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# Hierarchical Graph Based Segmentation and Consensus based Human Tracking Technique

Sunitha Madasi Ramachandra* , Haradagere Siddaramaiah Jayanna** and Ramegowda***

## Abstract

Abstract: Accurate detection, tracking and analysis of human movement using robots and other visual surveillance systems is still a challenge. Efforts are on to make the system robust against constraints such as variation in shape, size, pose and occlusion. Traditional methods of detection used the sliding window approach which involved scanning of various sizes of windows across an image. This paper concentrates on employing a state-of-the-art, hierarchical graph based method for segmentation. It has two stages: part level segmentation for color-consistent segments and object level segmentation for category-consistent regions. The tracking phase is achieved by employing SIFT keypoint descriptor based technique in a combined matching and tracking scheme with validation phase. Localization of human region in each frame is performed by keypoints by casting votes for the center of the human detected region. As it is difficult to avoid incorrect keypoints, a consensus-based framework is used to detect voting behavior. The designed methodology is tested on the video sequences having 3 to 4 persons.

Keywords: Consensus Based Framework , Hierarchical Graph Based Segmentation , SIFT Keypoint Descriptor

## 1. Introduction

Computers are known to perform repetitive, computational and data-intensive task at a better scale and pace than human beings have become ubiquitous in our daily lives. Hence their capabilities can be used further to carry out many intellectual and higher level tasks like analysis of visual scenes, reasoning and logical analyses.

Image processing is a term which denotes the processing of an image, which is taken as an input and the result of processing may be a related parameter of an image. The major task of image processing is to observe objects and traits that are not visible to the naked eye. Moving object detection and tracking is a key task in the area of computer vision.

In recent years, human motion tracking and analysis are most popular aspects of research in image processing. For example, Dalal and Triggs [1] have described a variety of methods of human detection. They commented that the important part of human detection and motion analysis is human movement. The main aim is to detect human movement from the background of the video sequences. The detection and analysis of human movements in a video surveillance scene has gained attraction by multiple applications like human gait characterization, abnormal event detection, person identification, etc. This is due to great popularity in the field of video surveillance, traffic monitoring, and has remarkable commercial significance.

However, there are a lot of limitations in human tracking like videos with low resolution, change in the illumination and the background changes to monitor events in surveillance displays. For an efficient classification of objects, intellectual methods that detect and capture movement information about moving objects are therefore required. The detected object is then monitored for further analysis.

This paper aims to present a robust method to track humans that involve two stages: (i) human detection using hierarchical graph-based segmentation and (ii) human tracking based on consensusbased matching and tracking technique. Hierarchical graph segmentation is used to divide an image into many disjoint subsets, where every subset has a significant part of an image [2]. Generated parts are grouped together to form human blobs. Each human blob undergoes novel human validation scheme. For each human blob, histogram of optical flow (HOF) features [3] are calculated and matched knowledge-based features to identify a human. Later, consensus-based matching tracking (CMT) approach, which uses a human representation based on a key point, is used. These estimations are also used in the identification of objects.

The remainder of the paper is organized so that the literature survey of recent work is discussed in Section 2. Section 3 proposes a methodology and Section 4 discusses different results obtained after the conduction of experiments. Finally, Section 5 presents conclusions.

## 2. Literature Review

2.1 Tracking

Santner et al. [4] provided a manner to deal with the visible tracking problems like drifting as they rely on self-updates of online finding out approaches. This work showed that the augmenting online studying strategy with complementary monitoring approaches can lead to extra steady outcomes. Additionally, they used an easy template mannequin as a non-adaptive mannequin and an optical-flow based mean shift tracker for adaptive detail. A web-based random forest is used as reasonably adaptive appearance headquartered learner. A combo of all the three trackers in a cascade is done. Since all these components execute on multicore techniques, the method works well for real-time performance. Experiments showed a precision rate of 0.9 for publicly available standard data set.

Andrilukaet al. [5] presented a novel process which combined the advantages of each detection and tracking in a single framework. In entire body for each human, an approximate articulation is detected utilizing nearby facets that mannequin the appearance of person physique components. Utilizing a hierarchical Gaussian approach latent variable model (HGPLVM), prior talents on viable articulations and temporal coherency within a running cycle is modeled in this work. Experimental results for detecting and monitoring multiple humans in cluttered scenes with reoccurring and occlusions on TUD dataset are established.

Andriyenko et al. [6] formulated multi-target tracking as a discrete continuous optimization which handles each facet in its ordinary area and makes it possible for leveraging strong tactics for multi-model becoming. Information organization is carried out using discrete optimization with label charges, yielding close optimality. With an easy closed type resolution, which is utilized in turn, to update the label, trajectory estimation is posed as becoming a steady concerned. Accuracy and robustness of this method are validated on standard datasets and suffered with low precision rate.

A consensus based dispensed multi target monitoring algorithm was proposed to handle the problem of naivety in a digital camera network [7]. In this work they have jointly addressed the tracking estimation mistakes and data organization. The outcomes of naivety to the progress of an information weighted consensus technique, which is termed as the multi-target information consensus (MTIC). Additionally, it is stated that MTIC algorithm is made very powerful to false measurements clutter by the incorporation of the probabilistic technique for data association. But this method suffered with the problem of data association with an increase in targets.

Mixed discrete continuous conditional random field (CRF) system tracks multiple targets and handles inter-object exclusions at information association stage situated on non-sub modular constraints [8]. To derive suitable CRF potentials a statistical information evaluation was additionally used. Experiments had been carried out on PETS’09, TUD and ETH dataset benchmark videos with precision rate of 87.2% and when compared with modern-day approaches, outcome showed with a transparent growth from the simultaneous exclusion constraints.

In [9], a technique for quick monitoring of customary objects in a video sequence with the help of a detector was presented. This detector uses the Hough transform into pixel established descriptor. By means of a powerful model of segmentation, the system is equipped to monitor objects which bear flexibility deformations. It also takes cognizance of variations in appearance and shapes. This process gives better results than state-of-the-art methods for Babenko dataset video sequence and non-rigid dataset with less precision rate. The method was not able to change bounding box size and aspect ratio during tracking.

The N consensus algorithm was proposed to reduce consensus process rate to track objects of the target monitoring system with better efficiency [10]. To recognize all nodes from viewing nodes, inside twice the viewing range, the hop count is calculated based on viewing and communication stages. In contrast to normal consensus, the N consensus method no longer requires prior competencies of node connectivity. The reason is that, they use an elevated rapid covariance intersection algorithm during the consensus updating.

Babenko et al. [11] offered a multiple instances learning (MIL) algorithm that utilizes more than one circumstance to obtain a potential procedure with lesser parameter tweaks. This strategy is proposed to obtain superior results than state-of-the-art method. The drawback of the method is that it loses the track of the object that has left the scene. Experimental results show that the precision rate is about 90% of Sylvester video clippings.

2.2 Human Detection

Tian and Sclaroff [12] offered an algorithm for the recovery of the globally optimal 2D human figure detection with the usage of the loopy graph technique. This work has overcome the project of computation time due to the fact that the time complexity increases with the amount of the biggest clique in the graph. The awarded approach made use of BB (branch and bound) to recycle the DP (dynamic programming) tables related to the tree model to search the tree situated lower bound instead of recomputing the lower bound from the beginning. The method fails while tracking multi-objects in a scene. The proposed work is evaluated on an Iterative parsing dataset with detection accuracy of 56.4% and shows that it runs speedy empirically.

Ta et al. [13] awarded new facets known as pairwise feature (PWF), for action awareness to encode both appearance and the spatio-temporal interest point (STIP) relations of the regional features. They proposed a combo of two codebooks for video representation. It fails to capture complex geometric structures among local feature. Experiments carried out on standard datasets KTH with a detection rate of 93.0% and 83% on Weizmann dataset.

Yang et al. [14] integrated a submission of two types of systems. It employs the brute force search approach to evaluate each and every space-time area within the video with the aid of a binary classifier on whether a detailed occasion occurs. The other method uses the capabilities of human detection and performs tracking to restrict the high priced brute drive search. It evaluates the candidate space-time cubes by means of combining 3D convolution neural networks (CNN) and support vector machine (SVM) classifiers founded on bag-of-words local features for the detection of event presence. To cut down the detection cost rates (DCR) by way of thorough cross-validation of the progress set, they have chosen suitable combining weights and thresholds. These systems also accomplished higher performance with a detection rate of 98% on a CAVIAR dataset with a single person like ObjectPut, CellToEar, and Pointing videos.

A method was presented in [15] which detects human by augmenting frequently used region, color and texture features. It leads to an enormously excessive dimensional characteristic area of larger than one hundred seventy thousand dimensions. Partial least squares (PLS) is an effective dimensionality reduction method and conserve tremendous discriminative knowledge for the projection of the info onto a much slash dimensional subspace. But works well only for low dimensional subspace. This process obtained 75% of recall rate on INRIA and ETHZ pedestrian dataset.

Peng and Zhang [16] proposed a structure to evaluate the first-class of specified segmentation quantitatively with a couple of ground truth segmentation. They have adaptively built a ground truth that is locally viable for segmentation. They also continue the structural consistency within the ground truths for a given segmentation. For quantitative segmentation analysis, this work effectively presents a structure to adaptively mix more than one ground truths. The analysis was completed on the benchmark Berkeley segmentation database.

Breitenstein et al. [17] addressed the dilemma in robotically monitoring individuals using an uncelebrated camera. A new technique for human monitoring via detecting in a particle filtering structure was proposed. As a graded statement model in addition to final excessive self-belief detection, proposed technique used the continuous self-assurance of pedestrian detectors and on-line expert, example specific classifiers. Detection and tracking of big quantity of dynamically moving persons in an elaborate environment without relying background modeling was done leading to impose less restriction. Demonstration of the algorithm presented compared to other current approaches is achieved on an ETHZ dataset with the precision rate of 70%. The result shows that robustness during partial occlusion is comparatively less.

