^{(l)}, j = 1,2, \, l =1, \ldots, k-2,$ with
respect to the variables $\xi,\eta,$ and $\zeta$. This means that equations for approximations of all
orders (<ref>-<ref>) can be solved step by step.
§ SOLUTION OF THE EQUATIONS FOR $\ALPHA$
We have obtained the equations (<ref>) with constant coefficients, which describe the
behavior of the envelopes of the field. Now we are going to discuss the methods of solving them.
The simplest case arises if the field does not depend on one of the lateral coordinates, for
example, on $\eta$. In this case, the equations for $\alpha_1$ and $\alpha_2$ are separated:
\begin{equation}\label{twoD-al}
\begin{aligned}
&\frac{\partial^2 \alpha_1}{\partial\xi^2}\ddot{\omega}_{11*}^H + \frac{\partial^2
\alpha_1}{\partial \zeta^2}\ddot{\omega}^0_{33*} + 2
\frac{\delta\omega}{c}\alpha_1 = 0, \\
&\frac{\partial^2 \alpha_2}{\partial\xi^2}\ddot{\omega}_{11*}^E + \frac{\partial^2
\alpha_2}{\partial \zeta^2}\ddot{\omega}^0_{33*} + 2 \frac{\delta\omega}{c}\alpha_2 = 0,
\end{aligned}
\end{equation}
where the coefficients are defined in (<ref>) and are the derivatives of the
dispersion functions for TM and TE type waves, calculated at stationary points. Equations
(<ref>) can be elliptic or hyperbolic ones depending on the type of a stationary point. According to the Appendix 1, we have $\ddot{\omega}_{11*}^f>0$, $f=E,H$.
Equations (<ref>) are
elliptic if $\ddot{\omega}^0_{33*}
>0$, and hyperbolic if $\ddot{\omega}^0_{33*}<0$. We solve these equations by means of the Fourier
method performing the Fourier transform with respect to the variable $\xi$. By dividing the obtained equations by
the coefficient $\ddot{\omega}^0_{33*}$, we derive the following equations for the functions
\begin{equation}\label{eq1111}
\frac{\partial^2 \widehat{\alpha}_j}{\partial \zeta^2} -
\frac{\ddot{\omega}^j_{11*}}{\ddot{\omega}^0_{33*}}p_{\xi}^2 \widehat{\alpha}_j + 2\frac{\delta
\omega}{c\,\ddot{\omega}^0_{33*}}\widehat{\alpha}_j = 0, \quad {j=1,2},
\end{equation}
\begin{equation}\label{def12HE}
\ddot{\omega}^1_{11*} \equiv \ddot{\omega}^H_{11*}, \quad \ddot{\omega}^2_{11*} \equiv \ddot{\omega}^E_{11*}.
\end{equation}
Solutions of (<ref>) are determined by the integral
\begin{equation}
\alpha_j(\brho) = \frac{1}{2\pi} \int\limits_{\mathbb{R}} dp_{\xi} \,\, e^{ip_{\xi}\xi} \left[
\widehat{\alpha}^-_j(p_{\xi}) \, e^{ip_{j\zeta}\zeta} + \widehat{\alpha}^+_j(p_{\xi}) \,
e^{-ip_{j\zeta}\zeta} \right],
\end{equation}
where $\brho = (\xi,0,\zeta)$ and
\begin{equation}
p_{j\zeta} = \sqrt{-p_{\xi}^2 \frac{\ddot{\omega}^j_{11*}}{\ddot{\omega}^0_{33*}} + 2\frac{\delta
\omega}{c\,\ddot{\omega}^0_{33*}}}.
\end{equation}
If $\ddot{\omega}^0_{33*} > 0$ and $0 < p_{\xi}^2\ddot{\omega}^j_{11*}/\ddot{\omega}^0_{33*} <
2\delta \omega/c\,\ddot{\omega}^0_{33*}$, then the integral describes the propagating waves
governed by the elliptic equation. The components with such values of $p_{\xi}$ that
$p_{\xi}^2\ddot{\omega}^j_{11*} > {2}\delta \omega/c$ do not propagate. If $\ddot{\omega}^0_{33*}
< 0$, then introducing the notation $\sigma^2_j$, $q^2$, we obtain
\begin{equation}\label{sigma}
p_{j\zeta} = \sqrt{p_{\xi}^2\sigma^2_j + q^2}, \quad \sigma^2_j =
\frac{\ddot{\omega}_{11*}^j}{|\ddot{\omega}^0_{33*}|}, \quad q^2 = -2 \frac{\delta\omega}{c
\end{equation}
If $\delta\omega>0$, only the components with large $p_{\xi}$ propagate.
The equation (<ref>) turns to a Klein-Gordon-Fock type equation.
If $\delta\omega = 0$, the equations (<ref>) are the one-dimensional wave equations, where
the variable $\zeta$ plays the role of time, and $\sigma$ plays the role of the speed of wave
propagation. The solutions can be found by the D'Alambert method and read
\begin{equation}
\alpha_j = F_j(\xi - \sigma\zeta) + G_j(\xi + \sigma\zeta),
\end{equation}
where $F_j$ and $G_j$ are some functions. If the function $F_j$ is localized for $\zeta=0$
near $\xi=0$, it is localized for any $\zeta$ near the line $\xi - \sigma\zeta = 0$ and
propagates undistorted. This means that in the medium under consideration there is the possibility
of existence of non-distorting beams, all of them having the the same angle with $z$ axis equal
\begin{equation}\label{angle_dif}
\varphi^f = \pm\mathrm{arctg} \sqrt{\frac{\ddot{\omega}_{11*}^f}{|\ddot{\omega}^0_{33*}|}}, \qquad
f=H, E.
\end{equation}
We have discussed and studied this effect numerically in <cit.>.
Now we proceed to the general case, where the field depends on both lateral coordinates $\xi, \eta$.
We obtained equations (<ref>) and (<ref>) for the functions $\tau_1, \tau_2$ defined by
(<ref>). Let us take the derivative of the equation (<ref>) with respect to $\xi$ and
of the equation (<ref>) with respect to $\eta$ and then calculate the difference. We obtain
an equation for $\tau_1$. To get the equation for $\tau_2$, we differentiate (<ref>) with
respect to $\eta$ and equation (<ref>) with respect to $\xi$ and take the sum of the results.
Thus, we find the equations for the functions $\tau_1, \tau_2$
\begin{equation}
\begin{aligned}\label{tau-3Deq}
\ddot{\omega}_{11*}^{H} \left( \frac{\partial^2 \tau_1}{\partial \xi^2} + \frac{\partial^2
\tau_1}{\partial \eta^2} \right) + \ddot{\omega}^0_{33*}\frac{\partial^2 \tau_1}{\partial \zeta^2}
+ 2\frac{\delta\omega}{c}\tau_1 = 0, \\
\ddot{\omega}_{11*}^{E} \left( \frac{\partial^2 \tau_2}{\partial
\xi^2} + \frac{\partial^2 \tau_2}{\partial \eta^2} \right) +
\ddot{\omega}^0_{33*}\frac{\partial^2 \tau_2}{\partial \zeta^2} +
2\frac{\delta\omega}{c}\tau_2 = 0.
\end{aligned}
\end{equation}
These equations are either hyperbolic or elliptic ones depending on the sign of
$\ddot{\omega}^0_{33*}$. Their solutions obtained by the Fourier transform read
\begin{equation}\label{tau_sol_four}
\tau_j(\brho) = \frac{1}{(2\pi)^2}\int\limits_{\mathbb{R}^2} dp_{\xi} dp_{\eta} \,
e^{ip_{\xi}\xi+ip_{\eta}\eta} \left[ \widehat{\tau}^{+}_j(p_{\xi}, p_{\eta}) e^{-ip_{j\zeta} \zeta}
+ \widehat{\tau}^{-}_j(p_{\xi}, p_{\eta}) e^{ip_{j\zeta} \zeta} \right],
\end{equation}
\begin{equation}
p_{1\zeta} = \sqrt{-\frac{\ddot{\omega}^H_{11*}}{\ddot{\omega}^0_{33*}}(p_{\xi}^2 + p_{\eta}^2) +
2\frac{\delta \omega}{c\,\ddot{\omega}^0_{33*}}}, \qquad p_{2\zeta} =
\sqrt{-\frac{\ddot{\omega}^E_{11*}}{\ddot{\omega}^0_{33*}}(p_{\xi}^2 + p_{\eta}^2) + 2\frac{\delta
\omega}{c\,\ddot{\omega}^0_{33*}}},
\end{equation}
and $\widehat{\tau}^{\pm}_j(p_{\xi}, p_{\eta})$ are such functions that the integrals
(<ref>) and the integrals, obtained by taking the derivative of (<ref>),
converge. For $\ddot{\omega}^0_{33*} < 0$, $\delta\omega >0$ only the components with large
$p_{\xi}^2+p_{\eta}^2$ propagate.
Now we are going to derive the equations for the original functions $\alpha_1, \alpha_2$. We take
the derivatives of two relations (<ref>) with respect to $\eta$ and $\xi$, respectively,
and take their sum and difference. We obtain
\begin{equation}
\triangle \alpha_1 = \frac{\partial \tau_1}{\partial \xi} + \frac{\partial \tau_2}{\partial \eta},
\quad \triangle \alpha_2 = \frac{\partial \tau_2}{\partial \xi} - \frac{\partial \tau_1}{\partial
\eta}, \quad \triangle \equiv \frac{\partial^2 }{\partial \xi^2} + \frac{\partial^2 }{\partial
\eta^2}.
\end{equation}
By the Fourier integral we get the following expressions for the functions $\alpha_1, \alpha_2$:
\begin{equation}\label{alpha_1alpha_2}
\begin{aligned}
\alpha_1(\brho) = \frac{-i}{(2\pi)^2} \int\limits_{\mathbb{R}^2} dp_{\xi}dp_{\eta} \,
e^{i(p_{\xi} \xi + p_{\eta} \eta)} \frac{p_{\xi}\widehat{\tau}_1 +
p_{\eta}\widehat{\tau}_2}{p_{\xi}^2 + p_{\eta}^2}, \\
\alpha_2(\brho) = \frac{-i}{(2\pi)^2} \int\limits_{\mathbb{R}^2} dp_{\xi}dp_{\eta} \,
e^{i(p_{\xi} \xi + p_{\eta} \eta)} \frac{p_{\xi}\widehat{\tau}_2 -
p_{\eta}\widehat{\tau}_1}{p_{\xi}^2 + p_{\eta}^2},
\end{aligned}
\end{equation}
\begin{equation}
\widehat{\tau}_j \equiv \widehat{\tau}^{+}_j(p_{\xi}, p_{\eta}) e^{-ip_{j\zeta} \zeta} +
\widehat{\tau}^{-}_j(p_{\xi}, p_{\eta}) e^{ip_{j\zeta} \zeta}.
\end{equation}
We also require the functions $\widehat{\tau}_j p_{\xi}/(p_{\xi}^2 + p_{\eta}^2)$ and
$\widehat{\tau}_j p_{\eta}/(p_{\xi}^2 + p_{\eta}^2)$ to be continuous and integrable. The
expressions for $\alpha_1, \alpha_2$ can also be rewritten in the polar coordinate system
$(p_{\rho},\gamma)$ as
\begin{equation}\label{alpha_1alpha_2_polar}
\begin{aligned}
\alpha_1(\brho) = \frac{-i}{(2\pi)^2} \int\limits_0^{+\infty} dp_{\rho} \int\limits_{0}^{2\pi}
d\gamma \, e^{i(p_{\rho}\xi \cos\gamma + p_{\rho}\eta \sin\gamma) } (\cos
\gamma \widehat{\tau}_1 + \sin\gamma \widehat{\tau}_2), \\
\alpha_2(\brho) = \frac{-i}{(2\pi)^2} \int\limits_0^{+\infty} dp_{\rho} \int\limits_{0}^{2\pi}
d\gamma \, e^{i(p_{\rho}\xi \cos\gamma + p_{\rho}\eta \sin\gamma) } (\cos\gamma \widehat{\tau}_2 -
\sin\gamma \widehat{\tau}_1),
\end{aligned}
\end{equation}
where $p_{\rho} = \sqrt{p_{\xi}^2 + p_{\eta}^2}$, $\cos{\gamma} = p_{\xi}/p_{\rho}$,
$\sin{\gamma} = p_{\eta}/p_{\rho}$.
We express also the principal approximation of the asymptotic solutions in terms of TM and TE solutions at the stationary point determined by the passage to the limit in every direction (<ref>).
Substituting solution of (<ref>) given by (<ref>) in formula (<ref>), we obtain the integral representation
\begin{equation}
\left( \begin{array}{c} {\mathbf E} \\ {\mathbf{H}} \end{array}
\right)(z, \brho) \simeq \frac{-i}{(2\pi)^2} \int\limits_0^{+\infty} dp_{\rho}
\int\limits_{0}^{2\pi} d\gamma \, e^{i(p_{\rho}\xi \cos\gamma + p_{\rho}\eta \sin\gamma) } ( \widehat{\tau}_1 \bPh^{H}_*(z) + \widehat{\tau}_2 \bPh^{E}_*(z) ),
\end{equation}
where $\bPh^{H}_*(z)$ and $\bPh^{E}_*(z)$ depend on the direction of propagation according to (<ref>).
§ CONCLUSIONS
We have elaborated a formal asymptotic approach for monochromatic electromagnetic fields in a layered periodic structure. The frequency of the field is close to that of a stationary point $\vp_*$ of one of the sheets of the dispersive surface $\omega=\omega^f(\vp)$. Here $f$ stands for the type of the polarization, which may be $H$
for TM- or $E$ for TE- polarization. The dispersive surfaces for waves of different polarizations are distinct, but the stationary points of them coincide.
For the conditions listed in Section <ref>, we found asymptotic series for the solutions of Maxwell equations (<ref>) by the two-scale expansions method.
The field is assumed to be a function of a fast variable $z$ and slow variables $\rho$; see (<ref>).
The field in the principal approximation is represented as a linear combination of Floquet-Bloch solutions of different
polarizations $\bPh_*^f(z),$ $f=X$ or $Y$ with slowly varying envelopes $\alpha_{j}(\brho),$ $j=1,2$. It reads
\begin{equation}\label{princip-final}
\bPs = \left(
\begin{array}{c} \mathbb{E} \\ \mathbb{H} \end{array} \right)(z, \brho) \simeq \alpha_{1}(\brho)\bPh^{X}_*(z) +
\alpha_{2}(\brho) \bPh^{Y}_*(z),
\end{equation}
\begin{equation}
\bPh_*^f(z) \equiv \bPh^f(z,\vp_*) = e^{ip_{z*}z}\bvph_*^f, \quad f=X,Y,
\end{equation}
where $\vp_* =(0,0, p_{z*})$, $p_{z*} = 0, \pm\pi/b.$ The functions $\bPh^f(z,\vp)$
are Floquet-Bloch solutions
of the system (<ref>). For $\vp=\vp_*$, they are expressed in terms of $(E_0, H_0)$ (see, (<ref>)), which are Floquet-Bloch solutions of the system (<ref>) with the quasimomentum $p_{z*}$.
The envelope functions $\alpha_j, \,\, j=1,2$, are defined by the
equations with constant coefficients (<ref>).
These coefficients are the second derivatives of the dispersion functions, i.e., the coefficients of the Tailor
expansion of $\omega=\omega^f(\vp)$ near the stationary points
\begin{equation}
\omega=\omega_*+ \frac{1}{2}\ddot{\omega}_{11*}^{f} (p_x^2+p_y^2) +
\frac{1}{2}\ddot{\omega}^0_{33*} (p_z-p_{z*})^2 + \ldots,
\end{equation}
where $\ddot{\omega}_{11*}^{f}$, $\ddot{\omega}^0_{33*}$ are the
second derivatives of the dispersion function $\omega$ with respect
to $p_x$ and $p_z$ calculated at the stationary point. Since the
problem is axially symmetric, the derivatives of the dispersion
functions with respect to $p_x$ and $p_y$ are equal and depend on
the type of the polarization, while the derivatives with respect to
$p_z$ do not depend on the type of the polarization.
The system (<ref>) can be split by introducing new functions $\tau_j$, $j=1,2,$ by means of (<ref>).
As was obtained in Section <ref>, the functions $\tau_1$ and $\tau_2$ satisfy the equations (<ref>).
The coefficients $\alpha_j, j=1,2,$ are expressed in terms of both $\tau_j, j=1,2,$ in (<ref>). The system can be split into two separated equations only if the field does not depend on one of the spatial coordinates. Such analysis was absent
in the papers of Longhi <cit.>, <cit.>.
An interesting particular case arises if $\ddot{\omega}^0_{33*} < 0$. Then the equations for $\tau_{1}$ and
$\tau_2$ are hyperbolic of the Klein-Gordon-Fock type. The coordinate
$\zeta$ stands for time. If additionally $\delta\omega = 0$, there are wave equations. The
effects that arise if the field depends only on one lateral coordinate were studied both
analytically and numerically in our paper <cit.>.
The investigation of qualitative consequences of the obtained results in the case of two lateral coordinates are out of scope of the present paper and will be discussed in the next publications. We expect that,
by choosing the localized solutions of envelopes, we can construct
beam-like solutions of the Maxwell equations.
The obtained formulas enable us also to find a change of polarization of the field in passing through the layered periodic structure.
The results may be generalized to another equations, which can be written in matrix form (<ref>).
§ APPENDIX 1
§.§ The dispersion relation
To make the paper self-contained, we obtain the dispersion
relation and find stationary points of the dispersive surfaces. To do this, we consider the Floquet-Bloch solutions of the
first subsystems (<ref>):
\begin{equation}\label{EH-Bl}
\left(\begin{array}{c} E_{\parallel}\\
H_{\perp} \end{array}\right) = e^{ipz} \left(\begin{array}{c} U_1\\
U_2 \end{array}\right),
\end{equation}
where $U_1, U_2$ are the periodic functions of $z$ with period $b$.
We accomplish the first of subsystems (<ref>) with the initial data
$E_{\parallel}=1, H_{\perp}=0$ and denote the solution of such a Cauchy problem by $e_1,h_1$. We
introduce another solution $e_2,h_2$, which satisfies (<ref>) and the initial data
$E_{\parallel}=0, H_{\perp}=1$. Both solutions are smooth in the intervals, where $\varepsilon,\mu$
are continuous, and continuous at the points of discontinuity of the parameters $\varepsilon,\mu$, but,
generally speaking, they are not periodic. These solutions depend on $p_{\parallel}^2, \omega$,
because the coefficients of (<ref>) depend on these parameters. These solutions are
linear independent and form a basis in the space of the solutions of the first subsystem of
(<ref>). We introduce the matrix $\mathbf{M}$ of the solutions $(e_1,h_1)^t$ and
$(e_2,h_2)^t$ as follows:
\begin{equation}
\mathbf{M}(z; p_{\parallel}^2, \omega) = \left(\begin{array}{cc} e_1 & e_2\\
h_1 & h_2 \end{array}\right)(z; p_{\parallel}^2, \omega).
\end{equation}
The matrix $\mathbf{M}(b; p_{\parallel}^2, \omega)$ is then a monodromy matrix:
\begin{equation}
\mathbf{M}(z+b; p_{\parallel}^2, \omega)=\mathbf{M}(z; p_{\parallel}^2, \omega)\mathbf{M}(b;
p_{\parallel}^2, \omega).
\end{equation}
We seek the Floquet-Bloch solutions in the following form:
\begin{equation}
\left(\begin{array}{c} E_{\parallel}\\
H_{\perp} \end{array}\right) = \mathbf{M}(z; p_{\parallel}^2, \omega) \left(\begin{array}{c} \beta_1\\
\beta_2 \end{array}\right),\label{App-Floq}
\end{equation}
where $(\beta_1, \beta_2)^t$ is the eigenvector corresponding to the eigenvalue $\lambda$ of the problem:
\begin{equation}\label{eigenvec}
\left(\mathbf{M}(b; p_{\parallel}^2, \omega) - \lambda \mathbf{I} \right)\left(\begin{array}{c}
\beta_1\\ \beta_2
\end{array}\right) = \left(\begin{array}{c} 0\\ 0
\end{array}\right),
\end{equation}
here $\mathbf{I}$ is the identity matrix. The equation for $\lambda$ is expressed in terms of
the determinant and the trace of the matrix $\mathbf{M}$. The determinant of the matrix
$\mathbf{M}(b; p_{\parallel}^2, \omega)$ is the Wronskian of the solutions $(e_1,h_1)^t$ and $(e_2,h_2)^t$ of the system (<ref>) at $z=b$. It is
constant by the Ostrogradskiy-Liuville theorem and, thus, can be calculated for any $z$. For $z=0$
it is equal to 1, so $\mathrm{det}\, \mathbf{M}(b; p_{\parallel}^2, \omega) = 1$ and $\lambda$
satisfies the quadratic equation
\begin{equation}\label{quadric}
\lambda^2 + {\rm Sp} \mathbf{M}(b; p_{\parallel}^2, \omega) \lambda + 1 = 0.
\end{equation}
In order to obtain the Floquet-Bloch solutions, we require $|\lambda|=1$ and we may assume
$\lambda_1 = e^{i p_z b}$, where $p_z$ is real-valued. Then $\lambda_2 = e^{-i p_z b}$ and
\begin{equation}
\label{disp1} M_{11}(b; p_{\parallel}^2, \omega) + M_{22}(b; p_{\parallel}^2, \omega) = 2
\cos{(p_z b)}.
\end{equation}
This equation establishes the relation between $p_z$, $p_{\parallel}^2$ and $\omega$, so we consider below only
two of these three variables as free parameters. We denote $\mathcal{F} \equiv M_{11}(b;
p_{\parallel}^2, \omega) + M_{22}(b; p_{\parallel}^2, \omega)$. The function $\mathcal{F} \equiv
{\cal F}( p_{\parallel}^2, \omega)$ depends on the problem parameters $\omega, p_{\parallel}$
analytically, because the coefficients of the system (<ref>) depend on these parameters
analytically if $\omega \neq 0$. Therefore the dispersion relation (<ref>) can be written in
the form of
\begin{equation}\label{disp2}
{\cal F}( p_{\parallel}^2, \omega) - \cos{(p_z b)}=0.
\end{equation}
The function ${\cal F}$ is an oscillating real-valued function of $\omega$, its minima and maxima
are larger than or equal $1$ (see, for example, <cit.>). The corresponding $p_z$
changes from $-\pi/b$ to $\pi/b$. If $|\mathcal{F}|$ is greater than $1$ for some interval of
$\omega$, then the bounded solutions do not exist and such interval is called the forbidden zone.
For each interval of $\omega$ (the allowed zone), where $|\mathcal{F}|\leq 1$ and the function $\mathcal{F}$ of $\omega$ is monotone, there exists an inverse
\begin{equation}\label{omegaH}
\omega=\omega_j^H(\vp).
\end{equation}
If $|\mathcal{F}|=1$ and $\partial \mathcal{F}/\partial \omega=0$, two intervals of monotone
behavior of $\mathcal{F}$ touch each other at this point. Further, we assume that $\partial
\mathcal{F}/\partial \omega \ne 0.$ This follows from the assumption that the system has one
bounded and one unbounded solutions on the boundary of the interval of the monotone behavior of $\cal{F}$;
see, for example, <cit.>.
Analogous considerations for the second subsystem of (<ref>) yield the dispersion relation
$\omega=\omega_j^E(\vp)$ for the wave of the TE polarization.
In other words, the problem under consideration can be treated as a
spectral problem for the operator
\begin{equation}
\left(-i\textbf{P}^{-1}\G \cdot \nabla + \textbf{P}^{-1}(\vp \cdot \G)\right) \bvph =
\frac{\omega}{c_0} \bvph, \quad \bvph\in\mathcal{M},
\end{equation}
where $\omega$ plays the role of the spectral parameter and $\vp$ is an external parameter of the
problem. For fixed $\vp$, the operator has a discrete set of eigenvalues $\omega=\omega_j$. For $\vp\in
\mathbb{R}^2\times (-\pi/b, \pi/b)$, $\omega_j(\vp)$ forms a sheet of number $j$.
§.§ The stationary points of the dispersive surface
We are going to find at least one stationary point of a single-valued function $\omega_j^H$, where
$j$ is the number of the sheet of the dispersive surface. Below we omit $j$ for the sake of brevity. We consider only such
zones, where $\partial \mathcal{F}/\partial \omega \neq 0$ on the entire zone. We take the
derivative of the implicitly defined function
\begin{equation}\label{deriv-omega}
\frac{\partial \omega^H}{\partial p_{\parallel}} = -2p_{\parallel}\left. \frac{\partial
\mathcal{F}}{\partial (p_{\parallel}^2)}\right/ \frac{\partial \mathcal{F}}{\partial \omega},
\qquad \frac{\partial \omega^H}{\partial p_z} = - b\sin p_zb\left/ \frac{\partial
\mathcal{F}}{\partial \omega}\right..
\end{equation}
We note that $\partial \omega^H/\partial p_z$ vanishes for $p_z
=0, \pm\pi/b$. The function $\omega^H(\vp)$ is a periodic function of $p_z$
with period $2\pi/b$. The derivative $\partial \omega^H/\partial
p_{\parallel}$ vanishes at least at the point $p_{\parallel}=0$
since $\mathcal{F}$ depends on $p_{\parallel}^2$. We denote the
stationary points with asterisk subscript: $\omega_* = \omega^H(\vp_*)$,
$p_{x*}=p_{y*}=0$, $p_{z*} =0, \pm\pi/b$. We note that such stationary points are the
bounds of the forbidden zones since $\cos p_{z*}b =\pm 1$, and hence
$\mathcal{F} = \pm1$.
Now we consider the second derivatives of $\omega$. First, we find the second derivative with
respect to $p_z$ from the formula (<ref>). At a stationary point it reads
\begin{equation}
\frac{\partial^2 \omega^H}{\partial p_z^2} = -b^2\cos p_{z*}b \left/ \frac{\partial
\mathcal{F}}{\partial \omega} \right. - b\sin p_{z*}b \frac{\partial }{\partial p_z} \, \left(
\frac{\partial \mathcal{F}}{\partial \omega} \right).
\end{equation}
The second term is equal to zero owing to (<ref>), and the first term has a constant nonzero
numerator and a denominator with the sign changing on each sheet of the function $\omega$. Now we
consider the second derivative of $\omega$ with respect to $p_x$ (or $p_y$). By (<ref>), we get
\begin{equation}\label{App1-omx}
\frac{\ddot{\omega}^H_{11*}}{c} u_*^{XX} = 2\left(\bvph_*^X,\G_1 \frac{\partial
\bvph^H_*}{\partial p_x}\right),
\end{equation}
\begin{equation} \left.\frac{\partial^2 \omega^H}{\partial p_x^2}\right|_{\vp_*} \equiv
\ddot{\omega}^H_{11*}.
\end{equation}
By the formula (<ref>) and the definition of the matrix (<ref>), we conclude that
the scalar product on the right-hand side of the formula (<ref>) is equal to the integral of
$|H_0|^2/(k\varepsilon)$ with respect to $z$ and hence it is positive if $\varepsilon>0$. Since
$u^{XX} >0$ for $\varepsilon>0,\mu>0$ , the second derivative $\ddot{\omega}^H_{11*} > 0$. This
means that the stationary points of $\omega$ are of two different types: hyperbolic and elliptic,
depending on the sign of the second derivative with respect to $p_z$.
§.§ The Floquet-Bloch amplitudes
We seek the Floquet-Bloch solutions in the form (<ref>), where $(\beta_1, \beta_2)^t$
is the eigenvector of the monodromy matrix (<ref>), which corresponds to the eigenvalue
$\lambda$. Up to an arbitrary constant factor, $\beta_1 = M_{12}, \beta_2 = M_{11} - \lambda_1$. We
denote the variable $U$ corresponding to the choice of the sign $\pm$ in the exponent $e^{\pm i p_z b}$ with a subscript $\pm$. We
consider TM-type solution and denote it by the superscript $H$. Thus,
\begin{equation}\label{sol}
\mathbf{U}_+^H = e^{-ip_{z}z} \left( M_{12}
\left(\begin{array}{c} e_1\\
h_1 \end{array}\right) + (M_{11} - \lambda_1)
\left(\begin{array}{c} e_2\\
h_2 \end{array}\right)\right).
\end{equation}
Here $(e_j,h_j)^t, j=1,2$ depends on the variable $z$ and on the parameters $p_{\parallel}^2,
\omega$. The monodromy matrix also depends on $p_{\parallel}^2, \omega$. We assume that the Floquet-Bloch solutions are expressed in terms of $\vp$.
We note that $p_{\parallel}^2 = p_x^2 + p_y^2$ and take into account the dispersion relation (<ref>). The formula (<ref>) takes a form:
\begin{equation}\label{Floq-Bl-App}
\left(\begin{array}{c} E_{\parallel}\\
H_{\perp} \end{array}\right)(z;\vp) = e^{ip_zz}
\mathbf{U}_+^H(z;p_z,p_{\parallel}^2,\omega^H(\vp)), \quad
\mathbf{U}_+^H = \left(\begin{array}{c} U_1\\
U_2 \end{array}\right).
\end{equation}
The second Floquet-Bloch solution is obtained by replacing $p_z$ by $-p_z$, and $\mathbf{U}_+^H$ by $\mathbf{U}_-^H$.
An exception is the case $p_z = p_{z*}$, $p_{z*}=0, \pm\pi/b.$ Each point $p_{z*}$ corresponds to the boundary of the forbidden zone, where $|\mathcal{F}(p_{\parallel}^2,\omega)|=1$. Then
the double root of the dispersion equation (<ref>) is $\lambda_1 = \lambda_2 = \pm 1$. We assume that $M_{12}\neq 0$. Formulas (<ref>), (<ref>) determine the unique, up to the constant factor, solution for $\vp=\vp_*$ (i.e., $p_{\parallel}=p_{\parallel*}=0$, $p_z = p_{z*}$).
We seek the second linearly independent solution in the form of
\begin{equation}\label{app-2}
\left(\begin{array}{c} E_{\parallel 2}\\
H_{\perp 2} \end{array}\right)(z; \vp_*) =
\mathbf{M}(z; 0, \omega_*) \left(\begin{array}{c} \gamma_1\\
\gamma_2 \end{array}\right),
\end{equation}
where the coefficients are found from the equation
\begin{equation}\label{adjoint}
\left( \mathbf{M}(b; 0, \omega_*) - \lambda \mathbf{I} \right)\left(\begin{array}{c} \gamma_1\\
\gamma_2
\end{array}\right) = \left(\begin{array}{c} \beta_1\\ \beta_2
\end{array}\right).
\end{equation}
Then the solution (<ref>) satisfies the relation
\begin{equation}
\left(\begin{array}{c} E_{\parallel 2}\\ H_{\perp 2}
\end{array}\right)(z+b; \vp_*) = \lambda\left(\begin{array}{c}
E_{\parallel 2}\\ H_{\perp 2} \end{array}\right)(z; \vp_*) +
\left(\begin{array}{c} E_{\parallel}\\
H_{\perp} \end{array}\right)(z; \vp_*),
\end{equation}
and can be written as
\begin{equation}
\left(\begin{array}{c} E_{\parallel 2}(z; \vp_*)\\ H_{\perp 2}(z; \vp_*)
\end{array}\right) = e^{ip_{z*}z} \left[\frac{z}{\lambda b}
\mathbf{U}^H_+(z; p_{z*},0,\omega_*) + \mathbf{Q}^H(z; p_{z*}, \omega_*) \right],
\end{equation}
where $\mathbf{U}^H_+$, $\mathbf{Q}^H$ are periodic functions of the variable $z$, the function $\mathbf{U}^H_+$ is defined
by (<ref>). We note that $\mathbf{U}^H_+$ and $\mathbf{U}^H_-$ are proportional at the point $\vp_*$.
For waves of the TE polarization, the Floquet-Bloch amplitudes are obtained analogously.
§ APPENDIX 2
Let $\mathcal{M}$ be a class of six-component vector-valued functions of $z$, which are periodic with period $b$,
piecewise smooth on the period, and their components with numbers $1,2,4,5$ are continuous.
(A piecewise function is a function that can be broken into a finite number of distinct pieces and on each piece both the function and its derivative are continuous, even though the whole function may have a jump discontinuity at points between the pieces.
Lemma 1.
The operator $\mathcal{A}_*$ defined by (<ref>) and (<ref>) on the
class of functions $\mathcal{M}$ is symmetric, i.e., for any $\w, \vv \in\mathcal{M}$ the following
relation is valid:
\begin{equation}
\left(\vv, \mathcal{A}_* \w \right) = \left(\mathcal{A}_* \vv, \w \right).
\end{equation}
This fact follows from the definition of the operator $\mathcal{A}_*$. Since $\mathbf{P}$
is a real-valued matrix and $\vv,\w \in \mathcal{M}$, we derive by integration by parts that
\begin{equation}\label{app1-1}
\left(\vv, \mathcal{A}_* \w \right) = \left(\vv, k_*\mathbf{P}\w + i\G_3\frac{\partial \w}{\partial z} - p_{z*} \G_3 \w
\right) = \left(\mathcal{A}_*\vv, \w \right) +i \sum\limits_{m=0}^M [\,<\vv,\G_3 \w>\,]_{m},
\end{equation}
where $<.,.>$ is a scalar product dependent on $z$; see (<ref>). By the definition of $\G_3$, see (<ref>), we get
\begin{equation}\label{app-jump}
<\vv(z),\G_3 \w(z)> = \overline{v_1(z)}w_5(z) - \overline{v_2}w_4(z) - \overline{v_4(z)} w_2(z) + \overline{v_5(z)}w_1(z).
\end{equation}
We denote a jump of any scalar function $h(z)$ at the point $z_m$ by
\begin{equation}\label{jump}
[\, h\,]_{m} = h(z_m+0) - h(z_m-0),\quad m \ne 0; \qquad [\, h\,]_0 = h(b) - h(0).
