text:
Eulers Method

## Description

eulers_method implements Euler’s method for finding a numerical solution of the first-order ODE $y′=f(x,y)$. The $x$ column of the table increments from $x_0$ to $x_1$ by $h$ (so $(x_1 − x_0)/h$ must be an integer). In the $y$ column, the new $y$-value equals the old $y$-value plus the corresponding entry in the last column. Euler's method is used primarily for pedagogical purposes. The following Sage commands use Euler's method to generate a solution for

(1)
\begin{align} \frac{dy}{dx} & = 5xy - 5 \\ y(0) & = 1, \end{align}

where $h = 1/2$ and $x$ ranges from $0$ to $1$.

## Sage Cell

#### Code

x,y = PolynomialRing(QQ,2,"xy").gens()
eulers_method(5*x+y-5,0,1,1/2,1)

## Options

#### Option

To generate a list of points instead of a table, we can use the following commands.

#### Code

x,y = PolynomialRing(QQ,2,"xy").gens()
eulers_method(5*x+y-5,0,1,1/2,1,algorithm="none")

#### Option

We can also generate solutions over the reals.

#### Code

RR = RealField(sci_not=0, prec=4, rnd='RNDU')
x,y=PolynomialRing(RR,2,"xy").gens()
eulers_method(5*x+y-5,0,1,1/2,1)

## Tags

Primary Tags: Differential Equations

Secondary Tags: Numerical Methods: Euler

A list of possible tags can be found at The WeBWorK Open Problem Library. For linear algebra tags see the Curated Courses Project.


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# What is the surface are of a colonial grain storage tank that has a height of 54 meters and a diameter of 18 meters? 1802m^2, 1781m^2, 3110m^2, 1548m^2.. thanks

The tank has the shape of a cone So, the area of the cone = π r²+ π r s where r is the radius                      s is the slant height = √(r²+h²)           h is the height given the diameter = 18 m and the height = 54 m ∴ r = 9 m  ,  h = 54 m ∴ s = √(r²+h²) = √(9² + 54²) = 54.745 m ∴ Area = π r²+ π r s              = π * 9² + π * 9 * 54.745 ≈ 1802.3 m² The correct answer is the first option 1802 m²

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Thoughts rearrange, familiar now strange.break flowersmore quotes
very clickable
thermodynamics + art
See how much you can see with my modern interpretation of the Snellen eye chart. Test yourself with mathematical operators, physical constants, chemical elements and nautical flags. There's even a version for the sightless.
The charts use my SnellenMK optotype font.

# It's Snowing in my CPU — a Snowflake catalogue

She was all stars and arrows, squares and triangles of ice and light, like a church window; she was like a flower with many shining petals; she was like lace and she was like a diamond. But best of all, she was herself and unlike any of her kind.
— Paul Gallico, Snowflake

sumery
leory
tonon
furfelt
sumiphpun
sicia
Arenh Jerrus of Bosmen

Somewhere in the world, it's snowing. But you don't need to go far—it's always snowing on this page. Explore light flurries, snowflake families and individual flakes. There are many unusual snowflakes and snowflake family 12 and family 46 are very interesting.

But don't settle for only pixel snowflakes—make an STL file and 3D print your own flakes!

Ad blockers may interfere with some flake images—the names of flakes can trigger ad filters.

And if after reading about my flakes you want more, get your frozen fix with Kenneth Libbrecht's excellent work and Paul Gallico's Snowflake.

This page uses code contributed by Alexander Pruss to generate an STL file for any of the flakes in the catalogue. The code is now hosted here to cross-origin requests.

## authentically-grown flakes for your tree

Here is the 3D printed snowflake morptel. Yes, they all have names—snowflakes are people too.

The physical manifestation of the snowflake $morptel$.

## Snowflake to STL conversion

It takes about 10–20 seconds to generate the STL file. You are about to generate STL for snowflake rolia (shown below).

To generate STL for a different flake, browse to the flake's page and click the Generate STL file for 3D printing link in the section SNOWFLAKE FILES AND RESOURCES.

Snowflake name mm mm mm cells


text:
A Decomposing the commutator norm into local parts

# Signatures of the Many-body Localized Regime in Two Dimensions

## Abstract

Lessons from Anderson localization highlight the importance of dimensionality of real space for localization due to disorder. More recently, studies of many-body localization have focussed on the phenomena in one dimension using techniques of exact diagonalization and tensor networks. On the other hand experiments in two dimensions have provided concrete results going beyond the previously numerically accessible limits while posing several challenging questions. We present the first large-scale numerical examination of a disordered Bose-Hubbard model in two dimensions realized in cold atoms, which shows entanglement based signatures of many-body localization. By generalizing a low-depth quantum circuit to two dimensions we approximate eigenstates in the experimental parameter regimes for large systems, which is beyond the scope of exact diagonalization. A careful analysis of the eigenstate entanglement structure provides an indication of the putative phase transition marked by the peak in the fluctuations of entanglement entropy in a parameter range potentially consistent with experiments.

Many-body localization (MBL) is a paradigm shift in out-of-equilibrium quantum matter. This novel phase of matter is characterized by the absence of thermalization Anderson (1958); Basko et al. (2006); Gornyi et al. (2005); Pal and Huse (2010); Oganesyan and Huse (2007); Nandkishore and Huse (2015). Thus, unitary quantum dynamics influences only a few degrees of freedom even in an interacting system while it can retain quantum coherence at high energies Bardarson et al. (2012). By localizing the excitations MBL can also protect certain forms of topological and symmetry-breaking orders in excited states and provides an opportunity to process quantum information in a system driven far from equilibrium Huse et al. (2013); Pekker et al. (2014); Bahri et al. (2015); Chandran et al. (2014). The quantum phase transition separating the MBL and thermal phases poses a major challenge for developing a theory of dynamical critical phenomena described by many-body entanglement in highly excited states Kjäll et al. (2014); Vosk et al. (2015); Potter et al. (2015); Grover (2014); Chandran et al. (2015a); Lim and Sheng (2016); Khemani et al. (2017); Dumitrescu et al. (2017).

It is well-known that dimensionality of real space affects single particle Anderson localization where in one and two dimensions (without spin-orbit coupling and broken time-reversal symmetry), the entire spectrum of single particle eigenstates is localized for arbitrary weak disorder Lee and Ramakrishnan (1985). Although the defining properties of MBL in one dimension are firmly established both theoretically and experimentally Imbrie (2016); Schreiber et al. (2015), the existence of the phenomenon in two and higher dimensions is much debated Chandran et al. (2016); De Roeck and Huveneers (2017); De Roeck and Imbrie (2017); Agarwal et al. (2017); Luitz et al. (2017); Ponte et al. (2017). Experiments in cold atoms measuring local and global equilibration have shown the persistence of quantum memory for long times providing indications of an MBL phase in two dimensions Choi et al. (2016); Bordia et al. (2017). On the other hand, theoretical criteria suggest that the lifetime of local memory is finite, albeit extremely long Chandran et al. (2016); De Roeck and Huveneers (2017).