A novel approach to conquer problems in tracking the formation of a swarm intelligence viewpoint was introduced in [18]. They presented a species-based particle swarm optimization process for a couple of objects monitoring via dividing global swarm into a couple of species matching the quantity of objects within the scene. Every species search for its objects and continues to track it. The occlusion relationship between one of a kind objects is absolutely determined by the vigor of all species. In this method object is not closely wrap by the bounding box, hence it does not suit for a subspace based tracking method.

Existing algorithms considered in the literature are having many advantages and disadvantages. Few existing methods show that the track of the object is lost if the object disappears from the scene. In the proposed method, the human is tracked every time by making use of the Consensus-based tracking approach. These tracked objects are again validated using features stored in the database. If the tracked human has disappeared from the scene for a few frames and reappear again, human will be recognized and re-tracked again. Since the validation of the human is done in different levels, false positive rate is relatively low. In [9], it is seen that the bounding box does not vary with regard to the size of the object through the tracking and is hardly ever initialized to enclose the entire object giving the accuracy of about 87.41%. But in the proposed system once the keypoints are obtained using scale-invariant feature transform (SIFT), the optical flow of each of these keypoint is measured using Lucas-Kanade technique. The optical flow of the keypoints is compared with the predefined threshold, so that the keypoints which are not within this range are considered and centroids of these keypoints are calculated. By considering the centroid point and the corner of the object, the bounding box is drawn. Hence, the bounding box is varied every time in accordance with the size of the object.

We have implemented an efficient detection and tracking algorithm by making use of hierarchical graph based segmentation and consensus based human tracking algorithm and compared their detection and tracking performance. The combination of these algorithms along with the novel validation system produces the most hopeful results in terms of detection and tracking quality.

## 3. Methodology

Proposed system has two phase called training and testing phases as shown in Fig. 1. Each step is described in detail in the rest of the paper.

A flow diagram of proposed human tracking system.

Training phase

In this phase, cropped human images are considered as input. All the images are resized in preprocessing stage. HOF features are extracted from these images and stored in a knowledge base. This knowledge base is further used for human validation.

Testing phase

The input for this phase is a video sequence from which frames are generated. Each frame is enhanced by applying a sharpening filter. The resultant enhanced image is passed to a human detection stage which is based on a hierarchical approach of graph-based segmentation. Part-level graph-based segmentation and object-level segmentation are two main steps of hierarchical graph-based segmentation algorithm as described in the below Section.

3.1 Human Detection

Humans in the input video sequence are detected by using hierarchical graph segmentation. First, color consistent clusters are obtained by using the segmentation of the part level on the pixels of the input frame using an efficient graph-based segmentation technique described in [19]. All color clusters are separated from the output of partial segmentation in object-level segmentation. Gabor features are calculated and transferred to SVM [20] for each cluster. The SVM will later classify clusters into human and non-human regions with these features.

3.1.1 Part level segmentation

Consider a graph [TeX:] $$G = ( V , E )$$ having vertices [TeX:] $$v _ { i } \in V$$, a set of pixels and [TeX:] $$\left( v _ { i } , v _ { j } \right) \in E$$ edges that match the neighboring vertices [21]. The goal of graph segmentation is partitioning the input frame into color clusters.

The input is shown in Figs. 2 and 3 gives the result obtained for input video after applying part level segmentation. First, a graph is constructed with vertices as input frame pixels and edges as the color differences among two adjacent vertices, [TeX:] $$v _ { i } \text { and } v _ { j } \text { as } W _ { P } \left( \left( V _ { i } , V _ { j } \right) \right) = \| I \left( V _ { i } \right) - I ( V ) ) _ { j } \|$$.

Input first frame.
Output of part-level segmentation.

Segmentation is initially carried out with each vertex vi in its own component, say Ci, and the edges are arranged in the ascending weight of wp. Region merging Dp(C1,C2) among C1 and C2 is computed using graph segmentation. Merging is done if the Dif(C1, C2) difference among the components is huge when compared to the inner difference with at least one of the components, int(C1) and int(C2). Later, two components are merged when

##### (1)
[TeX:] $$D _ { p } \left( C _ { 1 } , C _ { 2 } \right) = \left\{ \begin{array} { c c } { \text {True} } \ { \text { if } \operatorname { Dif } \left( C _ { 1 } , C _ { 2 } \right) > \operatorname { MInt } \left( C _ { 1 } , C _ { 2 } \right) } \\ { \text {False} } \ { \text { otherwise } } \end{array} \right.$$

where

##### (2)
[TeX:] $$\operatorname { Dif } \left( C _ { 1 } , C _ { 2 } \right) =\left. \right| \min _ { V _ { i } \in C _ { 1 } , V _ { J } \in C _ { 2 } , \left( V _ { i } , V _ { J } \right) \in E } W _ { P } \left( \left( V _ { i } , V _ { J } \right) \right)$$

and

##### (3)
[TeX:] $$\operatorname { MInt } \left( C _ { 1 } , C _ { 2 } \right) = \min \left( \ln t \left( C _ { 1 } \right) + \left( C _ { 1 } \right) , \operatorname { In } t \left( C _ { 2 } \right) + \left( C _ { 2 } \right) \right)$$

##### (4)
[TeX:] $$\operatorname { Int } ( C ) = \max _ { e \in M S T ( C , E ) } W _ { P } ( e )$$

where the component C’s minimum spanning tree is represented by the [TeX:] $$M S T ( C , E ) \text { and } ( C ) = \frac { k } { | c | }$$. Constant k refers to the value which control the merging and |c| depicts to the number of components.

3.1.2 Object level segmentation

To achieve this segmentation [TeX:] $$G ^ { \prime } = \left( V ^ { \prime } , E ^ { \prime } \right)$$, a graph is constructed having [TeX:] $$V _ { i } ^ { \prime } \in V ^ { \prime }$$ number of vertices, which are the parts of segmentation obtained by part level approach of segmentation [TeX:] $$\left( V _ { i } ^ { \prime } , V _ { j } ^ { \prime } \right) \in E ^ { \prime }$$. Edges are nothing but the weighted means of confidence values calculated from pairs of neighboring vertices as [TeX:] $$W _ { o } \left( V _ { i } ^ { \prime } , V _ { j } ^ { \prime } \right) = c _ { f } \left( R \left( V _ { i } ^ { \prime } \right) \right) + C _ { f } \left( R \left( V _ { j } ^ { \prime } \right) \right)$$ , where Cf represents the object confidence value obtained by classifiers and minimum rectangular region [TeX:] $$R \left( V _ { i } ^ { \prime } \right)$$ that consists of the pixels in the segment [TeX:] $$V _ { i } ^ { \prime }$$. Weight parameters obtained by the sizes of the regions [TeX:] $$as = \left| R \left( V _ { i } ^ { \prime } \right) \right| / \left( \left| R \left( V _ { i } ^ { \prime } \right) \right| + \left| R \left( V _ { j } ^ { \prime } \right) \right| \right) \text { and } \left| R \left( V _ { j } ^ { \prime } \right) \right| / \left( \left| R \left( V _ { i } ^ { \prime } \right) \right| + \left| R \left( V _ { j } ^ { \prime } \right) \right| \right)$$. The size of the rectangular region of V is represented as |R(V)|. After sorting the edge weights in a non-decreasing order, the region comparison predicts [TeX:] $$D _ { o } \left( S _ { 1 } , S _ { 2 } \right)$$ between two components S2 and S1 is calculated by checking whether the merged segment has a higher confidence than two segments as in Eq. (5). Later, two components are merged when [TeX:] $$D _ { o } \left( S _ { 1 } , S _ { 2 } \right)$$ is false.

##### (5)
[TeX:] $$D _ { o } \left( S _ { 1 } , S _ { 2 } \right) = \left\{ \begin{array} { l r } { \text { True } } \ { \text { if } \hat { C } _ { f } \left( S _ { 1 } + S _ { 2 } \right) < M \operatorname { Cof } \left( S _ { 1 } , S _ { 2 } \right) } \\ { \text {false} } \ { \text { otherwise } } \end{array} \right.$$

where

##### (6)
[TeX:] $$\operatorname { MCof } \left( S _ { 1 } , S _ { 2 } \right) = \max \left( C _ { f } \left( P \left( S _ { 1 } \right) \right) + C _ { f } \left( R \left( S _ { 2 } \right) \right) , C _ { o } \right)$$

##### (7)
[TeX:] $$\hat { C } _ { f } ( S ) = \operatorname { maxC } _ { f } ( R ( S ) + ) , \quad \sim \operatorname { Uniform } ( - B , B )$$

where R(S)indicates rectangular region contains all segmented pixels [TeX:] $$V _ { i } \in S . R ( S )$$ is denoted by four parameters, x, y, wt and ht of a rectangular box and is equivalent to [TeX:] $$\left| R \left( S _ { 1 } \right) \right| / \left( \left| R \left( S _ { 1 } \right) \right| + \left| R \left( S _ { 2 } \right) \right| \right)$$ and [TeX:] $$\left| R \left( S _ { 2 } \right) \right| / \left( \left| R \left( S _ { 2 } \right) \right| + \left| R \left( S _ { 2 } \right) \right| \right)$$. Where x, y indicates the initial point value of the rectangle, wt gives the width and ht specifies the height of the rectangle box. Gabor features [22,23] are extracted for these rectangular regions and given to SVM classifier [20] as input. By using minimum rectangular region confidence value, [TeX:] $$C _ { f } \left( R \left( S _ { 1 } + S _ { 2 } \right) \right)$$ is computed. Many region hypotheses are generated from a merged region [TeX:] $$R \left( S _ { 1 } + S _ { 2 } \right)$$ and location sizes of the regions are varied as in Eqs. (6) and (7) [21].The lower and upper bound is obtained and later uniformly the hypotheses are generated.

The comparisons of image segmented with gradient orientation pyramid (GOP) features which are stored in the knowledge base are done. An SVM classifier differentiates between human region and nonhuman region using trained GOP features to create a single blob. The whole procedure is demonstrated as in Fig. 4.