\end{equation}
The summation over all discontinuities of $\vv$ and $\w$ is implied in (<ref>).
Since $\vv, \w \in\mathcal{M}$, the function $<\vv(z),\G_3 \w(z)>$ is continuous, the terms outside the integral in (<ref>) vanish, and the
symmetry of $\mathcal{A}_*$ is proved.
Now we proceed to the proof of Lemma 2, which states that a solution from the class
$\mathcal{M}$ of the equation
\begin{equation}\label{problem-A}
\A_* \bphi =\vec{F}
\end{equation}
exists if and only if the following conditions are satisfied:
\begin{equation}
\left(\bvph^X_*,\vec{F}\right)=0,\quad \left(\bvph^Y_*,\vec{F}\right)=0,
\end{equation}
where the operator $\A_*$ is defined by the formula (<ref>), the functions
$\bvph^{X}_*(z),$ $\bvph^{Y}_*(z)$ are defined by (<ref>), (<ref>), and
(<ref>) and belong to ${\cal M}$.
In order to prove the lemma, we take the scalar product of (<ref>) and $\bvph_*^X$:
\begin{equation}\label{lemma-proof1}
\left( \bvph_*^X, \mathcal{A}_*\bphi \right) = \left(\bvph^X_*,\vec{F}\right).
\end{equation}
First, if a solution of (<ref>) $\bphi \in \mathcal{M}$, then the scalar product, by the
symmetry of $\mathcal{A}_*,$ can be rewritten as $\left( \bvph_*^X, \mathcal{A}_*\bphi \right) =
\left( \mathcal{A}_*\bvph_*^X, \bphi \right)$, and since $\mathcal{A}_*\bvph_*^X = 0$, it follows that the
scalar product on the right-hand side of the equation (<ref>) is also equal to zero.
Now let $\left(\bvph^X_*,\vec{F}\right)=0$. The equation (<ref>) for the
vector-valued function $\bphi=(\phi_1, \phi_2, \phi_3,$ $\phi_4, \phi_5, \phi_6)^T$ has the form
\begin{equation}
\begin{array}{ll} \left\{ \begin{array}{rcl} \displaystyle{ k\varepsilon \phi_1 +
i\frac{\partial \phi_5}{\partial z} -p_{z*}\phi_5 = F_1} \\ \\ \displaystyle{ k\mu \phi_5 + i\frac{\partial \phi_1}{\partial z} -p_{z*}\phi_1
= F_5} \\ \\ \displaystyle{ k\varepsilon \phi_3 = F_3}
\end{array} \right. & \qquad \left\{ \begin{array}{rcl} \displaystyle{ k\mu \phi_4 - i\frac{\partial \phi_2}{\partial z} + p_{z*}\phi_2 = F_4} \\ \\ \displaystyle{ k\varepsilon \phi_2 -
i\frac{\partial \phi_4}{\partial z} +p_{z*}\phi_4 = F_2} \\ \\ \displaystyle{ k\mu \phi_6 = F_6.}
\end{array} \right. \end{array}
\end{equation}
It splits into the pair of nonhomogeneous systems by the components with numbers 1,5 and 2,4,
respectively. The components with numbers 3 and 6 are found explicitly. We consider one of the
subsystems for the components 1,5, and the second system is considered similarly. The solution of
the subsystem can always be represented as a sum
\begin{equation}\nonumber%\label{sol_ful}
\left(\begin{array}{c} \phi_1\\ \phi_5 \end{array}\right) = A \mathbf{U}^H_+(z; p_{z*},0,\omega_*) +
B \left[ \frac{z}{b\lambda}\mathbf{U}^H_+(z; p_{z*},0,\omega_*) + \mathbf{Q}^H(z; p_{z*}, \omega_*)\right] +
\left(\begin{array}{c} \widetilde{\phi}_1\\ \widetilde{\phi}_5 \end{array}\right),
\end{equation}
where the first two terms are the Floquet-Bloch amplitudes satisfying
the homogeneous system, $A,B$ are some coefficients dependent on slow variables as on
parameters, and $( \widetilde{\phi}_1, \widetilde{\phi}_5 )^t$
indicates the particular vector solution of the nonhomogeneous equation,
which in general does not belong to the class $\mathcal{M}$. We show now that this
solution can be chosen continuous at all the points inside $(0,b)$. To get the continuity
inside $(0,b)$, we note that the solution $( \widetilde{\phi}_1, \widetilde{\phi}_5 )^t$ is always smooth on the intervals, where the parameters
$\varepsilon,\mu$ are continuous. It can have jumps at the boundary points of these intervals. At boundary points inside the period $(0,b)$, the jumps can be compensated by addition of the solutions of homogeneous equation with appropriate coefficients, which depend on intervals. However,
at the end of the period, the solution $( \widetilde{\phi}_1, \widetilde{\phi}_5 )^t$ may differ from the value at the
beginning of the period with jump $([\widetilde{\phi}_1]_0, [\widetilde{\phi}_5]_0)^t$; see (<ref>) for explanation of the notation. Integrating by parts, we obtain
the following expression:
\begin{eqnarray}\label{app2-2}
&&\left(\bvph^X_*\, \mathcal{A}_*\bphi \right) = \left(\mathcal{A}_*\bvph^X_*, \bphi \right) +
\overline{E_0(0;\vp_*)}\,\,[\bphi_5]_0 + \overline{H_0(0;\vp_*)}\,\,[\bphi_1]_0 \nonumber \\ &&= \overline{E_0(0;\vp_*)}\,\,[\bphi_5]_0 + \overline{H_0(0;\vp_*)}\,\,[\bphi_1]_0 =
\left(\bvph^X_*,\vec{F}\right)=0,
\end{eqnarray}
\begin{equation}
\left(\begin{array}{c}{[\phi_1]_0}\\ {[\phi_5]_0} \end{array}\right) = \frac{1}{\lambda}B\,\,\mathbf{U}^H_+(b; p_{z*},0,\omega_*) + \left(\begin{array}{c} {[\widetilde{\phi}_1]_0}\\ {[\widetilde{\phi}_5]_0} \end{array}\right).
\end{equation}
The jump of one of the components $\widetilde{\phi}_1$ or $\widetilde{\phi}_5$ of the particular
solution at the end of the period can always be compensated by an appropriate choice of the coefficient
$B$ of the nonperiodic solution of the homogeneous equation. The jump of the other component vanishes by
(<ref>) if $H_0(0;\vp_*)\neq 0$, $E_0(0;\vp_*) \neq 0$. Then the constructed solution to the problem
$\mathcal{A}_*\bphi=\vec{F}$ belongs to the class $\mathcal{M}$ and the lemma is proved. If $E_0(0;\vp_*)$ or
$H_0(0;\vp_*)$ is equal to zero, we take the beginning of the period at another point.
§ ACKNOWLEDGEMENTS
This work was supported by RFBR grant 140200624 (M.V.P. and M.S.S.) and by SPbGU grant 11.38.263.2014 (M.V.P).
|
1511.00253
|
In this paper, we construct a sequence of discrete time stochastic processes that converges in probability and in the Skorokhod metric to a COGARCH(p,q) model. The result is useful for the estimation of the continuous model defined for irregularly spaced time series data. The estimation procedure is based on the maximization of a pseudo log-likelihood function and is implemented in the package.
§ INTRODUCTION
The COGARCH(1,1) model has been introduced by <cit.> as a continuous time counterpart of the GARCH(1,1) process. The continuous time model preserves the main features of the GARCH model since the same underlying noise drives the variance and the return processes. For the COGARCH(1,1) case, different methods for its estimation have been proposed. For instance, <cit.> develop a procedure based on the matching of theoretical and empirical moments. <cit.> use an approximation scheme for obtaining estimates of parameters through the maximization of a pseudo-loglikelihood function while <cit.> develop a Markov Chain Monte Carlo estimation
procedure based on the same approximation scheme.The COGARCH(1,1) model has been generalized to the higher order case by <cit.> and <cit.>. Based on our knowledge, this is the only estimation method for higher order models and it is based on the matching of empirical and theoretical moments.In this paper, we construct a sequence of discrete time stochastic processes that converges in probability and in the Skorokhod metric to a COGARCH(p,q) model. Our results generalize the approach in <cit.> for building a sequence of discrete time stochastic processes based on a GARCH(1,1) model that converges in the Skorokhod metric to its continuous counterpart, i.e COGARCH(1,1) model.Results derived for a COGARCH(p,q) model in <cit.> are used in this paper for extending the estimation procedure based on the maximization of the pseudo log-likelihood function. This estimation method is then implemented in the package available on CRAN <cit.>. The outline of the paper is as follows. In Section
<ref> we review some useful properties needed in in Section <ref> where we introduce a discrete version of our process and prove the convergence to the COGARCH(p,q) model using the Skorokhod metric.
§ PRELIMINARIES
In this section we review useful results for obtaining a sequence
of discrete time processes that converges in Skorokhod distance <cit.> to a COGARCH(p,q) model.
The sequence of random vectors $Q_{n}$ is uniformly convergent in
probability to $Q$ if and only if:
\begin{equation}
\underset{\theta\in\Theta}{\sup}\left\Vert Q_{n,\theta}-Q_{\theta}\right\Vert \stackrel{P}{\rightarrow}0,\label{eq:ucp}
\end{equation}
where $\left\Vert \cdot \right\Vert $ is the Euclidean norm.
The definition holds also for any vector norm $\left\Vert \cdot \right\Vert _{A}$
induced by an invertible matrix $A$, i.e. $\left\Vert x\right\Vert _{A}$=$\left\Vert Ax\right\Vert $ where
$A$ is a non singular matrix.
Let $\mathcal{\left\Vert \cdot \right\Vert }$ be
a norm on $\mathcal{R}^{n}$, we introduce induced norm $\left\Vert \cdot \right\Vert_M $
as a function from $\mathcal{R}^{n\times n}$ to $\mathcal{R}_{+}$ defined
\[
\left\Vert A\right\Vert_M :=\underset{\left\Vert x\right\Vert \neq0}{\sup}\frac{\left\Vert Ax\right\Vert}{\left\Vert x\right\Vert}=\underset{\left\Vert z\right\Vert =1}{\sup}\left\Vert Az\right\Vert
\]
where $A\in\mathcal{R}^{q\times q}$.
The induced norm $\left\Vert \cdot \right\Vert_M $ satisfies the following
properties <cit.>:
1) $\left\Vert Ax\right\Vert \leq\left\Vert A\right\Vert_M \left\Vert x\right\Vert$
2) $\left\Vert \alpha A\right\Vert_M \leq\left|\alpha\right|\left\Vert A\right\Vert $
3) $\left\Vert A+B\right\Vert_M \leq\left\Vert A\right\Vert_M +\left\Vert B\right\Vert_M $
4) $\left\Vert AB\right\Vert_M \leq\left\Vert A\right\Vert_M \left\Vert B\right\Vert_M $, where $A\in\mathcal{R}^{q\times q}$, $B\in\mathcal{R}^{q\times q}$ and $\alpha$ is a scalar.
We have also that any induced vector norm satisfies the following inequality:
\begin{equation}
\left\Vert \frac{e^{At}-I}{t}-A\right\Vert_M \leq\frac{e^{\left\Vert At\right\Vert_M }-1-\left\Vert At\right\Vert_M }{\left|t\right|},\ t\in\mathcal{R}.\label{eq:InqInducedVectorNorm}
\end{equation}
Let $\left\Vert \cdot \right\Vert_M $ be the induced vector norm by the norm
$\left\Vert \cdot \right\Vert$ defined on $\mathcal{R}^{n}$, the logarithmic norm $\mu\left(A\right)$ <cit.> is defined as:
\[
\mu\left(A\right):=\underset{t\rightarrow 0^{+}}{\lim}\frac{\left\Vert I+At\right\Vert_M -1}{t}.
\]
For the logarithmic norm the following inequalities hold:
\[
\left\Vert e^{At}\right\Vert_M \leq e^{\mu\left(A\right)t}\leq e^{\left\Vert A\right\Vert_M t}
\]
Let $a_{n}$and $b_{n}$ be sequences of non-negative numbers for
$n=1,\ldots,N$. Define as a linear recursive equation the sequence
\[
\]
with initial condition $y_{0}=c$ where $c$ is a scalar.
If we have that $a_{n}\geq1$ and $b_{n}\geq0$, the sequence $y_{n}$ is
non decreasing with
\[
\]
\[
\]
§ MAIN RESULT
We recall the definition of a COGARCH(p,q) process, introduced in <cit.>, based on the following equations:
\begin{align}
\mbox{d}G_{t} & =\sqrt{V_{t}}\mbox{d}L_{t}\nonumber \\
V_{t} & =\alpha_{0}+\mathbf{a^{\top}}Y_{t-}\nonumber \\
\mbox{d}Y_{t} & =BY_{t-}\mbox{d}t+\mbox{e}\left(\alpha_{0}+\mathbf{a^{\top}}Y_{t-}\right)\mbox{d}\left[L,L\right]^{d}\label{eq:COGARCH_p_q}
\end{align}
where $B\in\mathcal{R}^{q\times q}$ is matrix of the form:
\[
0 & 1 & 0 & \ldots & 0\\
0 & 0 & 1 & \ldots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \ldots & 1\\
-b_{q} & -b_{q-1} & \ldots & \ldots & -b_{1}
\end{array}\right]
\]
and $\mathbf{a}$ and $\mathbf{\mathbf{e}}$ are vectors defined as:
\begin{align*}
\mathbf{a} & =\left[a_{1},\ldots,a_{p},a_{p+1},\ldots,a_{q}\right]^{\top}\\
\mathbf{e} & =\left[0,\ldots,0,1\right]^{\top}
\end{align*}
for $a_{p+1}=\ldots=a_{q}=0$.
As remarked in <cit.> the state process
$Y_{t}$ in a COGARCH(p,q) model is:
\[
Y_{t}=J_{s,t}Y_{s}+K_{s,t}\ s\leq t
\]
where $J_{s,t}\in\mathcal{R}^{q\times q}$ is a random matrix and $K_{s,t}\in\mathcal{R}^{q\times1}$ is a random vector.
In particular, if the driven noise is a Compound Poisson the matrices and vectors
in the state process have an analytical form. Let $N$ be the number
of jumps of a Compound Poisson in the interval $\left[0,t\right]$.
Define $\tau_{N}$ as the time of the last jump in this interval interval
and $Z_{N}:=\Delta L_{\tau_{N}}^{2}=\left(L_{\tau_{N}}-L_{\tau_{N}-}\right)^{2}$
the square of the jump at time $\tau_{N}$. The process $Y_{t}$ can
be rewritten as follows:
\[
Y_{t}=e^{B\left(t-\tau_{N}\right)}Y_{\tau_{N}}\ t\in\left[\tau_{N},\tau_{N+1}\right)
\]
where $Y_{\tau_{N}}$ is the state process at jump time $\tau_{N}$, i.e. the last jump of size less or equal to $t$, defined as:
\begin{equation}
\end{equation}
where the random coefficients $C_{N}$ and $D_{N}$ in (<ref>)
are respectively:
\begin{align}
C_{N} & =\left(I+Z_{N}\mathbf{ea}^{\top}\right)e^{B\Delta\tau_{N}}\nonumber \\
D_{N} & =\alpha_{0}Z_{N}\mathbf{e}.\label{eq:TRUE_D_C}
\end{align}
As in <cit.>, we construct a sequence of discrete
processes that converges to the COGARCH(p,q) model in (<ref>)
by means of the Skorokhod distance <cit.>. For each $n\geq0$
we consider a sequence of natural numbers $N_{n}$ such that $\underset{n\rightarrow+\infty}{\lim}N_{n}=+\infty$
and we obtain a partition of the interval $\left[0,T\right]$ defined
\begin{equation}
0=t_{0,n}\leq t_{1,n}\leq\ldots\leq t_{N_{n},n}=T.\label{eq:Partition}
\end{equation}
The mesh of this partition is:
\[
\Delta t_{n}:=\underset{i=1,\ldots,N_{n}}{\max}\Delta t_{i,n}\ \underset{n\rightarrow+\infty}{\rightarrow}0.
\]
Using the partition in (<ref>), we introduce the process
$G_{i,n}$ as follows:
\begin{equation}
G_{i,n}=G_{i-1,n}+\sqrt{V_{i-1.n}\Delta t_{i,n}}\epsilon_{i,n}\label{eq:G_i_n}
\end{equation}
where the innovations $\epsilon_{i,n}$ are constructed using the first
jump approximation method developed in <cit.> that we review here quickly.
Let $m_{n}$ be a strict positive sequence of real numbers satisfying the conditions:
\begin{align*}
& m_{n}\leq1 \ \forall n \geq 0,\\
& \underset{n\rightarrow+\infty}{\lim}m_{n}=0.
\end{align*}
We require the Lévy measure $\Pi$ to satisfy following property:
\[
\underset{n\rightarrow+\infty}{\lim}\Delta t_{n}\bar{\Pi}^{2}\left(m_{n}\right)=0
\]
where $\bar{\Pi}\left(x\right):=\int_{\left|y\right|>x}\Pi\left(\mbox{d}x\right)$.
We define the stopping time process:
\begin{equation}
\tau_{i,n}:=\inf\left\{ t\in\left[t_{i-1,n},t_{i,n}\right):\left|\Delta L_{t}\right|>m_{n}\right\} \label{eq:StoppingTimeTaui_n}
\end{equation}
and construct a sequence of independent random variables $\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)_{i=1,\ldots,N_{n}}$ with
\[
f\left(x\right)=\frac{\Pi\left(\mbox{d}x\right)}{\bar{\Pi}\left(m_{n}\right)}\left(1-e^{\Delta t_{i,n}\bar{\Pi}\left(m_{n}\right)}\right).
\]
We introduce the innovations $\epsilon_{i,n}$ defined as:
\begin{equation}
\epsilon_{i,n}=\frac{\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}-v_{i,n}}{\eta_{i,n}}\label{eq:InnovEpsilon_i_n}
\end{equation}
where $v_{i,n}$ and $\eta_{i,n}$ are respectively the mean and the variance of $\epsilon_{i,n}$.
The variance process $V_{t}$ in (<ref>) is approximated by the process $V_{i,n}$ as:
\begin{equation}
\end{equation}
where $Y_{i,n}$ is given by:
\begin{equation}
\end{equation}
with coefficients:
\begin{align}
C_{i,n} & =\left(I+\epsilon_{i,n}^{2}\Delta t_{i,n}\mathbf{ea}^{\top}\right)e^{B\Delta t_{i,n}}\nonumber \\
D_{i,n} & =\alpha_{0}\epsilon_{i,n}^{2}\Delta t_{i,n}\mathbf{e}.\label{eq:C_i_nAndD_i_n}
\end{align}
The couple $\left(G_{i,n},V_{i,n}\right)$ converges to the couple $\left(G_{t},V_{t}\right)$
in the Skorokhod distance. The Skorokhod distance between
two processes $U,V$ defined on $D^{d}\left[0,T\right]$, i.e. space
of càdlàg $\mathcal{R}^{d}$ stochastic processes on $\left[0,T\right]$,
\[
\rho\left(U,V\right):=\underset{\lambda\in\Lambda}{\inf}\left\{ \underset{0\leq t\leq T}{\sup}\left\Vert U_{t}-V_{\lambda\left(t\right)}\right\Vert +\underset{0\leq t\leq T}{\sup}\left|\lambda\left(t\right)-t\right|\right\}
\]
where $\Lambda$ is a set of increasing continuous functions with $\lambda\left(0\right)=0$
and $\lambda\left(T\right)=T$.First of all we need the following auxiliar result.
Let $N_{n}\left(t\right)$ be a counting process defined as:
\[
N_{n}\left(t\right):=\#\left\{ i\in\mathcal{N}:\ \tau_{i,n}^{\star}\leq t\right\}
\]
where $t\leq T$, $N_{n}\left(0\right)=0$, $N_{n}\left(T\right)=N_{n}$
and $\tau_{i,n}^{\star}=\min\left\{ \tau_{i,n},t_{i,n}\right\} $
with $\tau_{i,n}$ and $t_{i,n}$ in (<ref>) and (<ref>) respectively. Let $L_{t}$ be a Compound Poisson with finite second moment, the
positive process $H_{n}\left(t\right)$ defined as:
\[
\]
\[
C_{k,n}^{\star}:=\left(1+\epsilon_{k,n}^{2}\Delta t_{k,n}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}
\]
converges uniformly in probability on a compact interval $\left[0,T\right]$
(hereafter ucp) to the positive process $\tilde{H}_{n}\left(t\right)$
\[
\tilde{H}_{n}\left(t\right):=\prod_{k=1}^{N_{n}\left(t\right)}\tilde{C}_{k,n}
\]
\[
\tilde{C}_{k,n}:=\left(1+\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}
\]
\[
\underset{t\in\left[0,T\right]}{\sup}\left|H_{n}\left(t\right)-\tilde{H}_{n}\left(t\right)\right|\stackrel{p}{\rightarrow}0.
\]
For each fixed $n$, $\tilde{H}_{n}\left(t\right)$ is a non decreasing
striclty positive process in the compact interval $\left[0,T\right]$
such that $\forall t\in\left[0,T\right]$:
\[
\tilde{H}_{n}\left(t\right)\leq\tilde{H}_{n}\left(T\right)\leq e^{\left\Vert B\right\Vert _{M}T+\sum_{0\leq s\leq T}\ln\left(1+\Delta L_{s}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}
\]
We start from
\begin{align*}
\underset{t\in\left[0,T\right]}{\sup}\left|H_{n}\left(t\right)-\tilde{H}_{n}\left(t\right)\right| & =\underset{t\in\left[0,T\right]}{\sup}\left|\prod_{k=1}^{N_{n}\left(t\right)}C_{k,n}^{\star}-\prod_{k=1}^{N_{n}\left(t\right)}\tilde{C}_{k,n}\right|\\
& \leq e^{\left\Vert B\right\Vert _{M}T}\underset{t\in\left[0,T\right]}{\sup}\left|\prod_{k=1}^{N_{n}\left(t\right)}\left(1+\epsilon_{k,n}^{2}\Delta t_{k,n}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)-\prod_{k=1}^{N_{n}\left(t\right)}\left(1+\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)\right|\\
& =e^{\left\Vert B\right\Vert _{M}T}\underset{t\in\left[0,T\right]}{\sup}\left|e^{\sum_{k=1}^{N_{n}\left(t\right)}\ln\left(1+\epsilon_{k,n}^{2}\Delta t_{k,n}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}-e^{\sum_{k=1}^{N_{n}\left(t\right)}\left(1+\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}\right|.
\end{align*}
Observe that
\begin{align*}
L_{n} & :=\underset{t\in\left[0,T\right]}{\sup}\left|\sum_{k=1}^{N_{n}\left(t\right)}\ln\left(1+\epsilon_{k,n}^{2}\Delta t_{k,n}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)-\sum_{k=1}^{N_{n}\left(t\right)}\left(1+\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)\right|\\
& \leq\underset{t\in\left[0,T\right]}{\sup}\left|\sum_{k=1}^{N_{n}\left(t\right)}\left(\epsilon_{k,n}^{2}\Delta t_{k,n}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}-\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)\right|\\
& \leq\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\underset{t\in\left[0,T\right]}{\sup}\sum_{k=1}^{N_{n}\left(t\right)}\left|\left(\epsilon_{k,n}^{2}\Delta t_{k,n}-\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\right)\right|.
\end{align*}
As shown in <cit.>, we have that
\[
\underset{t\in\left[0,T\right]}{\sup}\sum_{k=1}^{N_{n}\left(t\right)}\left|\left(\epsilon_{k,n}^{2}\Delta t_{k,n}-\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\right)\right|\stackrel{p}{\rightarrow}0,
\]
that implies
\begin{equation}
\underset{t\in\left[0,T\right]}{\sup}\left|\sum_{k=1}^{N_{n}\left(t\right)}\ln\left(1+\epsilon_{k,n}^{2}\Delta t_{k,n}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)-\sum_{k=1}^{N_{n}\left(t\right)}\left(1+\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)\right|\stackrel{p}{\rightarrow0}.
\label{llll}
\end{equation}
Using result in (<ref>), we have
\[
\underset{t\in\left[0,T\right]}{\sup}\left|e^{\sum_{k=1}^{N_{n}\left(t\right)}\ln\left(1+\epsilon_{k,n}^{2}\Delta t_{k,n}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}-e^{\sum_{k=1}^{N_{n}\left(t\right)}\left(1+\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}\right|\stackrel{p}{\rightarrow0}.
\]
$\tilde{H}_{n}\left(t\right)$ is a non decreasing strictly
positive process since is a product of terms $\tilde{C}_{k,n}\geq1$
a.s. and if $s>t$ then $\tilde{H}_{n}\left(s\right)$ has at least
the same terms as in $\tilde{H}_{n}\left(t\right)$
. Moreover
\[
\tilde{H}_{n}\left(T\right)=e^{\left\Vert B\right\Vert _{M}T+\sum_{k=1}^{N_{n}}\ln\left(1+\mathbf{1}_{\tau_{k,n}<+\infty}\Delta L_{\tau_{k,n}}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}\leq e^{\left\Vert B\right\Vert _{M}T+\sum_{0\leq s\leq T}\ln\left(1+\Delta L_{s}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}
\]
since $\Delta L_{s}^{2}=\Delta L_{s}^{2}\mathbf{1}_{\left|\Delta L_{s}\right|\geq m_{n}}+\Delta L_{s}^{2}\mathbf{1}_{\left|\Delta L_{s}\right|<m_{n}}$.
We observe, from Theorem <ref>, that
\begin{equation}
H_{n}\left(t\right)\stackrel{ucp}{\rightarrow}\tilde{H}_{n}\left(t\right)\leq e^{\left\Vert B\right\Vert _{M}T+\sum_{0\leq s\leq T}\ln\left(1+\Delta L_{s}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}
\label{uffa}
\end{equation}
on an interval $\left[0,T\right]$. Moreover, the term on the right hand
side of the inequality in (<ref>) is bounded almost surely on the compact interval $\left[0, T\right]$
since $L_{t}$ is a Compound Poisson process.
The following theorem is established for the Compound Poisson driven noise case.
Let $L_{t}$ be a Compound Poisson process
with $E\left(L_{1}^{2}\right)<+\infty$. The Skorokhod distance computed on the
processes $\left(G_{t},V_{t}\right)_{t\geq0}$ and their discretized
version $\left(G_{i,n},V_{i,n}\right)_{i=1,\ldots,N_{n}}$ converges
in probability to zero, i.e.:
\[
\rho\left(\left(G_{i,n},V_{i,n}\right)_{i=1,\ldots,N_{n}},\left(G_{t},V_{t}\right)_{t\geq0}\right)\stackrel{P}{\rightarrow}0\ as\ n\rightarrow+\infty.
\]
The proof follows the same steps as in <cit.>
* Approximation procedure for the underlying process.
* Approximation procedure for the variance process.
* Approximation procedure for the COGARCH(p,q) model.
* Convergence of the pair in the Skorokhod distance.
Steps 1, 2, 4 are exactly the same as in <cit.>. To prove that
the discrete variance process $V_{i,n}$ converges ucp
on a compact time interval to the continuous-time process $V_{t}$ we
first need to show that $Y_{i,n}\stackrel{ucp}{\rightarrow}Y_{t}$. This
result is achieved through intermediate steps illustrated below. We introduce the
counting process $N_{n}\left(t\right)$ defined as:
\begin{equation}
N_{n}\left(t\right):=\#\left\{ i\in\mathcal{N}:\tau_{i,n}^{\star}\leq t\right\}
\label{eq:CountProcess}
\end{equation}
with $t\leq T$, $N_{n}\left(0\right)=0$ and $\tau_{i,n}^{\star}=\min\left\{ \tau_{i,n},t_{i,n}\right\} .$
$N_{n}\left(t\right)$ increases by 1 in each subinterval $\left(t_{i-1,n},t_{i,n}\right]$, $i = 1, 2, \ldots, n$, at
the first time the jump is of magnitude greater or equal to $m_{n}$ or
at $t_{i,n}$ if that jump does not occur.
Using the process $N_{n}\left(t\right)$ in (<ref>)
we construct the time process $\Gamma_{t,n}$ as:
\begin{equation}
\Gamma_{t,n}=\sum_{i=1}^{N_{n}\left(t\right)}\Delta t_{i,n}.\label{eq:Gamma_t_n}
\end{equation}
Now we want to show that the piecewise constant process $Y_{t,n}:=Y_{i,n}$ with
$t\in\left[t_{i,n},t_{i+1,n}\right)$ converges in ucp to the process
$\bar{Y}_{t,n}:=e^{B\left(t-\Gamma_{t,n}\right)}Y_{i,n}$ i.e.:
\begin{align*}
\underset{0\leq t\leq T}{\sup} & \left\Vert Y_{t,n}-\bar{Y}_{t,n}\right\Vert \stackrel{P}{\rightarrow}0.
\end{align*}
For each $t\in\left[0,T\right],$ we have:
\begin{align*}
\left\Vert Y_{t,n}-\bar{Y}_{t,n}\right\Vert & =\left\Vert e^{B\left(t-\Gamma_{t,n}\right)}Y_{i,n}-Y_{i,n}\right\Vert \\
& \leq\left\Vert e^{B\left(t-\Gamma_{t,n}\right)}-I\right\Vert _{M}\left\Vert Y_{i,n}\right\Vert \\
& =\left\Vert e^{B\left(t-\Gamma_{t,n}\right)}-I-B\left(t-\Gamma_{t,n}\right)+B\left(t-\Gamma_{t,n}\right)\right\Vert _{M}\left\Vert Y_{i,n}\right\Vert \\
& \leq\left(\left\Vert e^{B\left(t-\Gamma_{t,n}\right)}-I-B\left(t-\Gamma_{t,n}\right)\right\Vert _{M}+\left\Vert B\left(t-\Gamma_{t,n}\right)\right\Vert \right)\left\Vert Y_{i,n}\right\Vert
\end{align*}
using the inequality in (<ref>), we get:
\begin{align}
\left\Vert Y_{t,n}-\bar{Y}_{t,n}\right\Vert & \leq\left(e^{\left\Vert B\left(t-\Gamma_{t,n}\right)\right\Vert _{M}}-1\right)\left\Vert Y_{i,n}\right\Vert \nonumber \\
& \leq\left(e^{\left\Vert B\right\Vert _{M}\Delta t_{n}}-1\right)\left\Vert Y_{i,n}\right\Vert \label{eq:IneqA}
\end{align}
Since by construction $Y_{t,n}=Y_{i,n}$ with $t \in \left[t_{i,n}, t_{i+1,n}\right)$ and $Y_{t,n}$ has càdlàg paths,
it follows that $\underset{t\in\left[0,T\right]}{\sup}\left\Vert Y_{t,n}\right\Vert $
is almost surely finite and
\begin{align*}
\underset{t\in\left[0,T\right]}{\sup}\left\Vert Y_{t,n}-\bar{Y}_{t,n}\right\Vert & \leq\left(e^{\left\Vert B\right\Vert _{M}\Delta t_{n}}-1\right)\underset{t\in\left[0,T\right]}{\sup}\left\Vert Y_{t,n}\right\Vert \stackrel{P}{\rightarrow}0
\end{align*}
as $n\rightarrow+\infty$.