In this article we evaluate the eigenstates of bosons hopping in a disordered lattice in two dimensions with onsite interactions. We generalize the tensor network method developed earlier for one dimensional systems to approximate the eigenstates in the localized regime Wahl et al. (2017). The system sizes accessible by our method are much larger than those that can be currently achieved by exact diagonalization or quantum Monte Carlo Geraedts et al. (2016); Inglis and Pollet (2016). We study the model in parameter regimes directly relevant to the experiments. The approximate eigenstates evaluated by our method exhibit signatures of a phase with low entanglement. By studying the distribution of entanglement entropy of a single site, we probe the eigenstates in the localized regime at large disorder. On lowering the disorder strength the distribution of entanglement becomes bimodal with increasing weight at larger entanglement, indicating the transition into the thermal phase. We find a critical disorder strength which is reasonably close to the experimentally measured value. This is the first theoretical demonstration in a large-scale system of a transition between eigenstates with distinct statistical properties of entanglement in two dimensions.

Quantum memory in MBL
In one dimension, fully many-body localized (FMBL) systems are characterized by an extensive number of quasi-local operators, known as l-bits (Serbyn et al. (2013); Huse et al. (2014); Ros et al. (2015); Imbrie et al. (2017). These operators commute exactly with the Hamiltonian and also with each other. Therefore, all energy eigenstates are represented as a string of eigenvalues of the l-bits. Locality of the l-bits forces the eigenstates to satisfy the area law of entanglement Bauer and Nayak (2013). This feature greatly simplifies the spectral properties of an FMBL Hamiltonian. As a result, the unitary transformation () which diagonalizes the Hamiltonian can be efficiently approximated by a spectral tensor network (TN) Pekker and Clark (2017); Chandran et al. (2015b); Pollmann et al. (2016) where all eigenstates can be represented as matrix product states Friesdorf et al. (2015); Yu et al. (2017); Khemani et al. (2016). Throughout this work, we consider shallow quantum circuits, which are TNs which can be contracted efficiently in any dimension. They are composed of layers of local unitaries with each unitary acting on a finite contiguous block of spins Wahl et al. (2017). For the optimal set of local unitaries, the TN transforms the Hamiltonian into a predominantly diagonal form. The magnitude of the remaining off-diagonal matrix elements provides an estimate of the accuracy of the TN approximation. This method has been successfully implemented to evaluate the eigenstates of large systems in the fully many-body localized phase and estimate the location of the phase transition in one dimension Wahl et al. (2017). This method can also be used to construct the l-bits close to the MBL transition Kulshreshtha et al. (2017). Although exact diagonalization gives the entire spectrum of eigenstates, the construction of l-bits is non-trivial and the low-depth quantum circuit provides a systematic and efficient construction of the full set of operators.

In dimensions greater than one, the l-bit phenomenology can break down due to resonant interactions which destroy the exact conservation of the set of quasi-local operators Chandran et al. (2016); De Roeck and Huveneers (2017); De Roeck and Imbrie (2017). As a result, for a finite-size system thermalization can be restored at late times. However, if local operators remain approximately conserved, the system displays features analogous to MBL on experimentally relevant time scales. Unlike in one dimension where the l-bits are exactly conserved due to the suppression of resonances even for a finite system size, in higher dimensions the proliferation of resonances is believed to be exponentially sensitive to perturbations. Therefore, MBL giving rise to a stable quantum phase may only be well-defined as the system size tends to infinity, although there are arguments in favor of instability of MBL even in this asymptotic limit De Roeck and Huveneers (2017); De Roeck and Imbrie (2017). We use the semantics of quantum phase transitions to describe the features in our finite-size calculations, even though our numerical investigation may not capture this asymptotic limit. However, the error in our approximation provides a lower bound for the time scale over which local observables will decay in this system. Thus, our method approximates well the short-time dynamics. Hence, while not answering the question of the existence of exact l-bits in two dimensions, we expect our approach to describe the experimentally observed localization phenomena.

Note that even though the experimentally accessible time scales and system sizes are finite, there are still no theoretical descriptions of the observed phase diagram. Indeed, the conditons required to confirm the breakdown of conservation of these quasi-local operators likely require measurements for astronomically long times for a physical choice of parameters Choi et al. (2016). Nonetheless, our work sheds light in this experimentally relevant regime using our novel numerical technique.

In this article we evaluate eigenstates with an accuracy which grows with the range of the local unitaries. By studying the structure of entanglement in these eigenstates, we examine the system’s ability to serve as long-lived quantum memory over a range of disorder strengths which is consistent with state of the art experiments on MBL in two dimensions Choi et al. (2016); Bloch (). The low-depth quantum circuit captures the entanglement in local ensembles, but does not give the long-range entanglement of thermal systems.

Tensor network ansatz
We propose a shallow TN capable of approximating the two-dimensional localized phase. It encodes a unitary which approximately diagonalizes the localized Hamiltonian . We consider an lattice with periodic boundary conditions. The tensor network is composed of two layers of smaller unitaries acting on sites ( even) as shown in Fig. 1a. It is a natural extension of the ansatz in Ref. Wahl et al., 2017. The approximately conserved operators are given by ( is the spin- operator at site ). These operators commute mutually by construction. The approximates , which is itself expected to be only approximately conserved, but acts as a local memory on the experimentally relevant time scales. Hence, the tensor network can be optimized by minimizing the (squared) trace norm of the commutator with the Hamiltonian,

 f({ux,y,z}) =12N2+1N2∑i=1tr([H,~τzi][H,~τzi]†) =12N2N2∑i=1(tr(H2(~τzi)2)−tr((~τziH)2). (1)

corresponds to the tensor network contraction shown in Fig. 1b, which is non-trivial only in the region covered by the unitaries , . As a result, for a nearest-neighbor Hamiltonian, Eq. (1) decomposes into local parts,

 f ({ux,y,z})=const. −N/ℓ∑x,y=1fx,y(ux,y,1,ux,y,2,ux+1,y,2,ux,y+1,2,ux+1,y+1,2). (2)

The explicit representation of by tensor network contractions can be found in Appendix A. The overall computational cost scales as . In the general case, the length of the region within which is non-trivial is of order . Therefore, in the MBL regime, the error in representing is expected to decrease exponentially with . The same will be true for the error of expectation values of local observables. Hence, for fixed system size, this error is an inverse polynomial of the computational cost. The localization of the eigenstates in the MBL regime Chandran et al. (2016) implies that this error does not significantly increase with system size, making our approximation efficient. Note that even for , our approach can incorporate effects which are far beyond the reach of exact diagonalization: In that case, the operators are non-trivial within a region of . That implies non-vanishing correlations between a given site and other sites within a regime. Due to the strict locality of the operators, our approximation is not affected by the boundary conditions, i.e., it describes the MBL regime directly in the bulk.