A flow diagram of human blob generation.
Human patch generation.
Human detection by grouping human patches.

As shown in Fig. 5, the patches of human region marked with a rectangular box. All of these patches of same human regions form a human blob, as depicted in Fig. 6.

Human validation is done by comparing the HOF feature of the segmented human with feature stored in the knowledge base. Here the texture feature of the detected human was compared with the HOF features already stored in the knowledge base. During comparison, if the correlation of the features extracted is less when compared to the features already stored in the database then we match that person with the model. If the correlation difference obtained is more, then it is not considered as human. Texture features are chosen because they provide vital information for the classification of the segmented image. It is used in describing the high level semantics for the retrieval of human.

3.2 Human Tracking 3.2.1 Localization of keypoints using scale-invariant feature transformation

SIFT is a system for detecting and extracting local descriptors invariant for variation in scaling, illumination, rotation and little variations in viewpoints. Different phases of SIFT description are summarized below.

Scale space extrema detection

The initial part is a calculation that searches over entire image locations and scales. Efficient implementation is done by using different Gaussian functions to establish potential points of interest that can be invariant in scale and orientation. The point of interest for SIFT function corresponds to the extreme local difference of Gaussian filters on different scales.

Scale space is known by [TeX:] $$L ( x , y , \sigma )$$ and described as below

##### (8)
[TeX:] $$L ( x , y , \sigma ) = G ( x , y , \sigma ) * I ( x , y )$$

where I(x,y) is input image.

##### (9)
[TeX:] $$G ( x , y , \sigma ) = \frac { 1 } { 2 \Pi \sigma ^ { 2 } } e ^ { \frac { x ^ { 2 } + y ^ { 2 } } { \sigma ^ { 2 } } }$$

Eq. (9) is a Gaussian variable scale whose result converts an image with a Gaussian filter difference as below

##### (10)
[TeX:] $$D ( x , y , \sigma ) = L ( x , y , k \sigma ) - L ( x , y , \sigma )$$

This is unique from the Gaussian blurred images on both scales σ and kσ. The initial phase within the detection of interesting features is the convolution of the image with exclusive Gaussian filters. It produces a difference between Gaussian images and adjacent blurred images. The rotated images are grouped by octave and k value is chosen such that a set of blurred images per octave can be obtained. This ensures that the identical figure for variation of Gaussian images per octave is acquired. Once this difference of Gaussian is located, images are scanned for local extrema over space and scale.

Locating keypoints

Interesting points are identified as local minima or maxima of the difference of Gaussian (DOG) images across scales. Locating keypoints involves few steps as shown in Fig. 7. To find and describe the SIFT feature points, the steps given below must be followed:

Step 1: Input an image ranging from [0, 1].

Step 2: Use a Gaussian kernel variable scale G(x, y, σ) to generate scale space L(x, y, σ).

Step 3: Difference of Gaussian function is calculated as an approximation to the normalized Laplacian, because studies on this have shown that the normalized Laplacian does not change to the scale change.

Step 4: By comparing one of the pixels of its above, current and below scales in 3×3 areas maxima or minima of change of Gaussian performs value is found.

Step 5: By discarding points below a predetermined value, accurate keypoint’s locations are obtained.

##### (11)
[TeX:] $$D ( \hat { X } ) = D + \frac { 1 } { 2 } \frac { \partial \mathrm { D } ^ { T } } { \partial \mathrm { X } } \hat { X }$$

where [TeX:] $$\widehat { X }$$ is calculated by setting the derivative D(x, y, σ) to zero.

Step 6: The extremas of DOG have huge principle curvatures along edges that can be condensed by comparing

##### (12)
[TeX:] $$\frac { \operatorname { Tr } ( H ) ^ { 2 } } { \operatorname { Det } ( H ) } < \frac { ( r + 1 ) ^ { 2 } } { r }$$

H indicates a 2×2 Hessian matrix, r is the ratio between the highest magnitude and the lowest one.

Step 7: To obtain invariance to rotation, m(x,y) the gradient magnitude and [TeX:] $$\theta ( x , y )$$ orientation are recomputed as the equations given below.

##### (13)
[TeX:] $$m ( x , y ) = \sqrt { ( L ( x + 1 , y ) - L ( x - 1 , y ) ) ^ { 2 } + ( L ( x , y + 1 ) - L ( x , y - 1 ) ) ^ { 2 } }$$

##### (14)
[TeX:] $$\theta ( x , y ) = \tan ^ { - 1 } \frac { L ( x , y + 1 ) - L ( x , y - 1 ) } { L ( x + 1 , y ) - L ( x - 1 , y ) }$$

Step 8: This step is calculated taking a feature point and its 16×16 neighbors around it. Next dividing them into sub-regions of 4×4 and histogram of every sub-region with eight bins is calculated. After the keypoint orientation decision is made, the feature descriptor of orientation histograms on 4× 4 pixel neighborhoods is computed. The next step involves assigning orientation to each and every key factor to for reaching invariance to image rotation. This step is then adopted by using keypoint Matching.

Flow diagram of keypoints location approach.

Keypoint matching

In this step, keypoints among two images are in comparison through picking out their closest neighbors. The 2nd closest fit may also be near than the 1st in some circumstances. It is going to occur as a result of noise or various further causes. In such case, the ratio of distance closest to the 2nd closest distance is used. Distance is rejected if it is more than 0.8. Elimination of around 90% of false suits is completed even as discarding best 5% correct fits. So this concludes the summary of the SIFT descriptor for important points refer [24].

3.2.2 Consensus based matching and tracking

Consensus based matching and tracking method entails steps that are listed below. As in the Algorithm 1 it is seen that sequence of images [TeX:] $$A _ { 1 } , \ldots , A _ { n }$$ and an area [TeX:] $$b _ { 11 } , b _ { 12 } , \ldots , b _ { 1 n }$$ is initialised in A1. In every frame, improving pose of the interested object is carried out.

##### Algorithm 1

As given in the Algorithm 1 SIFT feature is applied to 1st frame i.e. A1 and to its entire selected regions [TeX:] $$b _ { 11 } , b _ { 12 } , \dots , b _ { 1 n }$$, resulting to get descriptors [TeX:] $$D _ { 1 } , \ldots , D _ { n }$$ and keypoints [TeX:] $$K _ { 1 } , \dots , K _ { n }$$. The object model is based on a set of keypoints, each of which represents an initialization of the location. SIFT feature is again applied to the entire second frame to get descriptor Df. Matching takes the first frame descriptor output and Df to get to matched descriptors [TeX:] $$M _ { 1 } , \dots , M _ { n }$$.

Flow chart for Lucas-Kanade method.

Tracking of this matched descriptors are done by considering first frame, second frame along with their descriptors to get output [TeX:] $$T _ { 1 , \dots , } T _ { n }$$. By employing the pyramidal variant of the Lucas-Kanade method optical flow is estimated as in Fig. 8. Matching and tracking of keypoints are done as in [25].

To locate the interested objects, each keypoint casts a single vote for the center of object resulting in set of votes [TeX:] $$V _ { 1 } , \ldots , V _ { n }$$. So voting approach presented in [25] is followed along with a Consensus approach to get [TeX:] $$V _ { 1 } ^ { c } , \ldots , V _ { n } ^ { c }$$. Fig. 9 gives an example of the object tracking result.

Human tracking result.

The combination of these algorithms along with the proposed novel validation method produces the better results in detection and tracking quality. Every tracked object is validated using features stored in the database so that if the tracked human is disappeared from the scene for a few frames and reappears. These reappeared humans are tracked again with same id.

## 4. Experimental Results

The performance of the presented methodology is analyzed by applying the proposed method on the videos generated from the frames available at Learning, Recognition & Surveillance Group dataset (https://www.tugraz.at/institute/icg/research/team-bischof/lrs/). The video sequences contain 3–5 persons and was generated in their laboratory. The videos differ in viewpoint (angle), i.e., each scene is viewed using three static cameras with portion of views being overlapped. Using these videos, 18 different video sequences are generated.

We have also tested our method on CAVIAR dataset which is publicly available [26]. It contains people moving randomly, entering and exiting shops in public places and shopping centers. Enough number of tests cases having different viewpoints are presented in following section.

4.1 Human Detection

The proposed algorithm is applied to the color image of size 256×256. The color-consistent clusters obtained after the part level segmentation on the first frame are shown in Fig. 10(a) and (b). After the separation of the color cluster using object-level segmentation of this output, the classification of the human region uses SVM as in Fig. 10(c). Fig. 10(d) gives grouping of these clusters to outline a human blob.

Fig. 11 depicts the result of part level segmentation and human detection result of frame_1 of video_6 considering three people in a video. Fig. 12 depicts the result of part level segmentation and human detection result of frame_1 of video_7.

For frame_1 of video_1. (a) Input frame, (b) part-level segmentation, (c) human patch generation, and (d) human detected by combining all human region clusters.
For frame_1 of video_6. (a) Input frame, (b) part-level segmentation, (c) human patch generation, and (d) human detected by grouping all human region clusters.
For frame_1 of video_7. (a) Input frame, (b) part-level segmentation, (c) human patch generation, and (d) human detected by grouping all human region clusters.
(a) Input frames 51, 173, and 429 of video_1, (b) human validation results of video_1, and (c) human tracking result of video_1.
4.2 Human Tracking

The result obtained from the proposed method of few video sequence of three different videos recorded from three different angles is presented here. Fig. 13(a), (b), and (c) depict the detection and tracking results of video_1 for frames 51, 173, and 429, respectively. In each frame, humans are detected successfully with different colored bounding box for all three people.

Fig. 14(a), (b), and (c) depict the result of frames 51, 147, and 353 of video_6, respectively. This video contains three people moving randomly by occluding one another many times. A proposed method successfully detected humans in each frame. Fig. 15(a), (b), and (c) depict the result of frame 51, 179, and 285 of video_7, respectively, which is the same sequence as video_6 in different camera location.