The next step is to show the convergence ucp of $\bar{Y}_{t,n}$
to $\tilde{Y}_{t,n}$ where the last process is defined as:
\begin{equation}
\tilde{Y}_{t,n}=e^{B\left(t-\Gamma_{t,n}\right)}\tilde{Y}_{i,n}\label{eq:Y_tilde_t_n}
\end{equation}
\begin{equation}
\tilde{Y}_{i,n}=\tilde{C}_{i,n}\tilde{Y}_{i-1,n}+\tilde{D}_{i,n}\label{eq:Y_tilde_i_n}
\end{equation}
where the random matrix $\tilde{C}_{i,n}$ and the random vector $\tilde{D}_{i,n}$
are respectively:
\begin{align}
\tilde{C}_{i,n} & =\left(I+\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}\mathbf{ea}^{\top}\right)e^{B\Delta t_{i,n}}\nonumber \\
\tilde{D}_{i,n} & =\alpha_{0}\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}\mathbf{e}.\label{eq:Ctilde_Dtilde}
\end{align}
We consider
\begin{equation}
\underset{t\in\left[0,T\right]}{\sup}\left\Vert \tilde{Y}_{t,n}-\bar{Y}_{t,n}\right\Vert \leq e^{\left\Vert B\right\Vert _{M}\Delta t_{n}}\underset{i=1,\ldots,N_{n}}{\sup}\left\Vert \tilde{Y}_{i,n}-Y_{i,n}\right\Vert \label{eq:SecondSup}
\end{equation}
and observe that, for $i=1,\ldots,N_{n}$, we have:
\begin{align}
\left\Vert \tilde{Y}_{i,n}-Y_{i,n}\right\Vert & \leq\left\Vert \tilde{C}_{i,n}\tilde{Y}_{i-1,n}-C_{i,n}Y_{i-1,n}\right\Vert +\left\Vert \tilde{D}_{i,n}-D_{i,n}\right\Vert. \label{eq:absdiffY_i_n_tilde_Y_i_n}
\end{align}
We analyze the second term in (<ref>) and get:
\begin{align}
\left\Vert \tilde{D}_{i,n}-D_{i,n}\right\Vert & =\left\Vert \alpha_{0}\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}\mathbf{e}-\alpha_{0}\epsilon_{i,n}^{2}\Delta t_{i,n}\mathbf{e}\right\Vert \nonumber \\
& \leq\left|\alpha_{0}\right|\left|\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}-\epsilon_{i,n}^{2}\Delta t_{i,n}\right|.\label{eq:diffD_i_n_tildeD_i_n}
\end{align}
The first term in (<ref>) can be bounded
by adding and subtracting the quantity $C_{i,n}\tilde{Y}_{i-1,n}$:
\begin{align}
\left\Vert \tilde{C}_{i,n}\tilde{Y}_{i-1,n}-C_{i,n}Y_{i-1,n}\right\Vert & =\left\Vert \tilde{C}_{i,n}\tilde{Y}_{i-1,n}-C_{i,n}\tilde{Y}_{i-1,n}+C_{i,n}\tilde{Y}_{i-1,n}-C_{i,n}Y_{i-1,n}\right\Vert \nonumber \\
& \leq\left\Vert \tilde{C}_{i,n}-C_{i,n}\right\Vert _{M}\left\Vert \tilde{Y}_{i-1,n}\right\Vert +\left\Vert C_{i,n}\right\Vert _{M}\left\Vert \tilde{Y}_{i-1,n}-Y_{i-1,n}\right\Vert \nonumber \\
& \leq\left\Vert \left[\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}-\epsilon_{i,n}^{2}\Delta t_{i,n}\right]\mathbf{ea}^{\top}e^{B\Delta t_{i,n}}\right\Vert _{M}\left\Vert \tilde{Y}_{i-1,n}\right\Vert \nonumber \\
& +\left\Vert C_{i,n}\right\Vert _{M}\left\Vert \tilde{Y}_{i-1,n}-Y_{i-1,n}\right\Vert \nonumber \\
& \leq\left|\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}-\epsilon_{i,n}^{2}\Delta t_{i,n}\right|\left\Vert \mathbf{ea}^{\top}\right\Vert _{M}e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}\left\Vert \tilde{Y}_{i-1,n}\right\Vert \nonumber \\
& +\left\Vert C_{i,n}\right\Vert _{M}\left\Vert \tilde{Y}_{i-1,n}-Y_{i-1,n}\right\Vert \label{eq:DiffCtildeC}
\end{align}
Substituting (<ref>) and (<ref>)
into (<ref>) we have:
\begin{align}
\left\Vert \tilde{Y}_{i,n}-Y_{i,n}\right\Vert & \leq\left\Vert C_{i,n}\right\Vert _{M}\left\Vert \tilde{Y}_{i-1,n}-Y_{i-1,n}\right\Vert \nonumber \\
& +\left|\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}-\epsilon_{i,n}^{2}\Delta t_{i,n}\right|\left(\left|\alpha_{0}\right|+\left\Vert \mathbf{ea}^{\top}\right\Vert _{M}e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}\left\Vert \tilde{Y}_{i-1,n}\right\Vert \right)\label{eq:DiffYtildeY2}
\end{align}
Since a.s.:
\begin{equation}
\left\Vert C_{i,n}\right\Vert _{M}\leq\left(1+\epsilon_{i,n}^{2}\Delta t_{i,n}\left\Vert \mathbf{ea}^{\top}\right\Vert _{M}\right)e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}:=C_{i,n}^{\star}\geq1\label{eq:CoeffGreaterThan1}
\end{equation}
and defining
\begin{equation}
K_{i-1,n}:=\left|\alpha_{0}\right|+\left\Vert \mathbf{ea}^{\top}\right\Vert _{M}e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}\left\Vert \tilde{Y}_{i-1,n}\right\Vert \label{eq:K_i_n}
\end{equation}
we have:
\begin{align}
\left\Vert \tilde{Y}_{i,n}-Y_{i,n}\right\Vert & \leq C_{i,n}^{\star}\left\Vert \tilde{Y}_{i-1,n}-Y_{i-1,n}\right\Vert \nonumber \\
& +\left|\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}-\epsilon_{i,n}^{2}\Delta t_{i,n}\right|K_{i-1,n}.\label{eq:DiffYtildeY3}
\end{align}
The right hand side in (<ref>) is a linear recursive
equation with random coefficients and condition (<ref>)
implies that:
\begin{align}
\underset{i=1,\ldots,N_{n}}{\sup}\left\Vert \tilde{Y}_{i,n}-Y_{i,n}\right\Vert & \leq\left[\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star}\right]\left\Vert \tilde{Y}_{0,n}-Y_{0,n}\right\Vert +\left|\left(\mathbf{1}_{\tau_{N_{n},n}<+\infty}\Delta L_{\tau_{N_{n},n}}\right)^{2}-\epsilon_{N_{n},n}^{2}\Delta t_{N_{n},n}\right|K_{N_{n}-1,n}\nonumber \\
& +\sum_{i=1}^{N_{n}-1}\left[\prod_{h=1}^{i}C_{N_{n}+1-h,n}^{\star}\right]\left|\left(\mathbf{1}_{\tau_{N_{n}-i,n}<+\infty}\Delta L_{\tau_{N_{n}-i,n}}\right)^{2}-\epsilon_{N_{n}-i,n}^{2}\Delta t_{N_{n}-i,n}\right|K_{N_{n}-1-i,n}.\label{eq:CrucialPoint1}
\end{align}
The term:
\[
\left[\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star}\right]\left\Vert \tilde{Y}_{0,n}-Y_{0,n}\right\Vert \geq0\ n\geq 1
\]
\[
E\left[\left(\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star}\right)\left\Vert \tilde{Y}_{0,n}-Y_{0,n}\right\Vert \right]=E\left[\left(\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star}\right)\right]\left\Vert \tilde{Y}_{0,n}-Y_{0,n}\right\Vert
\]
since $\tilde{Y}_{0,n}=Y_{0,n}$ we have:
\begin{equation}
E\left[\left(\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star}\right)\right]\left\Vert \tilde{Y}_{0,n}-Y_{0,n}\right\Vert =0\Rightarrow\left(\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star}\right)\left\Vert \tilde{Y}_{0,n}-Y_{0,n}\right\Vert =0\ \text{a.s.}\label{eq:PROVA}
\end{equation}
Condition (<ref>) becomes:
\begin{align}
\underset{i=1,\ldots,N_{n}}{\sup}\left\Vert \tilde{Y}_{i,n}-Y_{i,n}\right\Vert & \leq\left|\left(\mathbf{1}_{\tau_{N_{n},n}<+\infty}\Delta L_{\tau_{N_{n},n}}\right)^{2}-\epsilon_{N_{n},n}^{2}\Delta t_{N_{n},n}\right|K_{N_{n}-1,n}\nonumber \\
& +\sum_{i=1}^{N_{n}-1}\left[\prod_{h=1}^{i}C_{N_{n}+1-h,n}^{\star}\right]\left|\left(\mathbf{1}_{\tau_{N_{n}-i,n}<+\infty}\Delta L_{\tau_{N_{n}-i,n}}\right)^{2}-\epsilon_{N_{n}-i,n}^{2}\Delta t_{N_{n}-i,n}\right|K_{N_{n}-1-i,n}.\label{eq:CrucialPoint2}
\end{align}
\begin{align*}
Q_{n} & :=\left|\left(\mathbf{1}_{\tau_{N_{n},n}<+\infty}\Delta L_{\tau_{N_{n},n}}\right)^{2}-\epsilon_{N_{n},n}^{2}\Delta t_{N_{n},n}\right|K_{N_{n}-1,n}\\
& +\sum_{i=1}^{N_{n}-1}\left[\prod_{h=1}^{i}C_{N_{n}+1-h,n}^{\star}\right]\left|\left(\mathbf{1}_{\tau_{N_{n}-i,n}<+\infty}\Delta L_{\tau_{N_{n}-i,n}}\right)^{2}-\epsilon_{N_{n}-i,n}^{2}\Delta t_{N_{n}-i,n}\right|K_{N_{n}-1-i,n}.
\end{align*}
we observe that $Q_{n}$ can be bounded. Indeed, $\forall i=1,\ldots,N_{n}$:
\[
\prod_{h=1}^{i}C_{N_{n}+1-h,n}^{\star}\leq\prod_{h=1}^{N_{n}}C_{N_{n}+1-h,n}^{\star}
\]
and, from Theorem <ref>, the quantity $\prod_{h=1}^{N_{n}}C_{N_{n}+1-h,n}^{\star}$ converges in probability to a non negative r.v. that is a.s. bounded by:
\[
e^{\left\Vert B\right\Vert _{M}T+\sum_{0\leq s\leq T}\ln\left(1+\Delta L_{s}^{2}\left\Vert \mathbf{ea^{\top}}\right\Vert _{M}\right)}.
\]
Even $\underset{i=1,\ldots,N_{n}}{\sup}K_{i,n}$ is bounded a.s. $\forall n$. Consequently we have:
\begin{equation}
Q_{n} \leq \left[\prod_{h=1}^{N_{n}}C_{N_{n}+1-h,n}^{\star}\right]\left[\underset{i=1,\ldots,N_{n}}{\sup}K_{i,n}\right]\sum_{i=1}^{N_{n}}\left|\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}-\epsilon_{i,n}^{2}\Delta t_{i,n}\right|\label{eq:Prova2}
\end{equation}
Since $\underset{n\rightarrow +\infty}{\lim} \ \underset{i=1,\ldots,N_{n}}{\sup}K_{i,n}=M<+\infty$
a.s. and, as shown in <cit.>,
\[
\underset{t\in\left[0,T\right]}{\sup}{\sum_{i=1}^{N_{n}\left(t\right)}\left|\left(\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}\right)^{2}-\epsilon_{i,n}^{2}\Delta t_{i,n}\right|}\stackrel{p}{\rightarrow}0
\]
as $n\rightarrow + \infty$, then $Q_{n}\stackrel{p}{\rightarrow}0$
that implies $\bar{Y}_{t,n}\stackrel{ucp}{\rightarrow}\tilde{Y}_{t,n}$.
We observe that, since the driven noise is a Compound Poisson, we
have only a finite number of jumps in a compact interval $\left[0,T\right].$
We indicate with $\tau_{k}$ the time of the $k$-th jump. Since the
irregular grid becomes finer as $n$ increases and satisfies
the following two conditions:
\begin{align*}
\Delta t_{n} & :=\underset{i=1,\ldots,N_{n}}{\max}\Delta t_{i,n}\underset{n\rightarrow+\infty}{\rightarrow}0\\
T & =\sum_{i=1}^{N_{n}}\Delta t_{i,n},
\end{align*}
then exists $n^{\star}$ such that for $n\geq n^{\star}$, all jump times $\tau_{k}\in\left\{ t_{0,n},t_{1,n},\ldots,t_{N_{n},n}\right\}$. The
COGARCH(p,q) state process $Y_{t}$ in (<ref>) can
be defined equivalently $\forall\ n\geq n^{\star}$ as:
\begin{equation}
\end{equation}
with coefficients $C_{t_{i,n}}$ and $D_{t_{i,n}}$ defined as:
\begin{align*}
C_{t_{i,n}} & =\left(I+\Delta L_{t_{i,n}}^{2}\mathbf{ea}^{\top}\right)e^{B\Delta t_{i,n}}\\
D_{t_{i,n}} & =\alpha_{0}\Delta L_{t_{i,n}}^{2}\mathbf{e}.
\end{align*}
To show the ucp convergence of process $\tilde{Y}_{t,n}$ to $Y_{t}$,
we start observing that:
\begin{align}
\underset{t\in\left[0,T\right]}{\sup}\left\Vert Y_{t}-\tilde{Y}_{t,n}\right\Vert & =\underset{t\in\left[0,T\right]}{\sup}\left\Vert e^{B\left(t-\Gamma_{t,n}\right)}\left(Y_{t_{i,n}}-\tilde{Y}_{i,n}\right)\right\Vert \nonumber \\
& \leq e^{\left\Vert B\right\Vert _{M}T}\underset{i=1,\ldots,N_{n}}{\sup}\left\Vert \left(Y_{t_{i,n}}-\tilde{Y}_{i,n}\right)\right\Vert. \label{eq:LastInequalities}
\end{align}
We work on $\underset{i=1,\ldots,N_{n}}{\sup}\left\Vert \left(Y_{t_{i,n}}-\tilde{Y}_{i,n}\right)\right\Vert $
and for $i=1,\ldots,N_{n}$ and for fixed $n$ we have:
\begin{align}
\left\Vert \left(Y_{t_{i,n}}-\tilde{Y}_{i,n}\right)\right\Vert & \leq\left\Vert \left(C_{t_{i,n}}Y_{t_{i-1,n}}-\tilde{C}_{i,n}\tilde{Y}_{i-1,n}\right)\right\Vert +\left\Vert D_{t_{i,n}}-\tilde{D}_{i,n}\right\Vert .\label{eq:LastIneq2}
\end{align}
The term $\left\Vert D_{t_{i,n}}-\tilde{D}_{i,n}\right\Vert $ in (<ref>)
is bounded as follows:
\begin{equation}
\left\Vert D_{t_{i,n}}-\tilde{D}_{i,n}\right\Vert \leq\left|\alpha_{0}\right|\left|\Delta L_{t_{i,n}}^{2}-\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}^{2}\right|.\label{eq:DiffD_t_i_mDtilde}
\end{equation}
\begin{equation}
\Delta L_{t_{i,n}}^{2}:=\Delta L_{t_{i,n}}^{2}\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|\geq m_{n}}+\Delta L_{t_{i,n}}^{2}\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|<m_{n}},
\label{daric}
\end{equation}
the inequality in (<ref>) becomes:
\begin{align}
\left\Vert D_{t_{i,n}}-\tilde{D}_{i,n}\right\Vert & \leq\left|\alpha_{0}\right|\left|\Delta L_{t_{i,n}}^{2}\mathbf{1}_{0<\left|\Delta L_{t_{i,n}}\right|<m_{n}}\right|\nonumber \\
& \leq m_{n}\left|\alpha_{0}\right|\left|\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|>0}\right|.\label{eq:ABC}
\end{align}
Inserting (<ref>) into (<ref>), we have:
\begin{align}
\left\Vert \left(Y_{t_{i,n}}-\tilde{Y}_{i,n}\right)\right\Vert & \leq\left\Vert \left(C_{t_{i,n}}Y_{t_{i-1,n}}-\tilde{C}_{i,n}\tilde{Y}_{i-1,n}\right)\right\Vert +m_{n}\left|\alpha_{0}\right|\left|\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|>0}\right|\label{eq:LastIneq3}
\end{align}
We add and subtract the term $C_{t_{i,n}}\tilde{Y}_{i-1,n}$ into the
quantity $\left\Vert C_{t_{i,n}}Y_{t_{i-1,n}}-\tilde{C}_{i,n}\tilde{Y}_{i-1,n} \right\Vert $.
By exploiting the triangular inequality we obtain:
\begin{align}
\left\Vert C_{t_{i,n}}Y_{t_{i-1,n}}-\tilde{C}_{i,n}\tilde{Y}_{i-1,n} \right\Vert & \leq\left\Vert C_{t_{i,n}}Y_{t_{i-1,n}}-C_{t_{i,n}}\tilde{Y}_{i-1,n}\right\Vert +\left\Vert C_{t_{i,n}}\tilde{Y}_{i-1,n}-\tilde{C}_{i,n}\tilde{Y}_{i-1,n}\right\Vert \nonumber \\
& \leq\left\Vert C_{t_{i,n}}\right\Vert _{M}\left\Vert Y_{t_{i-1,n}}-\tilde{Y}_{i-1,n}\right\Vert +\left\Vert C_{t_{i,n}}-\tilde{C}_{i,n}\right\Vert _{M}\left\Vert \tilde{Y}_{i-1,n}\right\Vert \nonumber \\
& \leq\left\Vert C_{t_{i,n}}\right\Vert _{M}\left\Vert Y_{t_{i-1,n}}-\tilde{Y}_{i-1,n}\right\Vert \nonumber \\
& +\left|\Delta L_{t_{i,n}}^{2}-\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}^{2}\right|\left\Vert \mathbf{ea}^{\top}\right\Vert _{M}e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}\left\Vert \tilde{Y}_{i-1,n}\right\Vert .\label{eq:CDE}
\end{align}
\[
C_{t_{i,n}}^{\star\star}:=\left(1+\Delta L_{t_{i,n}}^{2}\left\Vert \mathbf{ea}^{\top}\right\Vert _{M}\right)e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}\geq\left\Vert C_{t_{i,n}}\right\Vert _{M},
\]
substituting (<ref>) into (<ref>) and using the same arguments as in (<ref>) and (<ref>), we obtain:
\begin{align*}
\left\Vert Y_{t_{i,n}}-\tilde{Y}_{i,n}\right\Vert & \leq C_{t_{i,n}}^{\star\star}\left\Vert Y_{t_{i-1,n}}-\tilde{Y}_{i-1,n}\right\Vert +m_{n}\left|\alpha_{0}\right|\left|\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|>0}\right|\\
& +\left|\Delta L_{t_{i,n}}^{2}-\mathbf{1}_{\tau_{i,n}<+\infty}\Delta L_{\tau_{i,n}}^{2}\right|\left\Vert \mathbf{ea}^{\top}\right\Vert _{M}e^{\left\Vert B\right\Vert _{M}\Delta t_{i,n}}\left\Vert \tilde{Y}_{i-1,n}\right\Vert .
\end{align*}
Using $K_{i,n}$ in (<ref>), we have:
\begin{align}
\left\Vert Y_{t_{i,n}}-\tilde{Y}_{i,n}\right\Vert & \leq C_{t_{i,n}}^{\star\star}\left\Vert Y_{t_{i-1,n}}-\tilde{Y}_{i-1,n}\right\Vert +m_{n}K_{i-1,n}\left|\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|>0}\right|.\label{eq:LastIneq4}
\end{align}
We introduce a stochastic recurrence equation on the grid $\left\{ t_{i,n}\right\} _{i=0,\ldots,N_{n}}$ defined as
\[
\zeta_{i,n}=C_{t_{i,n}}^{\star\star}\zeta_{i-1,n}+m_{n}K_{i-1,n}\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|>0}
\]
with initial condition $\zeta_{0,n}:=\left\Vert Y_{t_{0,n}}-\tilde{Y}_{0,n}\right\Vert =0$
a.s.. Since $\forall i\ C_{i,n}^{\star\star}\geq1$ and $m_{n}K_{i-1,n}^{\star}\mathbf{1}_{\left|\Delta L_{t_{i,n}}\right|>0}\geq0$
a.s., $\zeta_{i,n}$ is a non decreasing process that is an upper
bound for $\left\Vert Y_{t_{i,n}}-\tilde{Y}_{i,n}\right\Vert $ for each fixed $i$ then:
\begin{align}
\underset{i=1,\ldots,N_{n}}{\sup}\left\Vert Y_{t_{i,n}}-\tilde{Y}_{i,n}\right\Vert & \leq\left[\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star\star}\right]\left\Vert Y_{t_{0,n}}-\tilde{Y}_{0,n}\right\Vert \nonumber \\
& +m_{n}\left\{ \sum_{i=1}^{N_{n}-1}\left[\prod_{h=1}^{i}C_{N_{n}+i-h,n}^{\star\star}\right]\mathbf{1}_{\left|\Delta L_{t_{N_{n}-i,n}}\right|>0}K_{N_{n}-1-i,n}+\mathbf{1}_{\left|\Delta L_{t_{N_{n},n}}\right|>0}K_{N_{n}-1,n}\right\} \label{eq:LastIneq5}
\end{align}
The right-hand side in (<ref>) is non-negative as a summation
of non-negative terms. We split it into two parts:
\begin{align*}
G_{n} & :=\left[\prod_{i=0}^{N_{n}-1}C_{N_{n}-i,n}^{\star\star}\right]\left\Vert Y_{t_{0,n}}-\tilde{Y}_{0,n}\right\Vert \\
W_{n} & :=m_{n}\left\{ \sum_{i=1}^{N_{n}-1}\left[\prod_{h=1}^{i}C_{N_{n}+i-h,n}^{\star\star}\right]\mathbf{1}_{\left|\Delta L_{t_{N_{n}-i,n}}\right|>0}K_{N_{n}-1-i,n}+\mathbf{1}_{\left|\Delta L_{t_{N_{n},n}}\right|>0}K_{N_{n}-1,n}\right\}.
\end{align*}
Using the same arguments as in (<ref>), we can say that:
\[
G_{n}=0\ \texttt{a.s.}\ \forall n\geq0
\]
\[
\]
since for $n\rightarrow+\infty$, the quantity
\[
\sum_{i=1}^{N_{n}-1}\left[\prod_{h=1}^{i}C_{N_{n}+i-h,n}^{\star\star}\right]\mathbf{1}_{\left|\Delta L_{t_{N_{n}-i,n}}\right|>0}K_{N_{n}-1-i,n}+\mathbf{1}_{\left|\Delta L_{t_{N_{n},n}}\right|>0}K_{N_{n}-1,n}
\]
is composed by a finite number of terms and then it finite a.s. for
the same arguments in (<ref>). In conclusion we have:
\[
\underset{i=1,\ldots,N_{n}}{\sup}\left\Vert Y_{t_{i,n}}-\tilde{Y}_{i,n}\right\Vert \leq G_{n}+W_{n}\underset{n\rightarrow+\infty}{\rightarrow}0
\]
that implies
\begin{equation}
\end{equation}
where $Y_{t,n}$ is the constant piecewise process associated to
the process $Y_{i,n}$ defined in (<ref>). From (<ref>)
we obtain the ucp convergence of process $V_{i,n}$ to the COGARCH(p,q)
variance process $V_{t}$. The remaing part of the proof follows the
same steps as in <cit.>
The result can be generalized to any COGARCH(p,q) model driven by a finite variation Lévy process since, as shown in <cit.>, a COGARCH(p,q) driven by a general Lévy can by approximated by the same COGARCH(p,q) process driven by a Compound Poisson. Then using the triangular inequality, the discrete process $\left(G_{i,n},V_{i,n}\right)$ converges in the Skorokhod metric and in probability to any COGARCH(p,q) model.
§ MAXIMUM PSEUDO-LOGLIKELIHOOD ESTIMATION FOR THE COGARCH(P,Q) PROCESS
In this Section we show how to extend the maximum pseudo-loglikelihood
estimation procedure developed in <cit.> for
the COGARCH(1,1) model to the higher order case. We use the approximation scheme proposed in Section <ref>
as a generalization of the approach in <cit.> and used
recently also in <cit.> for the GRJ-COGARCH(1,1) model. First of all, we recall the variance of the integrated COGARCH(p,q)
model on the interval $\left[t_{i-1},t_{i}\right]$ <cit.>.
On the irregular grid
\begin{equation}
\end{equation}
we consider the increment of a COGARCH(p,q) process defined as:
\[
\Delta G_{t_{i}}:=G_{t_{i}}-G_{t_{i-1}}=\int_{t_{i-1}}^{t_{i}}V_{u}\mbox{d}L_{u}
\]
As shown in <cit.>, the conditional first moment and the conditional variance are
\begin{equation}
\begin{array}{l}
E\left[\Delta G_{t_{i}}\left|\mathcal{F}_{t_{i-1}}\right.\right]=0\\
Var\left[\Delta G_{t_{i}}\left|\mathcal{F}_{t_{i-1}}\right.\right]=E\left[L_1\right]\left[\frac{\alpha_{0}\Delta t_{i}b_{q}}{b_{q}-a_{1}\mu}+\mathbf{a}^{\top}e^{\tilde{B}\Delta t_{i}}\tilde{B}^{-1}\left(I-e^{-\tilde{B}\Delta t_{i}}\right)\left(Y_{t_{i-1}}-E\left(Y_{t_{i-1}}\right)\right)\right]
\end{array}\label{eq:VarianceCondForPseudo}
\end{equation}
where $\tilde{B}:=B+\mu\mathbf{ea}^{\top}$, $\mu=\int_{\mathcal{R}}y^{2}\mbox{d}\nu_{L}\left(y\right)$
and $\nu_{L}\left(y\right)$ is the Lévy measure of the process $L_{t}$
for simplicity we require the underlying process to be centered in zero
with unitary second moment $\mu=E\left(L_{1}\right)=1$. Under
the assumption that guarantees the existence of the stationary mean of process $Y_{t}$
<cit.> we have:
\[
\vdots\\
\end{array}\right].
\]
On the discrete grid in (<ref>) we construct
the discrete process $G_{i,n}$ introduced in (<ref>), in
particular we rewrite the state process $Y_{i,n}$ in (<ref>) as follows:
\begin{align}
Y_{i,n} & =\left(I+\Delta t_{i,n}\epsilon_{i,n}^{2}\mathbf{ea}^{\top}\right)e^{B\Delta t_{i,n}}Y_{i-1,n}+\alpha_{0}\Delta t_{i,n}\epsilon_{i,n}^{2}\mathbf{e}\nonumber \\
& =\left(I+\frac{\left(G_{i,n}-G_{i-1,n}\right)^{2}}{V_{i-1,n}}\mathbf{ea}^{\top}\right)e^{B\Delta t_{i,n}}Y_{i-1,n}+\alpha_{0}\frac{\left(G_{i,n}-G_{i-1,n}\right)^{2}}{V_{i-1,n}}\mathbf{e}\nonumber \\
& =\left(I+\frac{\left(G_{i,n}-G_{i-1,n}\right)^{2}}{\alpha_{0}+\mathbf{a}^{\top}Y_{i-1,n}}\mathbf{ea}^{\top}\right)e^{B\Delta t_{i,n}}Y_{i-1,n}+\alpha_{0}\frac{\left(G_{i,n}-G_{i-1,n}\right)^{2}}{\alpha_{0}+\mathbf{a}^{\top}Y_{i-1,n}}\mathbf{e}.\label{eq:DynamicYinForPseudoLogLik}
\end{align}
Using the results (<ref>) and (<ref>),
we are able to generalize the pseudo-likelihood estimation procedure
in <cit.> for the case of the COGARCH(p,q) model. The
idea behind the pseudo-loglikelihood is based on the markovian property
of the pair $\left(G_{t},V_{t}\right)$ and the substitution of the
real transition density with the normality assumption with mean and
variance determined as in (<ref>). Therefore
the maximum pseudo-loglikelihood estimates are obtained as solution
of the following optimization problem:
\[
\begin{array}{l}
\underset{\mathbf{a},\alpha_{0},B\in\Theta}{\max}\mathcal{L}_{N}\left(\mathbf{a},\alpha_{0},B\right)\\
\texttt{s.t.}\\
\left\{ \begin{array}{l}
Y_{i,n}=\left(I+\frac{\left(G_{i,n}-G_{i-1,n}\right)^{2}}{\alpha_{0}+\mathbf{a}^{\top}Y_{i-1,n}}\mathbf{ea}^{\top}\right)e^{B\Delta t_{i,n}}Y_{i-1,n}+\alpha_{0}\frac{\left(G_{i,n}-G_{i-1,n}\right)^{2}}{\alpha_{0}+\mathbf{a}^{\top}Y_{i-1,n}}\mathbf{e}\\
\end{array}\right.
\end{array}
\]
\[
\mathcal{L}_{N}\left(\mathbf{a},\alpha_{0},B\right)=-\frac{1}{2}\sum_{i=1}^{N}\left(\frac{\left(\Delta G_{t_{i}}\right)^{2}}{Var\left[\Delta G_{t_{i}}\left|\mathcal{F}_{t_{i-1}}\right.\right]}+\ln\left(Var\left[\Delta G_{t_{i}}\left|\mathcal{F}_{t_{i-1}}\right.\right]\right)\right)-\frac{N\ln\left(2\pi\right)}{2}
\]
and the set $\Theta$ contains the model parameters that ensure
the stationarity, the existence of the mean of the state process $Y_{t}$ and the non-negativity of process $V_t$.
urlstyle[Behme et al.(2014)Behme, Klüppelberg, and Mayr]behme2014
A. Behme, C. Klüppelberg, and K. Mayr.
Asymmetric cogarch processes.
J. Appl. Probab., 51A:0 161–173, 12 2014.
P. Billingsley.
Convergence of Probability Measures.
Wiley, New York, 1968.
[Brockwell et al.(2006)Brockwell, Chadraa, and Lindner]brockwell2006
P. Brockwell, E. Chadraa, and A. Lindner.
Continuous-time garch processes.
Ann. Appl. Probab., 160 (2):0 790–826, 05
[Brouste and Iacus(2013)]BrousteIacus2013
A. Brouste and S. M. Iacus.
Parameter estimation for the discretely observed fractional
ornstein-uhlenbeck process and the yuima r package.
Comput Stat, 28:0 1529–1547, 2013.
[Brouste et al.(2014)Brouste, Fukasawa, Hino, Iacus, Kamatani, Koike,
Masuda, Nomura, Ogihara, Y., Uchida, and N.]Brousteetal2013
A. Brouste, M. Fukasawa, H. Hino, S. M. Iacus, K. Kamatani, Y. Koike,
H. Masuda, R. Nomura, T. Ogihara, S. Y., M. Uchida, and Y. N.
The yuima project: A computational framework for simulation and
inference of stochastic differential equations.
Journal of Statistical Software, 570 (4):0
1–51, 2014.
E. Chadraa.
Statistical Modelling with COGARCH(P,Q) Processes., 2009.
PhD Thesis.
[Desoer and Vidyasagar(1975)]BookDesoer1975
C. A. Desoer and M. Vidyasagar.
Feedback systems: input-output properties.
Electrical science. Academic Press, New York, 1975.
[Haug et al.(2007)Haug, Klüppelberg, Lindner, and Zapp]Haug2007
S. Haug, C. Klüppelberg, A. Lindner, and M. Zapp.
Method of moment estimation in the cogarch(1,1) model.
Econometrics Journal, 100 (2):0 320–341,
[Iacus and Mercuri(2015)]IacusMercur2015
S. M. Iacus and L. Mercuri.
Implementation of lévy carma model in yuima package.
Computational Statistics, pages 1–31, 2015.
[Iacus et al.(2015)Iacus, Mercuri, and Rroji]iacus2015estimation
S. M. Iacus, L. Mercuri, and E. Rroji.
Estimation and simulation of a cogarch (p, q) model in the yuima
arXiv preprint arXiv:1505.03914, 2015.
[Klüppelberg et al.(2004)Klüppelberg, Lindner, and
C. Klüppelberg, A. Lindner, and R. Maller.
A continuous-time garch process driven by a lévy process:
Stationarity and second-order behaviour.
Journal of Applied Probability, 410 (3):0
601–622, 2004.
[Maller et al.(2008)Maller, Müller, Szimayer,
et al.]maller2008garch
R. A. Maller, G. Müller, A. Szimayer, et al.
Garch modelling in continuous time for irregularly spaced time series
Bernoulli, 140 (2):0 519–542, 2008.
G. Müller.
Mcmc estimation of the cogarch (1, 1) model.
Journal of Financial Econometrics, 80 (4):0
481–510, 2010.
D. Serre.
Matrices : theory and applications.
Springer, New York, Berlin, Paris, 2002.
T. Strom.
On logarithmic norms.
SIAM Journal on Numerical Analysis, 120 (5):0
pp. 741–753, 1975.
[Szimayer and Maller(2007)]szimayer2007finite
A. Szimayer and R. A. Maller.
Finite approximation schemes for lévy processes, and their
application to optimal stopping problems.
Stochastic Processes and their Applications, 1170
(10):0 1422–1447, 2007.
|
1511.00149
|
Max Planck Institute for the Structure and Dynamics of
Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
I propose a method for ultrafast switching of ferroelectric
polarization using mid-infrared pulses. This involves selectively
exciting the highest frequency $A_1$ phonon mode of a ferroelectric
material with an intense mid-infrared pulse. Large amplitude
oscillations of this mode provides a unidirectional force to the
lattice such that it displaces along the lowest frequency $A_1$
phonon mode coordinate because of a nonlinear coupling of the type
$g\qp\qir^2$ between the two modes. First principles calculations
show that this coupling is large in perovskite transition-metal
oxide ferroelectrics, and the sign of the coupling is such that the
lattice displaces in the switching direction. Furthermore, I find
that the lowest frequency $A_1$ mode has a large $\qp^3$ order
anharmonicity, which causes a discontinuous switch of electric
polarization as the pump amplitude is continuously increased.
§ INTRODUCTION
Ultrafast switching of polarization in ferroelectrics is of great
interest for potential application in non-volatile memory
devices. FLASH memories, which at present are the most commonly used
non-volatile memory devices, have an operating speed of milliseconds.
Because of their slow speed, they are not considered as candidates for
future memory applications <cit.>. Other emerging non-volatile
memory technologies that utilize phase or resistance change have write
and erase times of nanoseconds. Therefore, development of a switching
mechanism at sub-picosecond timescales has the potential to
revolutionize the field.
Non-destructive readout of the electric polarization at sub-picosecond
timescales has recently been demonstrated by analyzing the THz pulse
waveforms radiated after illumination of a ferroelectric sample by
femtosecond laser pulses at optical wavelengths <cit.>. This makes ferroelectric materials an exciting prospect for
memory applications if switching can be achieved at similarly
ultrashort timescales.
Ferroelectric materials exhibit remnant polarization even at zero
external electric field because the cations and anions in these
materials are asymmetrically displaced in the equilibrium structure.
To switch the polarization, the relative displacement between the
cations and anions need to be reversed. This can be achieved by
applying a (quasi-)static electric field because such an electric
field imparts a unidirectional force to the cations and anions. An
arbitrary light pulse, whose oscillating electric field integrates to
zero by definition, imparts a zero total force to the electric dipole
present in the material. Therefore, an ultrashort light pulse cannot
in general be used to switch the polarization of a ferroelectric
material. Nevertheless, there have been several proposals for
switching the polarization of ferroelectric materials using ultrashort
light pulses by controlling their soft phonon modes<cit.>.
In this paper, I propose a method for using mid-infrared pulses that
are resonant with the highest frequency infrared-active phonon mode of
a perovskite transition-metal oxide ferroelectric to switch its
polarization. This involves controlling the dynamical degrees of
freedom of the lattice and requires four main ingredients. First, I
notice that there always exists a low frequency fully symmetric $A_1$
phonon mode in the ferroelectric phase that involves the motion of the
cations and anions of the material in a way that changes the electric
polarization. Second, I find that this phonon mode couples to the
highest frequency infrared-active $A_1$ phonon mode of the material
with a large $g \qp \qir^2$ coupling, where $g$ is the coupling
constant and $\qp$ and $\qir$ are the normal mode coordinates of the
lowest frequency and highest frequency $A_1$ normal mode coordinates,
respectively. Third, I find from first principles calculations that
the sign of the coupling is such that the excitation of the highest
frequency $A_1$ mode provides a displacive force along the $\qp$
normal mode coordinate in the direction that switches the
polarization. Fourth, I find that the $\qp$ mode has a strong $\qp^3$
order anharmonicity, which facilitates an abrupt switch of electric
polarization as the $\qir$ amplitude is continuously increased.