MBL phase in 2D
In the following, we use our TN approach to investigate the MBL phase found experimentally Choi et al. (2016) in a two-dimensional optical lattice. The phase transition was determined by measuring the loss of memory of the initial conditions of bosonic Rubidium-87 atoms in a random optical potential. The dynamics of the atoms can be modeled by the disordered Bose-Hubbard Hamiltonian

 Hexp=−J∑⟨i,j⟩(a†iaj+aia†j)+U′2∑ini(ni−1)+δini, (3)

where the nearest neighbor links are counted once for each pair , () are bosonic annihilation (creation) operators and is the particle number operator. is random variable chosen from a Gaussian distribution with full-width half-maximum , on-site interaction and we fix the energy scale through . Note that we neglect the trapping potential, as we are interested in the phase transition in the bulk of the system, i.e., the center of the trap. The local bosonic Hilbert space is a priori infinite dimensional. We truncate the on-site occupation number to , which is an approximation suitable to describe the experiments where the higher occupation number states are rarely observed. (Experimentally, only up to 7 % of sites are occupied with doublons, and no triplons are observed Bloch ().) We perform calculations for (corresponding to hard core bosons or spin- particles) and (corresponding to spin- particles). As the Hamiltonian is particle number conserving and real, the unitaries can be parameterized as the exponentials of antisymmetric matrices, which also conserve particle number Wahl et al. (2017) (for details, see Appendix B). We verified numerically that no significant loss of accuracy occurs compared to a full parameterization.

A hallmark of the localized phase is the locality of entanglement even in the presence of interactions. A single spin is only weakly coupled to its environment and its reduced density matrix in an eigenstate (where represents the local spin and is the rest of the system), remains close to a pure state. On the other hand in the thermal phase, the single spin is maximally entangled with its environment. For a two level system (spin-), this maximal entanglement is equal to and is for a three-level system (spin-). We will first focus on the results. These come closest to the experiments Choi et al. (2016); Bloch (), which were carried out at an average filling . Since the overall boson number is conserved, the experimentally probed transition refers only to half filling. In contrast, our calculations describe a transition occurring at all occupation numbers (given the on-site truncation). For , the majority of simulated eigenstates are close to half-filled, whereas for , most eigenstates have on average approximately one boson per site.

The approximate eigenstates represented by the TN have a local structure which is reliable as the approximate integrals of motion given by the TN have a small commutator with the Hamiltonian. The degree of the commutation increases with increasing disorder strength (see Appendix E, Fig. 8b).

Signatures of the MBL transition
We calculated the on-site entanglement entropies () for 30 disorder realizations on a lattice as a function of disorder . Those entropies can be computed efficiently as explained in Appendix F. Their distribution displayed in Fig. 2 shows a sharp peak at . The distribution has a tail to large values of entanglement which decays quickly to zero. The bimodal entanglement entropy distributions with a sharp peak at along with a broad peak at lower values of for provide evidence for the phase transition from the MBL to the thermal regime. The bimodality indicates the co-existence of highly localized states with low entanglement and delocalized states with larger entanglement in the critical regime. This resembles the behavior around the critical point of one-dimensional models exhibiting MBL Yu et al. (2016). In contrast, for , the distribution is mostly concentrated in a single peak at , where delocalized states are absent. Note that the peak at for appears because the map from the reduced density matrix to the entropy has a maximum at ( is diagonal due to symmetry.). Therefore, transforming the probability distribution of to produces a singularity due to the derivative of tending to zero at the maximum. As expected there is no peak in the histogram at (not shown).

We also computed the reduced density matrices of all plaquettes (using the contraction scheme presented in Appendix F) and extracted their entanglement spectra. The entanglement spectra are given by the eigenvalues of the entanglement Hamiltonian defined via . The corresponding distributions are shown in Fig. 2c,d. They have a double-peak structure throughout with a narrow maximum at small entanglement energies and a broad maximum at higher entanglement energies . The former moves towards zero and the latter towards infinite entanglement energies with increasing . The behavior of the narrow maximum is consistent with findings using exact diagonalization in one dimension Geraedts et al. (2017). However, in that reference, the broad maximum is observed only in the MBL phase and close to the transition. It is also found to move to larger entanglement energies with increasing disorder strength. It is thought to be a remnant of the critical regime even deep in the localized phase. We obtain a broad maximum for small , too, presumably because the optimized tensor network does not fully capture the delocalized phase, and instead displays critical features there also.

In order to locate the critical point more precisely, we evaluated the variance of the on-site entanglement entropies with respect to different disorder realizations. Finally, we averaged over the site positions to improve smoothness Wahl et al. (2017). The results for the entanglement entropy fluctuation are shown in Fig. 3. For , it peaks at which is potentially consistent with the latest experimentally measured transition point Bloch (). Note that in Ref. Choi et al., 2016, the MBL-to-thermal transition was measured at for ; however there are several experimental conditions which can lead to a lower estimate of the measured critical point which continue to be under intense investigation.

As already mentioned above, for , we do not obtain the same transition point as in the experiments, which were not carried out at average filling 1. The entropy fluctuations using 30 disorder realizations and lattices are also shown in Fig. 3. They indicate MBL-to-thermal transitions at and , respectively. Future experiments initialized with a charge density wave of doublons might verify this prediction. The trend towards lower transition points with decreasing interaction strength is consistent with Anderson localization at for

Conclusion
We have presented the first large-scale numerical study of eigenstates in the MBL regime of the disordered Bose-Hubbard model in two dimensions using shallow tensor networks (quantum circuits). The novel variational ansatz approximates the full spectrum of eigenstates as area-law entangled in this regime. By characterizing the statistics of entanglement as a function of disorder, the location of the transition into the thermal phase has been estimated. The distribution of the entanglement spectrum of these eigenstates exhibits the characteristics of MBL which further supports our observations.