We have also demonstrated our results for the LeftBag and EnterExitCrossingPaths videos chosen from CAVIAR database [26]. These video clips consist of humans walking around, gathering, and picking and leaving bags, entering, exiting shopping malls, and so on. Fig. 16(a), (b), and (c) depict another result of frames 150, 250, and 870 of video_19 (LeftBag), respectively.

Each person in tracking phase is assigned with a unique id to represent the person. As shown in Fig. 16(c), detected persons are tracked with unique id (identification number) which is displayed above the bounding box. Each person is assigned with unique color bounding box and id.

(a) Input frames 51, 147, and 353 of video_6, (b) human validation results of video_6, and (c) human tracking result of video_6.
(a) Input frames 51, 179, and 285 of video_7, (b) human validation results of video_7, and (c) human tracking result of video_7.
(a) Input frames 150, 250, and 870 of video_19, (b) human validation results of video_19, and (c) human tracking result of video_19.
(a) Input frames 55, 73, and 90 of video_20, (b) human validation results of video_20,(c) human tracking result of video_20.

In Fig. 16, a person with green shirt detected with id3 has left the scene from 250th frame. After a few frames, he reenters to the view and the method proposed can track the person with the same id3. This is because the proposed method uses human validation phase, which stores features of each person along with id into the database as soon as the person is detected. Once the person re-enters the scene, features are extracted and validated. If the features are matched with already existing features in knowledge base, same id is assigned. If the feature does not match, then it is treated and assigned with new id.

Fig. 17(a), (b), and (c) depict the result of frames 55, 73, and 90 of video_20 (EnterExitCrossingPaths), respectively. This video contains two humans walking slowly towards the camera and these humans are occluded by humans crossing them. Our method detects and tracks all the humans successfully as shown in Fig. 17 even in occlusion condition.

The algorithm is again evaluated on one more indoor video which is shown in Fig. 18. In this video, three people are moving around the area by occluding one another many times. The proposed method can track the person even in occlusion condition as presented in Fig. 18(a), (b), and (c) for frames 220, 275, 380, and 390 of video_21, respectively.

The computed accuracy for tracking and detection for different videos are shown in the Table 1 using the formula given below:

##### (15)
[TeX:] $$\operatorname { accuracy } = \frac { T P + T N } { T P + T N + F P + F N } * 100$$

where TP indicated true positives, TN indicated true negative, FP indicated false positive, and FN indicates false negative.

(a) Input frames 220, 275, 380, and 390 of video_21, (b) human validation results of video_21, and (c) human tracking result of video_21.
Detection and tracking accuracy for video_1, 6, 7, 19, 20, and 21

Fig. 19 presents the comparison graph for different existing methods and proposed methods for detection and tracking rate as in the Table 2. Graph signifies detection and tracking rate obtained for proposed method is more compared to other methods in [27]. The work in [27] also has considered the similar type of indoor videos as used in the proposed methodology, making it more suitable for the comparison of overall accuracy.

Recall rate and precision values for human detection and tracking of proposed method for video_1, 6, 7, 19, 20, and 21
Comparison graph of proposed method with state-of-the-art method: (a) detection rate and (b) tracking rate.

Fig. 19 depicts the comparison graph for detection and tracking of different state-of-the-art tracking methods and our method. Graph shows that our method gives better detection and tracking rate than state-of-the-art method. Table 3 shows that tracking recall and precision average rate is 92.82% and 92.29% and detection average recall is 89.45% and precision rate is 92.44%. Detection and tracking rate of proposed method is better when compared to existing methods as in Table 3.

Comparison of precision and recall rate of HOG, SIFT, and combined feature method

The recall and precision values are calculated using the formula given below:

##### (16)
[TeX:] $$Recall = \frac { T P } { T P + F N }$$

##### (17)
[TeX:] $$Precision \ Rate = \frac { T P } { T P + F P }$$

The proposed method uses HOG and SIFT features to track humans in the scene. Table 3 shows that when single feature is used to track an object, it gives lesser precision and recall rate than proposed combined feature tracking method. When more features are added to track objects, even if one feature fails in certain situation, other feature may still work well to track the object. Figs. 20 and 21 present the precision and recall comparison graph of proposed method with single feature method. Comparison graph shows that combined feature gives better precision and recall rate than single feature methods.

Precision graph comparison of HOG, SIFT, and combined HOG and SIFT.
Recall graph comparison of HOG, SIFT, and combined HOG and SIFT.

## 5. Conclusion

We have proposed a method for multi object detection and tracking using combined features. The major challenges for proposed tracking technique are unreliable measurements in case of tracking the missed detection and false positives. The contribution of our approach is to discover how this unreliable source of information can be used for tracking multiple human efficiently. Our proposed object tracking method for video images uses hierarchical graph based segmentation and consensus based human tracking. Our approach uses hierarchical part level approach of segmentation, which aims at dividing an image into color consistent parts and object level segmentation to determine the objects with the help of previously trained SVM Classifier values. Proposed method is an effective approach involving SIFT key point descriptor method and consensus based object tracking technique with validation phase. This work gives good accuracy by detecting and tracking objects even in different conditions, sizes and shapes. Simulation results for frame sequences verify the suitability of the method for robust object tracking. Evolutions on datasets with different camera angle showed that the algorithm gives a good performance on huge number of application scenario compared to other existing methods.

## Biography

##### Sunitha Madasi Ramachandra
https://orcid.org/0000-0001-5678-552X

She received B.E. in Computer Science and Engineering from Kalpataru Institute of Technology, Bangalore University in 1997 and M.Tech. degree in Computer Science and and her Ph.D. in the Department of Computer Science and Engineering, Siddaganga Institute of Technology, Visveswaraya Technological University. She has published many research papers in national and international conferences and journals. Currently she is working as Professor in the department of Computer Science and Engineering, Adichunchanagiri Institute of Technology, Chikkamagaluru, Karnataka, India. Her research interests include Image processing, algorithms and cryptography. Hierarchical Graph Based Segmentation and Consensus based Human Tracking Technique

## Biography

##### Haradagere Siddaramaiah Jayanna
https://orcid.org/0000-0002-4342-9339

He received the B.E. and M.E. degrees from Bangalore University in 1992 and 1995, respectively, and Ph.D. degree from prestigious Indian Institute of Technology, Guwahati, India, in 2009. He has published a number of papers in various national and international journals and conferences apart from guiding a number of UG, PG and research scholars. Currently, he is working as a Professor in the Department of Information Science and Engineering, Siddaganga Institute of Technology, Tumkur, Karnataka, India. His research interests are in the areas of speech, limited data speaker recognition, image processing, computer networks and computer architecture.

## Biography

##### Ramegowda
https://orcid.org/0000-0002-9417-2465

He obtained multidisciplinary degrees such as B.E. from Mysore University (1983), M.E. (1987) and LL.B. from Bangalore University, MCA from IGNOU, New Delhi, and obtained Ph.D. in the field of image processing and analysis applied to geotechnical engineering from Visvesvaraya Technology University, Belagavi, Karnataka, India, in 2010. He has published a number of papers in various national and international journals and conferences apart from guiding a number of UG, PG and research scholars. Currently, he is working as Principal, Bahubali College of Engineering, Shravanabelagola (Jainakashi), Karnataka, India. His research interests are in the areas of soft-computing, image processing and analysis, GIS and remote sensing.

## References

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• 2 B. Peng, L. Zhang, D. Zhang, "A survey of graph theoretical approaches to image segmentation," Pattern Recognition, vol. 46, no. 3, pp. 1020-1038, 2013.doi:[[[10.1016/j.patcog.2012.09.015]]]
• 3 R. Chaudhry, A. Ravichandran, G. Hager, R. Vidal, "Histograms of oriented optical flow and binet-cauchy kernels on nonlinear dynamical systems for the recognition of human actions," in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Miami, FL, 2009;pp. 1932-1939. custom:[[[-]]]
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• 11 B. Babenko, M. H. Yang, S. Belongie, "Robust object tracking with online multiple instance learning," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 8, pp. 1619-1632, 2011.doi:[[[10.1109/TPAMI.2010.226]]]
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Table 1.

Detection and tracking accuracy for video_1, 6, 7, 19, 20, and 21
 Video Accuracy (%) Detection Tracking Video_1 86.33271 91.49125 Video_6 95.27831 94.01868 Video_7 88.84615 94.73541 Video_19 89.39876 92.65957 Video_20 87.20430 92.17191 Video_21 89.65000 91.82770 Average 89.45171 92.81742

Table 2.

Recall rate and precision values for human detection and tracking of proposed method for video_1, 6, 7, 19, 20, and 21
 Video Detection Tracking Recall Precision Recall Precision Video_1 95.26552 96.81404 94.01427 93.66999 Video_6 88.84615 88.84615 94.73790 94.45710 Video_7 89.40336 92.66953 92.66285 92.12272 Video_19 87.22581 97.12644 92.18508 91.56051 Video_20 89.66667 92.83865 91.83439 91.16466 Video_21 89.45670 92.43792 92.82380 92.28931 Average 86.33271 86.33271 91.50831 90.76087

Table 3.