Coherent displacement along Raman mode coordinates utilizing nonlinear
phonon couplings by resonantly exiting the highest frequency infrared
mode of various centrosymmetric oxides has previously been
demonstrated <cit.>. Therefore, the method
proposed here to switch ferroelectric polarization at ultrafast
timescales using mid-infrared pulses is experimentally feasible.
§ COMPUTATIONAL DETAILS
I illustrate the proposed mechanism for the case of PbTiO$_3$. The
phonon frequencies and eigenvectors and the nonlinear couplings
between two phonon modes were obtained using density functional theory
calculations with plane-wave basis sets and projector augmented wave
pseudopotentials <cit.> as implemented in the vasp software package <cit.>. The interatomic force
constants were calculated using the frozen-phonon method
<cit.>, and the phonopy software package was used to
calculate the phonon frequencies and eigenvectors <cit.>. Total
energy calculations were then performed as a function of the
lowest frequency $\qp$ and high-frequency $\qir$ coordinates to obtain
energy surfaces. The nonlinear couplings between the two modes were
obtained by fitting the calculated energy surface to the polynomial
shown in Eq. <ref>.
I used the experimental values of $a$ = 3.9039 and $c$ = 4.1348
Å for the tetragonal lattice parameters but relaxed the atomic
positions. The calculations were performed within the local density
approximation. A cut-off of 600 eV was used for the plane-wave basis
set expansion, and an $8 \times 8 \times 8$ $k$-point grid was used in
the Brillouin zone integration.
§ RESULTS AND DISCUSSIONS
The zone-center phonon frequencies and
their irreducible representation and optical activity of
ferroelectric PbTiO$_3$.
frequency (cm$^{-1}$) irrep optical activity
80 $E$ IR + Raman
149 $A_1$ ($\qp$) IR + Raman
187 $E$ IR + Raman
273 $E$ IR + Raman
290 $B_1$ Raman
356 $A_1$ IR + Raman
491 $E$ IR + Raman
655 $A_1$ ($\qir$) IR + Raman
The ferroelectric phase of PbTiO$_3$ exists in the $P4mm$ structure
with one formula unit per unit cell. This gives rise to 12 zone-center
optical normal modes with the decomposition $\Gamma_{\textrm{optic}} =
3A_1 + B_1 + 4 E$. The $A_1$ and $E$ modes are both Raman and infrared
active, whereas the $B_1$ is only Raman active. The calculated zone
center phonon frequencies and their symmetries are given in Table
<ref>. I find that the coupling between the lowest frequency
$E$ and the highest frequency $A_1$ mode is weak. Therefore, I focus
on the coupled dynamics of the lowest and highest frequency $A_1$
(Color online) Displacement patterns of the (a) lowest
frequency (149 cm$^{-1}$) $\qp$ and (b) highest frequency (655
cm$^{-1}$) $\qir$ modes of the ferroelectric phase of
PbTiO$_3$. Both modes belong to the $A_1$ irreducible
The atomic displacement pattern of the lowest frequency $A_1$ mode
(denoted by $\qp$) is shown in Fig. <ref>(a) with $z$ axis
chosen as the polarization axis. A finite magnitude of this mode
involves the motion of Pb$^{2+}$ and O$^{2-}$ ions along the $z$ axis
in the opposite direction, and a displacement of the lattice along the
coordinate of this mode modifies the electric polarization of the
material. In the convention used in this paper, a large negative value
of this normal mode coordinate would reverse the polarization. The
arrows in Fig. <ref>(a) indicate the movements of ions for
such a negative value of $\qp$. One can see that such a movement
reverses the relative displacement between the Pb$^{2+}$ and O$^{2-}$
ions. However, it should be noted that a displacement along this mode
does not bring the structure to the symmetrically equivalent ground
state with opposite polarization because the eigenvector of this mode
is not in general equal to the eigenvector of the unstable infrared
mode of the paraelectric phase that is responsible for the
ferroelectric instability. A further relaxation of the lattice, in
addition to the large negative value of the $\qp$ coordinate that
reverses the polarization, would take the structure to the
symmetrically equivalent switched ground state. I confirmed this by
starting with a structure that was displaced by a value of $-$8 Å$\sqrt{\textrm{amu}}$ along the $\qp$ coordinate and relaxing the
atomic positions by minimizing the forces. I found that the structure
indeed relaxes to the symmetrically equivalent switched phase rather
than going back to the initial ferroelectric equilibrium state that
was used as a starting point to displace along the $\qp$ coordinate.
Therefore, a coherent displacement of the lattice along this low
frequency $\qp$ phonon mode coordinate is a viable route for ultrafast
ferroelectric switching. In fact, Qi et al. have proposed a
method for switching the polarization by driving large amplitude
oscillations of this phonon mode using multiple THz pulses with an
asymmetric electric field profile <cit.>.
Here I propose a method of switching the polarization using a light
pulse that does not directly drive the low frequency $\qp$
mode. Instead, this involves exciting the high frequency $\qir$
infrared mode of the material (shown in Fig. <ref>(b)) by an
intense mid-infrared pulse that in turn provides a displacive force
along the $\qp$ coordinate in the switching direction due to a
nonlinear coupling of the type $g\qp\qir^2$ between the two modes.
Furthermore, the presence of a large $\qp^3$ order anharmonicity
causes a sudden increase in the displacement along the $\qp$
coordinate as the $\qir$ amplitude is continuously increased, and this
causes an abrupt reversal of the electric polarization without the
magnitude of the polarization going to zero.
(Color online) Total energy as a function of the $\qp$ and $\qir$
normal mode coordinates of the ferroelectric PbTiO$_3$. Top: energy
surface. Bottom: few energy curves that illustrate the behavior of
the $\qp$ mode as a function of $\qir$ mode.
I calculated the total energy as a function of the lowest frequency
$\qp$ and highest frequency $\qir$ infrared mode coordinates from
first principles using density functional theory calculations. The
calculated energy surface of ferroelectric PbTiO$_3$ is shown in
Fig. <ref>, and it fits the following expression:
\begin{eqnarray}
V(\qp,\qir) & = & \frac{1}{2} \omp^2 \qp^2 + \frac{1}{2} \omir^2
\qir^2 + \frac{1}{3} a_3 \qp^3 \nonumber \\
& & + \frac{1}{4} a_4 \qp^4 + \frac{1}{3} b_3 \qir^3 + \frac{1}{4}
b_4 \qir^4 \nonumber \\
& & + g\, \qp \qir^2 + h\, \qp^2 \qir + i\, \qp^3 \qir \nonumber \\
& & + j\, \qp \qir^3 + k\, \qp^2 \qir^3 + l\, \qp \qir^4.
\label{eq:energy}
\end{eqnarray}
A fit of the above expression to the calculated energy surface
determines ab initio the nonlinear couplings between the two
modes up to all significant orders of the two phonon coordinates. The
values of the coefficients of the coupling terms obtained from such a
fit are given in Table II. The calculated energy surface exhibits
complex features, and this is due to the presence of both even and odd
order nonlinearities. However, there are some salient features. When
$\qir = 0$, the energy curve of the $\qp$ mode has one minimum at
zero, as one would expect for a stable ground-state structure. The
energy increases rapidly for positive values of $\qp$, but the
increase is less rapid for negative values of $\qp$. In fact, for
negative values of $\qp$, the slope of the energy curve has a minimum
near $-$5 $\textrm{\AA}\sqrt{\textrm{amu}}$, where the energy curve is
shallow and the restoring force is small, before it shows an upturn
around $-$8 $\textrm{\AA}\sqrt{\textrm{amu}}$. This asymmetric nature
of the energy curve of the $\qp$ mode is due to the presence of a
large $a_3\qp^3$ term in the polynomial expression of the
energy surface. The physical reason for the asymmetric nature of the
$\qp$ energy curve is the presence of a state with reversed
polarization near a $\qp$ value of $-$8
$\textrm{\AA}\sqrt{\textrm{amu}}$ that is symmetrically equivalent to
the ferroelectric state at a $\qp$ value of zero.
The anharmonic terms and nonlinear
couplings of the $\qp$ and $\qir$ modes of ferroelectric PbTiO$_3$
determined from a fit to the energy surface calculated from first
coefficient value
$a_3$ (meV/amu$^{3/2}$/Å$^3$) 21.80
$a_4$ (meV/amu$^{2}$/Å$^4$) 1.89
$b_3$ (meV/amu$^{3/2}$/Å$^3$) 1567.65
$b_4$ (meV/amu$^{2}$/Å$^4$) 631.80
$g$ (meV/amu$^{3/2}$/Å$^3$) 70.32
$h$ (meV/amu$^{3/2}$/Å$^3$) $-$12.40
$i$ (meV/amu$^{2}$/Å$^4$) $-$0.79
$j$ (meV/amu$^{2}$/Å$^4$) 52.14
$k$ (meV/amu$^{5/2}$/Å$^5$) 2.29
$l$ (meV/amu$^{5/2}$/Å$^5$) 7.61
The energy surface is also asymmetric in the $\qir$ coordinate because
of the presence of odd order terms. The presence of both even and odd
order terms is consistent with the fact that this high frequency mode
also has the $A_1$ representation that does not break any crystal
symmetry and is both infrared and Raman active. Although the energy
surface is asymmetric in the $\qir$ coordinate, both positive and
negative $\qir$ displacements move the minimum of the $\qp$ coordinate
towards the negative direction. A negative value of $\qir$ displaces
the lattice towards the negative $\qp$ direction by a modest amount.
However, a positive value of $\qir$ does not continuously shift the
energy minimum along the negative $\qp$ direction. It raises the
energy curve near $\qp \approx 0$ and creates an energy minimum at a
large negative value of the $\qp$ coordinate such that a state with
reversed polarization is energetically favored. This abrupt shift of
the minimum of the $\qp$ mode is due to the presence of a large
$a_3\qp^3$ term.
If the oscillations along the $\qir$ coordinate are integrated out,
the average potential experienced by the lattice has a minimum at a
negative value of the $\qp$ coordinate with the magnitude of the
displacement along $\qp$ coordinate depending on the integration
cut-off (i.e., the amplitude of the $\qir$ oscillations). This is
because the coupling constant $g$ of the term $\qp \qir^2$ has the
largest magnitude, and it implies that the lattice will experience a
large unidirectional force $-\partial V / \partial \qp = -g \qir^2$
along the $\qp$ coordinate when the $\qir$ mode is being
driven. Furthermore, a large $a_3\qp^3$ term ensures that
the displacement of the lattice along the $\qp$ coordinate abruptly
increases as the $\qir$ amplitude is continuously increased, which
causes the electric polarization to switch discontinuously.
We can achieve a better understanding of the dynamics of the lattice
when the $\qir$ mode is pumped externally by a mid-infrared pulse by
treating the $\qp$ and $\qir$ modes as classical oscillators and
studying their coupled equations of motion. In this picture, the two
oscillators experience a force deriving from the calculated energy
surface, and the $\qir$ mode is additionally driven by a term $F(t) =
F \sin(\Omega t) e^{-t^2/2\sigma^2}$, where $F$, $\sigma$, and
$\Omega$ are the amplitude, width, and frequency of the mid-infrared
pulse, respectively. By treating the expression in Eq. <ref>
as the potential, the coupled equations motion are
\begin{eqnarray}
\ddot{Q}_{\textrm{IR}}+\omir^2\qir&=& - b_3 \qir^2 -
b_4 \qir^3 - 2 g\,\qp\qir \nonumber \\
& & - h\, \qp^2 - i\, \qp^3 - 3 j\, \qp \qir^2 \nonumber \\
& & - 3 k\, \qp^2 \qir^3 - 4 l\, \qp \qir^3 + F(t), \nonumber \\
\ddot{Q}_{\textrm{P}} + \omp^2 \qp & = & - a_3 \qp^2 - a_4 \qp^3
- g\, \qir^2 \nonumber \\
& & - 2 h\, \qp\qir - 3 i\, \qp^2 \qir - j\, \qir^3 \nonumber \\
& & - 2 k\, \qp \qir^3 - l \, \qir^4.
\end{eqnarray}
Dynamics of the $\qp$ mode for three different pump
amplitudes. Left panels: Displacements along $\qp$ coordinate as
function of time delay. Right panels: Fourier transform of the
positive time delay oscillations.
The results from numerical integration of the coupled equation of
motions for different pump amplitudes are shown in
Fig. <ref>. In these calculations, I have used a pump pulse
with a symmetric Gaussian profile and a width of $\sigma = 250$ fs,
which correspond to typical pump pulses used in mid-infrared
excitations <cit.>. A pump frequency of $\Omega =
1.03\,\omir$ was used in the simulations, which is chosen to be
slightly off-resonance with the frequency of the $\qir$ mode to
demonstrate that this method is efficacious even if the pump pulse is
not precisely resonant with the $\qir$ mode.
Even for small pump amplitudes that cause a change in the Ti–apical O
distances of a few percent along the $\qir$ coordinate, the $\qp$ mode
oscillates at a displaced position in the negative $\qp$ direction
(see the left panel of Fig. <ref>(a)). This is consistent with
the analysis of the $g\qp\qir^2$ coupling presented in
Refs. fors11 and sube14. In
Ref. sube14, it was shown that the $\qp$ mode experiences
an effective force $-g\qir^2 \propto -gF^2\omir^2\sigma^6(1 - \cos
2\omir t)$ when the $\qir$ mode is pumped by an external driving term
$F(t)$, and the time average of this forcing field has a rectified
non-zero value. The oscillation of the $\qp$ mode at a rectified
position is also seen in the Fourier transform of the time evolution
of the $\qp$ mode at positive time delays as shown in the right panel
of Fig. <ref>(a). The Fourier transform shows a peak at zero
frequency that is due to the displacement of the lattice along the
$\qp$ coordinate. In addition, there is a peak at $\omp$ and a
negligible presence of higher-harmonics, which shows that the dynamics
of the coupled oscillators are determined by the $g\qp\qir^2$ term,
and other nonlinearities only play a marginal role at small pump
amplitudes. At small pump amplitudes, the displacement along the
negative direction in the $\qp$ coordinate is small. Therefore,
although the electric polarization is reduced, the reversal of the
polarization has not occurred.
It is also noteworthy that the cubic terms in the energy potential
$a_3\qp^3$ and $b_3\qir^3$ are large. These also
impart unidirectional forces $-\partial V / \partial \qp = -a_3\qp^2$
and $-\partial V / \partial \qir = -b_3\qir^2$ to their respective
coordinates when the amplitude of the oscillations are large, and the
lattice will be rectified along these coordinates for the reasons
described in the previous paragraph. From my first principles
calculations of the energy surface, I find that the coefficients of
these terms are such that the lattice is rectified along the direction
that reverses the electric polarization.
As the pump amplitude is increased, the effects of the abovementioned
nonlinearities start to become noticeable. The oscillations of the
$\qp$ mode start to show higher frequency components due to the
presence of various nonlinear terms. Further increase of the pump
amplitude takes the dynamics to a highly nonlinear regime. In this
regime, the frequency at which the $\qp$ mode oscillates also changes.
Interestingly, there are two different mechanisms that change the
effective frequency of the $\qp$ mode. First, a large amplitude
oscillation of the $\qp$ mode causes a change in the effective spring
constant due to the $\frac{1}{4}a_4\qp^4$ nonlinearity in the energy
potential. In the equation of motion, this nonlinearity acts to modify
the frequency of the $\qp$ mode with a term $-\partial V / \partial
\qp = -a_4\qp^3$. The effective frequency
$\omp^{\textrm{eff}}\to\omp^2 \left( 1 + a_4 \qp^2(t)/\omp^2 \right)$
changes because the time-averaged $\omp^{\textrm{eff}}$ has a value
different from $\omp$ when the $\qp$ amplitude is large. The
rectification of the lattice along the $\qp$ and $\qir$ coordinates
provides the second cause for the modification of the effective
frequency. For example, the rectification along the $\qp$ coordinate
changes the effective frequency to $\omp^{\textrm{eff}} \to
\omp^2\left(1 + a_3\qp(t)/\omp^2\right)$ due to the
$a_3\qp^3$ term in the energy potential. The $h\qp^2\qir$
term in the energy potential similarly changes the effective frequency
as $\omp^{\textrm{eff}} \to \omp^2\left(1 + 2h\qir(t)/\omp^2\right)$.
Fig. <ref>(b) shows the results of the numerical integration
of the equations of motion for a pump amplitude that rectifies the
lattice close to a point where the polarization switches.
Near the polarization reversal, the slope of the potential for the
$\qp$ mode is less steep, and the $\qp$ mode oscillates at a smaller
In this regime, the pumped $\qir$ mode is oscillating at a displaced
position with an amplitude that changes the two Ti-apical O bond
lengths by 0.4 and 0.7 Å. These are large amplitude oscillations,
but they are comparable to the change in the Ti-apical O distance of
0.6 Å when a polarization switch occurs.
Fig. <ref>(c) illustrates the case where the lattice moves to
a far distance in the negative direction along the $\qp$ coordinate,
and this signals that the polarization has been switched. The pump
strength used in this instance causes the $\qir$ mode to oscillate at
a displaced position with an amplitude that changes the two Ti-apical
O bond lengths by 0.5 and 0.8 Å. In this regime, the oscillations
about the displaced position exhibit a strong nonlinear behavior with
the presence of a wide range of frequency components. Nevertheless,
the frequency component that has the largest spectral weight stiffens
once the displacement along $\qp$ coordinate advances through the
point of polarization reversal, although the frequency is still
smaller than $\omp$.
I find that the displacement along the $\qp$ coordinate shows a sudden
jump when the externally pumped $\qir$ amplitude is continuously
increased. This is consistent with the behavior of the energy
potential discussed above where a large $a_3\qp^3$ term causes an
abrupt change in the position of the minimum of the $\qp$ mode as
$\qir$ is continuously increased. As a function of the pump amplitude,
the displacement along the $\qp$ coordinate continuously increases
from a value of zero to $\sim$$-$1.5 Å$\sqrt{\textrm{amu}}$. However, a further increase of the pump
amplitude causes the $\qp$ mode to oscillate about a displaced
position of $\sim$$-$9.0 Å$\sqrt{\textrm{amu}}$. This indicates
that the electric polarization switches in an abrupt, discontinuous
manner when the $\qir$ mode is externally pumped. Such a behavior can
also be gleaned from the change in the frequency of the $\qp$ mode as
the polarization reversal happens. I find that the frequency of the
$\qp$ mode decreases by up to 60% as it is displaced along this
coordinate. But it does not soften completely to zero as the
polarization switch occurs and the frequency starts to increase again.
In the study presented here, nonlinear couplings between two phonon
modes and their dynamics when the higher frequency mode is externally
pumped has been used to predict that ferroelectric materials can be
switched using mid-infrared pulses. However, in real materials there
are additional dynamical degrees of freedom, and this has two main
implications. First, scattering with other degrees of freedom will
cause the phonon modes to be damped. Therefore, the rectifying force
along the $\qp$ coordinate exists only as long as the $\qir$ mode is
being externally pumped. Second, other degrees of freedom also respond
to the displacement of the lattice along the $\qp$ coordinate. If the
pump pulse is long enough, other degrees of freedom relax relative to
the switched state, and this forms an energy barrier that prevents the
lattice to move back to the initial state even in the absence of the
A more detailed theoretical study based on molecular dynamics
simulations would be required to ascertain the time it would take to
form the energy barrier.
As mentioned above, I performed a full relaxation of the lattice
starting from a structure that corresponds to a $\qp$ displacement of
$-$8.0 Å$\sqrt{\textrm{amu}}$ and found that the lattice indeed
relaxes to the symmetrically equivalent switched state. The relaxation
of the whole lattice to the symmetrically equivalent state with
reversed polarization provides a mechanism for repeated switching
because the lattice again experiences a unidirectional force in the
switching direction along the $\qp$ coordinate when the lattice is
excited anew by mid-infrared pulse.
In addition to PbTiO$_3$, I investigated the nonlinear couplings
between the lowest and highest frequency $A_1$ modes of BaTiO$_3$ and
LiNbO$_3$. I find a large $g\qp\qir^2$ coupling between the lowest
frequency $A_1$ mode ($\qp$) and the highest frequency $A_1$ mode
($\qir$) in these materials as well. The sign of the coupling is such
that the electric polarization of these materials could also be
switched by pumping the $\qir$ mode with a mid-infrared
pulse. Therefore, the method illustrated here seems applicable in
general to all perovskite transition-metal oxide ferroelectrics.
§ SUMMARY
In summary, I have illustrated that the polarization of PbTiO$_3$ can
be switched by exciting the highest frequency infrared active $A_1$
phonon mode of this material with a mid-infrared pulse. A large
amplitude oscillation of this mode provides a unidirectional force
along the lowest frequency $A_1$ phonon mode coordinate due to a
nonlinear coupling of the type $g\qp\qir^2$. A displacement of the
lattice along the $\qp$ coordinate changes the electric polarization
and can bring the system near the symmetrically equivalent switched
state. From my first principles calculations, I find that sign of the
coupling is such that the oscillations of the $\qir$ mode displaces
the lattice along the $\qp$ coordinate in the switching direction. I
also find that the switching occurs discontinuously because of the
presence of a large $a_3\qp^3$ term in the energy
potential, which abruptly moves the minimum of the $\qp$ mode as the
$g\qp\qir^2$ term gets continuously larger.
In addition to PbTiO$_3$, I find the presence of a similar
$g\qp\qir^2$ coupling and a large $a_3\qp^3$ anharmonicity
in BaTiO$_3$ and LiNbO$_3$, and this type of nonlinear coupling seems
to be universally present in perovskite transition-metal oxide
ferroelectrics. Therefore, a selective excitation of the $\qir$ mode
using a mid-infrared pulse can be a general method to switch the
polarization of perovskite transition-metal oxide ferroelectrics.
§ ACKNOWLEDGMENTS
I am indebted to Yannis Laplace for valuable discussions. I also
acknowledge Antoine Georges, Roman Mankowsky, Srivats Rajasekaran, and
Andrea Cavalleri for helpful comments and discussions.
jeon12 D. S. Jeong, R. Thomas, R. S. Katiyar, J. F. Scott,
H. Kohlstedt, A. Petraru, and C. S. Hwang, Rep. Prog. Phys. 75, 076502 (2012).
taka06 K. Takahashi, N. Kida, and M. Tonouchi,
Phys. Rev. Lett. 96, 117402 (2006).
talb08 D. Talbayev, S. Lee, S.-W. Cheong, and A. J. Taylor,
Appl. Phys. Lett. 93, 212906 (2008).
fahy94 S. Fahy and R. Merlin, Phys. Rev. Lett. 73,
1122 (1994).
qi09 T. Qi, Y.-H. Shin, K.-L. Yeh, K. A. Nelson, and
A.M. Rappe, Phys. Rev. Lett. 102, 247603 (2009).
fors11 M. Först, C. Manzoni, S. Kaiser, Y. Tomioka,
Y. Tokura, R. Merlin, and A. Cavalleri, Nature Phys. 7, 854
fors13 M. Först, R. Mankowsky, H. Bromberger, D. M. Fritz,
H. Lemke, D. Zhu, M. Chollet, Y. Tomioka, Y. Tokura, R. Merlin,
J. P. Hill, S. L. Johnson, and A. Cavalleri, Solid State
Commun. 169, 24 (2013).
sube14 A. Subedi, A. Cavalleri, and A. Georges, Phys. Rev. B
89, 220301 (2014).
mank14 R. Mankowsky, A. Subedi, M. Först, S. O. Mariager,
M. Chollet, H. T. Lemke, J. S. Robinson, J. M. Glownia,
M. P. Minitti, A. Frano et al., Nature 516, 71 (2014).
bloc94 P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
kres99 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758
kres96 G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
parl97 K. Parlinski, Z-.Q. Li, and Y. Kawazoe,
Phys. Rev. Lett. 78, 4063 (1997).
togo08 A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106 (2008).
|
1511.00197
|
We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $ \frac{\partial f}{\partial t}=\Lap f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.
§ INTRODUCTION
The elliptic Allen-Cahn equation $\Lap f=f^3-f$, with $|f|\leq 1$ is a very popular non-linear PDE, which gave rise to an interesting conjecture formulated by De Giorgi in <cit.>:
Suppose that $f$ is an entire solution to the Allen-Cahn equation
\begin{align}\label{A-C-elliptic}
\Lap f-f^3+f=0, \hspace{1cm} |f|\leq 1, \hspace{1cm} x=(x',x_n)\in\mathbb{R}^n
\end{align}
satisfying $\frac{\partial f}{\partial x_n}>0$ for $x\in\mathbb{R}^n$. Then, at least for $n\leq 8$, the level sets of $f$ must be hyperplanes.
Equivalently this can be stated as follows: any entire solution which is monotone in one direction should be one-dimensional. The conjecture was proved for $n=2$ by Ghoussoub-Gui <cit.>, for $n=3$ by Ambrosio-Cabre <cit.> and for $4\leq n\leq 8$ by Savin, under an extra assumption (see <cit.>). The equation has been deeply investigated, see for example: <cit.>, but still offers many interesting questions. Note that De Giorgi's conjecture is false for $n\geq 9$ (see, for example, <cit.>).
Originally, the equation appeared in the study of the process of phase separation in iron alloys, including order-disorder transitions (see <cit.>). Moreover, it is related to the study of minimal surfaces, so it is an interesting topic for geometry too.
In this paper we are focusing on the parabolic Allen-Cahn equation:
\begin{align}\label{A-C-parabolic}
\begin{cases}
\frac{\partial f}{\partial t}=\Lap f-(f^3-f) \\
\end{cases}
\end{align}
where $f:\mathbb{R}^n\times[0,\infty)\to \mathbb{R}$.
Some solutions have interfaces that travel in one direction with a constant speed $c$. Without loss of generality, we may assume that the travel direction is $x_n$. If the solution is of the form $f(x,t)=p(x_1,...,x_{n-1}, x_n-ct)$, for some function $p:\mathbb{R}^n\to\mathbb{R}$, then $p(x)$ is a traveling wave solution with speed $c\geq 0$. Substituting $f$ with $p$ in (<ref>) we obtain the following equation:
\begin{align}\label{A-C-travel}
c\frac{\partial p}{\partial x_n}+\Lap p =p^3-p
\end{align}
In the case when $c=0$, if we impose the condition $\lim\limits_{x_n\to\pm\infty} p(x)=\pm 1$, for all $x\in\mathbb{R}^n$, then the traveling wave (in this case, a standing wave) satisfies the elliptic Allen-Cahn equation (<ref>). Therefore, by obtaining estimates on the solution of the parabolic equation, it is possible to relate them to the solution of the elliptic one, by using the standing wave as an intermediary. For a nice introduction about traveling waves of the Allen-Cahn equation see <cit.>.
The focus of this paper is to use geometric methods to obtain a Harnack inequality for the solution of the parabolic Allen-Cahn equation. The techniques only work for positive solutions and as such cannot be directly applied to get a bound on the elliptic equation. We nevertheless formally compare the standing wave solution to Modica's gradient estimates and show that they have similar polynomial bounds.
The main result of this paper is the following theorem, which is a differential Harnack inequality for the solution of equation (<ref>) on a closed manifold:
Let $M$ be a compact Riemannian manifold without boundary of dimension $n$. Assume the Ricci curvature $\Ric\geq -k$, where $k\geq 0$. Consider $f$ to be a solution to the parabolic Allen-Cahn equation (<ref>), such that $0<f<1$. Denote with $u(x,t)=\log f(x,t)$. Define the Harnack quantity $h(x,t)$ to be
\[h(x,t)=\Lap u+\alpha |\nabla u|^2+\beta e^{2u}+\phi(t)\]
where $\alpha\in(0,1)$ and $\beta\leq\min\left\{-\frac{n(\alpha+2)}{2\alpha^2-2\alpha+3n}, -\frac{nk}{4\alpha}, -\frac{n}{2(1-\alpha)}\right\}$ and $\phi(t)=\frac{-2q\coth(-2qt)+b}{2a}$ ($a,b,d$ are positive real numbers that only depend on $\alpha$, $\beta$). Then $h(x,t)\geq 0$ for any $t>0$.
This is a Li-Yau-Hamilton type Harnack inequality, the study of which began in the seminal paper by Li-Yau <cit.>, and which has played a strong role in the study of geometric flows (see, for example, <cit.>). In the present paper we apply a method previously used in the study of Ricci flow and other heat-type equations (see, for example, the work of Cao <cit.>). The method both determines the Harnack quantity and proves that it's non-negative, using the maximum principle. The method has proven to be able to recover classical results for other non-linear parabolic equation (see <cit.>) or for the curve shortening flow (<cit.>). The procedure has its roots in the study of Ricci flow, but recently its efficiency has been observed for other heat-type equations also (see, for example, <cit.> for a non-linear heat equation, or <cit.> for the Endangered Species Equation).
By integrating the above along a space-time path, we obtain the following classical Harnack inequality:
Let $M$ be a compact Riemannian manifold without boundary of dimension $n$. Assume the Ricci curvature $\Ric\geq -k$, where $k\geq 0$. Consider $f$ to be a solution to the parabolic Allen-Cahn equation (<ref>), such that $0<f<1$. Then for $\alpha,\beta$ satisfying the conditions of theorem <ref>:
\[\frac{f(x_2,t_2)}{f(x_1,t_1)}\geq e^{-\frac{d(x_1,x_2)^2}{4(1-\alpha)(t_2-t_1)}}\cdot\left(\frac{e^{-2qt_2}-1}{e^{-2qt_1}-1}\right)^{-1/a}\cdot e^{-\frac{4q+b}{2a}(t_2-t_1)}\]
for $a,b,q$ positive real numbers that only depend on $\alpha$, $\beta$.
The standing wave of this equation cannot remain in the interval $(0,1)$. This can be seen by comparing it to a radial solution in a ball with expanding radius and zero boundary condition, which is below the standing wave solution. By maximum principle the radial solution would remain below the standing wave, but as the radius goes to infinity, the radial solution would tend to $1$, thus the standing wave would also be $1$. Therefore the Harnack inequality cannot be used directly to estimate the elliptic solution. But we can at least show that the geometric method produces a similar polynomial bound as the classical PDE method. We will formally assume that the standing wave solution is in $(0,1)$ and obtain the following:
Let $p$ be a standing wave solution for the parabolic Allen-Cahn equation (<ref>) in $\mathbb{R}^n$ such that $0<p<1$. Then $p$ satisfies:
\begin{align}\label{stand-wave}
|\nabla p|^2\leq p^2[(2n-1)-(n-1)p^2]
\end{align}
This is reminiscent of Modica's gradient estimate (<cit.>):
Let $F\in C^2(\mathbb{R})$ be a non-negative function and $u\in C^3(\mathbb{R}^n)$ be a bounded entire solution to the equation $\Lap u=f(u)$, where $f=F'$. Then $|\nabla u|^2(x)\leq 2 F(u(x))$ for every $x\in\mathbb{R}^n$.
In our case, Modica's result would translate as $|\nabla p|^2\leq \frac{1}{2}p^2(p^2-2)+\frac{1}{2}$ (note that a constant is added in order to obtain a non-negative anti-derivative of $x^3-x$). Comparing the two gradient estimates, one can notice that both bounds have a degree 4 polynomial and while the present result depends on the dimension of the underlying space, it does offer a polynomial ($x^2(2n-1-(n-1)x^2)$) that does not need to be non-negative everywhere. Moreover, as it will be shown in the last section, the new bound is an improvement on parts of the interval of solutions.
The paper is structured as follows: in section 2 we describe the setting of the problem and summarize the procedure we use to construct the Harnack quantity. In section 3 we determine the exact expression of the Harnack quantity and show that it's non-negative, thus proving theorem <ref>, while in section 4 we prove the classical Harnack inequality <ref>. The corollary <ref> involving the traveling wave and the comparision with Modica's result is proven in section 5.
Acknowledgements The author would like to thank Prof. Xiaodong Cao for suggesting applying this method to the Allen-Cahn equation and for fruitful discussions.
§ BACKGROUND AND SETTING
We consider an $n$-dimensional orientable complete compact Riemannian manifold $M$, without boundary and with Ricci curvature $\Ric\geq -k$ for some $k\geq 0$. Let $f:M\times[0,\infty)\to \mathbb{R}$ be a solution to the parabolic Allen-Cahn equation:
\begin{align}\label{A-C-parabolic-bis}
\begin{cases}
\frac{\partial f}{\partial t}=\Lap f-(f^3-f) \\
\end{cases}
\end{align}
Moreover, we assume that that $f_0(x)<1$. By maximum principle, if the function $f\in(0,1)$ at the beginning, then it will stay in $(0,1)$ at any time $t>0$ (this is because $f-f^3=f(1-f^2)$ stays non-negative for $0<f<1$). Therefore, $f(x,t)\in(0,1)$ from now on, for any $x\in M$ and $t\geq 0$.
The method that we will use to determine the Harnack quantity and prove that it's non-negative is summarized as follows: let $f>0$ be the function for which one needs to prove a Harnack inequality, where $f$ satisfies a heat-type equation. If $u=\log f$, then $u_t=\triangle u+|\nabla u|^2$+other terms. Inspired by the expression of $u_t$, one can define the Harnack quantity $h=\alpha\triangle u+\beta|\nabla u|^2+\text{possibly other terms}+\phi(x,t)$. $\phi$ is a function that has to go to infinity as $t\to 0$ and will be determined at the end. Assuming there is a first point (first, with respect to time) when $h(x,t)\leq 0$. As $\phi$ is very large at time $0$, it means that $\lim\limits_{t\to 0^+} h=\infty$, i.e. $h$ is positive close to $t=0$, so $t_1>0$. Moreover at this point, the time derivative of $h$ has to be negative. At that particular time, $h(x,t_1)$ is a local space minimum on the whole manifold, so if we apply the maximum principle, $\Lap h\geq 0$ and $\nabla h=0$. Therefore the quantity $h_t-\triangle h -2\nabla u\nabla h$ has to be non-negative (the fact that $M$ is closed allows for this to be done globally). We impose conditions on $\alpha, \beta$ and possibly other constants and restrictions for $\phi$ (from which one builds $\phi$) such that the expression turns out positive, contradicting the initial assumption. Therefore, we simultaneously determine $h$ and show that it's non-negative.