Our work has important consequences for experiments with ultra-cold atoms studying quench dynamics in the presence of disorder. It provides an estimate of the MBL transition point in two dimensions as a function of interaction strength, which can be verified in these experiments. The current experiments can access the transition using local dynamical properties of local observables in a finite system, which we expect to be well-described by our approximate eigenstates. This also opens the door to study the phenomenology of MBL in two dimensions, where the time-tested method of exact diagonalization is computationally prohibitively expensive. Large-scale simulations with higher will further elucidate the nature of the MBL regime in two dimensions.

Acknowledgments
We are grateful to Anushya Chandran and Chris Laumann for stimulating discussions and a careful reading of the manuscript, and to Antonio Abadal, Immanuel Bloch, Jae-Yoon Choi, and Christian Gross for detailed discussions related to the experiments. We would also like to thank David Huse, John Imbrie, Vadim Oganesyan, Wojciech De Roeck, Antonello Scardicchio, Shivaji Sondhi, and Thomas Spencer for fruitful discussions. S.H.S. and T.B.W. are both supported by TOPNES, EPSRC grant number EP/I031014/1. S.H.S. is also supported by EPSRC grant EP/N01930X/1. T.B.W. acknowledges usage of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work, http://dx.doi.org/10.5281/zenodo.22558. A.P. is supported by the Glasstone Fellowship and would like to thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, and Simons Center for Geometry and Physics, Stonybrook University for their hospitality, where part of this work was performed. T.B.W. is grateful for support by the European Commission under the Marie Curie Programme. The contents of this article reflect only the authors’ views and not the views of the European Commission. Statement of compliance with EPSRC policy framework on research data: This publication is theoretical work that does not require supporting research data.

## Appendix A Decomposing the commutator norm into local parts

Here, we describe the contraction scheme to evaluate the commutator norm as defined in Eq. (1). We assume that the Hamiltonian is a sum of nearest neighbor terms, . In that case, the two terms in the final expression of Eq. (1) are

 tr(H2(~τzi)2)=∑k,ltr(~U(Σzi)2~U†hkhl). (4)

and

 tr((~τziH)2)=∑k,ltr(~UΣzi~U†hk~UΣzi~U†hl), (5)

respectively. Using the representation for shown in Fig. 1b (which is non-trivial only in the region covered by the unitaries , ), the right side of Eq. (5) can be evaluated as shown in Fig. 4. The right side of Eq. (4) can be written as a similar tensor network contraction (this is most obvious if one inserts between and leading to an expression which is formally equivalent to Eq. (5)). Therefore, can be expressed as in Eq. (2).

The tensor network contraction of Fig. 4 is carried out most efficiently as follows. First, one combines unitaries and their adjoints (shown in the same color in Fig. 4) to new tensors. If there is no other tensor between them, this obviously results in the identity. In contrast, will be combined with and to form a new tensor . The other tensors might have to be combined with the Hamiltonian terms and , respectively. The resulting tensor network is shown in Fig. 5. A special case is the one of with the two Hamiltonian terms being connected by a separate line if they do not lie entirely within the region covered by the unitaries . This squares the dimension of the index contraction corresponding to one of the lines in Fig. 5. The leading contribution to the computational cost stems from the multiplication of matrices, i.e., it scales as . As the number of tensor network contractions of the type shown in Fig. 5 is independent of the system size, each term contributes to the overall computational cost, which is thus of order .

The derivative of our figure of merit can be calculated easily by cutting out the unitary matrix the derivative is taken of in Figs. 4 and 5, respectively, and contracting the tensors in such an order that the removed unitary would come last, cf. Ref. Wahl et al., 2017. This comes at a subleading increase of computational cost.

## Appendix B Optimization method

We take advantage of the symmetries of the considered model in order to reduce the number of parameters: The Hamiltonian is real and possesses symmetry, i.e., particle number conservation in the experimental model Eq. (3) and spin- conservation in its spin representations and the anti-ferromagnetic Heisenberg model, . In general, such global symmetries can be imposed on the individual tensors which make up the tensor network without much loss of accuracy Sanz et al. (2009); Pérez-García et al. (2010). We do so by taking real unitary matrices , i.e., orthogonal matrices, which on top of that preserve the spin- component individually, . Both taken together imply that the unitaries must have a block diagonal form (in the spin- basis),

 ux,y,z=⨁beA(b)x,y,z, (6)

where , a skew-symmetric matrix. In each minimization run, the unitaries are initialized as identities and optimized using a quasi-Newtonian routine supplied by the derivative with respect to the parameters. We optimize one unitary at a time and sweep alternately over all the columns and all the rows of the system (starting each sweep at site ).

## Appendix C Comparison with exact diagonalization

We benchmark our approach by comparing with exact diagonalization for the hard-core Bose-Hubbard model, Eq. (1), on a lattice with open boundary conditions for . To that end, we place additional unitaries in the top layer, which pairwise connect the ones of the bottom layer as shown in Fig. 6. The open boundary conditions are meant to prevent artificial delocalization (as compared to periodic boundary conditions) for such a small system. In Fig. 7, we show the energy errors and overlaps of the approximate compared to the exact eigenstates for and 30 disorder realizations. More than 99.9 % of all approximate eigenstates have at least 50 % overlap with the actual ones, which is a very high accuracy given that the dimension of the Hilbert space is . Because of the local structure of the tensor network, it follows that the true eigenstates are relatively well localized for this disorder strength. If they stay well localized for larger system sizes, the tensor network approximation remains accurate and is able to capture local observables with a constant accuracy as the system size is increased. The average optimized figure of merit was . We also calculated the variance averaged over eigenstates,

 ΔH2 =12N2∑i1…iN(⟨ψi1…iN|H2|ψi1…iN⟩ −⟨ψi1…iN|H|ψi1…iN⟩2). (7)

Its mean over disorder realizations was .

## Appendix D Increasing the system size

As the figure of merit decomposes into a sum of local parts, the expectation is it increases linearly with the system size on average. We confirmed this by studying 30 disorder realizations at for as a function of system size for periodic boundary conditions, see Fig. 8a.

In the MBL regime, eigenstates can be described by effective spin degrees of freedom Chandran et al. (2016) , which have exponentially decaying non-trivial matrix elements as a function of distance and commute with the Hamiltonian up to exponentially small corrections in the system size. Therefore, for sufficiently large systems, we gain the following picture: All effective spin degrees of freedom which are located at a distance from a given region contribute to local properties of that region by an amount of order . Hence, their contribution vanishes exponentially with distance and the error made by approximating the eigenstates in terms of strictly short-range tensor network -bits is independent of the system size. In other words, local observables can be approximated with a constant accuracy using our TN if the system size is increased. Note that at the same time the computational cost grows linearly with system size.