Comparison of precision and recall rate of HOG, SIFT, and combined feature method
 Video Human validation With HOG With SIFT With SIFT & HOG Recall Precision Recall Precision Recall Precision Sample 001 (video_1) 81.32344 81.32344 84.35374 84.35374 86.33271 86.33271 LeftBag (video_9) 84.40452 89.0401 87.39922 91.22385 89.40336 92.66953 EnterExitCrossing (video_20) 82.32258 95.7958 85.29032 96.63743 87.22581 97.12644
A flow diagram of proposed human tracking system.
Input first frame.
Output of part-level segmentation.
A flow diagram of human blob generation.
Human patch generation.
Human detection by grouping human patches.
Flow diagram of keypoints location approach.
Flow chart for Lucas-Kanade method.
Human tracking result.
For frame_1 of video_1. (a) Input frame, (b) part-level segmentation, (c) human patch generation, and (d) human detected by combining all human region clusters.
For frame_1 of video_6. (a) Input frame, (b) part-level segmentation, (c) human patch generation, and (d) human detected by grouping all human region clusters.
For frame_1 of video_7. (a) Input frame, (b) part-level segmentation, (c) human patch generation, and (d) human detected by grouping all human region clusters.
(a) Input frames 51, 173, and 429 of video_1, (b) human validation results of video_1, and (c) human tracking result of video_1.
(a) Input frames 51, 147, and 353 of video_6, (b) human validation results of video_6, and (c) human tracking result of video_6.
(a) Input frames 51, 179, and 285 of video_7, (b) human validation results of video_7, and (c) human tracking result of video_7.
(a) Input frames 150, 250, and 870 of video_19, (b) human validation results of video_19, and (c) human tracking result of video_19.
(a) Input frames 55, 73, and 90 of video_20, (b) human validation results of video_20,(c) human tracking result of video_20.
(a) Input frames 220, 275, 380, and 390 of video_21, (b) human validation results of video_21, and (c) human tracking result of video_21.
Comparison graph of proposed method with state-of-the-art method: (a) detection rate and (b) tracking rate.
Precision graph comparison of HOG, SIFT, and combined HOG and SIFT.
Recall graph comparison of HOG, SIFT, and combined HOG and SIFT.


text:
Wrong terminology [Design Issues]

Hi ElMaestro,

» The sum of residuals for a fitted normal linear model will be zero. Not asymptotically.

Sorry, I did not mean asymptotically (central limit theorem ) but approximately (due to numeric issues). Theoretically the sum should be zero, of course.
IIRC, in old versions of lm() fitting data without errors (like your first example) threw an error and there was even a warning in the man-page.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

format: plain


text:
# What factors are to be considered while deciding standards?

Dear Student,
the following are some of the factors that should be kept in mind while setting the standards in management:
i. Practical- The standards so decided should be practical in nature i.e. they should be obtainable in the given time limit.
ii. Measurable- An important factor while deciding the standards is that they should be measurable. In other words, the managers should set the standards in such a manner that the standards and the actual work done in comparison to the standards can be easy measured.
iii. Adaptable- The standards so set should be adaptable and flexible in nature. That is, they should be adaptable to the necessary changes when and wherever required.

• 5
What are you looking for?


text:
12:04 AM
Okay. Any insights to here are appreciated math.stackexchange.com/questions/3406440/….

The introduction of numbers as coordinates is an act of violence. -- Hermann Weyl

Hello, quick question: Can a model interpret a formula both true and false or not interpret it in first order logic? Looking at the definition, it looks like the model is inheriting the internal logic of the meta-object/model. Just one or two sentence of guidance would be nice.

I've had to defind my posts lately
you can see my recent posts
People assume I'm wrong or something...
Perhaps it does indeed form a ring... go figure
I actually PROVE that it does, and still it backfires
That's what makes MSE shitty at least for me
The proof is elegant and not flawed
1

Let $X$ be a finite, commutative, boolean monoid with a $0$ element. Let $A = \{ x \subset X : 0 \in x$ and $x^2 \subset x\}$ where $x^2 = \{ab: a,b \in x\}$. Define for $x, y \in A$, $x + y = x \Delta y \cup \{0\}$. Then since $x^2 \subset x$ for $x \in A$, we actually have $x \subset x^2$ as...

These shitters man

12:23 AM
@KonformistLiberal a formula must be either true or false (and not both) in a model

@LeakyNun Where I am confused: Don't we define an interpretation of a formula recursively, so doesn't your statement needs a proof, it's not a direct property through definition?

sure
for a formula we recursively evaluate it
$$\bigcup_{u \in \mathcal U} U_u$$

@LeakyNun So can't we get a case where a formula is both true and false? I am trying to imagine that a contradiction in the model can lead to a case where the interpretation is both true and false.

a model can't have any contradictions

why not?

12:34 AM
because $v(\bot) = 0$

oh, right
so by exactly this part of the definition, we argument that an interpretation of a formula under a model cannot be both true and false, and because this interpretation is for every formula defined, we know that it must be true or false, am I right?

genau

Thank you! I want to clarify one more thing, when we say that a formula is independent of an axiom system, we mean that it has models which model this system and the formula and also this system and the negation of the formula. Am I right?

@KonformistLiberal well that is a consequence

@LeakyNun What do you mean? That it's not the definition of independence?

12:47 AM
independence means that the system can neither prove it nor refute it

@LeakyNun Aren't they the same thing? A system can prove a formula if this formula is satisfied in every model of this system, right? So, we also cannot prove the negation of this formula and this directly implies that there must be models for each case. Am I right?

but that's a theorem

That, they are the same thing?

yeah, it's called the completeness theorem

Alright, so, now, my question would be where the "law of excluded middle" lives in this context?

12:52 AM
in the metatheory used to prove the completeness theorem

So we put law of excluded middle to this kind of first order logic through using it in our metatheory if necessary, one example being completeness theorem?
And I read a phrase "we believe that first-order logic is consistent". What does it mean?

wo hast du es gelesen?

7

Truth is something specific to a particular structure, or a particular model of a theory. Sometimes, when a theory has a canonical model, we abuse the terminology and when we say "true" what we mean is "true in the canonical model". This is the case with the natural numbers and with the real numb...

well it means that we believe that first order logic is consistent

is this a mathematical statement?

1:02 AM
sure
it means that we can't prove $\bot$

So, it says empty set is satisfiable?

yes

But can't we show that by giving a finite example?
Actually, doesn't any model of any formula satisfy empty set?

well if you believe that sets exist then sure you can just use any model

If x is in a baire space and there's an injection function mapping one set of natural numbers omega to another set of natural numbers omega...is it open? I think it is because there is an unique mapping so we have interior points and open balls and neighborhood

1:07 AM
Woo, on arxiv

@Semiclassical congrats

Yayyyy

@usukidoll how many sets of natural numbers are there...???

Given I to be $x \in ^{\omega} \omega$ in the Baire spare there is an injection from $\omega$ to $\omega$ so there must be two sets of natural numbers
Is it open or closed? Meep. I say it's open due to the unique mapping and...interior points meep

either I don't understand what "Baire space" means, or I don't understand what "natural numbers" mean

1:11 AM
Baire space is $^{\omega} \omega$
Set of natural.numbers.is.$\omega I could upload a screenshot so$\omega^{\omega^{\omega^{.^{.^{.}}}}}$? please do upload a screenshot ok that's the same set All I can see is that we have an injection and surjection function$\omega$is the set of natural numbers$\omega = \{0,1,2,3,\cdots\}$1:14 AM With x being in the Baire Space and yes$\omega$is the set of natural #s there are not two sets of natural numbers I was very confused so$x$is a sequence of natural numbers I get that part... Injection function is a unique mapping of elements isn't it?$x = (x_0, x_1, \cdots)$You have your set and I have mine, @Leaky. Yup 1:15 AM it can also be viewed as a function$\omega \to \omega$mapping$n$to$x_n$Oh it's Ted o.o Hiiii Where? Hi @TedShifrin But isn't it open because there exists interior points and open balls of such things? For that mapping? Oh oh, it's @topologicalmagician. 1:15 AM haha @TedShifrin how are you today? I was wondering whether I am drunk I'm dying in set theory 💀 Cooking dinner, so not here too long. @Leaky: Do you get drunk? Cuz this is the only thing to do lel 1:16 AM Let$f: [a,b] \rightarrow \mathbb{R}$be continuous. Suppose$f(x)\geq 0$for each$x\in [a,b]$and$f(c)>0$for some$c\in (a,b)$. Then$\int^{b}_{a}f>0$My Attempt: Suppose$f$is continuous for for$\epsilon=f(c)>0\existsdelta_1>0$such that whenever$x \in (c-\delta_1,c+\delta_1)$, we have$f(x)>f(c)$. Now, how do I show that$\int^{b}_{a} f(x)\geq \delta_1 f(c)