We note here that it's possible to obtain a local Harnack estimate, in the case when $M$ is complete, non-compact, but this is more technical and it is being done in a subsequent paper.
§ THE HARNACK QUANTITY
In this section we will determine the exact expression of the Harnack quantity and prove that it's positive for any time $t>0$.
Let $f$ be a solution to (<ref>). Assuming that $0<f(x,y)<1$, we are able to introduce $u=\log f$ (and hence $-\infty<u<0$). It follows that $u$ satisfies:
\begin{align}\label{u-Allen-Cahn}
u_t=\Lap u+|\nabla u|^2+1-e^{2u}
\end{align}
Let $h(x,t)$ be the following generic Harnack quantity:
\begin{align*}
h(x,t):=\Lap u +\alpha|\nabla u|^2+\beta e^{2u}+\varphi(x,t)
\end{align*}
where $\varphi(x,t)=\phi(t)+\psi(x)$ ($\phi$ is a time-dependent function, while $\psi$ is space-dependent). In the present situation, since we are working on a compact manifold, we won't need the spatial function. However, we keep it for the time being, for completeness of the method.
Our goal is to compute the time evolution of $h$ and then use the maximum principle to establish the positivity of $h$ given a particular choice of $\alpha$, $\beta$ and $\phi(t)$. Moreover, $\phi(t)$ has to be very large at $t=0$, in order to dominate the other terms and ensure that at time close to $0$ the Harnack quantity is positive.
Applying the heat operator to each term of $h(x,t)$, one obtains:
\begin{align*}
(\partial_t-\Lap)u &= |\nabla u|^2+1-e^{2u} \\
(\partial_t-\Lap)\varphi & = \phi_t-\Lap\psi \\
(\partial_t-\Lap)(\beta e^{2u}) & = 2\beta e^{2u}-2\beta e^{4u}-2\beta e^{2u}|\nabla u|^2=2\beta(e^{2u}-e^{4u}-e^{2u}|\nabla u|^2) \\
(\partial_t-\Lap)(\Lap u) & = \Lap |\nabla u|^2-2(\Lap u)e^{2u}-4|\nabla u|^2e^{2u}\\
(\partial_t-\Lap)(\alpha|\nabla u|^2) &= 2\alpha\nabla u\cdot \nabla(\Lap u)+2\alpha\nabla u\cdot\nabla(|\nabla u|^2)-4\alpha|\nabla u|^2e^{2u}-\alpha\Lap(|\nabla u|^2)\\
2\nabla u\cdot\nabla h & = 2\nabla u\cdot\nabla(\Lap u)+2\alpha\nabla u\cdot \nabla|\nabla u|^2+4\beta|\nabla u|^2e^{2u}+2\nabla u\cdot\nabla\psi
\end{align*}
Recall Bochner's identity:
\[\Lap|\nabla u|^2=2|\nabla\nabla u|^2+2\nabla u\cdot\nabla(\Lap u)+2\Ric(\nabla u,\nabla u)\]
Putting everything together yields the following time evolution for $h(x,t)$:
\begin{align*}
(\partial_t-\Lap)h-2\nabla u\cdot\nabla h & = 2(1-\alpha)|\nabla\nabla u|^2-2(\Lap u)e^{2u}-|\nabla u|^2(4e^{2u}+4\alpha e^{2u}+6\beta e^{2u}) \\
& +2(1-\alpha)\Ric(\nabla u,\nabla u) +2\beta(e^{2u}-e^{4u})\\ & +\phi_t-\Lap\psi-2\nabla u\nabla\psi
\end{align*}
Using the fact that $|\nabla\nabla u|^2\geq\frac{1}{n}(\Lap u)^2$ and $\Ric\geq -k$, one gets the following inequality:
\begin{align*}
(\partial_t-\Lap)h-2\nabla u\cdot\nabla h & \geq \frac{2}{n}(1-\alpha)(\Lap u)^2-2(\Lap u)e^{2u}-|\nabla u|^2(4e^{2u}+4\alpha e^{2u}+6\beta e^{2u}) \\
& -2(1-\alpha)k|\nabla u|^2 +2\beta(e^{2u}-e^{4u})+\phi_t-\Lap\psi-2\nabla u\nabla\psi
\end{align*}
$\Lap u$ can be replaced with $h-\alpha|\nabla u|^2-\beta e^{2u}-\phi-\psi$, which leads to the expression:
\begin{align}\label{Harnack-time-evolution}
(\partial_t-\Lap)h-2\nabla u\cdot\nabla h \geq & \nonumber\\
& h\left[\frac{2(1-\alpha)}{n}h -\frac{4(1-\alpha)}{n}(\alpha|\nabla u|^2+\beta e^{2u}+\phi+\psi)-2e^{2u}\right] \nonumber\\
& +\left[\frac{2(1-\alpha)}{n}(\alpha^2|\nabla u|^4+2\phi\psi)-2k(1-\alpha)|\nabla u|^2+\frac{4\alpha(1-\alpha)}{n}\phi|\nabla u|^2\right. \nonumber \\
& +\left.|\nabla u|^2e^{2u}\left(\frac{4\alpha\beta(1-\alpha)}{n}-6\beta-2\alpha-4\right)\right] \\
& \left[e^{4u}\cdot\frac{2\beta^2(1-\alpha)}{n}+e^{2u}\left(\frac{4\beta(1-\alpha)}{n}\phi+2\phi+2\beta\right)+\right. \nonumber\\
& \left.+\frac{2(1-\alpha)}{n}\phi^2+\phi_t\right] \nonumber\\
& +\left[\frac{4\alpha(1-\alpha)}{n}\cdot \psi|\nabla u|^2-2\nabla u\nabla\psi+e^{2u}\psi\left(\frac{4\beta(1-\alpha)}{n}+2\right)\right.\nonumber \\
&\left. +\frac{2(1-\alpha)}{n}\psi^2-\Lap\psi \right] \nonumber
\end{align}
Heuristically, we group terms that involve $|\nabla u|^2$, powers of $e^{2u}$ and $\psi$ separately. To make the computation easier to follow, we will denote each expression as follows:
\[P_1=\frac{2(1-\alpha)}{n}h -\frac{4(1-\alpha)}{n}(\alpha|\nabla u|^2+\beta e^{2u}+\phi+\psi)-2e^{2u}\]
\begin{align*}
P_2& =\frac{2(1-\alpha)}{n}(\alpha^2|\nabla u|^4+2\phi\psi)-2k(1-\alpha)|\nabla u|^2+\frac{4\alpha(1-\alpha)}{n}\phi|\nabla u|^2\\
& +|\nabla u|^2e^{2u}\left(\frac{4\alpha\beta(1-\alpha)}{n}-6\beta-2\alpha-4\right)
\end{align*}
\[P_3=e^{4u}\cdot\frac{2\beta^2(1-\alpha)}{n}+e^{2u}\left(\frac{4\beta(1-\alpha)}{n}\phi+2\phi+2\beta\right)+\frac{2(1-\alpha)}{n}\phi^2+\phi_t\]
\[P_4=\frac{4\alpha(1-\alpha)}{n}\cdot \psi|\nabla u|^2-2\nabla u\nabla\psi+e^{2u}\psi\left(\frac{4\beta(1-\alpha)}{n}+2\right)+\frac{2(1-\alpha)}{n}\psi^2-\Lap\psi\]
We have thus shown that
\[(\partial_t-\Lap)h-2\nabla u\cdot\nabla h \geq h P_1+P_2+P_3+P_4.\]
Assume there is a first time when $h\leq 0$. Since $\varphi(x,t)$ will be constructed such that it goes to infinity at time $0$, $h$ has to be positive close to the starting time. But since the solution is smooth (it's a heat-type equation), $h$ is also smooth, so there has to be a first time $t_0>0$ when $\min_M h(x,t)=0$. Recall that $M$ is a compact manifold, so by applying the maximum principle $h_t\leq 0$, $\nabla h=0$ and $\Lap h\geq 0$ at that time (the function $h(x,t_0)$ has a spatial minimum on the whole $M$ at $(x_0,t_0)$). Moreover, at this point $\Lap u=-\alpha |\nabla u|^2-\beta e^{2u}-\phi-\psi$. Therefore at $(x_0,t_0)$:
\[0\geq P_2+P_3+P_4.\]
We want to obtain a contradiction, so all we need to do is find specific values for $\alpha,\beta$ and $\varphi$ such that $P_2+P_3+P_4>0$.
In the present situation, the manifold is closed, so we don't have to worry about boundary issues (the minimum will be spatially global), therefore we don't need the spatial component for $\varphi$, hence $\psi=0$ or $\varphi(x,t)=\phi(t)$. Thus, $P_4=0$.
All we are left to do is find $\alpha,\beta, \phi(t)$ such that $P_2$ and $P_3$ are non-negative and at least one of them positive.
We start by analyzing $P_2$. Recall that
\begin{align*}
P_2& =\frac{2(1-\alpha)}{n}\alpha^2|\nabla u|^4-2k(1-\alpha)|\nabla u|^2+\frac{4\alpha(1-\alpha)}{n}\phi|\nabla u|^2\\
& +|\nabla u|^2e^{2u}\left(\frac{4\alpha\beta(1-\alpha)}{n}-6\beta-2\alpha-4\right)
\end{align*}
Notice that since $\psi=0$, the term involving $\phi\psi$ disappeared. Imposing that $\alpha\in(0,1)$ assures that the first is non-negative. To ensure the positivity of the sum of the second and third term, we need $\phi\geq \frac{nk}{2\alpha}$. Finally, for the last term to be non-negative, we need $\frac{4\alpha\beta(1-\alpha)}{n}-6\beta-2\alpha-4\geq 0$, which is equivalent to
\[\beta\leq -\frac{n(\alpha+2)}{2\alpha^2-2\alpha+3n}<0\]
Note that this means $\beta$ has to be negative.
Next we analyze $P_3$. Recall that:
\[P_3= e^{4u}\cdot\frac{2\beta^2(1-\alpha)}{n}+2e^{2u}\left[\left(\frac{2\beta(1-\alpha)}{n}+1\right)\phi+\beta\right]+\frac{2(1-\alpha)}{n}\phi^2+\phi_t\]
By applying the inequality $p x^2+2q x\geq -\frac{q^2}{p}$ for $x=e^{2u}$, we obtain:
\begin{align*}
P_3&\geq -\frac{\left[\left(\frac{2\beta(1-\alpha)}{n}+1\right)\phi+\beta\right]^2}{\frac{2\beta^2(1-\alpha)}{n}}+\frac{2(1-\alpha)}{n}\phi^2+\phi_t\\
\end{align*}
At this point, we need to solve a differential inequality, and there are two situations: first, if $\beta\geq -\frac{n}{4(1-\alpha)}$ (which is smaller than $-\frac{n(\alpha+2)}{2\alpha^2-2\alpha+3n}$ for any $n\geq 3$), then both expressions $\left(1+\frac{n}{2\beta(1-\alpha)}\right)$ and $\left(1+\frac{n}{4\beta(1-\alpha)}\right)$ are negative. This means that we want $\phi$ to satisfy:
\[-a \phi^2+b\phi -c+\phi_t\geq 0\]
for some positive numbers $a,b,c>0$.
Second, if $\beta\leq -\frac{n}{2(1-\alpha)}$ (which is also smaller than $-\frac{n(\alpha+2)}{2\alpha^2-2\alpha+3n}$ for any $n\geq 1$ and $\alpha\geq \frac{1}{2}$ or for any $n\geq 2$) then both of those expressions are positive, so $\phi$ would satisfy:
\[a \phi^2-b\phi -c+\phi_t\geq 0\]
for some positive numbers $a,b,c>0$.
Since the second case gives a solution for a larger range of values for $\beta$ and $n$, we will focus on it. For an arbitrarily small $d>0$, if we determine $\phi>0$ such that
\[a \phi^2-(b+d)\phi -c+\phi_t=0\]
then we will ensure that $P_3>0$.
The above equation has an exact solution, given by:
\[\phi_d(t)=\frac{-2q\coth(-qt)+(b+d)}{2a}=\frac{[(b+d)-2q]-e^{2qt}(b+d+2q)}{2a(1-e^{2qt})}\]
where $a=-\frac{2}{\beta}\left(1+\frac{n}{4\beta(1-\alpha)}\right)$, $b=2\left(1+\frac{n}{2\beta(1-\alpha)}\right)$, $c=\frac{2n}{1-\alpha}$ and $q=\sqrt{ac+\frac{1}{4}(b+d)^2}$. Notice that, in fact, $c$ depends on $a$ and $b$ since $2+\frac{c}{2\beta}=b$ and $2+\frac{c}{4\beta}=-a\beta$. Moreover, $q$ is well defined, since $a,c>0$ and thus the expression under the root is positive.
The expression of $\phi_d(t)$ from above is positive for all $t$ and its limit, as $t\to 0^+$ is $\infty$ ($a,b,q,c,d>0$). These are the desired properties for $\phi_d(t)$ that guarantee that $P_3>0$.
Recall that in order for $P_2>0$, $\phi(t)$ must satisfy $\phi\geq \frac{nk}{2\alpha}$, which in turns imposes the condition $\frac{b+d+2q}{2a}\geq \frac{nk}{2\alpha}$. By replacing $q$ with the expression above, we get the following condition:
\[\frac{b+d+2\sqrt{ac+\frac{1}{4}(b+d)^2}}{2a}\geq\frac{nk}{2\alpha}.\]
Since $ac$ is positive:
\[\frac{b+d+2\sqrt{ac+\frac{1}{4}(b+d)^2}}{2a}\geq\frac{b+d}{2a}.\]
If one imposes that $\frac{b}{a}\geq \frac{nk}{2\alpha}$, since $d,a$ are positive, then the condition on $\phi$ is satisfied. We will show that this leads to an upper bound on $\beta$.
First, $\frac{b}{a}=-\beta\frac{2\beta(1-\alpha)+n}{4\beta(1-\alpha)+n}\geq -2\beta$ since the quotient is less than $1$, which follows from the fact that $\beta<0$. Therefore, we only need the condition that $-2\beta\geq \frac{nk}{2\alpha}$, which is equivalent to saying that $\beta\leq -\frac{nk}{4\alpha}$.
In conclusion, for $\alpha\in(0,1)$, $\beta\leq\min\{-\frac{n(\alpha+2)}{2\alpha^2-2\alpha+3n}, -\frac{nk}{4\alpha}, -\frac{n}{2(1-\alpha)}\}$ and $\phi_d(t)$ having the above expression (with $a,b,d,q$ only depending on $\alpha$ and $\beta$), $P_2+P_3>0$. This contradicts the fact that at the point $(x_0,t_0)$, $P_2+P_3\leq 0$, hence there does not exist any point for which $h\leq 0$, hence $h_d(x,t)= \Lap u +\alpha|\nabla u|^2+\beta e^{2u}+\phi_d(x,t)> 0$ for any $t>0$.
The final step to prove theorem <ref> is to take $d\to 0$, which will turn $\phi_d(t)$ into $\phi(t)=\frac{-2q\coth(-qt)+b}{2a}=\frac{b-q-e^{2qt}(b+2q)}{2a(1-e^{2qt})}$, where $q=\frac{1}{2}\sqrt{b^2+4ac}$.
§ THE CLASSICAL HARNACK INEQUALITY
In this section we will obtain a classical Harnack inequality. The procedure will follow the classical method developed by Li-Yau and consists of integrating the differential Harnack estimate along a space time-curve.
We choose two points $(x_1,t_1)$ and $(x_2,t_2)$ and a space-time path connecting them $\gamma:[t_1,t_2]\to M$, such that $\gamma(t_1)=x_1$ and $\gamma(t_2)=x_2$. The value of the function $u(x,t)=\log f(x,t)$ along the path is given by $v(t):=u(\gamma(t),t)$. Its time derivative has the expression $v'(t)=u_t+\nabla u\cdot\frac{d\gamma}{dt}$, which is the same as
\[v'(t)=\Lap u+|\nabla u|^2+1-e^{2u}+\nabla u\cdot\frac{d\gamma}{dt}.\]
From the Harnack inequality, one can bound $\Lap u$ and get that $\Lap u\geq -\alpha|\nabla u|^2-\beta e^{2u}-\phi(t)$ which gives a lower bound for $v'(t)$:
\[v'(t)\geq (1-\alpha) |\nabla u|^2-(\beta+1) e^{2u}+1-\phi(t)+\nabla u\cdot\frac{d\gamma}{dt}.\]
Recall that $\alpha\in(0,1)$ and $\beta\leq\min\left\{-\frac{n(\alpha+2)}{2\alpha^2-2\alpha+3n}, -\frac{nk}{4\alpha}, -\frac{n}{2(1-\alpha)}\right\}$.
Using the inequality $ay^2+by\geq-\frac{b^2}{4a}$ for $y=|\nabla u|$, we obtain that
\[v'(t)\geq -\frac{1}{4(1-\alpha)}\left|\frac{d\gamma}{dt}\right|^2-(\beta+1)e^{2u}+1-\phi(t).\]
Since $f(x,t)\in(0,1)$, it follows that $e^{2u}=f^2\in(0,1)$, so $-e^{2u}\geq -1$. Moreover, $\beta$ is negative, so $-\beta e^{2u}\geq 0$, therefore one finally has:
\[v'(t)\geq -\frac{1}{4(1-\alpha)}\left|\frac{d\gamma}{dt}\right|^2-\phi(t).\]
First, we integrate in time from $t_1$ and $t_2$ and we obtain
\[v(t_2)-v(t_1)=\int\limits_{t_1}^{t_2}v'(t)\ dt\geq -\frac{1}{4(1-\alpha)}\int\limits_{t_1}^{t_2}\left|\frac{d\gamma}{dt}\right|^2\ dt-\int\limits_{t_1}^{t_2}\phi(t)\ dt.\]
If one chooses $\gamma$ to be a minimizing geodesic between the endpoints, the first integral becomes $\frac{d(x_1,x_2)^2}{t_2-t_1}$.
The second integral can be computed directly:
\[-\int\limits_{t_1}^{t_2}\phi(t)\ dt=-\int\limits_{t_1}^{t_2}\frac{-2q\coth(-qt)+b}{2a}\ dt=-\frac{4q+b}{2a}(t_2-t_1)-\frac{1}{a}\ln\left(\frac{e^{-2qt_2}-1}{e^{-2qt_1}-1}\right)\]
with $a=-\frac{2}{\beta}\left(1+\frac{n}{4\beta(1-\alpha)}\right)$, $b=2\left(1+\frac{n}{2\beta(1-\alpha)}\right)$, $c=\frac{2n}{1-\alpha}$ and $q=\frac{1}{2}\sqrt{b^2+4ac}$.
Finally, by noticing that $v(t_2)-v(t_1)=u(x_2,t_2)-u(x_1,t_1)=\ln\frac{f(x_2,t_2)}{f(x_1,t_1)}$ and taking the exponential we get the result of theorem <ref>.
If $M$ has non-negative Ricci curvature, one may choose $\alpha=\frac{1}{2}$ and $\beta=-n$. It follows that $a=1/n$, $b=0$, $c=4n$ and $q=2$. This turns the previous classical Harnack inequality into a simpler form:
\[\frac{f(x_2,t_2)}{f(x_1,t_1)}\geq e^{-\frac{d(x_1,x_2)^2+8n(t_2-t_1)^2}{2(t_2-t_1)}}\left(\frac{e^{-4t_2}-1}{e^{-4t_1}-1}\right)^{-n}\]
§ THE TRAVELING WAVE SOLUTION
We will now focus on the traveling wave solution, when when $M=\mathbb{R}^n$ (hence the Ricci curvature is $0$). Like at the end of the previous section, we choose $\alpha=\frac{1}{2}$ and $\beta=-n$, making $a=1/n$, $b=0$, $c=4n$ and $q=2$. The expression of $\phi(t)$ becomes:
\[\phi(t)=2n\frac{e^{4t}+1}{e^{4t}-1}\]
Notice that $\lim\limits_{t\to\infty}\phi(t)=2n$.
Theorem <ref> says that the quantity $h(x,t)=\Lap u+\frac{1}{2}|\nabla u|^2-n e^{2u}+\phi(t)$ is always non-negative. Replacing $u$ with $\log f$, the above becomes:
\[\frac{f_t}{f}-\frac{1}{2}\frac{|\nabla f|^2}{f^2}+(1-n)f^2+\phi(t)-1\geq 0.\]
Assume that $f$ is in fact a traveling wave with speed $c\geq 0$, i.e. there is a function $p:\mathbb{R}^n\to\mathbb{R}$ such that $f(x,t)=p(x_1,x_2,...,x_n-ct)$. In that case the above inequality becomes:
\[-c\frac{\partial_n p}{p}-\frac{|\nabla p|^2}{2p^2}+(1-n)p^2+\phi(t)-1\geq 0\]
Finally, in the case when the traveling wave is in fact stationary, i.e. $c=0$ (recall that, in that case, $p$ satisfies the elliptic Allen-Cahn equation (<ref>)), the above inequality becomes:
\[-\frac{|\nabla p|^2}{p^2}+(1-n)p^2+\phi(t)-1\geq 0\]
Since the wave is stationary, it will look the same at any time, so the inequality is true as $t\to\infty$, which leads to:
\[-\frac{|\nabla p|^2}{p^2}-p^2(n-1)+2n-1\geq 0\]
Therefore we obtain
\[|\nabla p|^2\leq p^2[2n-1-(n-1)p^2]\]
proving Corollary <ref>.
Comparison with Modica's result Recall that Modica's result asserts that a solution of the equation $\Lap u=F'(u)$ satisfies $|\nabla u|^2\leq 2 F(u)$, for a non-negative function $F$. In the Allen-Cahn equation case, this translates to $|\nabla u|^2\leq \frac{1}{2}u^2(u^2-2)+\frac{1}{2}$. A visual comparison between the two degree 4 polynomials suggests that our estimate would be an improvement around $0$, if proven to hold for $f\in[-1,1]$. This is because on the interval $[-1,1]$ the graph of $x^2(3-x^2)$ sits mostly below the graph of $\frac{1}{2}x^2(x^2-2)+\frac{1}{2}$ (we consider $n=2$, and notice that for larger $n$ the estimate is worse).
Figure 1: Comparision between $\frac{1}{2}x^2(x^2-2)+\frac{1}{2}$ (blue) and $x^2(3-x^2)$ (red)
|
1511.00417
|
The mathematical modeling and numerical simulation of semiconductor-electrolyte systems play important roles in the design of high-performance semiconductor-liquid junction solar cells. In this work, we propose a macroscopic mathematical model, a system of nonlinear partial differential equations, for the complete description of charge transfer dynamics in such systems. The model consists of a reaction-drift-diffusion-Poisson system that models the transport of electrons and holes in the semiconductor region and an equivalent system that describes the transport of reductants and oxidants, as well as other charged species, in the electrolyte region. The coupling between the semiconductor and the electrolyte is modeled through a set of interfacial reaction and current balance conditions. We present some numerical simulations to illustrate the quantitative behavior of the semiconductor-electrolyte system in both dark and illuminated environments. We show numerically that one can replace the electrolyte region in the system with a Schottky contact only when the bulk reductant-oxidant pair density is extremely high. Otherwise, such replacement gives significantly inaccurate description of the real dynamics of the semiconductor-electrolyte system.
Semiconductor-electrolyte system, reaction-drift-diffusion-Poisson system, semiconductor modeling, interfacial charge transfer, interface conditions, semiconductor-liquid junction, solar cell simulation, nano-scale device modeling.
82D37, 34E05, 35B40, 78A57
§ INTRODUCTION
The mathematical modeling and simulation of semiconductor devices have been extensively studied in past decades due to their importance in industrial applications; see <cit.> for overviews of the field and <cit.> for more details on the physics, classical and quantum, of semiconductor devices. In the recent years, the field has been boosted significantly by the increasing need for simulation tools for designing efficient solar cells to harvest sunlight for clean energy. Various theoretical and computational results on traditional semiconductor device modeling are revisited and modified to account for new physics in solar cell applications. We refer interested readers to <cit.> for a summary of various types of solar cells that have been constructed, to <cit.> for simplified analytical solvable models that have been developed, and to <cit.> for more advanced mathematical and computational analysis of various models. Mathematical modeling and simulation provide ways not only to improve our understanding of the behavior of the solar cells under experimental conditions, but also to predict the performance of solar cells with general device parameters, and thus they enable us to optimize the performance of the cells by selecting the optimal combination of these parameters.
One popular type of solar cells, besides those made of semiconductor p-n junctions, are cells made of semiconductor-liquid junctions. A typical liquid-junction photovoltaic solar cell consists of four major components: the semiconductor, the liquid, the semiconductor-liquid interface and the counter electrode; see a rough sketch in Fig. <ref> (left). There are many possible semiconductor-liquid combinations; see for instance, <cit.> for ${\rm Si/viologen}^{2+/+}$ junctions, <cit.> for n-type ${\rm InP/Me_2Fc}^{+/0}$ junctions, and <cit.> for a summary of many other possibilities. The working mechanism of this type of cell is as follows. When sunlight is absorbed by the semiconductor, free conduction electron-hole pairs are generated. These electrons and holes are then separated by an applied potential gradient across the device. The separation of the electrons and holes leads to electrical current in the cell and concentration of charges on the semiconductor-liquid interface where electrochemical reactions and charge transfer occur. We refer interested reader to <cit.> for physical principles and technical specifics of various types of liquid-junction solar cells.
Charge transport processes in semiconductor-liquid junctions have been studied in the past by many investigators; see <cit.> for a recent review. The mechanisms of charge generation, recombination, and transport in both the semiconductor and the liquid are now well understood. However, the reaction and charge transfer process on the semiconductor-liquid interface is far less understood despite the extensive recent investigations from both physical <cit.> and computational <cit.> perspectives. The objective of this work is to mathematically model this interfacial charge transfer process so that we could derive a complete system of equations to describe the whole charge transport process in the semiconductor-liquid junction.
Left: Sketch of main components in a typical semiconductor-liquid junction solar cell. Middle and Right: Two typical settings for semiconductor-electrolyte systems in dimension two. The semiconductor $\sfS$ and the electrolyte $\sfE$ are separated by the interface $\Sigma$.
To be specific, we consider here semiconductor-liquid junction with the liquid being electrolyte that contains reductant $r$, oxidant $o$ and some other charged species that do not interact with the semiconductor. We denote by $\Omega\subset\bbR^d$ ($d\ge 1$) the domain of interest which contains the semiconductor part $\Omega_{\sfS}$ and the electrolyte part $\Omega_{\sfE}$. We denote by $\Sigma \equiv \partial\Omega_\sfE\cap\partial\Omega_\sfS$ the interface between the semiconductor and electrolyte, $\Gamma_\sfC$ the surface of the current collector at the semiconductor end, and $\Gamma_\sfA$ the surface of the counter (i.e., auxiliary) electrode. $\Gamma_\sfS=\partial\Omega_\sfS\backslash(\Sigma\cup \Gamma_\sfC)$ is the part of the semiconductor boundary that is neither the interface $\Sigma$ nor the contact $\Gamma_\sfC$, and $\Gamma_\sfE=\partial\Omega_\sfE\backslash(\Sigma\cup \Gamma_\sfA)$ the part of electrolyte that is neither the interface $\Sigma$ nor the surface of the counter electrode $\Gamma_\sfA$. We denote by $\bnu(\bx)$ the unit outer normal vector at a point $\bx$ on the boundary of the domain $\partial\Omega=\Gamma_\sfC\cup\Gamma_\sfS\cup\Gamma_\sfA\cup\Gamma_\sfE$. To deal with discontinuities of quantities across the interface $\Sigma$, we use $\Sigma_-$ and $\Sigma_+$ to denote the semiconductor and the electrolyte sides, respectively, of $\Sigma$. On the interface, we use $\bnu^-(\bx)$ and $\bnu^+(\bx)$ to denote the unit normal vectors at $\bx\in\Sigma$ pointing to the semiconductor and the electrolyte domains, respectively.
The rest of the paper is structured as follows. In the next three sections, we present the three main components of the mathematical model. We first model in Section <ref> the dynamics of the electrons and holes in the semiconductor $\Omega_\sfS$. We then model in Section <ref> the dynamics of the reductants and oxidants, as well as other charged species, in the electrolyte $\Omega_\sfE$. In Section <ref> we model the reaction-transfer dynamics on the interface $\Sigma$. Once the mathematical model has been constructed, we develop in Section <ref> some numerical schemes for the numerical simulation of the device in simplified settings. We present some numerical experiment in Section <ref> where we exhibit the benefits of modeling the complete semiconductor-electrolyte system. Concluding remarks are offered in Section <ref>.
§ TRANSPORT OF ELECTRONS AND HOLES
The modeling of transport of free conduction electrons and holes in semiconductor devices has been well studied in recent decades <cit.>. Many different models have been proposed, such as the Boltzmann-Poisson system <cit.>, the energy transport system <cit.> and the drift-diffusion-Poisson system <cit.>. For the purpose of computational efficiency, we employ the reaction-drift-diffusion-Poisson model in this work. Let us denote by $(0, T]$ the time interval in which we interested. The bipolar drift-diffusion-Poisson model can be written in the following form:
\begin{equation}\label{EQ:DDP}
\begin{array}{rcll}
\partial_t \rho_n + \nabla\cdot\bJ_n
&=& -R_{np}(\rho_n,\rho_p)+\gamma G_{np}(\bx), &\mbox{ in }\ (0, T]\times\Omega_\sfS\\
\partial_t\rho_p + \nabla\cdot\bJ_p
&=& -R_{np}(\rho_n,\rho_p)+\gamma G_{np}(\bx), &\mbox{ in }\ (0, T]\times\Omega_\sfS\\
-\nabla\cdot(\eps_r^\sfS\nabla \Phi)&=&
\dfrac{q}{\eps_0}[C(\bx)+\alpha_p\rho_p+\alpha_n\rho_n],
&\mbox{ in }\ (0, T]\times\Omega_\sfS.
\end{array}
\end{equation}
with the fluxes of electrons and holes are given respectively by
\begin{equation}\label{EQ:DDP Current}
\begin{array}{ll}
\bJ_n = -D_n \nabla \rho_n -\alpha_n \mu_n \rho_n \nabla \Phi, \qquad
\bJ_p = -D_p \nabla \rho_p -\alpha_p \mu_p \rho_p \nabla \Phi.
\end{array}
\end{equation}
Here $\rho_n(t,\bx)$ and $\rho_p(t,\bx)$ are the densities of the electrons and the holes, respectively, at time $t$ and location $\bx$, and $\Phi(t,\bx)$ is the electrical potential. The notation $\partial_t$ denotes the derivative with respect to $t$, while $\nabla$ denotes the usual spatial gradient operator. The constant $\eps_0$ is the dielectric constant in vacuum, and the function $\eps_r^\sfS(\bx)$ is the relative dielectric function of the semiconductor material. The function $C(\bx)$ is the doping profile of the device. The coefficients $D_n$ (resp., $D_p$) and $\mu_n$ (resp., $\mu_p$) are the diffusivity and the mobility of electrons (resp., holes). These parameters can be computed from the first principles of statistical physics. In some practical applications, however, they can be fitted from experimental data as well; see, for example, the discussion in <cit.>. The parameter $q$ is the unit electric charge constant, while $\alpha_n=-1$ and $\alpha_p=1$ are the charge numbers of electrons and holes, respectively. The diffusivity and the mobility coefficients are related through the Einstein
relations $D_n=U_\cT\mu_n$ and $D_p=U_\cT\mu_p$ with $U_\cT$ the thermal voltage at temperature $\cT$ given by $U_\cT=k_B\ \cT/q$, and $k_B$ being the Boltzmann constant.
§.§ Charge recombination and generation
The function $R_{np}(\rho_n,\rho_p)$ describes the recombination and generation of electron-hole pairs due to thermal excitation. It represents the rate at which the electron-hole pairs are eliminated through recombination (when $R_{np}>0$) or the rate at which electron-hole pairs are generated (when $R_{np}<0$). Due to the fact that electrons and holes are always recombined and generated in pairs, we have the same rate function for the two species. To be specific, we consider in this work the Auger model of recombination that is based on interactions between multiple electrons and holes, but we refer interested readers to <cit.> for discussions on other popular recombination models such as the Shockley-Read-Hall (SRH) model and the Langevin model.
The Auger model is relevant in cases where the carrier densities are high (for instance in doped materials). It is expressed as
\begin{equation}\label{EQ:Auger}
R_{np}(\rho_n,\rho_p) = (A_n \rho_n +A_p \rho_p)(\rho_{isc}^2-\rho_n\rho_p),
\end{equation}
where $A_n$ and $A_p$ are the Auger coefficients for electrons and holes respectively. For given materials, $A_n$ and $A_p$ can be measured by experiments. The parameter $\rho_{isc}$ is the intrinsic carrier density that is often calculated from the following formula (see <cit.>):
\begin{equation}\label{EQ:NI}
\rho_{isc} = \sqrt{N_c N_v}\Big(\dfrac{\cT}{300}\Big)^{1.5} e^{-\frac{E_g}{2k_B \cT}}
\end{equation}
where the band gap $E_g = E_{g0}-\alpha \cT^2/(\cT+\beta)$ with $E_{g0}$ the band gap at $\cT=0 K$ ($E_{g0}=1.17q$ for silicon for instance), $\alpha = 4.73\ 10^{-4}q$, and $\beta = 636$. The parameters $N_c$ and $N_v$ are effective density of states in the conduction and the valence bands, respectively, at $\cT=300 K$.