## Appendix E Accuracy as a function of disorder strength

We also investigated the accuracy of our approximation as a function of for 30 disorder realizations. For different , only the overall strength of the random magnetic fields was adjusted to enhance comparability. The optimized figure of merit is shown in Fig. 8b. We find a strong, monotonic increase of the error with decreasing , especially below . Notwithstanding, a concrete picture of the phase transition can only be gained from the distribution of entanglement entropies.

## Appendix F Calculation of the entanglement entropies and spectra of 2×2 blocks of sites

In the following, we explain how the reduced density matrices of plaquettes of approximate eigenstates given by our tensor network can be calculated efficiently. The computation of single-site reduced density matrices is a special case thereof. We will focus on the case of plaquettes connecting four unitaries of the upper layer, which is the hardest one to evaluate. Other positions of the blocks can be treated in a simplified way.

A reduced density matrix of a given approximate eigenstate can be graphically represented by taking the tensor network in Fig. 1a, fixing its lower indices according to the -bit configuration of that approximate eigenstate and contracting it with its adjoint from above, leaving only the legs in the chosen region open. After cancelling unitaries with their Hermitian conjugates, one obtains the tensor network contraction shown in Fig. 9a (for ). Even though this tensor network is non-trivial in a Hilbert space of dimension ( Hilbert space dimension of a single site), its tensors can be contracted in such an order that the computational time is only of order , cf. Fig. 9b.

Finally, in Fig. 10 we present the data for the distribution of entropies for (after tracing out all but a single site) along with the histogram over the entanglement spectra for plaquettes. The data is based on 30 disorder realizations on a lattice and samples over 10,000 eigenstates per site.

### References

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electron-builder vs electron-packager [closed]

What is main difference between electron-builder and electron-packager. Is there some example projects for both? Which is better for standalone .exe application build?

closed as too broad by Bhargav Rao♦Dec 15 '18 at 7:30

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

electron-packager was (update: changed in this commit: "refactor: merge electron-packager") used by electron-builder. This article states

If you really want to go to town, take a look at electron-builder which takes the packages produced by electron-packager and creates automated installers.

Update August 2019

I've been using electron-builder for more than a year now, packaging my app for macOS, Windows 10, and Ubuntu Desktop. The maintainer, develar, added a great docs site earlier this year, and has been staying on top of gatekeeper changes to macOS to ensure installers and updaters still work.

Update November 2018

The primary developer of electron-builder has continued to put substantial work into the project, including support for Electron 3, new artifact publishing destinations, and releases regularly with bug fixes. The prior maintainer stepped down from electron-packager, and they still don't have a windows maintainer, and still doesn't have support for signed windows binaries. For reference:

electron-packager ( 72 releases,  17 open issues,   639 closed)
electron-builder  (651 releases, 399 open issues, 2,603 closed)

Update December 2017

Electron-builder has been robustly worked on over the past year, pushing out new versions with features and bugfixes weekly. The documentation at https://www.electron.build/ is excellent, and following the "Quick Setup Guide" (on the first page, several sections down) includes a boilerplate project and step-by-step instructions.

Update June 2017

electron-builder does both the packaging step (like what electron-packager does), and builds platform-specific installers (like pkg or Mac AppStore for Mac, NSIS or Windows Store for windows, and AppImage for linux).

For my existing project, electron-forge failed with a "missing dependency" that was present in my package.json and node_modules).

It's not well documented, but electron-builder switched from Squirrel.windows to NSIS earlier this year as the default windows installer. NSIS makes substantially smaller installer packages (30mb down from 90mb, in my project's case).

Original post

As of March 2017, in large part, they've got feature parity. I've found the following:

• They are both in the "electron-userland" org, so they're both "legit."
• electron-packager doesn't have a windows maintainer
• electron-builder has more contributors, and many more closed issues, so there may have been more bugs, or more users. electron-packager has more stars.
• electron-builder supports substantially more target formats than electron-packager
• electron-forge is based on electron-packager, so that might be an easier "out of the box" experience.

electron-packager packs source code and generates executable file.

electron-builder can generate installer with “auto update” support in electron.

http://electron.atom.io/docs/api/auto-updater/

Ok, I know this thread is old but I would like to suggest that anyone / everyone uses electron-forge, it will make your lives better and I think you will feel more intelligent by using it. I personally had waking nightmares confronting packaging woes. It has a dead-simple cli, install it globally and your dreams may come true.

I sure hope it helps! Don't worry, I didn't make it but it helps me make desktop apps.

• I wish I'd have found this comment before wasting the last two days messing with both builder and packager. – codestr Nov 28 '17 at 20:21
• Are you able to arrange auto-updates on all three platforms with electron-forge, like you can in electron-builder? – jayarjo Mar 5 at 5:32

You can use this https://github.com/SimulatedGREG/electron-vue/ to generate template using vue and electron, it support both electron-packager and electron-builder.

For me, I always prefer electron-builder because it is much simpler for me and can generate binary using travis ci and appveyor and pulish to github release, like this repo I made https://github.com/gengjiawen/android-studio-helper/releases.


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## G = Q8⋊8SD16order 128 = 27

### 2nd semidirect product of Q8 and SD16 acting through Inn(Q8)

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — Q8⋊8SD16
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — Q82 — Q8⋊8SD16
 Lower central C1 — C2 — C2×C4 — Q8⋊8SD16
 Upper central C1 — C22 — C4×Q8 — Q8⋊8SD16
 Jennings C1 — C2 — C2 — C2×C4 — Q8⋊8SD16

Generators and relations for Q88SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c3 >

Subgroups: 408 in 196 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C41D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×C4○D4, C4×SD16, C8×Q8, C4⋊SD16, D42Q8, C4.4D8, Q86D4, Q82, Q88SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, 2- 1+4, Q85D4, C22×SD16, D4○D8, Q88SD16

Smallest permutation representation of Q88SD16
On 64 points
Generators in S64
(1 51 59 25)(2 52 60 26)(3 53 61 27)(4 54 62 28)(5 55 63 29)(6 56 64 30)(7 49 57 31)(8 50 58 32)(9 20 33 46)(10 21 34 47)(11 22 35 48)(12 23 36 41)(13 24 37 42)(14 17 38 43)(15 18 39 44)(16 19 40 45)
(1 46 59 20)(2 21 60 47)(3 48 61 22)(4 23 62 41)(5 42 63 24)(6 17 64 43)(7 44 57 18)(8 19 58 45)(9 51 33 25)(10 26 34 52)(11 53 35 27)(12 28 36 54)(13 55 37 29)(14 30 38 56)(15 49 39 31)(16 32 40 50)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 33)(2 36)(3 39)(4 34)(5 37)(6 40)(7 35)(8 38)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(41 52)(42 55)(43 50)(44 53)(45 56)(46 51)(47 54)(48 49)