Mmk *eats @TedShifrin 's food$LOL, @usukidoll: I'm a good cook. @topologicalmagician partition Yup, what @Leaky said. Show that a lower sum for some partition is at least that big. No, you don't know$f(x)>f(c)$. That's no good. oh that wasn't what I had in mind but ok :D 1:17 AM What is the correct statement? I am so lost as to what this is asking :/ I assumed contradiction, forgot to mention that @Leaky, well then I'm not sure what you had in mind. Besides functions.yhey no wait 1:18 AM Contradiction is not a good approach. @usukidoll do you see how an element of the Baire space can be viewed as a function$\omega \to \omega$or maybe you need some rest if you're lost in symbols @TedShifrin proof by undergrad: assume A is false. [proof of A]. this contradicts with the initial assumption that A is false. so A must be true. This boogy is due tomorrow . I did most of it though. I spend more of my time with Python Cuz I wanna boost my resume ;) @TedShifrin that's the correct statement: Let$f: [a,b] \rightarrow \mathbb{R}$be continuous. Suppose$f(x)\geq 0$for each$x\in [a,b]$and$f(c)>0$for some$c\in (a,b)$. Then$\int^{b}_{a}f>0$@usukidoll the question is asking whether the set of injections$\omega \to \omega$is open in the set of all functions$\omega \to \omega$lmao @topologicalmagician he means what is the correct version of$f(x) > f(c)$in your proof Which it is...... Aren't there open balls in there? 1:19 AM well if you can prove it then it is! hello. i have a confusion. can R treated as an R-module break up into a direct sum of other R-modules? i felt like any submodule of R as an R-module should be free, but then it would have to have lower rank than 1, so must be trivial? R doesn't have to be commutative Damn... There is an open neighborhood too yipeee so... you're asking to check? Yes then present your proof @tigre yes. for example, R=Z/6Z, then R = Z/3Z + Z/2Z 1:20 AM |f(x)-f(c)|<f(c)$\impliesf(x)>f(c)$@tigre: Why must it have lower rank? oh wait lol LOL is right ! print('I am doomed') doom_meter = input('How doomed are you?') yes 1:22 AM oh it didn't have to have lower rank oops thanks I don't even know what rank means is it the maximum cardinality of an independent subset? well the submodule was to be free, so number of generators Hey hot cats. well not any submodule of a free module is free! I can't prove for thisssss ughhhhhh 1:24 AM unless you're in a PID print('Python is better') @usukidoll what are you working on? if user.input == "print('Python is better')": break This boogy about Baire Space, omegas, and injection and surjection functions Getting yeeted less memes, more math 1:25 AM @anakhro the only thing that is driving me nuts I'm glad I'm working on cooking dinner instead of this stuff. @TedShifrin what are you cooking? I made a calzone tonight. Chicken piccata, wild rice, veggies. @anakhro here it is x.x i.stack.imgur.com/FKcco.png print('Ted is a good cook') usukidoll is printing that only because she wants me to throw her a scrap. 1:26 AM :V Wow. I have no idea what :V means. it means :S Feed me LOL. Oh. wait it doesn't mean :S? 1:27 AM Idk lol okay, so$f(x)>\frac{f(c)}{2}$for all x in a delta neighbourhood of c, so would the right way of approaching it would be by constructing a partition such$a_0,a_1,a_2....,b$such that$a_i-a_{i-1}$<$\delta$for each i? @usukidoll what does the notation$^\omega\omega$mean? It's the Baire Space$\omega$is the set of natural numbers OK, @topologicalmagician. Why not take a partition like$a<c-\delta/2<c+\delta/2<b$? 1:28 AM So what does it mean? There are many Baire spaces. @anakhro it's the functions$\omega \to \omega$it's the Baire space THE Baire space? I had never heard of the Baire space heretofore. Yeah there's a unique mapping aka injection Dhiwnwuwmqhq d if G is a finite group, a group ring k[G] for k a field is artinian, so has a decomposition into a direct sum of indecomposable projective modules. how do you actually find these modules? in practice i mean 1:29 AM This is why I prefer Python. I rather code -_- @TedShifrin oh wow. That cleared things up @tigre that's the content of representation theory @tigre: You should page @MatheinBoulomenos or one of our algebra czars. Sooooo it's hard to find the modules 1:30 AM makes much more sense than the i was trying to construct you can use Brute force thank you thank you thank you thank you Am I doomed? so find a random element there, move it around using k[G], hope that you get a smaller module People coming out of low-level calculus think everything has to be a uniform partition, @topologicalmagician. 1:31 AM @usukidoll so which part are you stuck on? OK, back to the kitchen for me. @LeakyNun you mean take a random element, and treat it as a free module, and hope it doesnt generate the group ring? bye ted don't treat it as a free module generate a free module with it i mean look at the submodule it generates 1:31 AM I said that I is open because there is a unique mapping and there exists interior points and open balls @anakhro Meep the module generated wouldn't be free @Ted Shifrin , just out of curiosity, what do you mean by low-level calculus? oops i did it again it will be free if Ann(a) = 0? This is alllllll I can think offfffff it will be free if it is the whole ring! 1:32 AM @usukidoll unique mapping where? From omega to omega Unique injection you mean? I is an injection function Ya I is a set, not a function. @anakhro you'll never be able to decipher the question from her words 1:33 AM :S I doubt @usukidoll understands the question herself Well let me see how far I can get instead of discouraging people, @LeakyNun @tigre you should stop concerning yourself with whether modules are free I is a set? @usukidoll Yes. 1:34 AM So I is a set of injective functions Set of injective functions with unique mappings @tigre let's look at$\Bbb R[C_2]$for a simple example I still think this is open Yes, precisely the set of all injective functions which are elements of$^\omega\omega$. can you describe its elements? Open for salad 1:35 AM @usukidoll where are you seeing "unique"? @LeakyNun it's elements are$\{a[0] + b[1]\mid a,b\in R\}$where$[0],[1]$are the elements of$C_2\cong(\Bbb Z/2\Bbb Z,+)$I thought injection functions map one element to the next element without repeat @tigre now pick your favourite element$[0]+[1]$what do you get if you hit it with the elements of the group? (that's a strategy: take the G-span, and then take the K-span) 1:38 AM @usukidoll if I have a function$f\colon X\to Y$, what is the formal definition for$f$being injective? Damn if f(a) =f(b) then a=b Great. Or in this case if f(x)=f(y) then x=y i would have said the free module generated by$[0]+[1]$, but explicitly $$(a[0]+b[1])([0]+[1])=a[0] + b[1] + a[1]+b[0]=(a+b)[0]+(a+b)[1]$$ which i think is still a free module generated by$[0]+[1]$and i could even treat it as a R-module of rank 1 if i wanted Now whaaaaaat? 1:40 AM @tigre it's not a free module Alright, so we have that$I = \{f\colon\omega\to\omega \mid f\text{ is injective}\}$, does it make sense what$I$is now? it's not a free$\Bbb R[C_2]$-module it is a free$\Bbb R$-module (i.e.$\Bbb R$-vector space) since every vector space is free I is the set of injective functions with the set of natural numbers @tigre well you'll find it easier if you take the G-span and then the K-span instead of both at once @usukidoll what do you mean by "with the set [...]" Do you mean "on the set [...]"? 1:43 AM @tigre so the$\Bbb R$-span of$[0]+[1]$is an$\Bbb R[C_2]$-submodule of$\Bbb R[C_2]$can you see a complement? @LeakyNun oh, its not free because (a[0]-a[1])([0]+[1]) = 0 precisely righttttt ok am i right though that if ann(a)=0, then the module generated by a is free? yes that should deal with the other part of linear independence (which i was ignoring in the case there was one generater :S ) 1:44 AM but you should be able to prove that the only free submodule of K[G] is K[G] itself and the zero submodule hint: it's a basic linear algebra argument @anakhro ya @usukidoll it's important to be clear about what you mean when it comes to mathematics. So what is an open set in the Baire space? @usukidoll It has interior points And open balls 1:48 AM "it has interior points" describes a lot of sets, many of which are not open. So what constitutes an "open ball" in the Baire space? Infinite sequences? Whoa wait. Many sets are not open...so there are exterior points :/ and the complement is open if it's closed What is your definition of an open set? The first one that you were given in your notes. Not any of the equivalent ones. Open interval @usukidoll "open interval" is the definition of "open set"? I'm stressing outtttt xc 1:53 AM Why are you stressing out? Too many similar definitions... Ufhhwjjwjwjjw I feel like guessing but I shouldn't do that since this is the only exercise left x.x Dagh To determine if something is open/closed in$^\omega\omega$you need to know what sets constitute as open/closed sets in$^\omega\omega$. That's really the bottom line. I have it in my notes but both cases involve trees and I know the problem isn't asking for trees in here ;/ How do you know the problem isn't asking for trees? I don't tbh. I thought the problem didn't involve trees 2:00 AM You are saying that the exercise at the end of a section titled "Trees" in fact doesn't involve "trees"? Right....I think I am sooo thinking of this problem wrong because I know the second one involved trees so I used one of the lemmas @anakhro Poggers @LeakyNun ? :/ Pogges is the Pepe frog meme dude every module has a generating set (take all the elements themselves, which may be very large (uncountable etc)) right? 2:03 AM @usukidoll Let me give an example: suppose we are considering sets of$\mathbb R$. Is the set$(0,1)\cup (3,4)$open? @tigre correct and sadly, finitely generated modules can have non-finitely generated submodules that is true and when a module has only finitely generated submodules we call it noetherian. is there some nice conditions of R that make R-modules always finitely generated? That has to be open because if we take the union of two open sets then the union is also open because (0,4) 2:05 AM @tigre every any nonzero ring R has a not-finitely-generated module Great, so you notice how you found that this set is open because you knew what "open sets" look like in$\mathbb R$? @LeakyNun by just taking $$\bigoplus_{i=1}^\infty R$$ ? precisely ok good point was k[G] a noetherian module over itself for finite G? yes 2:07 AM @anakhro right @usukidoll so that's a good place to start with this question. Figure out what the open sets in$^\omega\omega$look like, and then you can better understand whether or not$I$is open or not. But what does open look like for Baire Space? There's my issue right there meep That's why you go to your notes and look at the definition for open set in the Baire space. Families of basic open sets @LeakyNun even if k is characteristic p, and p may divide |G|? 2:16 AM yes how do i show this? Basic open sets sounds like a good place to start, @usukidoll. What does a basic open set look like? @tigre k[G] is fin. dim. over k any k[G]-submodule is a k-subspace so take a k-basis this will be a k[G]-generating set wow thats smart Everything is open... Unions are open and intersections are open 2:19 AM Is [0,1] open in R? finite intersections? I just read a wiki article which claimed the following statement: Noooo... [0,1] is closed and bounded...compact @usukidoll why is it not open? Closed interval... Closed and bounded....compact 2:21 AM Oh wait there’s a typo let me try again Closed sets can also be open. Nuggh @usukidoll the point is that you really need to learn what a basic open set looks like. 2:25 AM It's a basis @usukidoll What is a basis? XC A collection of subsets That's all? Yeah because I am getting burned out just reading past ebooks and what not x.x Well I can assure you that that is not the definition of a "basis". But since you are burnt out, you should go to sleep. 2:29 AM that's what I have been saying from day 1 @usukidoll you write in the passive voice too much. responses are more confusing if they are written in the passive voice @WilliamSun still waiting for your question This is...not really a well-known fact I guess? I mean I haven’t seen it in any analysis or point set topology book... you're asking us to sample the population to find out how many people know it? or you're asking if its true, because you haven't seen a reliable reference Oh I am asking for its applications. Is this just a useless formulation so people forget about it? 2:34 AM useless??? What constitutes as "useful"? It is much cleaner than the standard definition I mean. To prove something? For example using this we can prove something interesting that does not follow directly from the usual definition. For example use the universal property of tensor products to prove the commutativity instead of using the construction. Then I call this an “application” of universal properties. It's an "if and only if" If you have "A iff B" and then A-->X or B-->Y, then you automatically get both A-->Y and B-->X for free. So ok here is the usual definition for limit of a function: @LeakyNun because when you treat k[G] as a module over itself, this corresponds to the standard k-linear representation of G, and this decomposition is somehow related to the decomposition of that representation into its irreducible subrepresentations? yes 2:43 AM those subrepresentations for some reason correspond to projective modules? and the irreducibility corresponds to the indecomposability And in this paragraph it doesn’t mention about the isolate points. > For the group algebra of a finite group, the (isomorphism types of) projective indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules: the socle of each projective indecomposable is simple (and isomorphic to the top), and this affords the bijection, as non-isomorphic projective indecomposables have non-isomorphic socles. @WilliamSun I am not sure what you are asking anymore. the simplest example would be$K[C_2]$when char(K)=2 where the whole ring is projective indecomposable and the simple module is the span of$[0]+[1]$i guess i should study$k[C_2],k[C_3],k[C_2\times C_2],K[S_3]$in$\text{char}(k)=2$, and$k[C_3],k[S_3]$in$\text{char}(k)=3$2:50 AM sure how about$k[C_n]\$ in char(k)=0