When the semiconductor device is illuminated by sunlight, the device absorbs photon energy. The absorbed energy creates excitons (bounded electron-hole pairs). The excitons are then separated into free electrons and holes which can then move independently. This generation of free electron-hole pairs is modeled by the source function $G_{np}(\bx)$ in the transport equation (<ref>). Once again, due to the fact that electrons and holes are always generated in pairs, the generating functions are the same for electrons and holes. We take a model that assumes that photons travel across the device in straight lines. That is, we assume that photons do not get scattered by the semiconductor material during their travel inside the device. This is a reasonable assumption for small devices that have been utilized widely <cit.>. Precisely, the generation of charges is given as
\begin{equation}\label{EQ:Illu Source}
\begin{array}{rl}
\sigma(\bx) G_0(\bx_0)e^{-\int_0^{s}\sigma(\bx_0+s'\btheta_0)ds'},& \mbox{if}\ \bx=\bx_0+s\btheta_0\\
0,& \mbox{otherwise}
\end{array}\right.
\end{equation}
where $\bx_0\in\Gamma_\sfS$ is the incident location, $\btheta_0$ is the incident direction, $\sigma(\bx)$ is the absorption coefficient (integrated over the usable spectrum), and $G_0(\bx_0)$ is the surface photon flux at $\bx_0$. The control parameter $\gamma \in\{0,\ 1\}$ in (<ref>) is used to turn on and off the illumination mechanism, and $\gamma=0$ and $\gamma=1$ are the dark and illuminated cases, respectively.
§.§ Boundary conditions
We have to supply boundary conditions for the equations in the semiconductor domain. The semiconductor boundary, besides the interface $\Sigma$, is split into two parts, the current collector $\Gamma_\sfC$ and the rest. The boundary condition on the current collector is determined by the type of contacts formed there. There are mainly two types of contacts, the Ohmic contact and the Schottky contact.
Dirichlet at Ohmic contacts. Ohmic contacts are generally used to model metal-semiconductor junctions that do not rectify current. They are appropriate when the Fermi levels in the metal contact and adjacent semiconductor are aligned. Such contacts are mainly used to carry electrical current out and into semiconductor devices, and should be fabricated with little (or ideally no) parasitic resistance. Low resistivity Ohmic contacts are also essential for high-frequency operation. Mathematically, Ohmic contacts are modeled by Dirichlet boundary conditions which can be written as <cit.>
\begin{equation}\label{EQ:DDP BC Dirich}
\begin{array}{lccl}
\rho_n(t,\bx)=\rho_n^e(\bx), & & \rho_p(t,\bx)=\rho_p^e(\bx), & \mbox{on}\ \ (0, T]\times\Gamma_{\sfC},\\
\Phi(t,\bx) =\varphi_{bi}+\varphi_{app}, & & & \mbox{on}\ \ (0, T]\times\Gamma_{\sfC},
\end{array}
\end{equation}
where $\varphi_{bi}$ and $\varphi_{app}$ are the built-in and applied potentials, respectively. The boundary values $\rho_n^e$, $\rho_p^e$ for the Ohmic contacts are calculated following the assumptions that the semiconductor is in stationary and equilibrium state and that the charge neutrality condition holds. This means that right-hand-side of the Poisson equation disappears so that
\begin{equation}\label{EQ:Equi}
\end{equation}
Thermal equilibrium implies that generation and recombination balance out, so $R_{np}=0$ at Ohmic contacts. This leads to the mass-action law, between the density of electrons and holes:
\begin{equation}\label{EQ:Mass}
\rho_n^e\rho_p^e-\rho_{isc}^2=0.
\end{equation}
The system of equations (<ref>) and (<ref>) admit a unique solution pair $(\rho_n,\rho_p)$, which is given by
\begin{equation}\label{EQ:Ohmic}
\begin{array}{l}
\rho_n^e(t,\bx)=\dfrac{1}{2}(\sqrt{C^2+4\rho_{isc}^2}+C),\qquad
\rho_p^e(t,\bx)=\dfrac{1}{2}(\sqrt{C^2+4\rho_{isc}^2}-C) .
\end{array}
\end{equation}
These densities result in a built-in potential that can be calculated as
\begin{equation}
\varphi_{bi} = U_\cT \ln(\rho_n^e/\rho_{isc}).
\end{equation}
Note that due to the fact that the doping profile $C$ varies in space, these boundary values are different on different part of the boundary.
Robin (or mixed) at Schottky contacts. Schottky contacts are used to model metal-semiconductor junctions that have rectifying effects (in the sense that current flow through the contacts is rectified). They are appropriate for contacts between a metal and a lightly doped semiconductor. Mathematically, at a Schottky contact, Robin- (also called mixed-) type of boundary conditions are imposed for the ${\rm n}$- and ${\rm p}$-components, while Dirichlet-type of conditions are imposed for the $\Phi$-component. More precisely, these boundary conditions are <cit.>:
\begin{equation}\label{EQ:DDP BC Robin}
\begin{array}{lccl}
\bnu\cdot\bJ_n(t,\bx)=v_n(\rho_n-\rho_n^e)(\bx), & &
\bnu\cdot\bJ_p(t,\bx)=v_p(\rho_p-\rho_p^e)(\bx), & \mbox{on}\ \ (0, T]\times\Gamma_{\sfC},\\
\Phi(t,\bx) =\varphi_{Stky}+\varphi_{app}, & & & \mbox{on}\ \ (0, T]\times\Gamma_{\sfC}.
\end{array}
\end{equation}
Here the parameters for the Schottky barrier are the recombination velocities $v_n$ and $v_p$, and the height of the potential barrier, $\varphi_{Stky}$, which depends on the materials of the semiconductor and the metal in the following way:
\begin{equation}\label{EQ:Schottky Height}
\varphi_{Stky} = \left\{
\begin{array}{ll}
\Phi_m-\chi, & \mbox{n-type}\\
\frac{E_g}{q}-(\Phi_m-\chi), & \mbox{p-type}
\end{array}
\right.
\end{equation}
where $\Phi_m$ is the work function, i.e., the potential difference between the Fermi energy and the vacuum level, of the metal and $\chi$ is the electron affinity, i.e., the potential difference between the conduction band edge and the vacuum level. $E_g$ is again the band gap. The values of the parameters $v_n,\ v_p,\ \Phi_m$, and $\chi$ are given in Tab. <ref> of Section <ref>.
Neumann at insulating boundaries. On the part of the semiconductor boundary that is not the current collector, it is natural to impose insulating boundary conditions which ensures that there is no charge or electrical currents through the boundary. The conditions are
\begin{equation}\label{EQ:DDP BC Neumann}
\begin{array}{lccl}
\bnu\cdot D_n\nabla\rho_n(t,\bx)=0, & &
\bnu\cdot D_p\nabla\rho_p(t,\bx)=0, & \mbox{on}\ \ (0, T]\times\Gamma_{\sfS},\\
\bnu\cdot \eps_r^\sfS \nabla\Phi(t,\bx) =0, & & & \mbox{on}\ \ (0, T]\times\Gamma_{\sfS}.
\end{array}
\end{equation}
In solar cell applications, part of the boundary $\Gamma_\sfS$ is where illumination light enters the semiconductor.
We finish this section with the following remarks. It is generally believed that the Boltzmann-Poisson model <cit.> is a more accurate model for charges transport in semiconductors. However, the Boltzmann-Poisson model is computationally more expensive to solve and analytically more complicated to analyze. The drift-diffusion-Poisson model (<ref>) can be regarded as a macroscopic approximation to the Boltzmann-Poisson model. The validity of the drift-diffusion-Poisson model can be justified in the case when the mean free path of the charges is very small compared to the size of the device and the potential drop across the device is small (so that the electric field is not strong); see, for instance, <cit.> for such a justification.
§ CHARGE TRANSPORT IN ELECTROLYTES
We now present the equations for the reaction-transport dynamics of charges in an electrolyte. To be specific, we consider here only electrolytes that contain reductant-oxidant pairs (denoted by $r$ and $o$) that interact directly with the semiconductor through electrons transfer (which we will model in the next section), and $N$ other charged species (denoted by $j=1,...,N$) that do not interact directly with the semiconductor through electron transfer. We also limit our modeling efforts to reaction, recombination, transport, and diffusion of the charges. Other more complicated physical and chemical processes are neglected.
We model the charge transport dynamics in electrolyte again with a set of reaction-drift-diffusion-Poisson equations. In the electrochemistry literature, this mathematical description of the dynamics is often called the Poisson-Nernst-Planck theory <cit.>. Let us denote by $\rho_r(t,\bx)$ the density of the reductants, by $\rho_o(t,\bx)$ the density of the oxidants, and by $\rho_j$ ($1\le j\le N$) the density of the other $N$ charge species. Then these densities solve the following system that is of the same form as (<ref>):
\begin{equation}\label{EQ:DDP Redox}
\begin{array}{ll}
\partial_t \rho_r + \nabla\cdot \bJ_r = +R_{ro}(\rho_r,\rho_o),
&\text{in }\ (0, T]\times\Omega_\sfE,\\
\partial_t \rho_o + \nabla\cdot \bJ_o = -R_{ro}(\rho_r,\rho_o),
&\text{in }\ (0, T]\times\Omega_\sfE,\\
\partial_t \rho_j + \nabla\cdot \bJ_j = R_{j}(\rho_1,\cdots,\rho_N),\ \ 1\le j\le N
&\text{in }\ (0, T]\times\Omega_\sfE,\\
-\nabla\cdot\eps_r^\sfE\nabla\Phi= \dfrac{q}{\eps_0} (\alpha_o\rho_o+\alpha_r\rho_r+\sum_{j=1}^N\alpha_j\rho_j),
&\text{in }\ (0, T]\times\Omega_\sfE.
\end{array}
\end{equation}
with the fluxes given respectively by
\begin{equation}\label{EQ:DDP Redox Current}
\begin{array}{lcl}
\bJ_r = -D_r \nabla\rho_r - \alpha_r\mu_r \rho_r \nabla \Phi, & &
\bJ_o = -D_o \nabla\rho_o - \alpha_o\mu_o \rho_o \nabla \Phi\\
\bJ_j = -D_j \nabla\rho_j - \alpha_j\mu_j \rho_j \nabla \Phi, & & 1\le j\le N
\end{array}
\end{equation}
where again the diffusion coefficient $D_r$ (resp., $D_o$ and $D_j$) is related to the mobility $\mu_r$ (resp., $\mu_o$ and $\mu_j$) through the Einstein relation $D_r=U_\cT \mu_r$ (resp., $D_o=U_\cT \mu_o$ and $D_j=U_\cT \mu_j$). The parameters $\alpha_o$, $\alpha_r$ and $\alpha_j$ ($1\le j\le N$) are the charge numbers of the corresponding charges species. Depending on the type of the redox pairs in the electrolyte, the charge numbers can be different; see, for instance, <cit.> for a summary of various types of redox electrolytes that have been developed.
Let us remark that in the above modeling of the dynamics of reductant-oxidant pair, we have implicitly assumed that the electrolyte, in which the redox pairs live, is not perturbed by charge motions. In other words, there is no macroscopic deformation of the electrolyte that can occur. If this is not the case, we have to introduce the equations of fluid dynamics, mainly the Navier-Stokes equation, for the fluid motion, and add an advection term (with advection velocity given by the solution of the Navier-Stokes equation) in the current expressions in (<ref>). The dynamics will thus be far more complicated.
§.§ Charge generation through reaction
The reaction mechanism between the oxidants and the reductants is modeled by the reaction rate function $R_{ro}$. Note that the elimination and generation of the redox pairs are different from those of the electrons and holes. An oxidant is eliminated (resp., generated) when a reductant is generated (resp., eliminated) and vice versa. This is the reason why there is a negative sign in front of the function $R_{ro}$ in the second equation of (<ref>). The oxidation-reduction reaction requires free electrons which are only available through the semiconductor. Therefore this reaction occurs mainly on the semiconductor-electrolyte interface. We thus assume in general that there is no oxidation-reduction reaction in the bulk electrolyte; that is,
\begin{equation}
R_{ro}(\rho_r,\rho_o)=0,\quad \mbox{in}\ \ (0, T]\times\Omega_\sfE .
\end{equation}
This is what we adopt in the simulations of Section <ref>. The oxidation-reduction reaction on the interface $\Sigma$ will be modeled in the next section.
The reactions among other charged species in the electrolyte are modeled by the reaction rate functions $R_j$ ($1\le j\le N$). The exact forms of these rate functions can be derived following the law of mass action once the types of reactions among the charged species presented in the electrolyte are known. We refer to <cit.> for the rate functions for various chemical reactions.
§.§ Boundary conditions
It is generally assumed that the interface of semiconductor and electrolyte is far from the counter electrode. Therefore the values for the densities of redox pairs on the electrode $\Gamma_\sfA$ are set as their bulk values. Mathematically, this means that Dirichlet boundary conditions have to be imposed:
\begin{equation}\label{EQ:DDP Redox BC}
\begin{array}{lll}
\rho_r(t,\bx)=\rho_r^\infty(\bx), & \rho_o(t,\bx)=\rho_o^\infty(\bx), & \mbox{on}\ \ (0, T]\times\Gamma_{\sfA},\\
\rho_j(t,\bx)=\rho_j^\infty(\bx),\ 1\le j\le N, & \Phi(t,\bx) =\varphi_{app}^\sfA, & \mbox{on}\ \ (0, T]\times\Gamma_{\sfA},
\end{array}
\end{equation}
where $\rho_r^\infty$ and $\rho_o^\infty$ are the bulk concentration of the respective species, and $\varphi_{app}^\sfA$ is the applied potential on the counter electrode. The values of these parameters are given in Tab. <ref> in Section <ref>.
On the rest of the electrolyte boundary, $\Gamma_\sfE$, we impose again insulating boundary conditions:
\begin{equation}\label{EQ:DDP Redox BC Neumann}
\begin{array}{lll}
\bnu\cdot D_r\nabla \rho_r(t,\bx)=0, & \bnu\cdot D_o\nabla\rho_o(t,\bx)=0, & \mbox{on}\ \ (0, T]\times\Gamma_{\sfE},\\
\bnu\cdot D_j\nabla \rho_j(t,\bx)=0,\ 1\le j\le N, & \bnu\cdot \eps_r^\sfE\nabla\Phi(t,\bx) =0, & \mbox{on}\ \ (0, T]\times\Gamma_{\sfE},
\end{array}
\end{equation}
§ INTERFACIAL REACTION AND CHARGE TRANSFER
In order to obtain a complete mathematical model for the semiconductor-electrolyte system, we have to couple the system of equations in the semiconductor with those in the electrolyte through interface conditions that describe the interfacial charge transfer process.
§.§ Electron transfer between electron-hole and redox
The microscopic electrochemical processes on the semiconductor-electrolyte interface can be very complicated, depending on the types of semiconductor materials and electrolyte solutions. There is a vast literature in physics and chemistry devoted to the subject; see, for instance, <cit.> and references therein. We are only interested in deriving macroscopic interface conditions that are consistent with the dynamics of charge transport in the semiconductor and the electrolyte modeled by the equation systems (<ref>) and (<ref>). Without attempting to construct models in the most general cases, we focus here on oxidation-reduction reactions described by the following process,
\begin{equation}
\mbox{Ox}^{|\alpha_0|+} + e^{-}(\mbox{\sfS}) \rightleftharpoons \mbox{Red}^{|\alpha_r|-},
\end{equation}
with $\alpha_o-\alpha_r=1$, $\mbox{Ox}$ and $\mbox{Red}$ denoting respectively the oxidant and the reductant, and $e^-(\sfS)$ denoting an electron from the semiconductor. Experimental studies semiconductor-electrolyte interface with this type of reaction can be found in <cit.> for ${\rm Si/viologen}^{2+/+}$ interfaces and in <cit.> for n-type ${\rm InP/Me_2Fc}^{+/0}$ interfaces.
The changes of the concentrations of the redox pairs on $\Sigma_+$, after taking into account the conservation of $\rho_r+\rho_o$, can be written respectively as:
\begin{equation}
\dfrac{d{\rho_r}}{dt}={k_f}{\rho_o}-{k_b}{\rho_r}
\qquad \mbox{and} \qquad
\dfrac{d{\rho_o}}{dt}={k_b}{\rho_r}-{k_f}{\rho_o},
\end{equation}
where $k_f$ and $k_b$ are the pseudo first-order forward and backward reaction rates, respectively. The forward reaction rate $k_f$ is proportional to the product of the electron transfer rate $k_{et}$ through the interface and the density of the electrons on $\Sigma_-$. The backward reaction rate $k_b$ is proportional to the product of the hole transfer rate $k_{ht}$ and the density of the holes on $\Sigma_-$. More precisely, we have <cit.>:
\begin{equation}
k_f(t,\bx) =k_{et}(\bx)(\rho_n-\rho_n^e) \ \ \mbox{and} \ \
k_b(t,\bx) =k_{ht}(\bx)(\rho_p-\rho_p^e),\ \ \mbox{on}\ \ (0,T]\times\Sigma .
\end{equation}
The changes of the concentrations lead to, following the relations $\bnu^+\cdot\bJ_r =-\frac{d{\rho_r}}{dt}$ and $\bnu^+\cdot\bJ_o =-\frac{d{\rho_o}}{dt} $, fluxes of redox pairs through the interface that can be expressed as follows <cit.>:
\begin{equation}\label{EQ:Interf E}
\begin{array}{ll}
\bnu^+\cdot\bJ_r ={k_{ht}(\bx)(\rho_p-\rho_p^e)}{\rho_r(t,\bx)}-{k_{et}(\bx)(\rho_n-\rho_n^e)}{\rho_o}(t,\bx), & \mbox{on}\ \ (0,T]\times\Sigma\\
\bnu^+\cdot\bJ_o =-{k_{ht}(\bx)(\rho_p-\rho_p^e)}{\rho_r(t,\bx)}+{k_{et}(\bx)(\rho_n-\rho_n^e)}{\rho_o(t,\bx)}, & \mbox{on}\ \ (0,T]\times\Sigma
\end{array}
\end{equation}
where the unit normal vector $\bnu^+$ points toward the electrolyte domain.
The fluxes of the redox pairs from the interface given in (<ref>) consists of two contributions: the flux induced from the transfer of electrons from the semiconductor to the electrolyte, often called the cathodic current after being brought up to the right dimension, $\bnu^+\cdot\bJ_n$, and the flux induced from the transfer of electrons from the electrolyte to the conduction band, often called the anodic current after being brought up to the right dimension, $\bnu^+\cdot\bJ_p$:
\begin{equation}\label{EQ:Interf S}
\begin{array}{rcll}
\bnu^+\cdot\bJ_n=(-\bnu^-)\cdot\bJ_n &=& - k_{et}(\bx)(\rho_n-\rho_n^e) \rho_o(t,\bx),& \mbox{on}\ \ (0,T]\times\Sigma\\
\bnu^+\cdot\bJ_p=(-\bnu^-)\cdot\bJ_p &=& - k_{ht}(\bx)(\rho_p-\rho_p^e)\rho_r(t,\bx), & \mbox{on}\ \ (0,T]\times\Sigma
\end{array}
\end{equation}
The interface conditions (<ref>) and (<ref>) can now be supplied to the semiconductor equations in (<ref>) and the redox equations in (<ref>) respectively. The values of the electron and hole transfer rates, $k_{et}$ and $k_{ht}$, in the interface conditions can be calculated approximately from the first principles of physical chemistry <cit.>. Theoretical analysis shows that both parameters can be approximately treated as constant; see, for instance, <cit.> for a summary of various ways to approximate these rates. The dependence of the forward and reverse reaction rates, $k_f$ and $k_b$, on the electric potential $\Phi$ is encoded in the electron and hole densities (i.e. $\rho_n$ and $\rho_p$) on the interface. In fact, we can recover the commonly used Butler-Volmer model <cit.> from our model as follows. Consider a one-dimensional semiconductor-electrolyte system (just for the purposes of presentation). At the equilibrium of the system, the net reaction rate is zero, i.e $\bnu^+\cdot\bJ_o=-\bnu^+\cdot\bJ_r=\bnu^+\cdot \bJ_n-\bnu^+\cdot \bJ_p=k_f\rho_o-k_b\rho_r=0$ on $\Sigma_+$. This leads to, by the expression for the fluxes (<ref>), the following relation between densities of redox pairs and the electric potential:
\begin{equation}
\rho_r=\rho_r^e \exp\left(\frac{\Phi-\Phi^e}{U_\cT}\right), \qquad
\rho_o=\rho_o^e \exp\left(\frac{\Phi-\Phi^e}{U_\cT}\right)
\end{equation}
where we have used the Einstein relations $D_r=U_\cT\mu_r$ and $D_o=U_\cT\mu_o$, $\Phi^e$ is the equilibrium potential of the electrolyte, and $\rho_r^e$, $\rho_o^e$ are the corresponding densities. Therefore at the system equilibrium the forward and reverse reaction rates satisfy:
\begin{equation}
\frac{k_f}{k_b}=\dfrac{\rho_o}{\rho_r}=\frac{\rho_o^e}{\rho_r^e} \exp\left(\frac{-2(\Phi-\Phi^e)}{U_\cT}\right).
\end{equation}
This implies that
\begin{equation}
-\dfrac{1}{2} U_\cT \left(\dfrac{d \ln k_f}{d\Phi}-\dfrac{d \ln k_b}{d\Phi}\right)=1.
\end{equation}
If we define the reductive symmetry factor (related to the forward reaction) $\alpha= -\frac{U_\cT}{2} \frac{d \ln k_f}{d\Phi}$, the above relation then implies that $\frac{U_\cT}{2} \frac{d \ln k_b}{d\Phi}=1-\alpha$. Moreover, the definition leads to $k_f=k^0\exp(-\alpha\eta(\Phi-\Phi^e))$ and $k_b=k^0\exp((1-\alpha)\eta (\Phi-\Phi^e))$ with $\eta=2/U_\cT$, where $k^0$ is the standard rate constant. We can therefore have the following Butler-Volmer model <cit.> for the interface flux:
\begin{equation}\label{EQ:BV}
\bnu^+\cdot\bJ_o = -\bnu^+\cdot\bJ_r =k^0\big[e^{-\alpha\eta(\Phi-\Phi^e)}{\rho_o}(t,\bx) -e^{(1-\alpha)\eta (\Phi-\Phi^e)} \rho_r(t,\bx)\big].
\end{equation}
§.§ Interface conditions for nonredox charges
For the $N$ charged species in the electrolyte that do not interact directly with the semiconductor through electron transfer, we impose insulating boundary conditions on the interface:
\begin{equation}\label{EQ:Interf Non Redox}
\begin{array}{lll}
\bnu^-\cdot \bJ_j = 0,& 1\le j\le N, & \mbox{on}\ \ (0, T]\times\Sigma
\end{array}
\end{equation}
We need to specify the interface condition for the electric potential as well. This is done by requiring $\Phi$ to be continuous across the interface and have continuous flux. Let us denote by $\Sigma_+$ and $\Sigma_-$ the semiconductor and the electrolyte sides, respectively, of $\Sigma$; then the conditions on the electric potential are given by
\begin{equation}\label{EQ:Interf Phi}
[\Phi]_{\Sigma}\equiv\Phi_{|\Sigma_-}-\Phi_{|\Sigma_+}=0, \qquad
[\eps_r\frac{\partial\Phi}{\partial\nu}]_{\Sigma}\equiv \Big(\eps_r^\sfE\dfrac{\partial\Phi}{\partial\nu}\Big)_{|\Sigma_-}-\Big(\eps_r^\sfS\dfrac{\partial\Phi}{\partial\nu}\Big)_{|\Sigma_+}=0, \quad \mbox{on}\ \ (0, T]\times\Sigma.
\end{equation}
Note that these continuity conditions would not prevent large the electric potential drops across a narrow neighborhood of the interface, as we will see in the numerical simulations.
The interface conditions that we constructed in this section ensure the conservation of the total flux $\bJ$ across the interface. To check that we recall that the total flux in the system is given as <cit.>
\begin{equation}\label{EQ:Total Current}
\bJ(\bx)=\left\{
\begin{array}{cl}
\alpha_p\bJ_p+\alpha_n\bJ_n, & \bx\in\sfS\\
\alpha_o\bJ_o+\alpha_r\bJ_r+\sum_{j=1}^N\alpha_j\bJ_j, & \bx\in\sfE
\end{array}\right.
\end{equation}
Using (<ref>), (<ref>), (<ref>) and the fact that $\alpha_o-\alpha_r=1$, we verify that $\bnu^+\cdot\bJ_{|\Sigma_-}=\bnu^+\cdot\bJ_{|\Sigma_+}$.
§ NUMERICAL DISCRETIZATION
The drift-diffusion-Poisson equations in (<ref>) and (<ref>), together with the boundary conditions given in (<ref>) (resp., (<ref>)), (<ref>), (<ref>), (<ref>), and the interface conditions (<ref>), (<ref>), (<ref>), and (<ref>) form a complete mathematical model for the transport of charges in the system of semiconductor-electrolyte for solar cell simulations. We now present a numerical procedure to solve the system.
§.§ Nondimensionalization
We first introduce the following characteristic quantities in the simulation regarding the device and its physics. We denote by $l^*$ the characteristic length scale of the device, $t^*$ the characteristic time scale, $\Phi^*$ the characteristic voltage, and $C^*$ the characteristic density. The values (and units) for these characteristic quantities are respectively (see <cit.>),
\begin{equation}\label{EQ:Characteristic}
l^*=10^{-4}\ {\rm (cm)},\ \ \ t^*=10^{-12}\ {\rm (s)},\ \ \ \Phi^*=U_\cT\ {\rm (V)},\ \ \ C^*= 10^{16}\ {\rm (cm^{-3})} .
\end{equation}
We now rescale all the physical quantities. For any quantity $Q$, we use $Q'$ to denote its rescaled version. To be specific, we introduce the rescaled Debye lengths in the semiconductor and electrolyte regions, respectively, as
λ_=1/l^*√(Φ^*ϵ^/q C^*), λ_=1/l^*√(Φ^*ϵ^/q C^*).
We also introduce the following rescaled quantities,
\begin{equation}
\begin{array}{l}
t'=t/t^*,\ \ \bx'=\bx/l^*,\ \ \Phi'=\Phi/\Phi^*,\ \ \rho_z' =\rho_z/C*,\ \ z\in\{n,p,r,o,1,\cdots, N\}\\
R_{np}'(\rho_n',\rho_p')=\dfrac{t^*}{C^*}R_{np}(C^*\rho_n',C^*\rho_p'),\ \ \ R_{ro}'(\rho_r',\rho_o')=\dfrac{t^*}{C^*}R_{ro}(C^*\rho_r',C^*\rho_o')\\
R_{j}'(\rho_1',\cdots,\rho_N')=\dfrac{t^*}{C^*}R_{j}(C^*\rho_1',\cdots,C^*\rho_N'),\ \ \ 1\le j\le N\\
G_0'=\dfrac{t^*}{l^* C^*}G_0,\ \ \ D_z'=D_z \dfrac{t^*}{{l^*}^2},\ \ \ \mu_z' =\mu_z\dfrac{t^*\Phi^*}{{l^*}^2},\ \ z\in\{n,p,r,o,1,\cdots, N\}\\
T'=T/t^*,\ \ C'=C/C^*,\ \ A_p'=t^*{C^*}^2 A_p',\ \ A_n'=t^*{C^*}^2 A_n,\ \ \rho_{isc}=\rho_{isc}/C^*
\end{array}
\end{equation}
for the model equations (<ref>) and (<ref>), and the following rescaled variables,
\begin{equation}
\begin{array}{llll}
k_{et}'=k_{et} t^* {C^*}/l^*, & k_{ht}'=k_{ht} t^* {C^*}/l^*, & v_n'=v_n t^*/l^*, & v_p'=v_p t^*/l^* \\
\varphi_{bi}'=\varphi_{bi}/\Phi^*, & \varphi_{app}'=\varphi_{app}/\Phi^*, & \varphi_{Stky}'=\varphi_{Stky}/\Phi^*, & \varphi_{app}^{\sfA'}=\varphi_{app}^\sfA/\Phi^*\\
%\rho_r^{'\infty}=\rho_r^\infty/C^*, & \rho_o^{'\infty}=\rho_o^\infty/C^* & \rho_j^{'\infty}=\rho_j^\infty/C^*, & \\
\rho_n^{e'}=\rho_n^e/C^*, & \rho_p^{e'}=\rho_p^e/C^*, & \rho_z^{'\infty} =\rho_z^\infty/C^*, & z\in\{r,o,1,\cdots, N\}\\
\end{array}
\end{equation}
for the boundary and interface conditions described in (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>).
We can now summarize the mathematical model in rescaled (nondimensionalized) form as the following,
\begin{equation}\label{EQ:DDP ND}
\begin{array}{ll}
\partial_{t'} \rho_n' - \nabla\cdot (D_n'\nabla \rho_n' + \alpha_n \mu_n' \rho_n'\nabla\Phi') = -R_{np}'(\rho_n',\rho_p')+\gamma G_{np}', &\mbox{in}\ \ (0,T']\times\Omega_\sfS,\\
\partial_{t'} \rho_p' - \nabla\cdot (D_p'\nabla \rho_p' + \alpha_p\mu_p' \rho_p'\nabla\Phi') = -R_{np}'(\rho_n',\rho_p')+\gamma G_{np}', &\mbox{in}\ \ (0,T']\times\Omega_\sfS,\\
-\nabla\cdot{\lambda_\sfS^2}\nabla\Phi'= C' +\alpha_p\rho_p'+\alpha_n\rho_n', &\mbox{in}\ \ (0,T']\times\Omega_\sfS\\
\Phi' = \varphi_{bi}'+\varphi_{app}' (\mbox{or}\ \ \Phi' = \varphi_{Stky}'+\varphi_{app}'),\ \ \rho_n'=\rho_n^{e'}, \ \ \rho_p'=\rho_p^{e'} & \mbox{on}\ (0,T']\times\Gamma_{\sfC}\\
\bnu\cdot\eps_r^\sfS\nabla\Phi' = 0,\ \ \bnu\cdot D_n'\nabla\rho_n'=0,\ \ \bnu\cdot D_p'\nabla\rho_p'=0 & \mbox{on}\ (0,T']\times\Gamma_{\sfS}\\
\Phi' = \varphi_{app}^{\sfA'}, & \mbox{on}\ (0,T']\times\Gamma_{\sfA}\\
\bnu\cdot\eps_r^\sfE\nabla\Phi' = 0, & \mbox{on}\ (0,T']\times\Gamma_{\sfE}
\end{array}
\end{equation}
for the transport dynamics in the rescaled semiconductor domain $\Omega_\sfS$, as the following,
\begin{equation}\label{EQ:DDP Redox ND}
\begin{array}{ll}
\partial_{t'} \rho_r' - \nabla\cdot (D_r'\nabla \rho_r' +\alpha_r \mu_r' \rho_r'\nabla\Phi') = +R_{ro}'(\rho_r',\rho_o'), &\mbox{in}\ \ (0,T']\times\Omega_\sfE,\\
\partial_{t'} \rho_o' - \nabla\cdot (D_o'\nabla \rho_o' +\alpha_o \mu_o' \rho_o'\nabla\Phi') = -R_{ro}'(\rho_r',\rho_o'), &\mbox{in}\ \ (0,T']\times\Omega_\sfE,\\
\partial_{t'} \rho_j' - \nabla\cdot (D_j'\nabla \rho_j' +\alpha_j \mu_j' \rho_j'\nabla\Phi') = R_{j}'(\rho_1',\cdots,\rho_N'), &\mbox{in}\ \ (0,T']\times\Omega_\sfE,\\
-\nabla\cdot{\lambda_\sfE^2}\nabla\Phi'= \alpha_o\rho_o'+\alpha_r\rho_r'+\sum_{j=1}^N \alpha_j\rho_j', &\mbox{in}\ \ (0,T']\times\Omega_\sfE\\
\rho_r' = \rho_r^{'\infty},\ \ \rho_o' = \rho_o^{'\infty},\ \ \rho_j'=\rho_j^{'\infty}, & \mbox{on}\ \ (0,T']\times\Gamma_{\sfA}\\
\bnu\cdot D_r'\nabla \rho_r' =0,\ \ \bnu\cdot D_o'\nabla\rho_o' = 0,\ \ \bnu\cdot D_j'\nabla \rho_j'=0, & \mbox{on}\ \ (0,T']\times\Gamma_{\sfE}
\end{array}
\end{equation}
for the transport dynamics in the (rescaled) electrolyte domain $\Omega_\sfE$, and as the following,
\begin{equation}\label{EQ:DDP Interf ND}
\begin{array}{ll}
{[}\Phi'{]}_{\Sigma}=0,\ \ {[}\eps_r\frac{\partial\Phi'}{\partial\nu}{]}_{\Sigma}=0,\ \ \bnu\cdot D_j'\nabla \rho_j'=0 & \mbox{on}\ (0,T']\times \Sigma\\
\bnu^+\cdot\bJ_n'=-k_{et}'(\rho_n'-\rho_n^{e'})\rho_o',\ \ \bnu^+\cdot\bJ_p'=-k_{ht}'(\rho_p'-\rho_p^{e'})\rho_r', & \mbox{on}\ (0,T']\times\Sigma\\
\bnu\cdot\bJ_r'=-\bnu\cdot\bJ_o'=k_{ht}'(\rho_p'-\rho_p^{e'})\rho_r'-k_{et}'(\rho_n'-\rho_n^{e'})\rho_o', & \mbox{on}\ (0,T']\times\Sigma
\end{array}
\end{equation}
for the dynamics on the interface $\Sigma$. Here the rescaled Auger generation-recombination rate $R_{np}'$ takes exactly the same form as in (<ref>), i.e.,
\begin{equation}\label{EQ:Auger ND}
R_{np}'(\rho_n',\rho_p') = (A_n' \rho_n' +A_p' \rho_p')(\rho_{isc}^{'2}-\rho_n'\rho_p'),
\end{equation}
and the rescaled photon illumination function $G_{np}'$ takes the form
\begin{equation}\label{EQ:Illu Source ND}
\begin{array}{rr}
\sigma'(\bx') G_0'(\bx_0')\exp\Big(-\int_0^{\bar s}\sigma'(\bx_0'+s\btheta_0)ds\Big),& \mbox{if}\ \bx'=\bx_0'+\bar s\btheta_0\\
0,& \mbox{otherwise}
\end{array}\right.