G:=sub<Sym(64)| (1,51,59,25)(2,52,60,26)(3,53,61,27)(4,54,62,28)(5,55,63,29)(6,56,64,30)(7,49,57,31)(8,50,58,32)(9,20,33,46)(10,21,34,47)(11,22,35,48)(12,23,36,41)(13,24,37,42)(14,17,38,43)(15,18,39,44)(16,19,40,45), (1,46,59,20)(2,21,60,47)(3,48,61,22)(4,23,62,41)(5,42,63,24)(6,17,64,43)(7,44,57,18)(8,19,58,45)(9,51,33,25)(10,26,34,52)(11,53,35,27)(12,28,36,54)(13,55,37,29)(14,30,38,56)(15,49,39,31)(16,32,40,50), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,33)(2,36)(3,39)(4,34)(5,37)(6,40)(7,35)(8,38)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49)>;

G:=Group( (1,51,59,25)(2,52,60,26)(3,53,61,27)(4,54,62,28)(5,55,63,29)(6,56,64,30)(7,49,57,31)(8,50,58,32)(9,20,33,46)(10,21,34,47)(11,22,35,48)(12,23,36,41)(13,24,37,42)(14,17,38,43)(15,18,39,44)(16,19,40,45), (1,46,59,20)(2,21,60,47)(3,48,61,22)(4,23,62,41)(5,42,63,24)(6,17,64,43)(7,44,57,18)(8,19,58,45)(9,51,33,25)(10,26,34,52)(11,53,35,27)(12,28,36,54)(13,55,37,29)(14,30,38,56)(15,49,39,31)(16,32,40,50), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,33)(2,36)(3,39)(4,34)(5,37)(6,40)(7,35)(8,38)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49) );

G=PermutationGroup([[(1,51,59,25),(2,52,60,26),(3,53,61,27),(4,54,62,28),(5,55,63,29),(6,56,64,30),(7,49,57,31),(8,50,58,32),(9,20,33,46),(10,21,34,47),(11,22,35,48),(12,23,36,41),(13,24,37,42),(14,17,38,43),(15,18,39,44),(16,19,40,45)], [(1,46,59,20),(2,21,60,47),(3,48,61,22),(4,23,62,41),(5,42,63,24),(6,17,64,43),(7,44,57,18),(8,19,58,45),(9,51,33,25),(10,26,34,52),(11,53,35,27),(12,28,36,54),(13,55,37,29),(14,30,38,56),(15,49,39,31),(16,32,40,50)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,33),(2,36),(3,39),(4,34),(5,37),(6,40),(7,35),(8,38),(9,59),(10,62),(11,57),(12,60),(13,63),(14,58),(15,61),(16,64),(17,32),(18,27),(19,30),(20,25),(21,28),(22,31),(23,26),(24,29),(41,52),(42,55),(43,50),(44,53),(45,56),(46,51),(47,54),(48,49)]])

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4H 4I ··· 4O 4P 4Q 4R 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 8 8 8 8 8 ··· 8 size 1 1 1 1 8 8 8 2 ··· 2 4 ··· 4 8 8 8 2 2 2 2 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 SD16 C4○D4 2- 1+4 D4○D8 kernel Q8⋊8SD16 C4×SD16 C8×Q8 C4⋊SD16 D4⋊2Q8 C4.4D8 Q8⋊6D4 Q82 C4⋊C4 C2×Q8 Q8 Q8 C4 C2 # reps 1 3 1 3 3 3 1 1 3 1 8 4 1 2

Matrix representation of Q88SD16 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 0 1 0 0 16 0
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 13
,
 12 5 0 0 12 12 0 0 0 0 0 4 0 0 13 0
,
 0 16 0 0 16 0 0 0 0 0 0 13 0 0 4 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[12,12,0,0,5,12,0,0,0,0,0,13,0,0,4,0],[0,16,0,0,16,0,0,0,0,0,0,4,0,0,13,0] >;

Q88SD16 in GAP, Magma, Sage, TeX

Q_8\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("Q8:8SD16");
// GroupNames label

G:=SmallGroup(128,2094);
// by ID

G=gap.SmallGroup(128,2094);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations

׿
×
𝔽


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# Scalar product of vectors over polynomial rings

I have been studying about homomorphic encryption (HE) lately, and I am trying to perform simple computations over encrypted data. Specifically, I am interested in the scalar product between two vectors.

I came across this presentation about HElib (an implementation of HE) and on slide 32 there's this particular way of encoding vectors that gives use the scalar product after we multiply the polynomials that represent them:

Given $v=[1, 2, 3]$, $u=[4,5,6]$

If we make two polynomials such as $V(x) = 1 + 2x + 3x^2$, $U(x)=4 + 5x + 6x^2$

But $V(x)U(x) = 4 + 13x + 28x^2 + 27x^3 + 18x^4$

Change a little bit $Û(x) = 6 + 5x + 4x^2$

$V(x)Û(x) = 6 + 17x + 32x^2 + 23x^3 + 12x^4$

$32 = \langle v, u \rangle$

So my question is: How to choose the polynomial encoding of the vectors in order to obtain the scalar product value from a particular coefficient after multiplication? In other words, why inverting the representation of $U(x)$ to $Û(x)$ gave us the scalar product in the third coefficient?

## 2 Answers

To encode vectors following that technique, you just have to generalize it.

For any two $(d+1)$-dimensional vectors $a = (a_0, a_1, ..., a_d)$ and $b = (b_0, b_1, ..., b_d)$, define the polynomials $a(x) = a_0 + a_1x + ... + a_dx^d$ and $b(x) = b_d + b_{d-1}x + b_{d-2}x^2 + ... + b_0x^d$.

Then, when multiplying $a(x)$ by $b(x)$, among all the coefficients that we multiply, we will multiply $a_0$ by $b_0x^d$, also $a_1x$ by $b_1x^{d-1}$, $a_2x^2$ by $b_2x^{d-2}$, and so on... Notice that all that multiplications are of the form $a_kx^k \cdot b_kx^{d-k}$, which result in $a_kb_kx^d$. Therefore, if we look to the $d$-th coefficient of the product polynomial we will see

$$\left( \sum_{k=0}^d a_kb_k \right) x^d$$

and that summation is exactly the inner product between the vectors $a$ and $b$.