Looking for n-ality transformation examples
i.e. a generalisation of dual mathematical objects


text:
8 months ago

## Homework Websites For Middle School Students hacker heros decaler

Homework Websites For Middle School Students

homework help websites for middle school students

homework websites for middle school students


text:
I am a PhD student in the équipe-projet COSMIQ (previously SECRET) at Inria Paris under the supervision of Nicolas Sendrier.

I am interested in Moderate Density Parity Check (MDPC) [doi] [eprint] codes, more specifically I am interested in their decoders. This type of codes is defined similarly to Low Density Parity Check (LDPC) codes, the main difference is the number of ones in the rows of the parity check matrix: a small constant for LDPC but in $$\mathcal{O}(\sqrt{n})$$ for MDPC (with $$n$$ the length of the code). This difference in density allows for a McEliece-like public key cryptosystem whose secret key attacks and message attacks reduce to hard problems on codes.

The NIST PQC candidate BIKE is based on the quasi-cyclic MDPC codes.


text:
A car accelerates from 31.0m/s to rest over a displacement of 130.0m. What is the car's acceleration, and how long did it take to stop?

Oct 11, 2017

$a = - 3.70 m {s}^{- 2} \left(2 \mathrm{dp}\right)$

$t = 8.39 s \left(2 \mathrm{dp}\right)$

Explanation:

we have

$u = 31 m {s}^{- 1}$

$v = 0 m {s}^{- 1}$

$s = 130 m$

to find $a \text{ & } t$

$\textcolor{b l u e}{{v}^{2} = {u}^{2} + 2 a s}$

$0 = {31}^{2} + 2 \times a \times 130$

$0 = 961 + 260 a$

$\therefore a = - \frac{961}{260} = - 3.70 m {s}^{- 2} \left(2 \mathrm{dp}\right)$

$\textcolor{b l u e}{v = u + a t}$

$0 = 31 - 3.70 t$

$t = \frac{31}{3.70}$

$t = 8.39 s \left(2 \mathrm{dp}\right)$


text:
Research

# A note on the existence and uniqueness of the solutions to SFDES

Yeol JE Cho1, Sever S Dragomir2 and Young-Ho Kim3*

Author Affiliations

1 Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Republic of Korea

2 School of Computer Science and Mathematics, Victoria University of Technology, P.O. Box 14428, MCMC Melbourne, VIC 8001, Australia

3 Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

For all author emails, please log on.

Journal of Inequalities and Applications 2012, 2012:126  doi:10.1186/1029-242X-2012-126

 Received: 26 March 2012 Accepted: 7 June 2012 Published: 7 June 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we establish a new proof of the existence and uniqueness of solution to stochastic functional differential equations with infinite delay at phase space BC((-∞, 0]: Rd) which denotes the family of bounded continuous Rd-valued functions φ defined on (-∞, 0] with norm ||φ|| = sup-∞ < θ≤0 |φ(θ)|.

Mathematics Subject Classification (2000): 60H05, 60H10.

##### Keywords:
existence; uniqueness; stochastic functional differential equations; infinite delay.

### 1. Introduction

Stochastic differential equations (SDEs) are well known to model problems from many areas of science and engineering. For instance, in 2006, Henderson and Plaschko [1] published the SDEs in Science and Engineering, in 2007, Mao [2] published the SDEs, in 2010, Li and Fu [3] considered the stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks.

In recent years, there is an increasing interest in stochastic functional differential equations(SFDEs) with finite and infinite delay under less restrictive conditions than Lipschitz condition. For instance, in 2007, Wei and Wang [4] discussed the existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, in 2008, Mao et al. [5] discussed almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, in 2008, Ren et al. [6] considered the existence and uniqueness of the solutions to SFDEs with infinite delay, and in 2009, Ren and Xia [7], discussed the existence and uniqueness of the solution to neutral SFDEs with infinite delay. Furthermore, on this topic, one can see Halidias [8], Henderson and Plaschko [1], Kim [9,10], Ren [11], Ren and Xia [7], Taniguchi [12] and references therein for details.

On the other hand, Mao [2] discussed d-dimensional stochastic functional differential equations with finite delay

(1.1)

where xt = {x(t + θ): -τ θ ≤ 0} could be considered as a C([-τ, 0]; Rd)-value stochastic process. The initial value of (1.1) was proposed as follows:

Furthermore, Ren et al. [6] also considered the stochastic functional differential equations with infinite delay at phase space BC((-∞, 0]; Rd) to be described below.

(1.2)

where xt = {X(t + θ): -∞ ≤ θ ≤ 0} could be considered as a BC((-∞, 0]; Rd)-value stochastic process. The initial value of (1.2) was proposed as follows:

(1.3)

Following this way, now we recall the existence and uniqueness of the solutions to the Equation 1.2 with initial data (1.3) under the non-Lipschitz condition and the weakened linear growth condition. In this paper, we will give some new proof of the existence and uniqueness of the solutions to ISDEs under an alternative way.

### 2. Preliminary

Let | · | denote Euclidean norm in Rn. If A is a vector or a matrix, its transpose is denoted by AT; if A is a matrix, its trace norm is represented by . Let t0 be a positive constant and , throughout this paper unless otherwise specified, be a complete probability space with a filtration satisfying the usual conditions (i.e. it is right continuous and contains all P-null sets). Assume that B(t) is a m-dimensional Brownian motion defined on complete probability space, that is B(t) = (B1(t), B2(t), ..., Bm(t))T. Let BC((-∞, 0]; Rd) denote the family of bounded continuous Rd-value functions φ defined on (-∞, 0] with norm ||φ|| = sup-∞ < θ≤0 (θ)|. We denote by the family of all -measurable, ℝd-valued process ψ(t) = ψ(t, w), t ∈ (-∞, 0], such that . And let is the family of Rd-valued -adapted processes {f(t)}atb such that .

With all the above preparation, consider a d-dimensional stochastic functional differential equations:

(2.1)

where xt = {x(t + θ): -∞ < θ ≤ 0} can be considered as a BC((-∞, 0]; Rd)-value stochastic process, where

be Borel measurable. Next, we give the initial value of (2.1) as follows:

(2.2)

To find out the solution, we give the definition of the solution of the Equation 2.1 with initial data (2.2).

Definition 2.1. [6]Rd-value stochastic process x(t) defined on -∞ < t T is called the solution of (2.1) with initial data (2.2), if x(t) has the following properties:

(i) x(t) is continuous and is -adapted;

(ii) and ;

(iii) , for each t0 t T,

(2.3)

A solution x(t) is called as a unique if any other solution is indistinguishable with x(t), that is

The integral inequalities of Gronwall type have been applied in the theory of SDEs to prove the results on existence, uniqueness, and stability etc. [10,13-16]. Naturally, Gronwall's inequality will play an important role in next section.

Lemma 2.1. (Gronwall's inequality). Let u(t) and b(t) be non-negative continuous functions for t α, and let

where a ≥ 0 is a constant. Then

Lemma 2.2. (Bihari's inequality). Let u and b be non-negative continuous functions defined on R+. Let g(u) be a non-decreasing continuous function on R+ and g(u) > 0 on (0, ∞). If

for t R+, where k ≥ 0 is a constant. Then for 0 ≤ t t1,

where

and G-1 is the inverse function of G and t1 R+ is chosen so that

for all t R+ lying in the interval 0 ≤ t t1.

The following two lemmas are known as the moment inequality for stochastic integrals which will play an important role in next section.

Lemma 2.3. [2]. If p ≥ 2, such that

then

In particular, for p = 2, there is equality.

Lemma 2.4. [2]. If p ≥ 2, such that

then

### 3. Existence and Uniqueness of the Solutions

In order to obtain the existence and uniqueness of the solutions to (2.1) with initial data (2.2), we define and x0(t) = ξ(0), for t0 t T. Let , n = 1, 2, ... and define the Picard sequence

Now we begin to establish the theory of the existence and uniqueness of the solution. We first show that the non-Lipschitz condition and the weakened linea growth condition guarantee the existence and uniqueness.

Theorem 3.1. Assume that there exist a positive number K such that

(i) For any φ, ψ ∈ BC((-∞, 0]; Rd) and t ∈ [t0, T], it follows that

(3.1)

where κ(·) is a concave non-decreasing function from ℝ+ to ℝ+ such that κ(0) = 0, κ(u) > 0 for u > 0 and .

(ii) For any t ∈ [t0, T], it follows that f(0, t), g(0, t) ∈ L2 such that

(3.2)

Then the initial value problem (2.1) has a solution x(t). Moreover, . We prepare a lemma to prove this theorem.