\end{equation}
In the rest of the paper, we will work on the numerical simulations of the semiconductor-electrolyte system based on the above nondimensionalized systems (<ref>) and (<ref>).
§.§ Time-dependent discretization
For the numerical simulations in this paper, we discretize the time-dependent systems of reaction-drift-diffusion-Poisson equations (<ref>) and (<ref>) by standard finite difference method in both spatial and temporal variables. In the spatial variable, we employ a classical upwind discretization of the advection terms (such as $\nabla\Phi'\cdot\nabla\rho_n'$) to ensure the stability of the scheme. To avoid solving nonlinear systems of equations in each time step (since the models (<ref>) and (<ref>) are nonlinear), we employ the forward Euler scheme for the temporal variable. Since this is a first-order scheme and is explicit, we do not need to perform any nonlinear solve in the solution process, as long as we can supply the right initial conditions. We are aware that there are many efficient solvers for similar problems that have been developed in the literature; see, for instance, <cit.>.
To solve stationary problems, we can evolve the system for a long time so that the system reaches its stationary state. We use the magnitude of the relative $L^2$ update of the solution as the stopping criterion. An alternative, in fact more efficient, way to solve the nonlinear system is the following iterative scheme.
§.§ A Gummel-Schwarz iteration for stationary problems
This method combines domain decomposition strategies with nonlinear iterative schemes. We decompose the system naturally into two subsystems, the semiconductor system and the electrolyte system. We solve the two subsystem alternatively and couple them with the interface condition. This is the Schwarz decomposition strategy that has been used extensively in the literature; see <cit.> for similar domain decomposition strategies in semiconductor simulation. To solve the nonlinear equations in each sub-problem, we adopt the Gummel iteration scheme <cit.>. This scheme decomposes the drift-diffusion-Poisson system into a drift-diffusion part and a Poisson part and then solves the two parts alternatively. The coupling then comes from the source term in the Poisson equation. Our algorithm, in the form of solving the stationary problem, takes the following form.
GUMMEL-SCHWARZ ALGORITHM.
[1] Gummel step $k=0$, construct initial guess $\{\rho_n^{'0}, \rho_p^{'0}, \rho_r^{'0}, \rho_o^{'0}, \{\rho_j^{'0}\}_{j=1}^N\}$
[2] Gummel step $k\ge 1$:
* Solve the Poisson problem for $\Phi^{'k}$ in $\Omega_\sfE\cup\Omega_\sfS$ using the densities $\rho_n^{'k-1}, \rho_p^{'k-1}, \rho_r^{'k-1}$, $\rho_o^{'k-1}$, and $\{\rho_j^{'k-1}\}_{j=1}^N$:
\begin{equation}\label{EQ:DDP ND Poisson}
\begin{array}{ll}
-\nabla\cdot{\lambda_\sfS^2}\nabla\Phi^{'k}= C' +\alpha_p\rho_p^{'k-1}+\alpha_n\rho_n^{'k-1}, &\text{in}\ \Omega_\sfS \\
-\nabla\cdot{\lambda_\sfE^2}\nabla\Phi^{'k}= \alpha_o\rho_o^{'k-1}+\alpha_r\rho_r^{'k-1}+\sum_{j=1}^N\alpha_j\rho_j^{'k-1}, &\mbox{in}\ \Omega_\sfE\\
\Phi^{'k} = \varphi_{bi}'+\varphi_{app}',\ \ \mbox{or}\ \ \Phi^{'k} = \varphi_{Stky}'+\varphi_{app}', & \mbox{on}\ \Gamma_{\sfC}\\
\bnu\cdot\eps_r^\sfS\nabla\Phi^{'k} = 0, & \mbox{on}\ \Gamma_{\sfS}\\
\Phi^{'k} = \varphi_{app}^{\sfA'}, & \mbox{on}\ \Gamma_{\sfA}\\
\bnu\cdot\eps_r^\sfE\nabla\Phi^{'k} = 0, & \mbox{on}\ \Gamma_{\sfE}\\
{[}\Phi^{'k}{]}_{\Sigma}=0,\ \ {[}\eps_r\frac{\partial\Phi^{'k}}{\partial\nu}{]}_{\Sigma}=0,& \mbox{on}\ \Sigma
\end{array}
\end{equation}
* Solve for $\rho_n^{'k},\rho_p^{'k}$, $\rho_r^{'k},\rho_o^{'k},\{\rho_j^{'k}\}_{j=1}^N$ as limits of the following iteration:
[i] Schwarz step $\ell=0$: construct guess $(\rho_n^{'k,0},\rho_p^{'k,0},\rho_r^{'k,0},\rho_o^{'k,0},\{\rho_j^{'k,0}\}_1^N)$
[ii] Schwarz step $\ell\ge 1$: solve sequentially
\begin{equation}\label{EQ:SchI}
\begin{array}{ll}
\nabla\cdot (-D_n'\nabla \rho_n^{'k,\ell} + \alpha_n\mu_n' \rho_n^{'k,\ell}\nabla\Phi^{'k}) = -R_{np}'(\rho_n^{'k,\ell},\rho_p^{'k,\ell})+\gamma G_{np}', &\mbox{in}\ \Omega_\sfS\\
\nabla\cdot (-D_p'\nabla \rho_p^{'k,\ell} +\alpha_p \mu_p' \rho_p^{'k,\ell}\nabla\Phi^{'k})= -R_{np}'(\rho_n^{'k,\ell},\rho_p^{'k,\ell})+\gamma G_{np}', &\mbox{in}\ \Omega_\sfS\\ \rho_n^{'k,\ell}=\rho_n^{e'}, \ \ \rho_p^{'k,\ell}=\rho_p^{'e}, & \mbox{on}\ \Gamma_{\sfC}\\
\bnu\cdot D_n'\nabla\rho_n^{'k,\ell}=0,\ \ \bnu\cdot D_p'\nabla\rho_p^{'k,\ell}=0, & \mbox{on}\ \Gamma_{\sfS}\\
\bnu^+\cdot\bJ_n^{'k,\ell}=-k_{et}'(\rho_n^{'k,\ell}-\rho_n^{e'})\rho_o^{'k,\ell-1},\ \ \bnu^+\cdot\bJ_p^{'k,\ell}=-k_{ht}'(\rho_p^{'k,\ell}-\rho_p^{e'})\rho_r^{'k,\ell-1}, & \mbox{on}\ \Sigma
\end{array}
\end{equation}
\begin{equation}\label{EQ:SchII}
\begin{array}{ll}
\nabla\cdot(-D_r'\nabla \rho_r^{'k,\ell} + \alpha_r\mu_r' \rho_r^{'k,\ell}\nabla\Phi^{'k}) = +R_{ro}'(\rho_r^{'k,\ell},\rho_o^{'k,\ell}), &\mbox{in}\ \Omega_\sfE\\
\nabla\cdot(-D_o'\nabla \rho_o^{'k,\ell} +\alpha_o \mu_o' \rho_o^{'k,\ell}\nabla\Phi^{'k}) = -R_{ro}'(\rho_r^{'k,\ell},\rho_o^{'k,\ell}), &\mbox{in}\ \Omega_\sfE\\
\nabla\cdot(-D_j'\nabla \rho_j^{'k,\ell} +\alpha_j \mu_j' \rho_j^{'k,\ell}\nabla\Phi^{'k}) = R_{j}(\rho_1^{k,\ell},\cdots,\rho_N^{'k,\ell}), &\mbox{in}\ \Omega_\sfE\\
\rho_r^{'k,\ell} = \rho_r^{'\infty},\ \ \rho_o^{'k,\ell} = \rho_o^{'\infty},\ \ \rho_j^{'k,\ell}=\rho_j^{'\infty}, & \mbox{on}\ \Gamma_{\sfA}\\
\bnu\cdot D_r'\nabla \rho_r^{'k,\ell} =0,\ \ \bnu\cdot D_o'\nabla\rho_o^{'k,\ell} = 0,\ \ \bnu\cdot D_j'\nabla \rho_j^{'k,\ell}=0, & \mbox{on}\ \ \Gamma_{\sfE}\\
\bnu\cdot\bJ_r^{'k,\ell}=-\bnu\cdot\bJ_o^{'k,\ell}=k_{ht}'(\rho_p^{'k,\ell}-\rho_p^{e'})\rho_r^{'k,\ell}-k_{et}'(\rho_n^{'k,\ell}-\rho_n^{e'})\rho_o^{'k,\ell}, & \mbox{on}\ \Sigma\\
\bnu\cdot D_j'\nabla \rho_j^{'k,\ell}=0,\ \ 1\le j\le N & \mbox{on}\ \ \Sigma.
\end{array}
\end{equation}
[iii] If convergence criteria satisfied, stop; otherwise, set $\ell=\ell+1$ and go to [ii].
[3] If convergence criteria satisfied, stop; Otherwise, set $k=k+1$ and go to step [2].
Note that since (<ref>) and (<ref>) are solved sequentially, we are able to replace the $\rho_n^{'k,\ell-1}$ and $\rho_p^{'k,\ell-1}$ terms in (<ref>) in the boundary conditions on $\Sigma$ with $\rho_n^{'k,\ell}$ and $\rho_p^{'k,\ell}$ respectively. This slightly improves the speed of convergence of the numerical scheme.
If the mathematical system is well-posed, the convergence of this iteration can be established following the lines of work in <cit.>. The details will be in a future work.
§.§ Computational issues
Initial conditions. To solve the time-dependent problem, we need to supply the system with appropriate initial conditions. Our initial conditions are given only for the densities. The initial condition on the potential is obtained by solving the Poisson equation with appropriate boundary conditions using the initial densities. That is done by solving problem (<ref>).
Stiffness of the system. One major issue in the numerical solution of the mathematical model is the stiffness of the system across the interface, caused by the large contrast between the magnitudes of the densities in the semiconductor and electrolyte domains. Physically this leads to the formation of boundary layers of charge densities local to the interface; see, for instance, discussions in <cit.>. To capture the sharp transition at the interface, we have to use finer finite difference grids around the interface.
Computation of stationary states. The Gummel-Schwarz iteration is attractive when stationary state solutions are to be sought since it is computationally much more efficient than the time-stepping method. However, due to the nonlinearity of the system, multiple solutions could exist, as illustrated, for instance, in <cit.> in similar settings. Starting from different initial guesses for the nonlinear solver could lead us to different solutions (and some of them could be unphysical). The time-stepping scheme, however, will lead us to the desired steady state for any given initial state. We can combine the two methods. For a given initial state, we use the time-stepping scheme to evolve the system for some time, say $\tilde T'$ (which we select empirically as a time when the fastest initial evolution of the solution has passed); we then use the solution at $\tilde T'$ as the initial guess for the Gummel-Schwarz iteration to find the corresponding steady state solution. In our numerical simulations, we did not observe the existence of multiple steady states. However, this is an issue to keep in mind when more complicated (for instance multidimensional) simulations are performed.
§ NUMERICAL SIMULATIONS
We present some numerical simulations based on the mathematical model that we have constructed for the semiconductor-electrolyte system.
§.§ General setup for the simulations
To simplify the numerical computation, we assume some symmetry in the semiconductor-electrolyte system so that we can reduce the problem to one dimension. We showed in Fig. <ref> (middle and right) two typical two-dimensional systems where such dimension reductions can be performed. In the first case, if we assume that the system is invariant in the direction parallel to the current collector, then we have a one-dimensional system in the direction that is perpendicular to the current collector. In the second setting, we have a radially symmetric system that is invariant in the angular direction in the polar coordinate. The system can then be regarded as a one-dimensional system in the radial direction.
In all the simulations we have performed, we use an $n$-type semiconductor to construct the semiconductor-electrolyte system. We also select the electrolyte such that the charge numbers $\alpha_o-\alpha_r=1$. Furthermore, we assume in our simulation that the reductants and oxidants are the only charges in the electrolyte. Therefore the equations for $\rho_j'$ $(1\le j\le N)$ are dropped. In Tab. <ref>, we list the values of all the model parameters that we use in the simulations. They are mainly taken from <cit.> and references therein. These values may depend on the materials used to construct the system and the temperature of the system (which we set to be $\cT=300K$). They can be tuned to be more realistic by careful calibrations.
Parameter Value Unit Parameter Value Unit
$q$ $1.6 \times 10^{-19}$ [${\rm A\ s}$]
$k_B$ $8.62\times 10^{-5}\ q$ [${\rm J\ K^{-1}}$]
$\eps_0$ $8.85\times 10^{-14}$ [${\rm A\ s\ V^{-1}\ cm^{-1}}$]
$\eps_r^\sfS$ $11.9$
$\mu_n$ $1500$ [${\rm cm^2\ V^{-1}\ s^{-1}}$]
$\mu_p$ $450$ [${\rm cm^2\ V^{-1}\ s^{-1}}$]
$A_n$ $2.8\times 10^{-31}$ [${\rm cm^6\ s^{-1}}$]
$A_p$ $9.9\times 10^{-32}$ [${\rm cm^6\ s^{-1}}$]
$N_c$ $2.80\times 10^{19}$ [${\rm cm^{-3}}$]
$N_v$ $1.04\times 10^{19}$ [${\rm cm^{-3}}$]
$v_n$ $5\times 10^6$ [${\rm cm\ s^{-1}}$]
$v_p$ $5\times 10^6$ [${\rm cm\ s^{-1}}$]
$\Phi_m$ $2.4$ [${\rm V}$] $\chi$ $1.2$ [${\rm V}$]
$k_{et}$ $1\times 10^{-21}$ [${\rm cm^4\ s^{-1}}$]
$k_{ht}$ $1\times 10^{-17}$ [${\rm cm^4\ s^{-1}}$]
$\mu_o$ $2\times 10^{-1}$ [${\rm cm^2\ V^{-1}\ s^{-1}}$]
$\mu_r$ $0.5\times 10^{-1}$ [${\rm cm^2\ V^{-1}\ s^{-1}}$]
$\eps_r^\sfE$ $1000$ $G_0$ $1.2\times 10^{17}$ [${\rm cm}^{-2} {\rm s}^{-1}$]
Values of physical parameters used in the numerical simulations. The numbers are given in unit of cm, s, V, A.
We perform simulations on two devices of different sizes. The two devices are designed to mimic a large (Device I) and small (Device II) nano- to micro-scale solar cell building block, as follows.
Device I. The device is contained in $\Omega=(-1.0, 1.0)$ with the semiconductor $\Omega_{\sfS}=(-1.0, 0)$ and $\Omega_{\sfE}=(0, 1.0)$ separated by the interface $\Sigma$ located at $x'=0$. The semiconductor boundary $\Gamma_{\sfS}$ is thus at the point $x'=-1.0$, while the electrolyte boundary $\Gamma_{\sfE}$ is at the point $x'=1.0$.
Device II. The device is contained in $\Omega=(-0.2,0.2)$ with the semiconductor $\Omega_{\sfS}=(-0.2, 0)$ and $\Omega_{\sfE}=(0, 0.2)$ separated by the interface $\Sigma$ located at $x'=0$. The semiconductor boundary $\Gamma_{\sfS}$ is thus at the point $x'=-0.2$, while the electrolyte boundary $\Gamma_{\sfE}$ is at the point $x'=0.2$.
§.§ General dynamics of semiconductor-electrolyte systems
We now present simulation results on general dynamics of the semiconductor-electrolyte systems we have constructed. We perform the simulations using the first system, i.e., Device I. General parameters in the simulations are listed in Tab. <ref>. We consider three different cases.
Case I (a) in dark environment ($\gamma=0$). Left column: charge densities (black solid for $\rho_p'$, black dashed for $\rho_n'$, red solid for $\rho_o'$, and red dashed for $\rho_r'$), electric field, and flux distributions at time $t'=0.05$. Right column: charge densities, electric field, and flux distributions at the stationary state.
Case I (a). In this simulation, we show typical dynamics of the semiconductor-electrolyte system in dark and illuminated cases. We consider the case when the densities of the reductant-oxidant pair are very high compared to the densities of the electron-hole pair. Specifically, we take $\rho_r^{'\infty}= 30.0$, $\rho_o^{'\infty}=29.0$. The system starts from the initial conditions: $\rho_n^{'0}=2$, $\rho_p^{'0}=1.0$, $\rho_r^0 = \rho_r^{'\infty}$, and $\rho_o^{'0} = \rho_o^{'\infty}$. We first perform simulation for the dark case, i.e., when the parameter $\gamma=0$. In the left column of Fig. <ref>, we show the distributions of the charge densities, electric field and total fluxes in the semiconductor and the electrolyte at time $t'=0.05$. The corresponding distributions at stationary state are shown in the right column.
Case I (a) in illuminated environment ($\gamma=1$). Left column: charge densities (black solid for $\rho_p'$, black dashed for $\rho_n'$, red solid for $\rho_o'$, and red dashed for $\rho_r'$), electric field, and flux distributions at time $t'=0.05$. Right column: charge densities, electric field and flux distributions at the stationary state.
We now repeat the simulation in the illuminated environment by setting the parameter $\gamma=1$. In this case, the surface photon flux is set as $G_0'=1.2\times 10^{-7}$ to mimic a solar spectral irradiance with air mass 1.5. The simulation results are presented in Fig. <ref>. In both the dark and illuminated cases shown here, the applied potential bias is $19.3$, and Ohmic contact (i.e. Dirichlet condition) is assumed at the left end of the semiconductor.
Case I (b) in dark environment ($\gamma=0$). Left column: charge densities (black solid for $\rho_p'$, black dashed for $\rho_n'$, red solid for $\rho_o'$, and red dashed for $\rho_r'$), electric field, and flux distributions at time $t'=0.05$. Right column: charge densities, electric field and flux distributions at the stationary state.
Case I (b). We now perform a set of simulations with the same parameters as in the previous numerical experiment but with a lower bulk reductant and oxidant pair densities: $\rho_r^{'\infty} = 4.0$, $\rho_o^{'\infty}= 5.0$. The initial conditions for the simulation are: $\rho_n^{'0}=2.5$, $\rho_p^{'0}=1.0$, $\rho_r^{'0} = \rho_r^{'\infty}$ and $\rho_o^{'0} = \rho_o^{'\infty}$. The results in the dark and the illuminated environments are shown in Fig. <ref> and Fig. <ref>, respectively.
Case I (b) in illuminated environment ($\gamma=1$). Left column: charge densities (black solid for $\rho_p'$, black dashed for $\rho_n'$, red solid for $\rho_o'$, and red dashed for $\rho_r'$), electric field, and flux distributions at time $t'=0.05$. Right column: charge densities, electric field, and flux distributions at the stationary state.
We observe from the comparison of Fig. <ref> and Fig <ref> with Fig. <ref> and Fig. <ref>, that when all other factors are kept unchanged, lowering the density of the redox pair leads to significant change of the electric field across the device, especially at the semiconductor-electrolyte interface. The charge densities in the semiconductor also change significantly. In both cases, however, the charge densities in the electrolyte remain as almost constant across the electrolyte. There are two reasons for this. First, the charge densities in the electrolyte are significantly higher than those in the semiconductor (even in Case I (b)). Second, the relative dielectric constant of the electrolyte is much higher than that in the semiconductor, resulting in a relatively constant electric field inside the electrolyte. If we lower the relative dielectric function, we observe a significant variation in charge density distributions in the electrolyte, as we can see from the next numerical simulation.
Case I (c) in dark environment ($\gamma=0$). Shown are: charge densities (black solid for $\rho_p'$, black dashed for $\rho_n'$, red solid for $\rho_o'$, and red dashed for $\rho_r'$), electric field, and flux distributions at time $t'= 0.05$ (left); and charge densities, electric field, and flux distributions at the stationary state (right).
Case I (c). There are a large number of physical parameters in the semiconductor-electrolyte system that control the dynamics of the system. To be specific, we show in this numerical simulation the impact of the relative dielectric constant in the electrolyte $\eps_r^\sfE$ on the system performance. The setup is exactly as in Case I (b) except that $\eps_r^\sfE=100$ now. We perform simulations in both the dark and illuminated environments. We plot in Fig. <ref> the densities, the electric field, and the flux distributions at time $t'=0.05$ and the stationary state. The corresponding results for illuminated case are shown in Fig. <ref>. It is not hard to observe the dramatic change in all the quantities shown after comparing Fig. <ref> with Fig. <ref>, and Fig. <ref> with Fig. <ref>. More detailed study of the parameter sensitivity problem will be reported elsewhere.
Case I (c) in illuminated environment ($\gamma=1$). Shown are: charge densities (black solid for $\rho_p'$, black dashed for $\rho_n'$, red solid for $\rho_o'$ and red dashed for $\rho_r'$ respectively), electric field and flux distributions at time $t'=0.05$ (left column); and charge densities, electric field, and flux distributions at the stationary state (right column).
§.§ Comparison with Schottky approximation
In simulations done for practical applications, it is often assumed that the densities for the reductants and oxidants in the electrolyte are extremely high, so that reductant-oxidant dynamics changes very little compared to the electron-hole dynamics in the semiconductor; see, for instance, <cit.>. In this case, it is usually common to completely fix the electrolyte system and only evolve the electron-hole system. This is done by treating the electrolyte system as a Schottky contact and thus replacing the semiconductor-electrolyte interface conditions with Robin-type of boundary conditions such as (<ref>); see more discussions in <cit.>
Our previous simulations, such as those shown in Fig. <ref> and Fig. <ref>, indicate that such a simplification indeed can be accurate as the densities of the redox pair are roughly constant inside the electrolyte. We now present two simulations where we compare the simulations based on the our model of the whole system with the simulations based on Schottky approximation. Our simulations focus on Device II, which is significantly smaller than Device I. Simulations for Device I show similar behavior which we omit here to avoid repetition.
Due to the fact that the Schottky boundary condition contains no information on the parameters of the electrolyte system, it is impossible to make a direct quantitative comparison between the simulations. Instead, we will perform a comparison as follows. For each configuration of the electrolyte system, we select the parameters $\Phi_m$, $\chi$ and $E_g$ such that total potential on the Schottky contact, $\varphi_{Stky}'+\varphi_{app}'$, equals the applied potential on the counter electrode, $\varphi_{app}^{\sfA'}$.
Case II (a): comparison between simulations from whole-system (red) and Schottky approximation (black) for Device II in dark ($\gamma=0$, left column) and illuminated ($\gamma=1$, right column) environments.
We performed two sets of simulations on Device II. In the simulations with Schottky approximation, the Schottky contact locates at $x'=0$.
Case II (a). In the first set of simulations, we compare the whole system simulation with the Schottky approximation when the bulk reductant-oxidant pair density are more than $20$ times higher than that of the electron-hole pair density. Precisely, the bulk densities for reductant and oxidant are respectively $\rho_r^{'\infty}=30.0$ and $\rho_o^{'\infty}=35.0$. The results in the semiconductor part are shown in Fig. <ref>.
We observed that in this case, the simulations with Schottky approximation are sufficiently close to the simulations with the whole system. This is to say that, under such a case, replacing the electrolyte system with a Schottky contact provides a good approximation to the original system.
Case II (b). In the second set of simulations, we compare the two simulations when the bulk reductant-oxidant pair density is much lower than that in Case II(a). Precisely, $\rho_r^{'\infty} = 2.0$ and $\rho_o^{'\infty} = 3.0$. The results for the semiconductor part are shown in Fig. <ref>.
While the qualitative behavior of the two systems is similar, the quantitative results are very different. We were not able to find a set of parameters in the boundary condition (<ref>) that produced identical behavior of the semiconductor system. In this type of situations, simulation of the whole semiconductor-electrolyte system provides more accurate description of the physical process in the device.
Case II (b): comparison between simulations from whole-system (red) and Schottky approximation (black) for Device II in dark ($\gamma=0$, left column) and illuminated ($\gamma=1$, right column) environments.
The results above show that even though one can often replace the electrolyte system with a Schottky contact, modeling the whole semiconductor-electrolyte system offers more flexibilities in general. In the case when the bulk densities of the reductant and the oxidant are not extremely high compared to the densities of electrons and holes in the semiconductor, replacing the electrolyte system with a Schottky contact can cause large inaccuracy in predicted quantities.
§.§ Voltage-current characteristics
In the last set of numerical simulations, we study the voltage-current, or more precisely the voltage-flux (since all quantities are nondimensionalized), a characteristic of the semiconductor-electrolyte system. We again focus on simulations with Device II, although we observe similar results for Device I.
Case II$^\prime$ (a) in dark ($\gamma=0$) environment: comparison of charge densities (top) and the corresponding electric fields (bottom) in applied forward (left column) and reversed (right column) potential bias.
Case II$^\prime$ (a). We first compare simulations with forward potential bias with simulations with reverse potential bias in a dark environment. We take a device with low bulk reductant-oxidant densities ($\rho_r^{'\infty}=4.5$, $\rho_o^{'\infty}=5.0$) and high relative dielectric constant in electrolyte ($\eps_r^\sfE=1000$). Shown on the left column of Fig. <ref> are electron-hole and redox pair densities and the corresponding electric fields for applied forward potential bias of $35.0$. The same results for the applied reverse bias of $-35.0$ are presented in the right column of the same figure. Aside from the obvious differences in the densities and electric field distributions, the fluxes through the system with forward and reverse biases are very different, as can be seen later on Fig. <ref>.
The simulations are repeated in Fig. <ref> in the illuminated environment. It is clear from the plots in Fig. <ref> and Fig. <ref> that illumination dramatically changes the distribution of charges (and thus the electric field) inside the device as we have seen in the previous simulations.
Case II$^\prime$ (a) in illuminated ($\gamma=1$) environment: comparison of charge densities (top) and the corresponding electric fields (bottom) in applied forward (left) and reverse (right) potential bias.
Case II$^\prime$ (b). We now attempt to explore the whole voltage-flux (I-V) characteristics of Device II. To do that, we run the simulations for various different applied potentials and compute the flux through the system under each applied potential. We plot the flux data as a function of the applied potential to obtain the I-V curve of the system. The parameters are taken as $\rho_r^{'\infty}=35$, $\rho_o^{'\infty}=30$), and $\eps_r^\sfE=1000$ to mimic those in realistic devices.
Case II$^\prime$ (b): voltage-flux (I-V) curves for an $n$-type semiconductor-electrolyte system in dark (dotted line with circles) and illuminated (solid line with dots) environments. The intersections of the dash-dotted vertical lines with the x-axis show the maximum power voltage ($\Phi_{\rm mp}$, red) and open circuit voltage ($\Phi_{\rm oc}$, blue). The intersections of the dashed horizontal lines with the y-axis show the maximum power flux ($J_{\rm oc}$, red) and short circuit flux ($J_{\rm sc}$, blue).
Fig. <ref> shows the I-V curve obtained in both dark (line with stars) and illuminated (solid line with circles) environments. The applied potential bias lives in the range $[-58.0,\ 73.5]$. The intersections of the dash-dotted vertical lines with the $x$-axis (horizontal) show the maximum power voltage ($\Phi_{\rm mp}$, red) and open circuit voltage ($\Phi_{\rm oc}$, blue). The intersections of the dashed horizontal lines with the y-axis show the maximum power flux ($J_{\rm oc}$, red) and short circuit flux ($J_{\rm sc}$, blue).
§ CONCLUSION AND REMARKS
We have considered in this paper the mathematical modeling of semiconductor-electrolyte systems for applications in liquid-junction solar cells. We presented a complete mathematical model, a set of nonlinear partial differential equations with reactive interface conditions, for the simulation of such systems. Our model consists of a reaction-drift-diffusion-Poisson system that models the transport of electron-hole pairs in the semiconductor and an equivalent system that describes the transport of reductant-oxidant pairs in the electrolyte. The coupling of the two systems on the semiconductor-electrolyte interface is modeled with a set of reaction and flux transfer types of interface conditions. We presented numerical procedures to solve both the time-dependent and stationary problems, for instance with Gummel-Schwarz double iteration. Some numerical simulations for one-dimensional devices were presented to illustrate the behavior of these devices.
Past study on the semiconductor-electrolyte system usually completely neglected the charge transfer processes in the electrolyte <cit.>. The mathematical models developed thus only cover the semiconductor part with the interface effect modeled by a Robin type of boundary condition. The rationale behind this simplification is the belief that the density of the reductant-oxidant pairs is so high compared to the density of the electron-hole pairs in the semiconductor such that the density of the redox pair would not be perturbed by the charge transfer process through the interface. While this might be a valid approximation in certain cases, it can certainly go wrong in other cases. For instance, it is generally observed that due to the strong electric field at the interface, there are dramatic change in the density of charges (both the electron-hole and the redox pairs) near the interface. What we have presented is, to our knowledge, the first complete mathematical model for semiconductor-electrolyte solar cell systems that would allow us to accurately study the charge transfer process through the interface.
There are a variety of problems related to the model that we have constructed in this work that deserve thorough investigations. On the mathematical side, a detailed mathematical analysis on the well-posedness of the system is necessary. On the computational side, more detailed numerical analysis of the model, including convergence of the Gummel-Schwarz iteration, efficient high-order discretization, and fast solution techniques, has to be studied. On the application side, it is important to calibrate the model parameters with experimental data collected from real semiconductor-electrolyte solar cells. Once the aforementioned issues are addressed, we can use the model to help design more efficient solar cells by, for instance, optimizing the various model parameters. We are currently investigating several of these issues.
§ ACKNOWLEDGEMENTS
We would like to thank the anonymous referees for their constructive comments that helped us improve the quality of this work. We also benefited greatly from fruitful discussions with Professor Allen J. Bard and Charles B. Mullins (both affiliated with the Department of Chemistry at the authors' institution). YH, IMG and HCL are partially supported by the National Science Foundation through grants CHE 0934450 and DMS-0807712. KR is partially supported by NSF grants DMS-0914825 and DMS-1321018.
A. M. Anile, W. Allegretto, and C. Ringhofer, Mathematical Problems
in Semiconductor Physics, Lecture Notes in Mathematics, Springer-Verlag,
Berlin, 2003.
A. M. Anile, V. Romano, and G. Russo, Extended hydrodynamical model
of carrier transport in semiconductors, SIAM J. Appl. Math., 61 (2000),
pp. 74–101.
A. J. Bard and L. R. Faulkner, Electrochemical Methods:
Fundamentals and Applications, Wiley, New York, 2nd ed., 2000.
M. Z. Bazant, K. Thornton, and A. Ajdari, Diffuse-charge dynamics in
electrochemical systems, Phys. Rev. E, 70 (2004).
J. Bell, T. Farrell, M. Penny, and G. Will, A mathematical model of
the semiconductor-electrolyte interface in dye sensitised solar cells, in
Proceedings of EMAC 2003, R. May and W. F. Blyth, eds., Australian
Mathematical Society, 2003, pp. 193–198.
N. Ben Abdallah, M. J. Cáceres, J. A. Carrillo, and F. Vecil, A
deterministic solver for a hybrid quantum-classical transport model in
nanoMOSFETs, J. Comput. Phys., 228 (2009), pp. 6553–6571.
N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models
for semiconductors, J. Math. Phys., 37 (1996), pp. 3306–3333.
P. M. Biesheuvel, Y. Fu, and M. Z. Bazant, Electrochemistry and
capacitive charging of porous electrodes in asymmetric multicomponent
electrolytes, Russian J. Electrochem., 48 (2012), pp. 580–592.
K. F. Brennan, The Physics of Semiconductors : with Applications to
Optoelectronic Devices, Cambridge University Press, New York, 1999.
M. Burger and R. Pinnau, A globally convergent Gummel map for
optimal dopant profiling, Math. Models Methods Appl. Sci., 19 (2009),
pp. 769–786.
C. Capellos and B. Bielski, Kinetic Systems: Mathematical
Description of Chemical Kinetics in Solution, Wiley-Interscience, New York,
J. A. Carrillo, I. Gamba, and C.-W. Shu, Computational macroscopic
approximations to the one-dimensional relaxation-time kinetic system for
semiconductors, Physica D, 146 (2000), pp. 289–306.
J. A. Carrillo, I. M. Gamba, A. Majorana, and C.-W. Shu, A
WENO-solver for the transients of Boltzmann-Poisson system for
semiconductor devices: Performance and comparisons with Monte Carlo
methods, J. Comput. Phys., 184 (2003), pp. 498–525.
C. Cercignani, I. M. Gamba, and C. D. Levermore, High field
approximations to a Boltzmann-Poisson system and boundary conditions in a
semiconductor, Appl. Math. Lett., 10 (1997), pp. 111–118.
height 2pt depth -1.6pt width 23pt, A drift-collision
balance asymptotic for a Boltzmann-Poisson system in bounded domains, SIAM
J. Appl. Math., 61 (2001), pp. 1932–1958.
S. Chen, W. E, Y. Liu, and C.-W. Shu, A discontinuous Galerkin
implementation of a domain decomposition method for kinetic-hydrodynamic
coupling multiscale problems in gas dynamics and device simulations, J.
Comput. Phys., 225 (2007), pp. 1314–1330.
Y. Cheng, I. Gamba, and K. Ren, Recovering doping profiles in
semiconductor devices with the Boltzmann-Poisson model, J. Comput. Phys.,
230 (2011), pp. 3391–3412.
K. A. Connors, Chemical Kinetics, The Study of Reaction Rates in
Solution, VCH Publishers, Weinheim, Germany, 1990.