It is worth noting that if this summation is greater than the module of the plaintext ring (usually a parameter $t$ or $p$), than the decoding will not work. Also, that representation is only possible if that value $d$ is smaller than the degree of the polynomial used to define the quotient ring (usually a cyclotomic polynomial).

• Thank you very much for the reply! I was having trouble to generalize it, but it is clear now. – mshcruz Aug 7 '16 at 6:53
• You're welcome. By the way, you have to study if this encoding is the best one for your application. If you are going to perform other operations in addition to the inner product, it may be better to encode each component of the vector into a plaintext... – Hilder Vitor Lima Pereira Aug 7 '16 at 18:53

Polynomial multiplication corresponds to convolution of coefficients. Given two polynomials $$U(x)=u_0+u_1 x+\ldots+u_n x^n$$ and $$V(x)=v_0+v_1 x+\ldots+v_m x^m$$ the product $$U(x)V(x)=\sum_{k=0}^n a_k x^k$$ satisfies $$a_s=\sum_{k=0}^s u_k v_{s-k},\quad 0\leq s \leq n+m,$$ making the "reversal" clear.

As you can imagine, thinking formally with no convergence issues which is irrelevant in this case, you can consider $$U(x)V(x) ~mod~ (x^n-1)$$ to obtain a cyclic convolution for two vectors of length $n$ which is useful in some other contexts related to cryptography.


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# HowTo:Term Orderings

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

This page is an introduction into term orderings in CoCoA-5 or ApCoCoA-2. In order to understand this topic, we assume that the reader is familiar with the concept of term orderings on polynomial rings.

## Mathematical definition

Let K be a field, let ${\displaystyle P=K[x_{1},\ldots ,x_{n}]}$ be the polynomial ring over K in n indeterminates and let ${\displaystyle \mathbb {T} (P)=\{x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}\mid \alpha _{1},\ldots ,\alpha _{n}\in \mathbb {N} _{0}\}}$ be the set of all power products in P.

Let ${\displaystyle \sigma }$ be a total order on T(P). For a pair ${\displaystyle (t_{1},t_{2})\in \sigma }$, we write ${\displaystyle t_{1}\leq _{\sigma }t_{2} is called a term ordering on T(P) iff it is multiplicative, i.e. ${\displaystyle t_{1}\leq _{\sigma }t_{2}}$ implies ${\displaystyle st_{1}\leq _{\sigma }st_{2}}$ for each ${\displaystyle t_{1},t_{2},s\in \mathbb {T} (P)}$, and we have ${\displaystyle 1\leq _{\sigma }t}$ for all terms ${\displaystyle t\in \mathbb {T} (P)}$.

For a polynomial ${\displaystyle f\in P}$, we can write ${\displaystyle f=c_{1}t_{1}+\cdots +c_{s}t_{s}}$ where ${\displaystyle t_{i}\in \mathbb {T} (P)}$ and ${\displaystyle c_{i}\in K^{\times }}$ for each i such that ${\displaystyle t_{1}>_{\sigma }\cdots >_{\sigma }t_{s}}$. Then we use the following notation

• ${\displaystyle {\rm {LT}}_{\sigma }(f)=t_{1}}$ is the leading terms of f,
• ${\displaystyle {\rm {LC}}_{\sigma }(f)=c_{1}}$ is the leading monomial of f and
• ${\displaystyle {\rm {LM}}_{\sigma }(f)=c_{1}t_{1}}$ is the leading coefficient of f.


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Hardcover | $34.00 Short | £27.95 | 280 pp. | 6 x 9 in | 31 b&w illus. | September 2017 | ISBN: 9780262036733 eBook |$24.00 Short | October 2017 | ISBN: 9780262342209
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## Waste Is Information

Infrastructure Legibility and Governance
Foreword by Carlo Ratti

## Overview

Waste is material information. Landfills are detailed records of everyday consumption and behavior; much of what we know about the distant past we know from discarded objects unearthed by archaeologists and interpreted by historians. And yet the systems and infrastructures that process our waste often remain opaque. In this book, Dietmar Offenhuber examines waste from the perspective of information, considering emerging practices and technologies for making waste systems legible and how the resulting datasets and visualizations shape infrastructure governance. He does so by looking at three waste tracking and participatory sensing projects in Seattle, São Paulo, and Boston.

Offenhuber expands the notion of urban legibility—the idea that the city can be read like a text—to introduce the concept of infrastructure legibility. He argues that infrastructure governance is enacted through representations of the infrastructural system, and that these representations stem from the different stakeholders’ interests, which drive their efforts to make the system legible. The Trash Track project in Seattle used sensor technology to map discarded items through the waste and recycling systems; the Forager project looked at the informal organization processes of waste pickers working for Brazilian recycling cooperatives; and mobile systems designed by the city of Boston allowed residents to report such infrastructure failures as potholes and garbage spills. Through these case studies, Offenhuber outlines an emerging paradigm of infrastructure governance based on a
complex negotiation among users, technology, and the city.


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Search: All articles in the CJM digital archive with keyword local index theorem

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1. CJM Online first

Carey, Alan L; Gayral, Victor; Phillips, John; Rennie, Adam; Sukochev, Fedor
 Spectral flow for nonunital spectral triples We prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a $C^*$-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow. Keywords:spectral triple, spectral flow, local index theoremCategory:46H30

2. CJM 2000 (vol 52 pp. 695)

Carey, A.; Farber, M.; Mathai, V.
 Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von~Neumann algebras as developed in \cite{CFM}. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic $L^2$ torsion, which shows that it is {\it not\/} in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic $L^2$ torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles. Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost KÃ¤hler manifolds, von~Neumann algebras, determinant linesCategories:58J52, 58J35, 58J20


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MORE IN Geotechnical Engineering 1
VTU Civil Engineering (Semester 5)
Geotechnical Engineering 1
May 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Critically define the terms void ratio, porosity and water content with phase diagram.
4 M
1(b) Derive the relation $$\gamma _{d}=\dfrac{G.\gamma w}{1+e}$$ with usual notations.
6 M
1(c) In an earthen embankment under construction, the bulk unit weight is 16.5kN/m3 at water content 11%. If the water content has to be increased to 50%, compute the quantity of water to be added per cubic meter of soil. Assume no change in void ratio. Also determine 'e' at this water content taking G = 2.65.
10 M

2(a) Define relative density of sand and list its importance in geotechnical engineering.
4 M
2(b) Describe consistency of soil. List and define consistency limits.
6 M
2(c) The following results are obtained by conducting liquid limit test on clayey soil in the laboratory:
 No. of blows (N) : 34 23 18 12 Water content (W%): 44.6 49.4 51.4 55.6

Plot flow curve. Determine Liquid limit, Toughness index. Assume plastic limit = 20%.
10 M