Lemma 3.2. Let the assumption (3.1) and (3.2) of Theorem 3.1 hold. If x(t) is a solution of equation (2.1) with initial data (2.2), then

where c2 = c1 + Eǁ ξ ǁ2, c1 = 3E ǁξ ǁ2 + 6(T - t0 + 1)(T - t0)[K + a]. In particular, x(t) belong to .

Proof. For each number n ≥ 1, define the stopping time

Obviously, as n → ∞, τn T a.s. Let xn(t) = x(t τn), t ∈ (-∞, T]. Then, for t0 t T, xn(t) satisfy the following equation

Using the elementary inequality when p ≥ 1, we have

By the Hölder's inequality and the moment inequality, we have

Hence, by the condition (3.1) and (3.2) one can further show that

Given that κ(·) is concave and κ(0) = 0, we can find a positive constants a and b such that κ(u) ≤ a + bu for u ≥ 0. So, we obtains that

where c1 = 3E||ξ||2 + 6(T - t0 + 1)(T - t0)[K + a]. Noting the fact that , we obtain

where c2 = c1 + E||ξ||2. So, by the Gronwall inequality yields that

Letting t = T, it then follows that

Thus

Consequently the required result follows by letting n → ∞. □

Proof of Theorem 3.1. Let x(t) and be any tow solutions of (2.1). By Lemma 3.2, x(t), . Note that

By the elementary inequality (u + v)2 ≤ 2(u2 + v2), one then gets

By the Hölder's inequality, the moment inequality, and (3.1) we have

Since κ(·) is concave, by the Jensen inequality, we have

Consequently, for any ϵ > 0,

By the Bihari inequality, one deduces that, for all sufficiently small ϵ > 0,

(3.3)

where

on r > 0, and G-1(·) be the inverse function of G(·). By assumption and the definition of κ(·), one sees that limϵ↓0 G(ϵ) = -∞ and then

Therefore, by letting ϵ → 0 in (3.3), gives

This implies that for t0 t T, Therefore, for all -∞ < t T, a.s. The uniqueness has been proved.

Next to check the existence. Define and x0(t) = ξ(0) for t0 t T. For each n = 1, 2, ..., set and define, by the Picard iterations,

(3.4)

for t0 t T. Obviously, . Moreover, it is easy to see that , in fact

Taking the expectation on both sides and using the Hölder inequality and moment inequality, we have

Using the elementary inequality (u + v)2 ≤ 2u2 + 2v2, (3.1), and (3.2), we have

Given that κ(·) is concave and κ(0) = 0, we can find a positive constants a and b such that κ(u) ≤ a + bu for u ≥ 0. So, we have

where c1 = 3E||ξ||2 + 6(T - t0)(T - t0+ 1)[K + a]. It also follows from the inequality that for any k ≥ 1,

where c2 = c1 = 6b(T - t0)(T - t0+ 1) E||ξ||2. The Gronwall inequality implies

Since k is arbitrary, we must have

(3.5)

for all t0 t T, n ≥ 1.

Next, we that the sequence {xn(t)} is Cauchy sequence. For all n ≥ 0 and t0 t T, we have

Next, using an elementary inequality and the condition (H3), we derive that

On the other hand, by Hölder's inequality, Lemma (2.4), and the condition, one can show that

(3.6)

where β = (T - t0)/α + 4/(1 - α). Let

From (3.6), for any ϵ > 0, we get

By the Bihari inequality, one deduces that, for all sufficiently small ϵ > 0,

where

on r > 0, and G-1(·) be the inverse function of G(·). By assumption, we get Z(t) = 0. This shows the sequence {xn(t), n ≥ 0} is a Cauchy sequence in L2. Hence, as n → ∞, xn(t) - x(t), that is E |xn(t) - x(t)|2 → 0. Letting n → ∞ in (3.5) then yields that

for all t0 t T. Therefore, . It remains to show that x(t) satisfies Equation 2.3. Note that

Noting that sequence xn(t) is uniformly converge on (-∞, T ], it means that

as n → ∞, further

as n → ∞. Hence, taking limits on both sides in the Picard sequence, we obtain that

on t0 t T. The above expression demonstrates that x(t) is the solution of (2.3). So, the existence of theorem is complete.

Remark 3.1. In the proof of Theorem 3.1, the solution is constructed by the successive approximation. It shows that how to get the approximate solution of (2.1) and how to construct Picard sequence xn(t). For SFDEs, we know that a weakened linear growth condition imposed on Theorem 3.1 is rigorous for our discussion. In papers [6,7], the proofs of the assertions are based on some function inequalities. Recall that the procedures has become more and more complicated in those proofs. For this reason, although analogous problem is studied here, the proof of the assertion in the Theorem is completely different with respect to the ones from [7]. Moreover, our new proof in this paper is completed by Bihari's inequality and more simple than that reported in [7].

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript.

### Acknowledgements

The authors wish to thank the anonymous referees for their endeavors and valuable comments. Also, this research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2011-0023547)

### References

1. Henderson, D, Plaschko, P: Stochastic Differential Equations in Science and Engineering. World Scientific Publishing Co., Singapore (2006)

2. Mao, X: Stochastic Differential Equations and Applications. Horwood Publication, Chichester (2007)

3. Li, X, Fu, X: Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks. J Comput Appl Math. 234, 407–417 (2010). Publisher Full Text

4. Wei, F, Wang, K: The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J Math Anal Appl. 331, 516–531 (2007). Publisher Full Text

5. Mao, X, Shen, Y, Yuan, C: Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching. Stoch Process Appl. 118, 1385–1406 (2008). Publisher Full Text

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text:
# Checkpoint / restart¶

Enzo-E can perform checkpoint dumps to save the current state, and can read them at initialization to continue a simulation where it left off. Setting up checkpointing involves including methods in the Method parameter group, and provided a schedule for them. Restarting requires adding a small number of parameters to a parameter file to turn on restart, and where the checkpoint files are located.

## Checkpoint¶

Method {
...
list = [ "order_morton", "check", ... ];
order_morton {
schedule { var = "cycle";  start = 5;  step = 5;  }
}
check {
schedule { var = "cycle";  start = 5;  step = 5;  }
dir = [ "Check-%02d", "cycle" ];
num_files = 4;
ordering = "order_morton";
}
...
}

This example writes checkpoint dumps every 5 cycles, starting with the 5th cycle.

See the input/Checkpoint/test_cosmo-checkpoint.in parameter file for a working example of writing checkpoint directories.

Note

Currently, there is a restriction that the domain blocking must be square (2D) or cubical (3D), have a power-of-two blocking along the axes, and the Adapt:min_level parameter must be set such that the coarsest (negative) level is a single block. For example, if Mesh:root_blocks = [ 16, 16, 16], then Adapt:min_level must be initialized to 4 ( $$= log_2 16$$ ).

## Restart¶

To restart from a checkpoint directory, edit a copy of the original parameter file to add two parameters, Initial:restart and Initial:restart_dir. For example, to restart the problem in the previous “Checkpoint” section at cycle 50, one would add the following:

Initial {

restart = true;
restart_dir = "Check-50";
}

Note that it’s also possible to rerun with some parameters modified, for example different linear solver convergence criteria or different mesh refinement criteria. The main limitation is that the simulation data cannot change on restart: you cannot add new fields, new particle types or attributes, or change the number of root-level blocks or block sizes. You can restart with more (or fewer) processors, though the constraint that the number of processors must be at most the number of root-level blocks must still be satisfied.

The parameter file input/Checkpoint/test_cosmo-restart.in is the “restart” counterpoint to the “checkpoint” example mentioned in the previous section.


text:
# What are the standard styles to format LaTeX source code?

In any other programming languages, we have several ways to format our source code. For example, some of the standard styles are:

1. Allman(ANSI)
2. K&R
3. GNU
....

Within LaTeX, I couldn't find a neat way to clean my code. Most of the time, I usually make them align with each other like so:

An example from TikZ Automaton

\begin{tikzpicture}[shorten >=1pt, node distance=3cm,auto,on grid,initial text=, every state/.style={minimum size=3mm,draw=blue!50,very thick,fill=blue!20}]
\node[state]            (q_1)                           {$q_1$};
\node[state]            (q_2)   [right=of q_1]          {$q_2$};
\node[state,accepting]  (q_3)   [right=of q_2]          {$q_3$};

\path[->]
(q_1) edge                      node             {a}        (q_2)
(q_1) edge  [loop left]         node             {b}        (q_1)

(q_2) edge  [bend right]        node[yshift=4mm] {a}        (q_1)
(q_2) edge                      node             {b}        (q_3)

(q_3) edge  [bend left]         node             {a}        (q_1)
(q_3) edge  [loop right]        node             {b}        (q_3)
;
\end{tikzpicture}

Needless to say, manually doing this is very time-consuming and tedious. Additionally, since it does not follow any rule, the format won't be consistent and it could vary from file to file. So my question is, is there a standard style for formatting .tex source code?

-
The formatting you give in the example is inconsistent itself: sometimes a space but in other cases not before square brackets, random spaces inside the options themselves, random amount of spaces (or did you use tabs?) between columns. Besides, formatting TikZ is a whole different story, opposed to formatting TeX. –  Pieter Jul 30 '11 at 7:43
Thanks for the quick response. Yes, I often use tabs between columns, but sometimes I use spaces when the line gets too long. –  Chan Jul 30 '11 at 7:46
Please let questions long enough open so that other people can post answers as well. There is no need to accept the first answer quickly. –  Martin Scharrer Jul 30 '11 at 10:23
@Martin Scharrer: I really apologize for being rush. I will wait a bit more next time. Feel free to post your answer since I think other people including me definitely appreciate it. Thank you. –  Chan Jul 31 '11 at 1:17
@Chan: Thanks. I wasn't talking about a potential answer of mine in this case but meant it as a general rule. See the Should we wait a bit before accepting answers? post on TeX - LaTeX Meta for details. –  Martin Scharrer Jul 31 '11 at 9:01
show 1 more comment

+1 I must agree with your answer. TeX is more close to human languages and require us to improve its readability based on the context. Thank you. –  Chan Jul 30 '11 at 9:50