C. de Falco, M. Porro, R. Sacco, and M. Verri, Multiscale modeling
and simulation of organic solar cells, Comput. Methods Appl. Mech. Engrg.,
245 (2012), pp. 102–116.
P. Degond and B. Niclot, Numerical analysis of the weighted particle
method applied to the semiconductor Boltzmann equation, Numer. Math., 55
(1989), pp. 599–618.
A. Deinega and S. John, Finite difference discretization of
semiconductor drift-diffusion equations for nanowire solar cells, Comput.
Phys. Commun., 183 (2012), pp. 2128–2135.
J. C. deMello, Highly convergent simulations of transport dynamics
in organic light-emitting diodes, J. Comput. Phys., 181 (2002),
pp. 564–576.
C. R. Drago and R. Pinnau, Optimal dopant profiling based on
energy-transport semiconductor models, Math. Models Methods Appl. Sci., 18
(2008), pp. 195–241.
B. Eisenberg, Ionic channels: Natural nanotubes described by the
drift diffusion equations, Superlattices and Microstructures, 27 (2000),
pp. 545–549.
A. M. Fajardo and N. S. Lewis, Free-energy dependence of
electron-transfer rate constants at Si/liquid interfaces, J. Phys. Chem.
B, 101 (1997), pp. 11136–11151.
W. R. Fawcett, Liquids, Solutions, and Interfaces: From Classical
Macroscopic Descriptions to Modern Microscopic Details, Oxford University
Press, New York, 2004.
F. Filbet and S. Jin, A class of asymptotic-preserving schemes for
kinetic equations and related problems with stiff sources, J. Comput. Phys.,
229 (2010), pp. 7625–7648.
J. M. Foley, M. J. Price, J. I. Feldblyum, and S. Maldonado, Analysis of the operation of thin nanowire photoelectrodes for solar energy
conversion, Energy Environ. Sci., 5 (2012), pp. 5203–5220.
J. M. Foster, J. Kirkpatrick, and G. Richardson, Asymptotic and
numerical prediction of current-voltage curves for an organic bilayer solar
cell under varying illumination and comparison to the Shockley equivalent
circuit, J. Appl. Phys., 114 (2013).
S. Gadau and A. Jüngel, A three-dimensional mixed finite-element
approximation of the semiconductor energy-transport equations, SIAM J. Sci.
Comput., 31 (2008), pp. 1120–1140.
M. Galler, Multigroup equations for the description of the particle
transport in semiconductors, World Scientific, 2005.
Y. Gao, Y. Georgievskii, and R. Marcus, On the theory of electron
transfer reactions at semiconductor electrode/liquid interfaces, J. Phys.
Chem., 112 (2000), pp. 3358–3369.
Y. Gao and R. Marcus, On the theory of electron transfer reactions
at semiconductor electrode/liquid interfaces. II. A free electron model,
J. Phys. Chem., 112 (2000), pp. 6351–6360.
A. Glitzky, Analysis of electronic models for solar cells including
energy resolved defect densities, Math. Methods Appl. Sci., 34 (2011),
pp. 1980–1998.
T. Grasser, ed., Advanced Device Modeling and Simulation, World
Scientific, 2003.
M. Grätzel, Photoelectrochemical cells, Nature, 414 (2001),
pp. 338–344.
height 2pt depth -1.6pt width 23pt, Dye-sensitized solar
cells, J. Photochem. Photobiol. C, 4 (2003), pp. 145–153.
H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of
Semiconductors, Springer-Verlag, Berlin, 1996.
B. Hille, Ionic Channels of Excitable Membranes, Sinauer
Associates, Sunderland, MA., 3nd ed., 2001.
T. L. Horng, T. C. Lin, C. Liu, and B. Eisenberg, PNP equations
with steric effects: A model of ion flow through channels, J Phys Chem B.,
116 (2012), pp. 11422–11441.
J. W. Jerome, Analysis of Charge Transport: A Mathematical Study of
Semiconductor Devices, Springer-Verlag, Berlin, 1996.
J. Jin, K. Walczak, M. R. Singh, C. Karp, N. S. Lewis, and C. Xiang, An experimental and modeling/simulation-based evaluation of the efficiency
and operational performance characteristics of an integrated, membrane-free,
neutral pH solar-driven water-splitting system, Energy Environ. Sci., 7
(2014), pp. 3371–3380.
S. Jin and L. Pareschi, Discretization of the multiscale
semiconductor Boltzmann equation by diffusive relaxation schemes, J.
Comput. Phys., 161 (2000), pp. 312–330.
A. Jüngel, Transport Equations for Semiconductors, Springer,
Berlin, 2009.
P. V. Kamat, K. Tvrdy, D. R. Baker, and J. G. Radich, Beyond
photovoltaics: Semiconductor nanoarchitectures for liquid-junction solar
cells, Chem. Rev., 110 (2010), pp. 6664–6688.
B. M. Kayes, H. A. Atwater, and N. S. Lewis, Comparison of the
device physics principles of planar and radial p-n junction nanorod solar
cells, J. Appl. Phys., 97 (2005).
M. S. Kilic, M. Z. Bazant, and A.Ajdari, Steric effects in the
dynamics of electrolytes at large applied voltages: II. modified
Nernst-Planck equations, Phys. Rev. E, 75 (2007).
R. Kircher and W. Bergner, Three-Dimensional Simulation of
Semiconductor Devices, Berkhäuser Verlag, Basel, 1991.
A. A. Kulikovsky, A more accurate Scharfetter-Gummel algorithm of
electron transport for semiconductor and gas discharge simulation, J.
Comput. Phys., 119 (1995), pp. 149–155.
D. Laser and A. J. Bard, Semiconductor electrodes: VII digital
simulation of charge injection and the establishment of the space charge
region in the absence of surface states, J. Electrochem. Soc., 123 (1976),
pp. 1828–1832.
height 2pt depth -1.6pt width 23pt, Semiconductor
electrodes: VIII digital simulation of open-circuit photopotentials, J.
Electrochem. Soc., 123 (1976), pp. 1833–1837.
N. S. Lewis, Mechanistic studies of light-induced charge separation
at semiconductor/liquid interfaces, Acc. Chem. Res., 23 (1990),
pp. 176–183.
height 2pt depth -1.6pt width 23pt, Progress in
understanding electron-transfer reactions at semiconductor/liquid
interfaces, J. Phys. Chem. B, 102 (1998), pp. 4843–4855.
J. Li, Y. Chen, and Y. Liu, Mathematical simulation of metamaterial
solar cells, Adv. Appl. Math. Mech., 3 (2011), pp. 702–715.
P.-L. Lions, On the Schwarz alternating method. I., in Domain
Decomposition Methods, SIAM, Philadelphia, 1988, pp. 1–42.
height 2pt depth -1.6pt width 23pt, On the Schwarz
alternating method. II. Stochastic interpretation and order properties.,
in Domain Decomposition Methods, SIAM, Philadelphia, 1989, pp. 47–70.
W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems
for ion channels with multiple ion species, J. Diff. Eqn., 246 (2009),
pp. 428–451.
B. Lu and Y. C. Zhou, Poisson-Nernst-Planck equations for
simulating biomolecular diffusion-reaction processes II: Size effects on
ionic distributions and diffusion-reaction rates, Biophysical Journal, 100
(2011), pp. 2475–2485.
S. Mafé, J. Pellicer, and V. M. Aguilella, A numerical approach to
ionic transport through charged membranes, J. Comput. Phys., 75 (1988),
pp. 1–14.
A. Majorana and R. Pidatella, A finite difference scheme solving the
Boltzmann-Poisson system for semiconductor devices, J. Comput. Phys., 174
(2001), pp. 649–668.
P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor
Equations, Springer, New York, 1990.
S. R. Mathur and J. Y. Murthy, A multigrid method for the
Poisson-Nernst-Planck equations, Int. J. Heat Mass Transfer, 52 (2009),
pp. 4031–4039.
R. Memming, Semiconductor Electrochemistry, Wiley-VCH, New York,
S. Micheletti, A. Quarteroni, and R. Sacco, Current-voltage
characteristics simulation of semiconductor devices using domain
decomposition, J. Comput. Phys., 119 (1995), pp. 46–61.
J. Nelson, The Physics of Solar Cells, Imperial College Press,
London, 2003.
J. Newman and K. E. Thomas-Alyea, Electrochemical Systems,
Wiley-Interscience, New York, 3rd ed., 2004.
B. Niclot, P. Degond, and F. Poupaud, Deterministic particle
simulations of the Boltzmann transport equation of semiconductors, J.
Comput. Phys., 78 (1988), pp. 313–349.
A. J. Nozik and R. Memming, Physical chemistry of
semiconductor-liquid interfaces, J. Phys. Chem., 100 (1996),
pp. 13061–13078.
J. H. Park and J. W. Jerome, Qualitative properties of steady-state
Poisson–Nernst–Planck systems: Mathematical study, SIAM J. Appl. Math.,
57 (1997), pp. 609–630.
M. Penny, T. Farrell, and C. Please, A mathematical model for
interfacial charge transfer at the semiconductor-dye-electrolyte interface of
a dye-sensitised solar cell, Solar Energy Materials and Solar Cells, 92
(2008), pp. 11–23.
M. A. Penny, T. W. Farrell, G. D. Will, and J. M. Bell, Modelling
interfacial charge transfer in dye-sensitised solar cells, J. Photochem.
Photobiol. A, 164 (2004), pp. 41–46.
K. E. Pomykal and N. S. Lewis, Measurement of interfacial
charge-transfer rate constants at n-type InP/CHOH junctions, J. Phys.
Chem. B, 101 (1997), pp. 2476–2484.
G. Richardson, C. Please, J. Foster, and J. Kirkpatrick, Asymptotic
solution of a model for bilayer organic diodes and solar cells, SIAM J.
Appl. Math., 72 (2012), pp. 1792–1817.
C. A. Ringhofer, Computational methods for semiclassical and quantum
transport in semiconductor devices, Acta Numer., 6 (1997), pp. 485–521.
A. Rossani, A new derivation of the drift-diffusion equations for
electrons and phonons, Physica A, 390 (2011), pp. 223–230.
A. Schenk, Advanced Physical Models for Silicon Device Simulation,
Springer, Wien, 1998.
D. Schroeder, Modelling of Interface Carrier Transport for Device
Simulation, Springer-Verlag, Wien, 1994.
Z. Schuss, B. Nadler, and R. S. Eisenberg, Derivation of PNP
equations in bath and channel from a molecular model, Phys. Rev. E, 64
S. Selberherr, Analysis and Simulation of Semiconductor Devices,
Springer, Vienna, 1984.
N. Seoane, A. J. García-Loureiro, K. Kalna, and A. Asenov, Impact
of intrinsic parameter fluctuations on the performance of HEMTs studied
with a 3d parallel drift-diffusion simulator, Solid-State Electronics, 51
(2007), pp. 481–488.
A. Singer and J. Norbury, A Poisson-Nernst-Planck model for
biological ion channels-an asymptotic analysis in a three-dimensional narrow
fun, SIAM J. Appl. Math., 70 (2009), pp. 949–968.
D. Singh, X. Guo, J. Y. Murthy, A. Alexeenko, and T. Fisher, Modeling of subcontinuum thermal transport across semiconductor-gas
interfaces, J. Appl. Phys., 106 (2009).
H. Steinrück, A bifurcation analysis of the one-dimensional
steady-state semiconductor device equations, SIAM J. Appl. Math., 49 (1989),
pp. 1102–1121.
M. J. Ward, L. G. Reyna, and F. M. Odeh, Multiple steady-state
solutions in a multijunction semiconductor device, SIAM J. Appl. Math., 51
(1991), pp. 90–123.
P. Y. Yu and M. Cardona, Fundamentals of Semiconductors: Physics and
Materials Properties, Springer, Berlin, 3rd ed., 2003.
S. Zhao, J. Ovadia, X. Liu, Y.-T. Zhang, and Q. Nie, Operator
splitting implicit integration factor methods for stiff
reaction-diffusion-advection systems, J. Comput. Phys., 230 (2011),
pp. 5996–6009.
|
1511.00405
|
Exploration of Beyond the Standard Model Physics at the ILC
Stefania Gori
The International Linear Collider (ILC) will advance our understanding of fundamental physics, through a program for precision measurements of the Higgs boson and of the top quark properties. An additional crucial goal of the ILC will be the direct discovery of light New Physics (NP) particles.
This proceeding gives an overview of the ILC direct discovery potential.
Special emphasis is dedicated to the discovery prospects for light hidden Supersymmetric particles, such as Higgsinos and staus. Furthermore, it will be extremely promising to have analyses aimed to the discovery of light hidden NP particles produced from new decay modes of the Higgs boson.
The ILC low background environment, as well as the precise knowledge of the initial-state beams are also ideal for the precise measurements of the spectrum and interactions of these NP particles, once discovered.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
With the LHC discovery of the 125 GeV Higgs boson, it is critically important to determine the properties of this new particle as comprehensively and as accurately as possible. The International Linear Collider (ILC) will cover a special role in measuring the various Higgs couplings with a precision at the (sub)percent level.
The ILC precision program will be complemented by a rich program of searches for direct production of New Physics (NP) particles. In particular, the ILC will have in several cases a unique sensitivity in searching for NP signals arising from the electroweak production of new particles, assuming these are kinematically accessible. This is especially important, in view of the null-results in the search for $SU(3)$ charged particles at Run I LHC. Very interesting examples are searches for light electroweak Supersymmetric (SUSY) particles, like Higgsinos and staus, as well as searches for exotic decays of the 125 GeV Higgs boson into light singlet-state particles.
At Run II LHC (LHC14), the production cross section of a pair of Higgsinos and a pair of staus is relatively sizable: $\sim 300$ fb for 200 GeV charged Higgsinos and $\sim 20$ fb for 200 GeV staus with an equal left and right-handed component. Nevertheless, these particles are quite challenging for the LHC, due to their hidden signatures, leading to background limited LHC searches.
Higgsinos and staus may well be observable in a
clean environment as at the ILC, as long as they are kinematically accessible (Sec. <ref>).
Furthermore, in addition to the precise measurement of the Standard Model (SM) Higgs couplings, the determination of the eventual Higgs couplings to light ($m_h>M_{\rm{NP}}/2$) NP particles through Higgs exotic signatures represents a crucial physics program to unveil physics beyond the Standard Model (BSM) and, in particular, to test hidden sectors (see <cit.> for a review). In Sec. <ref>, we will discuss the complementarity between possible LHC and ILC measurements of Higgs exotic decays.
§ LIGHT HIDDEN SUPERSYMMETRY
In Supersymmetric extensions of the Standard Model, Higgsinos are preferred to have masses at around the electroweak scale by naturalness arguments, since the $\mu$ parameter enters the tree-level Higgs potential. In SUSY spectra with Higgsinos Lightest Supersymmetric Particle (LSP) and with multi-TeV gaugino masses, the
mass splittings between charged and neutral Higgsino-like states are relatively small and of the order of 1-5 GeV. For much heavier gaugino masses, radiative corrections to the Higgsino-like chargino and neutralino masses are relevant and lead to a splitting of the order of $\sim 200$ MeV.
At colliders, Higgsino production proceeds through the s-channel exchange of a $Z$ boson, a photon or a $W$ boson: $pp(e^+e^-)\to\tilde \chi_2^0\tilde\chi_2^0,\tilde\chi_1^\pm\tilde\chi_1^\mp,\tilde\chi_2^0\tilde\chi_1^\pm$, followed by the decay of $\tilde\chi_2^0$ and $\tilde\chi_1^\pm$ into (soft) visible particles and missing energy, carried in great part by the LSP $\tilde\chi_1^0$. This type of spectra is notoriously challenging for the LHC due to the small amount of missing energy and to the low $p_T$ spectrum of the visible particles. One known way to improve the search sensitivity to small mass-gap regions is to
utilize a relatively hard initial state (ISR) jet, that results in an overall boost of the $\tilde\chi_2^0\tilde\chi_2^0,\tilde\chi_1^\pm\tilde\chi_1^\mp$ and $\tilde\chi_2^0\tilde\chi_1^\pm$ systems. Nevertheless, even using this technique, LHC analyses based on Run I data could not push the reach on Higgsinos significantly beyond the LEP reach, if the mass splitting is below (20-30) GeV <cit.>.
At the ILC, thanks to the precisely known initial state, an ISR photon can be used very efficiently to test squeezed Higgsino scenarios[Note that this applies generically to squeezed electroweak spectra.]. This results in a possible quick discovery of Higgsinos once they are kinematically accessible (see the left panel of Fig. <ref> for the discovery reach in the $\mu-m_{1/2}$ plane).
Furthermore, the ILC has the opportunity to precisely determine the masses of the Higgsinos, once discovered. The chargino mass can be reconstructed by fitting the distribution for the reduced center-of-mass energy $\sqrt{s^\prime}$ ($s^\prime=s-2\sqrt s E_\gamma$) near the endpoint region (see the red line in the right panel of Fig. <ref>). This gives a determination of the chargino mass at the level of $\mathcal O(1\%)$. Additionally, the mass splitting between the chargino and the LSP can be determined by fitting the energy distribution of the pion arising from the Higgsino decay with an uncertainty at the level of $\mathcal O(5\%)$ <cit.>.
Left panel (from <cit.>): ILC discovery reach for Higgsino pair production (black contours). Also shown the reach for gluino pair production at Run I LHC
(solid blue contour), and the region accessible to LHC14 searches of gluino cascade decays with 300 fb$^{-1}$ data (dashed and dot-dashed blue contours). Right panel (from <cit.>): distribution of the reduced center-of-mass energy of the system recoiling against the ISR photon for the benchmark dM770 with $m_{\tilde\chi_1^\pm} = 167.36$ GeV, $m_{\tilde\chi_1^0} = 166.59$ GeV, $m_{\tilde\chi_2^0}= 167.63$ GeV.
Higgsinos are not the only SUSY particles that can be hidden to the LHC. Light staus represent another theoretically well motivated example, since they arise e.g. in gauge-mediated supersymmetry breaking models, as well as in Dark Matter (DM) models with stau-DM co-annihilation. At the LHC, direct searches for promptly decaying staus are notoriously difficult (see for example <cit.>), due to
the challenge in disentangling a di-tau plus missing energy signature ($pp\to\tilde\tau_1\tilde\tau_1\to 2\tau+$MET) from the W+jets and di-boson SM backgrounds. Present LHC searches based on Run I data could not push the reach beyond the one obtained by LEP <cit.>.
Due to the cleaner ILC environment, the ILC could probe staus up to masses very close to the kinematical reach, if the mass splitting between the stau and the LSP is larger than $\sim 20$ GeV (see the red contour in the left panel of Fig. <ref>). Additionally, also squeezed spectra with $m_{\tilde\tau_1}-m_{\rm{LSP}}=\mathcal O({\rm{few \,GeV}})$ could be probed for stau masses up to $\sim 200$ GeV, at $\sqrt s=500$ GeV. Once a stau is discovered, the ILC will be able to perform a large set of precision measurements of the stau spectrum. In particular, the stau mass can be determined using threshold scans with an uncertainty at the level of $\mathcal O(1)$ GeV (see for example <cit.>). Finally, one can measure the left-right mixing of $\tilde\tau_1$ and the Higgsino/gaugino mixing of the LSP
by measuring both the cross section of $\tilde\tau_1$ production (see right panel of Fig. <ref> for the dependence of the cross section on the stau mixing angle $\theta_\tau$) and the average polarization of the $\tau$ from the $\tilde\tau_1$ decay <cit.>.
Left panel (from <cit.>): exclusion bound (in red) and discovery reach (in black) for a $\tilde\tau_1$ NLSP with 500 fb$^{-1}$ at $\sqrt s=500$ GeV. Right panel (from <cit.>): Cross section for $\tilde\tau_1\tilde\tau_1$ production as a function of the stau mixing angle $\theta_\tau$, for two different electron polarizations $P_{e^-}$.
§ DARK MATTER AND HIGGS
Dark Matter dominates the matter density in our Universe, but very little is known about it. One of the prime tasks of the LHC and of future colliders is to identify the nature of Dark Matter.
WIMP Dark Matter can be searched for by looking for mono-jet or mono-photon events. These searches are typically interpreted in terms of bounds on higher dimensional effective operators mediating the interactions of DM with SM particles and suppressed by a NP scale $\Lambda$. For WIMP masses below
the kinematic limit, the ILC reach in the effective operator scale is rather independent of the WIMP mass and is $\Lambda\sim 3$ TeV in the case of vector (D5) and axial-vector (D8) operators <cit.> (see also <cit.>).
Beyond the canonical WIMPs, also alternative models for DM have been developed. In particular, DM can belong to a hidden sector and carry its thermalization thanks to its interactions with BSM particles, not charged under the SM gauge group. Such particles could easily have escaped past
experimental searches, since they interact only feebly with ordinary matter.
The Higgs boson represents a unique opportunity to test hidden sectors, as it can serve as a portal, being the $HH^\dagger$ operator the lower dimensional singlet that can be written out of SM fields. As an example, several well motivated BSM models contain a hidden sector scalar that couples to the Higgs through the Higgs portal operator $ \xi |H|^2|S|^2$. If the scalar is lighter than $m_h/2$, this operator will generically lead to Higgs exotic decays of the type $h\to S S$ and therefore to a variety of new Higgs signatures, depending on the decay mode of the scalar $S$.
These exotic decays can be looked for at the ILC in a twofold way: either indirectly, through a precise measurement of the total Higgs width, or directly, looking for the new Higgs decay modes. The former method will be able to unravel the existence of new Higgs decay modes with branching ratios at the level of 3.8$\%$ at an Initial Phase and of 1.8$\%$ with the full data set[As presented in <cit.>, the Initial Phase will have 500 fb$^{-1}$ at 500 GeV, 200 fb$^{-1}$ at 350 GeV, and 500 fb$^{-1}$ at 250 GeV. The luminosity-upgraded phase will have an additional 3500 fb$^{-1}$ at 500 GeV and 1500 fb$^{-1}$ at 250 GeV.],
using the recoil technique that allows the measurement of the ZH cross section without reconstructing the decays of the Higgs
boson <cit.>.
Additionally, present and future colliders like the ILC can look directly for exotic decays of the Higgs boson. The LHC will produce $\sim 1.5\times 10^7$ Higgs bosons ($\sim 1.5\times 10^8$ at the High-Luminosity (HL) stage), offering a unique opportunity to probe tiny branching ratios for clean decay modes, as for example for the Higgs decay to dark photons ($Z_D$), $h\to Z_DZ_D\to 4l$, for which the reach is at the level of $\mathcal O(10^{-7})$ at the HL-LHC <cit.>.
On the contrary, the ILC will produce $\sim 3\times 10^5$ Higgs bosons during the Initial Phase and $\sim 2 \times 10^6$ Higgs bosons with the full data set, potentially probing branching ratios into NP particles as small as $\mathcal O({\rm{few}}\times 10^{-6})$.
In spite of the smaller number of Higgs bosons produced, the ILC can play a significant role in
searching for those exotic decays that are more challenging for the LHC, because background limited. Typical examples are
$h\to SS\to 4\tau$ and $h\to SS\to 4b$, that arise naturally in many theories beyond the Standard Model, as for example in the Next to Minimal Supersymmetric Standard Model (NMSSM). In the case of $h\to SS\to 4\tau$, Ref. <cit.> shows a possible reach at the level of $\mathcal O({\rm{few}}\times 10^{-3})$. It will be very interesting to perform a dedicated study to determine the ILC reach for $h\to SS\to 4b$.
§ CONCLUSIONS
An $e^+e^-$ collider like the ILC offers a unique opportunity to discover new phenomena indirectly, through the precise measurements of Higgs and top couplings.
At the same time, as we have stressed in this talk, the ILC
is a machine
for direct discoveries. In particular, the ILC will be able to test $SU(3)$-singlet new physics particles that are hidden to the LHC, but nevertheless extremely well motivated, such as Higgsinos and staus, if they are within the kinematical reach. Furthermore, to have a complete program for Higgs characterization, it will be extremely promising to have ILC analyses aimed to the discovery of exotic Higgs decay modes into light new physics particles, that are otherwise hidden to hadron
I am grateful to Michael Peskin for discussions.
D. Curtin et al.,
Phys. Rev. D 90, no. 7, 075004 (2014)
[arXiv:1312.4992 [hep-ph]].
G. Aad et al. [ATLAS Collaboration],
arXiv:1509.07152 [hep-ex].
H. Baer, V. Barger, P. Huang, D. Mickelson, A. Mustafayev, W. Sreethawong and X. Tata,
arXiv:1306.3148 [hep-ph].
M. Berggren, F. Brümmer, J. List, G. Moortgat-Pick, T. Robens, K. Rolbiecki and H. Sert,
Eur. Phys. J. C 73, no. 12, 2660 (2013)
[arXiv:1307.3566 [hep-ph]].
M. Carena, S. Gori, N. R. Shah, C. E. M. Wagner and L. T. Wang,
JHEP 1207, 175 (2012)
[arXiv:1205.5842 [hep-ph]].
H. Baer, M. Berggren, J. List, M. M. Nojiri, M. Perelstein, A. Pierce, W. Porod and T. Tanabe,
arXiv:1307.5248 [hep-ph].
M. M. Nojiri,
Phys. Rev. D 51, 6281 (1995)
G. Belanger, O. Kittel, S. Kraml, H.-U. Martyn and A. Pukhov,
Phys. Rev. D 78, 015011 (2008)
[arXiv:0803.2584 [hep-ph]].
T. Barklow, J. Brau, K. Fujii, J. Gao, J. List, N. Walker and K. Yokoya,
arXiv:1506.07830 [hep-ex].
Y. J. Chae and M. Perelstein,
JHEP 1305, 138 (2013)
[arXiv:1211.4008 [hep-ph]].
K. Fujii et al.,
arXiv:1506.05992 [hep-ex].
D. Curtin, R. Essig, S. Gori and J. Shelton,
JHEP 1502, 157 (2015)
[arXiv:1412.0018 [hep-ph]].
T. Liu and C. T. Potter,
arXiv:1309.0021 [hep-ph].
|
1511.00202
|
James Gilbertjames.gilbert@physics.ox.ac.uk0000-0001-5065-2101University of OxfordDepartment of PhysicsOxfordOxfordshireOX1 3RHUK
Gavin [email protected] of OxfordDepartment of PhysicsOxfordOxfordshireOX1 3RHUK
Ian [email protected] of OxfordDepartment of PhysicsOxfordOxfordshireOX1 3RHUK
WEAVE is a 1000-fiber multi-object spectroscopic facility for the 4.2 m William Herschel Telescope. It will feature a double-headed pick-and-place fiber positioning robot comprising commercially available robotic axes. This paper presents results on the performance of these axes, obtained by testing a prototype system in the laboratory. Positioning accuracy is found to be better than the manufacturer's published values for the tested cases, indicating that the requirement for a maximum positioning error of 8.0 microns is achievable. Field reconfiguration times well within the planned 60 minute observation window are shown to be likely when individual axis movements are combined in an efficient way.
§ INTRODUCTION
The WHT Enhanced Area Velocity Explorer (WEAVE) is a next generation wide field fiber-fed multi-object spectroscopic facility for the 4.2 m William Herschel Telescope (WHT). Details on its design and motivation are discussed in <cit.>.
The WEAVE fiber positioner system will enable the deployment of 1000 individual optical fibers or 20 integral field units across a flat focal plane 410 mm in diameter (2 on sky). The design employs a pick-and-place robotic gantry system with exchangeable field plates, a format chosen in light of its successful implementation in the 400-fiber 2dF instrument at the Anglo-Australian Telescope <cit.>.
WEAVE advances the 2dF concept with the addition of a second robot in order to reduce configuration times and thus allow full field reconfigurations within a nominal 60 minute observation window. WEAVE will use commercial off-the-shelf (COTS) robotic axes as a means to minimize costs and development time. The performance of these devices with regard to the instrument requirements is the focus of this paper.
§.§ Fiber positioner requirements
The performance requirements for the WEAVE positioner are shown in Table <ref>. The requirement on accuracy flows down from a top-level tolerance of 0.2 arcmin for fiber-target misalignment, split equally between the optical and positioner subsystems. Given the nominal observation window of 60 minutes, an average move time of 3.4 seconds is calculated with the assumptions that i) every fiber must be parked at the perimeter of the field before repositioning, and ii) the use of two robots is 90% efficient.
Top level fiber positioner requirements
Specification Requirement
Total allowable fiber-target misalignment (RMS) $\leq$0.2" (11.4 )
1eM Contribution from prime focus optics $\leq$8.0
1eM Contribution from positioner $\leq$8.0
Field reconfiguration time $\leq$60 min
1em Implied average time per move $\leq$3.4 s
§.§ Positional error sources
The achieved positioning accuracy will be the sum of various distinct error sources. Table <ref> lists the most significant ones, combined as the root of sum of squares (RSS) for simplicity. This gives a total `bottom-up' error estimate of 9.0 microns when taking axis performance as the manufacturer's specification. However, this reduces to 6.0 microns with the actual measured values presented in the following sections.
Bottom-up analysis of RMS positioning errors for the most significant sources, showing best estimates before and after testing
Error source Initial estimate () Revised estimate ()
Gantry non-orthogonality calibration 4.0 4.0
Metrology camera calibration 1.2 1.2
Fiber measurement error (0.2 pixels) 0.75 0.75
Robot axis accuracy ($x$+$y$) 7.1 3.6
Gripper jaws release repeatability 3.0 0.91
Gantry flexure variability 2.0 2.0
RSS total 9.0 6.0
§ PROTOTYPE POSITIONER ROBOT
A prototype positioner has been assembled in order to verify the performance of the COTS motors chosen for WEAVE. It is a simplified version of the full design, matching the format of the $y$-, $z$-, $\theta$-axes and gripper assembly. This allows pick-and-place positioning of magnetic fiber terminals, called `buttons', along a steel strip spanning the $y$-axis. Metrology is provided by a camera within the gripper assembly.
§.§ Performance tests
The following sections present the results of positioning accuracy and speed tests for the prototype robot described above. This initial evaluation of the chosen COTS motors indicates accuracies better than published values, and an estimated positioning time well within the maximum 60 minutes.
Linear axis repeatability
Axis accuracy was measured with a simple test of positional repeatability. The robot was instructed to position 500 mm away from its origin and then return. This routine was repeated 500 times and the apparent location of a static reference fiber extracted for each cycle. The scatter in this measurement provides an upper limit for positioning accuracy of the axis.
The results (Figure <ref>) show a scatter of only 2.55 microns along the active axis, which is substantially less than the manufacturer's published value of 5 microns.
[scale=0.6]JGilbert-test_axis_rep.epsplot_axis_repHistogram plot of the actual position of the $y$-axis when instructed to go to the same position 500 times
Gripper release repeatability
The gripper jaws were tested for repeatability when releasing fibers. A fiber was picked up, placed back down, and released. This was repeated 200 times, using two fiber buttons and two different locations along the $y$-axis. The difference in fiber location between gripped and released states was measured for every cycle.
The results (Figure <ref>) show a mean lateral shift of order 10 microns when releasing a button, mainly caused by the prototype field plate not being orthogonal to the $z$-axis of the robot gantry. Crucially, however, the scatter of this offset is sub-micron. This indicates that, as hoped, subtraction of the mean offset for each fiber button during positioning is feasible.
JGilbert-test_gripper_rep_1_small.epsJGilbert-test_gripper_rep_2_small.epsplot_gripper_repHistogram plots of the measured movement of a fiber when released by the robot gripper, for two different fibers at two different locations on the field plate
Timing estimates
The expected field reconfiguration time for WEAVE has been significantly improved by parallelizing the pick-and-place operation.
Table <ref> shows an expected pick-and-place time of 4.5 seconds when each action in the procedure is done sequentially. This would result in a total reconfiguration time of 79 minutes, exceeding the required maximum of 60 minutes (3.4 seconds per move).
By allowing some motor actions to begin before others have ended, the time spent waiting for moves to complete is significantly reduced. Table <ref> lists the results of a test of this scheme, giving an expected time of 2.2 seconds per pick-and-place operation. This reduces the estimated total reconfiguration time by more than half, to 39 minutes.
It is important to note that these estimates do not include any time spent waiting for the gantry structure to settle after a move, and that this cannot be tested until the full system has been assembled for the final instrument. Since the total estimated positioning time now occupies only 65% of that available, it seems reasonable to assume that mechanical settling delays will not result in a failure to meet the requirement.
A comparison of pick-and-place timings for the mean move distance of 205 mm. Safely overlapping moves significantly reduces the total positioning time.
Operation requiring wait Mean wait, sequential (s) Mean wait, overlapped (s)
Movement of robot axes (pick) 1.9 0.71
Movement of gripper jaws (pick) 0.25 0.17
Fiber metrology (pick) 0.10 0.10
Movement of robot axes (place) 1.9 0.86
Movement of gripper jaws (place) 0.25 0.25
Fiber metrology (place) 0.10 0.10
Entire pick-and-place operation 4.5 2.2
§ CONCLUSIONS
Testing of a prototype positioner has verified the published performance values for the COTS systems chosen for WEAVE. The performance of the axis and gripper is significantly better than specified, indicating that the requirement for a maximum positioning error of 8.0 microns is achievable.
The estimated total reconfiguration time for WEAVE has been improved by parallelizing the pick-and-place operation. Although mechanical settling times are not yet included, 45% of the available time window is still available for such delays. Therefore it is reasonable to believe that full fields can be reconfigured to the necessary accuracy within the allowed 60 minutes.
Positioner performance at different inclinations is accounted for by the use of counterweights on all gantry axes. Significant changes in behavior are, therefore, not anticipated with telescope movement. This will be confirmed in the next testing phase.
[Dalton et al.(2014)]jgilbert_weave2014 Dalton, G., Trager, S., Abrams, D. C., et al. 2014, SPIE, 9147
[Lewis et al.(2002)]jgilbert_2df2002 Lewis, I. J., Cannon, R. D., Taylor, K., et al. 2002, MNRAS, 333, 279
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.