3(a) Explain with the help of typical particle size distribution curve, well graded, poorly graded and gap gtaded soil.
4 M
3(b) With the neat sketch, explain structure of clay minerals.
6 M
3(c) Following results are obtained from the laboratory tests conducted on two soil samples :
 Soil sample A B Liquid limit 85% 45% Plastic limit 50% 12%

Show the positions of these soils on plasticity chart and classify as per I.S. system.
10 M

4(a) Derive the relation between co-efficient of permeability and percolation with usual notations.
7 M
4(b) Explain Quick - sand phenomena and list its importance during construction.
5 M
4(c) A falling head permeater accommodates a soil sample of 6cm length and 500cm2 in area. The permeability of sample is expected to be 1 × 10-4 cm/sec. Head of water in the standpipe falls from 30cm to 10cm in 40 minutes. Determine the size of the stand pipe which should be used.
8 M

5(a) Explain sensitivity and thixotropy of clayey soil.
4 M
5(b) List the factors affecting shear strength of soil.
4 M
5(c) A direct shear test results are obtained as follows :
 Normal stress (kN/m2) : 100 200 300 Shear stress (kN/m2) : 130 185 240

Determine shear parameters graphically. Also draw Mohr's circle corresponding to second test result and report major and minor principal stresses.
12 M

6(a) List the factors affecting compaction of soil and explain any 2 in detail.
5 M
6(b) List and explain various types of field compaction equipments.
5 M
6(c) The results of standard compaction test conducted in the laboratory are tabulated as follows:
 Water content% : 5 10 14 20 25 Bulk density kN/m3 17.7 19.8 21 21.8 21.6

Find MDD and OMC with usual notaions by plotting comoaction curve. Also draw ZAVD -line assuming G = 2.65.
10 M

7(a) State the assumptions of one - dimensional Terzaghi's theory of consolidation. Also write standard / general differential one - dimensional consolidation equation with usual notation.
8 M
7(b) Explain with neat sketch, determination of co-efficient of consolidation by square - root of time fitting method.
6 M
7(c) A layer of soft clay is 6m thick and lies under newly constructed building. The weight of sand overlying the clayey layer produces a pressure of 260 kN/m2 and this new construction increases the pressure by 100kN/m2. If the compression index is 0.5, compute settlement of soil layer given water content 40% and G = 2.65.
6 M

8(a) Critically discuss limitations of direct shear test.
4 M
8(b) Explain Vane shear test with neat sketch along with relations.
6 M
8(c) A vane apparatus test. A torque of 45N-m was applied at which failure took place. Subsequently, vane instrument rotated rapidly, so as to get reolded soil sample. This remolded soil was sheared at a torque of 18 N-m. Then calculate sensitivity of clayey soil.
10 M

More question papers from Geotechnical Engineering 1


text:
# Homework Help: Cartesian Vectors

1. Sep 18, 2016

### Robb

1. The problem statement, all variables and given/known data

Three forces act on the bracket, the resultant force is directed along the x' axis and has a magnitude of 9.8kN. Determine the magnitude of F. Determine the direction theta of F.

2. Relevant equations

3. The attempt at a solution

I'm really not sure how I am doing with. Any advice would be much appreciated!

2. Sep 18, 2016

### Staff: Mentor

Robb,

You'll get more responses and better help if helpers can see clear and commented work that's easy to quote in a response. A not-too-clear image of handwritten calculations without comments will deter many. It is better to insert a description of what you are doing and why at each step. The very best thing to do, of course, is to type out your math using LaTeX so that helpers can directly quote and comment...

3. Sep 18, 2016

### kuruman

In your drawing (that is hard to read) I see that you have attempted to write the forces using unit vector notation. That is good. However, these forces are in two dimensions in the plane of the screen. You seem to have written the forces in three dimensions. Write all three forces in unit vector notation using $\hat{i}$ and $\hat{j}$ only, add them and see what is needed to get a resultant in the direction x'.

On edit: Try to follow gneill's recommendation.

4. Sep 18, 2016

### Robb

Sorry for the confusing drawing. I guess I was looking at x' as a third axis/dimension. No?

5. Sep 18, 2016

### kuruman

No, x' is in the same plane as the 4kN, the 6kN and the resultant. It has to be in the same plane because it is the direction of the resultant. Maybe this will make it easier for you to find the answer.

6. Sep 18, 2016

### Robb

F1= 6i
F2= F(2x)i + F(2y)j
F3= (-4sin15)i + ( 4cos15)j= -1.03i = 3.86j
Fr- (9.8cos30)i +(9.8sin30)j = 8.49i + 4.9j

8.49= 6-1.03 + F(2x)
F(2x)= 3.52kNi
F(2y)= 4.9 - 3.86= 1.04kNj

F2= 3.67kN

I assume for the direction theta, they are referring to the one shown between the y axis and the vector F2? I labeled the vectors F1, F2, F3 starting with the x axis and moving counter clockwise.

7. Sep 18, 2016

### kuruman

Yes, that's what they mean. Find it and you are done. I didn't run your numbers but they look OK.

8. Sep 18, 2016

### Robb

arctan(1.04/3.52) = 16.46

Looking at the drawing this angle would make sense but which angle is tangent measuring here? Typically I would say between the x axis and F2 but that number doesn't make sense looking at the drawing.

9. Sep 18, 2016

### kuruman

Drop a perpendicular from the tip of F to the y-axis. You have a right triangle and the perpendicular you just drew is opposite to θ. What is tanθ?

10. Sep 18, 2016

### Robb

arctan (1.04/3.52)= 16.46

I looked at it that way to but I also wondered about dropping a perpendicular from the tip of F to the x axis. NVMD! Just saw it. Thanks for the help!

11. Sep 18, 2016

### Robb

So, the answer turned out to be theta= 73.5. Seems to me, from looking at the drawing, like that makes more sense if it were measured from the x axis, ccw.

12. Sep 18, 2016

### kuruman

The drawing shows that θ is measured with respect to the y-axis CW. You have to find the angle that the problem is asking you to find and 73.5o is the complementary angle of θ. Not all angles are measured with respect to the x-axis CCW.

13. Sep 18, 2016

### Robb

That's what I though but masteringengineering.com says the answer for theta is 73.5. I entered 16.5 first and it said I was wrong. I really hate that site for homework!

14. Sep 18, 2016

### kuruman

I would hate it too. You have learned all you need to learn from this problem. I suggest that you show your solution to your instructor and ask why it is marked wrong. He/She needs to know and, if enough people complain, perhaps a different homework platform will be chosen next time the course is taught.